[Vladimir P. Krainov, Howard R. Reiss, Boris M. Sm(BookZZ.org)

RADIATIVE PROCESSES IN ATOMIC PHYSICS

Radiative Processes in Atomic Physics. V. P. Krainov, H. R. Reiss, B. M. SmirnovCopyright © 1997 by John Wiley & Sons, Inc.ISBN: 0-471-12533-4

RADIATIVE PROCESSESIN ATOMIC PHYSICS

VLADIMIR P. KRAINOVHOWARD R. REISSBORIS M. SMIRNOV

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC.

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Copyright © 1997 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyondthat permitted by Section 107 or 108 of the 1976 UnitedStates Copyright Act without the permission of the copyrightowner is unlawful. Requests for permission or furtherinformation should be addressed to the Permissions Department,John Wiley & Sons, Inc., 605 Third Avenue, New York, NY10158-0012.

Library of Congress Cataloging in Publication Data:

Krainov, V. P. (Vladimir Pavlovich), 1938-Radiative processes in atomic physics / by Vladimir P. Krainov,

Howard R. Reiss, Boris M. Smirnov.p. cm.

Includes index.ISBN 0-471-12533-4 (cloth : alk. paper)1. Atomic spectroscopy. 2. Radiative transitions I. Reiss,

H.R. (Howard Robert), 1929- . II. Smirnov, B. M. (Boris Mikhallovich), 1938-III. Title.QC454.A8K73 1997539.2—dc21 96-47468

CIP

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

CONTENTS

Preface ix

1 Radiative Transitions of Atomic Electrons 1

1.1 Principal Quantities of Single-Photon Transitions, 31.2 Oscillator Strength, 101.3 Selection Rules and Rates of Electron Transitions in Atoms, 18

1.4 Selection Rules and Rates for Vibrational and Rotational Transitionsin Molecules, 31

1.5 Polarization of Radiation, 40

1.6 Radiative Transitions with Two Photons, 461.7 Polarizability of Molecular Gas, 55

2 Radiative Transitions between Discrete States in Atomic Systems 61

2.1 Radiative and Doppler Broadening of Spectral Lines, 622.2 Collision Broadening of Spectral Lines, 702.3 Quasi-Static Theory of Spectral Line Broadening, 872.4 Cross Sections for Absorption and Induced Emission of Photons:

Absorption Coefficient, 992.5 Cross Sections for Scattering and Raman Scattering of Photons, 1032.6 Two-Photon Absorption, 110

vi CONTENTS

3 Atomic Photoprocesses Involving Free Particles 119

3.1 Decomposition of Atomic Systems, 1193.2 Photoexcitation of Rydberg States of Atoms, 1373.3 Autoionizing States of Atoms, 1433.4 Bremsstrahlung from Scattering of an Electron by Atoms and

Ions, 1473.5 Photorecombination of Atomic Systems, 165

4 Coherent Phenomena in Radiative Transitions 177

4.1 Polarization Effects in Radiative Transitions in a Uniform MagneticField, 177

4.2 Interference of States During Radiation, 183

4.3 Resonance Transitions in Two-Level Systems, 1884.4 Resonance Transitions in Multilevel Atoms, 197

5 Atoms in Strong Fields 203

5.1 Properties Useful for Strong-Field Methods, 2045.2 Qualitative Strong-Field Properties, 216

A Angular Momentum 229

A. 1 Projection of Angular Momentum, 229

A.2 Square of the Angular Momentum, 230A.3 Commutation Properties of Angular Momentum Operators, 231A.4 Eigenvalue of the Squared Angular Momentum Operator, 232

B Clebsch-Gordan Coefficients 235

B.I Properties of Clebsch-Gordan Coefficients, 235B.2 Evaluation of Certain Clebsch-Gordan Coefficients, 237B.3 Wigner 3j Symbols, 240

C Rotation Functions 241

C. 1 Matrix Elements of Rotation Functions, 243

D Wigner 6j Symbols 247

D.I Properties of 6j Symbols, 248D.2 Partial Values of 6j Symbols, 249D.3 Matrix Elements in Addition of Momenta, 250

CONTENTS vii

E Fractional Parentage Coefficients 253

F Atomic Properties 257

G Vibrational and Rotational States of a Molecule 261

H Equation for the Density Matrix 265

I Atomic Units and Measures of Intensity 269

LI Atomic Units, 269

1.2 Electromagnetic Field Quantities, 270

J Properties of the Generalized Bessel Function 273

K Ionization Potentials 277

L Parameters of the Hydrogen Atom 283

L. 1 Angular Wave Functions of the Hydrogen Atom, 284L.2 Radial Wave Functions of the Hydrogen Atom, 285L.3 Algebraic Expressions for Expectation Values of Powers of the

Radial Coordinate in the Hydrogen Atom:{r») = £rl{r)r"J»dr, 286

L.4 Expectation Values of Powers of the Radial Coordinate in theHydrogen Atom (in Atomic Units), 286

L.5 Oscillator Strengths and Lifetimes for Radiative Transitions in

the Hydrogen Atom, 287

References 289

Bibliography 291

Index 293

PREFACE

This book has evolved from lectures by the authors on radiative processes in atomicphysics. The intent of the book is to help students and investigators in this field toextend and to make contemporary their knowledge of this important branch of atomicphysics. It is envisioned that both advanced students and active researchers in this fieldwill find it useful. Much material is contained herein that is not to be found elsewhere.Radiative processes, while constituting a fundamental area in atomic physics, havenot been the focus of many books.

A novel method has been adopted for the presentation of the subject matter.The material is presented as a succession of problems. These problems are statedsuccinctly, solved using basic principles of atomic physics, and then the resultsare discussed in qualitative terms. It has been our experience that a possible initialdiscomfort with this unfamiliar structure gives way to an appreciation of its importantadvantages: First, different aspects of a single topic are treated in separate problems,which makes possible a progressive deepening of the understanding of the subject.Second, by considering limited cases of a general topic, it is possible to simplify theunderlying mathematics so as to highlight the fundamental concepts. Third, althoughsome of these problems build progressively on the results of those that precede it, itis also possible to enter into the subject matter at any point. Fourth, there is the veryimportant feature that the problem/solution format reinforces in the reader the abilityto analyze the content of a physical problem and to apply the suitable mathematicaland physical tools to solve it. Finally, the qualitative discussion of the outcome of thesolution aids in the development of physical intuition.

It is presumed that the reader is already acquainted with the mathematical apparatusof quantum mechanics. These matters are not explicitly developed in the body ofthe book, but a set of appendices is provided to give a concise overview of those

ix

X PREFACE

methods of quantum mechanics most often employed in the treatment of radiativeproblems in atomic and molecular physics. The appendices review the basic tools ofthe quantum treatment of angular momentum, including rotation functions, Clebsch-Gordan coefficients, and the Wigner 3j and 6j symbols. There are summaries of aidsto calculation in atomic physics, such as the density matrix method and fractionalparentage coefficients. Two sections are devoted to quantities that arise in strong-field,nonperturbative problems. Finally, there are several sections that present some of thebasic data of atomic physics (such as ionization potentials) as well as a compilationof the parameters of the hydrogen atom, including angular and radial wave functions,expectation values of various positive and negative powers of r for the lower-lyingstates, and oscillator strengths and lifetimes for radiative transitions.

V. P. KrainovH. R. Reiss

B. M. SmirnovMoscow and Washington

1996

RADIATIVE PROCESSES IN ATOMIC PHYSICS

1RADIATIVE TRANSITIONS OFATOMIC ELECTRONS

We devote our attention here to the study of interaction processes between the radi-ation field and atomic electrons. As a result of these processes, the atomic electronswill experience transitions between states available to them. If the radiation field isregarded as a collection of the elementary radiation quanta—or photons—then thenumber of photons changes as a consequence of such transitions.

In this book we consider transitions in which the valence electrons of an atom ormolecule participate or in which vibrational or rotational molecular states change (seeAppendix G). In the case of valence electron transitions, a characteristic transitionenergy can be constituted from fundamental atomic constants: electron mass m,Planck constant h, and electron charge e. The value of this basic energy unit canbe estimated by combining these atomic constants in such fashion as to producethe dimensions of energy. This process yields me4/h2. Vibrational and rotationalmolecular transition energies are smaller in an essential way than this estimate, sincethe molecular mass is large as compared to the electron mass.

The atorhic electron velocity v corresponding to atomic energies is found frommv2 ~ me4/h2, so v is of the order of e1 /h. Comparing this velocity with thevelocity of light, c, we find that v/c ~ e2 /he ~ -^ < 1, so that one can use thenonrelativistic theory of radiation. The applicability of nonrelativistic theory leads toessential simplifications in our general analysis.

The interaction between the electromagnetic field and the atomic system is de-scribed by the interaction Hamiltonian (Ref. 1)

2 RADIATIVE TRANSITIONS OF ATOMIC ELECTRONS

where A(r, t) is the vector potential of the electromagnetic radiation field, and p is themomentum operator of an atomic electron that participates in the radiation transition.We presume that the radiation wavelength is large compared to a typical atomicdimension (i.e., the Bohr radius). This statement can be written as c/co > h2/me2,where co is the circular frequency of the radiation. This inequality can be rearrangedto read hco/mc2 < e2 /he, so it is again equivalent to a nonrelativistic condition sincee2 /he < 1 as cited above. Since the radiation frequency co is of the order of thefrequency of an atomic transition me4/h3, then we see that the dipole approximationis applicable (see, e.g., Ref. 2). That is, since the vector potential (and electric field)is nearly the same over the entire radius of the atom, we may set A(r, i) ~ A(t). Inconsequence of this approximation, the Hamiltonian written above for the interactionbetween the radiation field and the atomic electron can be written in the simpler form:

V = - E D. (1.1)

Here E is the electric field strength of the radiation field being considered, and D is thedipole moment operator of the atomic electron. We see that the dipole approximationis a consequence of the inequality v/c < 1.

Through most of this book, we shall also assume that the energy of interactionbetween the radiation field and the atomic system is small compared to the transitionenergy. This justifies the description by perturbation theory of radiation and absorptionprocesses involving photons of the radiation field. Perturbation theory is applicable ifthe electric field strength E is small compared to a characteristic internal electric fieldof the atom. We can take this internal field to be of the order of the Rydberg energydivided by the product of the Bohr radius and the electron charge, or of the order ofm2e5 /h4. Hence, we presume that E < m2e5 /h4. When this inequality holds true,then the operator (1.1) used for investigation of radiative transitions between statesof an atom can be considered to be a small perturbation.

Thus the most important emission or absorption processes involve a single photon,if that is permitted by quantum selection rules and by energy conservation. Unlessvery high-powered lasers are involved, the rates of processes in which more thanone photon participate are small compared to single-photon rates, and this small-ness is determined by the size of the small parameter of perturbation theory. Thesecircumstances allow one to make essential simplifications in the solution of problems.

The interaction Hamiltonian (1.1) will be used through most of this book. Suc-ceeding terms in the expansion in powers of v/c (quadrupole approximation, et seqq.)are important only if the selection rules forbid dipole transitions.

The electric field can be represented as the sum over monochromatic waves:

E = ^T Re[Ew exp(ia>01a)

where Ew is the complex strength of the electric field component of frequency co.We denote by 8o) the difference between neighboring wave frequencies and presumethat all differences are the same, a situation usually realized in laser radiation. Thewidth of the spectrum is designated by Aco. We presume that the value of Ew changes

PRINCIPAL QUANTITIES OF SINGLE-PHOTON TRANSITIONS 3

only slightly within this width, and we also impose the inequality Aco > 8a>. Thatis, we require that there be many frequency components in the spectrum. If thephases of the amplitudes Ew are statistically independent, then we can neglect theinteractions between different waves when we treat the most prominent radiativetransitions in lowest-order perturbation theory. This interaction contributes only tothe next higher order of perturbation theory. Another situation in which the interactionbetween component waves can be neglected arises when the frequency difference 8cobetween neighboring modes is sufficiently large, that is, when 8co > w, where w isthe transition rate. (See Problem 1.1.)

Radiative transitions will be considered in all but the last two chapters withinthe framework of time-dependent perturbation theory. We first consider radiativetransitions in which only one photon is emitted or absorbed, so that we employfirst-order perturbation theory.

The Appendices at the end of this book contain a summary of many basic results,such as the properties of angular momentum in quantum theory and the ways in whichangular momenta can couple. These Appendices will be referred to frequently.

1.1 PRINCIPAL QUANTITIES OF SINGLE-PHOTON TRANSITIONS

Problem 1.1. Find the transition rate for an atomic system for the absorption of onephoton and also give the expression for the intensity of the absorption.

SOLUTION. Our goal is the calculation of the radiative transition rate in the contextof first-order perturbation theory. This rate is small due to the small magnitude of theinteraction between the radiation field and the atom, as discussed in the introductionto this chapter. Then the inverse quantity, the transition lifetime, or elapsed timebetween successive transitions will be large as compared to characteristic atomictimes. The atomic time scale can be estimated as the period of the motion of anelectron around the atomic core, which is h3/(me4).

In the introduction to this chapter, it was pointed out that the interaction between anatomic electron and each monochromatic component of the electromagnetic wave canbe considered separately. Therefore we shall examine the operator for the interactionof an atomic electron with a single component of the field, which is

V = - D Re[Ew exp(iGtf)]. (1.2)

The Hamiltonian of an atomic system in the absence of the electromagnetic field willbe denoted as Ho. This Hamiltonian can describe, for example, an electron in theeffective mean potential of the atomic core. It is this electron that changes its statein the radiative transition. If the radiative transition is one involving a molecule, thenthe quantity Ho will refer to the entire molecule, describing its vibrations or rotations,if they change in the process of the radiative transition of the molecule. We do notdetail here the form of Ho; that will be done in subsequent problems.

We denote by ijjk the system of eigenfunctions of the Hamiltonian Ho, where sk

are the corresponding energy eigenvalues. These functions obey the time-independentSchrodinger equation Ho ijjk = sk\\)k. The expansion of the wave function of the systemin terms of the eigenfunctions \\tk is

M*1 = V^Cfcife(r)exp(—iskt/h). (1-3)

Here r is the set of coordinates of the atomic electron, or those coordinates thatdetermine that vibrational or rotational state of the molecule which changes in theradiation transition. We insert Eq. (1.3) for the wave function of the atomic electroninto the Schrodinger equation,

dt

where H — Ho + V is the total Hamiltonian of the system. Then we multiply theresulting equation by t//*p integrate over the coordinates of the atomic electron, anduse the orthogonality conditions for the eigenfunctions ij/k of the unperturbed atomicelectron. We thus obtain for the coefficients ck the equation

ih—£- = Y2 VmkCk exP0'wm*O> (1.4)

where o)mk = (sm — sk)/h, and Vmk is the matrix element of the time-dependentinteraction operator, Eq. (1.2), taken between states of the atomic electron. Usingfirst-order perturbation theory, we suppose that the perturbation is not present untilthe time t = 0, before which the electron was in the state with index 0. Then in zeroapproximation we have c(

k0) = 8k0.

We.take the radiation field to be linearly polarized, which means that Ew is areal quantity. Furthermore, we replace Ew by E. This replacement does not changethe essential character of the results. When we substitute the explicit form of theexpression (1.2) for the interaction operator,

V = -D-Ecoscof,

we obtain from Eq. (1.4) the first approximation

- <o)t] 1 - exp[i(a)k0 + <o)t] \

(1.5)2h { o)ko — o) cx)ko + o) J

The strongest transitions correspond to those states k for which the resonance condi-tion a)ko « co is fulfilled. In this case, we need to retain only the first term in Eq. (1.5).The probability for transition to the state k is

s)2 sin2[(co£0 - o))t/2]

o>? > ( L 6 )

where s is the unit vector in the direction of the electric field vector, the so-calledfield polarization vector. For small a and large t, the function in Eq. (1.6) of the formsin2(at)/a2 can be replaced by irt8(a), where 8(a) is the delta function. The deltafunction has the properties that 8 (a) = 0 when a =£ 0, 8 (a) = oo at a = 0, and

fJ — o

8(a)da = 1.

The function sin2 (at)/(ira2t) has essentially all of the properties just listed for 8(a),and is, in fact, often employed as a representation of the delta function when t —> oo.Taking account of these considerations, we obtain the transition rate

1w — -

tSX) E 2^. n (Dfco ' s) o(co^o ~~ (**)' (1-^)

In1

This formula is valid even when t is not extremely large, as long as wt < 1 andperturbation theory is applicable.

We now take into account the multimode properties of the radiation. To sumEq. (1.7) over all modes of the field, we make the identification

i- [dco,

where 8co is the frequency difference of neighboring modes (see above), and thequantity 1/Sco is the energy density of the states. In addition, we express the rate win terms of the number of field photons n^ in the state with frequency co, instead ofthe electric field strength E. We recall that the quantity

(H2/STT) _ E^

represents the mean value of the energy density of the electromagnetic field in asingle mode of the field. In arriving at this result, we replace the time-averaged valueof cos2 cot by ^, and we take into account that the electric field strength |E| is equalto the magnetic field strength |H| in a monochromatic plane wave. We can thus writethe energy density of modes in the frequency range [to, co + dco] as (E2/87r)(dco/8a>).From another point of view, this quantity is also given by

(2TT)3 TT2C3 '

where k is the photon wave vector, dk/(27r)3 is the number of states contained ina unit volume and the range of the frequencies co considered (Ref. 1), nw is thenumber of photons in a single state, and the factor 2 takes into account the twopolarization states of the photon (Ref. 3). The wave number k is connected with thephoton frequency co by the relation co = cky where c is the speed of light. Theseconsiderations establish the connection between the electric field strength E and thecorresponding number of photons nw.

RADIATIVE TRANSITIONS OF ATOMIC ELECTRONS

From the last two expressions we find

E =E = rV 7TC3 )

When we substitute E2 into Eq. (1.7), and integrate over the frequency range in theneighborhood of the transition frequency, we obtain the rate for absorption of photonswith polarization s as

w = ^rG>*o • s)2 (o) = <ok0). (1.8)

Such an integration, employing the delta function property f™x 8 (a) da = 1, is correctwhen the frequency range A co (the width characteristic of the radiation spectrum) islarge compared to typical values of the difference a)k0 — co, which, in turn, is of theorder off"1. Since wt < 1, we obtain the inequality Aco > w.

Thus, the result (1.8) is valid for a weak field with a broad frequency spectrum andat a time of observation that is not extremely large. We also require the incoherenceof waves with different frequencies.

Equation (1.8) is derived for the case n(o> \. We have implied that the electric fieldstrength does not change as a result of the radiation transition. Such a change doestake place in that the number of photons is diminished by one upon the absorption ofa photon. Nevertheless, Eq. (1.8) is correct at any value of nw. Indeed, from the natureof photon absorption, it follows that the photon absorption probability is proportionalto the number of photons in the volume under consideration. Thus, the relation (1.8)is also correct when one introduces the quantum properties of the electromagneticfield.

We find the intensity of the absorption by multiplying Eq. (1.8) by ho) to obtain

Although this expression does not contain the Planck constant, it becomes classicalonly when n^ > 1. To go to the classical limit, we replace the matrix element of thedipole moment, tyto, by the Fourier component of the time-dependent dipole momentcorresponding to the transition frequency a>£0 = (s* ~~ £o)/^ (see, e.g., Ref. 4). Thisis the form taken by the Bohr correspondence principle in this case.

We can average Eq. (1.8) over the polarization direction s by taking the solidangle average of cos2 fl, where 6 is the angle between the vectors D and s. Since thisaverage value is cos2 6 = | , we obtain the transition rate in the form

WOk = w(Of flu -> k, Ha, ~ 1) = —~z | D ^ 0 | 2 ^ « W . (1.9)3nc*

Here we introduce the factor g^, which is the statistical weight of the final state. Thatis, gk is the number of substates for this final level. Unlike Eq. (1.8), the expression(1.9) does not depend on the field polarization.


The quantity \/w represents the time between two consecutive photon absorptionevents. The time for the absorption of one photon is much less; it is determined bythe uncertainty principle, and is of the order of l/a^0 .

It can be seen from Eq. (1.6) that at very small times the transition rate is propor-tional to t2. However, this rate is so small that it is experimentally unobservable. Onthe other hand, the solution we have obtained will be incorrect at times so large thatwkOt ^ 1. In this case, the rate will depart from linear dependence. (See Chapter 2.)

Problem 1.2. Determine the photon emission transition rate between states of anatom.

Remark: Unlike the case of photon absorption, radiation from an excited atomicstate consists of two parts. One is the so-called spontaneous transition rate, cor-responding to radiation in the absence of the external field. Spontaneous emissionoccurs due to the interaction of the charged atomic electron with the electromagneticvacuum field (see Refs. [1, 4-6]).The second part, corresponding to production ofphotons stimulated by the external field, is called induced radiation. Thus the totalrate for the emission of radiation from the excited state k is equal to

wk0 = w(k, rioj -> 0, nw + 1) = Ak0 + B^n^ (1.10)

where the A term refers to spontaneous emission and B to induced emission. Theinduced emission rate is proportional to the number of photons in the cavity n^ sincethese photons produce the transition being considered. The quantities Ak0 and Bko arecalled Einstein coefficients. In this problem, the Einstein coefficients will be derived.We presume that a gas of atomic electrons is in thermodynamic equilibrium with theradiation field.

SOLUTION. We shall determine the Einstein coefficients starting from the equilibriumcondition for atomic electrons in a gas in an enclosure we shall refer to as a cavity.At equilibrium, the mean rate at which excitation transitions take place is balancedby the mean rate at which transitions occur in the opposite direction, that is,

NowOk = NkwkQ = NoAokJi^ (1.11)

The quantities No and Nk are the number densities of atomic electrons in the states0 and k, respectively. The bar over a quantity denotes an average over photon states,since different numbers of photons nM are possible at a fixed temperature in the cavity.The quantity n^ is the mean number of photons with the frequency co. The densitiescan be related to each other by the expression

No nw + 1 go

where go and gk are statistical weights of the states 0 and k, respectively. This relationfollows from the fact that the number of photons increases by one in the deexcitationfrom state k to state 0. Also, the number of electrons in the given state is proportional


to the statistical weight of the corresponding state. When we insert Eq. (1.12) intoEq. (1.11), and take (1.10) into consideration, we find, for arbitrary nw, the relation

Ako - Bko = AOk—.8k

Thus the transition rate for a process with the emission of a photon is of the form

vv*0 = w(fc n^ -> 0, nw + 1) = ^ 3 |D*o|2£o(^ + 1). (1.13)

Equation (1.9) has been employed here.In the derivation of Eq. (1.13), we did not specify any particular distribution

function for 7zw. In the case of thermal equilibrium, it is the well-known Planckdistribution. We have used only the fact that photons are bosons, so that any arbitrarynumber of them can be in a given state. The fact that thermal equilibrium has beenspecified has been used only to establish the connection between the rates of the directand inverse processes. Thus, the formula (1.13) is valid even without averaging. Wecan rewrite it for the case of radiation of a photon with definite polarization s [as inEq.(1.8)]as

4o)wk0 = ^|D*(rs|2so(«a, + 1). (1.14)

The radiation transition rate given by Eq. (1.8) or (1.14) contains a square of amatrix element of the dipole moment operator. This matrix element is nonzero onlyfor transitions with a certain change in the quantum numbers of the atom. The rulesgiving these possible transitions are called selection rules. Some of these rules willbe considered below.

The factor (n^ + 1) in Eq. (1.14) gives two terms in the transition rate. The termproportional to «w gives the rate for induced radiation. In this case, the radiatedphoton has the same wave vector and the same polarization s as the photons causingthe transition. The second term in Eq. (1.14), the one independent of rcw, correspondsto spontaneous radiation, since the transition can occur even in the absence of theexternal field. That is, it occurs even if nM = 0. The spontaneous radiation rate witharbitrary polarization is given by the second term on the right side of Eq. (1.13).

We have found that all three Einstein coefficients AOk, Ak0, and Bk0 are intercon-nected, and the rates of all radiation transitions can be expressed in terms of one ofthese coefficients. We have expressed these Einstein coefficients as functions of theparameters of the electromagnetic field and of the atomic state.

Problem 1.3. Derive a classical formula for the intensity of spontaneous dipoleradiation from the general quantum mechanical expression (1.14).

SOLUTION. We can state the intensity of spontaneous radiation as the energy radiatedper unit time. From Eq. (1.14), this radiation intensity for a photon of polarization s,

stemming from the radiative transition from an excited state A: to a lower state m, is

cm ' S| gm,

where hcokm is the energy difference between the states k and m. When we averageover the polarization of the emitted photon, and assume for simplicity that gm = 1,we obtain from Eq. (1.13) the total intensity of spontaneous radiation

3c3

In the classical case we should sum Ikm over all states m below the state k in orderto determine the lifetime of the state k. That is, we require the intensity

m,(okm>0

This expression can be simplified. Since the matrix element Dkm is proportional toaccording to general principles of quantum mechanics, we can then write

When we employ this in Eq. (1.15), we obtain

h = A y I(D)> I'- (1.16)3 ^ 3 / J \\/km\ v /

m,(x)km>0

We can estimate the sum in Eq. (1.16) in the classical approximation. From matrixmultiplication rules we have

EI (fl)J2 = (*)*•m

Furthermore, we know that the quasi-classical sum in Eq. (1.16) is dominated by thosestates m that are c lose in energy to state k. In fact, the values of the matrix elements(D)jkm calculated w ith quasi-classical wave functions decrease quickly with increasingenergy difference between states k and m, due to oscillations in the integrand whichgrow rapidly with energy difference. Now we take into account that states m bothabove and below state k are very similar to those that lie very close to k. We thusobtain

= Em, a)^m > 0 m, (Ohm < 0

The diagonal matrix element of an operator representing a physical quantity givesthe mean value of that operator. That mean value, in turn, yields the classical valuefor the physical quantity when the classical limit is taken. When we implement thisobservation, we obtain from Eq. (1.16) the total radiation intensity of an atom in the


classical limit,

The classical dipole moment D in this equation should be regarded as a given functionof time.

We conclude that, in the quasi-classical limit, the lifetime of a given state isdetermined by spontaneous transitions to nearby states.

1.2 OSCILLATOR STRENGTH

Problem 1.4. The oscillator strength for radiative transitions in an atom is introducedby the relation

fok = ^o>ko\(Dz)k0\2, (1.18)

nel

where 0 and k are the initial and final states, a)^ is the radiation frequency corre-sponding to the transition between these states, and (Dz)^o is the matrix element takenbetween these states of the projection of the dipole operator. Obtain the formula

where the sum is taken over all possible states, and N is the total number of electronsin the atom.

SOLUTION. The concept of the oscillator strength arises from a model of the electricproperties of matter in which we suppose that the atomic electrons are in equilib-rium positions and react elastically to small perturbations. That is, the electrons inthis model are caused by a low-intensity electromagnetic field to execute simpleharmonic motion around their equilibrium positions. Actually, electrons do not havefixed equilibrium positions in atoms. Instead, it is the statistical distribution of suchpositions that is realized, as determined by the squared modulus of the wave function.However, in classical language, a portion of the integral of this squared modulus ofthe wave function corresponds to each spectral frequency, and it is this part thatdetermines the oscillator strength for that frequency.

The oscillator strength fOk has been introduced in Eq. (1.18). The radiation transi-tion rate given in Eq. (1.14) can be written in terms of the oscillator strength as

rs 2 2

^ , + 1). (1.19)me5

The coordinate system is such that the z axis lies along the direction of the polarizationvector s. Equation (1.19) shows that the oscillator strength is the dimensionlessparameter that measures the intensity of the given radiative transition.

OSCILLATOR STRENGTH 11

The Hamiltonian of a system of atomic electrons consists of the sum of the electronkinetic energies and the potential energies of their interactions with each other andwith the atomic nucleus. The commutator of the projection of the momentum operatorPjZ of the electron bearing the index j , and the coordinate projection Zj in thisdirection, is given by the standard result

[pjz,Zj] = h/i. (1.20)

It follows from this that the commutator of the Hamiltonian H and Zj arises from thekinetic energy part only, with the outcome

For the commutator of the Hamiltonian H and the projection of the total dipolemoment, we obtain the expression

J

The matrix element of this relation taken between the states 0 and k gives

ei Y ^ xvz Ok m ^ jz Ok'

j

When we insert this expression into (1.18) for the oscillator strength, we arrive at

fok - -^ }](Pjz)0k(Dz)k0>j

which can be rewritten as

fok ~ 'YhJ

upon using the properties of the operators pjz and Dz. We can now express theoscillator strength as half the sum of the two preceding expressions to give thesymmetrized form

fok = T 2_J\(Pjz)Qk(Zji)kO ~ (ZjOok(Pjz)ko}'JJ'

When this expression is summed over all final states k, we obtain

Since the momentum operator of each electron commutes with the position operatorof any other electron, then the double sum above reduces to the case j = jf only.Furthermore, using the commutation relation (1.20), we find

yk=N, (1.21)

where N is the total number of electrons in the atomic system. If the sum is restrictedonly to those transitions in which valence electrons take part, then N is the numberof valence electrons in the atom.

The sum in Eq. (1.21) is, in fact, equal to the sum over all the states of the discretespectrum and the integral over the continuum states. It is seen from (1.21) that thissum is independent of the quantum numbers of the initial atomic state 0. Equation(1.21) is a generalization of the Thomas-Reiche-Kuhn sum rule.

If the state 0 is a highly excited state, then we face the problem of how to calculatethe sum semiclassically. We treated an analogous problem in Problem 1.3 for theintensity of the radiative transition. However, that method does not apply in thepresent case, where we would obtain zero instead of N on the right-hand side ofEq. (1.21) because all commutators are zero in the classical limit.

The semiclassical approach thus does not give the correct result. A partial ex-planation for this comes from the fact that those terms in the series in Eq. (1.21)corresponding to highly excited states with energies more than s0 and those for en-ergies less than s0 contribute values of opposite sign to the sum. Each term is muchgreater than the right-hand side of Eq. (1.21). The correct result arises as a smalldifference of large terms. For example, we can verify for the harmonic oscillatorpotential or for the Coulomb potential that the most important terms on the left sideof Eq. (1.21) are of the order of n, where n is the principal quantum number ofthe state 0. As a result of compensation of terms with opposite signs, the value ndisappears from the right side of Eq. (1.21).

Problem 1.5. Express the static polarizability of an atom in terms of the oscillatorstrengths for radiative transitions.

The static polarizability «o of the state 0 in the constant electric field E is givenby the relation

—— = -OQE, (1.22)

where 8s0 is the static Stark shift of the energy of the state 0 due to the electric field E.

SOLUTION. The energy shift of the state 0 resulting from the perturbation (1.1) iscalculated by second-order perturbation theory to be

8£ = ^ V" l(Dz)ok^h , <*>ok '


Hence the quantum mechanical expression for the static polarizability of an atom isof the form

2 ^ \(Dz)Ok\2 e2 ^ fOk . . .

ao = ~T } ^ = — > ^ —r> (1-23)

when we make use of Eq. (1.18) for the oscillator strength.From a classical viewpoint, the static polarizability of a harmonic oscillator with

frequency o) is determined by the same Eq. (1.22), where 8SQ is the energy of theoscillator in the electric field. This quantity is clearly of the form

e2

a =

From a comparison of these last two expressions, we see that the polarizability ofan atom is given by the sum of the polarizabilities of atomic oscillators, each ofwhich appears with a relative importance measured by the oscillator strength fOk.This explains the origin of the term oscillator strength.

We note that when state 0 is a highly excited state, we can repeat all of theconclusions just reached.

Problem 1.6. Calculate sums of the form

where n = 0, 1, 2, 3, and 4. These are examples of what are called sum rules. Allsums include the complete basis set of eigenstates, that is, the quantity S^ includesa sum over all discrete bound states and an integral over all continuum states.

SOLUTION. We consider first the simplest case, n = 0. We have

^=§El^ol2=^2)oo. d.24)

In the classical limit, the quantity (r2)oo is the mean value r2(t).For the case n = 1, the simplest avenue to the result is to use the solution of

Problem 1.4. We set N = 1 in Eq. (1.21), and obtain

3 - ^ , w - d-25)k

Now we shall evaluate the sum rule for n = 2. We use the relations

;i(p)*oP = (A)o = 2m(E -

and

(P)*o = im(x)k0(r)k0

to find

So2' = j E 4 ) l ( r ) * o l 2 = \(M - )oo- d-26)k

In the classical limit, this expression is equal to ( |)T(0, where T(t) is the mean valueof the electron kinetic energy.

We next consider the sum with n = 3. According to the Ehrenfest theorem ofquantum mechanics, we have

d ^;r(P)*o = i<at

Using these relations, we find

The sum with n = 3 is therefore

Since AV = Airep, where p is the density of the positive charge producing thepotential energy V, we obtain

For example, p = Z^5(r) for a hydrogenlike atom, and so Eq. (1.27) yields the result

Since linv_+0 iAo(/) ~ rl, where / is the orbital angular momentum quantum numberof the state 0, then we can conclude that S^3) = 0 for / =£ 0.

Finally, we consider the case n = 4. Again using the Ehrenfest theorem, we have

We thus obtain

S04) = ^ E <4Mo(r)«>l2 = ~ ((W)2)00. (1.28)

k

For example, a hydrogenlike atom has (V V)2 ~ 1/r4, so Eq. (1.28) gives

This integral diverges in the case / = 0, while it is finite for / =£ 0.Sum rules are useful for estimating matrix elements in asymptotic limits. For

example, it follows from Eq. (1.27) that, when sk —> oo, part of the sum is of the form

o>lo\(r)k0\2da)k0.

Since this integral converges, we conclude that the matrix element (r)fco has the simplebehavior

(r)*o ~ <*ko>

where p > 2.

Problem 1.7. Using the shell model of an atom, determine the effect on sum rulesof the Pauli principle (i.e., electrons cannot occupy states that are already filled).

SOLUTION. On first inspection, we see that the Pauli principle should influence thesum rule (1.21), which states that the sum of the oscillator strengths is equal to thenumber of electrons in an atom. In fact, when we found this sum rule (see Problem1.4), we summed (1.21) over all possible states of an electron in the self-consistentfield of the atomic core. According to the Pauli principle, we must exclude the filledelectron states from this sum. However, we shall show that we can neverthelessneglect the Pauli principle in the calculation of the sum of the oscillators.

The sum of the oscillator strengths for a given electron (in the single-electronapproximation) in state j can be written in a form taking into account the Pauliprinciple by writing the sum as

where the / index refers to the empty electron states. We showed in Problem 1.4 that

k

where k refers to all possible electron states. It follows from this that

j.k


and

where j and j ' are filled states, and N is the number of electrons. It follows fromEq. (1.18) that the oscillator strengths have the property fjy = —fyy Hence we findthat

j.j'

so that

Thus the Pauli principle does not affect the sum rule for the oscillator strengths. Ananalogous conclusion can also be reached for odd values of n in the sums

treated in Problem 1.6.

Problem 1.8. In a transition of an atom to a highly excited state, find the dependenceof the oscillator strength on the principal quantum number of this state.

SOLUTION. As shown in the solution to Problem 1.4 [see Eq. (1.18)], the oscillatorstrength is determined by the radial dipole matrix element,

fnl.n'l fJO

Rni(r)Rn'i'(r)r3 dr

Here Rn\ and Rn>i> are radial solutions of the Schrodinger equation for initial (nl) andfinal highly excited (n'V) states. The other factors in Eq. (1.18) for /„/,„///, includingsn - sni do not depend on n' due to the condition |^n| > \sni\. Thus, in order tofind the dependence of /n/>n/// on the principal quantum number n' > 1 of the highlyexcited state, we must estimate the functions Rni and Rniv, and also we must establishthe most important values of r in the integral for the dipole matrix element.

In the solution of this problem, we employ the atomic unit system in whichm — e = h = 1. These units make possible a substantial decrease in the apparentcomplexity of equations, with a concomitant improvement in clarity of the meaning.When n' > 1, the radial wave function Rnni can be written using the semiclassicalapproximation in the form


where

kn,v = 2(En, - Vv)x/\

and Eni = — \/(2n12) is the energy of the term with the principal quantum numbern1. This term is hydrogenlike due to the condition n! > 1. The quantity

is the effective potential for the radial motion of an electron. The index /' is theorbital quantum number of the final state, and r\ is the left turning point for thecorresponding classical motion.

We predicate n, I ~ 1 for the initial atomic state. Then we find that the essentialregion of r for the wave function Rni(r), and hence for the integrand in the radialmatrix element of the dipole moment, is of the order r ~ 1. Furthermore, it followsfrom the dipole selection rule (see Section 1.3) that at / ~ 1 we also have /' ~ 1. Inthe region r ~ 1 we have kniV ~ 1, and, consequently, Rn'v{r) ~ a.

We now wish to calculate the normalization factor a of a radial wave function fora highly excited final state. It is found from the normalization condition

/ R2n,v(r)r2 dr = 1.

Jo

In the semiclassical approximation this condition is of the form

a2 n dr_ _

From this relation we find

dr= rJri 2{-( (2/r) -

where r2 is the right turning point for the corresponding classical motion of anelectron. Since the important region in the integrand is where r ~ n12 > 1, and/ ; ~ 1, then we can neglect the contribution of the centrifugal potential in thenormalization integral. We thus obtain

i = r ± = B/3 ra* L 2[-(l/n'2) + (2/r)]i/2 J 2[(2A) -

Our final result is that a ~ l/n'3/2.In the essential region of the integrand in the expression for /„;,„';/, which is of the

order of r ~ 1, we have Rni ~ 1 and Rnqi ~ a. This leads to the result

/„/,„'/' ~ \Rn'i-\2 ~


The coefficient in this dependence is of order unity (in atomic units). For example,the explicit calculation for the ground state of hydrogen (n = 1, / = 0, V — 1) yields

1.3 SELECTION RULES AND RATES OF ELECTRONTRANSITIONS IN ATOMS

In this section we consider electron transitions in atoms. We limit ourselves to lightatoms, and examine selection rules for quantum numbers of the states involved intransitions. We also consider the connection between radiative transition rates andsymmetry properties of the wave function.

Problem 1.9. Consider a resonantly excited atom in a P state interacting with anotheratom of the same type in an S state. The distance R between the atoms remains fixedin the process of radiating a photon. Assume that the interaction between the atomsmay be regarded as a dipole-dipole interaction. Find the selection rules and calculatethe radiative transition rate for P —• S.

SOLUTION. A quasi molecule consisting of the two atoms in the problem has asymmetry plane perpendicular to the axis connecting the atomic nuclei and positionedat the midpoint of this axis. The wave functions of the system can be even or odddepending on the preservation of sign or change of sign, respectively, upon reflectionin the plane of symmetry. We shall compose the wave function of the system withdue regard to parity. Then we shall determine the rates of spontaneous transitions thatfollow from Eq. (1.13).

The wave functions of even (gerade, or g) states and odd (ungerade, or u) statesof the quasi molecule are of the form

where i// is the wave function of an atom in its ground (5) state, and (pm is the wavefunction of the excited (P) state. The atoms are labeled by the indices 1 and 2, and m isthe projection of the angular momentum of the excited P state on the axis connectingthe atomic nuclei (m = — 1, 0, +1). The wave function of the ground-state molecularterm is \\j\4fi-

Substituting these wave functions into the relation (1.13), we find the spontaneoustransition rate for the quasi molecule, averaged over the polarizations of the emittedphoton, to be

S ^ 2 (1.29)

Here D is the dipole moment operator of the atom, and co is the frequency of theatomic transition P —• S. It is seen that wum = 0 for an odd state. That is, the radiative

SELECTION RULES AND RATES OF ELECTRON TRANSITIONS IN ATOMS 19

transition between an odd excited state of a quasi molecule and its ground state isforbidden. This follows from the fact that the wave function of the ground state of thequasi molecule and the total dipole moment operator are even functions with respectto permutation of the atoms, while wave functions of odd excited states of the quasimolecule are odd functions with respect to such a permutation. Therefore the matrixelement of such a transition vanishes.

For even excited states of the quasi molecule, we first consider the case in whichan atom in a P state has zero projection of the angular momentum on the axis zconnecting the atomic nuclei. It is seen that

We now introduce the notation

Dz = ((p®\D]z\ilf\),

in terms of which Eq. (1.29) yields

S"\l£>zl2. (1.30)

When we perform the angular integrations in the expression for Dz (see AppendixA), we obtain

Dz = D c o s 2 0 s i n 0 d 0 — = —^,

where the quantity

D = e f Rp(r)Rs(r)r3 drJo

is the radial part of the integral in the matrix element of the dipole moment, and Rp(r)and Rs(r) are the radial wave functions. Equation (1.30) can thus be rewritten in theform

wg0 = ^D2. (1.31)

The lifetimes of the excited states with m = +1 and m = - 1 are plainly the same.In this case, Dz = 0 and

fn ? f ( 3 \ dipDy = Dx = D s n r 0 sin 0 dO / cos cp exp(/<p) — 7

Jo Jo V877"/ (4T7) 1 /Z

D

so that

The expressions obtained for the radiative transition rates are valid when R < A,where R is the interatomic distance and A is the wavelength of the light (A = 2TTC/G)).

In the opposite case, where R > A, the decay of the excited atom in the P state isindependent of the atom in the S state, and the radiative transition rates are of theform

4co3 ?

This result also holds when R > a, where a is a dimension typical of the atom; butwhen R is not much larger than a, then the wave functions are strongly distortedbecause of interactions between the atoms.

Problem 1.10. When two atoms are separated by a distance large as compared tothe radius of either atom, show that the interaction potential between the atoms canbe written as

V = ^ [ D , - D 2 - 3 ( D 1 - n ) ( D 2 - n ) ] ,

where the subscripts identify the two atoms, D is the atomic dipole moment, R is thedistance between the atomic nuclei, and n is the unit vector along the axis connectingthe atomic nuclei (R = nR).

SOLUTION. This form of the interaction operator corresponds to an expansion validin the limit of large distances between the atoms.

We can represent the Hamiltonian of the electrons of the interacting atoms in theform

H=H0~Y y^—: ~ Y r ^ ^ + Y 1 r (1.33)

= #o + V,

where HQ is the Hamiltonian of the noninteracting atoms. The index k refers toelectrons of the first atom, which has a nucleus of charge Z\; and the index / refers tothe electrons of the second atom, which has a nucleus of charge Z2; and r/, r^ are theradius vectors of each of the electrons with respect to their own nuclei. A standardresult of classical electrodynamics (see, e.g., Ref. 7) is that the interaction energybetween two dipoles p! and p2 separated by a distance R is

Wi,2 = ^3 [pi-p2 " 3(prn)(p2-n)], (1.34)

obtained from a multipole expansion of the total electrostatic energy between the twocharge distributions whose dipole moments are p! and p2. This multipole expansionis identical to an expansion of V in Eq. (1.33) in terms of the small quantities rk/Rand n/R. When the atomic dipole moment operators Dj = J2krk> D2 ~ J2i r/>are substituted for p t and p2 in Eq. (1.34), we obtain the desired interaction energyoperator between two atoms,

V = ^ [D! • D2 - 3(D! • n)(D2 • n)]. (1.35)

Problem 1.11. Under the conditions of the preceding problem, calculate the spectralshift of the radiation line compared to its unperturbed value hco - eP - ss.

SOLUTION. The dipole-dipole interaction between atoms produces a spectral shift inthe radiation line. The operator of this interaction is of the form

V= • ^ [ D 1 - D 2 - 3 ( D 1 - n ) ( D 2 - n ) ] ,

where n is the unit vector in the direction of the interatomic axis. We shall treat thisinteraction in the context of first-order perturbation theory, which will be correct forsufficiently large distances between the atoms (R > a). We treat the distance R as agiven quantity. The interaction potential between the atoms in the quasi molecule inan even excited state is of the form

Ugm(R) = (V?\V\V?). (1.36)

Obviously, the interaction potential for the quasi molecule in the ground S state iszero, so that the shift of the spectral line comes entirely from (1.36). We note thatthis shift vanishes when R > A, since the diagonal matrix element from the dipoleoperator goes to zero for the excited state <pm.

The quantity m in the expression (1.36) is the projection of the angular momentumof an atom in a P state onto the axis connecting the atoms. Simplifying Eq. (1.36),we find

U*m(R) = ^ 2

We introduce the radial matrix element D as in the preceding problem and calculatethe angular integrals. When m = 0, we obtain

n|<p°>|2 = D2 /3,

so that

U*(R) = -2D2/3R3.

Similarly, when m = 1 and m = — 1, we find

K^IDI^1)!2 = MDx\<pl)\2 + MDy\<pl)\2 = D2/3.

We also have the result that

IMD • iV>| 2 = MDz\<pl)\2 = 0,

so that

U8±l(R) =

The excited term of odd parity is also shifted. The interaction potential has thesame amplitude as the corresponding even-parity interaction potential Ufn(R), but hasthe opposite sign. We cannot, however, seek such a shift in the spontaneous transition,since it is forbidden by the dipole selection rule.

The shift of the even excited term is observed as a splitting of the spectral line forthe P —• S radiative transition. This splitting is asymmetrical, with the energy of them = ± 1 terms increasing by half the amount by which the line for m = 0 decreases.Also, we see from the solution of the preceding problem that the intensities of thetransitions are different. According to Eq. (1.32), the intensity of the m = ±1 line istwice that of the intensity of the m = 0 line.

All of the above considerations are valid if the interaction potential Um(R) betweenthe atoms is small compared to the transition energy hco, that is, when the shift ofthe transition line is small. This is true when R> a and allows us to represent thewave function of a quasi molecule as a combination of the wave functions of thenoninteracting atoms in the zero order of perturbation theory.

It should be noted that the shift of the spectral line is inversely proportional tothe cube of the distance R between the atoms. That is, it is large compared to theusual van der Waals interaction, which is a phenomenon of second-order perturbationtheory, and is proportional to the inverse sixth power of R.

Problem 1.12. Find the selection rules for the single-electron radiative dipole tran-sition between states of a light atom.

The atomic level scheme of a light atom is based on the picture that the orbitalangular momenta of the electrons couple to form the total angular momentum L ofthe atom, and the spins of the electrons couple to form the total spin momentum S(the so-called LS coupling). Relativistic effects, including spin-orbit interactions, aresmaller and are not included here.

SOLUTION. We designate by ML the projection of the total angular momentum on afixed direction, and Ms is the projection of the atomic spin on the same direction. Weare interested in the final states of a system when the matrix element

(LMLSMsa\Dg\LfMl

LS'Msa')


is nonzero, where a, a7 are those quantum numbers of the initial and final states,respectively, that are necessary to define the states in addition to the angular momen-tum quantum numbers. The quantity Dq is a component of the dipole moment vectoroperator in spherical vector coordinates. (See Appendix C.) These components are

DO = DV D±1 =(Dx±iDy)/y/2.

In the general form Dq for q = - 1 , 0, 1, we have

Dq = erYlq(£l),

where r, fl are spherical electron coordinates. We assume that only one electron takespart in the radiative transition.

Since the dipole operator does not depend on spin, it is nonzero only when theselection rules

5' = S, M's= Ms

are fulfilled. From Eq. (C.9), we have

{LMLa\Dq\L'M'La') ~ (LI, MLq\L'M'L)(L\y 00|L'0>.

The Clebsch-Gordan coefficient (LI, MLq\L'M'L) is nonzero only when ML = M[9

M'L ± 1. Thus, selection rules for the radiative transition being examined are of theform (for single-electron transitions)

L - L ' = ± l , 5 - 5 ' = 0, ML-M'L = 0, ±1, Ms - M's = 0. (1.37)

It can be shown in the case of many-electron atoms that the rule

L-Lf = 0

should be added to Eq. (1.37), as well as the requirement that the parity of the stateshould change in the transition. (This was fulfilled automatically above.) See Problem1.17 for details.

Problem 1.13. A light atom undergoes a radiative transition from a state with orbitalangular momentum L and spin S to a state with orbital angular momentum L1 andspin S1. Fine structure of the atom in the initial state is determined by the total angularmomentum J and its projection M on a fixed direction. Find the relative probabilityof the atom to be in the final state J1 after the transition.

SOLUTION. As in the preceding problem, we presume LS coupling to hold true. Thatis, states are determined not only by the total angular momentum J but also by theorbital angular momentum L and the spin angular momentum S. Since the dipolemoment Dq is an orbital quantity, its matrix elements are diagonal with respectto S. From Eq. (1.13) the relative transition probability to the given final state is

proportional to the quantity

] T MJMLSc^DqU'M'L'Sa1)?,qM'

where q = — 1, 0, 1, and J, M, L, S are, respectively, the total angular momentum,its projection, and the orbital and spin angular momentum quantum numbers of theatom in its initial state. Further, a and a' are all other quantum numbers necessary todescribe the initial and final states. We assume that the transition is a single-electronprocess.

We have Dq = erDlq0, where Dl

q0 = Yiq(Cl) is the rotation function. Using theformula (D.12) for the matrix elements of Dq, we obtain

Y^ $JMLSa\Dq\J'MfL'Saf)\2

qM'

= (2Lr + 1)(2/' + 1)

xY^<L%W\W)2(U',qM'\JArflLj J jX \{L'a'\er\La)\2

qM' ^ J

= (2L' + $(2J< + 1)<L'1, 00|L0>2 | ^ I j \ \{L'a'\er\La)\\

Selection rules on L —• L' were established in Problem 1.12. The same rules followalso from the Clebsch-Gordan coefficient (Z/l, 00|L0) in the expression above. Fromthe same expression, the selection rules on J follow from the properties of the Wigner67 symbol

W 1j s .

This symbol is nonzero if / - Jf = 0, ± 1 (except for the transition / = 0 —• J1 — 0).The relative probability for transition to the state with a given angular momentum

J1 is of the form

(2// + 1 ) { y I J ] 2 n> 1 iV^ )—1=QL+Y){2J'+\)r J, %\ ,

l! 1 L\ [J S J )

J S J'j

obtained with the use of relation (D.4). We thus find

w(J,L -^ J1,L') = (2L + 1)(2J' + 1) I . „ .,} w{J,L -> L1), (1.38)

where w is the rate for the radiative transition between the states given in the paren-theses attached to the w.

We now wish to average this expression over the total angular momentum J of theinitial state. The probability of finding an atom in the initial state with a given value


of J is proportional to the statistical weight of this state. That is, it is equal to

27 + 1(25 + 1)(2L + 1)'

which is the ratio of the statistical weight of this state to the statistical weight of theterm (SLa). We then find that

From Eqs. (1.38) and (1.39) we obtain

i!>{l' > $«,.-» (,40,

We now sum this quantity over the initial total momentum 7 to obtain

w(L - • J1, L') = Y^ wCA L -> 7', L1)j

27' + 1(2Lf + 1)(25

w(L-*L'). (1.41)

This formula is obvious in that it says that the sum of the intensities of all the lines ofa spectral multiplet with the same final level is proportional to the statistical weightof this final level, which is

27' + 1

(2V + 1)(25 + 1)

For example, consider the transition with L1 = L — 1. When we make the ap-propriate quantum number entries in the Wigner 6j coefficient in the expression forw(J, L —» Jf, V) and then make the simplifying assumption that J > 1, we obtainthe ratios

w(JL -» J - 1, L - 1) : w(JL -+ J, L - 1) : w(JL -> J + 1, L - 1)

= [(/ + Lf - S2}2 : 2| [(/ + Lf - S2] [S2 - (J - L)2} \ : [(J - L)2 - S 2 ) \

Thus, among the lines of a multiplet, the most intense is the line with A J = AL,the so-called principal line. The line with A7 = AL+ l i s approximately J2 timesweaker, since at / > 1 we have J ~ L. Finally, the line with AJ = AL + 2 isapproximately J4 times weaker than the principal line. These last two lines are calledsatellites. The greater the value of the total angular momentum / , the stronger theprincipal line is as compared to the satellite lines.

If the condition / > 1 cannot be fulfilled, the analysis is generally greatly com-plicated, except that relatively simple expressions can nevertheless be obtained foralkali atoms. We consider the atom to consist of a single electron in the field of


an atomic core. We then have S = \. The term La splits into two subterms withJ = L±\, and the term (L - l ) ^ ' also splits into two subterms, but with J1 = L-\and J1 = L - | . The selection rules on 7 allow three transitions from the term La tothe term (L — l)a'. Their relative probabilities are found from the formula obtainedabove to be in the ratios

w (J = L + \ -> J1 = L - \) : w (J = L - \ -± J1 = L - \)

:w(j = L - ± -> J1 = L~l)

= (L + 1)(2L - 1) : 1 : (L - 1)(2L + 1).

The fourth transition with 7 = L + ^ - * 7 = L - | i s forbidden since it wouldrequire A 7 = 2, in contradiction with the selection rules on 7 found above. If wesuppose that L > 1, then we see that A 7 = AL = 1 for the first and third transitions.Thus, both these lines are principal lines, as they have comparable intensities. Wecan see this also from the result obtained for the relative intensities of these lines.For the second line, A7 = AL — 1 = 0, so this is a satellite line with an intensityapproximately L2 times weaker than the principal lines.

We can generalize this example as well as the previous one, where transitionsfrom a fixed initial state of a multiplet with a definite value of 7 were considered.The conclusion can be reached that when one takes account of all possible transitionsbetween states of two multiplets with 7 > 1, the most intense are lines with A 7 = AL,which are principal lines. The greater the difference A 7 — AL, the weaker is theintensity of the corresponding line.

In conclusion, we note that, physically, averaging over initial or final states ofa multiplet is realized, for example, when we consider radiation in a gas with atemperature much greater than the fine structure spacing, but much less than theenergy distance between the multiplets considered.

Problem 1.14. Solve Problem 1.13 when 7, J1 ~ L, L1 > 1.

SOLUTION. We can simplify the expression for the 6j symbol in Eq. (1.40) using thegeneral expression (D.7). We shall estimate the quantity

L1 1 L\J S J1}'

Suppose first that 7, J', L, L1 > S > 1. We now examine some particular cases.

a. The case L1 = L — 1 has the three possibilities

(L-l 1 L } 2 (L + 7 + S)2(L + 7 - 5)2({ 7 S 7 - 1 / (8L7)2

- S)(L + S - 7)(7 + S - L)(L-l 1 L\2 _ 2(L + 7 + S)(L + 7\ 7 S Jj (8L7)2


fL -1 1 L V =\ J S 7 + 1J

(L-^-S- 7)2(7 + S-L)2

(8L7)2

The ratio of the rates for these transitions are given in Problem 1.13.b. The case L1 = L, which is realized in many-electron atoms, gives

(L 1 L 1 2=

\J S J - l f2(L + 7 + 5)(L + 7 - 5)(L + 5 - 7)(7 + S - L)

L 1 L\2 = 4(/2 + L2 - S2)2

J S J\ (SLJ)2

2|L 1 L V\ / 5 J+lj

2(L + 7 + S)(L + 7 - 5)(L + 5 - 7)(7 + S - L)(8LJ)2

Consequently, we obtain the ratios of transition rates

w(J,L-+J-l,L):w(J,L-> 7, L) : w(7,L -> 7 + 1,L)

= [(7 + L)2 - 52] [52 - (7 - L)2] : 2(72 + L2 - 52)2

: [(7 + L)2 - 52] [52 - (7 - L)2].

c. The case L1 = L + 1 contains the possibilities

(S + L- 7)2(5 + 7 - L)2fL+1 1 L \ =

\ J 5 7 - 1 /

/ L + l 1 L l 2 =

{ J S Jj

(8L7)2 '

2 2(L + 5 + 7)(L + 7 - 5)(L + S - 7)(7 + 5 - L)(8L7)2

L + l 1 L \2 (L + 7 + S)2(L + 7 - S)2

7 5 7 + 1 J (8L7)2

This gives the transition rate ratios

w(7, L -> 7 - 1, L + 1) : w(7, L -> 7, L + 1) : w(7, L -> 7 + 1, L + 1)

= [S2 - (L - 7)2]2 : 2| [(7 + L)2 - 52] [S2 - (J - L)2} | : [(L + 7)2 - 5 2 ] 2 .

In this case, the line with A 7 = AL = 1 is the principal line, a satellite withA7 = AL— 1 = Ois approximately 72 times weaker, and a satellite with A 7 =AL — 2 = — 1 is y4 times weaker than the principal line.

We have completed the treatment of transitions with large spin momentum S > 1.Now we consider small values of S, such that 5 — 1 , with the conditions that7, J', L,L' > 1. We need to retain ± 1 terms when they are in combination with 5, butwe can neglect 5 in combinations with 7 + L. We again examine several particularcases.


a. When L' = L - 1, we obtain

w(J, L -> 7 - 1, L - 1) : w(7, L -> 7, L - 1) : w(J, L -> 7 + 1, L - 1)

= (7 + L)4 : [(7 + L)2(L + 5 - 7)(7 + S - L + 1)]

: [(S + L - 7)(5 + L - 7 - 1)(5 + 7 - L + 1)(S + 7 - L 4- 2)].

b. The case V — L gives

w(7, L -> 7 - 1, L) : w(/, L -> 7, L) : w(J, L -+ J + 1, L)

= [(7 + L)2(7 + 5 - L)(L + 5 - 7 + 1)] : (7 + L)4

: [(7 + L)2(7 + 1 + S - L)(L + S - J)].

c. Finally, when L1 = L — 1, the results are

w(7, L -» 7 - 1, L + 1) : w(7, L -^ 7, L + 1) : w(7, L -> 7 + 1, L + 1)

= [(7 + S - L - 1)(7 + S - L)(L + S - 7 + 1)(L + 5 - 7 + 2)]

: |[(7 + L)2(L + 1 + 5 - 7)(7 + 5 - L)]| : (7 + S)4.

Problem 1.15. Find the relative transition probabilities for multiplets in the transi-tions 2S - • 2P, 2P -> 2 5, 2 P -> 2 A and 2D -^ 2P.

SOLUTION. All the transitions under consideration have spin of S = | . First we treatthe 2S —> 2P transition. Here, the relevant quantum numbers are L = 0, L1 = 1, and7 = | , 77 = ^ or | . According to Eq. (1.40), we obtain

1 1 0 2

f l 1 O 1 2 4

Now we consider the transition 2P —> 25. The quantum numbers here are L = 1,L' = 0, and 7 = | or ^, 7 / = 0. There are thus no multiplets in this case.

The next transition we consider is 2P —• 2D. The orbital angular momentumquantum numbers are L = 1,L' = 2. If 7 = j , then we must have Jf = | , and thereis no multiplet. However, if 7 = | , then 7 ; = | or | . Then Eq. (1.40) leads to

/ 2 1 1 \ 2

i 3 1 5 f1 2 2 2 J

Thus, the principal line corresponds to the transition with A7 = AL = 1. It is ninetimes stronger than the satellite line.

Now we turn to the last transition, 2D —> 2P. The orbital angular momentumquantum numbers are L = 2, L1 = 1. First we consider the case where J = | in theinitial state. Then the final state can have either J' = ^ or J' = | . For the ratios ofthe transition rates, we obtain

(I 1 2\2

w<3,2— 5, 1) _ 2 \ | 3 | j _ 15

vK|,2-i l) 4 n ! 212 21 1 2\2

1 L If2 2 2)

The principal line with AJ = AL = — 1 is 7.5 times stronger than the satellite line.If J = I in the initial state, then only J' = | is possible, and there is no multiplet.We have thus considered all the available options for the transitions being examined.

In conclusion, note that we observed in Problems 1.13 and 1.14 that the principallines are substantially stronger than the satellite lines when J > 1. We have nowfound in the present problem that even at moderate values of J there is a large factorin the ratios of the intensities of principal lines to satellite lines.

Problem 1.16. The total angular momentum F of an atom arises from a combinationof the total atomic electron angular momentum J and the nuclear spin /. Calculaterelative probabilities for radiative transitions with the change J —• J1 in the quantumnumbers of the atom between states with hyperfine structure F —> F1.

SOLUTION. Hyperfine splitting of atomic levels occurs as a result of interactions ofthe atomic electrons with the nuclear spin when that spin is nonzero. This problem isanalogous to Problem 1.13. Specifically, in Problem 1.13, the total angular momentumJ was a result of the combination of the orbital and spin angular momenta L and S,but the transition operator did not depend on the spin. In the problem we considernow, the total angular momentum F is a combination of the electron total angularmomentum J and the nuclear spin /, but the transition operator does not depend onthe nuclear spin. Therefore, in analogy to Problem 1.13, we find that the selectionrules on F are Ff = F, F ± 1.

As a consequence of the similarity of the two problems, we can rewrite the resultsof Problem 1.13 with the changes in notation

J->F, L -> 7, S - • /.

From Eq. (1.40), the radiative transition rate for the stated change in hyperfinestructure is given by the expression


The total transition rate from the given hyperfine state is given by Eq. (1.39) as

w(7, F -> J') = ] T w(J, F -> 7', F1)F>

IF + 1(27 + 1)(2/ + 1)

w(J

The radiative transition rate from all components of the hyperfine multiplet to thegiven hyperfine state is

w(J -> 77, F7) = 2 ^ vv(7, F -H.

F

2F' + 1

(27' + 1)(27 + 1)A

The statistical factors in these expressions have the simple explanation as the relativeprobabilities for filling the given states in the hyperfine structure.

We can treat as negligibly small the interactions between electron and nuclearspins. Therefore, in the transitions considered here, all selection rules from previousproblems with respect to electron total momentum 7 and electron parity remainvalid. In particular, electric dipole transitions between components of the hyperfinestructure of the same term are forbidden since they have the same parity.

In closing, we note that transitions of the type 7 = 0 —> J' = 0, forbidden in thedipole approximation, have the forbidden analog

F = 0 -> F1 = 0

in the present problem.

Problem 1.17. Clarify the differences in selection rules for dipole radiative transi-tions in many-electron atoms as compared to single-electron light atoms.

SOLUTION. The dipole moment is an orbital vector. Hence, matrix elements of allcomponents of this vector can be nonzero only for transitions in which orbital angularmomenta of the L electron shell change by ± 1 or 0, that is, L —» L, L ± 1. Also, thereis the additional selection rule that forbids transitions between states that both haveL - 0. This rule is a consequence of the spherical symmetry of states with L = 0,so that there does not exist a vector quantity representing the matrix element of thedipole moment vector.

The dipole moment operator

SELECTION RULES AND RATES FOR VIBRATIONAL AND ROTATIONAL TRANSITIONS 31

is an odd operator. Hence matrix elements of this operator between states of the sameparity vanish. Thus we obtain the so-called Laporte rule: Transitions between stateswith the same parity vanish.

The Laporte rule forbids transitions with AL = 0 in single-electron atoms, sincein the case of a single electron the orbital quantum number L determines the parityof the state through the parity of (— 1)L. This means that AL = 0 would requireunchanged parity in a dipole transition. However, for many-electron atoms the totalelectron orbital angular momentum L does not have a direct connection with parity,so the condition AL = 0 can be realized together with the requirement of a changeof parity in a dipole transition.

1.4 SELECTION RULES AND RATES FOR VIBRATIONAL ANDROTATIONAL TRANSITIONS IN MOLECULES

The selection rules and rates considered in this section are for radiative dipole transi-tions in molecules. Appendix G describes many of the properties of vibrational androtational molecular states we shall require here.

Problem 1.18. Consider a molecule consisting of two identical atoms. Show that ra-diative dipole transitions between vibrational states in such a molecule are forbidden.

SOLUTION. We shall prove that the matrix element of the dipole transition operatorvanishes. We first determine the dipole moment of the molecule, considering thenuclei to be in fixed positions, and viewing these nuclei as sources of a potential field.The molecule is symmetrical with respect to a plane that is a perpendicular bisectorof the straight line connecting the nuclei. The electron density in the molecules isdivided symmetrically by this plane. The electron density also has axial symmetrywith respect to the line joining the nuclei. Therefore the electron density p is invariantwith respect to inversion of all electrons. The dipole moment is thus

D = / ^2ertJ i

d r i . . . d r n = - 1 ^ 2 e v t P d v x . . . d x n = 0

for this case of fixed internuclear distance. Therefore the matrix element of thisoperator between vibrational states is zero, and there are no vibrational transitions.This conclusion holds true whether the two nuclei are of the same isotope or not.It is important only that the electric charges of the nuclei be the same. However,this statement is invalidated if molecular rotation influences the electron state. Thenthe inversion symmetry of the electron wave function is lost if the two nuclei havedifferent masses, that is, if they are of different isotopes of the given element.

On a more subtle level, the conclusion about vanishing of the dipole momentcan be violated even for identical nuclei because of the very weak interaction ofthe nuclear spins with the electrons. This interaction disrupts the symmetry of the

electron wave function due to the fact that the two nuclear spins may be differentlyoriented, and they will thus have different effects on the electron densities associatedwith each of the nuclei. The interaction of the electrons with the total nuclear spinwill lead to a weak mixing of electron states with opposite parity, and makes dipoletransitions possible.

Problem 1.19. Calculate the rates for radiative transitions between rotational statesof a linear dipole molecule, and determine the selection rules.

SOLUTION. For the sake of simplicity, we consider first only those terms in whichthe total molecular spin is zero. We denote by J the total angular momentum ofthe molecule in the initial state. It is composed of the orbital electron momentaand the rotational angular momentum of the nuclei. The projection of the totalangular momentum on a fixed axis is denoted by M. The projection of the orbitalangular momentum on the molecular axis is conserved due to the axial symmetry ofthe molecule. We label it A. Since the rotational angular momentum is perpendicularto the molecular axis, then the quantity A also represents the projection of thetotal angular momentum of the molecule onto its axis. All other quantum numbersnecessary to define the initial state are designated by a. Analogous quantities for thefinal state of the molecule are distinguished by a prime.

We wish to consider the transition JM —• J'M' between rotational states of themolecule for a given electron state, that is, for fixed quantum numbers A and a.The problem reduces to the calculation of the matrix element of the dipole momentoperator. From Eq. (C.ll), the matrix element of the component Dq (where q is aspherical component) of the dipole moment vector in the rest system is of the form

(JfMfAa\Dq\JMAa).

It can be related to the analogous matrix element in the coordinate system in whichthe z axis is along the direction of the molecular axis. The connection is given by

(j'M'Aa\Dq\JMAa) =

The index q takes on the values 0, ± 1.It is clear that the matrix element of the dipole moment operator does not depend

on the rotational quantum numbers in the coordinate system associated with themolecular axis. It is determined only by the electronic state of the molecule. Thusthis matrix element is diagonal for rotational transitions and is equal to the meandipole moment of the molecule,

D = <Aa|Dz|Aa).

Selection rules for dipole rotational transitions follow from the properties of theClebsch-Gordan coefficients contained in the expression above as


J-Jf = 0, ±1; M-M' = 4 = 0, ±1. (1.42)

Since the energies sj of the rotational states are determined by the well-knownformula

sj = BJ(J + 1),

then the spontaneous transition from the state with angular momentum J to the lowerstate is possible only with J1 = J — 1.

Using Eq. (1,13) for the rate of radiative processes, we obtain

^^^l±l{J'h A0\JA)2J£(J'l,M'q\JM)2.M'.q

This expression is averaged over polarizations of the emitted photon and integratedover the angle of emission. When we carry out the sum with the help of the rule (B.4)and use explicit values of the Clebsch-Gordan coefficients, we find the spontaneoustransition rate to be

w(J - J - 1) =

J2-A2 = 32B3D2 J2{J2 - A2)

3hc3 JilJ + 1) ~ 3hc3 2 7 + 1 '

Clearly, this result is valid for the transition considered only if the vibrational andelectronic states of the molecule do not change. (See Problem 1.20.) Also, it isobvious that in a diatomic molecule composed of identical atoms, symmetry con-siderations require D = 0. The above-cited transition rates then vanish. (See alsoProblem 1.18.)

If the spin of the molecule is nonzero, the results must be modified some-what. We consider two limiting cases. If the spin interaction with the molecularaxis (due to spin interactions of the electrons of both atoms) is large comparedto the spacing of rotational levels, then rotation does not destroy this coupling.Thus the projection of the total spin onto the molecular axis is conserved. Wedenote this quantity as 2 . Then the projection of the total angular momentum isH = A + S. In this case, Eq. (1.43) need be modified only by the replacement of Aby a .

In the opposite limiting case, when the interaction of the spin with the molecu-lar axis is small, rotation destroys the coupling of the spin with the axis. We canthen introduce the conserved sum K of the orbital and rotational angular momenta.The total angular momentum J is also conserved and is given by J = K + S,where S is the spin vector of the molecule. Each rotational level splits into amultiplet with 2 5 + 1 components, which have angular momenta ranging fromJ = K — S to J = K + S. If we do not specify the component of the multi-plet, then the total rotational transition rate is obtained from Eq. (1.43) by replacing


J with K. However, if we are interested in the relative probabilities for the in-dividual lines of the multiplet, then we obtain expressions analogous to those ofProblem 1.13. Specifically, in the present case, the angular momenta K and S areadded in analogy to the addition of momenta L and S in the fine structure of lightatoms. There is no coupling of the angular momenta K and S with the molecularaxis.

Problem 1.20. Find selection rules for radiative transitions between vibrationalstates of a molecule. The vibrational motion is to be considered in the harmonicoscillator approximation.

SOLUTION. Radiative transitions are possible between those vibrational states forwhich the dipole moment matrix element (v|D|i;') is nonzero. Here, v and v' arevibrational quantum numbers, and D is the dipole moment averaged over that partof the electron configuration that does not change during the vibrational transition.The electron state is thus presumed to be given. The distance between nuclei in theparameter D is regarded as a given parameter. Such an approach is based on theadiabatic approximation, justified by the fact that the motion of the nuclei is muchslower than the electron motion.

We assume that the amplitude of the vibrations of the nuclei is small as comparedto the distance between them. We can then employ the expansion

where the Qi are normal coordinates, and the index / enumerates the type of vibration.The quantity Do represents the dipole moment of the molecule at the equilibriumconfiguration of the nuclei. Derivatives of the normal coordinates are also evaluatedat the equilibrium configuration.

We consider the molecular vibrations within the framework of the harmonic os-cillator model. Then the matrix element of the normal coordinate Qt in the secondterm on the right-hand side of the last expression is nonzero only for transitions witha change of the vibrational quantum number by one. This matrix element is of theform

(v\Qi\v-l)=iCDi

where yn is the reduced mass of the molecule for the given type of vibrations, and co;is the frequency of this vibration in mode /. It is seen from the above formula for themean dipole moment that transitions with a change of vibrational quantum numberby two are possible. They correspond to the third term on the right-hand side of theexpression for the mean value of the dipole moment.

We now compare the expressions for the radiative transition rates with a changeof vibrational quantum number of two to those with a change of one. The ratio of

these radiative transition rates is given by

w(v -> v - 2)

w(v —> v-l)

(v\Dv -

(v\D\v -

h(v - 1)8

2)

1>

2

2-'I

\

' <?2D \]

KdQidQk) 0 O

- 2 )

1

2 r

. i

1/2

- 2

where we have used the rule of matrix multiplication that

{v\QiQk\v - 2) = (v\QAv - \){v - \\Qk\v - 2).

Let us estimate the value of cot. The potential energy fxicofr2/! of harmonic oscilla-tions is of the order of the electron energy ee\ when the oscillation amplitude r is ofthe order of the distances ae\ between atoms in a molecule. It follows from this that

ho)i - (m//x;)1/2£el,

where m is the electron mass. The derivatives of the dipole moment are of the orderof atomic quantities. Hence, we obtain the estimate for the ratio of probabilities

w(v —> v — 2) / ' m \

w(v —» v — 1) - v.

Numerically, this ratio is of the order of 10 2-10 3.Thus the most probable radiative transitions between vibrational states take place

with a change of vibrational quantum number of one. Transitions with a change oftwo in the vibrational quantum number are much less probable if the vibrationalquantum number v is not very large. The ratio of the rates becomes of order unitywhen the quantum number v is of the order of (ix/m)1^2 > 1. When this is true,the vibrational energy is of the order of se\, which would imply that the harmonicoscillator approximation is not valid. Consequently, such quantum numbers are notpresent in practice.

Transitions with v —• v — 2 also take place in the first-order term in the expansionof the dipole moment D if we take into account the anharmonicity of the oscillationsof the nuclei. To estimate the effect, we introduce the anharmonic term aQ3 intothe Hamiltonian describing vibrations. The value of a is of the order of an atomicunit. For simplicity, we shall consider only one type of vibration, so we may omitthe indexes / and k. In first-order perturbation theory, the correction to the harmonic

wave function t//°22 of the state with the quantum number v - 2 is of the form

-(0) _ .(0) %' •v' Sv-2 £v<

We now single out the term with v' = v - 1. Then we find that the matrix element(v\Q\v — 2) is of the order of

(v - l\Q3\v - 2)/ a h2v2

a (v\Q\v - 1) - 2~2-nco nco JJLZCOZ

Since hco ~ (m/jn)1/2£ei, we estimate the ratio of the radiative transition rates to be

wanharmO -> v - 2)w(v —> v — 1)

afl2V2V2

\ho)ji2o}2 J fiv

2

a2hv3 3

This ratio is of the order of unity at the quantum numbers v ~ (jLt/m)1/6, whenthe energy of the vibrational states, vhco ~ (ra/jn)1/3£ei is still small compared tothe electron energies se\. We find that the corrections to the transitions v —> v - 2due to anharmonicity of the nuclear vibrations are much more significant than thecorrections due to the dependence of the mean dipole moment on the distance betweenthe nuclei at large vibrational quantum numbers. However, both corrections are ofthe same order of magnitude for small quantum numbers v.

We have found that the rates for transitions with v —> v — 2 and v —• v — 1 areof the same order of magnitude if the correction to the wave function

\ 1/4

due to anharmonicity is comparable to the unperturbed harmonic wave functioni/>^2- ^ n e harmonic approximation is inapplicable under such circumstances, andsuch quantum numbers v are indeed absent even though the corresponding energiesare still small compared to the typical electron energies se\.

The conclusion can thus be reached that, in the range where the vibrational quan-tum number v is a "good" quantum number, the most effective radiative transitionstake place with a change in v of one.

Problem 1.21. Obtain the selection rules for rotational quantum numbers invibrational-rotational transitions of a linear molecule. Assume that the spacing ofrotational terms is small as compared to the spacing of the vibrational terms, andfind the relative transition probability to the state with a given rotational angularmomentum. (Take A = 0).


SOLUTION. This problem is a generalization of Problem 1.19, where the rotationaltransition rate without a change in the vibrational quantum number was determined.In that case, the matrix element of the projection of the dipole moment on themolecular axis was equal to the mean dipole moment of the molecule at the distancer = r0 between the nuclei, where r0 is the equilibrium distance. Such a matrixelement is zero for transitions associated with a change in the vibrational state as aconsequence of the orthogonality of vibrational wave functions. For this reason, weshould write the next term of the expansion of the mean dipole moment (in the senseof integration over electron coordinates) for small differences Q — r — r$. The termproportional to Q leads to matrix elements arising from the linear harmonic oscillatorcoordinate. It is well known that such matrix elements are nonzero for transitionsbetween neighboring vibrational states only. Thus the selection rule v' — v = ± 1 isvalid for the vibrational quantum number v.

Having established the change of vibrational quantum number, we now considerthe change of the rotational state. The dipole moment operator of the molecule isdirected along its axis. We define n to be the unit vector along this molecular axis. Inanalogy to Problem 1.19, the matrix element of the dipole moment is proportional tothe quantity

{JM\nq\J'M').

That is, nq bears the same relationship to n that Dq does to D. The spontaneoustransition rate of the molecule can be written in the form

w(v, J,M^ v't J'y M1) - \(JM\nq\j'M')\2,

where v, / , M are, respectively, the vibrational quantum number, the total angularmomentum of the linear molecule, and the projection of the angular momentum ona fixed axis in space. The primed quantities v\ Jf, M1 are the analogous quantumnumbers for the final state of the molecule.

The total transition rate for transitions to all rotational states is written as an inverselifetime, or

V w(v, J,M -> v1, J', M1) = - ./-^ TJ'M'q

Since a normalization condition is applicable,

J2 \(JM\n,\j'M')\2 = *£(JM\n2q\JM) = 1,

J'M'q q

then the last two expressions lead to

/, Af-> vf,Jf,Mf) = -\(JM\nq\j'M')\2,T

q = M' - M.

Now we calculate the matrix element of the operator

nq = I V 0, <p).

Using the relation (C.9), we obtain

O /M(7 /,M /) = \{JM\nq\J'M')\2

27' + 1= (7 1, M q\JM) (7 1, 00|70) ,

2 7 + 1where g takes the values 0, — 1, +1 . The second factor in this expression is nonzeroonly when J1 = 7 ± 1. Therefore the only transitions possible are those with achange in the rotational quantum number by one. Values of the function <£>jM(Jf, M1)for various values of J1 and M1 are given in Table 1.1.

The function <£>JM satisfies the sum rules

-> — / 2 7 + 1 "

They are obtained using the relation (B.3). It is seen also that

J'M'

Since energy increases with increasing vibrational quantum number v, then itis obvious that a spontaneous radiative transition will lead to the quantum numberv1 = v - 1. If the rate for this transition is averaged over projections M of themolecular angular momentum, then we obtain the spontaneous rate for emission of aphoton with an increase or decrease by one of the rotational quantum number 7 as

7 + 1 1w(v, J -> v - 1, 7 + 1) =

w(y, J -> v - 1, 7 - 1) =

27 + 1 T'

7 127 + 1 ?

TABLE 1.1. Relative Probabilities fc^cr, NT) for Molecular Vibrational-RotationalTransitions

M1

J1 M-\ M M + 1

(7 + M)(7+M- 1) J2-M2

7+ 1

2(27 - 1)(27 + 1) (27 - 1)(27 + 1) 2(27 - 1)(27 + 1)(7 - M + 1)(7 - M + 2) (7 + I)2 - M2 (7 + M + 1)(7 + M + 2)

2(27 + 1)(27 + 3) (27 + 1)(27 + 3) 2(27 + 1)(27 + 3)

Each of these expressions corresponds to the radiation of a photon with arbitrarypolarization. If we are interested in the radiative rate for the emission of a photon ineither of the two possible definite polarization states, these are equal to each other andthus each is one half of the total rate just cited. This fact follows from the observationthat, after averaging over projections of the total angular momentum, the rate foremission of a photon with a given polarization does not depend on the direction ofthis polarization.

Problem 1.22. Compare the rates for radiative transitions between rotational, vibra-tional, and electronic molecular states.

SOLUTION. Rate differences for radiative transitions are due to differences of transi-tion energies and to differences in the dipole moment operator matrix elements.

The ratio of photon energies for vibrational (ftcovib) and electron transitions (ft<oei)are of the order of magnitude (see Problem 1.20)

I)1/2

where /x is the reduced molecular mass. The matrix element ratio for the dipolemoment operators is of the order of magnitude (see also Problem 1.20)

(v\D\v - 1) l/2 / \ 1/4

where De\ is the matrix of the dipole moment operator for the electron transition, andv is the vibrational quantum number. Using these estimates, we obtain from Eq. (1.13)the ratio of the rates for radiative transitions with a change in the vibrational (wwlb)and electron (wei) states

w / a) \ 3 fm\ 1/2 I' m~ I 1 v I I ~ f (

It is evident that vibrational transition rates are significantly less than electronictransition rates.

Analogous estimates can be made for rotational transition rates (see Problem 1.19).Since the coefficient B in the rotational spectrum

sj = BJ(J + 1)

is of the order of

B -

we obtain for / > 1 the comparison

VVrot

Wei \V>


We conclude that rotational transition rates are much less than both electronic andvibrational transition rates.

Finally, it follows from the solution of Problem 1.21 that vibrational-rotationaltransition rates are of the order of the rates for vibrational transitions.

1.5 POLARIZATION OF RADIATION

Every photon is characterized by a direction of propagation that coincides with thewave vector k and by a polarization vector s that is perpendicular to the vector k.The polarization vector can be represented as a superposition of two vectors mutuallyperpendicular to each other and to the wave vector k. These two vectors determinetwo independent polarizations.

For induced radiation, the ejected photon has the same polarization as the initialphoton impacting on the atomic system. However, for spontaneous radiation, theproblem of the polarization of the emitted photon remains open for investigation.

Problem 1.23. Obtain the spontaneous emission rate from an excited atom of aphoton, expressed in terms of specific polarization and emission directions.

SOLUTION. The rate for the spontaneous transition k —• 0 is given by Eq. (1.14),where we must put rcw = 0. The statistical weight g0 is equal to (|)d 11/477, wheredfl is the solid angle element for the emitted photon, and the factor \ appears becauseof the two degrees of freedom for the photon polarization vector. We then obtain thespontaneous emission rate for a photon in a given direction with a given polarizationvector s as

dwk0 = ^ 2

We introduce the spherical components sq and Dq of the vectors s and D (see Ap-pendix C). Then this expression can be rewritten in the form

dwk0 = —— y2s-q(LMLa\Dq\L'M'La') dil, (1.44)

where L, ML, a are, respectively, the orbital momentum, its projection on the zaxis, and the remaining quantum numbers necessary to define the state. The samequantities with the primes refer to the final state 0. We also introduce the quantityAML =ML- M'L.

Only one term in the sum over q in Eq. (1.44) is nonzero for each of the threepossible transitions with AML = 0, ±1 . If AML = 0, then we obtain the rate foremission of a photon polarized in the plane defined by k and the z axis. We define 0to be the angle between the z axis and the direction n = k/k of the emitted photon.Then the direction of the vector s can be chosen to be either in the plane of n and

POLARIZATION OF RADIATION 41

the z axis or perpendicular to this plane. In the first case we have sz = sin 0, sincethe vectors s and n are normal to each other. In the second case sz = 0, since thevector s is perpendicular to the z axis. We thus obtain two spontaneous rates: onecorresponding to the emitted photon with polarization in the plane of n and the z axis;and the second rate corresponding to the emission of a photon with a polarizationperpendicular to this plane. These rates are

(1.45)

dw2 = 0.

Summing over the two nonzero terms and integrating over angles, we get the totalrate for photon emission polarized in the plane of n and the z axis,

w(LMLa->L'MLa') =

If ML = H= 1, then only one term is retained in the sum over q in Eq. (1.44), which is

sTX(LMLa\D±i\L'ML ± 1 a') = \{sx + isy)(LMLa\Dx ± iDy\L'ML ± 1 a1).

It determines the photon emission in the xy plane with right (left) circular polarizationcorresponding to the upper (lower) of the ambiguous signs.

If there is no physically imposed reason for selection of a z axis, then an atom willbe in each of its ML substates with equal probability. The transition rate dw(La —•L'a') is obtained by summing the expression for dw over M'L and averaging over ML.There is a simpler alternate way to arrive at the same result. We can elect to align thez axis in Eq. (1.44) with the direction of the polarization vector s. We obtain thereby

^ ^ \{LMLa\Dz\L'M'La')\2dClMLM[

a? 167rfic3 2L + 1 *-^

MLM'L

6<Trhc3 2L + 1\(La\D\L'a')\2dSl.

It is seen that the emission is isotropic. That this is so has the simple explanation thatall directions in space are equivalent. The quantity (La\D\L'a1) is the radial dipolematrix element. Integration over angles is equivalent to multiplication by 4TT becauseof the isotropy of the emission. That is, we have

wl2(La ^ L'a') = ^ 2L\ t \(La\D\L'a')\2.


This expression determines the total rate of photon emission with the fixed polar-ization direction, but it does not depend on this direction. When we sum over thetwo independent polarizations, we obtain the rate that establishes the lifetime of theparticular excited state k to be

w(La - Z.V) =-= A"\ \ |(La|D|LV)|2.r me5 2L + 1

Problem 1.24. The splitting of spectral lines that occurs when an atom is placed in amagnetic field is called the Zeeman effect. Calculate the relative intensity of Zeemanspectral lines both along and perpendicular to the direction of the magnetic field.

SOLUTION. Choose the z axis to be along the direction of the magnetic field vector.We consider first the emission along this axis. That is, the wave vector k is orientedalong the z axis. The polarization vector therefore lies in the xy plane. The photonemission rate, from Eq. (1.14), is proportional to the matrix element

\(LMLa\\y\L'M'La')s\2y

where L, MLy a and L'} M'L, a1 are, respectively, the orbital angular momentum, itsprojection on the z axis, and all remaining quantum numbers for the initial and finalstates. We choose two independent polarizations, Si = i* and S2 = iy, where i* and \y

are unit vectors along the x and y axes. The rate W||, summed over polarization statesof the photon, behaves as

wN ~ \(LMLa\Dx\LfMf

La')\2 + \(LMLa\Dy\LfMf

Laf)\2

~ ] T \(LMLa\Dq\L'M'La')\\q=±\

Recalling the solution of Problem 1.23, we can identify right circularly (AML = 1)and left circularly (AML = - 1 ) polarized light propagating along the z axis. Theirrelative intensities (called a components) correspond to terms in the above expressionwith q = - 1 and q = +1 , respectively. From Eq. (C.9), these relative intensitieshave the behavior

AML = 1 : W|{ ~ \(L, 1\ML, -\\L'fML - 1)|2,

- - 1 : w/jj" ~ \(L, 1;ML, \\L',ML + 1)|2.

Now we turn our attention to the observation of light in a direction perpendicularto the magnetic field vector. We can select this direction to be the y axis. Then thepolarization vector s of the emitted photon is in the xz plane. With the assignmentsSi = \z and S2 = i*, we find that the photon emission rate, after summation over thepolarizations, has the behavior

w± ~ \{LMLa\Dz\L'M'La')\2 + \(LMLa\Dx\LfMf

Laf)\2.


By definition, we have Dx = {D\ + D-\)/\J2, so then

2

; (Dq)k0

q=±\

2 ![Dq)k0\ + ~

9=±1

The last term of this expression must be zero, since for fixed states k and 0 bothmatrix elements (Di)kQ and (D-i)k0 cannot be nonzero simultaneously. If we haveAML = 1, then the matrix element (D-i)k0 is zero, and if we have AML = — 1, thenthe matrix element (D\)k0 is zero. We thus obtain

l(A*)fcol = ~ Z^ l(^M*ol •q=±i

The rate w± can thus be written in the form

wL ~ \{LMLa\Dz\L'M'La')\2 + \ T KLMLa\Dq\L'M[a')\2.

We again employ Eq. (C.9) in the calculation of the matrix elements. The inten-sity of the a components is half that of the intensity along the direction of lightpropagation, so that we have

AML = 1 : w+ ~ \\(Ly 1;ML, -\\L',ML - 1)|2,

\\{L, 1;ML,

The intensity of the component with A ML = 0 (the so-called TT component) isproportional to the quantity

w° ~\(L,UML,0\Lf,ML)\2.

Upon evaluation of the Clebsch-Gordan coefficients (see Table B.2), we find that therelative intensities of the various components can be as given in Table 1.2.

As a supplement to this table, we note that wf} = 0, wjj" = 2w~\_, and wjj" = 2w'L.Also, there can be no transitions with L1 = L for single-electron spectra, due to parityconservation (see Problem 1.17).

TABLE 1.2. Relative Intensities of the Components in Zeeman Splitting

wtw°±

wl

(L

(L

L' = l

+ ML){L

4(L2-

- ML)(L

1 - 1

+ ML~

-Mfr-ML-

1)

1)

(L +

(L-

V =

ML)(L -

AMIML){L •+

L

-ML^

-ML^

1-1)

M)

( i

( i

Z,'

-Mz ,+

4[(L-

+ Mi +

= L

1)(Z

M) 2

1)(£

+ 1

. - M i -

• + ML-

1-2)

1-2)


Problem 1.25. The splitting of spectral lines that occurs when an atom is placed inan electric field is called the Stark effect. Find the relative intensity of Stark spectrallines when the light is observed along the direction of the electric field vector, ascompared to the direction perpendicular to the field.

SOLUTION. A constant electric field causes both shifting and splitting of atomiclevels. If we do not consider the hydrogen atom, then such shifts and splittings aredetermined by second-order perturbation theory. As in Problem 1.24, the polarizationof the radiation is determined by the direction of observation. Let the z axis be alongthe direction of the electric field vector. When the light is observed along this direction,the emitted photons are polarized in the xy plane and correspond to transitions withMi —> Mi ± 1. Such components are called cr components as in the magnetic fieldcase.

In the direction perpendicular to the electric field vector, IT components are ob-served as well as a components. They correspond to transitions with Mi —>• ML, andare polarized in the xz plane. If ML = 0, then we can use the results of Problem 1.24,putting ML = 0.

If ML ± 0, then levels are doubly degenerate corresponding to the two signspossible for Mi. Hence the intensities of the IT components are double the resultsfound in Problem 1.24. The degeneracy arises from the fact that transitions withMi —• Mi and with — Mi —> — Mi have the same energies.

In the case of a components, we find that the energies of transitions with ML —>Mi — 1 and with — Mi —• 1 — Mi are the same. The first transition is associated withthe emission of right-handed circularly polarized light and the second to the emissionof left-handed circular polarization of the light. It is seen from Table 1.2 that theirintensities are the same, so that the sum of their intensities corresponds to a doubling.

Hence the factor 2 appears in all entries in Table 1.2, so this factor can be omittedaltogether for this case. In applying the results of Table 1.2 to the present problem,we should replace w+ by w(|ML| —> \ML - 1|), and w~L by w(|ML| —> \ML + 1|).

Problem 1.26. Alkali atoms are excited from the ground state to the resonant excitedstate 2Pj by circularly polarized light and then deexcited with the emission of aphoton. Neglecting atomic collisions, determine the spin polarization of atoms in theground state. That is, find the mean value of the atomic spin.

SOLUTION. According to the statement of the problem, an atom in an initial s state isexcited to a state with total angular momentum J with projection M onto a given axis.Polarization of an atom appears because the probability of excitation by circularlypolarized light depends on its spin polarization. After emission of a photon, the atomreturns to the same s state with the projection cr1 of its spin on the reference axis.This projection can take the values + ^ and — ^.

We denote by w(cr —> a1) the transition rate between states with spin projections aand cr1 in the process of excitation and deexcitation we have described. The numberdensities of atoms with spin projections + ^ and — ^ are labeled as N+ and N-,respectively. The rate equations for populations of these atomic spin states in the

initial s state are

dt V2

For a steady-state process, we neglect the left side of these equations, and find themean spin of the atom to be

N+-N- iw(-l^+l)-w(l^-l)

+ N-) 2w(-I->+i) + w(i->-i)

We now establish the relative probabilities of the radiative transitions. We takethe light to be right circularly polarized, so that the projection of the orbital angularmomentum increases by one upon excitation: m —• m + 1. The relative probabilityfor atomic excitation from the state with spin \ and spin projection a into the statewith total angular momentum J with projection M = a + 1 is, from Problem 1.12,

The relative probability for photon emission by an atom in this JM state for transitionback to an s state with spin projection &' is proportional to the squared Clebsch-Gordan coefficient

All of this leads to the transition probabilities

The conclusion is that the mean value of the spin, 5, is \. This signifies that theinitially unpolarized atomic system becomes polarized through interaction with thecircularly polarized light.

The physical meaning of this result is instructive. Suppose that J = | . Then onlyatoms with the initial value a — — \ are excited. They deexcite to both of the statesa' = \ and or' = - \ . However, reexcitation from the state a' = - \ can occur,whereas that from a1 = \ cannot. Therefore, the population of the a = - \ statewill diminish with time and all electrons will finally be in the a = \ state.

If the excited state has the total momentum J = | , then excitation of both thea = \ and the a — - \ initial states will take place. In the latter case, the excitedstate with M — | is produced. The selection rules allow deexcitation of this stateonly to the state with <J' = 5. As time passes, all electrons in the state with a = — \

will disappear, since transition from a = \ to the state with a' = - \ is forbidden.Hence all electrons will concentrate in the a' = \ state.

Problem 1.27. Establish the time required to achieve the spin polarization discussedin Problem 1.26.

SOLUTION. In contrast to the above problem where the steady-state situation wastreated, we now retain the time derivative in the rate equations. We can write theseequations as

dN+/dt = wN-, dN-/dt = -wN-,

where w stands for w{-\ —• | ) , since we are taking into account the conclusionfrom Problem 1.26 that w(\ —> - \) = 0. The initial conditions are

N-(0) - i, N+(0) = I,

since the initial s state has no polarization before the action of the external field. Thesolution of the rate equation is, thereby,

N+(t)= 1 - £ <

The time required to spin polarize the atoms is thus of the order of magnitude of 1/w.

1.6 RADIATIVE TRANSITIONS WITH TWO PHOTONS

We have presumed in the preceding discussion that typical times for two-photontransitions are large as compared to those for single-photon processes if the radiationfield interacts with a nonrelativistic atomic system. The ratio of the perturbationpotential — D • E to a typical transition energy is the small parameter that explainsthe negligible rates for two-photon transitions. This ratio is of the order of the ratioof a typical atomic velocity e2/h to the speed of light c in the case of spontaneoustransitions. However, if the single-photon transition is forbidden, then the radiativetransition is determined by the less effective two-photon process. A variety of two-photon transitions is presented in Table 1.3.

TABLE 1.3. Two-Photon Processes

Two-Photon Process Scheme of the Process

Emission of two photons A* —> A + k + k'Absorption of two photons A + k + k'—> A*Rayleigh scattering of light A + k —* A + k', k' = kRaman scattering of light A + k —+ A* + k;, k1 =£ k

RADIATIVE TRANSITIONS WITH TWO PHOTONS 47

In Table 1.3, A and A* are ground-state and excited atomic electrons, and k and kf

are the wave vectors of the two photon fields that take part in each of these processes.

Problem 1.28. Find the expression for the rate of a two-photon process.

SOLUTION. We use the same method for the treatment of two-photon processes as wasemployed earlier for single-photon transitions. (See Section 1.1.) We treat only thosetransitions that are not in competition with nonelectromagnetic processes. We employthe formalism of perturbation theory and assume that single-photon transitions areforbidden. We take the perturbation potential to be of the form - D • E.

The transition operator for the interaction of an atomic system with the electro-magnetic radiation field has the form

V= -Ei-Dcoscoif - E2-Dcosco2f, (1.46)

where Ei and E2 are the electric field vectors for the electromagnetic waves withfrequencies (L>\ and co2, respectively; and D is the dipole moment operator of theatomic system. The formalism of second-order perturbation theory then defines thetransition rate. The system of equations (1.4) gives, for the transition amplitude fromthe state 0 to the state ra,

= ~1T Ift Jo

Here, hcomk is the energy difference between states m and k, and the transitionamplitude of first-order perturbation theory is given in Eq. (1.5) by the expression

\t) = -l- [ Vk0(t')exp(i(ok0t')dt'«• Jo

In [ (ok0 - (D\ cok0 +

i? r* / P l > ( * o 2)] _L 1 - exp[i(G>*0 + i)t] \

2n y cok0 - 0)2 (i)k0 + (i)2 J

The two-photon transition is a resonant process, so that the relation

Wm0 = + ( ( 0 1 + (D2)

is almost exact, with the upper of the ambiguous signs referring to photon emissionand the lower sign corresponding to photon absorption. This relation is the energyconservation law for two-photon transitions and is analogous to the single-photonenergy conservation condition employed in Section 1.1.

Because of the resonance behavior, we retain in the two-photon transition am-plitude c{^\t) only those terms that contain the quantity a)m0 = T(o)\ + o^) in thedenominator. This denominator is zero when the energy conservation condition is

fulfilled exactly. We obtain

- exp[/(com0 ± a)! ±

where si and s2 are unit polarization vectors directed along the electric field vectorsEi and E2. The upper sign in Eq. (1.47) refers to the emission of a photon, and thelower sign is for the absorption of a photon.

We did not consider the case with Ei = E2 and (L>\ = co2, so that resonances withthe absorption or emission of two photons of the same electromagnetic field will notbe considered here.

In addition to the resonance terms explicitly exhibited in Eq. (1.47), additionalresonance terms exist, arising from terms in the expression for the amplitude ck

l\t).The terms we have reference to are those terms that contain com ± OJ\ or comfc ± co2 in thedenominator. They occur because of a sudden turn on of the perturbation. This suddenturn on of a perturbation is a mathematical device used to simplify the calculationsand is not a condition achieved in practice. In actuality, the perturbation is turned onsufficiently slowly that energy is conserved, and resonance denominators of the typeajmk ± cx)\ or (omk ± (x)2 do not occur. Such terms correspond to Fourier componentsof a perturbation with sudden onset and are not an attribute of a monochromaticelectromagnetic field. For these reasons, we give such terms no further consideration.

We define a transition rate as

wOm = lim \c%\t)\2/t,t—>cc

related to the use of the relation

sin2 atlim — = o(a)

in Section 1.1. From Eq. (1.47) we thus obtain the two-photon transition rate

(EXE2)2 ( ^ [(s2'Dm.)(srD.o) ^ (srDm.)(s2'D.0)l\2

g/W°m = 0*4 \ / ^ Z " + Z r O^mO ±(O\± C02).

S n q ^ L ^ k o ± o>\ (*>ko ± <*>2 J\ K J

(1.48)As previously, the upper sign corresponds to the emission of a photon and the lowersign to the absorption of a photon. The quantity hojk0 is the transition energy betweenthe two stated levels of an atomic system. Expressions (1.47) and (1.48) are applicableif the energy denominators co o — wi o r wfco — W2 a r e large compared to the width ofthe state k. If this is not true, we have a resonance transition, which is a significantlymore complicated problem.

We suppose also that the frequencies (x>\ and a>2 are sufficiently different fromeach other that waves from these two modes do not interact with each other (as in


the single-photon transitions considered in Problem 1.1). That is, we presume thatconditions for phases of the modes or frequency differences of neighboring modesare fulfilled.

Equation (1.48) has been written for the case of classical electromagnetic fields.To obtain the analogous expression when the radiation field is quantized, we use thecorrespondence principle. This takes the form of expressing the energy density of theelectromagnetic field in terms of the number of photons n^ in a state with energy hco.This semiclassical substitution is valid when n^ t> 1. However, we can generalizethis substitution by extending it to the situation where n^ ~ 1. In this way we cantake into account both induced and spontaneous properties.

In the solution of Problem 1.1, we found that

E2 d(D _ dk— —- nSlT 00)

We omit here a factor of 2 on the right-hand side, since we suppose that the photonpolarization is fixed. The quantity 8o) represents the frequency difference of neighbor-ing modes (see the beginning of Chapter 1). This equation is correct when n^ > 1, aswe have employed in its derivation that the field intensity and the number of photonsin a given state experience negligible change in a transition. In actuality, the numberof photons increases by one in emission and decreases by one in absorption of light.A more exact connection between the radiation field intensity and the photon numbercan be obtained from the results of Problems 1.1 and 1.2. Specifically, Eqs. (1.9) and(1.13) show that the quantity E2(do)/8o)) is equal to hcoin^ + \)dk/n2 for photonemission and is equal to tioon^dk/ir2 for photon absorption.

When we sum Eq. (1.48) over field modes (as in Problem 1.1) and make thesubstitution ^T^ —> do)/8o), we find

dwOm = (1.49)

/ 1 1\ / 1 1X («. . + 2 ± 2 J (""» + 2 ± 2

The upper of the ambiguous signs is used for absorption and the lower sign foremission of the corresponding photon. The number of photons in a given state isgiven by nw, and the photon polarizations S\ and S2 are taken to be fixed.

Equation (1.49) is the principal formula of the theory of two-photon radiativetransitions. It applies to all of the cases listed in Table 1.3. The problems that followcorrespond to the specific two-photon processes displayed in Table 1.3.

Problem 1.29. Calculate the lifetime for a two-photon spontaneous decay of anatomic state.

SOLUTION. The scheme of the process, as listed in Table 1.3, is

A* -> A + k! + k2.

From Eq. (1.49), the rate l / r 2 for two-photon spontaneous decay is given by

1 1 i •< < , , ^ , ^ ( 1 5 Q )

2

To arrive at this expression we have set n^ = ra^ = 0, arising from the absenceof radiation from the initial state; and, using a) = ck, we have employed the deltafunction in Eq. (1.49) to accomplish the integration over co2- The quantities dfl^ anddflkl in Eq. (1.50) refer to the solid angles for the two emitted photons. We haveassigned the index 0 to the initial state and the index m to the final state. The energyconservation law that follows from the delta function of Eq. (1.49) is

(*>0m = O>\ + G>2-

It should be kept in mind that the probability for emission of two photons withfrequencies o)\ and C02 is very small as compared to the probability of emission of asingle photon with the frequency co\ + C02. However, if the selection rules prohibit thesingle-photon transition, then only the two-photon process can occur. From Eq. (1.14),the lifetime for single-photon emission has the order of magnitude

n ~ (hc/e2)\a,

where ra is a typical atomic time, which is, in turn, of the order of magnitude

ra ~ h3/(me4).

Equation (1.50) then leads to the estimate

r2 ~ (hc/e2)6ra

for the lifetime with respect to two-photon decay. Therefore the lifetime for two-photon decay is (hc/e2)3 ~ 106 times longer than that for single-photon decayif both processes are allowed by the selection rules. In the following, we shallconsider transitions that are forbidden for single-photon decay. Important examplesare transitions between two states, both with angular momentum J = 0.

Averaging over polarization states and integrating over the directions of propaga-tion for the photons can be done in general. To accomplish this, we employ

where averaging is carried out both over polarizations and over photon propagationdirections. Summing over these quantities gives a result greater by a factor 2 X 4rr

than averaging. We find for the inverse lifetime the result

The factor o)\u>\ in the integrand has a sharp maximum at the point where co\ =(*>2 = wO m /2, since W] + co2 = wOm. If the state 0 is the first excited state, thenwe have <OJQ ^ 0 for all other states j . This implies that CJJQ =£ —<o\, — o>2. In thiscase there can be no resonance denominators in the above expression, and we can seto)\ = (x>2 — WQ W /2 in these energy denominators in accordance with the mean valuetheorem. Since we now have

we obtain

1T2 3l57rh2c6 4

yO + (Dk)mj(Vi)jQ

0>jO

where i, A; can take on the values - 1 , 0, +1 corresponding to vector spherical coor-dinates.

In summary, we see that two-photon emission will occur with both photons havingabout the same frequency and that this frequency corresponds to about half the totaltransition energy.

The initial and final states can be degenerate with respect to the magnetic quantumnumber. If this is the case, we should average the transition rate over the magneticquantum numbers of the initial state and sum over the magnetic quantum numbers ofthe final state. A typical example is the two-photon radiation from the 2sx/2 metastableexcited state of hydrogen. The transition is to the ls}/2 ground state of hydrogen. Thewave functions of the initial, final, and intermediate states are of the form

Vj = Rm(r)Ylm([iy9 M = -1, 0 - +1 ; zi = 2, 3, 4 , . . . ,

where Rni is the radial wave function of the hydrogen atom. We introduce the matrixelement of the radial dipole moment,

Dnnl = I RnJo

,0(r)Rn](r)r3 dr,

in terms of which we can write the two-photon transition rate from the 2s{/2 state ofhydrogen in the form


«2l/2

The sum over the principal quantum numbers can be evaluated only by numericalmeans. The sum contains both bound and continuum states. The primary term in thesum is the one corresponding to the 2p^/2 state (n = 2). If we retain only this term,we obtain the estimate T2 = 0.045 s for the lifetime, which is of the order of the exactvalue T2 = 0.15 s.

Problem 1.30. Calculate, within the constant energy denominator approximation,the lifetime for an excited atomic state to decay by two-photon spontaneous emission.

SOLUTION. Label an initial state with the subscript 0 and a final state with m. Thegeneral expression for the two-photon decay rate was given by Eq. (1.50) in thepreceding problem as

1 1 /

y ^ |~(D-Si)mj(D-s2);0 | (D-s2)mi(D-Si)>0

jO)2

We wish to carry out the summations in this expression within the constant energydenominator approximation. We accomplish this by the introduction of a fittingparameter representing the mean value of the energy differences in the denominatorsin the summation considered. By this means, we obtain

1

j0>k0

•[(D-s2)(D-Sl)]m0,

where we have used the completeness property of the eigenfunctions. The quantityo)£o is the fitting parameter for this sum. The approximation will be accurate if themost important terms in the sum are those that have nearly equal energies.

In this fashion, we obtain the two-photon transition rate in the form

1 1/ o)j 0)2

JmO

We wish now to carry out the integrations over the directions of photon emission.Since both photons are unpolarized and are emitted isotropically, we can replace thesolid angle of the ejected photon dflk by the solid angle dfis, where s is the unitvector for the photon polarization. Consequently, the integrals over the angles takethe form

In the interests of simplicity, we now specialize to a single-electron atom, so thatD = er. The wave functions of the initial and final states are 0 and tym, and wesuppose that these wave functions are real. The integral just written is equal to

e4 / ^ ( r ' ^ ^ o t r ' l t o W ^ f l s , d(lS2(r * si)(r' • Si)(r • s2)(r/ • s2)drdr'.

From the relation

cos 6r's = cos 6rr> cos 0rs + sin drr> sin 0rs cos <p,

we see that

/dn 8(r-s)(r /-s)=^-(rT /) ,

where the indices on the 0 angles identify the vectors between which the angle istaken, and (p is an azimuthal angle for the vectors r and s. With this result taken intoaccount, we find

/ •r')2)2 drdr1

EAltogether, we find that the lifetime is given by

1•£|<D<T2 97rf t2c6 Z-, - v — ^ ' J - 1 - 2 ^ w + Wl (Ok() + c

This expression should be multiplied by 4 to take into account the two polarizationstates of each of the two photons. We should also keep in mind that the energyconservation condition a)Qm = o)\ + o>2 is fulfilled in this result.

We can achieve notational economies by introducing the dimensionless variableof integration x, defined by

«i = (o>om/2)(l + x), or o)2 = (o>0m/2)(l - x); |x| < 1,


and representing the extent to which each of the photon frequencies deviates fromthe mean frequency. In these terms, the two-photon transition rate can be written inthe form

where the function / is

- x 2 ) 3

An important property of this function is that it is a smoothly varying function of itsargument, and varies only slightly over the entire range of the argument. For t —> oo,we have fit) = ^ « 0.457, while at the opposite limit, when t = 0, we havefit) — | ~ 0.667. For intermediate values of the argument t, the function fit) takeson values between the two limits shown.

In particular, for two-photon emission from the 2sx/2 excited state of hydrogento the Is 1/2 ground state, upon substitution of the explicit radial wave functions forhydrogen, the result is that

T2 2l7Th2C6/ RiO(r)R2o(r)r4 dr f (Wk2

\(O2\

where the indices 1 and 2 denote the ground and first excited s states, respectively.When we substitute / = 0.56, the mean of the extreme values of / , we obtainT2 = 0.12 s, which is close to the known exact result 0.15 s.

Problem 1.31. Find the selection rules for vibrational-rotational transitions in theRaman scattering of light by molecules.

SOLUTION. Raman scattering is a process in which light is scattered by a quantumsystem such that the initial and final quantum states of the system are different.

We use selection rules for vibrational and rotational transitions that were foundpreviously for single-photon transitions. To apply these rules to Raman scattering,we use them first for the transition from the initial to an intermediate state, and thenfor the subsequent transition from this intermediate state to the final state.

According to the solution of Problem 1.20, the vibrational quantum number vchanges by one in a single-photon transition. Therefore, the change of the vibrationalquantum number v between initial and final states in Raman scattering is

At; = 0, ±2.

In similar fashion, we found in Problem 1.19 the single-photon radiative transitionselection rule for a linear molecule that (A = 0)

A 7 = ± l , AAf = 0, ±1,

POLARIZABILITY OF MOLECULAR GAS 55

where A J and AM are the change in the rotational angular momentum and in its pro-jection on the molecular axis, respectively. When we apply these rules to the radiativetransition between initial and intermediate states, and then from the intermediate stateto the final state, we find

A7 = 0, ±2, AM = 0, ±1, ±2.

According to the solution of Problem 1.21, these same selection rules are also validfor vibrational-rotational transitions.

1.7 POLARIZABILITY OF MOLECULAR GAS

We define the tensor for the dynamic polarizability of a molecule in the scattering oflight by using Eq. (1.48). We obtain thereby

qq' * L h(<*>ko ~ G>I) h(a)ko + o)2) J '

where Dq is the component of the dipole moment in terms of projections q definedeither in the spherical vector coordinate system (q = — 1,0, +1) or in the rectangularCartesian coordinate system (q = x, v, z). The quantities co\ and a>2 are the frequenciesof the initial photon and the scattered photon. The polarizability tensor defines thepolarization properties of light upon being scattered by the molecules.

When we take the frequencies u>\ and co2 to be small as compared to typical electrontransition frequencies, we can neglect o)\ and co2 in Eq. (1.51). The concomitantconsequence of this condition is that light of such low frequency cannot exciteelectronic transitions. Therefore, the state m in Eq. (1.51) can be different from theinitial state 0 only in terms of vibrational or rotational quantum numbers.

Because of the small energy intervals in vibrational and rotational spectra, weconsider such scattering to be quasi-elastic, which allows us to denote both initialand final molecular states by the index 0. Furthermore, in the calculation of the quasi-elastic scattering rate, we should average this rate over final vibrational and rotationalsubstates with nearby energies. The small energy intervals in the vibrational androtational spectra permits such averaging to be done classically, that is to say, overorientations of the molecule in space.

From Eq. (1.48), and recalling the condition that photon energies are small ascompared to electron transition energies, we find that the scattering rate for a photonof fixed polarization has the behavior

w°kO

The averaging indicated in Eq. (1.52) by the superior bar refers to an averaging overfinal vibrational and rotational states at a fixed electron state 0. The sum over the


index k is taken over all intermediate molecular states, including electron states. Thepolarization state of the incident light is labeled by Si, while s2 is the polarization ofthe final scattered light.

From Eq. (1.51), the static polarizability tensor is of the form

XDq)Ok(Dql)k{)

Equation (1.52) then shows that we can express the transition rate in terms of thepolarization tensor as

(1.53)

We shall define the degree of polarization P. First, we select space axes so thatthe z axis lies along the polarization direction S\ of the incident light, and the x axisis oriented such that the wave vector k2 of the scattered light lies in the xz plane.The polarization of light is normal to the direction of its propagation, which is thedirection of the wave vector. Therefore the polarization vector s2 of the scattered lightcan be either in the xz plane or normal to this plane. In the latter case, it is along thedirection of the y axis. Then the degree of polarization is defined as

= w(s2 _L iy) - w(s2||iy)w ( s ± i ) + w ( s | | i ) ' l ' ;

The quantity P as defined by this relation is such that P = 1 corresponds to totalpolarization of the radiation (in the xz plane), while the value P = 0 corresponds toa lack of any polarization.

Problem 1.32. Calculate the degree of polarization of linearly polarized light scat-tered by a gas of linear molecules through the angle 0. This scattering angle is theangle between the wave vector of the incident light and the wave vector of the scat-tered light. The frequency of the scattered light is presumed to be much less thantypical frequencies for atomic electron transitions. The result should be expressed interms of the eigenvalues of the polarizability tensor.

SOLUTION. We take the polarization vector s2 of the scattered light to be in the xzplane. From Eq. (1.53), we find that the intensity of the light with polarization in thisplane is of the form

w(s2 _L iy) ~ (azz sin 6 + axz cos 0)2.

We now align the y axis with s2. The intensity of the light polarized in the directionnormal to its incident polarization has the behavior

i

Hence we obtain the degree of polarization of the scattered light in the form

(azz sin 0 + axz cos 6)2 - a2y

(azz sin 0 + axz cos 0)2 + a^

Now we take into account the symmetry properties that arise when averaging overthe rotational states is done. From the classical point of view, this averaging is overthe rotation of the molecule around the axis normal to the molecular symmetry axis.It follows from these considerations that

axy = axz = azy a n d azz<*xz = 0-

We are thus led to

P = («~ -™^ s i n

a 2s in 20 + a 2( l + cos2 Q)'

where we define

2 _ 2" 2 _ 2~ — 2~

When 0 = 0, then we have P = 0, while P has its maximum value at 0 = TT/2,where

_ op ~ a?

We denote by ay the eigenvalue of the polarizability tensor of the linear moleculealong its axis, so that

where £ is the axis along the moving molecular axis. We also define a± as theeigenvalue in the direction iq or in the direction £, the mutually perpendicular axesnormal to the molecular axis as well as to each other. This eigenvalue has theexpression

2(D7,)0fc(DT,)JM) _

and we note that both the 17 and £ axes are moving axes. From the definition of theeigenvalues, we find that

The next step is to express the degree of polarization P via these eigenvalues. Theangle between the molecular axis and the fixed axis z is denoted by 6. The z axisdefines the direction of polarization Si of the incident light. We elect to place themolecular axis in the xz plane. (This does not affect the end result, which would bethe same were we to place the axis in the yz plane.) We then find

Dz = Dc cos 0 + Dv sin 0, Dx = -Dc sin 0 + Dv cos 0.

It follows that

azz = ay cos2 0 + a i sin2 0,

axz = (a± — ay) sin 0 cos 0.

When these expressions are squared and averaged over the angle 0, we obtain

al = aycos4 0 + a2 sin4 0 + 2a±a\\sin2 Ocos2 0

^ f 3 a 2 ) ,

a? = (« i ~ c*||)2sin2 0cos2 6 = j^(ot± - a^)2.

In particular, the maximum degree of polarization is

6a2 + 8a||Q!x + a2

* max ~ZZ 9 '. ~Z TT •

5(0:2 + 2a!2)

This can achieve the value PmSLX = 1 when a^ = a±.

Problem 1.33. Express the polarizability tensor of a rotating linear molecule in termsof its eigenvalues in the moving coordinate system associated with the molecular axis.The photon frequencies are presumed to be small as compared to typical electrontransition frequencies.

SOLUTION. The polarizability tensor [see Eq. (1.48)] for Raman scattering can, ac-cording to Eq. (1.51), be written as

= y - $Pqi)mk(Pq)kX> + (Dq)mk(Dq')k0qq i L M<»ko ~ a>$ M<»ko + (02)

Here, Dq is the component of the dipole moment expressed in spherical vectorcoordinates, and u>\ and co2 are the frequencies of the incident and scattered light,respectively. Our goal is to separate out the rotational degrees of freedom and toexpress the polarizability tensor via its eigenvalues in the coordinate system movingwith the molecule.


The general formulas (C.ll) relate the matrix element of the dipole momentoperator in the rest system to the values in the moving coordinate system. We have

(Dq)ko = (J'M'K'k\Dq\JMhQ)

Yj-^j) (J'hMq\JM)(J'l, AVI/AXA'*|/)M|A0>,

where J, M are the initial state values of the rotational momentum and its projectionon the fixed direction, / ' , M1 are the values of the same quantum numbers in theintermediate state, and A and A' = A - /x are projections of the electron angularmomentum on the molecular axis in the initial and intermediate states, respectively.The indices 0 and k represent all other quantum numbers of the initial and intermediatestates, respectively.

When we account for the fact that the energy spacing of neighboring rotationalmolecular states is small as compared to energy differences for electronic and vibra-tional states, we find the polarizability tensor to be

(J'l,M'q\JiMi)(J'l,A'fi\JiA)(J'l,M'q'\JM)

In this expression, J\ and M\ are the final state rotational momentum and projectiononto the fixed direction, and D^ is the projection of the rotational-coordinate-systemdipole moment onto the molecular axis. In arriving at Eq. (1.55), we have taken intoaccount that the projection of the electron angular momentum onto the molecularaxis is unchanged from initial to final state, since we have that only the rotationalstate of the molecule changes in Raman scattering.

Equation (1.55) gives the connection between the polarizability component and itsvalues in the coordinate system where the symmetry axis coincides with the molecularaxis. In particular, if the photon frequency is small compared to the typical frequencyfor electron transitions, this equation allows us to express the polarizability tensorcomponent as a function of its eigenvalues as

a\\ = Z! h(D'kO

where D% and D^=Q are projections of the dipole moment operator onto the £ axisdirected along the molecular axis. In analogous fashion, we can express the other


component of the polarizability tensor as

/V A, IV

where Dv and D^ are projections of the dipole moment onto the TJ and f axes, whichare perpendicular to the molecular axis. The indices 0 and k refer to states of thenonrotating molecule.

In the limit of large (i.e., classical) rotational momenta of the molecule, wherej « j 1 « jx > i9 and recalling the condition that the photon frequencies are to besmall, Eq. (1.55) gives

,, 'l A'l\JA)(J'l,

+ (J'l, Af - l\JA)(J'l, A' - 1UIA>]}.

In deriving this result, we have made use of the approximation that

2J1 + 1[(27 + 1)(27} + I)]1/2

:=; 1

which is justified by the large rotational momenta premise, J ~ J1 ~ J\ > 1. Wehave also used Dg = 0, and D^ — D^ — D±\.

2RADIATIVE TRANSITIONSBETWEEN DISCRETE STATESIN ATOMIC SYSTEMS

The formalism for radiative transitions developed in Chapter 1 will now be applied totransitions between discrete states in a bound system, which is the simplest applicationfor this formalism. Transition energies as determined by energy conservation lawsare uniquely specified and can take on only discrete values. In application to realphysical systems, however, it is necessary to take into account the broadening ofspectral lines arising from various interaction mechanisms.

Equation (1.7) states that the probability per unit time for a radiative transitionfrom a state labeled 0 to another state labeled k has the form

w

where ha)k0 is the energy difference between the states. We introduce the distributionfunction aw for the absorption or emission of a photon of frequency co. That is,aajdco is the probability that the photon has a frequency in the interval from o) toa) + da). In accordance with the behavior of the transition probability given above,this distribution function has the form

~ co)dco, (2.1)

and it obeys the normalization condition

a(I)da> = 1. (2.2)/ •

It follows from Eq. (2.1) that, on a scale of the order of (ok0, the photon frequencydistribution function has the form of a delta function. This means that the frequencydifference between most photons and o)fc0 is small compared to cofc0. However, on


62 RADIATIVE TRANSITIONS BETWEEN DISCRETE STATES IN ATOMIC SYSTEMS

smaller scales, there will be a departure from the delta function form for the distribu-tion function. In other words, the distribution function does possess some structure.It is this function that determines the photon absorption and emission cross sectionsfor the atom. We shall now study the structure of the distribution function a^ and itsdependence on the mechanisms for the broadening of spectral lines.

2.1 RADIATIVE AND DOPPLER BROADENINGOF SPECTRAL LINES

The spontaneous emission of a photon by an atomic electron in an initial excitedstate causes this initial state to be nonstationary. Therefore the state is not sharplydefined in energy, and photons emitted from such states will have some distributionof frequencies. This broadening of spectral lines is called radiative broadening.

Doppler broadening of spectral lines arises from the motion of the radiating atomicsystem. The frequency of the radiation emitted depends upon the velocity of the sourceof emission relative to the observer, and the distribution of frequencies is determinedby the distribution of atomic velocities.

In contrast to the radiative broadening of spectral lines, Doppler broadening willoccur even if all photons have a single, well-defined frequency at the time of emission.A Doppler-broadened spectral line can be regarded as a sum of a large number ofnarrow lines. For this reason Doppler broadening is called nonuniform broadening,whereas radiative broadening is a property of any single emission, and is a uniformbroadening of the spectral lines.

Problem 2.1. Establish the shape of a spectral line if the broadening arises from afinite lifetime 7> of the initial state in the transition.

SOLUTION. A finite lifetime for an initial state k means that the state is nonstationary.We shall consider the single-photon transition between the initial excited state k andthe ground state 0. If w is the transition rate as established by Eq. (1.13), then fortimes t such that t > 1/w, the atom will be in its ground state with a probabilityapproaching unity. We wish to determine the distribution function a^ for the emittedphoton frequencies under these circumstances. Equations (1.4) for the coefficients ina perturbation expansion have the form

ih ck = ] P Vky0(Oc0oi exp [ i (w k 0 - <o)t], (2.3)(O

ih cQlo = Voc**c*exp[-/(G>K) - <o)t],

where c0(x) is the probability amplitude for the atomic electron to be in the state 0after the emission of a photon of frequency co. The initial conditions for the systemof equations in (2.3) are ck(0) = 1 and cOw(O) = 0.

We express the solution of (2.3) in the form ck(t) = exp(—wt/2), where w is aquantity to be determined. When we substitute this exponential decay behavior intothe second equation of (2.3), we obtain

RADIATIVE AND DOPPLER BROADENING OF SPECTRAL LINES 63

= yOa>,k 77 7—TIT • (2.4)n((0ko — co — iw/2)

The resonant character of this equation justifies the resonance approach we haveemployed in the interaction of an atomic electron with the electromagnetic field.Also, from Eq. (1.2), we have VOa)>k = -D0^-E/2. Substitution of Eq. (2.4) intothe first equation of (2.3) leads to the expression (1.13) for w with n^ = 0. Hencew is 1/Tfc, where rk is the lifetime of state k. Correspondingly, from Eq. (2.4), theprobability for the emission of a photon with frequency co is

a<o = koa,(°°)l2 = -z ,2 , n / , 0 .12. (2.5)27TTk (CO ~ Gift))2 + [ l / ( 2 7 * ) ] 2

Equation (2.5) satisfies the normalization condition (2.2). It is valid for cokQTk > 1.For frequencies such that co ~ cok0, Eq. (2.5) exhibits the behavior expressed inEq.(2.1).

We remark that we neglect a very small shift in the spectral line, which is knownas the Lamb shift. It is due to the interaction of an atomic electron with the "vacuum"electromagnetic field. This has reference to the effect of virtual electromagneticprocesses that can occur even in the absence of an externally applied field.

Equation (2.5) expresses the radiative broadening of spectral lines. The line shapeexhibited in Eq. (2.5) is of the Lorentz form.

Problem 2.2. Two sources of line broadening both lead to line shapes of Lorentzform, with widths of 1/TI and l / r2 . Show that the sum of the two processes alsoleads to a Lorentz line shape, and determine the width of the line.

SOLUTION. If we have two independent broadening mechanisms with distributionfunctions a$co) and a2(co), the total distribution function has the form

. 0 0

Gco — I a\(co')a2(co + coko — co')dco', (2.6)J-oo

where cok$ is the central frequency of the line being examined.For the circumstances defined in the statement of this problem, Eqs. (2.5) and

(2.6) give

/ :

dco1

/ _ {(CO' ~ COk0)2 + [1/(2T1)]2}{(CO " O)')2

The integrand in this expression can be rewritten as

1 1' — co — /co' — coko + i/ylT$ co' — co

1 'co' - cok0 - / / ( 2 T I ) CO' - co + i/(2r2)

rX - co - i/(27i) - i/(272)J [<0ko-(» + *V(27i) + i/(272)J '


The integral can then be accomplished in the complex plane by employing a path thatencloses the two poles in the upper half plane to give the result

( 2 ? )

where the notation is used such that

-L - ± + I . (2.8,

Thus the combined spectral line is Lorentzian, and its width is the sum of the widthsof the individual lines. This result can be generalized to the sum of any number ofindependent sources of the Lorentz broadening. That is, the total spectral line willbe of Lorentz form, and its width will be the sum of the widths of all the individuallines.

We can generalize Eq. (2.5) to the case where the initial state k can decay toother states in addition to the ground state. The expression ck(t) = exp(—wt/2) canbe retained, but now the quantity w characterizes the total width of the state k. Toaccommodate this extension of the theory, the right-hand side of the first element ofEq. (2.3) must have terms added to it to describe transitions to other states. Equation(2.5) will be obtained again, but with T> now representing the total lifetime of thestate, which can be composed of the partial lifetimes by means of Eq. (2.8).

Another generalization that can be made is if the lower state 0 is not the groundstate but can itself decay to other states, with the result that it has a lifetime T0. ThenEq. (2.7) is replaced by

" w = 2 ^ (co - «)k0)2 + [ 1 / ( 2 T , 0 ) ] 2 ' ( 2*9 )

where T^0 determines the so-called reduced lifetime, and the spectral width is givenby

- 1 = 1 + 1 (2.10)Tk0 Tk TO

replacing Eq. (2.8).

Problem 2.3. Find the spectral line shape that arises from the thermal motion ofatoms.

SOLUTION. We label as v the relative velocity of the emitter of radiation with respectto the system that receives the radiation. If the emitted frequency is to1, the Dopplerlaw gives a> for the apparent frequency of the received radiation, where

co = cof(l + vx/c). (2.11)

In this expression, c is the velocity of light, vx is the projection of the relative velocityof the radiator and receiver onto the line joining them, and vx > 0 corresponds to theradiator and receiver approaching each other.

We shall determine the frequency distribution of the emitted photons in the labo-ratory frame of reference. The probability that an atom has a velocity in the intervalfrom vx to vx + dx is f(vx)dvx, which must be normalized to unity, as in

f(vx)dvx = 1.

When we employ the connection (2.11) between the frequency of the radiation andthe velocity of the atom, we obtain the frequency distribution

a^dco = f(vx)dvx = —f c — dto. (2.12)

Specifically, the Maxwell velocity distribution function is

M \ 1 / 2 / ^ - 2 N

where M is the mass of the emitting atom and T is the temperature in energy units.Then Eq. (2.12) gives

U2

co1 \2TTT)

The Doppler function thus has a Gaussian form. From this expression, a typicalDoppler width for a spectral line can be estimated to beAw^ ~ cof[T/(Mc2)]l/2.The expression (2.13) describes a symmetrical spectral distribution centered at thetransition frequency co 0, with the width AcoD. For example, for atoms with massnumber A ~ 100 at room temperature, we have Acoo -~ 10~6co7.

Problem 2.4. Find the spectral line shape resulting from the simultaneous effects ofDoppler broadening and limited lifetime of the state.

SOLUTION. The spectral line shape in this problem is established by Eq. (2.9), butwith the central frequency shifted as a result of the atomic motion that gives rise tothe Doppler effect. The frequency distribution function found from Eq. (2.6) with thehelp of Eqs. (2.9) and (2.12), is

° dto1.^ (co - co')2 + [ 1 / ( 2 T , 0 ) ] 2 cok0

We take the f(v)in this equation to be the Maxwell velocity distribution. Ratherthan attempt the full analytical evaluation of the integral, we shall analyze certainlimiting cases. First we assume that the Doppler width is small compared with theLorentz width, which is to say that

/2T\l/2 1A(OD = (Ok0 —J < .

\Mc2 J 7>0


Then if we make the replacement

/ C -> 8(0)' - (Dk0)f

we obtain Eq. (2.9), which describes the Lorentz mechanism for the broadening ofspectral lines. The Doppler broadening can be neglected in this case.

The opposite limiting case is expressed by the relation

A 0)2)7*0 > 1.

Then for |co — co ol — Aco/), we can replace the 8 function by the Lorentz function inthe above formula. This step gives Eq. (2.13) for the distribution function aw, whichcorresponds to Doppler broadening of the spectral lines. A more complicated casearises from \co - (oko\ > Ao)£>. In this case, it is convenient to separate the integralinto the two regions co' — (ok0 ~ A(oD and a/ - a> ~ 1/T>O. The contribution fromthe first region gives

while the integral from the second region is equal to

-exp

Thus the Doppler form of the spectral line is preserved in the wings of the distributionif the condition

Tk0((x) - 0)k0)2

is fulfilled. If, on the other hand, the opposite relation

Tk0((O ~ COk0)2

holds true, then the distribution function has the form

_ 127TTk0(a) - (Dk0)

2 '

Thus, in the case of competition between Lorentz and Doppler broadening ofspectral lines, the wings of the spectral lines are always determined by the Lorentzmechanism, whereas the central part of the spectral line is determined by whichevermechanism is the stronger. In the intermediate region one can use the sum of Lorentzand Doppler contributions, because under the condition |o> — coko\ > Aa)D theircontributions to aw are independent.

In the general case, the spectral line is said to have the Voigt form. It is intermediatebetween the Lorentz and Doppler shapes.

Problem 2.5. Two sources of broadening lead to spectral lines of Gaussian shapewith widths of A! and A2. Prove that the total spectral line has a Gaussian form, anddetermine its width.

SOLUTION. From Eqs. (2.6) and (2.13), we have

/ e x p - ^

r ^ Aco —

exp

where A = (Aj2 + A|)1 / 2 . Thus the combined line is Gaussian, and its width is theroot mean square of the partial widths.

In addition to the Doppler effect, another broadening mechanism that leads to aGaussian form of the spectral line is the broadening that occurs with atoms embeddedin a crystal. Then fluctuations in local stresses cause frequency shifts in emittedphotons that lead to a Gaussian form of the line.

Problem 2.6. Determine the time-of-flight broadening of an absorption line arisingfrom the interaction of an atomic beam with an intersecting laser beam.

SOLUTION. In this case, the width of the absorption line is determined by the interac-tion time of the electrons with the electromagnetic field. This time is that required foran atom to cross the photon beam. Broadening takes place if the time of flight is shortcompared to the lifetime of the excited atoms. We assume the spatial distribution ofthe atoms to be Gaussian, in which case the electric field for a mode with frequencya) has the form

E(p, t) = EQ exp(-p2/<22) cos cot

for a beam of Gaussian width a. The coordinate p is the distance from the axis of thelaser beam. We take the atomic beam to be narrow as compared to the laser beam,and we assume the two beams intersect perpendicularly. If the atomic velocity is v,a characteristic interaction time of an atom with the laser beam is a/v. That is, thewidth of the absorption line goes as v/a. We shall find its form below.

The transition amplitude for single-photon absorption is, from the perturbationtheory expression (1.4),

4°/•00

= ^ /

In this expression, D^o is the matrix element of the dipole interaction operator, cok0

is the interaction frequency, and E is the electric field vector. We take the origin ofthe time coordinate to be that moment at which an atom crosses the axis of the laserbeam, so that p = vt. The transition amplitude is then

i r ( v2t2\4 ° = TEO*I>M) / cos cor exp ( -icok0t - —^- \ dt

The distribution function is proportional to |cj^|2, and assuming that \co - a>ko\ < co,we find

a^ ~ exp [-(a) - o)k0)2a2/(2v)2].

When we apply the normalization condition of Eq. (2.2) to this expression, we obtain

2v \-(co - ojk0)2a2

a = expa./ir (2v)2

Thus the time-of-flight broadening caused by a laser beam of Gaussian spatial distri-bution is itself Gaussian.

Now consider the laser beam to have a rectangular profile, and take the atomsto be in the laser field for a time T, where T = a/v. With the same cosine timedependence of the electric field as employed above, we find that

2 sin2 [(co - (%))/2]

Problem 2.7. An excited atom undergoes diffusive motion in a gas and experiencesmany collisions during its radiative lifetime. Determine the form of the spectral linecaused by the Doppler mechanism.

SOLUTION. Collisions in the gas cause the excited atom to change its direction manytimes before radiating. This problem is the reverse of Problem 2.3, where it is assumedthat the direction of motion of the atom is unchanged during radiation. We fix theorigin of coordinates on an atom at time t = 0, and then consider the probabilityW(xo, t) that the atom is to be found at JC0 at time t. This probability satisfies thediffusion equation

^ ^ (2 14)= Vdt dfi '

where T> is the diffusion coefficient of an atom in the gas. The probability W isnormalized by the condition

W(xo,t)dxo = 1.

The electric field strength for the electromagnetic wave emitted by the atom is

E = E o exp [ik(x - x0) - ia>kOt],

where k — o)ko/c. Averaging this field strength over the posi t ions of the radiat inga tom gives

E - / E W(xo, t) dx0 = Eo J(t) Qxp(ikx - icok0t),J -<x>


where

r7(0 = / exp(-ikxo)W(xo,t)dxo.

When we multiply Eq. (2.14) by exp(-ikx0) and integrate it over the positions JC0,we obtain

= V /d2w

dxQ.dt J-oo ^ dt2

Integration by parts, with application of the boundary condition W(x0 = ±00) = 0,gives

dt

The solution of this equation with the initial condition 7(0) = 1 is

7(0 = exp(-Vk2t).

From this it follows that the average strength of the electric field is

E = Eo cxp(ikx - iojk0t - Dk2t).

In a classical treatment, this field is generated by a dipole moment that oscillates witha frequency a^0, that is, the dipole moment behaves as

D ~ exp(-ja>H)f - T>k2t).

According to Eq. (1.13), the photon frequency distribution function, proportionalin the classical case to the square of a Fourier component of a dipole moment, is

k2T>(2A5)

Thus the distribution function has a Lorentz form with the width 2k2 ID. Then, inaccordance with Problem 2.2, taking account of the limited lifetime 7>0 of the initialstate in the transition leads to a Lorentz spectral line shape with the width 2k2 T) +

We remark that only the motion of the atom in the direction of propagationis significant, even though the diffusion is three dimensional. There is an analogywith Doppler broadening, where the transverse motion of the radiating atom is alsononessential. The validity criterion for Eq. (2.15) is that the time 7>o must be largecompared to the time for transit of a mean free path in the gas. Such a situationexists in masers. The opposite limit leads to the Doppler broadening considered inProblem 2.3.

2.2 COLLISION BROADENING OF SPECTRAL LINES

The collision mechanism (or impact mechanism) for the broadening of spectral linesfollows from the fact that an excited atom will sometimes experience collisionalinteractions with neighboring gas molecules. This interaction is strong, but it actsonly during a small portion of the excited atom's lifetime for photon emission. Theresult is the collision broadening of the emitted photon spectrum.

An excited atom will emit a monochromatic electromagnetic wave until such timeas it experiences interaction with other atoms in the gas. The result of collisionswith other atoms produces strong changes in the frequency emitted, but only fora time short as compared to the radiative lifetime. One can take this effect intoaccount by viewing each collision as causing a major shift in the phase of the emittedelectromagnetic wave. The emitted field will then consist of a succession of separatemonochromatic segments, with the phase of each segment shifted with respect toadjacent segments. The limited duration of each of these independent time segmentsin the emission determines the width of the resulting spectral line.

We decompose the wave functions of the initial state 0 and the final states k intostationary states with no broadening interaction present. For simplicity, we set h = 1in the equations that follow. The expansions of the wave functions are

ismt\ (2.16)

where i//rt, i//m are the spatial parts of the stationary state wave functions of the atom,and sn, sm are the energies of these states. The coefficients aon,akm are determinedby interactions with other perturbing atoms. Initial conditions for these coefficientsare aOn(O) = 80n, akm(0) = 8km.

We now introduce the quantities

fnmit) = aOn(t)alm(t\

and denote the special function fok(t) by f(t). In terms of the quantities cp(t) definedby

(2.17)

we can express the correlation function as

<£(T) = lim — / <p(t)<p*(t + T)dt.

Because the time dependences of aOn and akm are determined by phase factors, theamplitude of <p is simply \(p(t)\ = 1. With (p(—t) = <p*(0, and a change in the intervalof integration from 0 to T in the correlation function to the interval - 7/2 to 7/2, thecorrelation function can be expressed as

i rT/2

= lim - / <p(t)<p\t + T)dt.T^cc 1 J_T/2

rT/2

/ \ (2.18)_T/2

COLLISION BROADENING OF SPECTRAL LINES 71

Problem 2.8. Express the photon frequency distribution function a(co) in terms ofthe correlation function $(r) .

SOLUTION. The interaction (1.1) between an atomic electron and the radiation fieldis of dipole character, and so the transition amplitude is proportional to the matrixelement of the dipole moment operator between the initial and final states in thetransition, that is,

(2.19)

where Dnm = (i//jD|i//m). The Fourier transform of Eq. (2.19) gives the amplitudefor the emission of a photon of a given frequency. The probability for radiation of aphoton of frequency co is therefore proportional to

(2.20)

We are interested in the production of photons with frequencies near co o- Werestrict Eq. (2.19) to terms with m = k and n = 0, and with the assistance ofEqs. (2.16) and (2.17), we obtain

e~i0)t(p(t)dt (2.21)

This integral can be written in the form

au>~ dh I dt2<p(tl)<p*(t2)exip[ia)(t2 - f , ) ] .

In terms of new variables r = t\ — ti and t — tj, we have

aw ~ f dre-ia)T f dt <p*(t)<p(t + T).

On the basis of the definition of the correlation function in Eq. (2.18), we obtain

-™ dr, (2.22)

where we have employed the normalization condition (2.2) for the distribution func-tion aw. Equation (2.22) connects the emitted photon distribution function a^ withthe Fourier transform of the correlation function for the corresponding frequency ofthe radiated photon.

Problem 2.9. Determine the shape of a collision-broadened spectral line when itarises from collisions of an excited atom with atoms of a surrounding gas. Assumethe motion of the particles to be classical.

SOLUTION. The wave function of the atomic electron in the excited state k satisfiesthe Schrodinger equation

(2.23)7at

where Vkk is the diagonal matrix element of the interaction between the excitedelectron and the perturbing particles in the gas. We neglect inelastic collisions, sinceelastic collisions have the dominant line-broadening effect for a wide variety ofphysical conditions. On the basis of Eq. (2.23), we have the simple solution for theakk amplitude of Eq. (2.16),

akk(t) = exp Hf Vkk(tf)dtf (2.24)

An interaction between the excited atom and the perturbing gas atoms takes placeduring a very short interval of time. Times of collision ti are distributed randomly.Assume the interaction potential between colliding particles to be determined by thespatial locations of these particles, which is to say that we use a classical descriptionof their motion. We introduce the coordinates R/ of the colliding particles /, and wepostulate a certain time dependence R/(0- As a result of a collision with a particle,the wave function of the initial state acquires the additional phase

Thus Eq. (2.24) has the form

akk(t) = exp

where the step function r)(t) is defined by

(2.25)

(2.26)

•{'f 0, t < 0\ 1, t > 0 *

This function expresses the fact that the duration of a collision is small as comparedto a radiative lifetime.

Using an expression «oo(O similar to that for akk(t), we can write the function <p(t)ofEq. (2.17) in the form

q>(t) = exp ia)kOt + i -ti)

in which we have used the notation

Xi1 Z100

= *ik ~ K/O = £ / [Vtt(Rz) - Vbo(R/)] dt.

(2.27)

(2.28)

The quantity xi is the phase shift introduced by the difference between the interactionpotentials for the upper and lower states in the transition.

The correlation function in Eq. (2.18) can be written as

<f>(r) = <p*(t)<p(t + T) ,

where the superior bar indicates a time average. To find <£(T), we use the evolutionexpression

A(T) = $ ( T ) - exp(-^oAT)0(T + AT) (2.29)

= (p*(t)[<p(t + T) - txp(-io)k0Ar)(p(t + T + AT)] .

The time interval A r is assumed to be small compared to the time r characterizing thecollision broadening, which is given by the free-path flight time of the excited atom.However, AT is large compared to a collision time. Since we have the inequalityAT ^ r, there can be only one collision during AT, and that collision has smallprobability. With the insertion of Eq. (2.27), we have

A*(T) = <p*(t)<p(t + r)[l - exp{i'5^;tt[TKf + r + AT - *,) - v(t + r - *,)]}].

(2.30)The sum in this expression is taken over times in the interval between t + T andt + T + AT. We note that simultaneous collisions with two particles are neglectedbecause such collisions are rare.

Because of the random character of the collisions, we can do the averaging asif no previous collisions had occurred. The average value in Eq. (2.30) can thus beobtained as a product of individual averages, which gives

r + T < f / < / + r + Ar. (2.31)

The quantity p is the impact parameter of the collision associated with the value \from Eq. (2.28).

Averaging this expression over time amounts to averaging the collision times t\contained within some larger time interval. From a classical point of view the impactparameter of the collision is unambiguously connected with ti, so it is convenient tocarry out the averaging in terms of p. The collision probability per unit time in aninterval of impact parameters between p and p + dp is equal to lirpdpNv, so theaverage required for Eq. (2.31) is

r°°[1 - £?'XP)] = ATVN I lirpdp [1 - exp(i»], (2.32)

Jo

where TV is the number density of the perturbing particles and v is the relative velocityof the collision.

When we go to the limit A T -> 0 in Eq. (2.29) and take Eqs. (2.31) and (2.32)into account, we obtain

A * ( T ) = O(T) - (1 - IO>*0AT) O(T) + — AT

= - A T ! — - ia)k0&

/.CO

/ 2irpdp [l - exp(i»] .Jo

This is equivalent to

with the notation that

— - i(ok0® = -<3>Nv(a' + ia"),dr

af = /Jo

27rpdp(l -cos*),

. 0

a" = - /Jo

2TTPdp sin*.

The solution of Eq. (2.34) is

T>0

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

When we substitute Eq. (2.37) into Eq. (2.22), and employ the property4>*( -T) , we find the photon frequency distribution function to be

aM 2ir[(a> - (ok0

where we employ the notation

v = 2Nvaf, Av = Nva".

(v/2)2]'

(2.39)

We have found that the collision broadening of spectral lines exhibits theLorentzian form as in radiative broadening [see Eq. (2.9)]. In contrast to the broaden-ing due to a nonzero lifetime for radiative transitions, we now have both a broadeningof the spectral line and a significant shift in its central frequency. The shift in theradiative broadening case, the so-called Lamb shift, is small as compared to the linewidth, while for collision broadening the shift and width are comparable.

At room temperature and atmospheric gas pressure, a typical value for the collisionbroadened width is v ~ 1011 s"1. For radiative transitions in the visible range of thespectrum, we have (ok0 ~ 1015 s"1, so the ratio of width to frequency is v/a)k0 ~10~4. This is consistent with the approximations employed. We observe that the

radiative width 1/T>O ~ 108 - 10 9s - 1 is small compared with the above values.Also, the kinetic theory of gases identifies v as the time of flight for a mean free path.

Problem 2.10. Connect the parameters of the collision broadening of a spectral linewith the properties of elastic collisions of an excited atom with the surrounding atoms.Assume the interaction energy to be small as compared to the thermal energy of theatoms.

SOLUTION. Parameters describing collision broadening are connected with the phaseshift x(p)- Consider the case in which the interaction potential is small comparedwith the thermal energy of the gas atoms, that is, when the inequality holds thatU(R) < fiv2, where JLL is the reduced mass of the colliding particles and v is theirrelative velocity. From Eq. (2.28), we have

1 r°°

X(p)= £ / U{R)dt,

where R is the vector distance between the colliding particles and U(R) = V^(R) —Voo(R). The quantum theory of scattering gives the scattering phase shift for elasticcollisions in the form

8(p) = - ^

where U(R) is the interaction potential of the particles. Thus we have ^ = - 2 8 . Thetotal cross section for elastic scattering of atoms is

/.CO

at = Sir / sin2 8(p)pdp. (2.41)Jo

Using the relation x = - 2 8 , Eqs. (2.35) and (2.41) yield

a' = 4TT / sin2 8(p)pdp = —, (2.42)Jo 2

where the total cross section for elastic scattering refers to the interaction potentialU(R) = Vkk ~ Voo- F° r the cross section related to the spectral line shift, we have

a" = 2TT I sin[28(p)]pdp. (2A3)Jo

We can estimate the cross sections af and a". We have


This gives af ~ or" ~ 07 ~ p2,, where the parameter p0 is determined by

L ( 2 . 4 4 )

The quantity p0 is called the Weisskopf radius. It is the impact parameter of thescattering if the scattering phase shift is of order unity.

We now analyze the validity of the classical description of collisions for thephysical problem being treated. We introduce the collisional angular momentum/0 = i±pQv/fi, and rewrite Eq. (2.44) in terms of /0, which gives UOU(PO)/IJLV2] ~ 1.The classical description is valid if the collisional angular momentum is large, thatis, if /o > 1. The criterion for validity of a classical treatment is thus

This establishes the correctness of the conditions used in obtaining Eq. (2.40). In theclassical sense, we can consider the motions of the atoms to be free.

We now give expressions for classical cross sections if the difference of theinteraction potentials is approximated by the inverse power law dependence U(R) =CR~n. We then have

c r dt

JCTT1 / 2 T[(n -

2hvpn~l T(n/2) '

where F is the gamma function. It is seen that the total cross section for elasticscattering does not depend on the sign of the interaction potential, while the crosssection a" has the same sign as the interaction potential.

Inserting values of 5(p) into the expressions for the cross sections, we obtain

at = Sir pdp sin2 [ T ) = / y (n+1)/(rt 1}sin2yjy,Jo VP""1/ n-\ Jo

a" = -2rr [ pdpsm(^4r) = -2lTA j y-{n+l)An-l)sin2ydy,Jo \P J n- 1 Jo

with the definition

r[(/i -_

2hv T(n/2) '

With the integrals evaluated as

r^w^=^ ) c o s W 2 )

Jo

£


where /x = -2/(n - 1), we obtain

( I n\ \ 2/(n-1) / I ^1 \ 2/(n-1)

T- , \cr "\ = bn[xT

1\ , (2.45)fiv J \nv J

in terms of the quantitiesa" = ^ T { V r [ r ( ^ / 2 ) / 2 ] } lr[

2T«24/(«-D1)]l cos ( ^ l ) •(2.46)

277 f7rl/2r[(n-l)/2]\2/{n-i)\T[-2/(n-l)]\ . ( ir \bn n-l\ 2 IW2) / 2-2A»-D S i n l ^ T J -

In these expressions, we restrict n to n > 3. For n = 3, we have Z?n = oo. Theparameters an and Z?n are connected by the relation

an = 2bn cot\n — L

Table 2.1 lists values of an and bn for a variety of values of n.

Problem 2.11. Find the cross section for broadening of a spectral line if the inter-action potential is a sharply peaked function of the distance between atomic nuclei.Assume the particle motion to be classical.

SOLUTION. We employ the results found above for broadening cross sections, andtake into account the sharply peaked character of the interaction potential of thecolliding particles, using a classical context. The broadening cross section is given inEq. (2.42) as

a1 = —- = 4T7 / sin2 8(p)pdp,2 Jo

where the scattering phase shift is given by the expression

= __L r2h 7_c

u

where z = vt. We now wish to evaluate the integral containing the scattering phaseshift when the interaction potential has a sharp peak. The main contribution to theintegral comes from the vicinity of z = 0. We shall evaluate the integral by the

TABLE 2.1. Representative Valuesof the an, bn Quantities

n

an

K

4

11.49.8

6

8.12.9

71

8

.2

.7

12

6.60.96

16

6.40.68

method of stationary phase. We write the integral in the form

S(p) = -^- j exp [ln£/(A/p2 + z2)] dz.

and use a series expansion of the exponent near the stationary phase point at z = 0.We obtain

l n t / f ^ + z2) « l n £ / ( p ) + f ' - ' • ' -> • ~ i J l n t / ( P )

We now introduce the parameter

2p£/(p)

which is proportional to the logarithmic derivative of the interaction potential. Interms of this parameter, the scattering phase shift is

8(p) = --j— I exp(-azz)dz = - - r— - I - Inv Jo nv 2 \ a /

2 t/'(p)J •

We assume the integral to be convergent in a small region near z — 0 of extentz ~ a"1/2 < p, where p is a typical size that characterizes the behavior of theinteraction potential. From this it follows that the condition corresponding to theassumption of a sharply peaked interaction potential is

pU'{p)U(p)

1.

This is the criterion for the validity of these expressions.For example, for the power law interaction potential U(R) = CR~n, we obtain

a = n/(2R), and the validity condition has the form n > 2. For the power lawinteraction potential, the scattering phase shift is

We wish to compare this result with a precise expression from the preceding problem.When we employ the previous expression in the limit n —> oo? we obtain

C2hvpn~x r(n/2) n>\ hvp"-1 V In


where we have used Stirling's formula for the gamma function. Thus we see that thetwo results coincide.

The parameters an and bn of the preceding problem approach the limiting values

lim an = 2TT, lim bn = IT2/n.n—^oo n—>oo

These values are consistent with the entries in Table 2.1.We wish to evaluate the integral that gives the broadening cross section. The total

cross section for elastic scattering is

rat = Sir

Jo

pdp sinU(p) irpU{p)

hv V 2U'{p)

paverage value of \ because of the rapid oscillations of this function. This gives the

For small impact parameters, the square of the sin function can be replaced by its\

simple result

fp0fp

= 8TT /Jo

1pdp- =

where the end point value po is given by that p for which rapid oscillations of theargument of the sin function cease. This is determined by

£/(p) TTpU(p)p hv y iu'{p)

We check this general result using the example of the power law interactionpotential U(R) = CR~n, with n > 1. This leads to the Weisskopf radius

Po = I

and hence to the total cross section

at = 2TT

It follows from this that

77 \ ]

2nJ

and, because of the stated properties that n > 1 and j8 — 1, then j8 2/(" ]) —» 1, andthe dependence on n disappears. In the limit of large n, the quantity (7r/2n)1/(n~1)

can be replaced by unity, and we return to the asymptotic result an — 2TT.

Problem 2.12. Express the parameters for the collision broadening of spectral linesin terms of the characteristics of the collisions of excited atoms with the perturbingatoms of the gas. Take the collisions to be described in quantum terms, and assumethe excited atomic state to be nondegenerate.

SOLUTION. We assume for simplicity that the broadening arises only from the upperlevel of the radiative transition in the excited atom. Collisions with surrounding gasatoms lead to an abrupt change in the phase of the wave function for the upper state inthe transition, and this leads to the broadening of the radiated spectral line. However,in contrast to Problem 2.10, the collision process is not now described by classicallaws, although in both cases the process corresponds to a strong interaction betweenthe colliding bodies, acting during a short time. For the elastic scattering cross sectionit is now necessary to use the quantum expression

A

^ = -^^2(21 +l)sin28 /(* ), (2.47)

/=o

where 8J® is the elastic scattering partial-wave phase shift for the collisional angularmomentum /. The quantity K = fiv/h is the wave number for the colliding particles.When we repeat the considerations of Problem 2.10 up to Eq. (2.38), we obtain thesame connection between the broadening cross section and the total cross sectionfor elastic scattering as in the classical case: a1 = crt/2. This means that the totalcross sections for elastic scattering and for collision broadening of spectral lines inthe problem under consideration are determined by the same interaction.

We now take into consideration broadening involving both the upper and lowerstates in the transition. The Lorentz character of the broadening means that the overallwidth of the line will be a sum of the partial widths due to each of the interactions. Thebroadening cross section is then the average of the elastic scattering cross sections ofthese states, or

<r'=\ [o-<°> + of >] = g f > + 1) [sin2 5<°> + sin2 Sf > ] , (2.48)/=0

where the index 0 refers to the lower state in the transition. This result differs fromthe classical Eq. (2.41), which obtains for large collisional momenta. This differenceis the consequence of a quantum mechanical scattering phase interference.

Based on the analogy between quantum and classical results, we obtain the quan-tum cross section for the shift in the spectral line associated with collision broadening,when the broadening arises only from the upper state in the transition, as

/=0

If both upper and lower states participate in the shifting of the spectral line, thisexpression becomes

cr" = £ ]T(2/ + 1) {sin [28<°>] + sin W } . (2.49)/=o

Note that we have implicitly used asymptotic expressions for the wave functions.This leads to final expressions that involve elastic scattering cross sections and re-quires that the collision velocity must be large. The validity criteria for this step willbe considered in Problem 2.14.

Problem 2.13. An excited state of an atom is degenerate with respect to the projec-tions m of the orbital angular momentum. Determine the parameters for the collisionbroadening of spectral lines in this case, using the assumption that the broadening iscreated by collisions where only the excited states contribute.

SOLUTION. In this case the line broadening is a result of both the elastic collisions ofthe excited atom and of transitions with a change in the magnetic quantum number m.The wave function of the excited state is labeled ipm. Assume that at t = 0 theatom is in the nth angular projection sublevel, and transitions to other states occursubsequently. If anm(t) is the probability amplitude that the system is in the state n,the atomic wave function is

When we expand the wave function in the Legendre polynomials P/(cos 0), thegeneral expression for the scattering amplitude / is

( Z X ~ X ) P/(COS 0)' (2'50)

where Slnm is an element of the scattering matrix. Because of the isotropy of the

collisions, the expression (2.50) must be averaged over all directions. Equation (C.I)gives this averaged scattering matrix element as

where L is the angular momentum of an atomic electron in a given state with projec-tions labeled by n and m. The quantity £>^m(#) is the generalized spherical function,or rotation function, for the angle # between the collision direction and a fixedquantization axis. Using the relation (C.2),

we obtain

where n, m, i are magnetic quantum numbers of a degenerate excited state. In theabsence of degeneracy, we would obtain from this the usual S-matrix element ofscattering. When we substitute Eq. (2.50) into (2.51) and use the optical theoremconnecting the total cross section with the scattering amplitude for the forwarddirection, we obtain the broadening cross section

/=0 m=-L

We now wish to calculate a11. Equation (2.37) shows that a" appears in thecorrelation function in the combination a1 + icr". Hence one can obtain a" by takingthe real part of Eq. (2.50) and repeating the steps that led to Eq. (2.52). The result is

/=0 m=-L

Problem 2.14. Determine the parameters for broadening of a spectral line arisingfrom transitions from P states to S states in an atom immersed in a gas of like atoms.Use the framework of the collision broadening theory and the quasi-classical theoryof elastic collisions. The number density of atoms in the ground S state is N.

SOLUTION. To use quasi-classical methods we must have U < E, where U is theinteraction potential and E is the kinetic energy of the relative motion betweencolliding particles. Projections of the orbital angular momentum on a fixed axis canbe used as good quantum numbers for an excited atom as long as inelastic transitionsbetween sublevels of an excited atom can be neglected. The scattering S matrix isSl

nn = Qxp(2i8ln), where 5^ is the scattering phase shift for a sublevel n and collisional

orbital angular momentum /. The quasi-classical expression for this phase shift is

Sj = i r Undt,

where Un is the interaction potential for a P-state atom with an angular projection n.Cross sections for broadening and shifting are given by Eqs. (2.52) and (2.53).

Since the principal contributions to these sums come from large values of the collisionmomenta, the sums can be replaced by integrals over the impact parameter p, relatedto the angular momentum / by the relation / = Kp, where K is the magnitude of thewave vector of the relative motion of the colliding atoms. The index n has six values,arising from the three projections m = -1, 0, +1 for a P-state atom combined with

the parity of the state. Equations (2.52) and (2.53) give the broadening and shiftingcross sections

Interaction potentials for atoms in ground and excited states have the same values forgerade and ungerade states of interacting atoms, but different signs. For this reason,terms in the a" expression mutually cancel, leading to a" = 0. That is, there is noshifting of spectral lines in this case.

For calculation of af, we employ a fixed rectangular coordinate system whoseaxes are the impact parameter of collision p, the relative velocity of the atoms v,and the angular momentum M of the relative motion of the atoms. Take the wavefunctions of the excited states such that the origin of x corresponds to zero projectionof momentum on the axis p, the origin of y corresponds to zero projection of theatom momentum onto the axis v, and the origin of z corresponds to zero projection ofthe angular momentum onto the axis M that is perpendicular to the plane of motion.We take into consideration that one can regard the classical trajectories of the relativemotion of the atoms to be rectilinear.

The wave function of the ground state is designated by <p and that of the excitedstate by r//. Then the wave function of the quasi molecule—a system of collidingatoms—has the form (i//j<p2 — \\f2<P\)/\/2. Here the subscript labels the atom, theupper sign corresponds to the gerade state of the quasi molecule, and the lower signrefers to the ungerade state. The interaction operator of the atoms has the form (seeProblem 1.10)

V= ^ J [ D i - D 2 - 3 ( D 1 - n ) ( D 2 - n ) ] ,

where D\, D2 are the dipole moment operators of the atoms, and n is the unit vectordirected along the quasi molecule axis. Since the atoms are neutral, this interactionis most important for large distances between the atoms. In first-order perturbationtheory we have

U(R) = ^ 3 (Oh 92 ± <fe<Pi)|Di • D2 - 3(D! • n)(D2 • n

Using the selection rule for matrix elements of the dipole momentum operator, weobtain

U(R) = ±^3<i//i<p2|D1 • D2 - 3(Di • n)(D2 • n)|ifepi>,

where the plus and the minus refer to gerade and ungerade states, respectively, ofthe quasi molecule. Matrix elements of different projections of the dipole momentoperator are expressed by

dr,

where i//(r) and cp(r) are radial wave functions for excited and ground atomic states.This gives the value of the matrix element as

<<p|D-k|i/r> = Dcosd/y/3,

where k is some unit vector and 6 is the angle between this vector and the quantizationaxis, which is the line joining the two atomic nuclei.

In terms of the oscillator strength fOk of the S - P transition, given in Eq. (1.18),the matrix element D is expressed by

> f

where h COQ is the energy difference of the states. Table 2.2 gives D2 values in atomicunits (e2ak).

TABLE 2.2. Values of theSquared Dipole MomentMatrix Element

Atom (transition)

H(12S-+Li (22S ->N a ( 3 2 S ^

K(42S1 / 2-Rb(5251/2

Rb(52S1/2

Cs(62,S1/2

Cs(62Sl/2

He(21P -Mg(31P-Ca(4 J P-Sr(5 IP-^Ba(5'P -Zn(4lP -H g ( 6 1 F -H g ( 6 1 P -

22P)22P)

^22P)-* 42P1/2)-^ 52P1/2)- 52P3/2)^ 62P1/2)-+ 62P3/2)>2lP)•+3lP)

>4lP)5lP)

>5]P)>4!P)*6lP)^63P)

D2

1.66511.018.826.428.428.034.834.80.53

11.320.723.425.5

8.57.230.25

On the basis of the above expressions, we have

UXX(R) = ± ^ j Z ) 2 ( l 2

Uyy(R)= ±^D2(l

UZZ(R) = ±±D.

Other interaction operator matrix elements are zero. The indices x, y, z correspondto states with zero projection of the angular momentum vector of the P-state atomonto the p, v, M axes, respectively; 0 is the angle between the p axis and the R(quasi-molecular) axis; the plus sign corresponds to the gerade state of the quasimolecule, and the minus sign refers to the ungerade state.

The character of the atomic collision process must be analyzed. Simple elasticscattering is accompanied by a process in which the excitation energy of one atomis exchanged with that of the other. Atomic depolarization also occurs. This refersto the possibility that states with different angular momentum projections m becomemixed as a result of the scattering. This will influence the broadening cross section.

Transitions between states with different projections of the angular momentumcan be viewed as rotations of the interatomic axis of the quasi molecule. However,states with zero projection of the P-state atom on the axis perpendicular to the planedefined by the motion of the atoms are not mixed with other states. We representthe wave function of the system of colliding atoms with this angular momentumprojection as

where ip+z = (i/ru<P2* + <fe*<Piz)/\/2> ^-z = W\z<P2Z- *foz<P\z)/V^*with w a v efunc-tions as defined above, and where the index z characterizes the angular momentumprojection of an excited atom.

The Schrodinger equation ihdty /dt = HW has the form

ih a+z = (D2/3R3)a+z, ih d-z = -(D2/3R3)a-z,

where R2 = p2 + v2t2. Solving these equations for the free relative motion of theatoms gives the scattering matrix elements, or probability amplitude for survival ofthe corresponding state,

S+Z(p) = a+z(oo) = exp [-i2D2/(3hp2v)},

S-Z(p) = a-z(oo) = exp [i2D2/(3hp2v)] .

In evaluation of these scattering matrix elements, we impose the initial conditiona±z(-oo) = 1. We then obtain the results for the scattering matrix elements for

excitation exchange 5ex and for elastic scattering Sel:

The cross sections for excitation exchange, o"ex, and for elastic scattering, aQh are

(TeX = / 27TPdp\Sex\2 = —r—,

Jo *nv

CTC1= / 27TPdp\l-SQl\2= ^ - .

Jo 3™

The total cross section is the sum of these cross sections

O-tot = O-ex + O"el = 2TT2D2/(3hv),

and the broadening cross section is

<T[ = o-tot/2 = 7T2D2/(3hv).

Scattering matrices for other projections of the angular momentum cannot be foundanalytically. To simplify an approximate treatment of these matrix elements, weassume that there are no transitions between states with different angular momentumprojections in the excited state during collisions. In analogy with the case just treated,we find the equations for the coefficients of the wave functions to be

ih a+x = (1 - 3 cos2 6)(D2/3R3)a+Xy ih a-x= - ( 1 - 3cos2 6)(D2/3R3)a-x,

ih a+y = (1 - 3 sin2 6)(D2/3R3)a+y, ih a-y= - ( 1 - 3 sin2 6)(D2/3R3)a-r

The solutions of these equations give the scattering matrices

/ 2D2

S+X(p) = Slx(p) = exp i3hp2vJ'

S+y(p) = Sly(p) = 0.

These results give the average cross sections for excitation exchange and for elasticscattering,

<fex = Cfel = 2TT2D2/(9hv\

and the average broadening cross section,

arf = 2TT2D2/(9hv). (2.54)

QUASI-STATIC THEORY OF SPECTRAL LINE BROADENING 87

The width of the spectral line, a quantity that appears in Eq. (2.38), with the definitiongiven in Eq. (2.39), is

v = 2Nvaf = 4ir2D2N/(9h). (2.55)

We see that the spectral line width does not depend on the velocities of the atoms.The error associated with the approximation employed here can be appraised by com-paring with the outcome of a more accurate procedure. The accurate solution givesthe excitation exchange cross section o ex — 2.26TTD2 / (3hv) = 2.31D2 /(hv), whilethe approximation used in this problem gives ~aQX = 2TT2D2/(9hv) = 2A9D2/(hv).The precise value for the elastic scattering cross section is crel = 2.5<&ITTD2 / {3hv) —2.10D2/(hv), as compared to the outcome of the currently applied approximations ofOei = 2A9D2/(hv). Finally, the accurate broadening cross section is 2.53D2/(hv),versus the present approximation of 2.19D2 /(hv).

We wish now to assess the range of variables within which the current approxi-mation is useful. The validity criterion for the collision broadening of spectral linesfound in the following in Eq. (2.66) is that the Lorentz line form is appropriate in thefrequency range |co - a)k0\ when the condition v/yfa^1 > \o) - a)*ol is fulfilled. Thisimplies that

hv3/D2 >\(o- (ok0\2.

That is, the velocity of the atoms is constrained by the condition

This means that the broadening of the spectral line arises from separate and relativelyhigh velocity collisions of an excited atom with the surrounding ground-state atoms.In particular, for the main part of the spectrum where |co — co ol ~ v, this criterion is

hv3 D4N2 N2/3D2 ^ ^-W>~fr, or v>-r-. (2.57)

Collision broadening occurs, therefore, when the density of atoms is low.

2.3 QUASI-STATIC THEORY OF SPECTRAL LINE BROADENING

The quasi-static theory of spectral line broadening corresponds to physical conditionsopposite to those associated with the collision mechanism of line broadening. In thequasi-static case, broadening is created during times that are small compared to timesassociated with motion of the atoms. One can therefore consider the perturbing atomsto be motionless, and the spectral frequency shift resulting from the interaction of theradiating atom with the surrounding atoms is a sum of the shifts due to a pairwiseinteraction with each perturbing atom, regarding the spatial configuration of theseatoms to be fixed. This interaction energy is assumed to be small compared to typical

atomic energies. Perturbation theory is thus appropriate for the treatment of thisbroadening.

A general goal of the quasi-static theory of spectral line broadening is the determi-nation of the frequency distribution function of the emitted photons. It is connectedwith a spatial distribution of the perturbing atoms that depends on their interactionwith the radiating atom. To characterize the influence of this interaction on the distri-bution of the perturbing atoms in space, we introduce a function w(R), which is theratio of the probability for this atom to be at a distance R from the radiating atom tothe probability that the atom is at infinity.

Problem 2.15. Express the frequency distribution function a(co) of the emitted pho-tons in terms of w(R) within the framework of the quasi-static line-broadening theory.

SOLUTION. In lowest order perturbation theory, the shift in the frequency of an emittedphoton due to interaction of the excited atom with neighboring atoms is

<*ko ~ ** = \ Y, U(R™)- (2.58)m

Here, m is the index identifying the perturbing atom, Rm is the distance between theperturbing atom and the radiating atom, and U(Rm) is the difference between theinteraction potentials for the upper and the lower states in the radiative transition.Equation (2.58) is assumed to be averaged over the quantum states of both particles,that is, it is the diagonal matrix element of the pairwise interaction operator betweenthe atoms.

Equation (2.58) gives the dependence of the shift of the emitted frequency on thespatial locations of the perturbing atoms. We introduce the notation Um = U(Rm) andthe probability p(Um) dUm that the pairwise interaction potential of the atoms lies inthe interval between Um and Um + dUm. Then, according to the definition of aM, wehave

cia> dco = Ylp(Um) dUm at (ok0-a)=-m

It is assumed that the spatial location of any of the perturbing atoms is independentof that of the others.

When we introduce the spatial probability function w(R) described above, we have

p(Um)dUm = w(Rm)^} (2.60)

where fl is a normalization volume which includes the perturbing atoms and theradiating atom. If w = 1, the ratio dRm/il is the probability for the rath perturbingatom to be located in the element of space dRm.

We calculate first the Fourier transform of the frequency distribution function a^.From Eqs. (2.59) and (2.60), we have

l±{t) = I Qxp[it((o - ojko^a^daj (2.61)

•n/ exp( '•jrUm)p(Um)dUm.

We next define the quantity <&(/) by

which, according to Eq. (2.60), can be expressed as

*(0 = ^ /exp [ | ^ w | MR)dR. (2.62)

Equations (2.61) and (2.62) combine to give

If N is the number density of perturbing atoms, then the number of these atoms inthe normalization volume is Ml , and hence we have

ex P l fi

To obtain the frequency distribution function aw, we perform the inverse Fouriertransform and find the result

1 raOi = — / exp[-i7(o> -

(2.63)

The function f/(/?) in Eq. (2.63) corresponds to the U(R) in Problem 2.10 for thecollision broadening of spectral lines.

In general, the distribution function a^ has a complicated character and does notcorrespond to a Lorentzian. When the exponent Qxp(ivt/h) is expanded in a series,we obtain the Lorentz line shape (2.38) with zero width and the Stark shift

= - - N Ift J

U(R)w(R)dR. (2.64)

In particular, if we assume in Eq. (2.64) that w = 1 and R = vt (corresponding to astraight-line classical trajectory, since U < E), Eq. (2.64) leads directly to Eq. (2.36)with sin x replaced by x> Thus, within the context of perturbation theory, the collisiontheory and the quasi-static theory of spectral line broadening lead to the same result.

Second-order perturbation theory in Eq. (2.63) gives a nonzero addition to thebroadening cross section, but it is not of Lorentz form. The additional term is pro-portional to t2, whereas it must be proportional to t if it is to give rise to a Lorentzianline shape.

The validity criterion for perturbation theory, where the collision and quasi-statictheories of line broadening coincide, is x ^ 1- In this case the broadening crosssection is small compared to the line shift cross section because it corresponds to asecond-order perturbative effect. Equation (2.40) gives the validity criterion in theform

\U(R)R/(hv)\ < 1,

where R is a typical interatomic distance associated with the broadening cross section,and v is the relative velocity of the atoms.

Problem 2.16. Establish the validity criteria for the collision theory and for thequasi-static theory of spectral line broadening.

SOLUTION. We must analyze the physical conditions in which the collision or thequasi-static theory of spectral line broadening is valid. The basis of the collisiontheory is that one can find only one perturbing particle in the region of significantinteraction near the radiating atom, and the probability of the participation of twoperturbing atoms with the excited atom is small. The collision theory of broadeningcorresponds to a low-pressure gas situation where the effective radius of interactionis small compared to the average interparticle distance.

The collision time is of the order ofp/v, where v is the velocity of the collision andp is a typical impact parameter. The impact parameter can be estimated to be givenby the Weisskopf radius y/cri^ where crt is the total elastic scattering cross sectionfor the interacting particles. A typical collision time is thus of the order of y ^ / f .The rationale for collision broadening requires that this time be small as comparedto the flight time involved in a mean free path for the colliding particles, which is(Nva-t)~

l. The collision time should also be short compared to the detection time forthe frequency shift of the spectral line, which is given by the Heisenberg uncertaintyprinciple as (co — to£o)~l. Thus the collision theory of spectral line broadening requiresfor its validity that

> Max [Nvat, (CO - o)k0)]. (2.65)

We also deduce from the foregoing that the collision theory of broadening is validfor the wings of the spectral line when

\co-ajk0\< -£=. (2.66)

For the central part of the spectral line, we have |o> — co ol ~ Nvcrt, and so thecollision theory of line broadening is valid when

Naf/2 <$ 1. (2.67)

We can also conclude from Eq. (2.66) that the collision theory of broadening is notvalid in the far wings of the line.

We now turn to the validity criteria for the quasi-static theory of spectral linebroadening. This theory will work if a typical time (o> — coô)"1* during which thefrequency shift a> — co o is detected, is small compared to a characteristic time p/vfor the motion of a perturbing atom. If this constraint is satisfied, the perturbingatoms will not change their positions significantly during a broadening time. If theWeisskopf radius is used for p, the validity criterion takes the form

|co - cok0\ > - £ = . (2.68)

This requirement is the direct opposite of that found in Eq. (2.66) for the collisiontheory. For the central part of the spectral line, our new criterion is

Naf/2 > 1, (2.69)

which is also opposite to that found in Eq. (2.67) for the collision theory. Thus thecollision theory and the quasi-static theory are converse to each other. This conclusiondoes not contradict the earlier finding that the two theories coincide when perturbationtheory is valid. We remark in closing that according to the criteria (2.66) and (2.68),the far wings of spectral lines are described better by the quasi-static broadeningtheory.

Problem 2.17. Obtain an expression for the distribution function in the wings of aspectral line within the framework of the quasi-static theory of broadening.

SOLUTION. We start with the general expression (2.63) for the frequency distributionfunction of the emitted photons. If |co — aôl —> °°, the integral over t is determined byvery small times. In the opposite case the integrand would be a strongly oscillatingfunction, since typical times t are of the order of (co — oô)"1- We consider thatportion of the integrand given by

exp (iV j {exp [ ^ ] - 1} W{R)d^ , (2.70)

and estimate the exponent in this expression. This estimate can be represented inthe form Nr$, where r0 is a typical radius in the integral over R. The value of r0 isdetermined by the relation

U(r0) x

h((o — cofco)

The function U(R) decreases for large values of R when R > r0, and

exp [itU(R)/h] - 1 -> 0,

so that the contribution of such values ofR is small. The value of r0 can thus be foundfrom the relation

U(r0) ~ h(o) - (x)k0).

If (cu - cok0) —> oo, then r0 —> 0. Hence we have NrQ < 1 for sufficiently remote partsof the wings of the spectral line. The exponent (2.70) can therefore be expanded in aTaylor series, and Eq. (2.63) leads to

aM = — / w(R)dR / expf-iY(ft) - ^ 0 ) l exp {%-) - 1 \ dt.2TT J 7-OC L J L V h J J

Since (o) — co o) =£ 0, the second term in this expression is zero. The first term givesa delta function, and so we obtain

^ p | (R)dR (2.71)aa=N [sla-aw- ^ p | w

L dK J t/(/?)=^(f0-Ww)

This equation may be obtained by a simpler alternative. We have seen that thewing of the spectral line is produced by interactions at small distances, r0 —• 0. Theprobability of having two or more perturbing particles at small distances is vanishinglysmall, so we can neglect this possibility. The probability of finding one perturbingparticle in the range between R and R + dR is Nw(R) dR. Since the distance betweenthe perturbing atom and the radiating atom determines the frequency o) of the emittedphoton, we have

a^da) = Nw(R)dR.

Equation (2.58) allows us to write

doj = dU(R)/h,

and so we arrive again at Eq. (2.71).

In the special case where U(R) — C/rn and w(R) = 1, we find the explicitexpression for the frequency distribution function to be

47rhNRn+3

aw = (2.72)Cn

4ITN fC\Vn

h

We see that the wing of the spectral line is of Lorentzian form only if n = 3. If n > 3,then civ decreases more slowly than for a Lorentz profile. We can neglect the shift ofthe spectral line in this limit.

Problem 2.18. The center of the spectral line is described by the collision theoryof broadening. Examine how the transition from collision broadening to quasi-staticbroadening takes place as the frequency departs from the peak of the spectrum.

SOLUTION. Since the principal part of the spectral line is described by the collisiontheory of broadening, the criterion of applicability, Eq. (2.67), is Ncr^2 < 1. Nowwe consider the wing of the spectral line, where h\o) — co ol exceeds the typicalwidth Nvat of the line. Equations (2.66) and (2.68) indicate that the transition fromcollision broadening to quasi-static broadening takes place at frequencies such that

Since it is true that

/ ^ NV(TU

then the transition from collision broadening to quasi-static broadening occurs in thefar wings of the spectral line.

We now examine this region of the line. The quasi-static broadening theory,according to Eq. (2.71), gives the frequency distribution function of the photons inthe wing region of the line as

ciu = 47rhNR2w(R)(dU/dR)~K (2.73)

To estimate this quantity, we note that we have the behavior \o) — co ol ~ v/\/07> o r

U(R) ~ hv/yfcFt, in the transition region. We thus find that R ~ y/o^. Hence, thedistribution function (2.73) is of order

hNR3 Naf/2 Na2

U(R) \co — cofcol v

From another point of view, we can obtain from Eqs. (2.38) and (2.39) for thefar wings of the spectral line within the context of the collision theory, the orders of

magnitude

v Nvat Na2

w / \9 / 9 / \

(a> - o)k0)2 (vz/at) v

Thus, in the transition region where |w - o>£0| ~ v/y/a~t, the results of the collisionand quasi-static theories are of the same order of magnitude, as we would expect.Outside this region, the dependence of a^ on the frequency co, and the order ofmagnitude of aw are fundamentally different within the two theories of broadening.

Problem 2.19. Estimate the dependence on physical parameters of the width andof the frequency distribution function in the wings of a spectral line arising from aresonantly excited atom. This excited atom is in a gas of like atoms, with all othersbeing in the ground state. The excited atom interacts with the other atoms of the gasthrough dipole-dipole interactions.

SOLUTION. The dipole-dipole interaction is given in Problem 2.14. From Eq. (2.55),the width v of the spectral line is of the order of

v ~ ND2/ft (2.74)

in the collision-broadening approximation. The applicability criterion is given bythe inequality in Eq. (2.56). From Eq. (2.38), the collision approximation frequencydistribution function in the wings of the line is

2TT\(JO - cok0\z

(2.75)

where we have assumed that \a) — o)k0\ > v.We now make estimates in the quasi-static broadening theory. From Problem 2.17,

we have a^ ~ NR2 dR/dco. Within the dipole-dipole approximation, Eq. (2.58) gives

h(co - c^0) ~ D2/R\

so the quasi-static approximation distribution function behaves as

d(R3) ND2/h

a<o ~ N—— ~ 1 pr. (2.76)do) \(x) — eo£ol

When we compare Eqs. (2.75) and (2.76), we see that both theories of broadeninggive the same dependence on physical parameters for the far wings of the spectralline. For collision broadening, according to Eq. (2.56), the condition

ND2/h <\<o- a)k0\ < hl/2v3/2/D (2.77)

must be fulfilled. For the quasi-static theory, the condition given in Eq. (2.68) can berewritten with the help of Eq. (2.54) to give the requirement

\<o - o)k0\ > hl/2v3/2/D. (2.78)

Thus, with respect to the condition (2.57) we have two theories for the wingsof the spectral line with qualitatively the same form and the same dependence onparameters. Under conditions (2.77) the collision-broadening theory is valid, that is,Eq. (2.75) holds. In the more distant wings of the spectral line when Eq. (2.78) holdstrue, the quasi-static theory is valid; in other words, Eq. (2.76) is applicable to thephoton frequency distribution function.

The case opposite to Eq. (2.57) is when Eq. (2.69) is satisfied instead. This lastcondition can be written

N(D2/hv)3/2 > 1.

Then both the central part of the spectral line as well as its wings are correctlydescribed by the quasi-static broadening theory, and, in particular, Eq. (2.76) is validfor the wings of the line.

We conclude that the qualitative form (2.76) is always correct for the wings ofthe spectral line irrespective of the choice of collision or quasi-static broadeningmechanism. We point out that the region of the line wings is given by |co — coj-ol ^ND2/h, since the line width in the quasi-static limit is of the same order as the value(2.74) for collision broadening. This follows from Eq. (2.58), in that

|co - U(R)/h ~ D2/(hR3) ~ ND2/h.

Problem 2.20. Estimate the dependence on (w — c%)) of the antistatic wing of thespectral line when the signs of the difference (co - co*o) and of the interaction potentialU(R) are opposite. Employ the quasi-static theory of broadening, and consider thecase of the inverse power law potential U(R) = C/Rn .

SOLUTION. The expression (2.71) for the static wing of the spectral line was obtainedunder the condition that the quantities (co - o)k0) and U(R) have the same signs. Herewe suppose that (to - co o) and U(R) have opposite signs, that is, we consider theopposite wing of the line to that investigated in Problem 2.17. Since we consider thefar wing of the line, the expansion in power series of the exponent (2.70) remainsappropriate. Equation (2.63) thus leads to

/ dR / exp [-it(a) - % ) 4- itU(R)/h] dt.

However, we can no longer extract a delta function from this expression since wenow postulate that (co — cok0) and U(R) have opposite signs. In this case the values ofR* that supply the principal contribution to the integral can be found by the condition


of phase stationarity. The condition for stationary phase is cu - o)ko = U(R*), but thiscan be satisfied only for complex values of/?*.

In particular, for the potential U(R) = C/Rn (e.g., with C > 0), we find

R* = [C/(fi>«, - co)]1/nexp(7T/A).

The essential times t* that determine the integral over t in the expression for flw areof the order of t* ~ R*/v. Using the method of steepest descent (the saddle-pointmethod), we obtain for the exponentially small wing of the spectral line, the frequencydistribution function

where v is the relative velocity of the colliding particles.

Problem 2.21. Find the line shape for the transition nnin2 —> n!n\n!2 in a hydrogenatom in an ideal hydrogen plasma (where nx and n2 are the parabolic quantumnumbers). Broadening of the line is caused by the ions in the vicinity of the excitedatom. This is the so-called Holtzmark broadening.

SOLUTION. Let E be the total electric field produced by the ions around the atom beingexamined. This field can be regarded as a quasi-static potential since the ions are ofrelatively large mass and their velocities can be regarded as low. As is well known,the static electric field produces a linear Stark shift in an excited hydrogen atom. Thusthe upper and lower states of the radiative transition in the hydrogen atom are splitinto Stark components. These components are characterized by parabolic quantumnumbers. The quantum numbers for the initial and final states will be denoted bynn\n2 and nfn[nf

2, respectively. The value of the linear Stark shift is

ATnn]n2 = (l)n(n{ - n2)E.

Hence the transition frequency a)nn/ for the given radiative transition is shifted by theamount

A«W = ( | ) [n{nx - n2) - n\n[ - ri2)) E = xE.

Here and in the following we use the atomic system of units where e = h = m = 1.We are interested in the distribution function a^ for the photon frequencies a) that

are absorbed or emitted in the transition. We write this function in the form

a^do) = w{E)dE,

where the function w(E) is the probability that the field strength is E. The values ofa) and E are connected by the relation [see also Eq. (2.58)]

a) = conn/ + xEy

so that

dE/dco = x~\


We must therefore calculate the function w(E). The value of E is the sum of all thefield strengths of the surrounding ions. From Coulomb's law we have

where Rm is the radius vector of the rath ion in a coordinate system with the radiatingatom at the origin. For a fixed distribution of ions in space we obviously have

where 8(x) is the Dirac delta function. A change in the Rm vectors produces a changein the electric field strength E, and hence a shift in the transition frequency a)nn>. Thisproduces a broadening of the spectral line. To calculate the broadening, we averagethe function w(E) over the positions Rm of the ions.

The probability that the rath ion is in the volume dKm is dKm/fl [see alsoEq. (2.60)], where fl is the normalized volume of the plasma. Such probabilities forthe individual ions must be multiplied by each other, on the assumption that the ionsdo not interact with each other. The absence of such interaction is the underlyingprinciple of the ideal plasma.

We obtain in this fashion

where TV is the density of the ions and NCI is their total number. To calculate thisintegral, we employ the integral representation for the Dirac delta function given by

From the fact that each of the integrals over Rm is the same as all the others, theprobability w(E) can be written as

where R is any one of the Rm coordinates of the ions.We perform the limit M l -» oo in this expression by following the same procedure

as that employed in arriving at Eq. (2.63). That is, we add and subtract unity to

exp(-/R • r/ft3), and make use of / J R = 11 to arrive at the result

w(E) = / exp I N / J R exp

The normalization volume II vanishes, as it should. Evaluation of the integral over Ris done directly, leading to the result

dr\3'

We can evaluate the angular part of / dr and find that

where

Eo = 2TT(4N/15)2/3,

and

2 r° r /^\3/2iJ<(x) = — / tsinfexp - - dt.

irx JQ [ \xJ \

This is the so-called Holtzmark distribution.The function J-C(x) can be evaluated explicitly in the limit of small or of large

argument. If x < 1, then

If x > 1, then we obtain

The function J-C(x) has a single maximum at x = 1.7, with a value of approximately0.35.

The frequency distribution function a^ which determines the profile of the spectralline is of the form

d o JxE0

The Holtzmark width of this spectral line is xE0 ~ N2/3. The quantity Eo is the meanstrength of the ion field that acts on the excited atom under consideration. We see thatthe distribution function a^ decreases as (co - ow)~5/2« This agrees with the generalexpression (2.72) for the quasi-static wing of the spectral line at n = 2. Indeed, for

CROSS SECTIONS FOR ABSORPTION AND INDUCED EMISSION OF PHOTONS 99

the case we consider, Eq. (2.58) gives

(x) — conni ~ E ~ R 2

due to the Coulomb law. That is, we have n = 2. In this limit, broadening is producedonly by the nearest ion.

The frequency distribution function for the entire transition line for n —>• n' is ob-tained by simply summing the above partial distribution functions over the parabolicquantum numbers n\n2 and n[n'2 of the initial and final states in the transition. Thisfollows from the independence of the broadening of the separate Stark componentsof the levels of the hydrogen atom.

2.4 CROSS SECTIONS FOR ABSORPTION AND INDUCED EMISSIONOF PHOTONS: ABSORPTION COEFFICIENT

Our goal is the calculation of the parameters relevant to the fundamental interactionprocesses between radiation and atomic electrons. If the atomic transitions consist ofthe absorption or emission of a single photon, a basic parameter is the absorption oremission cross section. For sufficiently small intensities, the cross section does notdepend on the radiation intensity or on the density of atoms.

We want to define the photoabsorption cross section for transitions between thebound states 0 and k of the atomic electron. This process is, schematically,

AQ + nhco —>• Ak + in — \)hco>

where A is the atom, the index labels its state, and hco is the energy of a photon offrequency co. The photon absorption cross section is the ratio of the rate of photonabsorption wa to the flux of incoming photons j ^ in the given frequency range. Thatis, the cross section is

<Ta = Wa/U (2-79)

For induced emission, corresponding to the scheme

Afc + nhco —> AQ + (n + l)hco,

the cross section is the ratio of the induced emission rate wr to the photon flux densityjo, or

<rr = *>rlj». (2.80)

In terms of the absorption and induced emission cross sections, we introduce thequantity that determines the propagation of the radiation in a gas. We denote by/w the intensity of radiation of frequency co moving through a gas. This intensityvaries during propagation of the radiation as a result of absorption and inducedemission. Since the number of absorption or induced emission events is proportional


to the number of photons that take part in these processes, the change of the photonintensity with distance is proportional to this intensity, so that

dljdx = -kja, (2.81)

where x is the coordinate in the direction of propagation. The proportionality coeffi-cient ku is called the absorption coefficient.

Problem 2.22. Relate the absorption and induced emission cross sections to thephoton frequency distribution function. Obtain also the general expression for theabsorption coefficient.

SOLUTION. Consider first the photon absorption process. We take into account thatthis process does not really occur at a fixed value co 0 of the frequency, but rather itinvolves a range of frequencies in the vicinity of (ok0. Designate by aw the frequencydistribution function for the absorbed photons in the range [co, co + dco]. Previoussections of this chapter were devoted to the calculation of this function for a varietyof cases.

From Eq. (1.9), the photon absorption rate is

. Iwhere gk is the statistical weight of the final state k. We can relate the discreteabsorption spectrum to the continuous frequency spectrum by means of the frequencydistribution function aM by setting

dwa = wojfctfft, da). (2.82)

This rate is an average over the polarization states of the incoming photons.As in Eq. (1.13), we introduce the lifetime r of the state k with respect to sponta-

neous transition to the state 0 (setting /tw = 0), and obtain

>lo\Dk0\

2g0/3hc3,

where g0 is the statistical weight of the state 0. Then we can rewrite Eq. (2.82) in theform

dwa = -—n^dco. (2.83)

We must now find the photon flux impinging on a single atom, given n^ photonsin a given state. In a unit volume, the number of photons is

The factor 2 takes into account the two possible polarization states of the photons.Hence the photon flux for the frequency range [co, co + dco] is

= cdNw = n^day/iiTc)2. (2.84)

CROSS SECTIONS FOR ABSORPTION AND INDUCED EMISSION OF PHOTONS 101

Equations (2.79), (2.83), and (2.84) give

^ = 7 = (-)2--- <2-85>djoj \ CO / T go

Now we consider the process of induced emission. With Eq. (1.13) and the defini-tion of the frequency distribution function aw for the emitted photons, we can expressthe induced photon emission rate in the form

dwr = (l/^n^a^dco.

When we divide by the photon flux (2.84), we obtain according to Eq. (2.80), thecross section for induced emission

ar = (HE\2^ (2.86)\ CO / T

Now we calculate the absorption coefficient for a gas, associated with the radiativetransition between the states 0 and k of the atomic electron. We designate by No andNk the densities of the atoms in states 0 and k, respectively. From Eq. (2.81), thequantity k^ is that portion of the photon absorption that takes place in a unit lengthof the photon beam. The decrease of the photon population in the beam is given byNo<Ta, while the increase in the photon population in the beam is Nkcrr. Consequently,kc is

kM = Noaa - Nkcrr. (2.87)

Substituting Eqs. (2.85) and (2.86) into (2.87) gives the absorption coefficient

K = Nk (-Y — f ^ - - lV (2.88)

Problem 2.23. Calculate the resonance fluorescence cross section. Resonance fluo-rescence is the process in which a photon incident on an atom is absorbed, followedby photon emission with a return to the initial state. Take the photon energy to beequal to the excitation energy from one atomic energy level to another. Broadeningarises from spontaneous decay.

SOLUTION. Resonance fluorescence stems from the absorption by an atomic electronof a photon from the distribution function aw determined by the spontaneous widthl/xfc of the excited state k. The resonance fluorescence cross section is found fromEq. (2.85), in which is incorporated the statistical weight gk = (2Jk + 1) associatedwith the degeneracy of the excited state k with respect to projections of the angularmomentum Jk of this state. In like fashion, g0 = (270 + 1), where 70 is the angularmomentum of the state 0.

The quantity T in Eq. (2.85) represents the lifetime if of the state k with respectto spontaneous decay into the state 0. In the case when this is the only decay channel


open, we have the obvious relation T® = T . The function a^ in Eq. (2.85) is deter-mined by Eq. (2.9), with the width established by the total lifetime T> of the state k.From Eq. (2.85) we then obtain the photon absorption cross section

2Jt+l (l^\ (TkTkraa 2(2Jo + 1

The resonance fluorescence cross section as obtained from Eq. (2.89) is that partof the total cross section arising from the channel that returns the atomic electronback to the state 0 instead of to some other final state. Hence, we can write

= n = 2Jk + 1 7TC2(COTQ)-2

2(2/o + 1) (oko ~ Co? + [1/(27*)]*'

As an alternative, the factor rk/r® could be inserted into the statistical weight gk ofthe final state corresponding to the channel with return to the initial state 0.

The order of magnitude of the value of Eq. (2.90) is

df ~ TTC2/Q)2 ~ 7r(A/27r)2,

where A = 2irc/a> is the photon wavelength. We see that the cross section doesnot depend on the fine structure constant a — e2/he, in contrast to analogous crosssections in nonresonant cases. The resonance fluorescence cross section thus exceedsusual nonresonant cross sections for light scattering by a factor ((okQTk)

2.

Problem 2.24. Calculate the photoabsorption cross section integrated over all fre-quencies of the absorbed photons. Assume that the atomic electron is initially in itsground state.

SOLUTION. We first integrate the photoabsorption cross section in Eq. (2.85) overthose frequencies in the neighborhood of the fixed absorption line for the transition0 —> k. Using the normalization condition (2.2), we obtain

where we have set gk = g0 = 1. Integration over all frequencies of the absorbedphotons corresponds to performing a sum in Eq. (2.91) over all possible final statesof the atomic electron, including continuum states. When we use the sum rule (1.21)and take into account that

Itytol = 3|(Dz)ofcl ,

we obtain

2ir2e2naa dco = , (2.92)

mewhere m is the electron mass and n is the number of electrons in the atom.

CROSS SECTIONS FOR SCATTERING AND RAMAN SCATTERING OF PHOTONS 103

Problem 2.25. Calculate the maximum photoabsorption cross section in the atomictransition from state 0 to state k.

SOLUTION. The photoabsorption cross section in Eq. (2.85) is proportional to thedistribution function aw for the frequency a) of the absorbed photons. It follows fromthe definition of a^ that the minimum width of the spectral line corresponds to themaximum of aw, that is, to the maximum of cra. We have seen above that the minimumwidth occurs with spontaneous broadening. The function aw is then determined byEq. (2.5). If we set a)k0 - co = 0 in this equation, we can rewrite Eq. (2.85) in theform

4 — , (2.93)CO / 7£ IT go VCO/ 7£go

where we recall the meaning of gk and g0 as the statistical weights of the k and 0states of the atomic electron.

Although the cross section (2.93) is greater than the resonance fluorescence crosssection oy in Eq. (2.90), it is nevertheless of the same order of magnitude. Fluores-cence is one of the channels for deexcitation of an atomic electron.

2.5 CROSS SECTIONS FOR SCATTERING AND RAMANSCATTERING OF PHOTONS

We here regard photon scattering as a two-photon process in which the first photonof frequency co\ is absorbed and a second photon of frequency co-i is emitted, withthe concomitant transition of an atomic electron from the initial state 0 to the finalstate m. This process differs from fluorescence in that the final state m can differ fromthe initial state 0.

Problem 2.26. Calculate the cross sections for resonant and nonresonant Ramanscattering.

SOLUTION. Consider first the case of resonant Raman scattering, where the photonfrequency co\ is nearly the same as the excitation frequency c%) of the state k ofthe atomic electron. The photoabsorption cross section aa is given by Eq. (2.85).Subsequent steps in the solution of this problem parallel the solution of Problem 2.22about resonance fluorescence, except that Eq. (2.89) for the photoabsorption crosssection should be multiplied by the factor n/if, where if is the lifetime of the statek with respect to the transition into the final state. As a result, instead of Eq. (2.90)we find

2/* + 1 TTC2 (rFtfy1 . . . . .(Y = K K (0 94)

c 2(270 + 1) o>2 (a>*o - «)2 + [l/(2i>)]2"

In particular, the result at exact resonance is

2(270+ 1) co2 f f

As should be the case, this quantity is less than Eq. (2.93), which is the maximumvalue for the photoabsorption cross section. We observe that Eq. (2.94) is of the sameorder of magnitude as the resonance fluorescence cross section in Eq. (2.90).

We now treat nonresonant Raman scattering. We write the cross section as the ratioof the two-photon transition rate (1.46) to the incident photon flux. Both absorbedand emitted photons in Eq. (1.46) have definite polarizations (si and s2, respectively),so the appropriate flux is one half of Eq. (2.84). We can presume that there are noscattered photons in the incident beam, that is, n^ = 0, and we then obtain the crosssection for nonresonant Raman scattering,

£ COk0 - 0)1 0>lcO + (^2 JdVt2. (2.95)

In obtaining this expression from Eq. (1.46), we integrated over the frequency co2

of the emitted photon, using the energy-conserving delta function that appears inEq. (1.46). The sign conventions in Eq. (1.46) for co\ and co2 are chosen so as todescribe the absorption of a photon with frequency co\ and the emission of a photonwith frequency co2. The quantity £l2 is the solid angle of the scattered photon, and mis the index labeling the final state of the atomic electron. Energy conservation gives(omo = o)\ — co2. In particular, when the initial and final states are the same, thatis, when m = 0 and cox = co2, Eq. (2.95) gives the nonresonant fluorescence crosssection.

Problem 2.27. Calculate the cross section for photon scattering by a free electron,using the general expression (2.95).

SOLUTION. As everywhere above, we suppose that the energy of the incident photonhco is small as compared to the electron rest energy, that is, hco < me2. The photonmomentum is hco/c. The change of photon momentum in the scattering process andthe electron momentum after scattering are of the same order of magnitude: hco/c(excluding scattering through very small angles). The energy gained by the electronin the collision is of the order of (hco)2/(me2). It is seen that the energy gained issmall as compared to the rest energy, which is equivalent to the statement that thevelocity increment of the electron from the scattering is small as compared to c.The electron motion is thus nonrelativistic. Since the change of the energy of thephoton, also ~ (hco)2/(me2), is small compared to the initial photon energy hco, thephoton-electron scattering is quasi-elastic, which means co\ ~ co2.

We shall use Eq. (2.95) for the calculation of the cross section supposing thatco\ = co2 = co, and using the semiclassical approximation for the free electron statesdue to the semiclassical character of the initial continuum electron state 0. The dipole


operator of an electron is D = —er, where r is the electron coordinate. Equation(2.95) leads to

da =he2

uk0 -(D2d£l, (2.96)

where the index k identifies the free electron continuum quantum states.The principal contribution to the sum over k comes from states that have energies

near the energy of the initial state 0. Indeed, matrix elements ro^ are semiclassicallysmall for large differences in energy between the states k and 0. Consequently, wehave o) > a)k0, that is, the frequency of the scattered photon is large compared totypical electron frequencies. When (ok0 is neglected in the denominator of Eq. (2.96),we obtain the expression, independent of co,

2

da = dCl. (2.97)

We wish to evaluate the sum in Eq. (2.97). The coordinate axes are selected sothat Si is along the z axis and S2 is in the xz plane. The sum is then of the form

Six /2 ^kOXOkZkO + S2zk

The first of the sums in this expression is zero because of the odd parity of the productxoicZko- ^ changes sign when z —> ~z. The second sum can be calculated using thesum rule (1.21) for dipole transitions. The above expression then yields

Substituting this expression into Eq. (2.97) gives the cross section in the form

da = r2(sX'S2)2 dfi. (2.98)

The quantity re — e2/(me2) is the classical electron radius. Equation (2.98) is calledthe Thomson formula. It is a purely classical result, since the Planck constant doesnot appear.

To calculate the total cross section, we integrate Eq. (2.98) over the solid angle. Weselect the polar axis of a system of spherical coordinates to lie along the polarizationvector Si of the incident photon. The vector S2 lies in the plane determined by S\ andthe wave vector k2 of the scattered photon. For the direction normal to this plane, thescattering cross section is zero. (In this direction, Si and s2 are perpendicular to eachother.) Let 6 be the angle between S\ and k2. Since s2 and k2 are perpendicular, weobtain S\ -s2 = sin 6. The total cross section is thus

= r2 f (2.99)

Equations (2.98) and (2.99) are also attainable in classical radiation theory bysolving the Newtonian equations of motion for induced electron oscillations and

considering the emission of secondary waves with the same frequency. The classicalresults fail when the photon energy ha) is of the order of the electron rest energy me2

or greater. Then most of the incident photon energy is transferred to the electron, andthe scattering is therefore inelastic. In this case, relativistic and quantum effects willbe important simultaneously, and the electron spin will be an essential element in thedescription of the scattering.

Problem 2.28. Calculate the low-energy scattering cross section for a photon scat-tering from an atom with zero angular momentum. Evaluate the result in closed formfor the ground state of hydrogen.

SOLUTION. The frequency of the photon is taken to be small compared to typicalatomic frequencies, or co < co o- This limit is thus opposite to that considered inProblem 2.27 for a photon scattering off a free electron with co > co o- The smallfrequency condition allows us to simplify Eq. (2.95) to

da - V (2.100)

The initial state is specified to be an S state, so that its magnetic quantum numberis MQ = 0. The state k is therefore a P state in accordance with the dipole selectionrule, and so Mk = 0, ± 1. We take the axis of quantization z to lie along si. The vectorD is along the z direction, so it is along Si. In the opposite case, the quantity D • S\vanishes. Hence we have

We now define the polarizability tensor

(Di)Ok(Dj)kO

(see Section 1.7). It follows from the above considerations that atj is a diagonaltensor, so that atj = a8/ ;, and

2e2 sr^ l^ol2a = — y .ft ' J Wfco

When this result is substituted into Eq. (2.100), we find the scattering cross section

/ CO \ 4 -, i CO4Ot2 -.da = ( - I a2(s1'S2)

2dil = — ^ sin2 Sdtl, (2.101)\c / c4

where 6 is the angle between the polarization direction si of the incident photon andthe direction k2 of the wave vector of the scattered photon. If s2 is normal to theplane containing the vectors S\ and k2, then S\ • s2 = 0, since the directions S\ and

D coincide. In the opposite case, if the vector s2 is in the plane defined by S\ and k2,then si • S2 = sin 0, since s2 is perpendicular to k2. After integration over the angularcoordinates, we find the total cross section for photon scattering by an atom in thelow photon frequency limit to be

<r - ^ (2,02)

We now wish to solve the same problem by classical methods. From Eq. (1.17),the intensity of scattered light is

/ = 2D2/3c3,

where D is the induced dipole moment produced by the field of an electromagneticwave with an electric field given by E cos cot. By the definition of atomic polarizabilitya, we have D = —aE cos cot. We are thus led to the intensity of scattered lightexpressed as

/ = - ^ - a E c o s a t f .3c3

To find the cross section, we should divide this quantity by the energy flux ofthe incident radiation. This energy flux is given by the Poynting vector cE X H/4TT,

where E and H are the electric and magnetic fields of the electromagnetic wave. Inour case, the energy flux has the magnitude (CE2/4TT)COS2 cot. When we define thescattering cross section as the ratio of the intensity of scattered light to the energyflux of the incident radiation, we obtain

STTCO4 2

This agrees with the quantum result in Eq. (2.102). The advantage of the quantum-mechanical derivation is that it makes it possible to obtain the explicit expressionfor atomic polarizability. It is seen from the derivation that the scattering processconsidered is purely classical. A classical dipole moment radiates the same frequencythat is induced by the electromagnetic wave. Such scattering is called Rayleighscattering. It is interesting that both Rayleigh scattering (co < coko) and Thomsonscattering (co > cok0) are purely classical phenomena.

The maximum in the scattering of visible light by atoms with absorption fre-quencies in the ultraviolet range corresponds to the violet end of the spectrum, sincethe scattering cross section increases very strongly with frequency: as co4. The limitco < coko holds true nevertheless. This explains the blue color of the sky. Sunset is ofa red color for the same reason: the strong scattering of the violet part of the Sun'sspectrum in the direct flux of the Sun's rays leaves a predominance of red in theremaining part of the sunlight.

The static polarizability can be exactly calculated for the ground state of thehydrogen atom. The result is that a = ( f ) ^ where a0 = h2/(me2) is the Bohr

radius. Hence the cross section for the hydrogen ground state for low-energy photonscattering, with hco < ft2/{mafy, is given by

r?(^)\Sls2?<m. (2.103)

This expression describes accurately the elastic scattering cross section from zerofrequency up to the frequency of the first resonance when ha) = hu>i\ — 3me4/($h2).The indices 1 and 2 refer, respectively, to the ground and first excited states of thehydrogen atom.

An additional problem with accounting for degeneracy of the state 0 with respectto magnetic quantum numbers appears in the case of nonzero angular momentum.If the low-energy photon is scattered by an atom in an excited state, then Ramanscattering occurs as well as Rayleigh scattering, with the consequent transition of theatom to a lower lying state m.

Problem 2.29. Calculate the dependence of the intensity of induced Raman scatter-ing on the propagation distance of the photon beam in the gas.

SOLUTION. We consider Eq. (2.95) for the Raman scattering cross section dac whenan atomic electron makes a transition from the initial state 0 to the final state m. Ifwe denote by N/V the density of atoms, then the quantity

8 = ^CTC (2.104)

represents the number of photons with frequency co that is generated in a unit distancealong the photon beam. The total Raman scattering cross section, ac, is obtained fromEq. (2.95) by performing the integration over the angles of the emitted photons offrequency co2.

We now have n^ ^ 0, since the photons go from an incident beam of frequencyo)\ to photons of scattered light with frequency co2. If we select coordinates with thez axis along the propagation direction of the incident beam, then by the requirementthat each absorbed photon gives rise to a scattered photon, we have

n^iz) + n<O2(z) = const = /iWl(0), (2.105)

where nW] (0) is the initial number of photons in the incident beam. In the usualscheme of quantization, each mode of oscillation is contained in the volume V, sowe suppose that the typical characteristic length along the z axis is much greaterthan V1/3.

We can now write balance equations that determine the change with z of thequantities n^ (z) and n^iz). Equation (2.104) establishes the number of photons thatappear in a unit length along the photon beam, under the condition that there isone photon of frequency o)\ and none of frequency a>2- However, if we have at thecoordinate z the number nMl photons with frequency o)\ and n^ photons of frequency

a>2, then Eq. (1.46) says that the number of photons appearing in a unit length alongthe beam with frequency o>2 is

a(z) = gnai(z) [1 + n^iz)] . (2.106)

Hence, the balance equations are of the simple form

•7-"»,(*) = - - T n ^ ) = -<*(*)• (2.107)az dz

The solution of the system (2.107) under the conditions (2.105) is elementary. Wewrite it in the form

r 7 u L V (2-108)[exp(Gz)/nWl(0)] + 1

where G is defined as

G = g [1 + nWl(0)] = ^<rc [1 + nWl(0)] . (2.109)

The quantity G is called the increment coefficient. We see that at first the number ofscattered photons increases linearly with z. This corresponds to the general theorydeveloped in Problem 2.26. This linear increase occurs when Gz < 1. When Gz ~ 1,the linear increase becomes an exponential increase. Finally, when Gz > 1, saturationtakes place, so that all photons from the incident beam (a>0 are replaced by photonsin the scattered state (eo2), or rcW2(z) ~ nMx (0).

The intensity of induced Raman scattering for photons with frequency o)2 is givenby

72(z) =

where n^iz) is determined by Eq. (2.108). In the linear regime, Eq. (2.108) becomes

which is in good agreement with Eq. (2.95). If we take the volume V to have thelength z in the direction of the photon beam with frequency (Oi, where N the numberof atoms in this volume, then the cross section of the volume V is V/z. We nowcalculate the energy flux through this cross section for the photons of frequency OL>I,

and obtain

^ f e ) ^ ( 2 . 1 1 1 )

+ — chcoi +««),(0)J


Using Eq. (2.109) we rewrite the first term in Eq. (2.111) in the form

M

(2.112)

To calculate the cross section we divide Eq. (2.112) by the particle density N/V andby the photon flux for the incident photons. As should be expected, we obtain ac, theRaman scattering cross section given in Eq. (2.95).

In the nonlinear regime, the increment coefficient is more useful than the crosssection ac.

2.6 TWO-PHOTON ABSORPTION

We consider here the case where the transition frequency com0 of an atomic electronfrom state 0 to state m is approximately twice the frequency co of the incident photons.That is, we treat the case where como ~ 2co.

The electric field vector of the electromagnetic wave is taken to be of the form

E(0 = 2Ecosfecosatf. (2.113)

That is, the electromagnetic field is that of a standing wave. The notation here is thatk - co/c is the wave number, and z = vt, where v is the projection of the atomicvelocity on the z axis. Equation (2.113) can be expressed as a superposition of twotraveling waves with equal amplitudes, but with the different frequencies co + kv andco — kv. Specifically, we can write

E(0 = Ecos(co + kv)t + Ecos(w - kv)t. (2.114)

Radiative transitions with two photons were studied in Section 1.6. Problems inthis section are devoted to nonuniform broadening of spectral lines due to thermalmotion of the atoms.

Problem 2.30. Find the frequency dependence of the two-photon absorption coef-ficient when the absorbing atoms move in a resonator under the influence of thestanding light wave. The velocity distribution of the atomic velocities can be takento be Maxwellian. The density of atoms is presumed to be small, so that atomiccollisions can be neglected.

SOLUTION. We use Eq. (1.48) for the two-photon transition rate. Equation (2.114)gives the electric field for the standing wave of photons with frequencies co + kvand co — kv. The following processes can occur: (1) absorption of two photons withfrequency co + kv; (2) absorption of two photons with frequency co — kv; and (3)absorption of a photon of frequency co + kv followed by absorption of a co — kvphoton, or vice versa.

TWO-PHOTON ABSORPTION 111

The rate (1.48) contains the delta function

In the present problem, we have o>i = co ± kv and co2 = co + kv. We can alterEq. (2.9) to reflect the content of this delta function to obtain the photon frequencydistribution function

a' = (wm0 - CO! - a>2)2 + [l/(2rm0)]2 '

(2.115)

where rm0 is the reduced spontaneous lifetime for the transition m —> 0. This lifetimeis determined by single-photon spontaneous transitions into other atomic states, ratherthan by the two-photon spontaneous decay to the state 0.

To take Doppler broadening into account, we must combine the Lorentz line shape(2.115) with the Doppler line shape. First, we consider the absorption of photons fromdifferent beams. The distribution function (2.115) then does not depend on atomicvelocity. Consequently, averaging over the thermal distribution of velocities does notalter this function.

To calculate the corresponding two-photon transition rate according to Eq. (1.48),we suppose that both waves have the same polarization. We place the z axis alongthe common direction of the polarization vectors S\ and S2. The two-photon matrixelement is defined by the relation

m 0

The contribution to the two-photon transition rate is, from Eq. (1.48),

, ^ 2

6

U (2.117)

where a^ is determined by Eq. (2.115) with G)\ + a>2 = 2co.Now we consider the absorption of photons from one beam. The Lorentz profile

(2.115) that contains the velocity of the atom should be integrated with the Dopplershape. Actually, the Doppler width

l/1( 2T\l/1 ( T \——r = G)mQ ——-z

\Mcz ) \2Mcz )

1/2(A(OD = (X) ——r = G)mQ

\Mcz )

is large compared to the spontaneous width T~Q. We know from the solution ofProblem 2.4 that the resulting contour is Doppler-like, so we have

* = e xP2(O-

2AcoD

(2.118)

We shall now calculate the corresponding rate for the two-photon transition. InEq. (1.48) we have only one term instead of two in the two-photon matrix elementas a result of the similarity of the two photons. Consequently, the rate should bemultiplied by two because of the presence of two photon beams. When we employthe delta function 6(com0/2 — <o ± kv) in the Doppler shape (2.118), we obtain thecontribution to the two-photon transition

mO (2.H9)

The total two-photon transition rate is the sum of the individual contributions inEqs. (2.117) and (2.119). To obtain the absorption rate cra we divide this sum by theflux of incident photons cE2/(47rhco). From Eq. (2.87), we must then multiply aa

by the density of atoms No in the initial state 0 in order to obtain the two-photonabsorption coefficient

ch3 D\ 2al + at . (2.120)

Equation (2.120) describes a narrow resonance superimposed on a broad Dopplershape. The ratio of the maximum peak height to the height of the background is

The dependence of the absorption coefficient on co is thus of the form of a narrowhigh maximum amid a broad background. The width of the narrow resonance isdetermined by the reduced lifetime for the two-photon 0 —• m transition and isvery small. Determination of the position of this resonance requires Doppler-freehigh-resolution spectroscopy.

We have seen that the two-photon absorption coefficient depends on the intensityof the photon beam, unlike the single-photon absorption coefficient. In consequence,its value is significant only at the high intensities available with lasers.

We note, finally, that if we considered a traveling wave instead of the standingwave, the two-photon absorption is determined only by the broad Doppler shapecorresponding to the second term of Eq. (2.120).

Problem 2.31. Under the conditions of Problem 2.30, examine how an intermediateresonance level affects the two-photon absorption coefficient.

SOLUTION. Suppose that some level k is in resonance with the initial state 0, andthus also in resonance with the final state m. Both resonances are of single-photontype. Hence, in the sum over k in Eq. (2.116) for the two-photon matrix element, weshould retain only that one state that corresponds to the resonance. In the vicinity ofthe resonance we can write the resonant denominator in the form

cok0- co±kv - I'/(2TH)),

where the term i/(2rk0) takes into account the spontaneous lifetime of the state k. Webegin by considering the first term in Eq. (2.120). The quantity a^ does not containthe velocity v of the atom and is not altered by the Doppler averaging. However, theresonance factor

D(aw - a> ± kv)2 + 1/(2T>O)2'

when Doppler averaged under the condition Aco^o > 1, takes the form

27TTk0\(Dz)mk(Dz)k0\2a°, (2.121)

where the Doppler line shape a% is given by

The quantity AcoD coincides with that defined in Problem 2.30. In resonance, the firstterm in Eq. (2.120) is larger than in the nonresonant case by a factor of the order ofmagnitude

We now consider the second term in Eq. (2.120). Here the product of the lineshape (2.115) and the resonance factor in the two-photon matrix element

_ 1 1 \(DZ)mk(Dz)k0\2

om0 -2coT 2kv)2 + l/(2rm0)2 (<%> - co ± kv)2

is subjected to Doppler averaging. If we take into account that Ao)/> > r *, thisaveraging can be done by following the procedure employed in Problem 2.2, wherethe product of two Lorentz line shapes was averaged. In this way, we are led to theresult

\(Dz)mk(Dz)k0\2

(cok0 - wm0/2)2 + l/(2rm ,)2 V ll° 2rm0

where we use the definition

Tmk 2 r m 0 T/cQ

and where afL is given by Eq. (2.118).Summing the terms given in Eqs. (2.121) and (2.122), we find the two-photon

absorption coefficient in the resonant case to be

_ 7r2N0E2(Om0 2

y/ I -w -"iu D

(cok0 - (om0/2)2 + l /(2Tm 0)2 w

1- com0/2)2 + l / ( 2 i

We can draw a number of conclusion from Eq. (2.123). The first term correspondsto absorption of photons from different beams. As in the nonresonant case, it isof the form of a narrow peak superimposed on a broad background. However, thatpart of the second term corresponding to photon absorption from only one beamhas a height of the order of the height of the narrow peak when level k is exactly(to within an accuracy of the spontaneous width l/rm^) halfway between levels 0and m. Tuning of the intermediate level to resonance, on the one hand, increases thetwo-photon absorption coefficient, but on the other hand it also increases the heightof the Doppler background with respect to the narrow peak.

The second term in Eq. (2.123) is connected with two-photon absorption when,after the first photon is absorbed, the atom goes to the "real" state k. For sufficientlyexact tuning of the state k midway between the 0 and m states, the two-step absorp-tion process and the two-photon absorption process described by the first term inEq. (2.123) are of comparable importance in the two-photon absorption coefficient.

When the condition \tok0 - com0/2| <l Aa)£> is satisfied, the distribution functionsa® and a^ coincide. We observe that the inequalities |w^0 ~ wm0/2| > 1/T and\&ko ~ <^mo/2| < 1/T are both possible. However, if \o)k0 - como/2\ > ACL>£>, we thenobtain two Doppler profiles: one with the maximum at the frequency com0/2, and theother with a maximum at the frequency cx)k0. In the latter case, the height of the peakis much larger than the background.

Finally, from the first term of Eq. (2.123) we can conclude that when there is largedetuning from the resonant location of the state k at the midpoint between states 0and m, the background is asymmetrical, since the peak of the narrow resonance ata) = como/2 and the maximum of the background contour at co = o^o do not coincide.

Problem 2.32. Calculate the two-photon matrix element in Eq. (2.116) for transitionsbetween highly excited states under the conditions of Problem 2.30.

SOLUTION. We first consider a simplified case in which two-photon transitions occurbetween highly excited atomic states with zero orbital angular momentum quantumnumbers. We take the electromagnetic field to be linearly polarized. We adopt theatomic system of units in which e = h = m = 1.

We start by calculating the two-photon matrix element required for Eq. (2.116).That is, we need to evaluate the matrix element

where zmk, Zko are dipole matrix elements of s —• p transitions. For the quantum num-bers of the initial state 0, the intermediate state k, and the final state m, respectively,we have for state 0: n, 0, 0; for state k: n1, 1, 0; and for state m: n + 1, 0, 0. The firstquantum numbers in each set: n, n', n + 1, are the principal quantum numbers forthese highly excited states. The second number in each set is the orbital angular mo-mentum quantum number, and the third number in each set is the magnetic quantumnumber.

After the straightforward integration over angles, we express the dipole matrixelement in terms of the radial part,

_J_r>

The semiclassical expression for the radial dipole matrix element R^Q under theconditions nyn' ' > 1 and An = \n - n'\ < n, n1 (this constraint on the values of Anis essential in the sum over intermediate states) in a Coulomb potential is found withthe help of the Coulomb radial wave functions to be

*«'• 3'/6 v(2\ nn'\3

If A n = 0, we then obtain

R"nl = - 3 « 2 / 2 .

With the energy of the state with principal quantum number n denoted as sn, weintroduce the resonance detuning

and presume that £ < n~3 so that a two-photon resonance is possible. On the otherhand, we also impose the requirement that £ > n~4 so that we can neglect anhar-monicity effects. This last condition can be understood as follows. In the summationover n' in the two-photon matrix element, we have states symmetrically disposedabout the initial and final states of the transition. That is, we sum over states with theprincipal quantum numbers n! = n+An+1 and/i' = n—An, where An = 1, 2, 3,The contributions from these symmetrically placed states tend to cancel each other.This cancellation is broken both by the resonance detuning £ and by the anharmonic-ity of the highly excited atomic spectrum. We will require that the detuning £ shouldbe much less than the n~4 anharmonicity. If this condition is not satisfied, the calcu-lations are greatly complicated by the necessity of introducing corrections of the nextorder in the semiclassically small parameter \/n. We recall the analogous situationin the semiclassical evaluation of the sum rule in Problem 1.4.

The summation over the intermediate states n' gives the two-photon matrix elementin the form

x _ U +r ( 2 / 3 )

31/624/3 32/328/3 Z-^ [Arc(Arc + l)]5/3(An + 1/2)2 [ '

The first term within the brace in this expression comes from the contribution of theintermediate states n' = n and n1 = n + 1. If the two-photon resonance is exact(£ = 0), this two-photon matrix element vanishes within the framework of this orderof the semiclassical approximation. The sum in the second term within the brace inthis expression converges quickly. Furthermore, its magnitude is only about 5% ofthat of the first term in the brace. Our final result is

D^Q = 1.87 \n(n +1)] (sn+\ ~ £n ~ 2co).

This two-photon matrix element grows as about n10, a very strong dependence on theprincipal quantum number.

The results we have obtained can be applied also to complex atoms. In that case,n and n + 1 should be replaced by effective quantum numbers of the correspondingstates, deduced from experimental values of the atomic energy levels using thehydrogenlike formula sn = ~\n2. Corrections to this expression are the subjectof quantum defect theory.

We also point out that the results obtained are applicable only if the externalelectromagnetic field is sufficiently weak that it is valid to neglect the effects ofStark shifts and Stark splittings of highly excited states. These effects increase withgrowing values of n.

According to Eq. (2.120), the vanishing of the two-photon matrix element in theexactly resonant case strongly diminishes the height of the narrow resonance peakcompared to the broad Doppler background.

We now move on to the consideration of two-photon transitions between states withnonzero orbital angular momentum quantum numbers /. If / ~ n, the semiclassicaldipole moment is very small, and hence the two-photon absorption coefficient is verysmall. We may therefore confine our attention only to the case where / < n. We shallassume that the value of / is the same for initial and final states in the two-photontransition.

The matrix element zn^l~Xm is connected to the radial dipole matrix element bythe relation

7n'/-lm = 1 ^ ~ m~ I pn'l-lnlm [ (2 / -1 ) (2 /+ 1)J nl

where m is the magnetic quantum number of the initial (and final) state. The radialdipole matrix element is independent of / for small values of /, as we have observedabove.


By analogy, we can also write the matrix element z^/^lm. Then we must averagethe two-photon transition rate over the magnetic quantum number m of the initialstate. The factor by which the two-photon transition rate for / = 0 must be multipliedis

15(2/ -

For / = 0, we obtain the expected result /(0) = 1. The value of /( /) is near unityeven for / =£ 0.

Problem 2.33. Alkali atoms are excited by two oppositely directed laser beamsfrom the Sx/2 ground state to an excited D5/2 state via an intermediate P3 / 2 state. Thelifetime of the P3/2 state is T\ , and that of the D5 / 2 state is r2. Find the dependence ofthe photon absorption coefficient on the frequencies (O\ and co2 of the lasers for thefirst and second stages of the two-photon excitation.

SOLUTION. The frequencies o>i and a)2 are different by several Doppler widths fromthe quantity o)m0/2, where 0 is the S{/2 state and m is the D5/2 state. Two-photonabsorption from only one laser beam, with both photons of frequency o)\ or bothphotons of frequency cu2, is thus impossible. Only the absorption of a single photonfrom each of the beams can take place. We presume also that the energy denominatorsin the matrix element in Eq. (1.48) for two-photon transition rates are large comparedto the spontaneous widths of the atomic levels. Hence we can neglect spontaneousbroadening. Moreover, we suppose that resonance detunings are very large comparedto the Doppler width of the intermediate state. The two-photon matrix element isconstant in the presence of the stated conditions.

Since a)\ =£ w2, the linear Doppler effect is only partly compensated. We will fixone of the frequencies (a>2, e.g.), and consider the photon distribution function a^.From Eq. (2.115), it is of the form

^ ) 2 4 ] ' • (2-124)When this expression is averaged over a Maxwellian distribution of atomic velocitiesv, it produces the so-called Voigt spectral line (see Problem 2.4),

1/2 /»oo / » M 2

X

IT e

i - i

The Doppler width {(o\ — co2)\/T/(Mc2) is of the same order of magnitude as thespontaneous width 1/T2, SO that it cannot be neglected in this integral. We thus cannotgo to Lorentz or Doppler limiting distributions, unlike Problem 2.4. However, we do

assume that the quantity (2co! - com0) ~ (o>i - co2) is large compared to the Dopplerwidth in order to be able to neglect absorption of photons from only one of the beams.This is valid under the obvious condition T < Me2.

It is clear that the greater the difference \coi — (02I, the larger is the width ofthe spectral line. By the appropriate choice of the frequency o>2, we can achievea resonantly large two-photon matrix element. We are thus restricted to only oneintermediate state: In the present case, this is the state k = P^/2- The nearby stateP\/i does not contribute because the transition P\/i —* D5/2 is forbidden by thedipole selection rules. We conclude that the decrease of the absorption rate due to theincrease in line width that accompanies growth of the frequency difference \wi — (021can be compensated by resonantly increasing the two-photon matrix element. Thisphenomenon is called resonant increase of two-photon absorption.

The two-photon cross section can be of the same order of magnitude as the single-photon cross section for the allowed S —> P transition.

3ATOMIC PHOTOPROCESSESINVOLVING FREE PARTICLES

Radiative transitions between discrete levels of atomic systems were treated in Chap-ter 2. We now proceed to the study of radiative transitions when either the initialor the final state of the atomic system possesses a continuous spectrum. Treatmentof the intense-field environment will be deferred until Chapter 5, so we shall treathere only single-photon processes and only those problems in which the number ofparticles in initial and final states differ by no more than one.

3.1 DECOMPOSITION OF ATOMIC SYSTEMS

The process of photodecomposition can be described generically by

XY + hco —> X + Y,

where the initial state of the system XY is a bound state of the atomic particles X andY, whereas in the final state these particles are free. Examples of processes of thiskind are photodetachment of a negative ion,

A~ + ho) —> A + e\

photoionization of an atom (or of a positive ion),

A + hoo —> A+ + e\

and photodissociation of a molecule,

XY + h(o —> X + Y.

Radiative Processes in Atomic Physics. V. P. Krainov, H. R. Reiss, B. M. Smirnov **"Copyright © 1997 by John Wiley & Sons, Inc.ISBN: 0-471-12533-4

120 ATOMIC PHOTOPROCESSES INVOLVING FREE PARTICLES

The special feature of the photodecomposition process is that the transition isfrom a state in the discrete spectrum to a state in the continuum. It is thus necessaryto modify Eq. (1.9) for the absorption probability per unit time to take into accountthe fact that the frequency of the absorbed photon lies in a continuous range ofpossible values and is not confined to a unique frequency corresponding to thetransition between a pair of discrete levels. Because continuum wave functions arenot square-integrable in the conventional sense, it is necessary to define an alternativenormalization scheme for continuum wave functions. It is convenient to adopt thenormalization

f i//q(r)i//q*,(r) J r =(27r)3S(q - q'), (3.1)

where r is the distance between particles X and Y with wave vectors q and q',respectively, and i//q(r), ijjq>(r) are the wave functions that describe the motion ofthese particles.

We now consider the particles to be contained within a large spatial volume ft,and we wish to find the connection between the wave function i//q(r) and the wavefunction i/^(r), which is normalized to unity according to

<fc(r) I2 dv = 1.

These wave functions are the same to within a multiplicative constant C: ijjk = Ci//q.For the determination of C we use the orthonormalization conditions

= 8'km*

i\>t(T)ii>m{r)dr= 1.

If we relate the index k to the wave vector q, and the index m to the wave vector q7,these conditions yield

J2 J ^k(r)ilfm(r)dr = 1 = C2 J = C2ft.

This requires that C = f l"1 / 2 .The wave function *//q(r) is describable by a plane wave when the distance between

the atomic particles is so great that the interaction between them can be neglected.Then, with the normalization given by (3.1), one has

</rq(r) = exp(iq-r).

(For interactions that decline as slowly with distance as the Coulomb interaction, thisexpression is correct only to within a logarithmic term which must be added to thephase.)

DECOMPOSITION OF ATOMIC SYSTEMS 121

The corresponding plane wave limit for the wave function that is normalized tounity is

ife(r) = Cexp(iq-r),r—>oo

along with the normalization condition

/ | ife(r) \2 dr= 1 = C2 f dv = C2ft,

a a

so that we have C = ft~1/2 as before.

Problem 3.1. Obtain the expression for the photodetachment cross section of anatomic particle.

SOLUTION. Rewrite expression (1.9) for the absorption probability per unit time byusing the wave functions i//q(r) in the transition matrix element in place of i/fc(r). Thiscorresponds to a change of the matrix element | Do& |2 to | DOq I2 /ft, where thefinal-state wave function is normalized by condition (3.1). The statistical weight gk

of the final state is now given by

Cldq*

where dftq is the element of solid angle that characterizes the relative motion of thefinal-state particles. Now employ the condition of energy conservation

h(o = I + h2q2/(2ix), (3.2)

where / is the binding energy of the atomic particles X and Y in their initially boundstate, and JJL is the reduced mass of these two particles. This gives

qdq — — do).

When the new expression for the statistical weight of the final state is inserted intoexpression (1.9), the result obtained for the probability per unit time of radiativetransitions is

4co3 | DOq |2 Oq ix3nc5 LI $TT3 n

Naturally, the arbitrary reference volume ft disappears from this expression. Di-viding the probability dwi by the photon flux (2.84)

an expression for the photodetachment cross section is obtained that is independentof the number of photons nw,

DOq I2 <mq . (3.3)

Note that in this form the photodetachment cross section corresponds to a singleinternal state of the final particles X and Y.

Problem 3.2. Determine the photodetachment cross section for a negative ion, orthe atomic photoionization cross section, assuming that the field of the neutral orpositive atomic core in which the valence electron resides is spherically symmetrical.In addition, take the state of the atomic core to be unchanged after the radiativetransition of the valence electron.

SOLUTION. Use formula (3.3) for the photodetachment cross section. Because ofthe symmetry of the field of the atomic core, the state of the valence electron canbe described by the principal quantum number n, the angular momentum quantumnumber /, and the projection m of / in a given direction. Based on this description,the wave function of the initial bound state of the electron is

t//0(r) = -unl(r)Ylm(0, <p).r

Here r, 0, q> are the spherical polar coordinates of the electron, uni(r) is the radialwave function of the electron, normalized by the condition

oc

/u2

nl(r)dr = 1,

and Yim(6, <p) is a normalized spherical harmonic.The final state of the detached electron with momentum hq can be expressed in

terms of spherical functions in the form

1 °°*q(r) = - ^ ( 2 / + l)ilin(q, r)P/(cos 0qr).

r i=o

That is, the polar axis of spherical coordinates is determined by the direction of q.Here 0qr is the angle between the vectors q and r, and m(qf r) is the radial wavefunction of a free electron with angular momentum /. Far from the atomic core, thiswave function has the asymptotic form

ui(q, r) = - sin ( qr - — +

where 8t is the phase shift for scattering of an electron by the atomic core. Thisexpression is valid for photodetachment of a negative ion. In the case of atomicphotoionization, the phase shift 5/ becomes r dependent.


If the detached electron is sufficiently energetic, its interaction with the atomiccore can be neglected, and we can take

= exp(/q • r).

In this case,

ui(q,r) =

where 7/+(1/2) is the ordinary Bessel function, or

ui(q, r) = rji(qr)

in terms of the spherical Bessel function j t .The general form of the photodetachment cross section can be expressed by means

of integrals of radial wave functions. According to Eq. (3.3), the photodetachmentcross section has the form

qco= ~ fu«r (3.4)

Here a$ — h2/(/xe2) is the Bohr radius, and we used D = ex.Now we shall calculate the integral in (3.4).We introduce the unit vector s such that projection of the electron momentum in

a given direction in the initial state is zero for this particular direction. Then, sincem — 0, the angular wave function of the initial state simplifies to

y/0(ft tp) = [(21 + l)/(47r)]1/2P /(cos0rs),

where 0rs is the angle between the vectors r and s. With the above expressions forthe wave functions, the integral can be written in the form

juH? " /'«• / /

X ]P(2/' + \)il'uv(q, r)P//(cos 0qr)

Upon carrying out the square indicated in the integrand, we obtain

f

X l)P/'(cos 0qr)

X ?, r)uVi(q, r').

The integration over the solid angle / d£lq can be accomplished by using thetheorem for Legendre polynomials

rill

PP,(cos0qrO = PAcos0rr,)/>(cos0qr) + 2^2

X P/S(cos 0rr/)P^(cos dqr)cos[m(cpr -

/ | rOq |2 d£lq = J2 \R»'\2 I I d n r dnr, cos fa

J / / = 0 J J

The indices on the polar angle 0 identify the vectors whose angle of intersectiondefines 0, and <p is the azimuthal angle.

The sum over I" is carried out using the orthogonality of Legendre polynomialswith different indices,

fl 2/ d(cos 0)P/(cos 0)P//(cos 0) = ——-8W .

We obtain thereby

00

(cos 0rs)P/(cos 0r/s)

X P,,(cos eTiT)(2l + 1)(2/' + 1),

where the notation is used such that the radial matrix element is defined by

Rw = / uni(r)ruv(q, r) dr.

In order to make further use of the orthogonality of the Legendre polynomials, weintroduce the recursion relation

(21' + 1)*P//(JC) = (I' + 1)PZ/+1(JC) + l'PV-x(x).

This leads to the form00

~ " l)

X [(/' + 1)P//+1(COS 0r/r) + l'Pi/-i(C0S 0r/r)]P/(COS 0rs).

By using the summation theorem for the Legendre polynomials, we get

.,(21J J

+ I'Pi>-i(cos 0r/s)P//_i(cos 0rs)lP/(cos 0rs)P/(cos 0r/j

f | rOq I2 dOq = ] £ \Ru>\2 f f d£lr dilr,(2l + l)P/(cos 6rJ / /=0 J J

/ | rOq |2 <mq = Y, \Rn'\2 I I d^r dClrJ / /=0 J J

We now employ orthonormality for the Legendre polynomials to obtain the crosssection for photodetachment of a single-electron ion or atom in the form

' + ( / + 1 ^ ' + l ] - (3'5)

As is shown by the above manipulations, radiative transitions of electrons into thecontinuum require fulfillment of the selection rules that the electron orbital angularmomentum must change by unity, and the parity of the state must change.

The proportionality &[ ~ q is the threshold dependence of Wigner. It is the usualresult, explained by the statistical weight of the final state. When an increase of thephoton energy causes a departure from threshold conditions, the dependence of at

on q becomes complicated by a dependence of Rw on q. As will be seen below, thematrix elements Rw for the Coulomb potential depend strongly on q.

Note that formula (3.5) relates to the case of an absorbed photon of definitepolarization.

Later, in Section 3.5, in addition to the process of photodetachment just treated, weshall consider as well the inverse process—photorecombination. This is the processin which an electron makes a transition from a state in the continuum to a boundstate.

Problem 3.3. Determine the cross section for photoionization of an atom or pho-todetachment of a negative ion on the basis of the shell-model atom (see Appendix F).Adopt the condition that it corresponds to a transition into the continuous spectrumof one of TV valence electrons, and neglect spin-orbit interactions within the atom.

SOLUTION. This problem is the generalization of the preceding ones for the casewhen one has TV valence electrons in place of a single valence electron. Analyze firstthe initial state of the atom. The wave function of this state is determined by thecoordinates of TV valence electrons in an atom at rest. We represent this function inthe form

Here, L is the orbital angular momentum, ML is its projection in a given direction, Sis the spin of the atom, and Ms is its projection on the same direction as for Mi. Wewrite this function as

, . . . , TV) = P ] T GLL's

s(Lflfim \ LML)

X (\s(rms | SMS) </w(TV)^LVms(l, 2,..., TV - 1),

where ^v^sm* is the wave function of TV — 1 electrons, which are characterized byquantum numbers L1, JJL, S, ms\ and t/>/mo-(TV) is the wave function of the TVth valenceelectron of orbital angular momentum /, with projection m on the given direction,

and spin ^ with spin projection a. The operator P permutes the positions of theelectrons, G^'s

s is the fractional parentage coefficient, and the angle brackets (• • •) arethe Clebsch-Gordan coefficients, which account for the sum of the spin and orbitalangular momenta of a valence electron and the corresponding angular momenta ofthe remaining N — 1 electrons. Here we assume that all of these electrons have thesame n and / quantum numbers, that is, that they all relate to one electron shell.

Now consider the wave function of the final state. It is characterized by the releaseof an electron, and therefore is described by a parameter that is the momentum ofthis electron, hq. Because this electron departs to large distances from the remainderof the atomic system, we do not sum the momentum of this electron and that of theremnant atom. Thus the wave function of the final state has the form

U^W(1> 2,..., N -N

where i//q(r) is the wave function of a free electron that is normalized to unity bycondition (3.1), and Xa is the spin wave function of an electron with spin projectionor on a given direction.

The dipole moment operator of an atom is

where the index / numbers the electrons. Using the above expressions for the electrons,summing over final states (the projections of the orbital momenta and spins for thesystem of N — 1 electrons, and the spin of the free electron), and averaging over thepossible initial states (the projections of the orbital and spin momenta of the atom),we obtain

|DOq|2 = N (G^ * ^ (L'l^m I LML>2

m\xML

xJ2(s-msa\SMs) \(lm | r | q)|2mso- ^ '

e2N / ,/c\2<

2L +(3.6)

Here (Im \ r | q) is the matrix element of the dipole moment of the electron thatmakes the transition from the bound Im shell of the atom to the unbound state ofmomentum q.

Using the expression for the one-electron matrix element obtained in the precedingproblem, we find on the basis of the formulas (3.5) and (3.6) the photodetachmentcross section

2L


Problem 3.4. Determine the photodetachment cross section for an s electron from anegative ion in a lS state. Assume the interaction between the valence electron andthe atom to be important only in a region near the atom which is small compared tothe size of a negative ion.

SOLUTION. In a negative ion, there will be two valence electrons in s states. Thephotodetachment cross section for each of these can be obtained on the basis of theresults of the preceding problem by using the following parameters: I = 0,N =2, G^1/2) = 1. From (3.7) we have

l ^ l , (3.8)

This result can also be found directly from Eq. (3.5) by taking into account thefact that the photodetachment cross section for two valence electrons is twice thatobtained for a single electron.

Now we will calculate the radial matrix element. We first give the expression forthe radial wave function of a valence electron in a negative ion. The electron bindingenergy will be written as h2y2/(2m). The Schrodinger equation for the radial functionwo(r) with / = 0 has the form UQ = y2u0. In obtaining this equation, we neglect theinteraction between the detachment electron and the atom core because of the shortrange of this interaction. Thus for a short-range interaction between an electron andthe remaining atom, the wave function obtained as the solution of the Schrodingerequation is

Because of the absence of residual interaction between the ionized electron andthe remnant atom, the wave function of a free electron with / = 1 is the first harmonicin the expansion of the plane wave exp(/q • r) in Legendre polynomials

1 / sin qr \u\(q,r) = rjx{qr) = - I — — - cos qr 1.

Using this function, we obtain for the radial matrix element

#oi = / uo(r)rui(q, r)dr = .Jo (T + r)

This then gives the result for the cross section (3.8) for the photodetachment of anegative ion

where

hco = — (y2 + q2)

is the energy of the absorbed photon, and v = hq/m is the velocity of the detachedelectron. The validity of the expression obtained is based on the assumption that thesize 1 /y of the negative ion is large compared with the region of interaction of anouter electron with a neutral atom. Therefore it is valid for small binding energies ofan electron in a negative ion. The expression is not valid for large values of q becausethe principal contribution in the matrix element /?Oi implies electron distances fromthe atom ~ l/q, and these distances must be large compared with atomic sizes.

If q > y (but, nevertheless, q is not large), formula (3.9) for the photodetachmentcross section has the asymptotic form

_ 64TT e2 y1 q>y 3 hcq3'

Near the threshold, where q < y, this expression gives

_ 64TT e2 q3

1 q<y 3 he y5

The dependence 07 ~ q3 is determined by the fact that the final electron state is a pstate (not an s state), and at small values of q the wave function is proportional to q(while for an s state of an electron it does not depend on q).

Problem 3.5. Determine the photoionization cross section of the hydrogen atom inits ground state.

SOLUTION. According to formula (3.5), the photoionization cross section for theground state of hydrogen has the form

_ &<irq<o 2

3 a o c

and our goal is the calculation of the matrix element R0\ . For simplicity, we introducethe atomic system of units where h = m = e = 1. The radial wave function of anelectron with 1=1 and momentum hq in the continuum spectrum of a Coulombfield is

3 [1 ~ exp(-27r/#)

where F(a, y, x) is the confluent hypergeometric function. Using the known expres-sion for the radial wave function of the ground state of hydrogen, we obtain the matrixelement Roi'm the form

Ro\ = / 2re rux(q, r)dr.Jo

To evaluate this integral, we use the formula

y, kz)dz =fJo

This gives the result

exp[-(2/q)arctanq]

<72)5 V l

The photoionization cross section for the ground state of hydrogen is then

_ 29TT2 exp[-(4/<7)arctang]a' 3c ( l+^[ lexp(2i r /«) ] - ( }

Now consider some special cases of Eq. (3.10). For small values of q this formulaleads to the threshold value of the photoionization cross section

(3-11)297T

where a = e2/he is the fine structure constant, and we have departed from atomicunits to return to conventional units. It is seen that the threshold cross section doesnot approach zero as would follow from the Wigner threshold formula. The result canbe understood in terms of the behavior of the wave function of an electron movingslowly in a Coulomb field. Under these circumstances, the matrix element /?Oi tendsto infinity near the threshold, with q~xfl dependence. Indeed, near the threshold, wehave

where here e = 2.718... is the base of the natural logarithm. This dependencecompensates for the small statistical weight of a slow electron. By contrast, in thepreceding problem we had /?oi ~ q, which led to a threshold cross section 07 ~ q3.Thus the Coulomb interaction yields a nontypical result due to the singularity of theCoulomb potential at r —> 0.

For large q(q> 2TT), for the photon energy significantly in excess of the ionizationpotential / of ground-state hydrogen, that is,

he* = h2q2/(2m) > 4TT2I « 40/,

Eq. (3.10) gives

256TT aa2 256TT / / \ 7 / 2

i n 3 ' r s U = > > ' (312)

Note that/ = meA/(2h2).From (3.10) it follows that the photoionization cross section is maximal at thresh-

old. As the photoelectron momentum increases, the cross section falls linearly atfirst, behaving as 07/070 ~ 1 ~ f 2 (where we have reverted again to atomic unitsfor simplicity), with the threshold value of 07 designated as ai0. As (3.12) shows,the photoionization cross section for hydrogen falls rapidly as q increases. One


might expect that at h2q2/(2m) = I the photoionization cross section would beclose to its threshold value. In fact, it differs from the threshold value by a factorof e4/(25 sinhTr) ~ 0.15, which is about an order of magnitude. There is then avery extended transition to the region q > 1, and the behavior at ~ q'1 ~ co~7^2

of Eq. (3.12) holds true. The transition to the asymptotic dependence (3.12) is soextensive that only at [h2q2/(2rn)]/I « (IOTT)2 does at reach 90% of (3.12). Thebehavior of (7;/cr;o is illustrated in Figure 3.1, where the limiting analytical formsare shown.

The decrease of the photoionization matrix element and cross section accordingto a power law for q > 1 follows from the singularity of the Coulomb potentialat r —>• 0. In fact the matrix element RQ\ for q t> 1 is the Fourier component forlarge values of the argument. Without the singularity in the interaction potential, theFourier component would have an exponential dependence rather than a power-lawdependence on the argument.

Formula (3.12) is valid up to values of the frequency hco ~ me2, when relativisticeffects become essential.

Problem 3.6. Find the angular distribution of the photoelectrons from the ionizationof ground-state hydrogen.

SOLUTION. According to Eq. (3.12),

where n is the unit vector in the direction of emission of the ionized electron, and sis the unit vector along the direction of polarization of the absorbed photon. One candetermine the normalization factor by requiring that the integration over solid angle

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

-8q2/3

0 10 20 30 40 50 60 70 80 90 100

CJ (a.u.)

Figure 3.1. Cross section for the photoionization of hydrogen from the ground state, comparedto the threshold cross section. Behavior near threshold and at high photoelectron momenta areshown on the figure.

of this expression should give the total cross section (3.10). This then yields

3ddi = —o-/(n • s)2 dflq. (3.13)

The prediction of (3.13) is that most of the ionized electrons are emitted in a di-rection close to that of the polarization vector of the incident radiation. Perpendicularto this direction, including the direction of propagation of the incident radiation, thephotodetachment cross section is zero.

If the incident radiation is unpolarized, Eq. (3.13) must be averaged over polariza-tion directions s for a fixed direction n0 of the wave vector of the incident radiation.Then, from the relations cos 0ns = sin 6nno cos <ps and cos2 <ps = | , we have

da. = J L ^ n x n0)2dftq. (3.14)

In the case of photoionization from the innermost, or K shell, of an atom, one canuse the above formulas; but it is necessary to double the cross sections since the Kshell normally contains two electrons.

Sometimes the absorption coefficient k^ is used instead of the cross section [seeEq. (2.87)]. The absorption coefficient is found by multiplying the cross section bythe number density of atoms,

Values of the absorption coefficient can be obtained in this fashion from the abovecross-section formulas.

One can generalize the above formulas to hydrogenlike atoms or to the K shell ofany atoms that are similar to the hydrogen atom. Then it is necessary to change theBohr radius a0 to aQ/Z, where Z is the nuclear charge. Then, according to (3.11) wehave at ~ Z~2 near threshold, while from (3.12) it follows that 07 ~ Z5 for largephoton energies.

Problem 3.7. Determine the frequency dependence of the photoionization cross sec-tion of an atomic state with angular momentum / for large photon energy.

SOLUTION. As indicated by the preceding problem, it is necessary to analyze thematrix element of the radial dipole moment to solve the present problem. For thecase a* —> oo, the principal contribution to the matrix element will arise from smalldistances from the nucleus because r ~ 1/q, and q will be large for high photonfrequency. This can be seen from Problem 3.5 for a transition S —• P in the hydrogenatom. Let us analyze the radial wave function u\{q} r) of the continuous spectrum. Itis a standard result in quantum mechanics that the radial wave function of a particlewith angular momentum / at small distances r from the center has the dependenceu\ ~ rl. Therefore an additional (as compared with Problem 3.5) multiplier rl willgive an additional multiplier q~l in the matrix element. Thus, for determining thedependence of the photoionization cross section on the frequency, it is necessary to

multiply the result of Problem 3.5 [Eq. (3.12)] by the value q~21. As a result we have

(3.15)

It is seen that there is a strong decrease in the photoionization cross section withan increase in the angular momentum of a bound electron when that electron isionized by an energetic photon. The asymptotic limit (3.15) corresponds to very highfrequencies (hco > 40/), as follows from Eq. (3.12).

Problem 3.8. For atomic photoionization or negative ion photodetachment nearthreshold, find the dependence of the cross section on the momentum of the de-tached electron.

SOLUTION. According to the general formula (3.5), the solution of this problemrequires that one determine the dependence of the radial dipole moment matrixelement Riti±\ on the wave vector q (or the momentum hq) of a detached slowelectron.

The matrix element of the radial dipole moment is given by the relation

- / q, r)r dr,

and is determined by the behavior of the wave functions within a region of size r ~ 1(in the atomic system of units). It is such distances that characterize the behavior ofthe radial wave function un\ of the ground state. We obtain thereby the estimate

for the matrix element.First, we consider the photoionization process for a neutral atom or positive ion.

An ionized electron moves in the field of a positively charged ion. Because of thehypothesis that q < 1, for r ~ 1 the de Broglie wavelength has the form

A = [<72 + 2 / r - / ( / + l ) / r 2 ] ~ 1 / 2 ~ 1.

Therefore, at r ~ 1 we have

dX/dr ~ 1,

that is, one can use a quasi-classical approach for the determination of qualitativebehavior. The quasi-classical radial wave function of the final electron state has theform

ui±i(q,r)= -J^-sinf f kdr - <pY (3.16)

where the notation k = 1 /A is used. The normalization multiplier is found from theasymptotic behavior of the wave function (see Problem 3.2)

ui±i(q,r) = -sin(qr - <p),4

which corresponds to the normalization condition (3.1) for radial wave functions ofstates in the continuum.

For r ~ 1 we obtain the estimate w/±1 ~ q~{/2 > 1. Therefore, on the basis ofthis and of Eq. (3.5) we have Rij±\ ~ q~1^2 and 07 = const. Thus the thresholdphotoionization cross section is a constant for atoms and positive ions. This resultis corroborated by the work on the photoionization of hydrogen near the threshold,as obtained in Eq. (3.11). However, the photodetachment of a negative ion gives aqualitatively different result because of the short-range character of the interactionbetween the detached electron and the remaining neutral atom. Outside of this limitedrange, the wave function of the detached electron is described by a plane wave, andits decomposition into spherical harmonics (see Problem 3.2) gives the expression

11/fe r) = rjiiqr)

for the radial wave function, where ji is the spherical Bessel function. For q < 1 andr ~ 1 we obtain the estimate

Therefore,

q > Ru-\ ~ q •

Equation (3.5) thus leads to the threshold dependence for the photodetachment crosssection of a negative ion

/ q3 I = 0;/

In particular, for / = 0, this agrees with a special case of Problem 3.4.

Problem 3,9. Determine the general character of the electron angular distributionresulting from photoionization.

SOLUTION. According to Eq. (1.8) the photoionization cross section is determined bythe matrix element

(3.17)

Here, the vector s characterizes the polarization of the incident radiation, and q is thewave vector of the detached electron. Further, we select the direction q as the quantumaxis for the quantum numbers nlm of the initial state of the atom, and the differentialcross section must be averaged over all projections m of the angular momentum.

The angle between the vectors s and q is labeled #. Averaging over m is similar toaveraging over directions of the quantum axis r. We label with # ' the angle betweenr and q. Further, we use the summation theorem for trigonometric functions that herehas the form

r • s = r(cos #cos # ' + sin # sin ftf cos <p'). (3.18)

Using (3.18) in (3.17), and averaging over angles (with cos <pf = 0), the photoioniza-tion cross section is obtained in the form

dai = (a + bcos2 &) dflq. (3.19)

The fact that the photoionization cross section does not contain odd powers ofcos # means that it does not depend on the sign of the momentum of a detachedelectron, or on the sign of q. It can be obtained from general considerations. Indeed,a change of the sign of q is analogous to a change of the sign of r in t//q(r). But thesubstitution r —> — r leads to the multiplication of i//n/m by a factor (— I)1. Since thephotoionization cross section is a quadratic function of the dipole matrix element,the cross section does not change as the result of such a transformation.

If the initial state nl is an S state (/ = 0), the angular dependence of the photoion-ization cross section is simplified. In this case, we shall expand the wave functionof the final state in Legendre polynomials as a function of the angle # ' between thevectors q and r to obtain

00

</>q(r) = ^ Ufa, r)P,(cos #'). (3.20)1=0

The wave function of the initial state does not depend on angles. Since the function(3.20) does not depend on the angle <p;, the second term of Eq. (3.18) is zero due tothe relation

I•2-n-

cos (p'dip1 = 0.

Thus the dipole matrix element is proportional to cos #, and the cross section has theform

ddi = bcos2&dflq. (3.21)

This dependence for the photoionization cross section of the hydrogen atom was usedat the end of Problem 3.5.

Problem 3.10. Obtain the expression for the photodissociation cross section of adiatomic molecule, viewing this process as the photoexcitation of a diatomic moleculeto a repulsive state.

SOLUTION. The mechanism of the process we consider follows from Figure 3.2,where the electronic potential energy curves 1 and 2 for a diatomic molecule areshown as a function of the distance R between nuclei. Upon absorbing a photon offrequency co, a diatomic molecule makes a transition from the ground state 1 to arepulsive or antibonding state 2. The atoms would then continue to separate, leadingto dissociation of the diatomic molecule. We designate by D the dissociation energyof the molecule. According to the law of energy conservation, the value ho)-D mustbe equal to the sum of the kinetic energies of the atoms that are formed, h2q2/{2jx) (JJL

is the reduced mass of the atoms) and the value A £/(oo), which is the asymptotic (forR —> oo) difference of the energies for the bound and repulsive states of the diatomicmolecule. Note that the value hq is the momentum of the atoms in the center-of-masssystem. Thus, we have

ftV/W (3.22)

For determination of the photodissociation cross section we use Eq. (3.3). Theinitial and final state wave functions in the transition are given as the products ofelectron and nuclear wave functions. Then we have

DOq = <0|D12|q>,

where D i 2 is the matrix element of the dipole moment operator taken between electronwave functions, and the indices 0 and q correspond to nuclear wave functions for thelower and upper electron states of Figure 3.2.

When we take into account the small amplitude of the nuclear vibrations ascompared with typical electron motions, we conclude that the matrix element D12does not depend on the distance between the nuclei. Thus we obtain

D()q = ^ 1 2 ' ^0q»

where SOq is the integral of the overlap of the nuclear wave functions <£0, Oq, whichare, respectively, the nuclear functions for the initial and final states of the diatomicmolecule. They are not orthogonal because they relate to different Hamiltonians.These Hamiltonians have as interaction potentials the terms £/i(R) and ^ ( R ) shownin Figure 3.2. The value of SOq is nonzero in the range of q that corresponds to theclassical region of motion for the lower of the two potentials. See Figure 3.2. Outsideof this range the overlap of the nuclear wave functions declines sharply.

RFigure 3.2. Electron potential energy for a diatomic molecule as a function of the distanceR between the atoms. The curve labeled 1 corresponds to the ground state and that labeled 2represents an excited, repulsive state.

The bound-state wave function <J>o is normalized to unity, whereas the unboundwave function <J>q is normalized by the condition (3.1), that is,

/<Dq*,(R)<Dq(R)dR = (27r)36(q - q'),

and therefore it satisfies the relation

»*(R/)^>q(R)^q = (2TT)3S(R - R) .

We thus find the condition

J\SOq\2dq = j <t>o(R)<t>o(K)%(R)%(Rf

= (2TT)3 = f dqd(lq (3.23)

for the normalization of the overlap integral SOq-On the basis of the above relations, we rewrite the expression for the photodisso-

ciation cross section as

/I cPOq

The quantity

IT fa I l oqi d£lq. (3.24)nc J

/ l2=|^L|Dl2p (3.25)

is the oscillator strength for the electron transition [see Eq. (1.18)], m is the electronmass, and o>2i is the transition frequency for the equilibrium distance |R| betweenthe nuclei. This distance can be taken to have a definite value because the amplitudeof nuclear vibrations is so small as compared to typical electron motions.

We shall now obtain an integral relation for the photodissociation cross section ofa diatomic molecule. It is akin to a sum rule. We take into account that the oscillatorstrength does not depend on the nuclear coordinate R. We multiply expression (3.23)by dco, introduce the integral over a), and use the relation d(h(o) = hqdq/ \x, whichfollows from (3.22), to obtain

2 T T VaDda) = /12. (3.26)

me

This relation is a particular case of a general sum rule found in Section 1.2. Notethat the integrand as a function of oo is concentrated in the range of frequenciesnear the electron transition co2i because the photodissociation cross section falls sosteeply outside of this range. The sum rule (3.26) makes it possible to estimate thephotodissociation cross section under certain conditions.

PHOTOEXCITATION OF RYDBERG STATES OF ATOMS 137

3.2 PHOTOEXCITATION OF RYDBERG STATES OF ATOMS

Highly excited states of atoms have the behavior of the outermost electron determinedprimarily by the Coulomb field of the atomic core. For this reason, such highly excitedstates exhibit the same structure as the hydrogen atom. In particular, the energy ofa highly excited atom has the simple form s = — l/(2n2) to within an accuracy ofthe order of 1 /n3. Throughout this section we use the atomic system of units withh = m = e = 1. Atoms in highly excited states are usually called Rydberg atoms(see also Problem 1.8). In this section, we consider the photoionization of a Rydbergatom.

Problem 3.11. Find the connection between the excitation cross section for transitionof an atom to a Rydberg state and the photoionization cross section of this atom nearthreshold.

SOLUTION. According to the result of Problem 1.8, the oscillator strength for tran-sition to a highly excited state with principal quantum number n is estimated to beproportional to n~3. From Eq. (2.85), the photoexcitation cross section to such a statehas the form

that is,

1 3 2

or

Can = -,a(o) - a)n0).n5

Here a(co — a)no) is the distribution function for absorbing photons of frequency a>,and the quantity

_ _ _ 1 _ _

is the frequency (or energy) difference in initial and final states.The width of the distribution function is determined by the mechanism of spectral

line broadening. We use this concept in the limiting case that the width of the spectrallines significantly exceeds the distance between neighboring levels, in which casethe photoexcitation cross section coincides with the photoionization cross section.Indeed, in this case the discrete spectrum of excited electrons appears to an incomingphoton to be the same as a continuous spectrum, and the behavior of weakly boundelectrons and low-energy free electrons is similar in the field of the atomic core. Thuswe have

V^ ^ V ^ l < ^ n ^CFi = > (Tn — C > —zd(O) — (OnQ). {3'^t)

n n

Note that in this photoionization cross section those parts of the smoothed Rydbergstates that are connected with a particular orbital angular momentum correspond


directly to those parts of the continuum for the same orbital angular momentum. Todetermine the normalization constant C, we make use of the hypothesis that n is largeto replace the summation in Eq. (3.27) by an integration. Then we have

From condition (2.2) for normalization of the wave function, we see that the valueof this integral is unity, and so at = C. Because the photoionization cross sectionis approximately constant near threshold, the relation at = C is approximatelyvalid for any relation between the width of the spectral line and the distance betweenneighboring excited states of the Rydberg atom. Thus we have for the photoionizationcross section of a highly excited level

an = -!ra(a) - a)n0). (3.28)

Problem 3.12. Determine the dependence of the photoexcitation cross section onphoton frequency near the photoionization threshold. Assume Lorentz broadening ofthe spectral lines.

SOLUTION. Now allow the width of the spectral lines F to be either larger or smallerthan the distance n~3 between neighboring levels. The Lorentz frequency distributionof photons is

a(o) - o)n0) = —2TT2TT (OJ - o)n0)

2 + F 2 / 4

According to Eq. (3.28), the photoabsorption cross section expressed as a sum ofexcitations of high-lying states is

Y i iaa(co) = V an = a-/— V 3 , r 2 / / 1 . (3.29)

^—' 2TT ^ ^ n5 (a) — con0) + i 2 / 4

The main contribution to the sum in (3.29) can be written in terms of values rC (whichare not necessarily integers) for which a)n*0 = a). We expand con0 for n in the vicinityof n* in the Taylor series

con0 - (o = (n- rC)do).'nO

dn

Using (3.30) in (3.29), we have

n- n (3.30)

Because of the strong convergence of the series (3.31), we can extend the lower limitof the range of summation to — oo. The Mittag-Leffler theorem provides the result

00

^ n-x2

77 sinh(27ry)

y2 y[cosh(27ry) - COS(2TTX)]'

PHOTOEXCITATION OF RYDBERG STATES OF ATOMS

Using this, we can rewrite Eq. (3.31) as

cra(co) = disinh(7ra*3r)

cosh(7772*3r) - cos(27ra*)'

139

(3.32)

The quantity rf in this equation can be expressed in terms of the frequency of theabsorbed photon by

n = [2\s0 + co\]-1/2

with co < \&o\ .We shall now consider special cases of Eq. (3.32). If F > (rc*)~3 and the en-

ergy levels overlap, then aa(co) = o-,. In this case photoabsorption corresponds tophotoionization of an atom.

In the opposite special case F < (?z*)~3, we have

aa(co) = at (3.33)- cos(2im*) + (Tm*3F)2/2*

As n* changes, the photoabsorption cross section oscillates from the minimum value

at half-integer values of n* to the maximum value of the photoabsorption cross section

2

for integer n*. Figure 3.3 shows the dependence of cra(co) on frequency.

Figure 3.3. Cross section for excitation of a Rydberg state as a function of photon frequency,as calculated from Eq. (3.32). The maximum frequency indicated by the vertical dashed linecorresponds to the binding energy of the initial atomic state.

The values cr™x correspond to resonance excitation of discrete levels with aprincipal quantum number n*. In the vicinity of whole numbers n, where n — [n*] isthe largest integer contained in «*, we can write n* = n + (rt — n), and then

1 - cos(2™*) = 27T2(n - w*)2,

which gives the photoabsorption cross section

<ra(w) = 4 o FT7T- (3.34)n5 2 u (a) — (onoY + 1 / 4

This equation coincides with (3.28).After integration over the frequencies co, we obtain the sum rule

aa(o))da)=^. (3.35)

Problem 3.13. Determine the oscillator strength for transitions between the Rydbergstates n, I and the states n'yl ± 1 if An = n1 — n < n, n'.

SOLUTION. As a first step it is necessary to determine matrix elements for a giventransition. Because both initial and final states are quasi-classical, the correspondenceprinciple of quantum mechanics can be invoked, which gives these matrix elements asthe Fourier components of the corresponding time-dependent classical coordinates.For Coulomb bound states with principal quantum number n and orbital angular mo-mentum quantum number /, it is convenient to parameterize the electron coordinatesx(t) and y(t) for motion in the xy plane in terms of a variable £ such that

x = rc2(cos £ - s)\ y = n2y\ — £2sin£; t = rc3(£ -

This describes an ellipse with eccentricity e given by

s = v 1 — (l/ri)2.

We calculate first the matrix element of the y coordinate,

1 fT / x fiAnt\ J

ynn> = r / y(t) exp —=- dt,

T Jo V n3 Jwhere T = 27ra3 is the period of the motion. This integral is equal to

7

AnsIn the same fashion, the matrix element of the x coordinate is found to be

In the above expressions, J and J' denote the Bessel function and its derivative,respectively. The evaluation of the integrals is accomplished through an integration

PHOTOEXCITATION OF RYDBERG STATES OF ATOMS 141

by parts and the use of the integral representation of the Bessel function

rJo

The operator x + iy corresponds to a lowering operator for decreasing the magneticquantum number by unity. In the classical case, the orbital angular momentum thenalso decreases by unity. In the same way, x — iy is a raising operator for increasingthe orbital angular momentum by unity. Thus we have

± iy)n,l-n',lT\ = IT-

A c c o r d i n g t o E q . ( 1 . 1 8 ) , t h e o s c i l l a t o r s t r e n g t h f o r t h e t r a n s i t i o n n , / — > « , / ± 1 w h e nn,n'>\ and An = (n' - ri) < n, n', is given by

f(n,l^n',l± 1)= -<*„,„1

that is,

f{nt

iy)n,l;n',l±\

-JAnisAn) (3.36)

This expression can be summed over the final orbital angular momentum quantumnumbers V = / ± 1 to give

(3.37)fin, I -> n\ I1) =

It is seen from Eq. (3.37) that, for n > 1, one has

fin, I -> n\ I1) - n.

The oscillator strength decreases strongly with an increase of An, as is characteristicof classical matrix elements.

Problem 3.14. Determine the average oscillator strength for transition betweenRydberg states with principal quantum numbers n and n1. Assume that the initialstate has substates that are equally populated.

SOLUTION. In this case, we average the oscillator strength (3.37) over all the availablesubstates of the initial state with principal quantum number n. The expression for theoscillator strength then has the form

f{n ~n')=-2 ]T(2/ + I) fin, I -+ n\V = I ± 1).


We have used the fact that there is a (2/ + l)-fold degeneracy in terms of orbitalangular momentum quantum numbers /, and the total number of substates for aprincipal quantum number n is n2. Based on the large magnitude of n, we changefrom summation to integration, choose the variable e to replace /, and use the relationn2s ds = —Idl, which follows from the definition of s

e2 = l - l2/n\

to obtain

An fl F 1 — s 2 1

fin - „ ' ) = — jf * ['&(**") + —^JL (***)] ds.Now use the property of the Bessel functions

z \j'p\z) - \\ J2p(z)] = j z [zJP{z)J'p{z))

to find that

K An(3.38)

Expression (3.38) can be simplified if An = n' — n is sufficiently large. Then weobtain the asymptotic forms

Thus we have, from (3.38),

Using the property of the gamma functions

SinTTX

we obtain the expression for the averaged oscillator strength in the form

K An( 3 - 3 9 )

Problem 3.15. Determine the photoionization cross section for the Rydberg statesof an atom near the ionization threshold.

AUTOIONIZING STATES OF ATOMS 143

SOLUTION. We use the concept of Problem 3.11 that in the limit of a broad spec-tral line the photoabsorption cross section coincides with the photoionization crosssection near the ionization threshold. Therefore it is necessary to determine the pho-toabsorption cross section for the transition between two Rydberg states. We can thenuse Eq. (3.39) assuming An > 1. According to Eq. (2.85) the photoabsorption crosssection is

2 ,a(<o(0)aa = 2TTzf(n -+ n')

c

When we use formula (3.39) for the oscillator strength in this expression, we obtain

8TT

where we have used An = con3.For evaluation of the photoionization cross section &(, we can make use of

Eq. (3.35). When we integrate Eq. (3.40) over w, and account for the normaliza-tion condition (2.2) on the photon frequency distribution function a(co — (ontn), weobtain Kramers's formula for the photoionization process

This expression can be understood in terms of the fact that the state n1 is in a part ofthe spectrum where the width of the distribution function a(co — a)n/n) is greater thanthe distance between neighboring levels.

If the ionization potential of the final state is small as compared to that of theinitial state, then o) = l/(2n)2, and Eq. (3.41) gives

647re2 9

In this last equation, we have returned from atomic units to the usual units, and #o isthe Bohr radius. Thus the photoionization cross section for a highly excited atom nearthe ionization threshold, which is the maximum atomic photoionization cross section,changes proportionally to n as the principal quantum number increases, whereas thearea of a highly excited atom is proportional to n4al.

3.3 AUTOIONIZING STATES OF ATOMS

To this point, we have considered photoionization as a single-electron process, whilereal atoms typically contain other electrons in addition to the valence electron. Theseremaining electrons have ionization threshold energies larger than that of the valenceelectron. Nevertheless, the atomic core has excited states, and excitation energies forsome of these states may exceed the ionization potential of the atom. Such states arecalled autoionizing states. Autoionizing states are characterized by a certain lifetime,


and therefore the autoionizing levels have a width F corresponding to ionizationof the atom with the release of a valence electron. This width is determined bythe interaction of the valence electron with the remainder of the atom and is smallcompared to typical energies of the atom. The existence of autoionizing states leadsto resonances in the photoionization cross section at energies that correspond toexcitation of these states. We consider this phenomenon as well as giving examplesof the influence of autoionizing states on the character of photoionization processes.

Problem 3.16. Determine the photoionization cross section in the energy neighbor-hood of the excitation of autoionizing states.

SOLUTION. When autoionizing states exist, photoionization can take place in twochannels. In one, the atomic core is excited and this excitation transfers itself to thevalence electron. In the second case, direct photoionization of the valence electronoccurs. Since both processes involve the same initial and final states, they are coherentand the transition amplitudes add. The second process occurs in a single step and canbe determined within first-order perturbation theory, while the first photoionizationchannel includes an interaction with the external electromagnetic field as well as withthe internal electric field of the system.

It is essential that both channels possess the same final and initial states, andtherefore one can sum the amplitudes (but not the probabilities) of these two channels.The initial state of the valence electron will be given the label 0, and the final statewill be identified by hq, the momentum of the photoionized electron. Equation (3.3)gives the cross section for this photoionization process as

^ | 2 j a « - ( 3 - 4 3 )

Here, m is the electron mass, dflq is the differential solid angle describing the motionof the free electron, and DQq is the effective dipole moment of the transition. It has

the form Dgq = DOq + D ^ , where DOq is the dipole moment operator for the usual

photoionization transition, while D ^ is the dipole moment operator related to thetwo-step transition. Label by A and B the initial and final states of the atomic core,and the frequency of the transition between them as a)AB. In the case of the two-stepprocess, the external electromagnetic field first excites state B. Label by DAB thedipole moment matrix element for this transition. The following step is the transitionof the atomic core from state B to state A with transfer of the excitation energy to avalence electron. This electron then makes a transition from state 0 to the state withmomentum hq. The interaction matrix element between the valence electron and theion is labeled UQ£. Then, from Eq. (1.44), within the framework of second-orderperturbation theory we have

(wBA -to)

AUTOIONIZING STATES OF ATOMS 145

Expression (3.44) is correct in the general vicinity of the resonance, where pertur-bation theory is valid, but it is not correct on resonance. To obtain the proper result atresonance, it is necessary to consider the following corrections within the frameworkof perturbation theory. First consider the energy denominator of Eq. (3.44). Becauseof interaction with the continuum states, level B experiences a second-order shift bythe amount

1 \TJAB\2

o)BA

When 8 —> 0+ in the denominator, this corresponds to a path around the poleassociated with damping in level B. The real part of the correction to sB is small, andwe shall neglect it. The imaginary part of this energy has the form

lm SsB = - J Y, KO\28(^BA ~ <oq0)

= - f KoTp(q). (3-46)

Here, p(q) is the density of final states for a valence electron.The value (3.46) characterizes the width of the autoionizing level. The quantity

Im 8sB/h is the probability per unit time of transition from state B as a result ofinteraction with the continuum spectrum due to the potential U, when this value isobtained within the context of second-order perturbation theory. We obtain

This expression agrees with "Fermi's golden rule."It is evident that there is no imaginary correction to level A. Also, the probability

wB significantly exceeds the corresponding probability for radiative decay of this statebecause of the absence from this expression of the small parameter e1 /(he). Nowwe analyze the numerator of Eq. (3.44), where it is necessary to add a second-ordercorrection. This correction takes into account the two-step transition through state Balong with the direct one. According to Eq. (1.44), we have

'E Pft *-^ O)qo — 0) — 10

When we neglect real corrections and retain only the imaginary part, we obtain

DAB + iirDOqU$p(q)/fi. (3.47)

The procedure for obtaining this result is similar to that used to find Eq. (3.46).When Eqs. (3.46) and (3.47) are used in Eq. (3.44), and Eq. (3.44) is employed in

(3.43), we obtain for the photoionization cross section in the vicinity of an autoion-


izing state the result

_ mqo)6irh2c

[DAB(3.48)

Simplifying this expression, and labeling by daj0) the photoionization cross sectionfar away from resonance, Eq. (3.48) leads to

_ ^ ( Q ) \(Q>BA -Ji = dvf

j3|2

((OBA ~(3.49)

where wB = F/ft is the above probability per unit time for the decay of the autoion-izing state, and the parameter j8 is

'Oqy0q- (3.50)

A number of conclusions can be drawn from Eq. (3.49). If o)BA — o) = — /3, thenthe photoionization cross section vanishes. The maximum cross section occurs atthe frequency GJ = coBA — w|/(4/3), which gives a larger result than at resonance(a) = (OAB). The ratio dai/daf^ is shown in Figure 3.4. The maximum cross sectionexceeds the cross section far from resonance by the factor (2/3/wB)2. Evidently, itdisappears if DAB — 0. Figure 3.4 shows two different values of the parameter a =(2/3/wB)2. The larger the value of a, the more strongly the resonance is exhibited.The solid curve in the figure corresponds to a = 2 , and the dashed curve is fora =

T

dada(o)

- 2 - 1

Figure 3.4. Ratio of the photoionization cross section in the vicinity of the excitation of anautoionizing state to the corresponding cross section far from the resonance.

BREMSSTRAHLUNG FROM SCATTERING OF AN ELECTRON BY ATOMS AND IONS 147

3.4 BREMSSTRAHLUNG FROM SCATTERING OF AN ELECTRONBY ATOMS AND IONS

The photoabsorption processes studied so far correspond to the transition of an atomicelectron from a state in a discrete spectrum of energies to a final state that may beeither in a discrete or a continuum spectrum. Now we shall study radiative processesthat result from the interaction of free atomic particles. The radiation arising fromthe deflection of a free, charged particle by an external field is called bremsstrahlung.In particular, we consider the radiation resulting from the interaction of an electronwith atoms and ions. This is the principal bremsstrahlung process in atomic physics.

Problem 3.17. Determine the cross section for bremsstrahlung from the scatteringof an electron by a center of force.

SOLUTION. Consider an electron with the initial wave vector q that, after beingscattered from a center of force with the concomitant emission of a photon, has afinal wave vector q'. Conservation of energy yields the condition

where m is the electron mass. We can use the general expression (1.13) for theradiation frequency in the case under consideration, which, because of the absenceof photons in the initial state, has the form

Indices 1 and 2 refer to the initial and final states of the system, and fl is a volumein which the system is contained. (We note that this quantity must not appear in thefinal result.) Using wave functions normalized by condition (3.1), the matrix elementis

_ 1 1

where r is the electron coordinate. The flux of incident electrons is

and, far from the scattering center, the electron wave function is

i//q = exp(/q • r).

In addition, energy conservation gives

da) = —a1 da',m


The transition probability per unit time, when divided by the incident flux, gives thecross section for bremsstrahlung

dv±=eW£ [\r \2dil (3.52)dco 6TT fi c q J

where dClq/ is the element of solid angle into which an electron is scattered aftercollision.

Problem 3.18. Obtain the cross section for bremsstrahlung for an electron scatteringfrom a spherical atomic system.

SOLUTION. Our goal is to simplify Eq. (3.52) by using the symmetry of the problem,and to express the matrix elements of the dipole moment operator in terms of radialwave functions. We represent the electron wave function in the form of the sphericalharmonic decomposition

_ lA,,

A similar representation is used for the wave function i//q/. These relations are thenused in Eq. (3.52). For the quantity

1 = j\rqql\2d£lql,

we find the expression

I = Yl RwRnni(2l + 1)(2// + l)(2n + \)(2ri + 1)l,V,n,n'

IX / cos 0rr/P/(cos 0qr)P//(cos 0q/r)Pn(cos Oqr/)Pni(cos 6qir>) dilr dflri dflqi.

Here the subscript vectors associated with the angles 6 define each such angle as theangle between the vectors cited, dfl is the element of solid angle for the directionshown by the subscript vector, and the radial matrix element is

Rw = / ui(q, r)uv(q', r)r dr.Jo

The final result will be expressed in terms of these matrix elements.The integration over the solid angle flq/ is carried out with the help of the sum-

mation theorem for the Legendre polynomials

/V(cos 0q/r/) = Pn/(cos 0q/r)Pn/(cos 0rrO

2 x y*'-"*)!„,

and their orthogonality property. We obtain in this fashion

/ = 4 7 7 ^ ( 2 / 4- 1)(2/' + l)(2n + \)RiVRnv

X / cos 0rr/P/(cos 0qr)F//(cos 6rr/)Pn(cos 6qr

We integrate over the solid angle, use the recurrence relation for the Legendre poly-nomials

{21' + \)xPv(x) = (/' + 1)P//+I(JC) + l'PV-x(x),

and the summation rule for the Legendre polynomials. The result is

eqr)fIV,n

X RlltRnr [(lf ]

= 64TT3 ^ 5 ^ ^ / / ^ / [(lf + l)8nJI+l + 1*8^

/=0

/=o

When this expression is used in Eq. (3.52), we obtain the cross section for brems-strahlung by an electron as a result of its collision with a spherical structurelessatomic system as

do) 3hc3aoq f^

Problem 3.19. Determine the cross section for emission of bremsstrahlung photonsof long wavelength as a result of the scattering of an electron by an atom.

SOLUTION. Assume the energy of the emitted photon to be small compared to that ofthe electron, that is, assume q — q1 < q. Use the asymptotic expression for the radialwave function of a scattered electron far from the scattering center,

1 / irl \ui{q, r) = - s i n I q r - — + 5/ I ,

where 5/ is the scattering phase shift. In the case considered, the primary contributionto the bremsstrahlung corresponds to a large impact parameter (or large distance) of

the electron with respect to the scattering center. Then the radial matrix elements aredescribed by the expression

r i r 77/ c 1&U+1 = —7 sin tfr - — + 8i(q)\

Jo qqf L 2 JX sin \q'r — + 8i+\(qf)\ rdr.

L 2 JWhen we exclude from this expression the strongly oscillating term, we have

1 r r i 7T~\Ru+\ ^ —o / c o s l&iiq) ~ &i+\(q) ~^~ (q ~ q)r ~^~ w\r dr,

2ql Jo L 2 Jwhere we have used the fact that q and q' are almost the same. From the form of theintegral, it is evident that the main contribution to it comes from r ~ l/(q — q1).This is large enough to validate our procedure of using asymptotic expressions forthe radial wave functions. Thus we obtain

sin2q\q - q1)2

The same expression applies as well to Ri+\j. Using these expressions in Eq. (3.53),we obtain the cross section for bremsstrahlung emission

3 (q ~ q'YA

CliXJ ^-.~^ - r , , _ — ,

When the electron energy s = h2q2/(2m) is introduced, q — q1 is expressed in termsof the photon frequency a) — hq(q — q')/m, and the initial postulate hco < s is used,then we obtain

D/+U«in 2 (*-* + . ) . (3.54)

Note that if this cross section is integrated over w, the result diverges. This isbecause perturbation theory is violated for slow electrons. Let us estimate the min-imum emitted frequency a)m[n down to which expression (3.54) is valid. We haveassumed that the emission of a single photon is more probable than the emission oftwo photons. However, at small photon frequencies, co —> 0, this is not true. Equation(3.54) gives an estimate for the probability of the emission of one photon with a largewavelength as

w I o^ In .mc) tie no)


Therefore the condition

e2 e I" 1he

gives the limit of the applicability of Eq. (3.54), which is based on perturbation theory.

Problem 3.20. Determine the cross section for the bremsstrahlung produced by thescattering of an electron by an ion.

SOLUTION. We use Eq. (3.52) and replace the matrix element of the electron positionvector by the matrix element of the electron momentum,

For the determination of the matrix element pqq/, we employ the wave function in themomentum representation. Based on the premise of large electron velocity, we useperturbation theory for the wave function. This gives

«fo(qi) = «(q - qi) J"

where V(q — qO is the Fourier transform of the interaction potential. We can use thesame procedure to write the wave function for the electron in the final state. Withinthe restriction of first-order perturbation theory, we have

= - ( q - q')V(q - q'),0)

where

From this, we obtain the expression for the cross section of bremsstrahlung in theBorn approximation

dab _do)

For the problem under consideration, V = —Ze2/r, and its Fourier transform is


On the basis of this expression, we must determine the integral

J Iq - q'|2l V(q - q')\2dilql = (AirZe2)2 J ^ * _

0)

/_! q2 +q'2 -2qq'cosO

32TTVZ2

qq1 Inq- q1

Here 6 is the angle between the vectors q and q'. The result for the cross section forbremsstrahlung from an electron scattered by an ion is

da^ = 16Z2 fe2\3

dw 3q2a)q + q1

q-q1 (3.55)

The validity of the Born approximation for the scattering of a particle in a Coulombfield is better the higher is the velocity of this particle. The criterion for the validityof the Born approximation has the form

nv

where v is the electron velocity.Note that, as in the preceding problem, the cross section for the emission of soft

photons integrated over frequencies is divergent. In the special case q1 = 0, wherethe total electron momentum is transferred to the photon, then the cross section forthe process is zero. Outside the Born approximation, neither of these limiting casesgives these end results.

Expression (3.55) corresponds to bremsstrahlung produced by the scattering of afast electron by the nucleus of an ion with a charge multiplicity of Z. We may alsotake into account the bremsstrahlung produced by this fast electron in interaction withthe electrons bound to the ionic nucleus. This can be done by the same technique justemployed.

Now the interaction potential includes the terms

where the r, are the coordinates of the ionic electrons, and r is the coordinate of thefast electron that is being scattered. The Fourier component of this potential is

Aue2

Quantum-mechanical averaging of this Fourier component over the wave function^ ( r ] , r2 , . . . ) of the ionic electrons produces the so-called form factor

vx dv2... £ exp [i(q - q') • r{] \9(rlt r2,.. .)|2 •

It should be noted that exchange effects can be neglected in view of the high velocityof the incident electron as compared to the velocities of the ionic electrons.

Combining the contribution of the ionic electrons with that from the ionic nucleuscalculated above, we obtain the final result for the bremsstrahlung cross section

dah %q'

do)

It is seen that the form factor diminishes the effective ionic charge. That is, it describesthe screening of the ionic nucleus by its electrons.

All the above considerations apply as well to a neutral atom. In this case, at smallmomentum transfer when

Iq - q'lao ^ 1»

where a$ is the atomic dimension, the form factor is equal to Z. This inequality isfulfilled for a sufficiently small deflection angle of the fast electron,

0 ^ 6a = (qao)~K

It occurs when the impact parameter of the collision is large. We can concludethat small deflection angles 6 < 0a do not make an essential contribution to thebremsstrahlung cross section for neutral atoms. This can be explained simply. A fastelectron with a large impact parameter does not feel the electron structure of theneutral atom, and hence the incident fast electron experiences virtually no brakingacceleration from the atom, that is, there is no bremsstrahlung.

Problem 3.21. Determine the cross section for bremsstrahlung from the scatteringof a slow electron by an atom.

SOLUTION. Our task is the determination of the sum

/=0

in Eq. (3.53). The value of q - q' is not small now compared with q, so we can notuse the results of Problem 3.19. For the determination of the above sum, we musttake into account the fact that the main contribution to the matrix element /?/,/+1comes from distances r ~ \/q > 1. Then one can use asymptotic formulas forthe electron wave functions. These wave functions are close to the wave functions

of a free electron, for which we label the corresponding matrix elements asBecause electrons free of external influences do not radiate, we have

= 0. (3.56)/=o

Note that the maximum difference between the matrix elements Rf?+l and Rw+\relates to small values of /. At small electron energies, only / = 0 will give acontribution to the cross section, so we make this restriction. Further, because ofEq. (3.56), only differences between /?/,/+1 and Rf?+l will contribute to the requiredsummation. The sum is then

4, -/=o

Now we evaluate the matrix element R^. The relevant wave functions for theelectron are

prrF 1, r) = J—J\/2(qr) = rjo(qr) = - sinqr;

Ux{q'r) = v j3/2(q'r) = rjl{q'r) = i> \^~ ~ cosq'r

On the basis of these wave functions we have

q')r - sin(</ — q)r] >dr.

Each of the four terms in this integral gives zero after integration because of thestrong oscillations of the trigonometric functions. Thus, one obtains R$ = 0. Forevaluation of the matrix elements /?oi and R\Q, we use the asymptotic expression ofthe wave function WQ, which is

UQ{q, r) = - sin(qr + So),q

where 60 is the s-wave phase shift for the scattering of an electron by the atom. For asmall electron energy, this phase shift is 80 = -Lq, where L is the scattering lengthof an electron scattering from the atom. L does not depend on q. For the state with

/ = 1 we use the expression for a free electron. Thus we have

*oi = 7T1 I {(1 A ' ) [cos ((q - q')r + 50) - cos ((q + q')r + 80)]Lqq Jo

- r [sin ((q + q')r + 80) - sin ((qf - q)r - 80)]} dr

= _ 1 _ / - i . I" singQ _ s i ngo 1 + [ sin50 sin80

2 ^ 1 ^ U - ^ q + q'\ l(q + q')2 iq1~ q)2

_ 2qf sin 60 _ _

T(q2-qf2)2 (q2-q12)2'

The result for /?i0 can be obtained from ROi by the substitutions q —» q' and— <7. In this way, we have

and thus obtain

/=o v^ ^ 7

q2 + Q12

where cre = ATTL2 is the cross section for the elastic scattering of a slow electron byan atom.

Using these expressions in Eq. (3.53), we obtain for the cross section for theemission of bremsstrahlung arising from the scattering of a slow electron by an atom,the result

dab _ 32mco3qf q2 + q12

d(o 3hc3aoq ir(q2 — q/2)4

In this equation, we now replace q and q1, the electron wave vectors in initial andfinal states, by the energy of the emitted photon

and the energy of the incident electron s = h2q2/(2m). Then we have

dab 4 e2 (2 — h(o/s)y/l — hco/s __x

-7— = ^ T 7—7 °V (3.57)ao) 37rmcJ nco/s

The cross section for the emission of bremsstrahlung decreases monotonically asthe frequency co increases. For soft photons (hco < s), Eq. (3.57) gives

dab _ 8 e2e ae—— — - - . P'Jojdco 3TT me5 no)

This result follows also from Eq. (3.54) of Problem 3.19 if we use the same parameters.Note that, as in the preceding problem, the cross section for bremsstrahlung emissiongoes to zero if q1 = 0, when the entire initial electron energy is transformed into theenergy of the emergent photon.

Problem 3.22. Calculate the cross section for the emission of low-frequency brems-strahlung resulting from the scattering of fast electrons from an atom, that arises as aresult of the dynamic polarizability of the atom.

SOLUTION. In previous problems in this section, the bremsstrahlung studied was aresult of the deceleration of an electron as a consequence of scattering from the atom.For complex atoms, another mechanism exists that can produce bremsstrahlung. Theelectron incident on the atom can produce oscillations of the electrons bound to theatom. These induced electron oscillations can lead to the radiation of photons, whichcan then be regarded as bremsstrahlung. The goal of this problem is to calculatequantum mechanically the cross section for this process.

This problem will be solved on the basis of the Born approximation applied to theinteraction between the incident fast electron and the atomic electrons. We consideredsuch an interaction in Problem 3.20. The form factor calculated in the solution ofProblem 3.20 was diagonal with respect to the wave function of the ground state ofthe atomic electrons. Here we take into account that the atom can make a transition toan excited state and back; that is, we consider the nondiagonal part of the form factor,also called the inelastic form factor. Such transitions are taken into account in termsof the polarizability of the atom by going to second order in perturbation theory.

We express the induced dipole moment of an atom as D = /3(co)E, where /3(co) isthe atomic polarizability of the atom at frequency a> (the so-called dynamic polariz-ability), and E is the electric field strength produced by the fast electron. Accordingto the Coulomb law, we have E =er/r3, where r is the distance between the incidentfast electron and the atom. We suppose that the magnitude of this distance r is largecompared to the atomic dimension a^, so that the coordinates of the atomic electronsare not essential in the solution of the problem.

As in the solution of Problem 3.17 [see Eq. (3.52)], the bremsstrahlung crosssection is of the form

dat _ m2a)3q'

dco 67T3h3c3q

When we substitute in this equation the expression for the dipole moment D, we find

dat _ e2m2co3qfp2(co) r ' - - * 2

do) 67T3h3c3q r3/qq' dfl,q'.

Calculation of the Fourier component yields

Iq-qTHence the bremsstrahlung cross section can be written as

dat _ $e2m2(o3q'p2((o) f ddqf

J Iq-qi2'Unlike Problem 3.20, integration over angles of the wave vector q ; is limited

because of the restriction that the incident fast electrons experience only small de-flections from the original direction. Indeed, were it presumed that |q — q;|ao — 1>where a$ is a typical dimension of the atom, then such transferred momenta wouldimply distances between the fast electron and the atom of the order of r < a^.However, we have assumed that r f> a®, SO the foregoing presumption about thetransferred momenta cannot be correct. Thus the upper limit of integration over theangle of deflection 6 is of the order of % = (qao)~l < 1. This inequality followsfrom the assumption that the velocity of the fast electron is large as compared totypical velocities of atomic electrons. When the integration over the range of angles0 < 6 < 6Q is performed, the result is

J I q - q i 2 Jo <?2 +d(cos6)

q12 - 2qqfcos6

77 In1 Inqq1 (q - q1)2

In the numerator of the logarithm in this expression the second term is large comparedto the first, since qq'Q^ ~ %2, and

m2(x)2 _2

where co is the frequency of the radiated photon. The last relation follows fromconservation of energy during the emission of the low-frequency photon, which is

hq2 hq11 h2

The inequality

a) < hq/mao = v/ao,

where v is the velocity of the fast electron, represents the condition that the radiatedphoton should be soft, that is, of low energy. This condition is a requirement inthis problem since the argument of the logarithm would be of order unity at higherfrequencies, and the cross section would be very small. It should be noted that thephoton energy is negligibly small as compared to the energy of the fast electron, so

that the scattering of this electron is almost elastic. Hence, under the conditions statedfor this problem, we have (with q ~ q1)

I 277

Iq-qi2 q2

This calculation is, of course, correct only to within logarithmic accuracy, since thedimension of an atom a0 is expressed here only as an order of magnitude.

Finally, the bremsstrahlung cross section is, within logarithmic accuracy,

dat _ 16e2co3/32(co) _v_

da) 3hc3v2 coao

This type of bremsstrahlung is called polarization bremsstrahlung. It vanishes in thelow-frequency limit co —• 0. The polarization bremsstrahlung cross section becomeslarge at photon frequencies (o corresponding to frequencies of bound-bound transi-tions between ground and excited states of the atom, since at such frequencies thedynamic polarizability j3(co) is resonantly large.

It should be noted that, in addition to the polarization bremsstrahlung evaluatedhere, there also exists the usual bremsstrahlung due to the simple scattering of a fastelectron by an atom. This was calculated in Problem 3.20. The ratio of the polarizationbremsstrahlung to the usual bremsstrahlung [see Eq. (3.55)] is approximately

e2/(ma)2)

The denominator of this estimate expresses the polarizability of a free electron. Weconclude that the dynamic polarizability produces resonant dependence of the crosssection on the photon frequency a>. The influence of dynamic polarizability vanishesat small frequencies since polarization bremsstrahlung does not exhibit the infraredcatastrophe, unlike the usual bremsstrahlung.

It is important to note that conventional bremsstrahlung and polarization brems-strahlung do not interfere with each other. This can be explained by the fact that thefirst is nonzero in the range of scattering angles of a fast electron 6 > do, while theother is nonzero in the complementary range of angles 6 < 6Q.

Problem 3.23. Calculate the cross section for bremsstrahlung resulting from thescattering of fast electrons by a negative ion.

SOLUTION. On first inspection it might seem that the cross section for bremsstrahlungarising from the scattering of a fast electron should be the same if the scatterer iseither a positive or a negative ion, since according to the solution of Problem 3.20the cross section does not depend on the sign of the ionic charge. However, negativeions are distinguished from positive ions because they possess a large polarizabilityarising from the small binding energy of the excess electron. Therefore the solutionof the preceding problem is appropriate to this case, but only with some modifica-tions incorporated. If the energy of the radiated photon is large as compared to the


photodetachment energy / for the negative ion, then the scattering of the fast electronon the excess electron is analogous to the scattering of one free electron by another. Itwill be shown in Problem 3.25 that, in the dipole approximation, no bremsstrahlungarises from such a process. There will then exist only the small cross section forbremsstrahlung of a fast electron on a neutral atom.

The solution of the preceding problem can be used, after modification to the form

dat 16e2co3 f e2 ]2 vj3(co) + J In . (3.59)dco 3hc3v2 [ mo)2\ coa0

Here /3(co) is the polarizability of the negative ion, and the second term in the squarebrackets is associated with the Coulomb scattering arising from the excess negativecharge of the ion (albeit without taking its structure into account). This second term isfound from Eq. (3.55) with Z = - 1 , but with a different argument for the logarithmicfunction, since the impact parameter of the fast electron cannot be small.

It should be noted that we add the amplitudes of two processes and not the crosssections. This follows from the fact that the perturbation is the sum of two potentials inthe Born approximation: the potential of the interaction between the fast electron andthe induced dipole moment of the negative ion, and the potential of the interaction ofthe fast electron with the negative charge of the ion. That is, the transition amplitudeis the sum of two processes in first-order perturbation theory. If ha* > /, then

P(a>) = -e2/ma)2,

and the cross section vanishes, as was pointed out above.On the other hand, if co —> 0, then the contribution from ]8(a>) disappears, and

only the second term in the square brackets remains in Eq. (3.59).The bremsstrahlung cross section as a function of co has a maximum when the

dynamic polarizability is largest. This occurs in the frequency range correspondingto the photodetachment threshold, since that is where both the value of j3(co) and thebremsstrahlung cross section have broad maxima.

Problem 3.24. Obtain the expression for the power emitted in the classical limitfor the case of bremsstrahlung caused by an electron incident on a spherical atomicsystem.

SOLUTION. The value

/

characterizes the power radiated during the process. For small co, Eq. (3.58) givesthe behavior dab ~ dco/a). This means that the emitted power has a bounded value.From Eq. (3.52) we have

hcodcTt, =

and on the basis of the relations rqq/ = -cu2rqq/ and dco = hqfdqf/m, we obtain

We use here the usual notation that dq = dqxdqydqz = q2dqdClq. The integralstated above is then

[J

\fqqi\2dqf =qqi [

Jq'<q

Jallq.q1

(2TT) :

[ f dr

= 4TT3 [ \r(r')\2 W(r')dY', (3.61)

and the quantity

W(r') = \i\jq{v')\2

is the relative probability amplitude for the particle to be at the point r'.Assume the interaction potential of the electron with the atomic system to be

spherically symmetrical, and consider the classical motion of the electron flux relativeto the scatterer. We suppose that the electrons occupy a volume element 2irp dp dsfar from the scattering center, where p is the impact parameter and ds is the length ofa volume element. At some time the electrons will occupy a volume

lirp'dp'ds1 = W{Y')dv'.

If v' is the velocity of the electron at point r1, conservation of electron flux gives

v lirpdp = v1 lirp1 dp1,

where v is the electron velocity far from the scattering center. Also, ds1 = vfdt,where dt is the time required to move the distance ds'. From this we obtain

lirp'dp'ds1 = iTrpdpvdt =

and from Eqs. (3.60) and (3.61) we have

Jhcodab = ^ ^ J | r (t)\227TPdpdt. (3.62)

We shall explain below that q = mv/h.

We now Fourier analyze r(?), which gives

r(0 = / rae-ia*d(0\ rw = -?- / r(t)eia)t dt; r» = -co2rw.

This yields the result

I " \r(t)\2dt = ffT e*a>-t»l)t<o2r*aa>f2r<o,da>d(otdt

= 2TT (I 8((o - a)')<o2rlr<af(oa dco dco'

/»00 /»00

= 2TT \rj2 to4 dco = 4TT / Ir jVdco.

When we use this expression in Eq. (3.62), we obtain* o 2 rt*

hcodab = - — 47T / / \vj^co4 dco2irpdp. (3.63)

o

When we compare the expressions contained in the integral, we find that

dab _ SIKU I \Yj2l7Tpdp ( 3 6 4 )

3hc3 Jodco

The presence of the Planck constant in this equation does not imply a quantumcharacter for the result. In fact, hco dab/dco is a classical quantity that does not containquantum parameters. It is to be noted that Eq. (3.64) can be derived without usingthe apparatus of quantum mechanics. In a classical approach, one uses the classicalexpression for the intensity of radiation, and determines the total electron energy lostto radiation as a result of a single collision of the electron with the scattering center.The criterion for the applicability of the classical calculation is the condition thatthe photon energy is small as compared to the energy of the incident electron, thatis, that hco < s. However, the photon energy hco can be of the order of magnitudeof the kinetic energy mv2/2 of the impact electron. At first sight, it is thereforeunclear whether the electron described in terms of its trajectory r(t) in Eq. (3.62) andits Fourier components rw in Eqs. (3.63) and (3.64) is the electron with the initialvelocity v or the final velocity v'. Since these velocities are so different from eachother, then also the trajectories of the classical motion will be very different. Theresolution of this apparent ambiguity is that it is only the region of the hyperbolictrajectory near the perihelion (where the distance between the electron and the atomicsystem has its minimum) that is essential to the problem considered. It should be keptin mind that hyperbolic trajectories relate only to Coulomb scattering. In the generalcase, the trajectories are more complex. However, the trajectory near perihelion doesnot depend on the velocity of the electron and is determined only by the centrifugalpotential. Clearly, these considerations are correct only if the atomic potential does


not contain a strong singularity at small distances r, so that r(t) in Eq. (3.62) doesnot depend on the electron velocity in the essential range of the integration over thetime t.

Problem 3.25. Prove that there is no dipole radiation in the collision of two electrons.

SOLUTION. The operator for the dipole moment for two electrons has the form

e{YX + r2) =

where R is the coordinate of the center of mass of the electrons. We can express thecoordinates for the two-electron problem as center-of-mass coordinates and relativecoordinates r = ri — r2. We express the wave function as a product of center-of-mass wave functions and relative wave functions. Bremsstrahlung leads to a changeof the electron energy as viewed in the center-of-mass system, that is, the electronenergy as expressed in the relative coordinates. The bremsstrahlung does not lead to achange in R. That is, the matrix element of the dipole operator will vanish because thematrix element is between different (and hence orthogonal) states of the relative wavefunction, with no change in the dipole operator. Thus there is no dipole componentto bremsstrahlung from the interaction of two electrons.

Problem 3.26. Estimate the dependence of classical bremsstrahlung on the param-eters of the problem of an electron scattering from an ion. Assume that the energyof the incident electron is very much smaller than typical atomic energies, and theenergy of the photons emitted significantly exceeds the value hmv3 /e2.

SOLUTION. The conditions stated for this problem imply that the interaction potentialbetween the electron and the ion in the region where the radiation is created is verymuch larger than the electron kinetic energy far from the ion. Thus it is necessary toanalyze the region of strong interaction of an electron with the ion.

The classical cross section (3.64) for the bremsstrahlung cross section can berepresented by the analytical estimate

dab e2co3 2 r2

- j — ~ -T~yP ~~2- (3.65)da) nc5 coz

Here, p is the collision impact parameter, which is determined by the cross sectionfor bremsstrahlung for a given frequency co, and r is a typical distance between anelectron and an ion that gives rise to this radiation. Evidently we should choose as ra value rmjn that corresponds to the closest approach of the electron to the ion on agiven trajectory. The distance can be obtained by setting the interaction energy equalto the centrifugal energy, or

? 2 2

e mv p

where v is the electron velocity far from the ion. We find from this that rm[n ~ p2s/e2,where e = mv2/2 is the initial electron energy.

Next we estimate a value for the impact parameter that gives the principal contri-bution in the radiation of a photon of frequency o). This frequency has the order ofmagnitude co ~ fmax Amin, where fmax is the electron velocity at the distance of clos-est approach. We establish this value from the principle of momentum conservation

mvp = m^maxrmin, i.e., ^max = vp/r^xn.

Thus we have

vp e4v

which gives

/ 4 \ 1/3

and rmin r-4. v 1/3 7f \ p e

— I and r_:_ ~

Using these relations in Eq. (3.65), we obtain

4 \ 2/3 1 / 9 9 \ 2/3 A 9

e4v\ I (elvl\ ebvz

dco he3 \ £2(o I a)2 \ so)2 I hoj3s2

e6 ' V h2

f: ) ^ T - - (3-66)hojc3m2v2 \hcJ m2v2oj

This estimate for dab/doj is valid with the conditions that rmin mv3/e2.

This is a criterion for a lower bound on the frequency. The upper limit for thefrequency comes from the condition for the validity of classical mechanics. For thispurpose, we require that the main contribution to the bremsstrahlung should arisefrom large values of the collisional angular momentum, or / > I. That is, we demandthat

(me4\mpv mv / e4v\ (me4\

It follows from this that the frequency of the bremsstrahlung emitted must be muchless than typical atomic frequencies, or to < me4/h3. Thus the criteria for the validityof Eq. (3.66) are

me4/h3 > o) > mv3/e2.

From this it follows that e2/(hv) > I, that is, the electron velocity must be small.

Problem 3.27. Under the conditions of Problem 3.26, find the specific numericalcoefficients that belong in Eq. (3.66).

SOLUTION. The solution of this problem requires the evaluation of the Fourier trans-form rw for the Kepler scattering problem. As in the preceding problem, the electronmotion can be considered to be classical. The motion of two particles subject toan attractive Coulomb potential is equivalent to the motion of a single particle ofreduced mass describing a hyperbolic orbit under the influence of a fixed center offorce. Problem 3.26 provided us with all the dimensional information necessary forthe cross section, so we now simplify matters by using the special system of unitswith m = v = e = 1. Consider the particle with the reduced mass to be moving inthe xy plane. The coordinates for the motion can be written in the parametric form

v = psinh£;

where £ is a variable parameter. In addition, it is convenient to evaluate the Fouriertransforms of x and y rather than x and v. Because of the relation rw = -/corw, thecross section for bremsstrahlung given in Eq. (3.64) can be written

The Fourier transform of x is

— — — I sinh £ exp i<o (y/\ + p2 sinh £ - £ J d£

The physical problem we are considering is one in which the bremsstrahlunghas (o > 1. The impact parameters are such that p < 1. Therefore only values ofthe parameter £ < 1 will contribute significantly to the integral. This enables us tosimplify the integral to

In the same way, the expression for j w is

PHOTORECOMBINATION OF ATOMIC SYSTEMS 165

This integral is the Airy integral. It can be expressed on terms of modified Besselfunctions of the second kind (also referred to as Macdonald functions) Kx/3 as

7T\/3

The quantity xw can be evaluated in an analogous way to yield

x - ^ Kw 7? 2/3

7TV3Using these expressions in Eq. (3.67), we find

dab_ 16a, H 2 (<^\ 2 (°>P3

The dimensionless integrals are equal to, respectively, TT/33/2 and 2TT/33 /2 .

Restoring the dimensional factors m, e, v in the cross section for bremsstrahlung,we obtain

( ZT 7 = • (3.00)

doj \hcj 3 ^ 2 2 <>

Equation (3.68) is called Kramers's formula. It describes the principal part of theclassical bremsstrahlung spectrum resulting from the scattering of a charged particlein a Coulomb field.

In Eq. (3.68), the value Z is introduced, which represents the number of elementarycharges contained on the scattering center. In other words, the interaction parameteris now Ze2 instead of e2, and the cross-section expression depends on Z2.

The range of frequencies where Eq. (3.68) is valid is as determined in the previousproblem, and the classical case corresponds to low velocities, v < e2 /h,

For highly excited states the concepts of photoabsorption and photoionization areclosely linked, as analyzed earlier, and the cross sections are related as shown inEq. (3.28). In an analogous fashion, the bremsstrahlung process considered here isrelated to photorecombination when the electron passes through a highly excitedlevel. This connection will be analyzed further.

3.5 PHOTORECOMBINATION OF ATOMIC SYSTEMS

The process of photorecombination of atomic systems is inverse to that of photode-tachment and proceeds according to the scheme

X + Y -> XY + hco.

According to the principle of detailed balance, the ratio of the cross sections for theseprocesses has the form

ZL = 8LM9 (3.69)o-i gi j p

Here, gr, gt are the statistical weights of the final states for recombination and de-tachment processes, respectively, and yph, jp are the fluxes of photons and constituentatomic particles for the corresponding channels of the processes, normalized to onephoton per given volume.

Problem 3.28. Obtain the connection between the cross sections for photorecombi-nation and photodetachment of atomic particles.

SOLUTION. The statistical weights of the final states considered are

47rk2dk _ Airq2dqgr ~ 8XY2!2^-; gi~gxgYl^>

where gx, gy, gxY are statistical weights of internal states of the particles, k is thephoton wave vector, and q is the wave vector of the relative motion of the particles.The photon flux for the case where there is one photon in a volume fl is yPh = c/ft,and the particle flux in this case is jp = v/Cl. Here, v = fiq/yu is the relative velocityand /x is the reduced mass of the particles. From the dispersion relation for photonsk = co/c, and the law of energy conservation (3.2), Eq. (3.69) leads to

^ ^ L (3.70)r gxgv

From this it follows that ar < a, because k < q. For example, if the frequencyand wave vector of a photon in a photoionization process is of the order of a typicalatomic value, we can make the estimate

and therefore the photorecombination cross section is approximately four orders ofmagnitude smaller than the photodetachment cross section.

Equation (3.70) describes total cross sections. An analogous expression for thedifferential cross sections is

dar _ k2 gxY dendilk qz gxgy d{lq

Equation (3.71) applies to the radiation or absorption of photons of a given polariza-tion, and thus it does not have a factor of 2, in contrast to Eq. (3.70).

Problem 3.29. Using the principle of detailed balance, calculate the cross sectionfor the recombination of an electron with a hydrogenlike ion, ending in the groundstate of the atom.

SOLUTION. As a prelude to the examination of the recombination process, we considerthe cross section cr, for the ionization of the ground state of hydrogen with theproduction of an electron moving with the speed v. This is given by Eq. (3.10). Aswritten, this equation refers to hydrogen. It can be generalized to an atomic nucleusof charge Ze by replacing v with v/Z and replacing the Bohr radius a0 = h2 /me2

with h2/Zme2. We find

at = (29ir2/3cZ2)f(v), (3.72)

with the notation

Z8 exp[-(4Z/t,) arctan(t,/Z)](Z2 + t,2)4[l - exp(-2irZ/t/)]'

It is the cross section for the inverse process that is sought here, that is, photore-combination of an electron with speed v with transition to the ground state of an ionwith charge Z — 1. We can obtain this by using Eq. (3.70), which follows from theprinciple of detailed balance. The statistical weight of the electron and the ion arege — gi\ — 1> and the statistical weight of the ground ionic state is ga = n2 = 1. Weobtain thereby

or = (2k2/v2)ah

where k = w/c is the wave number of the photon radiated in the recombinationprocess. Substituting Eq. (3.72) in this expression, we find

^ _ 2W7T2O>2

Energy conservation gives

« = (Z2 + z,2)/2,

so Eq. (3.73) can be rewritten as

287T2Z6 7 4

where

exp[-(4Z/f) arctan(f /Z)]^V) ~ v2(Z2 + t/2)2[l - exp(-27rZ/i;)]"

In the WKB (Wentzel-Kramers-Brillouin) limit of slow electrons, when v < Z,Eq. (3.74) reduces to

crr = 287T2Z2/3e4cV,

where here e is the base of natural logarithms. It is seen that the cross section forrecombination into the ground state increases with a decrease in the electron velocity.

In the opposite limit of fast electrons, when v > Z, one can also simplify thegeneral expression (3.74). The result is

07 - 2 7 T T Z 5 / 3 C V .

It is seen that here also the photorecombination cross section increases with a decreasein the electron velocity, but much more sharply than in the slow-electron case.

Problem 3.30. On the basis of the cross section for bremsstrahlung, determine thecross section for the recombination of a slow electron and an ion into an atom in ahighly excited state.

SOLUTION. The states involved in the process under consideration have large statis-tical weights. We shall therefore use a classical description for the electron motion.Within the framework of classical electron motion, we consider this transition to be aresult of a bremsstrahlung process. The atomic system of units with m = e — h = 1will be used.

According to Eq. (3.68), the cross section for the process being considered is

16TT 1 dco „ „d r^—' (3.75)

c w

Energy conservation gives

co = \/{2n2) + v2/2,

and when we differentiate this, we get

dco = —dn/n3.

The cross section for photorecombination of a slow electron with an ion, leadingto a state with principal quantum number n, follows from these expressions with theuse of dn = 1. (This takes into account that highly excited states with closely similarvalues of n are approximately equally spaced in energy.) Thus we have

ar = dcrb(dn = 1) =

' • " ' (3-76,lC3 nco(2n2co — 1)

The boundary of photorecombination described in this way is determined by thecondition co > l/(2n)2, which corresponds at n > 1 to the condition co < 1 for the"softness" of photons. Note that the classical description of electron motion requiresthat the condition v < 1 [or e2/(hv) > 1 in usual units] be fulfilled.

Problem 3.31. Compare intensities of bremsstrahlung and photorecombination ra-diation of electrons in multicharged ions. Assume that the Maxwell distributionfunction holds true for the electron velocities.


SOLUTION. The cross section ar for photorecombination of an electron, with transi-tion to the ionic state with charge multiplicity Z - 1 and principal quantum numbern > 1, can be obtained from Eq. (3.76) by suitable introduction of Z2. The crosssection is

= 16TTZ2 1(Jr 33/2c3 n(o[(2n2co/Z2) - 1]' { ]

obtained by multiplication by Z2 and replacement of the hydrogenic energy l/2n2 byZ2 /In2 for the ion of charge multiplicity Z. The multiplier Z2 arises from a combina-tion of factors. There is a replacement of the cube of the coupling strength (e2/he)3

by (Ze2/hc)3, a replacement of the square of the Bohr radius a^ = (h2/me2)2 by(h2/Zme2)2, and a substitution of n/Z for n in the denominator. Altogether, thereis a multiplication by Z2. In this review of the origins of the extra factor we havedeparted from the atomic system of units for expository clarity, but Eq. (3.77) is againexpressed in atomic units.

The rate of the recombination transition is given by Nevar, where v is the electronvelocity and Ne is the electron concentration. At each transition, the photon energyco is radiated, so the radiation intensity for a single ion is Ne(vcrr(D), where thebrackets correspond to an averaging of the electron velocity in terms of the Maxwelldistribution. The radiation intensity S^n) per unit volume is obtained by multiplyingby the number Nt of ions in a unit volume, so that

S(rn) = NiNe{v(jrw). (3.78)

The cross section expression (3.77) can be rewritten with the help of the energyconservation expression

co = v2/2 + Z2/2n2,

so the factor in brackets in the denominator of (3.77) becomes

(2n2co/Z2) - 1 = rcV/Z2.

The recombination cross section then becomes

16TTZ4

°"r ~ 33/2c3n3a>v2'

This, together with Eq. (3.78) gives the radiation intensity in the form

(n) _

It is seen from this formula that radiation to the ground state (n = 1) predominates.However, it should be kept in mind that the WKB formula (3.76) of Kramers canproduce errors for the ground state of 10-15%.

We now evaluate the radiation intensity resulting from recombination to all of thebound states. For this, the mean value of the inverse velocity

{v~l) = (2/nT)i/2

is needed (T is the temperature), as well as the sum

1.202,

where £(JC) is the Riemann function. The summed radiation intensity is

n=l

The intensity of bremsstrahlung from a unit volume of a medium produced byscattering of electrons on ions, designated 5/(7), can be shown to be

. . ~ JeNh (3.80)

The ratio of this result to Sr is

St(T) IT

This expression indicates that if T > Z2, then bremsstrahlung predominates; but ifT < Z2, then recombination radiation predominates. However, it must be recalledhere that the condition for applicability of the WKB method, upon which bothEqs. (3.79) and (3.80) depend, is that the electron velocity should be small. Thisrequires that T < Z2 be fulfilled. Hence, photorecombination radiation predominatesin the region of applicability of the expressions obtained.

Problem 3.32. Obtain the expression (3.41) for the photoionization cross section ofa Rydberg atom on the basis of the photorecombination cross section (3.76) and theprinciple of detailed balance. Consider the accuracy of both formulas if the principalquantum numbers are not large.

SOLUTION. In order to use the principle of detailed balance, it is necessary to calculatethe statistical weights for the channels involved. Because there are no electron or ionspins in the problem, one can take the statistical weights of the electron and the ionto be unity: ge — gi — 1. The statistical weight of an excited state is ga = n2. Usingthese expressions in Eq. (3.76) along with the principle of detailed balance (3.70),we obtain

2k Sa

for the photoionization cross section of a Rydberg atom upon absorption of a photonof frequency co. This is Eq. (3.41) for the photoionization cross section of a Rydbergatom with ejection of a slow electron near the threshold of the process. Equations(3.76) and (3.81) are called Kramers's formulas.

Kramers's formulas are adequate even at small values of n. Let us compareEq. (3.81) with the accurate value of the threshold cross section for photoioniza-tion in the ground state (co = ^) given by Eq. (3.11),

cn,ac = 297T2/(3e4c). (3.82)

If we employ the values n = 1 and co = \ in Eq. (3.81), we obtain the ratio of thecross sections

<ri/*i,ac = e4/(&7T^) = 1.25.

As for the frequency dependence of the photoionization cross section near threshold,Eq. (3.81) gives o"; ~ co~3, while the accurate result according to Eq. (3.10) iso"i,ac ~ o>~8/3. Thus the threshold behavior of the quasi-classical cross section isquite similar to the accurate cross section for the ground state. The accuracy of thequasi-classical cross section increases with an increase in n.

With an increase of photon frequency co, the accuracy of quasi-classical formulasdeclines, while the applicability of the Born approximation, in which one neglects theinteraction of the detached electron with the remaining ion, improves. In particular,for n = 1, the comparison of quasi-classical and accurate photoionization crosssections for large photon frequencies (co > 1) gives

Problem 3.33. The recombination coefficient is defined as the recombination rateconstant averaged over electron velocities. Determine the temperature dependence ofthe recombination coefficient ar if the process leads to a highly excited state of theatom formed. Presume that the velocity distribution function for the electrons is theMaxwell distribution.

SOLUTION. The recombination coefficient can be found from Eq. (3.76) for the pho-torecombination of an electron with an ion, forming an atom in a Rydberg state. Wefind that

16TTar = var = —--^-r(vo)) K (3.83)

3 V 3 " c

Atomic units are used here, and the bar over a quantity means that an average is tobe taken over electron velocities.

With the expressions for the photon frequency

co = v2/2+ \/(2n2\

and the electron energy, s = v2/2, we obtain for the recombination coefficient

_ 16^277 1 r exp(-e/r)a J **With a simple change of variable, the integral converts to the exponential-integralfunction, and we obtain the form

Let us consider special cases of Eq. (3.84). If the binding energy for the Rydbergstate is large compared with the energy of the thermal electron, n2T < \ (the thresholdcase), we can use the asymptotic expression for the exponential-integral function,

Ei(-z) - • - - e x p ( - z )

and obtain thereby

For the opposite case of n2T > 1, we use the small-argument limit of the Ei function

Ei(-z) - ln(yz),z—•()

where y = exp(C) and C = 0 .5772. . . is Euler's constant. In this limit we have

n2

\ y16y/2iT 1 [2n2T\

ar = V In . (3.86)3 / 3 c3«3r3/2 \ y J

It is seen that the recombination coefficient decreases with an increase in theelectron temperature. Note that when n2T > 1, the Kramers formulas are not valid,and the Born approximation becomes applicable.

We shall now appraise the validity of the quasi-classical expression (3.85) for therecombination coefficient if we use this equation for the hydrogen atom in its groundstate. First we use Eq. (3.11) for the photoionization cross section near threshold toobtain the accurate expression (for n = 1)

297T2

Assuming the electron temperature to be small compared with the ionization potential,Eq. (3.70) gives the photorecombination cross section


where v is the velocity of a slow electron. This gives, for the case under consideration,the recombination coefficient

1 --r

Since v ~x = yj2/(jrT) for a Maxwell distribution function describing the velocitiesof the electrons, we obtain the final result

The quasi-classical value of the recombination coefficient ar corresponds toEq. (3.85) with n = 1. The ratio of this value to the accurate value (3.87) isur/oir,ac = e4/(STT\/3) = 1.25. Thus the quasi-classical approximation worksrather well even at values of parameters where its validity conditions are violated.This conclusion relates both to the recombination coefficient obtained by averagingover electron velocities and also if this averaging is not done.

Problem 3.34. Calculate the recombination coefficient for an electron on an ion withcharge multiplicity Z. Assume a hydrogen-like model for the ion.

SOLUTION. The photorecombination coefficient to a final state with the principalquantum number n for the case Z = 1 is given by Eq. (3.84). This expression can beextended in simple fashion to values Z > 1 by reference to the photorecombinationcross section given in Problem 3.29. Introduction of a general Z value in Eq. (3.84)leads to

3 3 / 2 c 3 b3n/2exV(bn)Ei(-bn), (3.88)

where the quantity bn is

bn = Z2/(2n2T).

If the condition Z2 > n2T is fulfilled (i.e., the electrons are slow), then the expressionfor ar simplifies to

_ 32(2TT) 1/2

" ~ 33/2 C

This result is the generalization of Eq. (3.85) to the case Z > 1. It is seen that ar

decreases slowly (as n~l) as n increases.Expression (3.88) for the photorecombination coefficient can be summed up to

values of the principal quantum number n0 ~ Z /7 1 / 2 , after which ar begins todecrease more sharply with further increases in n. According to Eq. (3.86), ar then

varies as n~3. With the result thatno j

Y^ - « ln(n0),n = \

we obtain (within logarithmic accuracy) the final expression for the photorecombina-tion coefficient summed over all levels of a hydrogenlike ion with charge multiplicityZ, that

, ^ 32(2TT) Z1

n=\

If the opposite inequality Z2 < T is fulfilled for the ground state, then it is certainlysatisfied for excited states with n > 1. In this case, the summed photorecombinationcoefficient is determined for all practical purposes entirely by the term with n — 1.Taking the appropriate limit in Eq. (3.88) gives

s _ 16(2ir) Z4 fyZ2

"' " 3 3 / V W2 n\2T

where y is given by y = exp(C), and C is Euler's constant, C = 0.5772

Problem 3.35. Determine the recombination coefficient for two colliding atoms as-suming that the recombination proceeds from a repulsive molecular state to highlyexcited vibrational levels of the final molecule formed.

SOLUTION. This process is inverse to that treated in Problem 3.10 and is describedby the scheme

X + Y -> (XY)* + hco.

The molecular energies involved in the transition are illustrated in Figure 3.2. As inProblem 3.10, we assume the motions of the nuclei to be classical. We now introducethe quantity w(R), defined as the probability per unit time for the spontaneous radiativerecombination of the atoms as a function of the distance R between nuclei. It is givenby Eq. (1.13) with JTZ — 1. Then the total probability for radiative recombination asa result of the collision of the atoms is

/»OO

W= w(R)dt. (3.89)J —oo

This value is assumed to be less than unity. This assumption is usually well ful-filled because typical times for spontaneous radiative transitions are less than atomiccollision times.

The probability W depends on the collision impact parameter p, so the crosssection for radiative transition can be written

ar = / 27rpdp / w(R)dt. (3.90)J0 J-oo


We evaluate the integral by using the classical connection between dt and dR,

dt = dR (3.91)vy/l-(P

2/R2)-U2(R)/s

Here v is the relative velocity of the nuclei of reduced mass /x, s = ixv2/2 is theirkinetic energy in the center-of-mass system, and U2(R) is the interaction potential ofthe atoms in the final channel of the process.

Using (3.91) in (3.90) and changing the order of integration, one can evaluate theintegral over the impact parameter. This yields

f dR /n^ rJ v Jo

pdp

- (p2/R2) ~ U2(R)/sAA r

= — / w(R)R2y/l - U2(R)/sdR, (3.92)v JRO

where Ro is the distance of closest approach for collision with zero impact parameter,that is, U2(RQ) = e.

The recombination coefficient averaged over a Maxwell distribution for the veloc-ity of the atoms follows from Eq. (3.92) as

The integral over s can be evaluated to give

Ju2(R)

When this result is used in Eq. (3.93), we obtain the photorecombination coefficient

ar =4rr exp - - ^ ^ w(R)R2 dR. (3.94)Jo L T J

The structure of this equation is such that the photorecombination coefficient isrepresented in the form of a probability per unit time as a function of the distance be-tween the nuclei, weighted by the corresponding Boltzmann factor exp[— U2(R)/T].Evidently, this result could have been obtained from very basic considerations.

Problem 3.36. Using the conditions of Problem 3.35, determine ar(a>), the spectralrecombination coefficient of atoms, defined such that ar{co) da) is the recombinationcoefficient corresponding to the emission of photons in the frequency interval fromco to co 4- dco.


SOLUTION. According to the definition of the spectral photorecombination coefficientav(a>), the above value of the recombination coefficient is related to it by

ar=Jar (a>) day.

From energy conservation and the classical character of the motion of the atoms, itfollows that a photon of frequency co can arise only at a particular distance betweenthe nuclei that satisfies the condition

U2(RJ ~ Ui(RJ = ha). (3.95)

Then, from (3.94) we have

We introduce the derivatives of the interatomic potential with respect to the distancebetween the atoms, F\2(R) = dU\f2(R), and arrive at the form

ar((o) = 4irv^(RJRih\Fl(Ra) - F2(RM)\-1 exp [~U2(RJ/T]. (3.96)

In a similar way, the photorecombination cross section 07(a>) for frequency co canbe evaluated from the relation

07 = / ar(a))do).

The spectral properties of the cross section are given by

J - F2(Ra,)\U2(Ra>)

4COHERENT PHENOMENA INRADIATIVE TRANSITIONS

Oscillations with the same frequency, and in a fixed-phase relationship are calledcoherent. Using lasers, we can produce mixtures of coherent oscillations with differentphases. The development in time of such coherent states results in interferencephenomena, and the investigation of these phenomena gives information about theproperties of coherent states. Some of these interference phenomena, usually calledcoherent spectroscopic phenomena, will be considered in this chapter.

4.1 POLARIZATION EFFECTS IN RADIATIVE TRANSITIONSIN A UNIFORM MAGNETIC FIELD

In this section we shall consider phenomena associated with the appearance of po-larization in the radiation scattered by an atomic electron due to the presence ofa constant, uniform magnetic field. These phenomena arise from the interaction ofatomic electrons with the radiation.

Problem 4.1. A beam of unpolarized light propagating in a direction normal to amagnetic field is resonantly absorbed by atoms. Transition from a ground s state toan excited p state takes place in this process. We wish to investigate the spontaneousradiation in the direction of the magnetic field and calculate the degree of polarizationof the radiation, expressed by

In this expression, Ix and Iy are the intensities for emission of photons with polariza-tions in the x and y directions, respectively, with the z axis taken to be in the direction


178 COHERENT PHENOMENA IN RADIATIVE TRANSITIONS

of the magnetic field vector H. Magnetic sublevels with projections ± 1 of orbitalangular momentum are excited in the transition. The energy spacing between thesesublevels is hcoH. The spontaneous lifetime of the excited p state of the atom is T andis presumed to be large compared to the duration of the radiation pulse. Calculatealso the mean value P, averaged over the observation time.

SOLUTION. The scheme of the experiment and the corresponding coordinate systemare shown in Figure 4.1. We take the y axis to be in the direction of propagation ofthe radiation. It can have polarization components along the z and x axes. The partof the radiation polarized along the z axis excites a magnetic sublevel of the p statewith m = 0. The spontaneous radiation from the atom is observed along the z axis,and can have polarization in the xy plane. The matrix element of the transition p —> sbetween sublevels with m = 0 resulting from the x or y operators is equal to zero.Therefore, in the process considered, only the part of the radiation propagating alongthe y axis and polarized along the x axis will contribute.

This part of the radiation excites sublevels of the p state with magnetic quantumnumbers ±1 . Both matrix elements from the operator x are the same, as is wellknown. The wave function of the p state at t = 0 is of the form ^(0) = i/>i + i/>-i,where 1//+1 are coordinate wave functions of the states considered. If the magneticfield is absent, then at t > 0, the wave function ^(t) contains a phase factor commonto the m = ±1 sublevels, since their energy is the same. The intensity of emission ofa photon with a given polarization s is

/ ^ K ^ o l s - D K ^ + i//-!))!2. (4.1)

Here D is the dipole moment operator of the atom, and "^0 is the wave function ofthe s state. It is seen that only the intensity component Ix is nonzero, since the matrixelements from the coordinate y between the states ^0> *Ai and between ^0> *A-i

H A magnetic field

pola

observedradiation

cell with atoms /'"\

izer

incidentradiation

Figure 4.1. Geometry of the Hanle experiment and the axes employed.

POLARIZATION EFFECTS IN RADIATIVE TRANSITIONS 179

the same moduli and opposite signs. Thus if the magnetic field is absent, then theradiation is emitted along the z axis, polarized only along the x axis. In this case, thepolarization coefficient is obviously equal to 1.

Now we consider the case where the magnetic field is turned on. The Zeemaneffect causes the sublevels with magnetic quantum numbers +1 and - 1 to splitsymmetrically to opposite sides of the initial level. If we denote the energy intervalbetween these sublevels as ha)H, then instead of Eq. (4.1), we have at t = 0,

Is ~ | ( % Is • D| [fa exp( - /oW2) + i//_! exp(/co//r/2)] >|2. (4.2)

In consequence, we have

Ix ~ \{9O\DX\ [faexp(-i(oHt/2) + i/MexpOW/2)])!2 ~ cos2(a>Ht/2),

since the matrix elements of the operator x are the same for both of the sublevels ofthe state. It follows from Eq. (4.2) that, for the intensity of radiation with polarizationalong the y axis, the result is

Iy - sin2(a)Ht/2),

since matrix elements of the operator y are the same in modulus but opposite in signfor the states fa and i//_ {. Hence, the polarization of the emitted radiation at time t is

c o s ( 0 (4.3)

It is seen from Eq. (4.3) that the polarization of the emitted light oscillates dependingon the time of emission of the photon. This is due to the interference of the twomagnetic sublevels. It can be viewed as arising from an oscillation of an atomicelectron in its classical motion in the field. The polarization returns to the initial valuewhen the orbital motion completes an integer number of cycles.

Now we average the polarization coefficient (4.3) over the observation time, takinginto account the exponential decrease of the population of the p state with time, toobtain

1(4.4)

Thus the polarization coefficient of the radiation decreases when the magnetic fieldis turned on. That is, depolarization of the radiation occurs. This phenomenon ofdepolarization of the spontaneous radiation of an atom in a magnetic field is calledthe Hanle effect. This effect was discovered in 1924, and it is one of the first of theinterference phenomena discovered in atomic systems.

We have shown that the decrease of the polarization of the light emitted in themagnetic field is explicable from the classical point of view of the different preces-sions of the orbital momentum in the magnetic field for the sublevels of the p stateof an atom.


Problem 4.2. Calculate the cross section for photon absorption under the conditionsof Problem 4.1 when the emitted photons are polarized along the x axis. Assume thatspontaneous decay from the p state to the s state is the only channel for radiativetransitions.

SOLUTION. Start with Eq. (2.89) for the photon absorption cross section in resonancefluorescence. The following modifications must be made to this expression to makeit applicable to the present problem:

1. Introduce the factor cos2(a)#£/2) in accordance with Eq. (4.2) for calculationof the cross section associated with the emission of photons along the x axis.

2. Replace the factor 2Jk + 1 by 1 in the numerator of Eq. (2.89), since the initialand final channels are determined by a fixed photon polarization.

3. Average over time t with the probability function ( 1 / T ) exp(—t/r) in analogywith the procedure in the preceding problem.

We obtain in this fashion

(Tr =

2(O2T

Here a^ is the Lorentz distribution function for the absorption of photons of frequency(o, with the width determined by the spontaneous lifetime of the p state.

In an analogous fashion, we find that the corresponding cross section for photonsemitted in the direction of the z axis with polarization along the y axis, is given by

10-v =

It follows from Eqs. (4.5) and (4.6) that, in the cross section for emission along the zaxis of unpolarized photons,

7T2C2

the interference effect disappears. This cross section is in agreement with the generalexpression (2.85) for the photon absorption cross section.

Now we shall take into account the motion of the atoms that is responsible for theDoppler effect. Since the Doppler width is usually large as compared to the sponta-neous width, then, in Eqs. (4.5) and (4.6), we should replace the Lorentz distributionfunction for the absorbed photons by the Doppler distribution of frequencies. Whenthese expressions are averaged over atomic velocities, it is found that the interferencestructure is retained.

POLARIZATION EFFECTS IN RADIATIVE TRANSITIONS 181

Problem 4.3. Obtain Eqs. (4.5) and (4.6) in the framework of a model of a dampedclassical dipole in a constant magnetic field.

SOLUTION. We posit as above that the incident radiation propagates along the y axisand is polarized along the x axis. In the classical model, the oscillations of an atomicelectron along the x axis take place during the excitation process. The interaction ofthis dipole with the emitted radiation is given by the expression — d • E, where d isthe dipole moment and E is the electric field vector. Hence, the emitted radiation isalso polarized along the x axis. As above, we take the direction of observation of theemitted radiation to be coincident with the x axis.

Now add to the above problem a constant magnetic field, with the magnetic fieldvector H along the z axis. After the dipole moment is excited, it begins to rotate in thexy plane, in accordance with the principles of classical electromagnetic theory. Thisis the plane perpendicular to the magnetic field vector. It is known that the precessionfrequency (the so-called Larmor frequency) is given by a)o = eH/(2mc). Let the timet = 0 be the moment of excitation of the dipole. As above, the time of duration ofthe incident radiation is presumed to be small as compared to the relaxation time ofthe dipole, that is, as compared to its lifetime for emission of spontaneous radiation.During the time t, the dipole d turns through the angle a)tf with respect to its initialdirection along the x axis. We designate by Ix(t) the intensity at time t of the radiationscattered along the z axis with polarization along the x axis. It is determined by thex projection of the dipole moment, dx. This projection is equal to d cos a^t. Hence,the contribution to the intensity Ix(t) will be decreased by the factor cos2 a^t, that is,

Ix(t) = Icos2 o)0t,

where / is the intensity of scattered radiation in the absence of the magnetic field(o>o = 0). We also take into account the exponential decrease with time of themagnitude of the dipole moment. The quantity /, which is proportional to the squareof the dipole moment, also decreases exponentially, so that

/ (0 = /0exp(-f/T).

Here 70 is the constant that determines the intensity at t = 0 of the scattered radiationpolarized along the x axis.

We now calculate the intensity averaged over the observation time, Ix, as the totalradiation energy JQ

X Ix dt for all times, divided by the damping time T, or

7x= - / Ixdt. (4.7)• = - rv Jo

In the absence of a magnetic field (co0 = 0), we have Ix = 70. Using the aboverelationships to calculate the integral in Eq. (4.7), we obtain

1 \

(4.8)

When we compare Eq. (4.8) to Eq. (4.5), we see that the time r is the natural lifetimefor the excited p state, and COM — 2COQ-

In an analogous fashion, we calculate the intensity of the radiation along thedirection of the z axis with polarization along the y axis. In this case, the projectiond sin (o0t onto the y axis replaces the projection d cos (o0t. Thus, instead of Eq. (4.8)we find

This result vanishes when the magnetic field is zero. The scattered radiation is totallypolarized along the y axis in the observation of radiation along the x axis.

Problem 4.4. Determine the directions of the polarization vector of scattered radi-ation for both the maximum and the minimum time-averaged intensity 7 under theconditions of the previous problem. Scattered radiation is observed along the directionof the magnetic field. Calculate the polarization coefficient of the scattered light

-p *max -*min

*max ' *min

Intensities are averaged as in Eq. (4.7).

SOLUTION. In analogy to the preceding problem, we take the magnetic field strengthvector H to be directed along the z axis, and the incident electromagnetic wavepropagates along the y axis with its polarization along the x axis. Calculate theintensity of the light emitted in the z direction such that its polarization vector lies inthe xy plane and is directed at an angle <p with respect to the x axis.

We solve this problem in a manner similar to the previous problem. Let us supposethat the dipole moment is excited at time t — 0, and at the time t the vector d rotatesto the angle coot with respect to the initial direction along the x axis. Here co0 is theLarmor frequency of precession in the magnetic field H. The projection of the dipolemoment onto the direction of the polarization vector of the scattered radiation at timet is given by d cos((o$t + <p). Hence, the averaged intensity of the scattered radiationfor a fixed value of <p is

(-) / exp (--) cos2(co0t + cp)dt.

Evaluating the integral, we obtain

cos [2cp —1 + (4.10)

As in the preceding problem, the quantity /Q determines the radiation intensity withphotons polarized along the x axis in the absence of the magnetic field (c o = <p = 0).In particular, at cp = Oandatcp = TT/2, Eq. (4.10) yields, respectively, the expressions

INTERFERENCE OF STATES DURING RADIATION 183

(4.8) and (4.9), as one would expect. Expression (4.10) has the maximum value

~1/2

at the angle

<p — | arctan (2O)QT) ,

and the minimum value

(/o/2)[l-(l+4Wo2T2)

at the angle

<p = \[TT + arctan (2CO0T)].

The directions for maximum and minimum intensities are mutually perpendicular, aswould be expected.

The polarization coefficient of the scattered radiation is

P = ( l + 4 o , 0 V ) " 1 / 2 . (4.11)

It is seen that this value is larger than that given by Eq. (4.4) with a)H = 2co0. Thequantity (4.11) changes as a function of H more slowly than Eq. (4.4). Equation(4.11) is the correct polarization coefficient for scattered radiation.

It is clear that, under the conditions of Problem 4.1 for absorption of light byatoms with a resulting transition from an s state to a p state, the expression (4.11) iscorrect, but then r is the lifetime of the atomic p state, and the quantity 2COQ shouldbe replaced by (X>H. The energy hcon is the interval between magnetic sublevels withprojections +1 and — 1 of the orbital angular momentum of the p state. This energyinterval is proportional to the magnitude of H.

4.2 INTERFERENCE OF STATES DURING RADIATION

In the previous section we considered the simplest phenomena associated with co-herent processes in radiative transitions. Now we wish to describe more complicatedproblems related to experiments where interference of atomic states during radiationis important.

Problem 4.5. Investigate the photon radiation from a damped classical dipole in aconstant magnetic field, where the radiation is to be examined in the direction of themagnetic field. Radiation is incident on the system with time dependence given by

7(0 = const [l + 6rcos (Clt + ijj)], s <^ 1.

The quantity s determines the degree of modulation of the incident radiation. Themodulation frequency fl is assumed to be small compared to the frequency co0 of theincident radiation.

SOLUTION. This problem is a generalization of Problem 4.4 to the case of radiationmodulated in time. We suppose again that the magnetic field is directed along the zaxis, and the incident radiation is polarized along x and propagates in the y direction.It should be noted (see Problem 4.1) that if the incident radiation is polarized alongthe z axis and propagates along y, then interference effects do not appear. We set theorigin of time, t = 0, at the time at which the dipole is excited.

Let us consider the radiation scattered in the direction z of the magnetic field, withpolarization in the xy plane. We designate by <p the angle between the polarizationvector and the x axis. Repeating the procedure of Problem 4.4, we obtain for theaveraged intensity

Tp = — / [l + scos(n* + i/0] e~t/r cos2 (co0t + cp)dt, (4.12)T Jo

in place of Eq. (4.10). This expression takes into account the intensity modulation ofthe incident radiation as given in the statement of this problem.

The first term in Eq. (4.12), independent of s, obviously produces Eq. (4.10).Therefore only the second term, which is proportional to the modulation strength s,is of interest here. Let us write this term in the form

871 = — / cos (fir + il/)e~t/T\l + cos[2(w0r + <p)l }dt. (4.13)2T JO

Expression (4.13) is represented again as a sum of two terms. The simplest to evaluateis the first term, since an analogous integral was calculated in the solution of Problem4.4 [see Eq. (4.10)]. We need only make the substitutions 2o>0 —> ft and 2<p —• ijj.Then, expressing 81^ in the form

we obtain for Sj/^ the expression

sin cos \ip — arctan(flT)]Si/9 = — Y/2— • (4.14)

It should be noted that Eq. (4.14) does not depend on the magnetic field strength H,nor on the polarization angle <p of the scattered radiation.

Physically, expression (4.14) represents the intensity modulation of the scatteredradiation. Indeed, if we set cp = 0 and H = 0, then the quantity /0 is modified as

It follows from Eq. (4.15) that the degree of modulation of the scattered radiation isapproximately (1 + H 2 T 2 ) 1 / 2 times less than the degree to which the incident radiationis modulated. In addition to this, the phase of the scattered radiation is delayed withrespect to the modulation phase \\s of the incident radiation by the angle arctan(flT),which depends on the depletion time r of a dipole.

Now let us consider the second term in Eq. (4.13), that is, let us calculate thequantity 82/<p. Rewriting it in the form

e~t/r [cos(2o)0r + 2<p - Of - i/0 + cos(2w0f + 2<p + fit

we see that it can be calculated in a fashion analogous to the way Eq. (4.14) wasderived. One need only make appropriate changes in the identity of the parametersin the integrand. In this fashion we obtain

c o s [2<p + i// — arctan [(2<D0 + H)T]]

cos [2<p - i// - arctan [(2co0 - H ) T ] ] I4- — V . (4.16)

[l + (2co0 - O)2 T2

Equation (4.16) describes interference oscillations that depend on the relation betweenthe frequencies ft and 2co0. The second term in Eq. (4.16) is resonantly large if thecondition O ~ 2co0 is fulfilled. In like fashion, the first term of Eq. (4.16) increasesresonantly under the condition (1 ~ -2o)0.

We refer back to Problem 4.1, where, instead of the quantity 2o>o, w e had thefrequency (oH, which measures the energy splitting between Zeeman sublevels ofpstates with quantum numbers +1 and — 1. We conclude that interference oscillationsare important when the frequency of these oscillations is approximately equal tothe transition frequency between the sublevels considered. A resonant increase ofintensity is thus achieved when the photon energy h£l coincides with the energysplitting of sublevels with magnetic quantum numbers +1 and — 1 of the atomic pstate. This is the usual resonance condition in the quantum theory of atomic radiation.

We emphasize that all interference effects disappear after averaging over the phasei// of the intensity modulation of the incident radiation.

Equation (4.16) can be simplified for a very strong magnetic field, that is, underthe condition that O)QT > 1 and the resonance condition il ~ 2coo (or O ~ —2COQ ).Then Sj/^ —> 0, and we obtain

( 4 J 7 )

It follows from this expression that

sT 7 2 f

2[l+(2wo-n)2T2] 1/2 f '

Finally, we find the polarization coefficient

/max + /min 2 [l + (2(O0 " ft)2T2] '

The polarization coefficient is largest at exact resonance. Then it is equal to Pmax =e/2, and does not depend on ft, i//, or r.

Problem 4.6. Calculate the radiation intensity along the direction of the magneticfield for a classical dipole placed in a magnetic field H. The quantity H is subjectedto a modulation in time of frequency ft and strength s. Investigate the phenomenonof paramagnetic resonance between the modulation frequency ft and the Larmorprecession frequency OOQ.

SOLUTION. According to the statement of this problem, we take the Larmor precessionfrequency in the magnetic field to be of the form

- ^ = o)(t) = co0 [1 + s cos (tit + i/0] . (4.19)

Here, a>o is the Larmor frequency in a constant magnetic field, i// is the modulationphase angle, and a(t) is the angle of dipole rotation in the case of a variable rotationfrequency co(t). After performing the integration over time in Eq. (4.19), we obtain

a(t) = coat + (t)(\— \sin(£lt + ii/) — sinib] .ft L J

We set the zero of time at the moment when the dipole is excited, so that a(0) = 0.The initial direction of the dipole is taken to coincide with the x axis. We wish to findthe radiation in the direction of the magnetic field strength, which is polarized in thexy plane. We designate by cp the angle between the polarization vector and the x axis.

In analogy to the solution of Problem 4.1, we obtain the intensity /^ of the emittedphotons as

Tp = - / e't/r cos2 {o)0t + <p + co0^- [sin(ftr + i//) - sin ip] j dt. (4.20)T Jo ^ ** '

In order to calculate the integral in Eq. (4.20), we introduce the generating functionfor the ordinary Bessel function Jn(^) as given by

n=—oc

When this relation is employed in Eq. (4.20), we find

t , V - T /2co06r\ cos [2<p + m// - (2oWft)sini// + |3]2 Jn\ ~ft~ I l/2

(4.21)

Here the notation is introduced that

/3 = arctan [(2co0 + nil) T] .

It follows from Eq. (4.21) that, under the condition nil = ±2co0, one term in thesum in Eq. (4.21) will be resonantly large. Such a resonance is called a parametricresonance. In the vicinity of the nth resonance, which is clearly observed whenco0r > 1, we can restrict the result to a single term of the above sum. We then obtain

V T 1 + Jn(-ns)— ——jz— } . (4.22)

Since the magnitude of the second term in Eq. (4.22) is less than unity, the entireexpression (4.22) is, of course, positive.

It is seen from Eq. (4.22) also that the amplitude of the resonance depends es-sentially on the modulation strength s. If the quantity s is of the order of unity,then the amplitude of the resonance is of the order of the nonresonance background,especially for small values of n. On the other hand, if s < 1, then the amplitude ofthe resonance is proportional to sn and decreases very rapidly with an increase inthe index n. Hence, parametric resonance consists of a series of resonances, and theintensity of these resonances decreases rapidly with increasing multiplicity n of theresonance.

Now we wish to determine from Eq. (4.22) the degree of polarization Pn of theemitted radiation. First we choose cp so that the cos function in the equation has thevalue +1 , and then we select <p so that the cos function has the value — 1. This givesus the expression

[l + (2 w0 + nil)1 T2 I1/2

for the degree of polarization. In the case of exact resonance (nil = —2O)Q ), we findthe maximum degree of polarization

^max = \Jn(-ne)\.

The degree of polarization decreases as the number n grows. It is proportional to sn

at e < 1. If s ~ 1, then the degree of polarization oscillates as a function of s.The condition OJQT > 1 is achieved in strong magnetic fields. In this case, the

above resonances are observed clearly. From the physical point of view, a resonancetakes place in a system if its inherent frequency 2co0 is equal to, or a multiple of, the"forcing" frequency il.

We can also conclude from Eq. (4.21) that even for very small modulation strengthss < 1, s still plays an important role if the inequality COQ H is fulfilled. In thiscase we have (sa)0/il) ~ 1, and the Bessel function cannot be expanded in a powerseries. Then the quantity I9 oscillates but is nonresonant.

In the above, we considered the electromagnetic radiation field to be a smallperturbation. The next sections will be devoted to investigation of processes in whichthe electric field strength E is sufficiently large that perturbation theory is inapplicable.We shall see later that the condition for a field to be strong is (D^ • Er/ft) > 1, whereD o is the dipole matrix element of the atomic transition k —> 0, and the quantityr is the broadening of this transition. We are restricted here to resonance processeswhere the radiation frequency is close to the frequency co^o of the atomic transitionbeing considered. Apart from this case, problems are fundamentally complicatedwhen the intensity of the radiation field increases into the nonperturbative domain.Nonperturbative problems will be treated in more detail in the next chapter.

4.3 RESONANCE TRANSITIONS IN TWO-LEVEL SYSTEMS

In this section, we consider problems in which the radiation frequency is approxi-mately equal to the frequency o)ko of some atomic transition. Relaxation of the excitedstate k is assumed to occur only by spontaneous emission to the ground state 0. We canthen exclude other atomic states from consideration, which simplifies in an essentialfashion the solution of problems with a strong electromagnetic field.

Problem 4.7. The radiation field with electric field strength E and field frequencyo) excites the atomic transition 0 <-• k, which has the transition frequency co o ~ <o.Relaxation of the excited state k is determined by the spontaneous transition back tothe ground state 0 with the lifetime 7>. Calculate the cross section for the absorptionof radiation for an arbitrary value of E.

SOLUTION. We shall find first the equilibrium distribution of atomic electrons inthe states 0 and k in the external electromagnetic field. We consider the system ofequations (H.7) and (H.9) for elements of the density matrix. Using the resonanceapproximation, we retain only the required exponents in the interaction potentialD • E cos cot. We then obtain the system of equations

1 /— -pkk + wr (®ko ' E ) (e l(O pok — ela> p k o ) ,

1 \ i ( 4 * 2 4 )

,. i *~*u + ^— I Pko + z r (tyfco * E) e l0)t (pkk - Poo); Pok — Pko-dt \ lik) 2n

It is clear from these equations that in the stationary case, which is realized atlarge times, we have

<Poo _ n dpkk _dt dt

and so

Poo, Pkk = const, p ^ = Ae~ia)t. (4.25)

RESONANCE TRANSITIONS IN TWO-LEVEL SYSTEMS 189

Instead of writing a differential equation for pkk, we can use the condition for theconservation of particle number

Poo + Pkk = 1. (4.26)

When we substitute Eq. (4.25) into the second equation of the system (4.24), and takeEq. (4.26) into account, we find

CYA = -£—jfQpkk-l). (4.27)

New quantities are introduced here for resonance detuning A = (o^o — u>\ width ofthe excited state k, T = 1 /2T* ; and the dimensionless parameter of the field intensity

G = (Dk0-E)rk/h.

Substituting Eq. (4.27) into the first equation of (4.24), we find

_ x r2( 4 2 8 )

The new dimensionless parameter introduced here is

The equilibrium value obtained in Eq. (4.28) for the population of the excited levelk is constant due to a balance between two processes: excitation of this state bythe external radiation field and the spontaneous decay to the ground state 0. Inequilibrium, both processes have the same transition rate, which is

18 k — Pkk = 21 pkk8k-

We determine the cross section for absorption of radiation as the ratio of the rateto the flux of incident photons. The flux is

1 cE2—=— cE2

7W = cosz cot —na> 4TT

Then we obtain

STThwXr2gk

aa c& [A2 + r2(i + x)]

for the absorption cross section. We now express the square of the matrix element ofthe dipole transition in terms of the lifetime 7> of the excited state k as

1 A(y> 0

- = 2 r = -^3-|(Dz)w |2go. (4-30)


Then the substitution of Eq. (4.30) into Eq. (4.29) gives the result

c2 2, r 2c2 2, rda = 2TT— ^ (4.31)

co2 go A2 + T2(l + x)

We see that the quantity x *s m e parameter that determines the influence of theradiation field on the transition cross section. The dependence of the absorption crosssection on the field strength E in the case x ^ 1 (mentioned in the preceding section)takes place when the transition time between states k and 0 under the influence of thewave field [which is of the order of h/ (Dk0E) ] is less than the time rk for spontaneousdecay.

In the case x ^ 1, we have

Pkk = POO = 2'

corresponding to equipartition between the two levels in the strong field (for resonancedetunings A which are not extremely large). In this case we obtain

aa = 2TT^^-. (4.32)

It is seen that the absorption cross section tends to zero. This phenomenon can bereferred to as a "transparency" of an atomic medium in a sufficiently strong field ofresonance radiation.

In the opposite limiting case x ^ 1, Eq. (4.31) coincides with Eq. (2.89), which wasobtained within the framework of perturbation theory. The corresponding statisticalweights are go = 270 + 1 and gk = 2Jk + 1, where Jo and Jk are the angular momentaof the initial and final states, respectively.

Expression (4.31) can also be converted to the frequency distribution function aM

in a strong field. After normalization, we obtain [see also Eq. (2.9)]

' V ' (4-33)

Problem 4.8. Under the conditions specified for the preceding problem, calculatethe cross section for coherent scattering of the radiation; that is, the cross section forphoton emission with frequency co in a strong field.

SOLUTION. According to Eq. (1.17), the intensity of coherent radiation Ik of an atomin the classical limit is

Ik = 2(Dz)2/c\

where Dz is the projection of the electric dipole moment onto the direction of theelectric field vector. In the case under consideration, the dipole moment is produced

by an external field. According to the definition of the density matrix, we have

Phj = (Pz)okPko + 0z)koPok (4.34)j

= 2Re(Dz)0kpk0.

Here Dz is the quantum mechanical dipole moment operator. We find here the corre-spondence between the classical and quantum approaches for radiation intensity.

According to Eq. (4.25), we have pk0 = Aexp(-icot). Hence the dipole momentalso oscillates with the frequency co, and will therefore emit photons with this fre-quency. Therefore the radiation being considered is coherent, that is, the scattering iselastic.

The quantity A is determined by expression (4.27), and pkk in turn follows fromEq. (4.28). Thus we find that

br" (435)

When we substitute this result into Eq. (4.34), we obtain for the intensity of theradiation emitted with the frequency oy (averaged over a period of the field, lir/coi),the result

c3 [4= + P ( l + xtfUpon division of this expression by the incident energy flux of the electromagneticradiation,

cE2

COS2 (tit = ,4TT 8TT

we obtain the cross section of the coherent radiation

«-**» *»(A' + r') ( 4 J 7 )

<»2go [A2 + r 2 ( i + #)]

We have used here the relation (4.30) as in Problem 4.7.The cross section for incoherent radiation corresponding to the scattering of ra-

diation with shifted frequency can be obtained as the difference between the totalabsorption cross section [see Eq. (4.31) in the preceding problem] and the coherentscattering cross section, Eq. (4.37). This difference is

at = 2 T T 4 (4-38)8o [A2 + T2(l l 2 *

Incoherent scattering can be explained physically by noting that, at high intensity,the emitted photons cover a wide frequency range, so that the energy of the emittedphotons can differ significantly from the energy of the incident photon.


The ratio of the cross sections for coherent and incoherent scattering can be foundfrom Eqs. (4.37) and (4.38) to be

ae A2

alr

It is seen from Eq. (4.39) that the scattering of resonance radiation by an atom ismostly coherent at small radiation intensity, while in the high-intensity limit (withmoderate resonance detunings), the scattering will be almost totally incoherent.

Problem 4.9. Calculate the absorption cross section of an atom for resonant radiationif the atom also experiences collisions with neighboring atoms.

SOLUTION. We generalize the system of equations for the density matrix given in(4.24) to the case where collisional broadening exists, making use of Eq. (H.12) toobtain

= \Pkk + hi(Do* •E) (e~iwtpok ~ eiu*Pk°)'

\/A ) + k(DE) ~**( ) (4'40)\/A v)pko + k(D*°'E) e~**(Pkk ~Poo) •PDA: = P*kO-

As in the solution of Problem 2.9 [see Eq. (2.39)], we have employed in Eqs. (4.40)the averages over the velocities v of the colliding atoms given by

v = INvu1, Av = Nva".

Here Af is the density of the perturbing atoms, and the quantities a1 and a" aredetermined by expressions (2.42) and (2.43), respectively. The quantity 8(p) in thoseequations represents the difference of phases for scattering in the excited (k) and theground (0) states.

We see that only the nondiagonal elements of the matrix element are modified.We solve the system (4.40) in a fashion analogous to that used in the solution of(4.24) in Problem 4.7. Let us find the stationary solution that does not depend on theinitial conditions. The time dependence of the density matrix elements is determinedby Eqs. (4.25). After substitution into the second equation of the system (4.40), weobtain in place of Eq. (4.27) the result

J ^ - 1). (4.41)=(A - iT

Here, A = A - A v is the resonance detuning taking into account the Stark shiftarising from atomic collisions. The quantity T = T + v/2 is the total width takinginto account both spontaneous and collision broadening. Other notations are takenfrom Problem 4.7, to wit: T = 1/(2T*), A = (ok0 - co, and G = (DkQ • E) r/h.

When we substitute p o as given in (4.25) into the first equation of the system(4.40), and use the value of A determined by Eq. (4.41), we obtain

- . (4-42)

which is the required generalization of Eq. (4.28) for p^. We have introduced herethe quantity

Now we wish to derive the rate of decay of the excited state k, given byDue to the stationary, or equilibrium, nature of this process, the required rate is equalto the excitation rate of the ground state 0. Proceeding in analogy to Problem 4.7,we introduce the absorption cross section as a ratio of the decay rate to the flux ofincident photons, or

Then the absorption cross section <ra can be written in the form analogous to Eq. (4.31)if the relation (4.30) is taken into account, giving

2 T'T<ra = 2 T T - ^ ^ ^ ^ . (4.43)

W2SOA2 + r2(i + x)

In the limit v = Ay = 0, Eq. (4.43) reduces to Eq. (4.31). We see that collisionsproduce additional broadening and Stark shifting of the resonance. Expression (4.43)is in agreement with Eq. (2.89). The numerator has the product of the total width bythe partial width that corresponds to the spontaneous decay of the state k.

Problem 4.10. Under the conditions of Problem 4.9, determine the cross sectionsfor coherent and incoherent scattering of resonance radiation by an atom.

SOLUTION. This problem is the generalization of Problem 4.8 to the case wherecollisions are included. When Eq. (4.42) is substituted into Eq. (4.41), Eq. (4.35) isreplaced by

A - iT A2 + T2(l +(4.44,

We continue by substituting Eq. (4.44) into Eq. (4.34) and find the generalization ofEq. (4.36) for the intensity of coherent radiation of photons to be

4co4 | (DZ)J2 G2r2(A2 + f2)k = ^ gk-^r- —_ —"7- (4-45)

The cross section for coherent radiation is determined by the generalization ofEq. (4.37) as

^ 4u2 go [A2

When we subtract this from the total absorption cross section (4.43), we find the crosssection for incoherent photon scattering,

(4*7)

Let us analyze the expression (4.47). In the weak field case when \ ^ 1, the crosssection (4.47) does not vanish but yields the result

4^^L_. (4.48)r 2 )

Thus we can conclude that collisions give rise to incoherence of the radiation. In theopposite limiting case where ^ > 1, the coherent scattering cross section has thevanishing limit

= 2 ^ ^ . (4.49)

x>\ a)2 g0 r vHence the total cross section in this case reduces to the incoherent part

2 i

<ra « cr; = 2 7 r ^ - . (4.50)x^i w go X

This also tends to zero with increasing intensity, but more slowly than the coherentpart.

Incoherent scattering is also dominant when collisional broadening is large com-pared to spontaneous broadening, that is, when v > T. In this case we get

£o 2(A2 + v2/4 + vxT/2)'(4.51)

The absorption of radiation is an incoherent process in this case, nearly independentof the intensity x °f m e incident radiation. It should be noted that, if x ~ 0(1),then v2 > vx^ in Eq. (4.51), and the cross section does not depend on x- Intensitydependence appears only at x ^ 1> when the cross section begins to decrease.

Problem 4.11. Under the conditions of Problem 4.9, estimate the order of magnitudeof the intensity of resonance radiation at which saturation appears, that is, when


Pkk ~ 1- Assume that broadening due to collisions is large as compared to thespontaneous width.

SOLUTION. Saturation corresponds to the case when the population pkk of the excitedstate is of the same order as the population pOo of the ground state. According toEq. (4.42) with v > T, we have

where we have used the relation T ~ v/2, and consequently x — X@T/v). We canestimate the photon flux to be

J = cE2/(87rhco)

for the case pkk — 1. It follows from Eq. (4.52) that then 2^F ~ v2, and

•A) -VCD2

ATTC2'

We can now rewrite Eq. (4.52) in terms of this flux as

= v2(J/Jp)P

It is seen that saturation is achieved on the edges of the spectral profile at much higherphoton fluxes than are required near the center of the line. In fact, we find

Earlier we found that the ratio of the spontaneous lifetime to the transition time inthe external field is

G = (Dk0-E)Tkh.

This parameter can be written in the form

r- l

Since v > T, we then find that G > \ near saturation conditions (7 ~ Jo). Thatis, transitions between levels take place due to induced transitions and absorption ofphotons only.

The rate of induced transitions between levels 0 and k is given by the relation

This expression gives the connection between the quantity ra and the spontaneouslifetime of the excited state rk, as well as with the photon flux / .


Problem 4.12. Obtain Eqs. (4.42) and (4.43) from solutions of the approximate rateequations that take into account only the variations in populations of atomic levels.Discuss the applicability of these equations to the derivation of the coherent scatteringcross section.

SOLUTION. Take wko to be the transition rate between the states 0 and k due to anexternal electric field E. Then we designate by l/rk the spontaneous rate for thetransition k —• 0. The quantity wko is determined by Eq. (1.7), where the 8 functionshould be replaced by the distribution function aM for the emitted photons [seeEq. (2.38)]

fla, = ~ ~ F ~ . (4.55)7TA2+T2

This distribution function takes into account both spontaneous and collision broad-ening of levels. Hence, according to Eq. (1.17), we have

J E)|2^. (4.56)

For the sake of simplicity, we set g0 = 8 k = 1 •The rate equation for the population of level k is of the obvious form

~ Poo)- (4.57)dt T*™

From conservation of the number of particles, it follows that

Pkk + POO = 1-

It should be noted that Eq. (4.55) does not take into account the influence of a strongfield on the distribution function aM. Indeed, the quantity vv o determines transitionrates on short time scales so that an atom cannot transit several times between states0 and k. However, we are interested in obtaining steady-state solutions of equationsover such long time intervals that many transitions between states 0 and k can takeplace. The steady-state solution of Eq. (4.57) has the simple form

1 (4.58)2 + (wkQrk)

l

We have taken into account the conservation of the number of particles. SubstitutingEq. (4.56) into (4.58) and again introducing the notation G = (D^o * E) rk/h, we find

1Pkk =

2+ (A2 + r2)/(G2rr)

This coincides with Eq. (4.42) if the notations T = 1/2T> and \ = 2G2T/T areintroduced again. Hence, Eq. (4.43) based on Eq. (4.42) correctly describes the photonabsorption cross section within the framework of the rate equation method.

RESONANCE TRANSITIONS IN MULTILEVEL ATOMS 197

This means that, for the calculation of the photon absorption cross section, we canuse the simple rate equation method instead of the more cumbersome density matrixmethod. However, the rate equation method does not permit the correct calculationof coherent scattering cross sections for photons since, according to the solution ofProblem 4.8, nondiagonal elements of the density matrix pkQ are required. The rateequation method applies only to diagonal matrix elements of the density matrix.

The solution of Problem 4.10 has shown that coherent scattering is negligiblysmall when v > F, and all scattering is inelastic. Thus, we can conclude that rateequations give correct descriptions of physical processes when collision broadeningis predominant, that is, when v > T. We see in the system of equations (4.40) thatthe condition v > T causes nondiagonal matrix elements p,o to vanish much morequickly than diagonal matrix elements, since we have the behavior p^ ~ exp(— vt)in contrast to p^, poo ~ exp(—Ft).

4.4 RESONANCE TRANSITIONS IN MULTILEVEL ATOMS

The approach of the preceding section is inapplicable if the initial state 0 is nota ground state or if the excited state k can experience spontaneous transitions toother atomic levels. The reason is that the previous approach was based on a con-servation law for the number of particles. Here we consider a situation where thespontaneous transition between resonant levels k and 0 is negligibly small. After theexternal radiation field is turned on, more and more electrons will disappear fromthe initially presumed two-level system. This changes the nature of the analysis. Thefollowing discussion is concerned with this modified physical situation. Problem 4.13is simplified in that it does not contain depletion. It is the so-called classical Rabiproblem.

Problem 4.13. Obtain an expression for the wave function of an atom perturbed bymonochromatic resonance radiation. Relaxation of the levels is assumed not to occur.

SOLUTION. We start from Eq. (1.4) for the amplitudes c0 and ck for an atomic electronwith a lower state 0 and an upper state k available to it. We shall write these equationsin resonance approximation, assuming that the resonance detuning A = cok0 — a> issmall compared to the field frequency co. We thus obtain

decsin—- = - 5 (Do* • E)exp(-/Af)Q, (4.59)

dck 1in— = ~ (D^o * E)exp(/A?)q>

at L

After the substitution

Co = exp(iAt)cQ,


system (4.59) reduces to the system of differential equations with constant coefficients

ih— = \ (DH) • E) c0, (4.60)

Upon eliminating c*, we obtain the second-order equation for CQ,

It has the two linearly independent solutions

c j = B± exp[z QA ± ft) t] . (4.62)

We have introduced here the quantity

which is known as the Rabi frequency.Substituting Eq. (4.62) into the second of the Eqs. (4.60), we obtain results for

ck. The coefficients B± are determined by the initial conditions. We suppose that att = 0 an atom is found in the state 0, so that

co(O) = 1, Q ( 0 ) = 0.

Then the wave function at time t is given as

¥(f) = [cosft^ + (/A/2ft) sinftf] exp(-/Af/2 - isQt/ti) %( r ) (4.63)

- i [(D^o • E) /2M1] sinftr exp(/Ar/2 - iskt/h)

We find that the population of the excited state k is

This oscillates monochromatically in time from zero to the maximum value

(4.64)

(4-65)

with the frequency of oscillation 2ft, that is, with twice the Rabi frequency.If A = 0 (exact resonance), then Eq. (4.64) takes the simpler form

1 f f lPkk(t) A=Q - 11 - cos - (D,o ' E) | . (4.66)

This probability changes periodically from zero to unity and back. Thus the atomicsystem oscillates between being entirely in state 0 and entirely in state k. Such


oscillations of population are explained by the suddenness of the turn on of theperturbation at time t = 0. They decrease with a more gradual turn on of theperturbation, and they vanish for a turn on that is adiabatically slow.

Problem 4.14. Under the conditions of Problem 4.13, take account of the broadeningof states 0 and k due to transitions to other states. Calculate the photon absorptioncross section and the absorption coefficient. Assume that the states have the samelifetime for spontaneous decay.

SOLUTION. We denote by r the spontaneous lifetimes of each of the states. Then,according to the theory of spontaneous radiation considered in Section 2.1, oneshould change the energy of each of the states by the replacement

£o -> so - H (2T) , sk^sk- il (2T) (4.67)

in all formulas of the preceding problem. Since the resonance detuning A and Rabifrequency i l depend only on the difference of energies of the states 0 and k, themodification (4.67) leaves the values of A and ft unaltered.

Thus all of the modification of the wave function (4.63) consists of attaching thefactors exp(-r/2r) to each of the functions ^o(T) and \Pfc(r). Hence, the populationof state k at time t is given by the generalization of Eq. (4.64),

(4.68)

The rate of decay of this state is given by Pkk(t)/r. When we integrate this over theentire time the perturbation is applied, we obtain the total probability of decay of thestate k,

The quantity Wk is also the photon absorption probability. That is, it is equivalent tothe stationary population p^(°°) of the state k found in Problem 4.7.

We now introduce the notations

r = 1/T, G = (Da, • E) r/h, X = G2. (4.70)

With this terminology, Eq. (4.69) becomes

Since, at t = 0, we have one electron in state 0, then the quantity Wo can be foundfrom the obvious conservation law

Wo + Wk = 1. (4.72)


The parameter x defined in Eq. (4.70) is called the saturation parameter. It gives thepopulation of the excited state k as a function of the intensity of the radiation field.

In a very strong field, when x ^ 1»me quantities Wk and WQ achieve their limitingvalues, which is ^ for both of them. In the opposite case, when x ^ 1, we obtain theperturbation theory result

w* = f^Tn- (4-73)

In this case, the photon absorption probability Wk is proportional to the intensity ofthe radiation field. A continuing increase of the field leads to a slowing of the growthof Wk. Finally, at x ^ 1 we obtain saturation, that is, Wk = \.

The close correspondence between Eqs. (4.71) and (4.28) should be noted. Thedifference is that, in Eq. (4.71), both k and 0 states have spontaneous widths, whilein Eq. (4.28) only the excited state k has a spontaneous width.

We shall now calculate the photon absorption cross section. The quantity TWk

determines the rate at which state k loses population. In the steady-state case, thisquantity is also the photon absorption rate. When we divide it by the flux of incidentphotons, J = cE2/(8Trh(x)), we obtain the photon absorption cross section

We have introduced in this expression the statistical weights go and gk of the initialand final states, respectively. The close correspondence of Eqs. (4.74) and (4.31)should be noted.

According to Eqs. (2.87) and (2.88), the absorption coefficient is given by therelation

kw=Naa(W0-Wk), (4.75)

where TV is the density of atoms. Substituting Eq. (4.74) for the absorption crosssection and Eq. (4.71) for Wk into Eq. (4.75), and taking account of Eq. (4.72), wefind

TTC2 gk r 2 ( A 2 + r 2 )

k» = V— ^ ^ . (4.76)w2 go [A2 + r 2 a + *)]

It is seen from Eq. (4.76) that the absorption cross section diminishes as the fieldintensity parameter x increases and tends toward zero for x ^ 1 •

It should be observed that the width F in this problem can be produced, forexample, by the ionization of the excited state by the given radiation field (or otherradiation fields), instead of arising from spontaneous relaxation of the state. In thiscase, the solution of this problem includes the solution of the problem of resonancetwo-photon ionization of an atom in a strong field.

Problem 4.15. Solve the preceding problem when states 0 and k have differentspontaneous lifetimes. Determine how the population of the excited state varies withtime.


SOLUTION. We refer again to the wave function of Eq. (2.63) and make the change

s0 -> s0 - i/ (2T0) , sk->ek- i/ (2T>) . (4.77)

With this modification, the population (4.64) of the state k takes the form

(4.78)

We have introduced here the notation

where r is given by

In addition, the quantity r in Eq. (4.78) is now determined by the relation

(4.80)

The probability for photon absorption through the action of the external field,integrated over all time, is

Wk = - pkk{t)dt. (4.81)Tk JO

Substituting Eq. (4.78) into Eq. (4.81), and carrying out an elementary but cumber-some integration, we find

L ^ v (4'82)w * T A 2 - u r ^ i -2rk A2 + T2(1

The terminology we employ here has the meaning

f + ) . (4.83)T 2 \ )

If the two spontaneous lifetimes are the same, then Eq. (4.82) coincides, as it should,with Eq. (4.71).

When x ^ 1> Eq- (4.82) gives the perturbation theory result

fft;is4- (4-84)As x increases, the growth of Wjt slows, and when x ^ 1, we obtain the saturatedprobability

W* = TIT-?-' (4 '8 5 )

1 + n/T

Therefore we can call the parameter x of Eq. (4.83) the saturation parameter. Theresult (4.85) can be obtained more directly if we make use of the facts that, at x ^ 1>the ratio Wk/Wo is simply TO/T^, and that Wk + WQ = 1 according to Eq. (4.72).

It follows from Eq. (4.83) that saturation can be approached not only in a verystrong field but even in weak fields if the inequalities Tk > T0 or T> < T0 are satisfied,that is, if one of the spontaneous widths is much less than the other.

We now derive the absorption cross section. By proceeding in a fashion parallelto the derivations in the preceding problem, we obtain

^ 4 A 2co2 go rk0 A2

Here the quantity 7>o refers to the partial lifetime of state k with respect to state 0.In conclusion, we wish to establish the time dependence of pkk(t), the population

of the state k. For simplicity, we restrict ourselves to the case of zero resonancedetuning, A = 0. It follows from Eq. (4.78) that for moderate fields, where Eq. (4.79)shows that

X- |(DW • E)| < JL, (4.87)

then the quantity II takes on imaginary values. The population pkk(t) is then of theform

pkk(t) ~ sinh2(|fi|i) exp(-r/T), (4.88)

that is, the population of state k decreases aperiodically. If, on the other hand, theinequality is reversed, so that

\ |(D«, • E)| > -Lthen we have from Eq. (4.78)

pkk(t) ~ sin2(|n|r) exp(-f/T),

and the population of the state oscillates with a decaying amplitude. Finally, in theintermediate case

E)| ,

the population of the state behaves as

pkk(t) - r2exp(-r/r) .

We have emphasized in the foregoing problem that the results are applicable whenthe broadening of the states is produced by mechanisms other than spontaneousrelaxation. In particular, they can be used for the solution of the problem of resonanceionization of an atom in a strong field. Then the quantity rk~

l will be the ionizationrate of the excited resonance state k.

As in the preceding problem, we can also derive the absorption coefficient.

5ATOMS IN STRONG FIELDS

The physical phenomena that can appear under strong-field circumstances are oftenqualitatively different from those familiar in the perturbative domain. This subjecthas recently received so much theoretical and experimental attention that it has nowbecome a recognized independent area of research within the larger field of atomicphysics. Although this chapter is not intended to present a complete treatment ofstrong-field atomic physics, a modern discussion of radiative processes in atomswould not be complete without an introduction to intense-field phenomena.

When the interaction energy of an atom with an externally applied field is suffi-ciently large, then a perturbation expansion to describe this interaction will not beconvergent. It is thus necessary to seek some method other than perturbation theory todescribe the behavior of the atom in the field. Some problems employing nonperturba-tive methods have appeared earlier in this book, but the great majority of the problemsare solved using perturbation theory, as has been true in atomic physics until veryrecently. We shall give here a brief survey of the physical environments that demandintense-field treatment and of some of the methods adapted to the strong-field regimein atomic physics. The emphasis here will be on methods that attain an analyticalsimplicity through the restriction that field strengths are well beyond the perturbationtheory limits. This is in contrast to perturbation theory, which achieves simplicitythrough the assumption of small field intensity. The intensity domain that marks thetransition between perturbation methods and strong-field methods is characterizedby considerable analytical complexity, and will not be treated here.

Radiative Processes in Atomic Physics. V. P. Krainov, H. R. Reiss, B. M. Smirnov ^03Copyright © 1997 by John Wiley & Sons, Inc.ISBN: 0-471-12533-4

204 ATOMS IN STRONG FIELDS

5.1 PROPERTIES USEFUL FOR STRONG-FIELD METHODS

We shall begin this chapter with some basic results that will set the stage for otherproblems exploring the methods of strong-field physics and their physical conse-quences. The problems in this section have universal applicability, but they areespecially relevant to the strong-field domain.

Problem 5.1. Find general criteria to judge when the limits of perturbation theorywill be exceeded.

SOLUTION. Several criteria for the applicability of perturbation theory can be statedon very general grounds. The violation of any one of these restrictions is adequategrounds for the conclusion that perturbation theory gives incorrect predictions for thestrong-field environment.

A perturbation theory of atomic phenomena is normally couched in terms of aninteraction Hamiltonian ///, whose magnitude is small as compared to that Hamilto-nian Ho that describes the unperturbed system. In the case of an atom subjected toa strong field, Ho includes the effects of the atomic binding potential, and /// is theadditional part of the total Hamiltonian arising from the externally applied electro-magnetic field. To associate a magnitude with the ratio |/// | / | Ho\, we may presumethat the transition caused by Ht is ionization of an atom from its ground state, so thatthe transition energy is of the order of £;, the ionization potential of the atom. Wetake the interaction Hamiltonian to be of the form

2/// = - — A - p + - ^ A2, (5.1)me 2mcz

where A is the vector potential of the externally applied field (which we shall, forconvenience, often refer to as the "laser field"). To assign a magnitude to the firstterm in Eq. (5.1), we replace the p operator by the commutator

£ = >'r]' (5-2)which will be valid when Hp contains the kinetic energy operator and potentials thatdepend only on spatial coordinates. The magnitude of the A • p term in Eq. (5.1) isthen eAoEiRo/ (he), where Ao is the amplitude of A, and Ro is a radius typical of thebound system. Furthermore, if we take the magnitude of Hp to be Ei9 and replace theradius Rp by the general expression Rp ~ h/ (mEt)l//2, then we find the ratio of theA • p term to the magnitude of Hp to be

1

\Ho\e— A p

me

eE0R0R0

he hco(5.3)

where the last result in this equation follows from the replacement of the vectorpotential Ap of the laser field of frequency co by Epc/(o, where Ep is the amplitude ofthe electric field. The quantity in Eq. (5.3) appears in the literature in two differentguises. The inverse of this quantity is widely known as the Keldysh parameter y,

PROPERTIES USEFUL FOR STRONG-FIELD METHODS 205

where

ho)y = -=rir- (5-4)

eERThe Keldysh parameter is also known as the adiabaticity parameter. If the quantityin Eq. (5.3) is squared, and the generic replacement is made that R^ = ti2/ (mEj),then we obtain the intensity parameter often designated z\, where

2UPz\ = -rr-> (5.5)

and Up is the ponderomotive potential

The angle bracket in Eq. (5.6) refers to a time average over a period of the applied laserfield. The name bound-state intensity parameter is sometimes applied to z\, since itrelates an energy characterizing the electromagnetic field to an energy characterizinga bound state.

Other implications of the significance of z\ come from a consideration of the"power broadening" of atomic energy levels, that is, the broadening due to theshortened lifetime of a level in the presence of a strong field. This broadening isapproximately |eE • r| ~ eEao, where a$ is the Bohr radius. When this broadeningapproaches the energy of a single photon of the field, it then becomes indefinite as tothe number of photons required to connect that level to others. This is nonperturbativebehavior, and so perturbation theory requires eEa$ < hco. If we square the ratio ofeEao to ho), and replace a\ by h2 / (2m/?oo), (where Roo is the Rydberg constant, equalto the binding energy of hydrogen), we find that we have (eEao/hco)2 =« z\ < 1.

Both from the need to have the interaction Hamiltonian of smaller magnitude thanthe unperturbed Hamiltonian, and to have the power broadening of atomic energylevels smaller than a photon energy, we conclude that a necessary (but not sufficient)condition for perturbation theory to be valid is that

y > 1 or z\ < 1. (5.7)

The ponderomotive potential of Eq. (5.6) is a quantity that appears frequently instrong-field atomic problems. It can be viewed as the minimum kinetic energy a freeelectron possesses as a result of its oscillations in a laser field, or equivalently, as thepotential energy of a free electron due to its interaction with the field.

Another intensity parameter that measures the need for intense-field treatment is

* — p — e /V2\ a Snfico 2mna)5 x '

The transition of an ionized electron to a detached state in which the motion of theelectron in the laser field is fully developed (as opposed to the case in which the elec-tron emerges into a simple continuum Coulomb state) is an explicitly nonperturbative

effect. That is, no small number of photons can place an ionized electron into thestate of oscillatory motion characteristic of classical electron motion in the externalfield when the energy of that motion is of the order of the photon energy. Perturbativetreatment of ionization thus requires the condition

z< (5.9)

The quantity z has been called the continuum-state intensity parameter to contrast itwith the bound-state intensity parameter z\. It has also been called the nonperturbativeintensity parameter, since, as will be seen shortly, it is usually the factor that is criticalin determining the limits of perturbation theory.

Yet another limitation on the application of perturbation theory comes from therequirement that the internal electric field in the atom should exceed the externallyapplied electric field. That is, in atomic units, the external electric field should beless than unity. In Gaussian units, this means that Eo < 2Roo/ (eao), where R^ isthe Rydberg unit of energy, and a0 is the Bohr radius. This yields the upper limit ofperturbation theory as

<m2e5

h4 or? 5

me

or(5.10)

where the value of Eo is interpreted as a root mean square for purposes of the limit onthe ponderomotive potential. Expressed in atomic units h = m = e2 = 1, Eq. (5.10)becomes EQ < 1.

The upper bounds on field intensity in Eqs. (5.7), (5.9), and (5.10) are all necessary,but not sufficient, conditions for the convergence of perturbation theory. At the veryleast, all of these conditions must be satisfied simultaneously. Each of these conditionshas a field frequency dependence different from the others. The resulting limitationas a function of frequency is illustrated in Figure 5.1. The range of laser frequencies

101 i

= 5

6

cCD

100]

1 0-2 i

1 0-3 i

1 0-4

1 0-5

1 0-6

Jo=L

perturbation domain

0.00 0.05 0.10 0.15 0.20 0.25Frequency (a.u.)

Figure 5.1. Limits on perturbation theory as a function of frequency. The vertical lines cor-responding to wavelengths of 10.6 and 0.248 i±m encompass the full range of frequencies forwhich intense-field lasers are available. Perturbation theory is valid below the curve at z = 0.1,labeled "Pert, lim."


for which nonperturbative intense fields can be attained extends from about 1 /xm to248 nm (excimer laser). As shown in the figure, it is the intensity parameter z thatmeasures the effective limit of perturbation theory. This conclusion remains true evenif the wavelength is extended to 10 jum, representing the CO2 laser.

Problem 5.2. Suppose that a particle of mass mx and charge q\ is bound by acentral potential to a particle of mass m2 and charge g2- Show how the two-particle Schrodinger equation separates into a center-of-mass equation and a relative-coordinate equation when an external electromagnetic field is present. This field isdescribed by a vector potential A(7) and/or a scalar potential of the form —qr • E(r),where E(t) is the electric field vector.

SOLUTION. The procedure we follow is the same as that normally employed to showhow the two-particle Schrodinger equation separates, except that we include thepresence of external fields. This will lead to the appearance of a reduced charge inaddition to the familiar reduced mass.

The new coordinates R and r for the location of the center of mass and for therelative coordinate are given in terms of the coordinates of the two separate particlesby

r = n-r2, (5.11)m\ + ra2

with inverse transformations

r i = R + _ ^ r , Vl = R - _ ^ r . (5.12)m\ + ra2 m\ + ra2

It is useful to define symbols for the total mass mt and total charge qt,

mt = mi + ra2, qt = q\ + #2- (5.13)

The gradient operators in the R, r coordinates are related to those in the r^ r2

coordinates by

V l = V r + ^ V * , V2 = - \ r + ^VR. (5.14)mt mt

Direct substitution of the transformation Eqs. (5.11)—(5.14) into the two-bodySchrodinger equation

I —ihV2 ~ — A(Ol ~ ^ r 2 * E(0 [> dib(rh r2, 02m2

(5.15)

leads to the new Schrodinger equation

M) = f 1 r _ « rf _[ 2ra, L c J

- ^r • E(0 + V(r) W R , r, 0-J+2mr J

(5.16)

In addition to the total mass mt and the total charge qt, Eq. (5.16) now contains alsothe reduced mass and reduced charge

mr = , qr = . (5.17)

Equation (5.16) is plainly separable. We introduce the product solution

i|f(Rr,0 = W R , 0 W r , 0 (5.18)

into Eq. (5.16), and divide by if/ in the form of Eq. (5.18) to obtain

= —— < ih—- — f — ihVr — — A j — qrx • E + V(r) i//r > . (5.19)

The left side of this equation is a function of R, t only. The right side is a function ofr, t only. Hence each side of Eq. (5.19) must be independent of either R or r and canbe set equal to some arbitrary function of time, f(t). Our final result is then the twoseparated equations:

ih = -— ( —iftVn — —A) — qtH • E + f(t)\ ipR (5.20)dt [2mt \ c J J

for the center-of-mass equation and

dt [2mr V r c ) J

for the relative-coordinate equation.These equations have a number of interesting features. The cm. (center-of-mass)

equation depends entirely on the total mass and charge, as one would expect. If weare concerned with a system such as a neutral atom, we have qt = 0, and Eq. (5.20)reduces to the free-particle equation

(2mf

For the neutral atom considered as a two-body system, we can set q\ equal to theelectron charge, q\ — —e, and so the nuclear charge is q2 — e. Equation (5.17)

for the reduced charge then gives identically qr — — e. If, however, we consider apositive ion like He+, the single electron again has q\ — — e, but the nuclear chargeis q2

= 2e, so the reduced charge is

(2m\ + ra2\qr = -e [ , (5.23)

V mi + m2 )

and the electron in the ion has a charge slightly modified from the usual —e. For anegative ion such as H~, considering the two-body problem to consist of a neutralcore atom and a single electron, the reduced charge will be

^ - ) • (5-24)

Ions will thus exhibit reduced charge effects in addition to reduced mass effects.There will be, for example, an isotopic effect on the apparent electron charge as wellas on the apparent electron mass.

The arbitrary function f(t) in Eqs. (5.20) and (5.21) may be viewed as the generat-ing function for a gauge change corresponding to an alteration of the scalar potential.A change in electromagnetic gauge accomplished by the generating function x(t)will not affect the vector potential but will change the scalar potential (/> according tothe prescription

\jt (5.25)

When changing gauge, the wave function has its phase altered such that

If we now associate x(t) with the f(t) in Eqs. (5.20) and (5.21), we can set

<fe = exp[i/(0/ft] ^ , </v = exp[-//(0A] «#, (5.26)

so that ifjRifjr = ^ i / / / . Equations (5.20) and (5.21) then become

dt \_2mt V c J J^ ) (5.27)dt \_2mt V c J J

and

^ \ ( Y l r- (5.28)

Problem 5.3. Repeat Problem 5.2 when the field is described by a vector potentialonly, and this potential has the specific form of a plane wave, where A = a cos(cot —k • r). That is, the dipole approximation is not used.

SOLUTION. It will be found that the two-particle Schrodinger equation does notseparate into uncoupled cm. and relative-coordinate equations, unlike Problem 5.2.

It will be instructive to examine the nature and extent of the approximation made inpresuming that separation does occur.

The Schrodinger equation for the two particles labeled by subscripts 1 and 2 is

= J i r _ £ . A ( r i > o i + V ( | _ r 2 | )c J

i, r2, t), (5.29)

which differs from Eq. (5.15) in that the Coulomb gauge is specified (so there isno scalar potential associated with the laser field), and there is no longer a dipoleapproximation employed for the vector potential of the electromagnetic field. Someassumptions must be introduced in Eq. (5.29) in order to proceed further.

The vector potential will be taken to have the explicit form

A = a cos (a>t — k • r) ,

and it will be assumed that |k • r| < 1 and |k • R| « 0. The change of variablesexpressed in Eqs. (5.11)—(5.14) puts Eq. (5.29) in the form

. , # ( R , r ( 0 f 1 r „ _ a a, -|2in = < — mvR — qt- cos cot — a r - k • r sin con

dt 12mt L c c \

1 r a a ~\ ^+ —ihVr — qr- cos cot — qe-k • r sin con

2mr L c c J

In addition to the total mass and charge of Eq. (5.23) and the reduced mass and chargeof Eq. (5.27), there is now also an effective charge qe given by

qe = — (q\mj + q2m\). (5.31)mt

It is plain that Eq. (5.30) cannot be separated. The term qr (a/c) (k • r)sinatfcontained in the same square bracket as -ihVR makes this impossible. We canestimate the importance of this term by comparing its energy to that associated withthe atom, which we take to be the atomic unit of energy, which is 2Roo. The notationis that Roo is the Rydberg unit, which is approximately 13.6eV. That is, we wish toestimate the dimensionless ratio

1 1

IRoc 2mt

-1c

Since |k| = co/c, and |r| ~ a0 = Bohr radius, we find that

1 1(5.32)


where E = aco/c is the amplitude of the electric field associated with the vectorpotential of amplitude a. From Eq. (5.32), we see that the coupling between cm. andrelative motions in the two-body system is proportional to the intensity of the laserfield. If we suppose that the laser is focused to an energy flux of 3.5 X 1016 W/cm2,this corresponds to the applied electric field equal to the internal Coulomb field ina hydrogen atom. Equation (5.32) is then % ~ Roo/mtc

2. For a hydrogen atom, thisis R ~ 10~8, and can be discarded as negligible. However, laser energy fluxes ofas much as 3 X 1021 W/cm2 are anticipated for the near future, which would make% ~ 10~3, which is still small but might be observable. With laser energy fluxes thishigh, the present nonrelativistic treatment is inadequate. Nevertheless, one would stillexpect this coupling of cm. and relative coordinates to occur at these extreme laserintensities.

Finally, we note the term qe (a/c) (k • r) sin a)t contained in the same squarebracket as —ihVr in Eq. (5.30). This is of the form to be expected as a first correctionto the leading dipole approximation term. The only unusual feature is that it occurswith the effective charge qe rather than the reduced charge qr. As long as one ofthe masses in the two-body system is of the order of a nuclear mass, with theother being the electron mass, then qe ~ qr ~ —e, and the difference in charge isunimportant.

Problem 5.4. Consider an atom subjected to an interaction Hamiltonian that is peri-odic in time. That is, the total Hamiltonian is of the form //(r, t) = Ho (r) + V (r, t),and V is periodic with period T, so that V (r, t + T) = V(r, t). Show that the com-plete set of solutions of the Schrodinger equation with such a Hamiltonian may bedivided into subsets of solutions, each of which is physically equivalent to all theothers. This is called the Floquet property.

SOLUTION. The Schrodinger equation is

ih—V (r, t) = H (r, t) V (r, t) = \H0 (r) + V (r, t)] * (r, t). (5.33)dt L J

From general symmetry principles, if H is a periodic function of time,

T) = H(t), (5.34)

one should be able to select a set of solutions of the Schrodinger equation that are alsoperiodic, with the same period T. We associate a frequency a> with this periodicityin the usual fashion, with o) — 2TT/T. Periodicity demands only that the value of thewave function be repeated after a period to within an arbitrary constant phase, so werequire

We (r, t + T) = exp (-ieT/h) ^ e (r, 0 , (5.35)

where e is a real parameter defined so as to have units of energy. It is called the quasienergy.

We can introduce functions <& (r, t), which are exactly periodic, by setting

Ve (r, t) = exp (-iet/h) $ e (r, t), <S>e (r, t + T) = $ e (r, t). (5.36)

Suppose now that we have states of different quasi energies e\ and €2- Hermiticityof the Hamiltonian requires that the inner product of any two states (^e,, M^) beindependent of time. Therefore, if e2 - e\ =£ nha), where n is an integer, then thestates must be orthogonal at any given time, 0P6l (0, ^e2(0) = 0. We also concludethat the quasi energy is defined only up to an additive constant nhco.

The conclusion from the above is that states <J>e) (t) and <f>€2 (t), which differ onlyin that 62 — €\ ± nha>, are physically indistinguishable. Nevertheless, they representdifferent states in the complete set of ^ e (t) states.

In particular, in an external monochromatic electromagnetic field, the wave func-tion can be represented in the form

* = exp (-i-r) ]T] exp (-mart) </>,,. (5.37)

Determination of the expansion coefficients <f>n constitutes the content of the Floquetmethod for the solution of strong-field problems in atomic physics.

Problem 5.5. Consider a system bound by a time-independent potential and sub-jected to a laser field that may be treated in the dipole approximation. Find a unitarytransformation to a moving coordinate system such that the transformation removesthe A • p and A2 terms from the Schrodinger equation. The electromagnetic field afterthe transformation will be manifested as a field-induced modification in the centralpotential term.

SOLUTION. The Schrodinger equation is

ifij^ (r, 0 = { ^ [-iftV - e-A (o] + V(r)\ 9 (r, t)

(5.38)

where the unperturbed Hamiltonian //0 and the interaction Hamiltonian Hj are givenby

^ f (5.39)

W () ^ L (5.40)c m 2mcl

Because the vector potential A is a function only of the time, the term e2A2/ (2mc2)can be removed by an energy shift operator we shall designate by CIE- The otherterm in /// contains the momentum operator —ihV multiplied by a quantity thatis a function of time only. This suggests removal of this term by the translation

operator flr. That is, we set

— J A(T)dT\.

Since fl^ commutes with all the terms on the right-hand side of Eq. (5.38), substitutionof Eq. (5.41) into Eq. (5.38) yields the equation for $ ,

^J ^ (T,t) + £i;lV (r)nr<& (r,t). (5.42)dt 2m

To simplify Eq. (5.42), we define

a = -— f A(T) JT , (5.43)me J^

so that

ftr = e x p ( - a - V). (5.44)

We now employ the theorem, often called the Baker-Hausdorff theorem, that

eBCe-B = c + [B C] + i_ ^ [ f t C]] + i_ [A [ f t [B C]]j + (5 4 5 )

where the square bracket indicates the commutator of the generally noncommutingquantities B and C. We associate B with H"1 and C with V (r), so that Eqs. (5.43)and (5.44) give

a;lV(r)ftr = V(r) + a • V + 1 (a • V)2 V(r) + ^ (a • V)3 V(r) + . . .

= V(r + a ) . (5.46)

Finally, the equation satisfied by <£ is

/ft—^ (r, t) = — (-ihV)2 + V(r + a) $(r , r) . (5.47)^ L ^ m J

The transformation given by Eq. (5.41), leading to the result (5.47), is generallycalled the Kramers-Henneberger transformation. It has been known since the earlydays of quantum mechanics.

To interpret the result physically, we observe that the a quantity defined inEq. (5.43) satisfies the equation

^ a = ~-^A(0 = eE(t), (5.48)dtL c at


where E (t) is the electric field. This is the classical equation of motion for thedisplacement a of a free electron from its position of equilibrium. Thus, V [r + a (t)]is an expression of the potential experienced by the electron as measured in a systemof coordinates oscillating with the electron, and O r gives the transformation to thatsystem. The quantity

e2A2

U ( 5 - 4 9 )

is the energy of the oscillatory motion of a free electron in an electromagnetic fielddefined by the vector potential A. That is, it is the potential energy of the motion ofa charged particle in this field. The transformation ftE displaces the energy in theproblem by the amount UE- The time average of UE over a period of the field gives theponderomotive energy Up of the electron in the field, encountered above in Eq. (5.6).That is, we have the connection

e2 (A2)

Only for the special case of circularly polarized laser light do we have Up = UE-

Problem 5.6. Find an exact solution (within the nonrelativistic dipole approxima-tion) for the motion of a free, charged particle interacting with an electromagneticfield.

SOLUTION. The solution can be found directly from the results of the precedingproblem. For a free particle in a laser field, the equation of motion is just Eq. (5.38)without the V(r) term. The transformation (5.41) leads to, from Eq. (5.47), simply

ih—* (r, 0 = T - (-ifrff $ (r, r),dt 2m

which has the solution

[ i ( ^ ) ] (5.51)

where C is a normalization constant. This free-particle solution is an eigenfunctionof the operator fl r, and so the Kramers-Henneberger transformation (5.41) appliedto Eq. (5.51) gives the required solution

(5.52)

Equation (5.52) is known as the Volkov solution, or the Gordon-Volkov solution.Though derived here for the case of the nonrelativistic dipole approximation case,the exact solution can also be stated for a relativistic Dirac or Klein-Gordon particle.

Problem 5.7. Devise a general system of calculating quantum transition rates, suitedto a nonperturbative environment associated with very intense laser pulses. A charac-teristic of such problems is that the laser field (i.e., the "perturbing" field that causesthe transitions) is present only during a short period of time. The system is initiallyprepared, and final measurements are made, in space and/or time domains in whichthe laser field is absent. The calculational scheme should be such that there is no needto rely on sharp turn-on or turn-off of the field, nor need there be any resort to "adi-abatic decoupling" schemes, ^-matrix methods, originally introduced in connectionwith scattering problems, are particularly adapted to this purpose.

SOLUTION. The initial and final states of the atomic system are states in whichno electromagnetic interaction exists. These states form the complete set {<&„} ofsolutions of the Schrodinger equation

ihd& = #0*, (5.53)

where Ho is given in Eq. (5.39), and the shorthand notation is introduced that dt =d/dt. The full Schrodinger equation containing all external field effects is

ihdiV = (H0 + HI)'V, (5.54)

where /// is the interaction Hamiltonian given in Eq. (5.40), and it is understood thatthe vector potential A satisfies the asymptotic condition that

lim A(0 = 0, => lim Hi = 0. (5.55)t±oo t±oot

We remark that the way in which this limit occurs is unimportant for the formulation,and also that the adjective "asymptotic" simply means before the laser pulse is onand after it is off. Thus t —> ±o° might refer to femtoseconds in some modern lasersystems.

The fully interacting atomic system is prepared initially in a well-defined nonin-teracting state <&,-, so that

lim ^ + ) = * f , (5.56)t

where the superscript (+) designates a state that satisfies the condition (5.56). Theresults of an experiment are evaluated by examining the end products, which is tosay that one finds the relative amplitude that the fully interacting state ^ / is to befound in a noninteracting state O/. This gives a transition amplitude that, to followthe convention established in the scattering community, will be called an S matrix.The probability amplitude that the state starting as 4>/ ends as O/ is

Sfi= lun^ff^+)). (5.57)

The two statements (5.56) and (5.57) can be combined as


(S - = Sf - 8 , -

The equations of motion (5.53) and (5.54), when employed in Eq. (5.58), give theresult

(S - (5.59)

The equations of motion in quantum mechanics are invariant under time reversal, soa fully equivalent transition amplitude comes from inquiring of the relative probabilityamplitude that a well-defined final state O/ can arise from some initial state. That is,Eqs. (5.56) and (5.57) are replaced by

lim

Sfi= ijm

These equations lead to

(5.60)

(5.61)

(5.62)

which is fully equivalent to Eq. (5.59).Equations (5.59) and (5.62) are exact as long as all their component quantities

are exactly stated. Since one normally does not know an exact expression for ^ + )

or ^f~\ some approximation must be employed for this purpose, as will be seen inlater problems.

Despite the use of the historically motivated terminology of calling the above tran-sition amplitudes S matrices, they are in no sense confined to free-free (or scattering)problems. Either the initial or final state can be either free or bound.

5.2 QUALITATIVE STRONG-FIELD PROPERTIES

There have been essentially five means of treating the response of an atom to theapplication of a laser field too strong to permit a perturbative treatment. One methodis purely numerical. The Schrodinger equation is solved directly on a computer. Thismethod has yielded many useful results, but it is inevitably limited by computer ca-pacity despite major recent advances in the speed of computers. For example, it hasnot been found possible to cope with low-frequency problems (even the commonlyused laser wavelength of 1.06 jim presents serious difficulties), only limited infor-mation can be gleaned about photoelectron spectra, the so-called stabilization regime

QUALITATIVE STRONG-FIELD PROPERTIES 217

can be treated only for very high frequencies, and there seems to be little realistichope for the treatment of problems in the relativistic domain. The main focus of thisbook is analytical, and so no further mention of purely numerical methods will bemade.

Methods based on the Floquet property have been developed, but they have provento be at least as computer-intensive as direct numerical treatment of the equation ofmotion. A third approach is the "high-frequency approximation," which followsfrom approximating the atomic potential V(r + a) [see Eq. (5.47)] by the first termin its Fourier expansion. This limits the method to high frequencies well beyondthe presently available range of laser frequencies. This leaves two methods whichare analytical and with a wide domain of applicability: tunneling methods and thestrong-field approximation, or SFA.

The SFA is similar to one of the tunneling methods, known as the Keldysh ap-proximation. Both the SFA and the Keldysh method use Eq. (5.62) as a startingpoint, the difference being that the SFA uses (in atomic units) the interaction Hamil-tonian Hj = —A • p /c + A 2 / (2c2), while the Keldysh approximation employs thegauge where /// = — E • r. Exact gauge invariance does not hold in these strong-field approximations, and so the two results are not identical. The analytical formof the transition amplitude and transition rate turns out to be significantly simplerin the SFA. Furthermore, the choice of — A • p /c + A 2 / (2c2) for the interactionHamiltonian blends smoothly into the corresponding relativistic theory.

The property of continuous connection to a relativistic treatment is an importantadvantage in a strong-field theory, since for sufficiently strong fields the quivermotion of a free electron in the laser field acquires relativistic velocities, and thenonrelativistic theory becomes inadequate purely as a result of high field intensity.Another way to view this physical situation is to note that a free electron in alinearly polarized plane-wave electromagnetic field acquires a figure-8 motion withthe long axis of the figure along the polarization direction and the short axis alongthe direction of propagation of the laser field. When the amplitude of this figure-8motion in the direction of the field propagation approaches the size of the atom,then the dipole approximation is no longer valid. This occurs at field intensitieswell below those found in Problem 5.3 to impede the separation of the Schrodingerequation into cm. and relative coordinates. The impossibility of using the dipoleapproximation represents a problem most easily solved by employing a relativistictheory. The connection of a theory with Hj = — E • r to non-dipole-approximationand/or relativistic theories is very difficult to establish.

The analytical difficulty of the Keldysh method has confined its application to thelimiting case of large photon numbers only. This leads to tunneling results. Since thetunneling case can also be obtained as a limit of much more general SFA results,it is the SFA that will be treated here. That is, the last two of the five theoreticalapproaches to strong-field theories as listed above are both accessible from the SFA.

To illustrate the principal properties of strong-field results with the minimum ofcomplexity, we shall examine one-dimensional problems.

As is the general practice in strong-field work, atomic units shall be used in theremainder of this chapter.


Problem 5.8. Find the general expression for the ionization rate of a one-dimensional"atom" when the interaction Hamiltonian is Hi — —pA/c + A2/(2c2) and the laservector potential A is so intense that the ponderomotive energy of the ionized electronin the laser field is much larger than the initial field-free atomic binding energy.

SOLUTION. The transition amplitude is given either by Eq. (5.59) or (5.62). The state^^ + ) required for Eq. (5.59) is, however, very difficult to approximate when the laserfield is strong. Although strong fields suggest an approximation in which we takethe laser field to be more important than the atomic binding potential, it is neverappropriate to simply neglect the effect of the atomic potential in the initial boundstate H?j+\ No laser field can be so strong that it can be dominant over the effect ofthe atomic field in the immediate vicinity of the center of the binding potential. Yet,by hypothesis, the laser electric field is so strong that it can be larger than the atomicfield at a Bohr radius from the atomic center, and so we cannot presume the laserfield to be dominated by the atomic field throughout the atom.

The above dilemma does not arise for the 1Pi~) state required for Eq. (5.62). Sincethe electron in the state MS~* is ionized, it is consistent to assume that a sufficientlystrong laser field will so dominate the effect of the atomic potential on the electronin the continuum that we can neglect the atomic potential altogether in the ionizedstate. The state Wi^ will then be replaced by the Volkov state ^ ~ ) v , which followsfrom Eq. (5.52). As written, Eq. (5.52) refers to a ^ + ) V state. The correspondingexpression for a one-dimensional ty(

f~)V state is, in atomic units,

where the normalization constant C of Eq. (5.52) is replaced by L 1/2, where L is thelength of a "box normalization" volume as often employed for continuum electronstates. We shall introduce a monochromatic field given by the one-dimensional vectorpotential

A = a cos cot (5.64)

associated with the electric field

E — EQ sin cot', EQ — aco/c. (5.65)

In this case, the Volkov solution is simply

Mfv = ~Yn e x P U [px - —t - z(ot + £ sin cot - - sin 2cot) , (5.66)L I \ 2 2 ) \

where

% (5.67)coc col 4cocz Aco5

The quantity z in Eq. (5.67) is the one-dimensional version of the intensity parameterz introduced in Eq. (5.8).

With the interaction Hamiltonian

H, = i-Y + f2' (5.68)c dx 2cl

and with Eq. (5.63) replacing ^{f\ the transition amplitude (5.62) becomes thestrong-field approximation, which we shall identify by a superscript SFA. We notethat the Volkov solution in Eq. (5.66) is an eigenvalue of the /// operator of Eq. (5.68).It is thus permissible to employ Hj with the -i(d/dx) operator replaced by the scalarquantity p.

The combination of Eqs. (5.62), (5.64), (5.66), (5.67), and (5.68) gives the transi-tion amplitude

(S - 1 )fk = -i f dt (¥ / (x, t), H& (x, 0)J -oo

= - ^ (eipx, (-cof coscof + 2cozcos2 a>f)<fc(*))

X / dtexp\i(?-t + zat- fat + | sin2atf ) eiE*'. (5.69)

We have used the usual stationary-state relation

cDy (x, 0 = (f>i (x) e~mit = $ (x) eiE*\ (5.70)

recognizing that the initial state is bound, and so

EB = l^l - -Eh (5.71)

We identify the Fourier transform of the spatial part of the initial state wave function(i.e., the momentum space wave function) as

UP) = (eip\ 4>, (x)) = I e~i'aUx)dx. (5.72)J -oo

Equation (5.69) becomes

X dt exp \i I — + z(o + EB \t — i£ sin cot + i - sin 2cot

X (-<o£cos cot + OJZ + (ozcoswt). (5.73)

The integral over time in Eq. (5.73) suggests an integration by parts. When this isdone, the surface terms ait = ±oo are simple harmonic terms with phases linear in t.They are thus representations of the zero distribution (generalized function) and can

be set to zero. We are left with

(S - 1)S/A = 7T7*<fc(p) [Zr+EB) / A exp i V + zco + EB )t

- iC sin at + £ sin 2a>t . (5.74)

This quantity can be expressed in terms of a generalized Bessel function, whoseprincipal properties are summarized in Appendix J. The generalized Bessel functioncan be defined as

Jn (M, V) = — / dd exp [i (u sin 6 + t; sin 20 - «0)]. (5.75)^ ^ J-7T

It enters into Eq. (5.74) because of the generating function for Jn (w, v\ which is

exp[—/(wsinjc + v sin2x)] = V^ 7W (w, iy)exp(—mx). (5.76)

Equation (5.74) then becomes

xr \ fp2 \ l/ dt exp / I — + zo) + EB - «co 1 r ,

J-oo L V / Jwhich yields the delta-function expression

o • °° / 2 \

(5 - 1 )fA = j ^ Yl J» (£ - \ ) * ( P ) f y + EB ) 8 & ~ no)^' (5-77>

in which we have introduced the definitions

T = P- + Za> + EB (5.78)

and

f _i_ |7 /, J\ /c newno = \Z + tB/o)j , (5.79)

where the brace { } in Eq. (5.79) means the smallest integer containing the quantitywithin the brace.

The delta function in Eq. (5.77) conveys important physical information. Fromthe meaning of T given in Eq. (5.78), 8(1 - nto) signifies that n photons fromthe laser field supply the atomic binding energy EB, the kinetic energy p2/2 of theemitted photoelectron, and the ponderomotive energy zo> = Up of the free electron ininteraction with the laser field. This last requirement is an important feature uniqueto strong-field physics. It does not arise in perturbation theories.


The delta function also leads to the imposition of a minimum order no in the sumover n, to make it possible to achieve a zero in the argument of the delta function. Thefact that the sum over n continues to larger values means that the formalism accountsfor contributions of n0, n0 + 1, n0 + 2,.,.. photons. We shall see that contributionsto the transition amplitude by greater than the minimum number of photons canbe very important, in sharp contrast to perturbation theory, where only the lowestpossible photon order is of any significance. The phenomenon where more than nophotons contributes to the ionization process has come to be called above-thresholdionization, or ATI. Furthermore, we see from Eq. (5.79) that the minimum numberof photons required for ionization will index upward as the value of the intensityparameter z increases. This occurs because the laser field must supply enough energyto sustain the field-induced quiver energy of the detached electron in the presence ofthe laser field. Each time the value of n0 indexes upward to the next integer becauseof an increase in field intensity, it is referred to as a channel closing.

To form a transition rate from the transition amplitude (S - 1 )^FA, we must eval-uate

| f A 2 (5.80)w= l imt->oo

In doing this, a square of the delta function arises, which can be viewed as

[8 (T - nwj\2 = 8 (T - no)) 8 (0). (5.81)

The delta function of zero argument can be replaced by the limiting form of theintegral representation of the delta function,

1 ft/2 16(0) = lim — / exp(/0)A = lim —-t, (5.82)

f*°° 2TT J-t/2 '*00 2TT

1 ft/2

— /2TT J-t/2

which yields

(5.83)n=n0

The total differential transition rate is then found by integration over the phase spaceavailable to the emerging photoelectrons, which, in this one-dimensional problem,gives

dW

- / * % •(5.84)

where the 2TT in the denominator is the volume (in atomic units) of a unit cell inthe phase space. Integration over solid angle in one dimension is simply a matter ofsumming over backward and forward directions, but the general notation dW/d£l isemployed in Eq. (5.84) to guide the three-dimensional procedure.

The delta function in Eq. (5.83) will be used to accomplish the integral in Eq. (5.84).In terms of the quantities

E = a) (n - z - eB); eB = EB/w, (5.85)

we can transform the delta function by the following steps:

" < 2 E ) " 2 ] < 5 J 6 )

The two terms in Eq. (5.86) represent the two possible directions of emission inone dimension. Each of the factors </>;(/?) and Jn(Eop/(o2, — z/2) in Eq. (5.83) hasdefinite parity with respect to the replacement p —• -p. It follows that each of thetwo emission directions gives an equal contribution to the differential transition ratein Eq. (5.83), so the result for the total transition rate is

(5.87)

with E = o) (n - z - eB) as given in Eq. (5.85).An alternative expression for Eq. (5.87) is to substitute the explicit value for p as

given by the definition of E to obtain

(5.88)

Problem 5.9. Find the low-intensity, single-photon limit of the SFA, and comparewith the first-order perturbation theory result.

SOLUTION. It is sufficient to examine only the transition amplitude, since passagefrom the amplitude to the total transition rate would follow the same series of stepsin perturbation theory as in the SFA.

The low-intensity limit allows us to retain only terms of first order in EQ or A, andthe single-photon condition means that co > EB and no = 1. The generalized Besselfunction in Eq. (5.77) can be approximated as

\ COl

Equation (5.77) then approaches the expression

m Eop fp2

7777 o ^r + EB ~ w 0/(/7). (5.89)

For the perturbation theory result, we can start with either of the general amplitudes(5.59) or (5.62), with ^ replaced by <&. The two terms in the expression for theinteraction Hamiltonian Hi are replaced by only the first-order term. That is, thefirst-order time-dependent perturbation theory (PT) transition amplitude is

(5.90)

where the p in the last line of Eq. (5.90) is the scalar /?, and no longer the operator.When we substitute A (t) = a cos cot, as in Eq. (5.64), use the ordinary free-particleplane wave solution for $ / and the stationary state (5.70) for <!>/, we find

(5.91)

Of the delta functions in Eq. (5.91), the first can never have a zero in the argument,and so we need retain only the latter. We have then

(S - 1)? = - ^ ^ 5 (£ + EB - o) UP), (5.92)

differing from Eq. (5.89) only in an irrelevant overall sign. (The change in sign is anartifact of the integration by parts that was done in deriving the SFA result.) Thus theSFA reduces to exactly the perturbation limit for large o) and low intensity.

Expressed as a transition rate, Eq. (5.89) or (5.92) gives

2 p=±[2(co- EB)\ \

Problem 5.10. Find the asymptotic form taken by the SFA transition amplitude inthe limiting case where the laser field is of very low frequency. This means thateB > 1. For ionization to occur at all, the field must be very intense, or z > 1. Theminimum photon order has n0 > z, so n0 > 1. It will also be assumed that z > eB.

SOLUTION. From Eqs. (5.5) and (5.8), it is seen that z\ = 2z/eB. The statement thatz> eB means that z\ > 1.

To evaluate the asymptotic behavior of the SFA transition amplitude in Eq. (5.77),it is necessary to examine the generalized Bessel function. From the integral repre-


sentation in Eq. (5.75), we can write

g (6) = - sin 6 - 4~ sin 20 - 0, (5.94)n In

where the n has been extracted as the large parameter necessary for the application ofasymptotic methods. We shall here take n0 = z + efi, which is equivalent to Eq. (5.79)for large orders. When the value of the momentum p allowed by the delta function isinserted into the definition of £ in Eq. (5.67), we find

£ = (8z) 1 / 2 (n-z- eB)xn = (8z) 1 / 2 (n - no)l/2. (5.95)

The ratio £/n in Eq. (5.94) will then always be extremely small, since the spread ofphoton orders will be much less than the lowest order. Since the other two terms inEq. (5.94) are of order unity, the first term will be neglected, and we shall henceforthnot distinguish between HQ and n. The saddle points in g (0) are at

4 | = - - c o s 2 0 - 1 - 0 .ad n

The saddle points so located are designated #o a nd are at

cos 20O = " - = -Z-^-^-. (5.96)z z

This means that the saddle points cannot be on the real axis. If we set do = x + iy,then the saddle points are at

x = ± y , y = - arcsinh ( l / ^ / 2 ) . (5.97)

Saddle points at a corresponding location in the upper half of the complex 6 planeare not considered because one cannot deform the initial path of integration along thereal axis into a path of steepest descent through the upper-half-plane points.

To evaluate the leading behavior of Jn(£, —z/2), we must find the value ofexp[mg(#o)L that is, we wish to find

ing (60) = - i | sin20O ~ ind0, (5.98)

where, for the saddle point at x = TT/2 we have

cos 0O = -i-rjT, sin 0O = - 1 + — 1 , 0O = — - / arcsinh

zY \ z\J 2

and for the saddle point at x = - TT/2 we have

1 / 1 \ ^ ^

cos 0O = —i-rrz, sin 0O = ( 1 + — ) , 0o = ~TT ~ /arcsinhzY V z \ ) 2

Since (l/z1/2) < 1, we can expand the inverse hyperbolic function as

arcsinh G )Both saddle points give the same real part in Eq. (5.98), and we obtain the asymptoticresult that

(5.99)

The transition rate is proportional to the square of this quantity, so when the argu-ment in Eq. (5.99) is doubled and converted to the directly physical parameters, theasymptotic result for the transition rate behaves as

2 ( 2 £ * ) V 2 ' (5.100)

Equation (5.100) has long been known as the behavior of the transition rate fortunneling of a bound particle through a potential barrier for both constant and slowlyvarying electric fields of amplitude Eo.

Problem 5.11. Apply the SFA formalism to find the total transition rate and photo-electron energy spectrum for ionization from a one-dimensional system bound by thedelta function potential V (x) = — (2£#)1//2 8 (x), where EB is the binding energy ofthe single bound state possessed by this system.

SOLUTION. In terms of the notation r\2 = 2£#, the normalized solution of theSchrodinger equation for the delta function potential is well known to be

(5.101)

The Fourier transform of the space part of (5.101) is

2T73/2

TJ2 + p2

The momentum conservation condition p = (2E)J^2 is associated with the SFAtransition amplitude of Eq. (5.88), where E is defined in Eq. (5.85). When this isemployed in the denominator of Eq. (5.102), one finds that TJ2 + p2 = (2o>) (n — z).The SFA transition rate is thus

W = \jn U - |L V 2

^ \n U | )~ z - €B)1/2 L V 2/i ( 5 1 0 3 )


The rate in (5.103) is for a monochromatic wave. It is a per-atom transition rate. Inpractical application to the description of realistic laser experiments, this elementaryrate is employed in the solution of a rate equation applied to the problem of finding thenumber density of ions formed from a collection of initially neutral atoms. The atomicrate enters into this procedure as the elementary atomic rate W [I(x, t)] dependent onthe field intensity distribution I(x, t) describing the space and time profile of an actuallaser pulse. When so employed, depletion effects in the initial collection of atomsare accounted for. This is important, since the large transition rates typical of veryintense field laser experiments means that depletion of the initial population of atomsplays a significant role in the measurements of ion yield and photoelectron spectra.Nevertheless, the monochromatic rate W given above is sufficient in itself to describemost of the qualitative features of real experiments.

Expression (5.103) gives the total transition rate for the formation of ions, butit also contains within it the information necessary to find the energy spectrum ofemitted photoelectrons. Each of the terms in the sum over the index n represents anelectron with a different energy of emission. As we have seen, the kinetic energy ofthe emitted electron is p2/2 = co(n - z~ e#), so each successively larger value ofn corresponds to the addition of another increment of energy co to the photoelectron.This spectrum can be very extensive in strong fields, and for really extended ATIspectra, the individual peaks for given n values can no longer be distinguished. Thespectrum appears to be continuous.

Figure 5.2 gives the spectrum predicted by Eq. (5.103) in a laser pulse of Gaussianshape in time, but without the spatial variation to be found in a focused laser in alaboratory experiment. No depletion effects are included. The Gaussian distributionhas a peak intensity of / = 1, EB has been set to \ (as in the hydrogen atom),

1 02 r

101

_CD

~6OlCD>

-t—>

DCD

01

1

1

11

1

100

Q —1

0 -2

0 — 3

0-4

10-51 0"1 1 00 1 01

Energy (a.u.)Figure 5.2. Spectrum of photoelectrons emitted from a one-dimensional "atom" bound by adelta function potential with a binding energy of 0.5 a.u., when the laser causing the ionizationhas a frequency of -^ a.u. and a pulse shape Gaussian in time with a peak intensity of 1 a.u.


and the wavelength is taken to be ^ , a typical value for a laser. It is seen that thespectrum extends to an energy of about 100 a.u. Since each photon has an energy ofj£ a.u., the spectrum spans about 1600 photon orders, which is an extreme exampleof the ATI phenomenon. Although this example refers to a very high intensity, reallaboratory experiments have yielded spectra of extent nearly that shown in the figure.This phenomenon plainly bears no resemblance to traditional atomic physics and hasno explanation within perturbation theory.

APPENDIX A

ANGULAR MOMENTUM

The angular momentum of a classical particle is given by

L = r X p,

where r and p are the radius vector and the linear momentum of the particle. Ifthe motion is in a central field, L is a conserved quantity. In quantum mechanics,it is no longer true that each component of the angular momentum vector is sepa-rately conserved. A more limited statement of conservation of angular momentumin quantum mechanics can be achieved, however, as a result of the separation ofradial and angular variables in the equation of motion, made possible by the centralforce property. The angular part of the Schrodinger equation is independent of theform of the central-field potential, and so the angular part of the equation of motionis connected only with angular momentum properties. The basic properties of theangular momentum of a particle in quantum mechanics will be summarized below.(See Ref. 8 for further details.)

A.I PROJECTION OF ANGULAR MOMENTUM

The angular momentum operator in quantum mechanics has the same expression asin classical physics,

L=fXp,

except for the operator nature of the physical variables. In the configuration repre-sentation, the linear momentum operator is

229Radiative Processes in Atomic Physics. V. P. Krainov, H. R. Reiss, B. M. SmirnovCopyright © 1997 by John Wiley & Sons, Inc.ISBN: 0-471-12533-4

230 ANGULAR MOMENTUM

The component of the angular momentum operator in the z direction is

h ( d dT h ( dL z = - [ x — - y

i \ dy dx

We now introduce a spherical coordinate system (r, 6, <p) with the polar directionalong the z axis. The angle 6 is the polar angle, that is, the angle between the radiusvector r and the z axis. The azimuthal angle cp is measured from the x axis to theprojection of the radius vector r onto the xy plane. In this coordinate system, we have

Lz = ? f . (A.1)I dip

The eigenfunctions 4> (<p) °f this operator satisfy the equation

Lz<\> = Lzcj),

and are of the form

4>(<p) = Cexp(iLz<p).

For this function to be single valued in coordinate space, we must have

so that

Lz/h = m,

where m is an integer. That is, the eigenvalue of the z projection of the angularmomentum operator is

Lz = hm. (A.2)

A.2 SQUARE OF THE ANGULAR MOMENTUM

We transform the square of the angular momentum operator,

L2 - (r X p) • (r X p)

so that the linear momentum operators are on the right. To do this, we must use thecommutation properties of the linear momentum operators with the coordinates,

~r--r~ = - 8/

where j , k are rectangular components of the vectors. Using the properties of thetriple scalar product of three vectors, we obtain

L2 = r2p2 - r2vl.

COMMUTATION PROPERTIES OF ANGULAR MOMENTUM OPERATORS 231

The p2 operator can be written in terms of the Laplacian V2 as

p2 = -h2V2,

and so, using the expression for the Laplacian in spherical coordinates, we find

( A 3 )

The Hamiltonian for a particle in a central field is

The squared angular momentum operator is seen to be the only part of the Hamiltonianthat relates to angular coordinates, and so it determines the angular state of the particle.

A.3 COMMUTATION PROPERTIES OF ANGULARMOMENTUM OPERATORS

From the commutation properties of the linear momentum and the coordinates,

^ _hPkrj ~ rjPk 7Ojk,

where the indices refer to projections onto x, yy z axes, we obtain the commutationrelations for the components of angular momentum,

\Ly,Lz\ = ihLx,

We now introduce the operators

L+ — LJX ~\~ iLy, L— — Lx iLy.

The commutation relations for these operators are

= hL+, [LZ,L_1 - -hi-.

We also find the commutators of the rectangular components with L2 to be

\p,lx] = [L2,L,] = [£2,LZ] =0.

232 ANGULAR MOMENTUM

The quantum state of a particle can be determined by the eigenvalues of a set ofmutually commuting operators. In the case of angular momentum, the square of theangular momentum and one of the x, y, x components of angular momentum formsuch a set. These operators commute with the Hamiltonian in Eq. (A.4) for a particlein a central field, as well as commuting between themselves. Therefore the state of theparticle can be characterized by the eigenvalues of the angular momentum squaredand of one of the projections onto a fixed axis of the angular momentum.

A.4 EIGENVALUE OF THE SQUARED ANGULARMOMENTUM OPERATOR

The value of an operator averaged over its values in a given state will be indicated bya superposed bar. With that notation, we can write

L2 = L2 + L2 + L2, L2 ^ 0, L2 > 0.

From this and Eq. (A.2), we conclude that

L2 - h2m2 > 0.

The eigenvalue of one of the projections of the angular momentum operator has arange of possible values, but they are bounded from above and below by the lastinequality. We introduce a quantum number / associated with the squared angularmomentum operator, which we can specify as

/ = max(|ra|).

We shall be able to express the eigenvalue of the squared angular momentum operatorexplicitly in terms of /.

The wave function that is a simultaneous eigenfunction of the operators L2 and Lz

will be written in terms of the quantum numbers as "$?im. From the properties of theoperator L+, it follows that

(ZZL+ -

and, since we know that

we can conclude that

Lz (2+rPfo,) =h(m+l)

The functions L+^/m are seen to satisfy the same equation as would ^ / m +i . We canthus write

L+Vlm = const X ^ V H .

EIGENVALUE OF THE SQUARED ANGULAR MOMENTUM OPERATOR 233

Since m cannot exceed /, we have

L+%j = 0. (A.5)

We now apply these results to the determination of the eigenvalues of the squaredangular momentum operator. We find

L2 = Lzx + If + L\ = L-L+ + L\ + hLz. (A.6)

If we operate with this expression on the state ^// and use Eq. (A.5), we obtain

The eigenvalue of the squared angular momentum operator does not depend on theeigenvalues of a projection of the angular momentum. We then conclude that

We now calculate matrix elements of the angular momentum operators. Since theangular momentum is a Hermitian operator, we have

l,m+ 1 Im) = (Im L- (A.8)

In view of the raising operator property of L+, this is the only nonzero matrix elementof this operator for the given values of / and ra. Equations (A.6) and (A.7) then leadto

2

/, m + 1 Im 1)].

We obtain, finally, the relations

(A.9)

APPENDIX B

CLEBSCH-GORDAN COEFFICIENTS

Clebsch-Gordan coefficients arise when two angular momenta are combined into atotal angular momentum. This will occur when the angular momentum of a systemis found as the combination of the angular momenta of two subsystems or when twotypes of angular momenta relating to the same particle are combined to find the totalangular momentum for that particle, as in the addition of orbital and spin angularmomenta to obtain a total angular momentum for the particle.

Let us find the total angular momentum j as a sum of the momenta ji and j 2 . Thewave function for the total momentum can be written as

(JU2, ™\™2 \jm) iphmi iphm2, (B.I)YYl\ ,TYli

where the indices on the wave functions characterize the angular momentum and itsprojection onto a fixed axis. The coefficients in this expansion are called Clebsch-Gordan coefficients. We now examine their properties.

B.I PROPERTIES OF CLEBSCH-GORDAN COEFFICIENTS

B.I.I Condition for Addition of Angular Momentum Projections

It follows from the conservation law for the sum of angular momentum projectionsthat

(JiJ2>mim2 \Jm) — 0 if m\ + m2 ^ m- (B.2)

Radiative Processes in Atomic Physics. V. P. Krainov, H. R. Reiss, B. M. Smirnov 235Copyright © 1997 by John Wiley & Sons, Inc.ISBN: 0-471-12533-4

236 CLEBSCH-GORDAN COEFFICIENTS

B.1.2 Orthogonality Condition

Orthogonality properties of the wave functions are expressed as

This leads to the orthogonality condition for the Clebsch-Gordan coefficients, whichis

2_] Oi72, m\m2 \jm) (j\j2, m\m2 jmf) = 8mm>, (B.3)

where 8mm> is the Kronecker delta symbol defined by

1, m — m1

0, mi- m1

B.1.3 Inversion Property

In the inversion transformation of the radius vector, r —> — r, the wave functions willchange sign or not, depending on their parity. That is, the wave functions transformas

\\r. v (— i \i~m \\r .

This leads to

so that we obtain

O"i7*2, -mh -m2 \j, ~m) = (-l)J~Jl~j2 (jiJ2,m\m2 \jm). (B.4)

B.1.4 Permutation Properties

It follows from the rules for the construction of the Clebsch-Gordan coefficients andthe rules for the addition of two angular momenta into a zero total angular momentumthat

{j\h,mxm2\jm) = (-l)Jl m' xj———(jljfmlt ~m\j2 - m2) (B.5)

72, — m, m2 \j\ —mi).

EVALUATION OF CERTAIN CLEBSCH-GORDAN COEFFICIENTS 237

B.2 EVALUATION OF CERTAIN CLEBSCH-GORDANCOEFFICIENTS

We consider first the evaluation of Clebsch-Gordan coefficients for the frequentlyoccurring case h ~ \- We begin by obtaining a relation for Clebsch-Gordan coef-ficients that is valid for any angular momenta. From the addition properties used toform j = ji + J2, we obtain the condition

j 2 ~ ji ~ J2 = 2ji j 2 = 2juJ2z + 7*1 + 72- + 71-7*2+.

When this operator acts on the wave function in Eq. (B.I), the result is

JU + 1) ~ 7i0*1 + 1) ~ 72(72 + 1) = y^0'i72> raim217m) p!72, m[m'2\jm)

X {^hm^hm^hzhz + 7*1 + 7*2-

+ 7*1-7*2+ l^,m>;2m2).

For the given value of m, the conservation rule for angular momentum projectionsgives

m\ = m — 1712, m[ = m — m2.

With the help of the momentum operator eigenvalues in Eqs. (A.2) and (A.9), we findthat

7(7 + D - 7 i ( 7 i + l ) - 72(72+ D

m2

- mQQ'i + mi + \)(j2 - m2 + l)(j2 + rn2)m2

x {j\J2,mxm2\jm)(jiJ2,mi + 1, m2 - l\jm)

^ 72 - m2)(j2 + m2 + 1)m2

x O'i7*2, rnxm2\jm){jxJ2y mi - 1, m2 + \\jm). (B.6)

Now we apply the above results to the particular case where j2 = \. For the givenvalues 7, 7i, there are two nonzero Clebsch-Gordan coefficients that we shall denoteas

X={h\,m-\,\\jm),

Using this notation, Eq. (B.6) for j2 = \ becomes

JU + 1) " 7i O'l + 1) " I = (m - \)X2 - [m + i ) Y

The normalization condition for the Clebsch-Gordan coefficients given in Eq. (B.3)is of the form

X2 + Y2 = 1.

When this is inserted into the preceding expression, we obtain

U - h)U + h + 1) - 3 = m (X2 - Y2) + 2\l(jx + | ) 2 - m*XY.

With the notation

we can rewrite our results as the system of equations

t(X2 - Y2) + 2 \ / l — f-XY = ±1, X2 + Y2 = \.

The ambiguous sign ± is such that the upper sign corresponds to j = j \ + \, whilethe lower sign is associated with j = jx — \.

The solution of the equations we have obtained is

for 7 = 7i + 5, and for j = j \ ~ \ it is

These results are summarized in Table B.I.

TABLE B.I. ( j \ \, m - <r, o-1 jm)

j - h

12

12

I2

j ji + m +

V 2h + 1

[h-m +V Vx + 1

i2

12

cr

_ l2

> —V 2 y , +

/ .( + m• i /

+" \1

+ i

EVALUATION OF CERTAIN CLEBSCH-GORDAN COEFFICIENTS 239

Other values for Clebsch-Gordan coefficients are generally more complicated toexpress. Thorough treatments of the relations between Clebsch-Gordan coefficientsand tables of values are available in the literature. See, for example, Refs. 9 or 10.Another relatively simple form taken by the Clebsch-Gordan coefficients occurswhen the projection of the angular momentum coincides with this momentum. In thatcase we have

0. . . . . v O/'i +m2,m

\j2,J\m2\jm) = y/U\ +72

C/2 ~ W2)! (7 + m)\ (2j + 1) (271)! (7 — 71 + 72)!

C/2 + m2)l (j ~ m)\ (7! - 72 + 7)! (7, + j 2 - 7) ! '

(B.7)

V(7i +72 + 7

x /(7i " mi)! (j + m)\ (2j + 1) (2j2)\ (Ji ~ h + 7)!

{]\j2,mxj2\jm) =

(7i + mx)\ (j - m)\ (j - j , + j2)\ (Ji + 72 ~ 7)!'

(B.8)

<7i72, "71^2 \M) = (-l)j]+h~j(JU2, 7i " m2 \j - m), (B.9)

<7i72, mi - j2 \jm) - {-\)^h~j^ {jxh -m[h y _ my ( E U 0 )

Values of the Clebsch-Gordan coefficients are given in Table B.2 for the casej2 = 1. If m2 = 1 or - 1 , then the coefficients can be calculated using the connectionsin Eqs. (B.9) and (B.10). The third possibility, m2 = 0, can be calculated using thevalues obtained for m2 = 1 and — 1, and the normalization condition (B.3) for theClebsch-Gordan coefficients.

TABLE B.2. (jil; m - m2, m2 I jm)

7 - 7 i 1

m2

,KjL-~*-~,xji--m) / ( 7 i - m + 1) (7 ! + m + 1) /(71 - i

(27! + 2) (27! + 1) V (27i + 1) (7i + 1 ) V (27i + 2 ) (27, + 1)

/(7i m + 1) (7i + m) mU — i

27,(7i

2 lUi-m+VUi-m) _ /O'i-m)Oi+w). /(71 + m + 1)0"i + ^)27i (27i + D V h (27i + D V 27, (27, + 1)


B.3 WIGNER 3j SYMBOLS

A quantity closely related to the Clebsch-Gordan coefficient is the Wigner 3 7 coef-ficient, designed to achieve maximum symmetry. It can be defined as

^ ) = (-l)j]~J2~m3(JiJ2>mim2 \j3, -m3). (B.ll)

The 3 7 symbol has the property that

= 0 if mi + m2 + m3 =£ 0,mx m2 - • \ i i >

in place of Eq. (B.2).We list the principal symmetry and orthogonality properties. Even permutation of

the columns leaves the 3j symbol unchanged, or

7i h h \ = (h h h\ = ( h h hmi m2 m3 I V m2 m-x nt\ J V m3 mi m2

Odd permutation of the columns, on the other hand, is equivalent to multiplicationby( - iy i + 7 2 + j 3 , so tha t

7i h h \ = (h h hmi m2 m3 J \m2 mi m3

= ( h h h \ = I h h J\\ mi m3 m2 I \ m3 m2 mi

Orthogonality properties of the Wigner 3j symbols are

h h h \ (J\ h h \ _ o cjI mi m2 m3 J \m[ m'2 m3

h h h \ ( h h ^ \ = hjfnvnjSUUlh)mi m2 m3 J \mi m2 m\ j 2/^ + 1£

where 8 (jij2j3) in Eq. (B.12) is a quantity defined as

8 (71/2/3) = { I [f[jl ". hl ~ h - h + h10 otherwise

The statement in Eq. (B.13) is called the triangular condition.

APPENDIX C

ROTATION FUNCTIONS

Rotation functions can be introduced as the coefficients of the transformation of awave function from one set of quantization axes to another:

The transformed wave function i///m is for the state with angular momentum j andangular momentum projection m in the new coordinate system, ifjjK is a wave functiondescribing the state with quantum numbers j and K in the old coordinate system,DJ

mK is the rotation function, and 6 represents the set of angles that defines thetransformation from the old to the new coordinate system. The rotation functions arealso referred to as generalized spherical functions, rotation matrices, D functions, orD matrices.

We shall now explore the basic properties of the rotation functions. From orthonor-malization of the wave functions and the relation (C.I), we find that the rotationfunctions satisfy the relation

" L E (L) * L 8KK>. (C.2)

Another property of these functions is connected with the parity of the wave functions,that is, with the transformation of the wave functions upon inversion of the coordinateaxes. When we also incorporate the relation between the rotation functions in directand reverse transformations, we find

[]* (C3)


242 ROTATION FUNCTIONS

Other important properties of the rotation functions follow from the action ofthe angular momentum operator on these functions. The rotation operator can bedescribed either in terms of the angular momentum operator or in terms of therotation functions. A simple relation exists, therefore, between these two quantities.We denote by x, y, x the axes in a stationary coordinate system, while the axes £ TJ, <;refer to a moving coordinate system. (See Fig. C.I.) The rotation functions establishthe connection between these two coordinate systems. These functions satisfy theeigenvalue relations

(C.4)

Equations (C.4) allow us to give an alternative interpretation to the rotation func-tions. They are eigenfunctions of the operators corresponding to simultaneous rota-tions with respect to both the stationary and moving coordinate systems. In particular,a rotation function represents an eigenfunction of the rotational state of a multiatomicmolecule with two equal moments of inertia. If the £ axis is the third direction, thenwe can describe the rotation of a molecule by the set of quantum numbers j , K, and m.These quantum numbers represent, respectively, the total rotational momentum, theprojection of this momentum on the moving axis £, and the projection of the totalmomentum on the fixed axis z. The motion of a molecule can be represented assuccessive rotations of the molecule with respect to its axis £, and the rotation of thisaxis in the stationary coordinate system. The rotation function is the eigenfunctiondescribing such a motion of the molecule.

Another example illustrating an application of the rotation function relates to alinear molecule rotated around an axis normal to the molecular axis. We assume thatthe interaction is weak between the rotation of the molecule and the motions of theelectrons in the molecule. Then the rotation function DJ

mK represents an eigenfunction

ys-

AFigure C.I. Relation between the fixed x, y, z axes and the rotating £, TJ, f axes.

MATRIX ELEMENTS OF ROTATION FUNCTIONS 243

of the state that includes both the rotation of the molecular axis and the rotation ofthe electrons in the molecule. The quantum number j is the total angular momentumcomposed of the sum of the rotational molecular and electron angular momenta, mis the projection of the rotational momentum on the stationary axis z, and K is theprojection of the electron momentum onto the molecular axis.

In both examples, when K = 0, the rotation function coincides with the wavefunction describing the rotation of a linear molecule and is of the form

(C.5)

Here Yjm (d, <p) is the normalized spherical harmonic corresponding to rotationalmotion with angular momentum j and projection m of this momentum on the fixedreference axis; 6 and <p are the polar and azimuthal angles, respectively; and Pj isthe associated Legendre function.

Since the rotation functions describe the motion of a symmetrical rotator, theysatisfy the addition rules

< * , (*)9U 2 (#) = J2 (Jl& m'm2 \Jm> X <JIJ* K^ \JK>DiK (#)>J=\J\~J2\

(C.6)

and

^ 2 , \ j m ) X {jlj2,KlK2\jK)D>jliKi ( # ) D ^ ( # ) . (C.7)

Here ft represents the set of angles that determine the position of the new coordinatesystem as a rotation with respect to the stationary coordinate system; (7172, m\ nt2 \jm)are Clebsch-Gordan coefficients; and, in accordance with the properties of thesecoefficients, we have m = m\ + m^ and K = K\ + K2.

C.I MATRIX ELEMENTS OF ROTATION FUNCTIONS

The next important property of the rotation functions is the integral of the tripleproduct of these functions,

o 2

hh,KxK2\jK)9 (C.8)2j + 1 v/1</

where dfl is the element of solid angle and integration is over all angles.

244 ROTATION FUNCTIONS

Using Eq. (C.8), we can now calculate matrix elements of the rotation functionstaken on rotational states of the symmetrical rotator. With j2 = m2 = K2 = 0, wefind that orthonormalized wave functions for the symmetrical rotator are

\jmK) =

where j , m, and K are, respectively, the angular momentum, its projection on thefixed axis, and its projection on the molecular axis. With the use of Eq. (C.8), we findthat the matrix elements of the rotation function are

j\tn\K\ jmK) =y zj t i

(C9)Using Eq. (C.9), we can relate matrix elements of a vector in the stationary and in

the rotating coordinate systems. If we express a matrix element in terms of its valuein the rotating coordinate system, we can thereby exclude the rotation of the axisof the symmetrical rotator and reduce the problem to the calculation of quantitiesthat depend only on the internal states of the rotator. We consider a vector A, withprojections AX9 Ay, Az on the stationary set of axes, and projections A^, A^, A^ on theaxes of the rotating coordinate system. Our goal is to connect the matrix elementsof these two sets of vector projections. It is simpler to find first the relationshipsbetween the sets of quantities Ax + iAy, Az, Ax — iAy and A^ + /A^, A , A^ - iAv.We shall denote the first set by Aq, where q = 1, 0, — 1, and the second set by A^,where /JL = 1,0, — 1. From the transformation rules for vectors, we have

(CIO)

where # is the set of angles defining the transformation from the stationary coordi-nates to the rotating coordinates.

From Eqs. (C.9) and (C. 10) we find the matrix elements of the vector componentsin the stationary system to be

{j'm'K'a1 \Aq\ jmKa) = ^ (j'm'K1 \Dlm\ jmK) (K'a1 \A^\ KOL)

2 . /+1

X (K'a'lA^Ka). (C.ll)

Here j , j ' are total angular momentum quantum numbers in initial and final states,m, m' are the angular momentum projections onto the stationary axes, K, K' areprojections of the angular momenta onto the molecular axes, and a, a' are all theother quantum numbers necessary to define the initial and final states, respectively.

In conclusion, we give explicit results for the rotation functions for the smallvalues of angular momentum j — \ and j = 1. We take the transformation between

MATRIX ELEMENTS OF ROTATION FUNCTIONS 245

the stationary and rotating systems to be such that the TJ axis coincides with the yaxis. That is, the transformation corresponds to a rotation around the r\ axis. Theangle between the 9 and z axes is labeled # (as in Fig. C. 1). The value of the rotationfunctions are given in Tables C.I and C.2.

TABLE C.I. D^(ft)

i cos(d/2)

- i sin(#/2)

TABLE C.2. DlmK{&)

K

i

0

- 1

1

1 + cos ^2

1 - cos # s2

m

~ 2

- sin(#/2)

cos(#/2)

m

0

sin# 1 -

os# - -

iin ft 1 +

-1

COS1^

2

cos#2

APPENDIX D

WIGNER 67 SYMBOLS

In this appendix we consider the addition of three angular momenta: j u j 2 , 73. Thetotal angular momentum is / , with projection M onto a fixed axis. Addition can becarried out in two ways. In the first, we add the angular momenta j \ and 72 to forman angular momentum 74, and then add 74 and 73 to obtain J. See Figure D.I. Werepresent by ^JfM a wave function formed in this manner. The second approach is toadd the angular momenta j \ and 73 to form 75, and then combine the angular momenta75 and 72 into the total angular momentum J. This wave function is denoted by ^j5

M.The overlap between these functions can be written as

) jf j4\. (D.I)J 73 J5)

The quantity contained in the brace is called the Wigner 6j symbol, and the quantity

j \ hJ h 75;;}

is called the Racah coefficient.Successive applications of Eq. (B.I) to write a wave function for added angular

momenta gives the Wigner 67 symbol in the form

73 75/ vx <7374, w3m4 \JM){jij3, W1W3 \j5m5) {j2j5, m2m5 \JM).

(D.2)Radiative Processes in Atomic Physics. V. P. Krainov, H. R. Reiss, B. M. SmirnovCopyright © 1997 by John Wiley & Sons, Inc. 2 4 7

ISBN: 0-471-12533-4

248 WIGNER 67 SYMBOLS

From the definition of the Wigner 67 symbols, we can obtain the sum rule for theproduct of three Clebsch-Gordan coefficients,

/ \JlJ2> Wl\Wl2 \J4WI4 ) \J3J4y WI3WI4 \JM ) \7l73> ^ 1 ^ 3 \J5^5 )

f jf\ (J2J5,m2m5 \JM), (D.3)^(274 + 1) (275 + I) [J Ji

which follows from Eqs. (D.2) and (B.3).From the definition of the Wigner 67 symbols, we find the sum of binary products

of the 67 symbols to be

h

j} h J4\J J6]

[J3 J 76 J

71 n 74 . I JL J5 JO I ( D 4 )

j h h :

(D.5)

D.I PROPERTIES OF 67 SYMBOLS

The principal symmetry property of the 67 symbols is that the permutation of anypairs

^ '" or M^J/in) [J4 n) \]\

leaves the value of the 67 symbol unaltered. In all, there are 48 such permutationspossible for each Wigner 67 symbol.

This property can be understood geometrically from the diagram in Figure D.I.The momentum represented by one side of a triangle is the vector sum of the othertwo sides. Choosing some sets of sides for the momentum triangles that definethe tetrahedron leads to equivalences with other schemes for defining the sametetrahedron. According to this property, the 67 symbol is the same for any possibleway to add the angular momentum vectors. We can calculate the number of suchcombinations by noting that we have 4! = 24 possibilities for the composition of thefour triangles in the given scheme of addition, and two ways in each variation forchoosing the first angular momenta that are added. We thus have 48 different ways toadd the angular momenta, which therefore leads to 48 different Wigner 67 symbolsthat have the same value but differ from each other by permutation of several of theelements.

PARTIAL VALUES OF 6; SYMBOLS 249

Figure D.I. Diagram showing alternative schemes for the coupling of three angular momenta.

We exhibit here another property of the Wigner 6 j symbol:

71 727 73

73 + 74 - js) \ (72 - 73 + 74 + 75)

73 ~ 74 + 75) \ (-72 + 73 + 74 + 75)(D.6)

Next we consider the simplest analytical expressions for the 67 symbols. Ananalytical form exists for the case when one of the angular momenta is an arithmeticsum of two of the others. In this case, the tetrahedron in Figure D.I degenerates intoa triangle in a plane. The expression for this case is

J7i 72 71+72I17 73 75 J

(271)! (272)! (71 + 7 2 + 73 + 1)!

X

I (271 +272 + I)! (7! +73 +75 + 1)!

(71 + 72 + 73 - 7')! (71 + 72 - 73 + 7)!(72 + 75 + j + 1)! (~7l ~ 72 + 73 + 7)! (71 + 73 - 75

X ( - 7 1 + 7 3 + 7')! ( -72+73+7 ' ) ! 11/2

(71 " 73 + 7 5 ) ! (72 +j~ 7 5 ) ! (72 - j + 7 5 ) ! J(D.7)

D.2 PARTIAL VALUES OF 6j SYMBOLS

If one of the angular momenta is zero, then the tetrahedron collapses to a triangle,with only three different nonzero angular momentum values. If we set j \ = 0, thenEq. (D.7) gives the Wigner 67 symbol in the form

fO 72 721 = (-lV2 + 7 3 + 7

V -/3 h\ V(272 + l) (273 + 1)'(D.8)

250 WIGNER 6; SYMBOLS

If one of the angular momenta has the value | , then the Wigner 6j symbol can beexpressed as

jji h JA\ _

X

X

+ 1)(274+ l)(27"s + 1)

O'I + h + JA + 1)!0"i + h + h + D!

(7i ~ h + 75)! (71 - h + U)\ (-71 + h + 74)! ] ' / 2

O'l + h - JsV- (7i + 72 - 74)! (-71 + 73 + 75)! J

(D.9)

D.3 MATRIX ELEMENTS IN ADDITION OF MOMENTA

Since the Wigner 6j symbol expressed in Eq. (D.2) is a sum of quaternary productsof Clebsch-Gordan coefficients, these symbols will appear in matrix element calcu-lations for a system consisting of two subsystems. We illustrate this assertion withan example of such a matrix element. Take J to be a total angular momentum arisingfrom the angular momenta j \ and J2 of two subsystems. We wish to calculate thematrix element of a quantity that depends on variables of the first subsystem. Weassume that this quantity is a spherical tensor, that is, its dependence on angles is ofthe form

Aiq=ADlq0($), (D.10)

where A is independent of the angles, Dlq0 is the rotation function, and d is the set of

angles. Our goal is to calculate the matrix element

(j'M'a'\Aiq\jMa), (DM)

where J, J1 are the angular momenta of the initial and final states, respectively; M, M'are their projections on the quantization axis; and a, a' are the sets of other quantumnumbers necessary to fully define the states.

We now calculate the matrix element (D. 11) using the fact that the operator (D. 10)depends only on variables of the first subsystem. Using Eq. (B.I) for the wave functionof the system, we can write the matrix element in the form

{j'M'a'\Alq\jMa)= ]T (jij2,mlm2\JM)m\m'lm2

X (j[j2,m{m2 \J'M') (./>{,/'a' |

MATRIX ELEMENTS IN ADDITION OF MOMENTA 251

We calculate the matrix element of the operator (D.IO) taken over states of the firstsubsystem, employ Eq. (C.9), and obtain

{j'M'a1 \Alq\ JMa) = ]T (jij2, mxm2 \JM) (j[j2, m[m2 \j'M')

x{j[j'a'\A\hJa).

The sums over projections of the angular momenta are accomplished using Eq. (D.3).The final result is

{J'M'a1 \Alq\ JMa) = (-\)h+J[+*h+i ^2j[ + l) (2jx + I) (j[l, 00 \j{0

(D.12)

We have thus explicitly determined the dependence of the matrix element on themagnetic quantum numbers. Equation (D.12) is called the Wigner-Eckart theorem.

APPENDIX E

FRACTIONAL PARENTAGECOEFFICIENTS

We consider the wave function ^ of N valence electrons with the same principalquantum number, bound in an atom. This wave function can be represented as aproduct of the wave function for the other valence electrons. We denote by le the orbital angular momentumof the electron thus singled out, with its angular momentum projection on a fixeddirection called /x, and with a as the electron spin projection in the same direction.We adopt the notation

for the wave function of this electron. In analogous fashion, we can describe theremaining valence electrons by the wave function

where /, s are the orbital and spin momenta of the other N - 1 valence electrons, withprojections mi, ms on the fixed direction.

We now introduce the Clebsch-Gordan coefficients

( \s , ams \SMS ) , < « /un/ \LML)

describing the addition of the electron spins into the total spin S of the atom, andalso the addition of the orbital angular momenta of the electrons into a total orbitalangular momentum L of the atom, with projections on the fixed direction of Ms andML, respectively.

Multielectron configurations are generally in several levels. We classify each ofthese levels by definition of the initial level. The initial level, in turn, is identifiedas a level that produces the given atomic state after addition of an electron. The

254 FRACTIONAL PARENTAGE COEFFICIENTS

specification of the initial level can be regarded as a definition of the parentage of thelevel. The total wave function ^LMLSMS (1, 2 , . . . , Af) is represented as a linear combi-nation of products corresponding to various initial levels. The coefficientsin this combination are denoted as Gff (le, N). They are called fractional parentagecoefficients. That is, we have the total wave function expressed as

(1, 2,..., AO = ? XI G^(/" N) &' a )lmisms

X (IJ, m i \LML) 4>lmisms (l,2,...,N-l) tp,^ (N),

where P is the operator for permutation of identical electrons.

TABLE E.I. Fractional parentagecoefficients for the case L = 1

p-Electron State Atomic State

P(2P)

p2([D)

p\2D)

P3(4S)

p\2P)

P4(3P)

p\lD)

p*CD)

p\2P)

P3(4S)

p\2D)

p\2P)

p3(2D)

p\2P)

P4([D)

pXD)

p\2P)

P5(2P)

P5(2P)

111

1

1A/2-1/V2

0

-i/v^FT

V T8

00

vT

V 40

00

_ 1212

1

FTV 5

l/\/3

l/v/15

FRACTIONAL PARENTAGE COEFFICIENTS 255

Fractional parentage coefficients are calculated by making use of the fact thatamong those states that may be constructed from tp and (/> by straightforward additionof angular momenta, there will be some that are excluded by the Pauli principle. Thatis, only some of the combinations of (p and 4> will satisfy the Pauli principle.

Fractional parentage coefficients are determined by incremental construction start-ing with N = 3. For example, the N = 3 wave function begins with a pair of electronsantisymmetrized in accordance with the Pauli principle. This is then combined withthe wave function of the third electron. The wave function thus obtained will changesign upon permutation of the first two electrons but will not have this property interms of permutations of each of the first two electrons with the third. If we alterthe scheme by which the angular momenta are combined, we can obtain a wavefunction that will change sign upon permutation of the second and third electrons.A linear combination of the original and permuted wave functions can be requiredto change sign upon the permutations 1 ^ 2 and 2 ^ 3 . This is then the requiredfully antisymmetrized wave function ^ for the three valence electrons. By analogousprocedures, we can obtain systems of equations to determine fractional parentagecoefficients in the case of four or more equivalent electrons.

Fractional parentage coefficients must also permit the normalization of the wavefunction *P, which means they must have the property

For the case of valence s electrons, the fractional parentage coefficient is unity.Fractional parentage coefficients for an electron system in a valence p shell are givenin Table E.I.

APPENDIX F

ATOMIC PROPERTIES

We shall treat an atom as a system comprised of electrons and a Coulomb centralpotential. Since an ion can be described in the same terms, the results obtained beloware valid both for neutral atoms and for ions.

Relativistic effects will be neglected. This is a valid assumption for atomic nucleiof small and moderate charge numbers. The Hamiltonian of the electron system is ofthe form

£2 7^2 2r, (F.I)

where / and k are indices enumerating the electrons, r; and r^ are position vectors ofthe corresponding electrons, and Z is the nuclear charge number. The wave functionof the electron system changes sign upon the permutation of any pair of electrons.

It is easily shown that the orbital angular momentum operator of the electrons,

commutes with the Hamiltonian. In this expression, 1/ is the orbital angular momentumoperator of the /th electron, with position vector r, and linear momentum operatorpi. Hence, the good quantum numbers describing a state of the atom are:

L, the orbital angular momentum of the atomML, the projection of this momentum on a fixed direction5, the spin angular momentum of the atom, since the Hamiltonian of Eq. (F.I)

does not depend on spin, and therefore commutes with the spin operator of theatom


258 ATOMIC PROPERTIES

Ms, the projection of the spin angular momentum on the same fixed directionrelating to ML

The spin state of an atom is determined by S and Ms only, although any arbitrarysuperposition of spins of various electrons commutes with the Hamiltonian. However,only that spin operator for an atom that sums the spins of all the electrons will have thenecessary antisymmetry property when acting on the atomic wave function. That is,this operator must produce a change in sign of the wave function upon the permutationof any two electrons.

We shall now replace the field acting on all the individual electrons by a self-consistent field depending on the coordinates of only that one electron being studied.This field is spherically symmetrical, so the introduction of a self-consistent field isequivalent to a replacement of the Hamiltonian (F.I) by the model Hamiltonian

h2

, hi = ~^i + V(n), (F.2)

where hi is the Hamiltonian of a single electron, and V is the potential of the self-consistent field. The variables are separated in the Schrodinger equation based on theHamiltonian in Eq. (F.2), and so the wave function for the system of electrons canbe written as a product of single-electron wave functions. Symmetry properties mustbe imposed on this wave function. The state of a single electron is determined bythe quantum numbers n> I, m, and o", where n is the principal quantum number, / isthe orbital angular momentum quantum number, m is the magnetic quantum number,and a is the projection of the electron spin on a fixed direction. The quantities n, /,and m are integers satisfying the inequalities n > 1, / < n — 1, - / < m < /, and ahas the two possible values ± \.

The energy of a single electron depends on the quantum numbers n and / only,which means that each electron state has a 2(2/ 4- l)-fold degeneracy. The energyof a single electron increases with an increase in the n and / quantum numbers, sothat states of the electron are enumerated beginning with the states of smallest n and/ values. We must now take into account the Pauli principle, which requires that nomore than one electron can be in a given state. This leads to a system of electronshells, each of which is characterized by fixed values of n and /, and each containing2(2/ + l)substates.

This model of the atom is called the shell model. Internal shells are filled. Electronsof the outermost atomic shell have smaller binding energies than the inner electrons,and are called valence electrons.

The distribution of the atomic electrons in these shells is described by a standardnotation that we summarize here. States with the values / = 0, 1, 2, 3,4,.. . are labeledwith the letters s, p,d, f, g, The principal quantum number is placed before thesymbol that identifies the orbital angular momentum state, and the number of electronsthat may occupy this shell is written as a superscript to the orbital angular momentumsymbol. For example, the electron shell configuration for the ground state of the

ATOMIC PROPERTIES 259

oxygen atom is written as

O(\s22s22p4).

The notation signifies that two electrons are in the state with n = 1, / = 0; two moreare in the state with n = 2, / = 0; and the state with n = 2, / = 1 is occupied byfour electrons.

We shall now construct a wave function within the shell model: a wave functionfor the system of electrons formed from products of single-electron wave functionsso composed as to obey the Pauli principle. This principle requires the wave functionto change its sign upon the permutation of any two electrons. If all electron shellsare filled, then only one combination of single-electron wave functions satisfies thiscondition. The wave function of an atom with k electrons is of the form

(1) 4>n2l2m2 (r2) V<r2 (2) . . .

The operator P includes all permutations of any two electrons, with each permutationchanging the sign of the wave function. The total number of such permutations is

so the wave function in Eq. (F.3) is the sum of C\ products of k single-electron wavefunctions. In Eq. (F.3), i/fn./.m. (r,) r)(Ti (i) is the space part of the wave function for the/th electron with quantum numbers nif lt, and m,-, and r)a. (i) is the spin wave functionfor this electron with the spin projection cr, on the fixed quantization axis. Equation(F.3) can be written in the form of a determinant for an atom with filled shells. Forexample, for the ground state of the helium atom we obtain

where the notation coincides with that for Eq. (F.3), and labels a spin projection of+ ^ by + and a spin projection of — \ by —. Expressions for atomic wave functionsin the form of Eq. (F.4) are called Slater determinants.

APPENDIX G

VIBRATIONAL AND ROTATIONALSTATES OF A MOLECULE

The major difference between the masses of nuclei and the mass of the electronmakes it possible to divide the problem of the description of molecular energy levelsinto two parts. We first determine electron energies at fixed positions of the nuclei,and then we consider the motion of the nuclei for a fixed state of the electrons. Wedenote by U(r) the potential function for the interaction between nuclei. We restrictourselves to the case of a diatomic molecule. The motion of the center of mass of themolecule is not of interest, and we confine our attention to the motion of the nucleiwith respect to the center of mass.

Within the framework of the adiabatic approximation of quantum mechanics,the interaction potential U(r) describes simultaneously the energy of the electronlevels. The motion of the nuclei in this central potential reduces to a one-dimensionalproblem in terms of the effective potential

where K = J — L is the angular momentum of the nuclei, J is the total angularmomentum of the molecule, and L is the total orbital angular momentum of the elec-trons. The second term in the expression for the effective potential is the centrifugalenergy. The average over electron states of the centrifugal energy operator at a fixeddistance r between the two nuclei is

The last term in this expression depends only on the state of the electrons and can beincluded in the definition of U{r). The same is also true for the second term, since the


262 VIBRATIONAL AND ROTATIONAL STATES OF A MOLECULE

angular momentum K is perpendicular to the axis of the molecule, while the vectoroperator L is directed along this axis since it is the only physically explicit directionin the problem. Thus, the effective potential energy takes the form

The motion of the nuclei in weak excited states can be considered as small oscil-lations with respect to the equilibrium position. A power series expansion about thisequilibrium position gives the lowest order term

tfeff = U(rc) + —J(J + 1) + -U"(re)(r - ref, (G.2)

where / = Mr] is the moment of inertia of the molecule, M is its mass, and re is themean distance between the nuclei.

The third term in Eq. (G.2) is the potential of a one-dimensional harmonic oscil-lator. We can therefore give the energies of the vibrational-rotational levels as

EvJ = JjJ(J+V + h"e (v 4- M , (G.3)

where v is the vibrational quantum number and J is the rotational quantum number.The quantity

coe =

is the frequency of classical oscillations. The second term in Eq. (G.3) is the vibra-tional energy of the molecule, and the first term is the rotational energy. From thedefinition of a)e, we see that it behaves as

in the atomic system of units. This quantity gives a measure of the intervals betweenvibrational levels. Since M > 1, this spacing is small as compared to the inter-vals between neighboring electron levels, which are of order unity in atomic units.Furthermore, since / ~ M, then the spacing of rotational levels can be estimated as

2/ M'

which is small compared to intervals between vibrational levels, which are themselvesclosely spaced by comparison with electron levels.

We conclude that vibrational motion splits an electron level into a manifold oflevels, and each vibrational level is split into a manifold of rotational levels.

Now we consider the classification of molecular levels. If the spin S is zero, thenthe classification is the same as described above. However, if S =£ 0, then a relativisticinteraction between spin and orbital angular momentum occurs. This interaction issmall as compared to vibrational intervals, and so it is very small as compared

VIBRATIONAL AND ROTATIONAL STATES OF A MOLECULE 263

to electron level intervals. However, the structure of rotational levels is usually soclosely spaced that spin-orbit interactions can be more energetic than this rotationallevel splitting (excluding light molecules). We shall now consider the classificationof such levels.

First, we neglect the rotation of the nuclei. In addition to the projection A of theelectron orbital angular momentum onto the molecular axis and the quantum numberv of the vibrational motion, we should also consider the projection of the total spinon the molecular axis. This quantity is usually given the designation 2 . Clearly, thevalue of 2 covers the range from - 5 , - 5 + 1 , . . . up to . . . , 5 - 1 , 5. The projectionof the total angular momentum is £1 = A + 2 . It can take the values

n = A + 5, A + 5, , A - 5.

An electron level with a projection of the orbital angular momentum A is split into2 5 + 1 sublevels, which differ in their fl values. This splitting is called^me structure,in analogy to such splitting in atoms.

Now we consider the energies of the levels. Since the spin-orbit potential isproportional to L • S, and the vector operator L is directed along the molecular axis,we shall write the spin-orbit interaction energy as A(r)2. For a fixed value of A, weobtain the electron energy as

U(r) + A(r)VL,

which means that the sublevels of the manifold are equidistant.Now we describe the rotational levels. The centrifugal energy operator has the

form

2Mr2

When we average this quantity over electron states, we obtain the effective potentialenergy

= U(r) + A(r)ft + ^ ^ [j2 " 2J* (£ + §) + L2 + 2L-S + S21 . (G.4)

When we associate the operators with their quantum numbers, we can make thereplacements

J2 = 7 ( 7 + 1 ) , L - An, S = 2n,

where n is the unit vector along the molecular axis. We also have the relationships

(h + sVn = (j - K) • n = J • n = U

so Eq. (G.4) becomes

tfeff = U(r) + A(r)Ct + - ^ \j(J + 1) - 2H2 + L2 + 2L-S + S21 .2Mrz L J

264 VIBRATIONAL AND ROTATIONAL STATES OF A MOLECULE

Averaging over electron states is done using wave functions that do not depend onspin, so we obtain

S2 = S(S + 1).

Furthermore, the averaging of the angular momenta L and S can be done indepen-dently, so we find

L-S = L - S = AS.

As above, the mean value L2 can be incorporated into U(r), since it does notdepend on 2 or J. The terms proportional to 2 can be included in the term A (r) ft.The final expression we obtain for the effective potential is

h2

Utf = U(r) + A(r)Sl +

The energies of the levels are obtained, as above, by means of an expansion oversmall oscillations of the nuclei to obtain

E=Ue+ Ae£l + hcoe fv + X- J + Y [J (J + 1) - in2]. (G.5)

The constants in this expression are taken to be at equilibrium values of the inter-nuclear distance.

APPENDIX H

EQUATION FOR THEDENSITY MATRIX

The density matrix is useful when a transition involves several states. The densitymatrix approach greatly simplifies the treatment of problems in which a relaxationprocess takes place among the states considered. In particular, the density matrixmethod is very useful for the description of relaxation due to collisions betweenparticles.

The wave function of a system can be expressed as a superposition of stationarystates by the expression

(H.I)

where the \pm are eigenf unctions of the Hamiltonian //0, and the Em are its eigenvalues,that is,

= Emifjm.

The probability density is given by

P = ^^=^2pmn^m€, (H.2)m,n

where

pmn = aman

(onm = (En - Em) /h.


266 EQUATION FOR THE DENSITY MATRIX

The matrix pmn is called the density matrix. According to its definition, the matrixelements satisfy the relation

Pmn ~ Pnm-

To derive the dynamical equation satisfied by the density matrix, we presume thattransitions in a quantum system take place due to a perturbing potential V. The totalHamiltonian is then H = Ho + V. The Schrodinger equation for the wave function is

ih^- = (Ho + v) V. (H.3)

When we substitute the expansion (H.I) into Eq. (H.3), multiply from the left byt//*, integrate over the variables that characterize the system, and use the orthogonalitycondition for the wave functions, we obtain the equation for the expansion coefficients,

iaj = - ]jP Vjm exp (i(ojmt) am. (H.4)m

The equation for the density matrix elements follows from Eq. (H.4) as

Pjk + iWjkPjk = T ^2 (VnkPj« ~ VJnPnk) • (H.5)Tn

In particular, for the diagonal elements of the density matrix, we have

Now we shall generalize Eq. (H.5) to apply to a density matrix appropriate whenthe states have a finite lifetime. Then the probability to find a system in the state mshould be multiplied by the factor exp(—t/rm), where rm is the lifetime of this state.This means that the probability amplitude am to find a system in the state m shouldbe multiplied by exp (—t/2rm). The density matrix pmn acquires therefore the factor

2rm 2rnJ'

and the quantity o)jk in Eq. (H.5) should be replaced by

2rm 2rn

Equation (H.5) for the density matrix has now become

pjk + iWjkPjk + ( — + — ) pjk= ^Yl iVnkPjn ~ VjnPnk) , (H.7)\ Z T 7 ATk J n

n

and Eq. (H.6) for the diagonal elements of the density matrix is now

PJJ + ^PjJ =J

EQUATION FOR THE DENSITY MATRIX 267

The diagonal element of a density matrix is the probability to find a system inthe state identified by the diagonal index. If transitions from other states to this statetake place, then it should be accounted for in the equation for the density matrix. Thequantity 1/T*/ is the transition rate from the state k to the state j . According to thedefinition of a state lifetime, we have

1 - V l

Tk . ?kj

Equation (H.8) for the diagonal density matrix elements thus takes the form

1 _ ^ 1 i

Equation (H.8) for the off-diagonal matrix elements has not been changed.We continue to generalize by specifying a part of the density matrix determined by

collisions between particles. Each collision constitutes a strong interaction betweencolliding particles and makes a fundamental change in the state of the molecule.These collisions are random, however, and collision times are small compared totimes for radiative transitions. We may therefore disregard the time duration of acollision. We now convert these qualitative considerations into terms in the densitymatrix equation arising from collisions.

We take rq to be the time at which some particular collision occurs betweenthe atom being examined and the surrounding atoms. The parameters defining thecollision are the impact parameter, the relative velocity of the collision, the anglebetween the relative velocity vector and a fixed direction in space that may be adirection of an external field or the direction of observation. We characterize thetransitions resulting from the collision by a transition S matrix, so that

ak (rq + At) = ^2 (Snk)an K - At) ,n

where A Ms a time much larger than a collision time for the atoms but much less thana typical time for an alteration of the ak (t) amplitudes under the action of a radiationfield or other external field. The change in the amplitude during a time r a s a resultof collisions is

ak (t) - ak (0) = V " (Snk - 8nk) an (0).

Hence the change in the density matrix due to the collision is

Pjk (0 - Pjk (0) = aj (t) 4 (t) - aj (0) 4 (0)

Snk — Smj&nk) Pmn-

This change takes place in a time much less than typical times for radiation processes.When we take the limit in which the collision time goes to zero, we obtain the time

268 EQUATION FOR THE DENSITY MATRIX

derivative of the density matrix due to collisions,

V dt /coll

where rq is the time at which a collision occurs.Since collisions take place chaotically and abruptly compared to radiative pro-

cesses, we can average Eq. (H.10) over collision parameters. In particular, if thecollision between the radiating atom and the perturbing atom occurs according to thelaws of classical mechanics, then the collision probability during the time dt with theimpact parameter in the range [p, p + dp] is given by

Nv lirpdpdt,

where N is the density of perturbing atoms, and v is the relative velocity for collisionsbetween the radiating and perturbing atoms. When we substitute this relation intoEq. (H.10) and integrate over collision time, we obtain

m,n

Nv / (SmjS*nk - 8mj8nk) lirpdppmn ) ,Jo /dt J coll t i \ Jo I

where the angle bracket symbolizes averaging over relative velocities of the atoms.When the collision contribution is inserted into Eq. (H.7) for the density matrix, weobtain

{ t ^) k Vnkpjn ~VinPnk) (R11)

r * \Nv 2irpdp (SmjS*nk - 8mj8nk) pmn ) .

Jo IWe now examine the special case where there is no inelastic scattering. All possible

channels are presumed to have elastic scattering only. The transition S matrix is thenof the simple form

Smj (p) = 8mj exp (2i8j),

where 8j (p) is the scattering phase for the channel j . Equation (H.I 1) simplifies to

-£- + (io)jk + — + — + Re vjk + 1 Im vjk ) pjk = - V (Vnkpjn - Vjnpnk),at \ ZTj LTk J n

(H.12)where the notation has been introduced that

vJk = (NV I 2irpdp{\ - exp [2i (Sj - 8k)}}\ .\ Jo I

The equation for the diagonal elements of the density matrix is not affected bycollisions in this case and continues to be expressed by Eq. (H.9).

APPENDIX I

ATOMIC UNITS AND MEASURESOF INTENSITY

Atomic units greatly simplify the appearance of expressions in atomic physics, andthese units are thus widely used. They are due to Hartree. The basic atomic unitsare summarized, and then various measures of electromagnetic field quantities areintroduced in terms of atomic units, and relations among these field quantities aresummarized.

1.1 ATOMIC UNITS

Atomic units are specified by the requirement that

h = 1, m = 1, e2 = 1, (1.1)

where m and e2 refer, respectively, to the mass of the electron and the squared chargeof the electron. These units then are applied in terms of the properties of the simplehydrogen atom based on infinite mass of the atomic nucleus. A single atomic unit ofthe most important physical quantities is then:

Unit of length: Bohr radius = a0 = h2 / (me2) = 1 a.u. = 5.29177249 X 10~u m.Unit of velocity: Velocity of an electron in the first Bohr orbit = v$ = e2/h =

la.u. = 2.18769142 X 106m/s.Unit of time: Time for an electron to travel a distance a0 at the velocity VQ = r0 =

ao/Vo = h3/me4 = 1 a.u. = 2.41888433 X 10~17s.Unit of frequency: Inverse of the unit of time = v0 = 1/T0 = me4/h3 = 1 a.u. =

4.13413732 X 1016Hz.Unit of energy: Twice the binding energy of hydrogen = Eo = 2Roo = me4/h2 =

la.u. = 27.2113962 eV.


270 ATOMIC UNITS AND MEASURES OF INTENSITY

1.2 ELECTROMAGNETIC FIELD QUANTITIES

1.2.1 Basic Field Quantities

Unit of electric field strength: Eo = E0/a0 = m2e5/h4 = 1 a.u. = 5.14220826 X10 u V/m.

Unit of energy flux (intensity): /0 - C£ 2 / (8TT) = 3.50944758 X 1016W/cm2.

1.2.2 Convenient Relations

Angular frequency in a.u. and wavelength in nm: a>(a.u.) = 45.5633526/A(nm).

Velocity of light and the fine structure constant: c— I /a .

1.2.3 Field Measures Useful in Strong-Field Atomic Physics

To set the convention used for the amplitudes, we shall regard a monochromaticlinearly polarized field as being of the form

E = E0esin((ot - k r), e • e = 1; (1.2)

and a circularly polarized monochromatic field as being of the form

E = —^ [ex sin (wt - k • r) ± ev cos (cot -k- r ) ] , (1.3)

e * e* = e • e = 1, ex • e = 0.

In Eq. (1.3), the presumption is that wave propagation is along the z axis, and e , eare unit vectors along the x and y axes, respectively. Whereas the use of the amplitudeEQ in Eq. (1.2) is universal, the definition employed in Eq. (1.3) is not. Some authorsomit the factor 2 - 1^2 . Equations (1.2) and (1.3) have the amplitude relations

. . _ ( E O t linear^ ' " \ £ 0 / 2 1 / 2 , circular '

( I 4 )|E|2 = l£g, circular ' ( L 4 )

where the angle bracket indicates a time average over a period.

Irradiance, or Energy FluxA physical quantity frequently employed to measure the "strength" of the field isoften expressed in units of Watts/cm2, or Joules/(s • cm2). This is a flux of energy,also known as irradiance. It has, however, now become quite common to refer to this

ELECTROMAGNETIC FIELD QUANTITIES 271

quantity simply as the intensity. We shall here use the designation / for the irradiance,given by the amplitude of the Poynting vector, or

/ (Gaussian units) = uc = —- (E2) = — EQ.

The atomic unit of irradiance /0, as indicated above, is that energy flux that corre-sponds to Eo = 1 a.u., so that 70 = C/(8TT) - 3.50944758 X 1016 W/cm2.

When all quantities are in atomic units, then

Ponderomotive EnergyThe ponderomotive energy is that energy that a free charged particle possesses becauseof its oscillation, or "quiver," motion when in an electromagnetic field. It is exactlythe kinetic energy of such oscillatory motion as expressed in the frame of referencein which the center of mass (cm.) of the charged particle is, on a time average, atrest. If a charged particle should emerge adiabatically from a region in which thereis an electromagnetic field into a region free of the field, this ponderomotive energyis converted into a directed kinetic energy. The ponderomotive energy Up is thussometimes called the ponderomotive potential. In atomic units,

Free-Electron Amplitude of MotionThe amplitude of motion of an electron in a plane wave field is often designated OLQ.As expressed in the frame of reference in which the cm. of the electron is at rest onthe average, this motion is a rectilinear oscillation along the polarization direction fora linearly polarized plane wave of moderate intensity, with

This result is actually the limit of the true relativistic motion of the electron, wherethe relativistic nature of the motion will arise as a result of the intensity of the field.The relativistic motion is in the form of a figure 8, with the axis of the lobes along thepolarization direction, and the plane of the figure 8 determined by the polarizationdirection and the direction of propagation.

For a circularly polarized electromagnetic field, the electron motion in the simplestframe of reference is a circle executed in the plane perpendicular to the direction ofpropagation. This circle has the radius

21/2ft)2"

272 ATOMIC UNITS AND MEASURES OF INTENSITY

Dimensionless Intensity ParametersThe Keldysh parameter, or adiabaticity parameter, is actually an inverse intensityparameter. It is given as

_ co(2Ei)l/2 _ co(2Ei)

l/2 _ E)/2

7 Eo 71/2

where Et is the ionization potential of the atom.The inverse square of the adiabaticity parameter is directly an intensity parameter,

which can be defined ab initio as twice the ratio of the ponderomotive energy to theionization potential of the atom. When used in this form, it is generally designatedz\, where

_E2

0 = f ((oao)2 / (2Et), linear2a)2Ei 2(i>2Ei \ (co«o)2 /Et, circular

This quantity has been called the bound-state intensity parameter.Another intensity parameter is the ratio of the ponderomotive energy of the electron

to the energy of a single photon of the field. That is, it is given by

_ Up _ ^ _ ^o _ / (OCCQ/4, linearco 4co3 4co3 \ coal/2, circular

This quantity has been called the intensity parameter for the final continuum state,and also the nonperturbative intensity parameter, since it is the quantity that is theprimary indicator of when perturbation theory will fail.

There is a free-electron parameter that always arises in the description of theinteraction of a free electron with an electromagnetic field. There is no universallyaccepted symbol for this parameter, although many authors have defined the verysame quantity. We here designate it Zf. It is most readily defined as twice the ratio ofthe ponderomotive energy of the electron in the field to the rest energy of the electron,or

s ^EL = l = I (**)2 = / |(<Wc)2, linearZf c2 2{cocf 2\coc) \(coa0/c)2, circular

By its definition, this parameter plainly is a measure of the appearance of relativisticeffects in the field-induced motion of an electron. It is the free-electron intensityparameter.

APPENDIX J

PROPERTIES OF THE GENERALIZEDBESSEL FUNCTION

The generalized Bessel function Jn{u, v) arises in connection with Volkov solutionsfor an electron in a linearly polarized electromagnetic field. Because the Volkovwave function is an exact solution of the Schrodinger equation for a free electron in asuperposition of unidirectional plane wave fields, it has been widely employed in thetreatment of strong-field problems whenever at least one of the states in the problemdescribes an unbound electron. The function Jn(u, t;) is thus frequently encountered inthe strong-field literature, but its properties are not to be found in any of the standardreference works on transcendental functions. The principal properties of Jn(u, v) aresummarized below.

It will be assumed that n is an integer, and u and v are real.

Integral representation:

1 f™Jn(u,v)=— / dOexp[i(usind + v sin20 - nd)].

^T* J -TT

Series representation:

Jn (M, V) =

Generating function:

00

exp[/ (u sin 6 + v sin 26)] = ^ ein6Jn (w, v).n—— oo


274 PROPERTIES OF THE GENERALIZED BESSEL FUNCTION

Sums of generalized Bessel functions:

V" (-\

Recursion relations:

Jn-\ (u, V) - Jn+i (W, V) = 2 — Jn(u, V),du

Jn-2 O, V) - Jn+2 {U, V) = 2 — ]n (M, V),

dv

2nJn (w, v) = u[jn-\ (u, v) + Jn+X (u, v)]

4- 2v [jn-2 (M, V) + Jn+2 (w, v)

— 7n(w, f) = 2—/„_! (w, v) + 2—7n+1 (u, v).

Symmetry relations:

jn(-u,v) = (-iyjn(u,v),

Jn(u, -v) = (-l)n7_n(w, f).

Sums of binary products:

JnTk (W, V) Jk {u1, t / ) = JTn (li ± U'', V ± t / ) ,

00

^ [7,(w,t,)]2 = 1,Jfc=-oo

k=-oo

^ /^ (W, V) J-k {u, v) = Jo {2U, 2V),

(-l)kJk(u,v)J-k(u,v) = 0.k=-oo

PROPERTIES OF THE GENERALIZED BESSEL FUNCTION 275

Limiting cases:

0, n odd '

Jn(u, V) « Jn(M) + \v[jn-2(u) - Jn+lW], \V\ < 1,

Jn(uyv)~ 1 - ^- (l - ^-J 7n/2O) , |wH 1, /i even,

Jn (U, V) « M [ / ( n - i ) / 2 (V) - ^(n+l)/2 (V)] , W\ < I, Yl Odd,

\imJn(ex/2ii,ev)

n even

xik+l/2

APPENDIX K

IONIZATION POTENTIALS

Ionization potentials of atoms and positive ions are given in Table K.I. The values70, / i , h, and 73 are, respectively, the ionization potentials of the neutral atom and thesingly, doubly, and triply ionized atoms. Potentials are given in eV. The values arethose listed in the Handbook of Chemistry and Physics (Ref.l 1).

TABLE K.l. Ionization Potentials of Atoms andPositive Ions

z123456789

10111213141516

Atom

HHeLiBeBCNOF

NeNaMgAlSiPS

/o.eV

13.59824.5875.3929.3238.298

11.26014.53413.61817.42321.5655.1397.6465.9868.152

10.48710.360

/i,eV

54.41875.64018.21125.15524.38329.60135.11734.97140.96347.28615.03518.82916.34619.77023.338

/2,eV

122.45153.9037.93147.8947.4554.93662.7163.4571.62080.14428.44833.49330.20334.79

/3,eV

217.72259.38

64.4977.4711.41487.1497.1298.91

109.27119.99

5.14251.44447.222

(continued)


277

278 IONIZATION POTENTIALS


z17181920212223242526272829303132333435363738394041424344454647484950515253545556575859

Atom

ClArKCaScTiVCrMnFeCoNiCuZnGaGeAsSeBrKrRbSrYZrNbMoTcRuRhPdAgCdInSnSbTeI

XeCsBaLaCePr

/o,eV

12.96815.7604.3416.1136.5616.8286.7466.7677.4347.9027.887.6407.7269.3945.9997.9009.8159.752

11.81414.0004.1775.6956.2176.6346.7597.0927.287.3617.468.3377.5768.9945.7867.3448.649.010

10.45112.1303.8945.2125.5775.5395.464

/i,eV

23.81427.63031.6311.87212.80013.5814.6616.4915.64016.18817.08318.16920.29217.96420.5115.93518.6321.1921.8124.36027.28511.03012.2413.1314.3216.1615.2616.7618.0819.4321.4916.90818.8714.63216.5318.619.13121.2123.1510.00411.110.810.6

/2,eV

39.6140.7445.8150.91324.75727.4929.3131.033.6730.6533.535.1936.8439.7230.734.228.430.8235.9036.9539.242.8820.52523.025.027.229.528.531.132.934.837.4828.030.5025.3227.9633.032.133.435.819.1820.2021.62

/3,eV

53.4759.8160.9167.373.4943.2746.7149.251.254.851.354.957.459.464.45.750.142.9547.352.552.656.2860.6034.3438.346.4

5440.7444.1637.41

45464749.939.7638.98

(continued)

IONIZATION POTENTIALS 279


z60616263646566676869707172737475767778798081828384858687888990919293949596979899

100101102

Atom

NdPmSmEuGdTbDyHoErTmYbLuHfTaWReOsIrPtAuHgTlPbBiPoAtRnFrRaAcThPaU

NpPuAmCuBkCfEsFmMdNo

/o.eV

5.5255.555.6445.6706.1505.8645.9396.0226.1086.1846.2545.4266.8257.897.987.888.739.058.969.226

10.4386.1087.4177.2898.4179.0

10.754.05.2785.26.15.896.1946.2666.066.06.026.236.306.426.506.586.65

/i,eV

10.710.911.111.2412.111.511.711.811.912.112.1813.914.9

18.5620.518.7620.4315.03216.7

10.1511.7511.5

11.9

/2,eV

22.122.323.424.920.621.922.822.822.723.725.0520.9623.3

3434.229.8331.9425.56

2020

20

/3,eV

40.441.141.442.744.039.841.442.542.742.743.645.2533.3

4346

42.3245.3

28.7

37

280 IONIZATION POTENTIALS

Table K.2 gives electron binding energies (electron affinities) in negative ions,labeled as EA in the table. The accuracy to which these affinities is known is indicatedin the table by the code: A-the accuracy is better than 1%, B-the accuracy is between1 and 3%, C-the accuracy is between 3 and 10%, D-the accuracy is lower than 10%.The entry "no" means that the stable negative ion of this element does not exist.

TABLE K.2. Electron BindingEnergies in Negative Ions

z

1234566789

1011121313141414151617181920212122232425262728293031

Ion, term

H-( ]S)He-

U~(lS)Be-

B-(3P)C~(45)C ( 2 D )

N"O-(2P)F~(}S)

Ne~Na~(lS)

Mg-A1"(3P)A\-(lD)sr(4s)Si"(2D)Si-(2P)P-(3P)S~(2P)

cr(ls)Ar"

KT(lS)Ca~(2P)Sc'CD)Sc~(3D)Ti-(4F)V-(5D)Cr-(6S)

Mn"Fe-(4F)Co-(3F)Ni-(2D)Cu-C1^

Zn~Ga"(2P)

Shell

\s2

2s2s2

2p2P

2

2p3

2p3

2 /2P

5

2 /3s3s2

3p3p2

3P2

3P3

3^3

3P3

3 /3P5

3p6

AsAs2

As2Ap3d4p3dAp3d3

3d4

3d5

3d6

3d7

3d*3d9

3d10

ApAp2

EA, (eV)

0.75416 (A)No

0.618 (A)No

0.28 (C)1.2629 (A)0.035 (C)

No1.4611 (A)3.4012 (A)

No0.5479 (A)

No0.441 (A)0.332 (B)1.394 (A)0.526 (B)0.034 (C)

0.7465 (A)2.0771 (A)3.6127 (A)

No0.5015 (A)0.018 (D)0.19 (C)0.04 (D)0.08 (D)0.53 (C)0.67 (B)

No0.151 (B)0.662 (A)1.15 (A)1.23 (A)

No0.5 (D)

(continued)

IONIZATION POTENTIALS 281

TABLE K.2. Electron BindingEnergies in Negative Ions

Z Ion, term Shell EA, (eV)

32 Ge~(45) V 1.233 (A)33 As~(3P) V 0.80 (C)34 Se~(2P) 4p5 2.0207 (A)35 Br^S) 4p6 3.3636 (A)36 Kr~ 55 No37 RbrC1^) 5s2 0.4859 (A)38 Sr~(2P) 5s25/7 0.026 (D)39 Y'C1/)) 4d5p 0.31 (C)39 Y^D) 4d5p 0.16 (D)40 Zr'(4F) 4d3 0.43 (B)41 Nb~(5D) 4d4 0.89 (B)42 Mo~(6S) 4d5 0.75 (B)43 TcT(5D) 4d6 1.0 (D)44 Ru~(4F) 4dn 1.5 (D)45 Rh~(3F) 4J8 1.14 (A)46 PcT(2D) 4d9 0.56 (B)47 A g ^ S ) 4d10 1.30 (A)48 CcT 5p No49 In~(3P) 5p2 0.3 (D)50 Sn'(4S) 5p3 1.112 (A)51 Sb ' ( 3 P) 5p4 1.05 (B)52 Te-(2P) 5p5 1.9708 (A)53 T^S) 5p6 3.0590 (A)54 Xe" 6s No55 Cs-(J5) 652 0.4716 (A)56 Ba- 5d No57 La ' ( 3 F) 5d2 0.5 (D)73 Ta-(5/)) 5J 4 0.32 (C)74 W"(65) 5d5 0.816 (A)75 Re'(5D) 5d6 0.15 (D)76 Os"(4F) 5J 7 1.4 (D)77 Ir"(3F) 5d* 1.57 (A)78 Pt"(2D) 5^9 2.128 (A)79 AirC1^) 5t/10 2.3086 (A)80 Hg" 6/7 No81 T1~(3P) 6p2 0.5 (D)82 Pb'(4S) 6p3 0.364 (A)83 Bi~(3P) 6/74 0.95 (B)84 Po~(2P) 6/?5 1.6 (D)

APPENDIX L

PARAMETERS OF THEHYDROGEN ATOM


284 PARAMETERS OF THE HYDROGEN ATOM

L.I ANGULAR WAVE FUNCTIONS OF THE HYDROGEN ATOM

/ m Ylm(8,<p)

0 0

1 0

1

4TT

3~cos 6

1 ±1 H-y — sin 6exp(±icp)

2 ±1 -h\l-^-sin 0 cos 6 Qxp(±iip)QTT

2 ±2 ^

3 0 - \ - (5cos3 0 - 3cos 0)4 y 77 v }

I—

3 ±1 + - W - sin 0(5 cos2 0 - l)exp(±i<p)

3 ±2 -W-—sin20cos0exp(±2/(p)4 V 2TT

3 ±3 +o\ —sin3 0exp(±3/(p)8 V TT

4 0 j A ( |

4 ±1 M/4 y 4TT

(7cos20 - 3)exp(±i<p)

4 ±2 -\ — sin2 0 (7cos2 0 - 10) exp(±2/<p)8 y 2TT V '

4 ±3 +T\/—sin30cos0exp(±3/(p)4 V 4TT

3 /354 ±4 - W — sin

RADIAL WAVE FUNCTIONS OF THE HYDROGEN ATOM 285

L.2 RADIAL WAVE FUNCTIONS OF THE HYDROGEN ATOM

State Rni

Is 2exp(-r)

2s —,

24

2

3r r2 r3 \ / r

4 / ^ ^ r 3 e x p ( " i

286 PARAMETERS OF THE HYDROGEN ATOM

L.3 ALGEBRAIC EXPRESSIONS FOR EXPECTATION VALUES OFPOWERS OF THE RADIAL COORDINATE IN THE HYDROGEN

Parameter Expression

(r)

2

y [5n2 + 1 -31(1+ 1)]

n2

— [35n2 (rc2 - l) - 30rc2 (/ + 2) (/ - 1) + 3 (/ + 2) (/ + 1) / (/ - 1)]

— [63rc4 - 35«2 (2/2 + 2/ - 3) + 5/ (/ + 1) (3/2 + 3/ - 10) + 12]

f)(/ + l)(/ + i ) / ( / - i )

L.4 EXPECTATION VALUES OF POWERS OF THE RADIALCOORDINATE IN THE HYDROGEN ATOM (IN ATOMIC UNITS)

State

Is2s2p3s3/73dAs4/74d4 /

(r)

1.565

13.512.510.524232118

(r2)

34230

207180126648600504360

(r3)

7.5330210

344228351701

1872016800131007920

<r4)

22.528801680

6.136 X4.420 X2.552 X5.702 X4.973 X3.629 X1.901 X

104

104

104

105

105

105

105

10.250.250.1110.1110.1110.06250.06250.06250.0625

(r~2)

20.250.08330.07410.02470.01480.03120.01040.006250.00446

(r~3)

—0.0417

—0.01230.0247

—5.21 X 10-3

1.04 X 10"3

3.72 X 10~4

<r~4>

——

0.0417—

0.01375.49 X 10-3

—5.49 X 10~4

2.60 X 10~4

3.7 X 10~5

OSCILLATOR STRENGTHS AND LIFETIMES FOR RADIATIVE TRANSITIONS 287

L.5 OSCILLATOR STRENGTHS AND LIFETIMES FOR RADIATIVETRANSITIONS IN THE HYDROGEN ATOM

Transition

\s-2p\s-3p\s-4p\s-5p2s-3p2s-4p2s-5p2p — 3s2p-3d2p-4s2p-4d2p-5s2p-5d3s-3p3s-5p3p-4s3p-4d3p-5s3p-5d3d -4p3d -4f3d -5p3d - 5 /4s-5p4p-5s4p-5d4d - 5 p4d - 5 /4 / -5d4/-5g

fif

0.41620.07910.02900.01390.43490.10280.04190.0140.6960.00310.1220.00120.0440.4840.1210.0320.6190.0070.1390.0111.0160.00220.1560.5450.0530.6100.0280.8900.0091.345

rfi,ns

1.65.4

12.424

5.412.424

16015.6

23026.5

3607012.424

23036.5

3607012.47324

14024

3607024

14070

240

REFERENCES

[1] W. Heitler, Quantum Theory of Radiation, Oxford University Press, Oxford, 1954.

[2] R. Loudon, Quantum Theory of Light, Oxford University Press, Oxford, 1983.

[3] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,Plenum, New York, 1977.

[4] A. B. Migdal and V. P. Krainov, Approximation Methods in Quantum Mechanics, Ben-jamin, New York, 1969.

[5] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Relativistic Quantum Theory,Pergamon, Oxford, 1971.

[6] J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, Reading, MA, 1967.

[7] J. D. Jackson, Classical Electrodynamics, Wiley, New York, 1975.

[8] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, Perga-mon, Oxford, 1977.

[9] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press,Princeton, 1960.

[10] M. E. Rose, Elementary Theory of Angular Momentum, Wiley, New York, 1957.

[11] Handbook of Chemistry and Physics, 76th ed., CRC Press, Cleveland, 1995/96.


BIBLIOGRAPHY

A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics, Interscience, New York,1965.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, New York, 1975.

S. Bashkin and J. O. Stoner, Atomic Energy Levels and Grotrian Diagrams, vol. I: Hydrogen

I-Phosphorus XV, North-Holland, Amsterdam, 1975.

S. Bashkin and J. O. Stoner, Atomic Energy Levels and Grotrian Diagrams, vol. II: Sulphur

I-TitaniumXXII, North-Holland, Amsterdam, 1978.

S. Bashkin and J. O. Stoner, Atomic Energy Levels and Grotrian Diagrams, vol. HI: Vanadium

I-Chromium XV, North-Holland, Amsterdam, 1981.

S. Bashkin and J. O. Stoner, Atomic Energy Levels and Grotrian Diagrams, vol. IV: Manganese

I-XXV, North-Holland, Amsterdam, 1982.

K. Blum, Density Matrix Theory and Applications, Plenum, New York, 1981.

M. Born and E. Wolf, Principles of Optics, Pergamon, Oxford, 1970.

B. H. Bransden, Atomic Collision Theory, Benjamin Cummings, Reading, MA, 1983.B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, Longman, Harlow, UK,

1983.P. G. Burke, Potential Scattering in Atomic Physics, Plenum, New York, 1977.

C. Cohen-Tannoudji, Atom-Photon Interactions, Wiley, New York, 1992.

E. U. Condon and H. Odabasi, Atomic Structure, Cambridge University Press, Cambridge,

1980.

E. U. Condon and G. H. Shortley, Theory of Atomic Spectra, Cambridge University Press,

Cambridge, 1967.

N. B. Delone, Interaction of Laser Radiation with Matter, Editions Frontieres, Gif-sur-Yvette,

1993.

N. B. Delone and V. P. Krainov, Atoms in Strong Light Fields, Springer-Verlag, Berlin, 1985.291


292 BIBLIOGRAPHY

N. B. Delone and V. P. Krainov, Fundamentals of Nonlinear Optics of Atomic Gases, Wiley,New York, 1988.

N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms, Springer-Verlag, Berlin,

1994.

J. H. Eberly, J. Javanainen, and K. Rzazewski, Above-Threshold Ionization, Physics Reports

204,332-383(1991).

F. H. M. Faisal, Theory of Multiphoton Processes, Plenum, New York, 1986.

U. Fano and L. Fano, Physics of Atoms and Molecules, University of Chicago Press, Chicago,

1972.

U. Fano and A. R. P. Rau, Atomic Collisions and Spectra, Academic, Orlando, 1986.

H. Friedrich, Theoretical Atomic Physics, Springer-Verlag, Berlin, 1991.

S. Gasiorowicz, Quantum Physics, Wiley, New York, 1974.

M. Gavrila, Atoms in Intense Laser Fields, Academic, New York, 1992.

G. Herzberg, Atomic Spectra and Atomic Structure, Dover, New York, 1944.

L. D. Landau and E. M. Lifshitz, Classical Theory of Fields, Pergamon, Oxford, 1979.

I. Lindgren and J. Morrison, Atomic Many-Body Theory, Springer-Verlag, Berlin, 1985.

W. H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973.

A. Messiah, Quantum Mechanics, Vol. I, Wiley, New York, 1965.

A. Messiah, Quantum Mechanics, Vol. II, Wiley, New York, 1966.

P. W. Milonni and J. H. Eberly, Lasers, Wiley, New York, 1988.

M. H. Mittleman, Theory of Laser-Atom Interaction, Plenum, New York, 1993.

R. G. Newton, Scattering Theory of Waves and Particles, Springer-Verlag, Berlin, 1982.A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions, Springer-

Verlag, Berlin, 1985.H. R. Reiss, S-Matrix and Keldysh Techniques for Strong-Field Processes, Progress in Quan-

tum Electronics, 16, 1-71 (1992).M. E. Rose, Multipole Fields, Wiley, New York, 1955.

Y R. Shen, Principles of Nonlinear Optics, Wiley, New York, 1984.

1.1. Sobelman, Atomic Spectra and Radiative Transitions, Springer-Verlag, Berlin, 1979.A. F. Starace, Theory of Atomic Photoionization in Encyclopedia of Physics, Vol. 31, Springer-

Verlag, Berlin, 1982, pp. 1-121.

C. H. Townes and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955.

INDEX

3y symbol, see Wigner6j symbol, see Wigner

above-threshold ionization, see ATIabsorption coefficient, 99-103, 117, 131,

199,202two-photon, 110, 112, 116

adiabatic approximation, 34Adiabaticity parameter, see Keldysh

parameterAiry integral, 165angular distribution, 130, 133angular momentum, 21, 22, 29, 122

classical, 163, 229collisional, 76, 80, 82commutation properties, 231eigenvalues, 233lowering operator, 141, 231LS coupling, 22matrix elements, 233molecular rotation, 32nuclear spin, 29, 31orbital, 14, 23, 30, 32, 81, 85, 114, 116,

125, 131, 137, 142, 178, 183quantum operator, 229raising operator, 141, 231relative atomic motion, 83spin, 23, 125, 126

squared eigenvalue, 233squared operator, 230total, 22, 33, 44, 101, 106,190

anharmonicity, 35, 115asymptotic limit, 15, 79, 81, 122, 128, 130,

132,135, 142,149, 172,215ATI, 221,226, 227atom

alkali, 44bound-bound transitions, 61-99ionization potential, 143, 277many-electron, 23, 31, 257multilevel, 197-202recombination, 125, 165-176shell model, 15,125,131,258spin states, 44, 258valence electron, 1, 12, 122, 125, 127,

143atomic stabilization, 216atomic time, 3atomic units, 16, 18, 84, 129, 143, 269-270autoionizing states, 143-146

Baker-Hausdorff theorem, 213Bessel function, 140, 186

integral representation, 141second kind, 165spherical, 123, 133


293

294 INDEX

Bohr correspondence principle, seecorrespondence principle

Bohr radius, 2, 107, 123, 131, 143, 167, 169Boltzmann factor, 175Born approximation, 151, 156, 159, 171bremsstrahlung, 147-165, 168, 170

classical, 162polarization, 158

broadening, 61, 63, 137, 202collision, 70-87, 90, 93, 192-196Doppler, 62, 64, 65, 68, 111, 113, 117, 180Holtzmark, 96, 98Lorentz, 63, 66, 69, 74, 80, 87, 93, 111,

113,138power, 205quasi-static, 87-99radiative, 62, 74, 101, 111, 114, 199time of flight, 67

cavity, 7channel, 101, 103, 144, 166, 170, 180, 221classical limit, 3, 8, 13, 60Clebsch-Gordan coefficient, 23, 24, 32, 43,

45, 126, 235, 243inversion property, 236orthonormality, 236permutation, 236sum of tertiary products, 248values, 238

commutation relation, 12, 230, 231confluent hypergeometric function, 128correlation function, 70-74, 82

Fourier transform, 71correspondence principle, 6, 49, 140cross section

absorption, 99-103, 140, 180, 188, 189,192, 194, 199multilevel atom, 200Rydberg atom, 143

bremsstrahlung, 147-165, 168broadening, 77, 80, 83, 86coherent radiation, 191, 194, 196elastic scattering, 75, 79, 80, 86, 103-110excitation exchange, 85incoherent scattering, 191, 194induced emission, 99-103nonresonant fluorescence, 104photoabsorption, see cross section,

absorption

photodetachment, 121, 122, 125, 127,132, 166

photodissociation, 134photoionization, 125, 128, 132, 144

angular dependence, 130, 133frequency dependence, 131Rydberg atom, 137,142,171

photon emission, 180photorecombination, 165-176resonance fluorescence, 101, 103, 180soft photon emission, 150, 152, 156spontaneous emission, 180total, 75, 80, 131, 166

D function, see rotation functionD matrix, see rotation functiondelta function, 5, 6, 50, 61, 92, 104, 111,

220,221,223,224potential, 225

density matrix, 188, 192, 197, 265diagonal elements, 266dynamical equation, 266, 268

depolarization, 85, 179detailed balance principle, 166, 170detuning, see resonancediffusion, 68dipole approximation, 2, 217dipole moment, 6, 18, 31, 35, 69, 94, 116,

131,135,162, 181,188effective, 144Fourier component, 69induced, 156, 159, 190operator, 2, 8, 23, 30, 37, 39, 67, 71, 83,

105, 126, 135, 144, 178, 188, 191two-electron, 162

dipole rotation, 186dispersion relation, 166distribution function, see frequency

distribution functionDoppler frequency shift, see broadening

effective charge, 210, 211Ehrenfest theorem, 14Einstein coefficient, 7, 8electric field, 2, 44, 67, 69, 96, 107, 110,

144, 156, 181, 188, 196,206,211electromagnetic field, 5, 68, 110, 144, 182,

188, 191, 204, 206, 207, 211, 214, 271energy density, 5, 49

INDEX 295

Poynting vector, 107, 271scalar potential, 207vector potential, 2, 204, 207, 209, 218

equipartition, 190Euler's constant, 172, 174exponential-integral function, 172

Floquet property, 211,217form factor, 153, 156Fourier component, 6, 48, 69, 130, 140, 152,

157, 161Fourier transform, 71, 88, 151, 164, 219, 225fractional parentage coefficient, 126, 254,

255free-electron amplitude, see intensity

parameterfree-electron motion, 217, 271frequency distribution function, 61, 63, 65,

71,74,88,89,91,93,96,101,111,117,137, 143, 190,196

collision approximation, 94Doppler, 65, 69, 180Gaussian, 65, 67Lorentz, 66, 69, 138, 180

frequency spectrum, see spectrum

gauge transformation, 209Gaussian pulse, 226generalized Bessel function, 220, 223,

273-275generalized spherical function, see rotation

functiongerade state, 18,83,85golden rule, 145Gordon-Volkov solution, see Volkov

solution

Hamiltonian, 11,20, 135anharmonic, 35central field, 231eigenfunction, 4energy eigenvalue, 4interaction, 1, 2, 4, 46, 204, 212, 219multi-electron atom, 257periodic, 211self-consistent, 258total, 4unperturbed, 204, 212

Hanle effect, 179

harmonic oscillator, 12, 34, 35high-frequency approximation, 217Holtzmark distribution, 98hydrogen, 96, 99, 106, 107, 108, 128, 130,

134, 167, 172angular wave functions, 284expectation values, 286lifetimes, 287oscillator strengths, 287radial wave functions, 285

hyperfine splitting, 29

impact parameter, 73, 76, 79, 82, 90, 149,153, 160, 163, 174, 268

incoherence, 6increment coefficient, 109induced emission, 99-103, 195infrared catastrophe, 158intensity, 99, 107-109, 112, 161, 169, 178,

181,200intensity parameter, 189

bound-state, 205, 272continuum-state, 205, 272free-electron, 271free-electron amplitude, 271Keldysh, 204, 272nonperturbative, 205, 272

interaction, 61, 63, 67, 75, 80, 99Coulomb, 129, 165dipole-dipole, 18,21,94Hamiltonian, 2, 20, 188inverse power law, 76, 78, 95operator, 20, 77, 144, 147, 151scattering, 75short-range, 127, 133van der Waals, 22

interference, 80, 177, 179, 183-188irradiance, see intensity

Keldysh approximation, 217Keldysh parameter, see intensity

parameterKramers's formula, 143, 165, 171Kramers-Henneberger transformation, 213,

214Kronecker delta, 236

Lamb shift, 63, 74Laporte rule, 31

296 INDEX

Larmor frequency, 181, 182, 186laser, 2, 67, 112, 117, 177, 204-206, 211,

215,226Legendre polynomial, 81, 134

orthogonality, 124, 149recursion relation, 124, 149summation theorem, 124, 148

lifetime, 9, 19, 62, 65, 68, 100, 111, 117,143, 188

spontaneous, 113, 178, 180, 182, 195,199-202

light, speed of, 5, 46line shape, see broadening; see also spectral

lineline shift, 90, 145

collisional, 74, 80Lamb, 63Stark, 96

line width, 74, 80, 87, 95, 101, 118, 137,143,145, 189

LS coupling, see angular momentum

magnetic field, 5, 42, 107, 177-188magnetic sublevel, see sublevelMaxwell velocity distribution, 65, 110, 117,

168, 171, 173, 175Mittag-Leffler theorem, 138modulation phase, 184modulation strength, 187molecule

centrifugal energy operator, 261classification of states, 262diatomic, 33photodissociation, 119, 134, 136effective potential, 261, 263fine structure, 263level energies, 264moment of inertia, 262polarizability, 55, 58recombination, 174rotation, 1,3,31,32,262spin, 33spin-orbit interaction, 263valence electron, 1vibration, 1,3,31,35, 174,262

multiplet, 28multipole, 21

negative ion, 158photodetachment, 119, 122, 125, 127, 132

nonrelativistic condition, 2, 104, 130normalization, 61, 65, 68, 120, 130, 132,

136, 138, 143, 190nuclear spin, see angular momentum

operatordipole moment, see dipole momentmomentum, 2, 11permutation, 126, 254, 259raising and lowering, 141

optical theorem, 82oscillator strength, 10, 13, 15, 84, 136, 137,

140-143, 287

parity, 18,22,83, 105, 125particle number conservation, 189, 196Pauli principle, 15, 255perturbation theory, 2, 62, 67, 83, 88, 144,

150, 159, 188, 200, 201limits of, 204, 207second order, 44, 144, 156

phase shift, see scatteringphotodecomposition, see photodissociationphotodetachment, 119-133, 158, 166photodissociation, 119-136, 142photoionization, 119-138, 165, 170-172

strong-field, 204, 218, 221, 223, 225photon

flux, 100frequency, 5, 7number, 7, 49polarization state, 5, 39, 42, see also

polarizationspontaneous emission, 7, 10, 18, 37, 40,

49wave number, 5

photorecombination, see recombinationPlanck

constant, 105, 161distribution, 8

polarizability, 12, 13, 55, 56, 58, 107, 156,158

negative ion, 159tensor, 55, 106

polarization, 8, 18, 40, 100, 104, 125, 130,133,177-183

average over, 100bremsstrahlung, 158circular, 41, 44, 270coefficient, 179, 183, 186

INDEX 297

degree of, 56, 177, 187linear, 4, 270scattered radiation, 182spin, 44vector, 5

ponderomotive energy, see potentialpositive ion, 119, 132, 158

ionization potential, 277potential

centrifugal, 17, 161Coulomb, 12, 115, 125, 129, 152, 164, 257effective, 17inverse power law, 95molecular, 143, 172, 204pair-wise interaction, 88perturbation, 46ponderomotive, 205, 214, 220, 271

precession frequency, see Larmor frequencyprobability, 61, 65, 68, 71, 73, 145, 174, 180

decay, 199photoabsorption, 120-122, 199, 201

quadrupole approximation, 2quantum defect, 116quasi molecule, 83quiver motion, see free-electron motion

Rabifrequency, 198, 199problem, 197

Racah coefficient, 247radiated power, 2radiation, 2

bremsstrahlung, 147, 168induced, 8, 40, 101intensity, 169, 191, 193modulated, 184multimode, 5polarization, 40, 44recombination, 168spontaneous, 8, 40, 101,111

radiative recombination, 174Raman scattering, see scatteringrate equation, 44, 196, 226Rayleigh scattering, see scatteringrecombination, 125, 165-176reduced charge, 207, 209-211reduced lifetime, 64, 111reduced mass, 34, 39, 75, 121, 135, 164,

166, 175, 207, 210

resonance, 47, 63, 104, 112, 140, 145, 146,158, 185, 188-202

condition, 4detuning, 115, 117, 189, 192, 197, 199fluorescence, 101, 104, 180parametric, 187scattering, 104two-photon, 115, 118

Riemann function, 170rotation function, 24, 81

addition rules, 243definition, 241eigenfunction of axis rotation, 242integral of triple product, 243inversion property, 241matrix elements, 244sum of binary products, 241values, 245

rotational levels, 33Rydberg

atom, 137, 170constant, 205energy, 2, 206, 210state, 137, 143, 170

S matrix, 82, 85, 86, 215, 267, 268saddle-point method, see steepest-descent

methodsaturation, 109, 195saturation parameter, 200, 202scattering, 103-110, 147-165, 170, 182,

192amplitude, 81coherent, 190, 194, 196Coulomb, 159elastic, 80, 86excitation exchange, 86incoherent, 191, 194Kepler, 164length, 154nonresonant Raman, 104phase shift, 73, 75, 77, 80, 82, 122, 149,

154, 268Raman, 46, 54, 58, 103-110Rayleigh, 46, 107resonance radiation, 192resonant Raman, 103Thomson, 105, 107

Schrodinger equation, 4, 16, 72, 85, 127,211,215,225,266,273

298 INDEX

Schrodinger equation (cont.)numerical solution, 216separation of variables, 207, 209, 217

selection rule, 2, 8, 22, 23, 30, 50, 54, 83,106, 125

dipole, 17, 118self-consistent field, 15, 258SFA, 217, 222, 223, 225shell model, see atomSlater determinant, 259solid angle, 6, 40, 53, 104, 121, 124, 130,

144spatial probability function, 88spectral line

broadening, see broadeningGaussian, 65, 67, 226principal line, 25-27, 29satellite, 25shape, 63-67, 111, 113, 117shift, 22splitting, 22Stark, 44Zeeman, 42

spectroscopic notation, 258spectrum, 2, 6, 120, 137, 143, 165

continuum, 12, 100, 119, 125, 128, 131discrete, 12highly excited, 116line shift, 225strong-field, 225visible, 74, 107

spherical harmonic, 122, 148, 243spherical tensor, 250spin, see angular momentumstabilization, see atomic stabilizationStark effect, 12, 44, 89, 96, 119, 192statistical weight, 6, 7, 25, 40, 100, 101, 102,

121,166-168,170,190,200steepest-descent method, 96, 224step function, 72

strong-field approximation, see SFAsublevel

magnetic, 82, 106, 108, 115-117, 178,183,185

sum rule, 12, 13, 15,38, 102, 115, 136, 140symmetry, 18, 122, 148

thermal equilibrium, 8Thomas-Reiche-Kuhn sum rule, see sum

ruleThomson formula, 105transition, 137, 145

molecular, 134, 136rotational, 37-39Rydberg, 137-143spontaneous, 18, 33, 38, 40, 50, 62, 100,

111, 117, 174, 177vibrational, 34, 37, 39, 174

two-photon processes, 46, 50, 52, 111,113-118,200

tunneling, 217, 225

uncertainty principle, 7, 90ungerade state, 18, 83, 85

valence electrons, 1, 12, 122, 125, 127,144

van der Waals interaction, 22Volkov solution, 214, 218, 273

wave number, 80, 110, 167Weisskopf radius, 70, 79, 90Wigner

3j symbol, 2406y symbol, 24, 247-250threshold condition, 125, 129

Wigner-Eckart theorem, 251WKB method, 167, 169

Zeeman effect, 42, 43, 179, 185

[Vladimir P. Krainov, Howard R. Reiss, Boris M. Sm(BookZZ.org)

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