9814619922

in Atomic and Molecular PhysicsBreaking Paradigms

9301_9789814619929_tp.indd 1 10/2/15 6:55 pm

May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PSTws

This page intentionally left blankThis page intentionally left blank

N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientific

Auburn University, USA

Eugene Oks

in Atomic and Molecular PhysicsBreaking Paradigms

9301_9789814619929_tp.indd 2 10/2/15 6:55 pm

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

BREAKING PARADIGMS IN ATOMIC AND MOLECULAR PHYSICS

Copyright 2015 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4619-92-9

In-house Editor: Song Yu

Typeset by Stallion PressEmail: [email protected]

Printed in Singapore

SongYu - Breaking Paradigms in Atomic.indd 1 26/2/2015 11:17:21 AM

March 4, 2015 9:49 Breaking Paradigms in Atomic. . . 9in x 6in b1984-fm page v

With love toAlex, Ellen, and Andrew

v

March 4, 2015 9:49 Breaking Paradigms in Atomic. . . 9in x 6in b1984-fm page vii

Contents

Chapter 1. Introduction 1

Chapter 2. Role of Singular Solutions of QuantalEquations in Atomic Physics 5

2.1 A Long-Standing Mystery of the High-EnergyTail of the Linear Momentum Distributionin the Ground State of Hydrogen Atomsor Hydrogen-Like Ions (GSHA) . . . . . . . . . . . . 5

2.2 Early, Unsuccessful Attempts to Explainthe Mystery . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Singular Solutions: General Results on Classesof Potentials to Which They are Applicable . . . . . 13

2.4 Engaging Singular Solutions for the SuccessfulExplanation of the Mystery . . . . . . . . . . . . . . 15

2.5 Opening a Way to Test Intimate Detailsof the Nuclear Structure by PerformingAtomic, Rather than Nuclear, Experiments . . . . . 17

Chapter 3. Classical Description of Crossings ofEnergy Terms and of Charge Exchange 19

3.1 Brief History and Importance of theCorresponding Studies . . . . . . . . . . . . . . . . . 19

3.2 Helical States of DiatomicRydberg Quasimolecules . . . . . . . . . . . . . . . . 22

vii

March 4, 2015 9:49 Breaking Paradigms in Atomic. . . 9in x 6in b1984-fm page viii

viii Breaking Paradigms in Atomic and Molecular Physics

3.3 Crossings of Classical Energy Termsof Diatomic Rydberg Quasimolecules . . . . . . . . . 28

3.4 Eects of a Static Magnetic Field: Stabilizationof Diatomic Rydberg Quasimolecules . . . . . . . . . 34

3.5 Eects of a Static Electric Field on DiatomicRydberg Quasimolecules: Enhancementof Charge Exchange and of Ionization . . . . . . . . 42

3.6 Eects of the Screening by Plasma Electronson Diatomic Rydberg Quasimolecules . . . . . . . . . 51

3.7 Applications to CL in Plasmas . . . . . . . . . . . . 63

Chapter 4. Classical Stationary Statesand non-Einsteinian Time Dilation:Generalized Hamiltonian Dynamics (GHD) 71

4.1 GHD for the Motion in the Coulomb Potential . . . 714.2 Extending GHD to the Motion in a Modied

Coulomb Potential . . . . . . . . . . . . . . . . . . . 82

Chapter 5. Underestimated Role of the SingularSpinSpin Interaction in the BindingEnergy of Two-Electron Atoms/Ions 93

Chapter 6. The Last Observed Line in the SpectralSeries of Hydrogen Lines in MagnetizedPlasmas: Revision of InglisTeller Concept 99

Chapter 7. Extrema in Transition Energies ResultingNot in Satellites But in Dips WithinSpectral Lines 107

7.1 Breaking the Paradigm and RevealingCharge-Exchange-Caused Dips (x-dips) . . . . . . . . 107

7.2 Classical Model of x-Dips . . . . . . . . . . . . . . . 1167.3 Advanced Quantal Theories of x-Dips . . . . . . . . 119

March 4, 2015 9:49 Breaking Paradigms in Atomic. . . 9in x 6in b1984-fm page ix

Contents ix

7.4 Practical Applications: ExperimentalDeterminations of the Rates of Charge ExchangeBetween Multicharged Ions UsingObserved x-Dips . . . . . . . . . . . . . . . . . . . . 127

7.5 Future Prospects . . . . . . . . . . . . . . . . . . . . 131

Chapter 8. Conclusions 137

Appendix A. Classical Description of Muonic-ElectronicNegative Hydrogen Ion in Circular States 141

Appendix B. Helical and Circular States of DiatomicRydberg Quasimolecules in a Laser Field 157

References 169

March 4, 2015 9:49 Breaking Paradigms in Atomic. . . 9in x 6in b1984-ch01 page 1

All truth goes through three steps. First, it is ridiculed. Second, it isviolently opposed. Finally, it is accepted as self-evident.

Arthur Schopenhouer

Chapter 1

Introduction

In this book, we present a number of counterintuitive theoreti-cal results. These fundamental results break several paradigms ofquantum mechanics and provide alternative interpretations of someimportant phenomena in atomic and molecular physics.

In Chapter 2, we revisit singular solutions of the Schrodingerand Dirac equations the solutions that were always rejected. Wedemonstrate that they should not have been always rejected: Theycan be legitimate and necessary for explaining some experimentalresults. Specically, they resolve a long-standing dispute concern-ing the high-energy tail of the linear momentum distribution in theground state of hydrogen atoms/hydrogen-like ions. Apart from thefundamental importance, these results also have a practical impor-tance. Namely, they open up a unique way to test intimate detailsof the nuclear structure by performing atomic (rather than nuclear)experiments and calculations. The results presented in Chapter 2have been published in paper [1.1].

In Chapter 3, we take a fresh look at one of the most impor-tant processes in atomic and molecular physics: Charge exchange.Charge exchange was always considered as an inherently quantalphenomenon see, e.g., [1.2, 1.3]. We demonstrate that chargeexchange is not really an inherently quantal phenomenon, but ratherhas classical roots. Charge exchange and crossings of correspondingenergy levels that enhance charge exchange are strongly connectedwith problems of energy losses and of diagnostics in high temperatureplasmas. Besides, charge exchange was proposed as one of the mosteective mechanisms for population inversion in the soft X-ray andVUV ranges.

1


2 Breaking Paradigms in Atomic and Molecular Physics

One of the most fundamental theoretical playgrounds for studyingcharge exchange is the problem of electron terms in the eld of twostationary Coulomb centers (TCC) of charges Z and Z separatedby a distance R. It presents fascinating atomic physics: The termscan have crossings and quasicrossings [1.4]. These rich features ofthe TCC problem also manifest in a dierent area of physics such asplasma spectroscopy: A quasicrossing of the TCC terms, by enhanc-ing charge exchange, can result in an unusual structure (a dip) inthe spectral line prole emitted by a Z-ion from a plasma consistingof both Z- and Z-ions, as was shown theoretically and experimen-tally. The paradigm was that the above sophisticated features of theTCC problem and its ourishing applications are inherently quantumphenomena. We disprove this paradigm by presenting a purely clas-sical description of the crossings of energy levels in the TCC problemleading to charge exchange.

The classical TCC systems represent one-electron diatomic Ryd-berg quasimolecules encountered in plasmas containing more thanone sort of ions. We also study how their classical energy terms areaected by magnetic or electric elds, or by the screening by plasmaelectrons.

Apart from the fundamental importance, these results also havea practical importance. We apply them to the problem of continuumlowering in plasmas, which plays a key role in calculations of theequation of state, partition function, bound-free opacities, and othercollisional atomic transitions in plasmas.

There are also two additional classical studies using (as the start-ing point) the same formalism as above, though not dealing withcrossings of energy terms: Muonic-electronic negative hydrogen ion,and diatomic Rydberg quasimolecules in a laser eld. These studiesare presented in Appendices A and B, respectively.

The results presented in Chapter 3 have been published in papers[1.51.13].

In Chapter 4, we turn to the most challenging problem of classicalphysics that led to the development of quantum mechanics: Thefailure of classical physics to explain the stability of atoms.Classically, electrons revolving around a nucleus should emit


Introduction 3

electromagnetic radiation, leading to the loss of the energy and thefall to the nucleus see, e.g., [1.14]. We show that this challengingproblem can be actually solved within a classical formalism fromrst principles: The fall of atomic electrons on the nucleus due to theradiative loss of the energy, which seemed to be classically unavoid-able, does not occur within Diracs generalized Hamiltonian dynam-ics (GHD) applied to atomic physics.

The GHD is a purely classical formalism for systems havingconstraints; it incorporates the constraints into the Hamiltonian.Dirac designed the GHD specically for applications to quantumeld theory [1.151.18]. In our application of the GHD to atomicand molecular physics, integrals of the motion are chosen as theconstraints. The result is the appearance of classical non-radiatingstates coinciding with the corresponding quantal stationary states.The underlying physics can be interpreted as a non-Einsteinian timedilation. As an application, we discuss advantages of this formalismover classical and semiclassical models employed in chemical physicsfor describing electronic degrees of freedom. The results presented inChapter 4 have been published in papers [1.19, 1.20].

In Chapter 5, we present counterintuitive results concerning therole of the spinspin interaction in two-electron atoms/ions. Usuallythis interaction was considered as an unimportant correction to thebinding energy. However, it turns out that the spinspin interactionactually makes a signicant contribution to the binding energy ifthe singular nature of this interaction is properly taken into account.For the antiparallel orientation of spins, the allowance is made for thenite nature of motion of the electrons in the region of very smallinterelectronic distances determined by the change of the congu-ration space by the magnetic elds of the spins. The paired stateformed in this case is described by the one-electron wavefunction fora particle with double mass and charge and principal quantum num-ber n = 2. As a result, a good agreement with experimental values ofthe ionization potential is obtained for a wide range of two-electronatoms/ions without resorting to variational procedures. The resultspresented in Chapter 5 have been published in paper [1.21].



In Chapter 6, we revisit the following problem. An isolatedhydrogen atom emits practically innite series of spectral lines, corre-sponding to radiative transitions from the upper level of the principalquantum number n to the lower level of the principal quantum num-ber n0. (Here by hydrogen atoms and hydrogen spectral lineswe mean atoms and spectral lines of hydrogen, deuterium, and tri-tium.) In other words, there is practically no restriction on how highthe number n can be. However, when a hydrogen atom is placed in aplasma, the observed spectral series (such as, e.g., Balmer or Paschenseries) terminates at some n = nmax. According to the InglisTellerconcept, developed as early as in 1939 [1.22] and widely used for diag-nostics of laboratory and astrophysical plasmas, nmax is controlledby the electron density Ne and thus can serve for measuring Ne.

However, InglisTeller concept breaks down in plasmas withsuciently large magnetic eld B. We show that in this case, nmaxis controlled by the product BT1/2, where T is the atomic temper-ature. This fundamental result has an important practical applica-tion. Namely, in magnetized plasmas, the number nmax of the lasthydrogen line observed in a particular experiment, can be used formeasuring the atomic temperature T , if the magnetic eld is known,or the magnetic eld B, if the temperature is known. We presentexamples of applications to the edge plasmas of tokamaks and to solarchromosphere. (Tokamaks are a type of plasma machines designedfor the research in the area of magnetically-controlled thermonuclearfusion leading to a practically-inexhaustible source of energy.) Theresults presented in Chapter 6 have been published in paper [1.23].

In Chapter 7 containing conclusions, we summarize the presentedcounterintuitive results. We especially discuss the role and impor-tance of classical models that provide an adequate picture of thereality.


Chapter 2

Role of Singular Solutions of QuantalEquations in Atomic Physics

2.1. A Long-Standing Mystery of the High-Energy

Tail of the Linear Momentum Distributionin the Ground State of Hydrogen Atoms

or Hydrogen-Like Ions (GSHA)

Tests of fundamentals of quantum theory had long ago passed thelevel of non-relativistic quantum mechanics and had moved threelevels higher (passing quantum electrodynamics and then quantumchromodynamics) to testing what might occur beyond the so-calledstandard model. However, at the level of non-relativistic quantummechanics, there still remains a fundamental dispute started over 35years ago.

The problem we are talking about is the distribution functionf(p) of the linear momentum p in GSHA. Hydrogen atoms, being thesimplest atoms, were always considered as a test-bench for checkingatomic theories versus experiments.

In 1935 Fock [2.1] derived non-relativistic wave functions fora hydrogen atom or a hydrogen-like ion in the momentum rep-resentation (here and below, for brevity we use simply the wordmomentum meaning linear momentum). As a particular resultfor the bound electron in the GSHA, he obtained a distributiondw = fquant(p)dp with the following distribution function f(p):

fquant(p) = 32p2p50/[(p20 + p

2)4], p0 Zme2/. (2.1)Here Z is the nuclear charge; p0/Z 1.9921019 gcm/s practicallycoincides with the atomic unit of the linear momentum; m is the

5



reduced mass of an electron in a hydrogen atom or hydrogen-like ion.From Eq. (2.1) it follows that the high-energy tail of the momentumdistribution (HTMD) has the form

fAsquant(p) fquant(p p0) 1/p6. (2.2)However, in the mid-60s, Gryzinski developed a classical free-

fall model of hydrogen atoms [2.2] and then its modied version [2.3](some authors later on suggested another advanced free-fall model[2.4]). In these models, the atomic electron moves along straight linestoward or away from the nucleus (with a possible exception of a smallvicinity around the nucleus), so that the angular momentum of theelectron is zero. The HTMD, corresponding to the free-fall models,has the form

fAsclass(p) 1/p4, (2.3)which is very dierent from the quantum HTMD given by Eq. (2.2) see also a later paper [2.5] comparing the quantal and classical distri-bution functions f(p) in detail. Based on the free-fall model, Gryzin-ski derived an amazing amount of well-working relations that yieldeda very good agreement with experiments for a great variety of col-lisional processes between atoms (including hydrogen atoms) andcharged particles (electrons and protons).

To avoid any confusion, we should note the following. Beforedeveloping the free-fall model, Gryzinski tried to eliminate a discrep-ancy between classical and quantal ionization cross-sections (IC) athigh incident energies. At that time it was known that for high inci-dent energies E, the classical IC falls of as 1/E, while the quantal ICfalls o as log(E)/E (see, e.g., [2.6]). Gryzinski has shown [2.7, 2.8]that it is possible to obtain a classical IC which falls o as log(E)/Eif the HTMD for the bound electron would be fAshyp(p) 1/pk, k = 3(here the sux hyp stands for hypothetical). He also mentionedin [2.8] that due to the approximate character of the theory, a valueof k slightly dierent from 3 cannot be excluded. In 1970 Percivaland Richards [2.9] showed that a classical IC, calculated by extend-ing Bohrs correspondence principle, falls o as log(E)/E regardlessof the HTMD of the bound electrons (for the latest development of


Role of Singular Solutions of Quantal Equations in Atomic Physics 7

the quantal-classical correspondence see a paper by Flannery andVrinceanu [2.10]). Besides, some additional approximations of a sec-ondary importance, used by Gryzinski in [2.8] and in his precedingpapers, were also criticized later on; corrections, extensions, and sim-pler derivations were provided by a number of authors for some ofthe relations he derived (see, e.g., reviews [2.112.13]).

However, the above criticism does not relate to purely classicalcalculations based on the free-fall model. The free-fall model wassuccessfully applied to various inelastic processes not only in hydro-gen atoms, but in many other atoms and molecules as well see,e.g., review [2.14] and references in the later paper [2.15].)

Moreover, attempts by the Percivals group [2.16, 2.17] to use inclassical calculations for the bound electrons the quantal distributionfunction from Eq. (1), having the HTMD 1/p6, resulted in about60% discrepancy with the experimental IC of atomic hydrogen byelectrons at relatively low incident energies (LIE), while the employ-ment of the free-fall model (where the HTMD 1/p4) for calculat-ing the same IC [2.182.20] yielded a very good agreement with theexperiments at the same range of energies. (At high incident energies,ICs calculated using either the HTMD from (2.2) or the HTMD from(2.3) agree well with each other and with the experiments.)

Later Kim and Rudd [2.21] introduced a modied classical binary-encounter model and achieved an agreement (as the Gryzinskismodel did) with the LIE part of the experimental IC of atomic hydro-gen by electrons. However, their model deals only with the averagedkinetic energy of the bound electron and therefore cannot distinguishbetween the two disputed forms of the HTMD. Besides, even if Kimand Rudd would have allowed for details of the HTMD, this wouldnot have claried the above dispute because the model from [2.21]has two other weak points. First, Kim and Rudd omitted one outof two terms which could be sensitive to the HTMD of the boundelectrons. Second, they chose an approximate coecient in front ofthe remaining term. Thus, in frames of the classical or semiclassicalapproaches, the above dispute remains unresolved up to now.

Concerning purely-quantal (hereafter called quantal) calcula-tions of the same IC, the situation is as follows. Early (perturbative)



quantal calculations of the same IC showed an even much more dra-matic disagreement with the experiments than the mixed classical-quantal calculations from [2.16, 2.17]: Various such calculationsexceed the corresponding experimental results by several times atthe LIE (see, e.g., Fig. 6 from review [2.14]). More recent (non-perturbative) quantal calculations by a distorted wave method [2.22]or by close coupling methods [2.23, 2.24] (as well as by a closelyrelated method using R-matrix with pseudo-states [2.25]) improvedthe situation. However, dierent versions of the distorted wavemethod yield dierent results [2.26]. Moreover, even within the sameversion there is a signicant ambiguity, leading to dierences up to50100% at the incident energies 20 eV compare, e.g., Fig. 1from [2.21] and Fig. 1 from [2.26]. As for the close coupling/R-matrixmethods, their good agreement with one out of two sets of the exper-imental IC might be overshadowed by (and questioned due to) a dra-matic disagreement up to a factor of 2 between the results ofthese methods for the width of the spectral line B II 2s2p [2.27] andthe corresponding benchmark experiments [2.28], while the latter isin a good agreement with semiclassical calculations [2.29]. We alsonote a hidden crossing theory [2.30], which reproduced very well theshape of the experimental IC at the LIE, but that did not reproducewell the absolute values of the experimental IC at the LIE.

The above does not diminish the ingenuity of the authors of[2.212.27, 2.30], but rather illustrates a well-known fact: The lowerthe incident electron energy, the stronger the electron correla-tions become, creating greater and greater computational dicultiesfor quantal calculations. Besides, it is unclear whether these non-perturbative quantal methods are sensitive enough to the HTMD.

So, one point we are trying to make is that the above fundamen-tal dispute still remains unresolved: The experiments seem to favora HTMD 1/pk, where k is at least 1.5 times smaller than in thequantum HTMD. This puzzle is even more mind-boggling due tothe following two facts. First, while the HTMD (2.3) corresponds torelatively large values of the linear momentum p p0, these valuesare still below the relativistic domain of p. Indeed, since for hydrogenatoms we have p0/mc = e2/c 1/137, there is a signicant



range of p, where it seems that the non-relativistic quantum theory(used for deriving Eqs. (1) and (2)) should remain valid. Second,a non-relativistic classical treatment of a hydrogen atom can deter-mine exactly both the energies and the wave functions, so that thedynamics of hydrogen atoms can be fully described in terms of thecorresponding classical orbits, as was shown by Kay [2.31].

Another point is that a possible resolution of the above disputemight be connected to the problem of singular solutions of quantalequations, as will be shown below. A role of singular solutions ofthe Schrodinger or Dirac equations is a fundamental problem in itsown right. With respect to the Dirac equation, such a study startedin 1945, when Pomeranchuk and Smorodinskii [2.32] introduced anite nuclear size into the Coulomb problem. This activity wasresumed about 40 years ago see, e.g., papers [2.332.35], as wellas textbooks [2.36, 2.37]. In these studies, the interaction potentialinside the nucleus was modeled either as a constant potential insidea charged spherical shell or as a potential of a uniformly chargedsphere. All the preceding works focused primarily at the range of thenuclear charge Z > 1/ 137 to determine the electrodynamiclimit on the nuclear charge of stable hydrogen-like ions. It was shownthat in the range of Z > 1/, for the above two model potentialsinside the nucleus, it is possible to match the regular interior solu-tion with both regular and singular exterior solutions. However, forZ < 1/, for the above two model potentials, it was found impossi-ble to match the regular interior solution with the singular exteriorsolution see, e.g., textbook [2.38]. As a result of these studies, theparadigm is that, even with the allowance for a nite nuclear size,singular solutions of the Dirac equation for the Coulomb problemshould be rejected for Z < 1/.

Here we break this paradigm. First, we derive a general conditionfor matching a regular interior solution with a singular exterior solu-tion of the Dirac equation for arbitrary interior and exterior poten-tials. Then we nd explicit forms of several classes of potentials thatallow such a match. Finally, we show that, as an outcome, the HTMDfor the GSHA acquires terms falling o much slower than the 1/p6-law prescribed by the previously adopted quantal result.



2.2. Early, Unsuccessful Attempts to Explainthe Mystery

For spherically-symmetric potentials, radial parts of non-relativisticwave functions both in the coordinate representation Rnl(r) and inthe momentum representation Pnl(p) are interrelated as follows (see,e.g., [2.39])

Pnl(p) = [r/p]1/2(il/) 0

drJ l+1/2(pr/)Rnl(r)r, (2.4)

where Jl+1/2(z) is the Bessel function.For the GSHA, the non-relativistic quantal distribution function

(2.1) is

fquant(p) = |P10(p)|2p2,

P10(p) = [2/()]1/2(1/p) 0

dr sin(pr/)R10(r)r, (2.5)

The set of equations (2.2) show: The fact that something could bewrong with the function P10(p) at large p (at p p0) translatesinto a statement that the radial part R10(r) of the correspondingcoordinate wave function could be incorrect at small r (namely, atr /p0 = 2/(Zme2), the latter quantity practically coincidingwith the Bohr radius divided by Z).

For hydrogen atoms or hydrogen-like ions, the functions Rnl(r)have the following behavior at small r(see, e.g., [2.39, 2.40]) Rnl(r) rl, so that R10(r) const at small r. Since for the GSHA, the exper-iments show that the true HTMD falls o much slower than the non-relativistic quantum HTMD (2.2), then a true radial part Rtrue10 (r) ofthe coordinate wave function should have a relatively strong singu-larity at small r: Rtrue10 (r) 1/rq, q 1.

Now we conduct a preliminary analysis that would help ndinga resolution of this challenging fundamental enigma. As the rstattempt, let us look more carefully at the textbook solutions of theSchrodinger equation for a hydrogen atom. After separating variablesand reducing the problem to a one-dimensional equation for Rnl(r),



it turns out that the latter equation actually allows two classes ofsolutions characterized by two dierent types of behavior of Rnl(r)at small r in addition to the regular Rnl(r) rl, it allows also asingular solution (see, e.g., [2.39, 2.40]): Rnl(r) 1/rl+1.

For l 1 it is easy to reject the singular solution because the nor-malization integral

0 |Rnl(r)|2r2dr diverges at r = 0. As for l = 0,

the normalization integral does not have any divergence at r = 0. Theoverwhelming majority of textbooks on quantum mechanics rejectsthe second option for l = 0 without any explanation. Only a coupleof textbooks [2.39, 2.41] provide some explanations, but one of themis questionable while another explanation is incorrect.

Indeed, in the book [2.39], the singular solution for l = 0 isrejected on the ground that it would make the integral expressingthe mean value of the kinetic energy diverge at r = 0. However,the original idea of Schrodinger was that limitations imposed on for-mal solutions of his equation should be only the requirements forthe wave function to be single-valued and to allow the normaliza-tion (the latter requirement being equivalent to imposing bound-ary conditions) [2.42]. Therefore, an additional requirement for themean kinetic energy to be nite lacks the elegance intrinsic to theSchrodingers idea. Besides, this additional requirement implies thatthe mean kinetic energy of the bound electron is an observable quan-tity, what might be questionable.

As for the authors of the book [2.41], while proving that the sin-gular solution for l = 0 cannot exist, they make an implicit assump-tion that such a solution can be expanded in the Laurent series atr = 0. However, only analytic functions can have the Laurent seriesexpansion (see, e.g., [2.43]), while nothing prevents a formal solutionto be a non-analytic function. Below we show that such a solutionis indeed a non-analytic function, so that the proof in the book[2.41] is indeed incorrect.

An explicit form of the second formal solution of the Schrodingerequation for the GSHA for the entire range of r was never pub-lished to the best of our knowledge. By working out thisproblem in a straightforward way, we found the second formal



solution to be

R10(r) = const{Ei(2r) [exp(2r)]/(2r)} exp(r), (2.6)where Ei(z) is the exponential integral function (here and below weuse the Coulomb units Z = = m = e = 1, unless Z, or , or m,or e appears explicitly). At r 0, this solution has the followingapproximate form

R10(r) const[1/r + log(1/r2)]. (2.7)Equations (2.6) and (2.7) show that this solution is indeed a non-analytic function at r = 0, but there is no reason whatsoever toreject this formal solution based on its behavior at r = 0. An actualreason for rejecting this solution comes from analyzing it at r .It turns out that at r this solution behaves as exp(2r), thusmaking impossible to normalize it.

As the second attempt, let us move to relativistic quantummechanics and look at the textbook solutions of the Dirac equationfor a hydrogen atom. Similarly to the case of the Schrodinger equa-tion, the Dirac equation for a hydrogen atom formally allows twoclasses of solutions characterized by two dierent types of behaviorof the radial part Rnj(r) of the coordinate wave function at small r(see, e.g., textbooks [2.36, 2.38, 2.44, 2.45])

Rnj(r) 1/r1+s, s = [(j + 1/2)2 (Z)2]1/2, e2/(c) 1/137, (2.8)

where n is the radial quantum number and j is the quantum numbercorresponding to the total (spin plus orbital) angular momentum. Forthe GSHA we have n = 0 and j = 1/2, so that Eq. (2.2) reduces to:R1/2(r) 1/rq, q = 1 [1 (Z)2]1/2.

We are interested only in the case of Z 1. In this case, thesolution corresponding to k = 1 [1 (Z)2]1/2 (Z)2/2 2.66105Z2 is accepted, because its singularity at r = 0 is weak anddoes not prevent the normalization (below we refer to this solutionas being regular). The solution corresponding to q = 1 + [1 (Z)2]1/2 2 is rejected because its strong singularity at r = 0prevents the normalization.



2.3. Singular Solutions: General Results on Classesof Potentials to Which They are Applicable

Thus, for the GSHA, both the Schrodinger and the Dirac equa-tions provide a formal solution having a relatively strong singularityat r = 0, which could be needed for resolving the dispute on theHTMD. However, these singular solutions cannot be normalized: Forthe Schrodinger equation it blows up at r , while for the Diracequation such a solution is too much singular at r = 0.

The above analysis did not take into account a finite nuclear size.In the previous studies that allowed for a nite nuclear size, theinteraction potential inside the nucleus was modeled either as a con-stant potential inside a charged spherical shell or as a potential ofa uniformly charged sphere see, e.g., papers [2.322.35], as wellas textbooks [2.36, 2.37]. All the preceding works focused primarilyat the range of the nuclear charge Z > 1/ 137 to determinethe electrodynamic limit on the nuclear charge of stable hydrogen-like ions. It was shown that in the range of Z > 1/, for the twoabove model potentials inside the nucleus, it is possible to matchthe regular interior solution with both regular and singular exteriorsolutions. However, for Z < 1/, for the two above model potentials,it was found impossible to match the regular interior solution withthe singular exterior solution see, e.g., textbook [2.38]. As a resultof these studies, the paradigm is that, even with the allowance forthe nite nuclear size, singular solutions of the Dirac equation forthe Coulomb problem should be rejected for Z < 1/.

Let us study the problem in the following, more general set-up.We consider an arbitrary spherically-symmetric interaction potentialV (r), which takes two dierent forms in the interior region r < Rand in the exterior region r > R. It is known from the previousstudies [2.322.37] that the matching of the solutions of the Diracequation at the boundary r = R reduces to the requirement thatthe ratio f(r)/g(r) (r) of the two radial components of the Diracbispinor should be continuous at r = R. Therefore, instead of usingtwo coupled dierential equations for f(r) and g(r), it would be moreconvenient to start directly from the following Ricatti-type equation



for (r) (see Eq. (5.22) from [2.36]). For the ground state it reads

= 2/r + [V (r) E + 1] + [V (r) E 1]2, (2.9)

where E is the total energy and d/dr . Here and below we usethe natural units c = = m = 1. Introducing u(r) (r)r2, wesimplify Eq. (9) to the form:

u = r2[V (r) E 1][u(r)]2 + [V (r) E + 1]r2. (2.10)

We study solutions of Eqs. (2.9) and (2.10) at r 1 for the caseof R 1. For a regular solution the interior region (r < R), fromgeneral properties of the solutions of the Dirac equation we shouldexpect that reg(r) 1, so that ureg(r)/r2 1. Therefore, we candisregard the rst term in the right side of Eq. (2.10) and obtain:

ureg(r) r0

V (r)r2dr + (1 E)r3/3. (2.11)

For a singular solution in the exterior region (R < r 1), fromgeneral properties of the solutions of the Dirac equation we shouldexpect that sing(r) 1, so that using(r)/r2 1. Therefore, wecan disregard the second term in the right side of Eq. (2.10) andobtain

dusing/u2sing [V (r) E 1]/r2, (2.12)

so that

using(r) {

r[V (r)/r2]dr (1 + E)/r

}1. (2.13)

Thus, the matching condition ureg(R) = using(R) takes the form:

R0

V (r)r2dr + (1 E)r3/3

{

R[V (r)/r2]dr (1 + E)/r

}1. (2.14)



2.4. Engaging Singular Solutions for the SuccessfulExplanation of the Mystery

Now we specically study the GSHA, so that V (r) = Z/r atr R and also (1E) 1. The latter condition allows disregardingthe second term in the left side of Eq. (2.14). It is easy to verifythat for the interior region (r < R) indeed neither a constant inter-action potential V = Z/R, corresponding to a charged spheri-cal shell, nor an interaction potential of a uniformly charged sphereV = (r2/R2 3)Z/(2R) satisfy Eq. (2.14).

However, from experiments on the elastic scattering of electronson protons it is well known that the charge density inside protonneither has a peak at the periphery nor is constant, but rather ithas a maximum at r = 0 [2.462.48]. This means that for r < R, amore realistic interaction potential should rise toward r = R muchfaster than for V = (r2/R2 3)Z/(2R). The central point is thatsuch interaction potentials can satisfy Eq. (2.14) and thus provide amatch of the singular solution for the Coulomb field in the exteriorregion with a regular interior solution. We illustrate this point by thefollowing two particular examples.

First, for the region r R we consider a class of interactionpotentials of the form:

V (r) = (Z/R) exp[(R r)/b], 0 < b R. (2.15)

For this class of potentials, Eq. (2.14) yields: (2Zb3/R) exp(R/b) 2/R + Z/(2R2). The latter equation has, e.g., the following solu-tions:

(A) R b ln(Zb3) R1, where b 1/ ln(Z) so that Z R1 1;

(B) R Z, provided that b Z/ ln(4Z44/5).Second, for the region r R we consider another class of inter-

action potentials:

V (r) = (Z/R)(Rm + bm)/(rm + bm), m 3, 0 < b R.(2.16)



For instance, for m = 6, Eq. (2.14) yields: ZR5/(6b3) 2/R +Z/(2R2). The latter equation has, e.g., the following solutions:

(A) R [12/(Z)]1/6b1/2 R2, where b 1/(Z)1/3, so thatZ R2 1;

(B) R Z, provided that b (Z77/15)1/3.Thus, for those interaction potentials in the interior region that

rise rapidly enough toward the boundary r = R, it is in fact possibleto match the singular solution for the Coulomb eld in the exteriorregion with a regular interior solution. Therefore, for r R, theradial part of the Dirac bispinor for the GSHA is a linear combinationof both the regular and singular solutions for the Coulomb eld.Then, by using the well-known forms of these regular and singularsolutions [2.342.36, 2.38, 2.45] and keeping only the leading termsin the pre-exponential factor for R r 1/(Z), we obtain for theGSHA:

f(r) {1/r1 + [(Z)2(2)2 ]12/r1+}(2)1/2+(1 E)1/2 exp(r),

g(r) {1/r1 + [(2 1)(Z)2(2)2 ]1/r}(2)1/2+(1 + E)1/2 exp(r),

(1 Z22)1/2, (1 E2)1/2, (E0 E), E0 (1 Z22)1/2. (2.17)

Here E0 is the unperturbed energy of the GSHA, E is the energy ofthe GSHA perturbed by a nite nuclear size.

Now we can calculate the HTMD fAsRQFN(p) for the solution fromEq. (2.17) for p0 p 1/R (here and below the sux RQ stands forrelativistic quantal and the sux FN stands for nite nucleus).This can be done analytically by using a known integral

0

xa[exp(r)] sin(px )dx= [(1 + a)/(p2 + 2)(1+a)/2] sin[(1 + a) arctan(p/)],

(2.18)



where (z) is the gamma-function. As a result, for the GSHA weobtain:

fAsRQFN(p) [1/p6 + (22)1/p4 + 3/2/p3], (Z)2 1, < 2. (2.19)

Therefore, if we would approximate the HTMD (2.19), which is validfor p0 p 1/R, by some eective power law fAs(p) 1/pk, wewould get the value of k noticeably smaller than 6.

We note that the mean kinetic energy has a nite value regardlessof Eq. (2.19), since Eq. (2.19) is not valid in the ultrarelativisticrange of p 1/R. A nite mean kinetic energy is a general featureof interaction potentials having a nite value and a zero slope at theorigin (see, e.g., [2.33]) and both classes of the interaction potentials(2.15) and (2.16) possess these properties.

2.5. Opening a Way to Test Intimate Details

of the Nuclear Structure by PerformingAtomic, Rather than Nuclear, Experiments

We broke the paradigm that, even with the allowance for the nitenuclear size, singular solutions of the Dirac equation for the Coulombproblem should be rejected for Z < 1/. We derived a general con-dition for matching a regular interior solution with a singular exte-rior solution of the Dirac equation for arbitrary interior and exteriorpotentials. We found explicit forms of several classes of potentialsthat allow such a match. We showed that, as an outcome, the HTMDfor the GSHA acquires terms falling o much slower than the 1/p6-law prescribed by the previously adopted quantal result.

The above might be considered as a resolution of a long stand-ing dispute between classical and quantal calculations of the HTMDfor the GSHA in favor of the presented, more sophisticated quantalcalculations. Besides, our results open up a unique way to test inti-mate details of the nuclear structure by performing atomic (ratherthan nuclear) experiments and calculations. This outcome is highlycounterintuitive.



Indeed, the HTMD can be subdivided into three ranges:non-relativistic (p0 p < 1), relativistic (1 < p < 1/R), andultrarelativistic (1/R < p). Obviously, modications of the potentialinside the nucleus (i.e., at r < R) aect the ultrarelativistic part ofthe HTMD (p > 1/R). However, we found that such modicationscould also result in noticeable/observable changes of the HTMD notonly in the adjacent relativistic range of the HTMD, but in the non-relativistic part of the HTMD as well.

Our results should not be perceived as casting any doubt on thecomputational methods used for quantal, fully-numerical calculationsfor electron ionization of hydrogen atoms. We also note that themodication of the wave function, predicted by Eq. (2.17) on the scaleintermediate between the atomic and nuclear sizes, might be probedin quantal numerical calculations not only for electron ionization ofhydrogen atoms, but for some other processes as well.

However, what we would really like to emphasize is that classicaltreatments of electron ionization should be sensitive to the modica-tions of the HTMD, predicted by Eq. (2.19). Therefore, we hope thatour results could revitalize classical calculations of electron ioniza-tion at a more sophisticated level, which would rival computationalmethods used in quantal numerical calculations for LIE.

While resolving this long-standing dispute in favor of quantal cal-culations, our results do not diminish the signicance of Gryzinskisworks for atomic physics. His works inspired us to perform this study.


Chapter 3

Classical Description of Crossingsof Energy Terms and of

Charge Exchange

3.1. Brief History and Importance of theCorresponding Studies

Charge exchange and crossings of corresponding energy levels thatenhance charge exchange are strongly connected with problems ofenergy losses and of diagnostics in high temperature plasmas see, e.g., [3.1, 3.2] and references therein. Besides, charge exchangewas proposed as one of the most eective mechanisms for popula-tion inversion in the soft X-ray and VUV ranges [3.33.6]. One ofthe most fundamental theoretical playgrounds for studying chargeexchange is the problem of electron terms in the eld of two sta-tionary Coulomb centers (TCC) of charges Z and Z separated bya distance R. It presents fascinating atomic physics: the terms canhave crossings and quasicrossings.

The crossings are due to the fact that the well-known NeumannWigner general theorem on the impossibility of crossing of termsof the same symmetry [3.7] is not valid for the TCC problem ofZ = Z as shown in paper [3.8]. Physically it is here a consequenceof the fact that the TCC problem allows a separation of variables inthe elliptic coordinates [3.8]. As for the quasicrossings, they occurwhen two wells, corresponding to separated Z- and Z -centers, havestates and , characterized by the same energies E = E, by thesame magnetic quantum numbers m = m, and by the same radialelliptical quantum numbers k = k [3.93.11]. In this situation, the

19



electron has a much larger probability of tunneling from one well tothe other (i.e., of charge exchange) as compared with the absence ofsuch degeneracy.

These rich features of the TCC problem also manifest in a dierentarea of physics such as plasma spectroscopy as follows. A quasicross-ing of the TCC terms, by enhancing charge exchange, can result inunusual structures (dips) in the spectral line prole emitted by a Z-ion from a plasma consisting of both Z- and Z -ions, as was showntheoretically and experimentally [3.123.17]. From the experimentalshape of these dips it is possible to determine rates of charge exchangebetween multicharged ions, which is a fundamental reference datavirtually inaccessible by other experimental methods [3.17].

Before year 2000, the paradigm was that the above sophisti-cated features of the TCC problem and its ourishing applicationswere inherently quantum phenomena. But then in year 2000, papers[3.18, 3.19] were published presenting a purely classical descriptionof both the crossings of energy levels in the TCC problem and thedips in the corresponding spectral line proles caused by the cross-ing (via enhanced charge exchange). These classical results wereobtained analytically based on rst principles without using anymodel assumptions.

In the classical studies, the TCC systems represent diatomicRydberg quasimolecules encountered, e.g., in plasmas containingmore than one kind of multicharged ions. Naturally, the classicalapproach is well-suited for Rydberg quasimolecules.

Later applications of the results from papers [3.18, 3.19] includeda magnetic stabilization of Rydberg quasimolecules [3.20], a problemof continuum lowering (CL) in plasmas [3.21] (which plays a key rolein calculations of the equation of state, partition function, bound-freeopacities, and other collisional atomic transitions in plasmas), andthe study of the classical Stark eect for Rydberg quasimolecules(paper [3.22] and Sec. 3.5).

In these studies a particular attention was given to circularRydberg states (CRS). Circular states of atomic and molecular sys-tems are an important subject in its own right. They have beenextensively studied both theoretically and experimentally for several


Classical Description of Crossings of Energy Terms and of Charge Exchange 21

reasons (see, e.g., [3.183.20, 3.233.37] and references therein):(a) they have long radiative lifetimes and highly anisotropic colli-sion cross-sections, thereby enabling experiments on inhibited spon-taneous emission and cold Rydberg gases, (b) these classical statescorrespond to quantal coherent states, objects of fundamental impor-tance, (c) a classical description of these states is the primary termin the quantal method based on the 1/n-expansion, and (d) they canbe used in developing atom chips.

As examples of experimental studies of Rydberg states, we referto paper [3.23] where such studies made in the last three decadeshave been enumerated. In particular, Day and Ebel [3.38] predictedtheoretically in 1979 that probability of a wake electron being cap-tured by fast-moving ions traversing a solid to a state with largeprincipal (n) and angular momentum (l) quantum numbers is quitehigh and much of the time the electron is captured CRS (l = n 1)distributed over a narrow band near nmax. Note here that l = n 1denes circular orbits, whereas the full qualication of CRS requires|ml| = l = n1. Day and Ebel proposed existence of an optical win-dow in ion velocity as a possible explanation for non-observabilityof the CRS in beam-foil spectroscopy work. Also, the CRS are bothlong lived with respect to radiative transitions and short lived withrespect to collisions, hence their observation requires a wide apertureand very good vacuum. Pegg et al. in 1977 [3.39] observed a strongcascade tails in the decay curves of Cu18+ in a beam-foil interactionand attributed it to the successive decay of long-lived CRS or yraststates. Note that CRS can radiate only to the next lower state,which leads to a chain of successive yrast transitions till they reachto the ground state. Recently, from the study of the time-resolvedbeam-foil X-ray spectra of projectile or projectile-like ions of 2p,2s 1s transitions in H-like Fe, Ni, Cu, and Zn at dierent delaytimes (in the range 2501600 ps). Nandi identied, in each case, asingle circular Rydberg and/or an elliptic Rydberg state cascadingsuccessively to the 2p or 2s level.

The paradigm was that the above sophisticated features of theTCC problem and its ourishing applications are inherently quantumphenomena. We disprove this paradigm. Here we present a purely



classical description of the crossings of energy terms in the TCCproblem leading to charge exchange. Our classical description is basedon rst principles and does not use any model assumptions.

In this chapter, we also present detailed studies of modicationsof the classical energy terms, their crossings, and charge exchangeunder various external factors. Those factors, such as a magneticeld, or an electric eld, or the screening by plasma electrons, leadto quite dierent physical outcomes.

There are also two additional classical studies using (as the start-ing point) the same formalism as above, though not dealing withcrossings of energy terms. One of these studies portrays classicallymuonic-electronic negative hydrogen ion, i.e., the system where anelectron and a muon are bound to a proton. It employs also theseparation of rapid and slow subsystems. This study is presented inAppendix A.

In another work, the object of a classical study is diatomic Ryd-berg quasimolecules in a laser eld. It reveals laser-eld-caused satel-lites of spectral lines emitted by the quasimolecules. This study ispresented in Appendix B.

3.2. Helical States of Diatomic Rydberg Quasimolecules

We consider a TCC system, where the charge Z is at the origin andthe Oz axis is directed to the charge Z, which is at z = R (here andbelow the atomic units = e = me = 1 are used). In the cylindricalcoordinates (z, , ), the Hamiltonian of the system has the form

H = (p2z + p2 + p

2/

2)/2 + U(z, ), (3.1)

where the potential energy is

U(z, ) = Z/(z2 + 2)1/2 Z /[(R z)2 + 2]1/2. (3.2)The relation between the momenta and the corresponding velocitiesfollows from the Hamiltonian equations of the motion:

dz/dt = H/pz = pz, d/dt = H/p = p,

d/dt = H/p = p/2. (3.3)



Since H does not depend on , the corresponding momentum isconserved:

p = 2d/dt M = const. (3.4)Physically, the separation constant M is a projection of the angularmomentum on the internuclear axis. Thus, the z- and -motions canbe determined separately from the -motion. Then the -motion canbe found from the -motion via Eq. (3.4).

The Hamiltonian for the z- and -motions can be represented inthe form

H = (p2z + p2)/2 + Ue(z, ), (3.5)

where an eective potential energy (EPE) is:

Ue(z, ) = M2/(22) + U(z, ). (3.6)

We introduce scaled (dimensionless) variables w and v, a scaled pro-jection of the angular momentum m, as well as a ratio of the nuclearcharges b:

w z/R, v /R, m M/(ZR)1/2, b Z /Z. (3.7)Then the EPE can be re-written as

Ue (Z/R)ue(w, v,m, b),ue(w, v,m, b) = m2/(2v2) (w2 + v2)1/2 b[(1 w)2 + v2]1/2.

(3.8)

Now we seek equilibrium points of the EPE. We equate to zeroits derivative with respect to w

Ue/w = (Z/R)ue/w = 0, (3.9)

and nd a relation

b/[(1 w)2 + v2

]3/2= w/

[(1 w)(w2 + v2)3/2

], (3.10)

which determines a line v0(w) in the plane (w, v), where the equilib-rium points are located:

v0(w, b) ={[w2/3(1 w)4/3 b2/3w2]/[b2/3 w2/3/(1 w)2/3]

}1/2.

(3.11)



Fig. 3.1. The dependence of the equilibrium value of the scaled radius v /R of theelectron orbit on the scaled axial coordinate w z/R of the electron for the ratio ofnuclear charges Z/Z = 3.

For b < 1, the equilibrium value of v exists for 0 w < b/(1 + b)and for 1/(1 + b1/2) w 1. For b > 1, the equilibrium value of vexists for 0 w 1/(1 + b1/2) and for b/(1 + b) < w 1. For b = 1,the equilibrium value of v exists for the entire range of 0 w 1.Below we refer to these intervals as the allowed ranges of w. As anexample, Fig. 3.1 shows the dependence v0(w) for b = 3.

By equating to zero the derivative of the EPE with respect to v

Ue/v = (Z/R)ue/v = 0 (3.12)

and then substituting v0(w, b) from Eq. (3.10) instead of v, we nd:

m = v20(w, b)/{(1 w)1/2[w2 + v20(w, b)]3/4} m0(w, b). (3.13)While deriving Eq. (3.13), we used Eq. (3.10) to eliminate an explicitdependence on b, so that b enters Eq. (3.13) only implicitly asan argument of the function v0(w, b). In a number of subsequentderivations, we will also use Eq. (3.10) for the same purpose withouta further notice.

For each set of (b,m), Eq. (3.13) determines one or more val-ues of wi(b,m) and then Eq. (3.11) determines the corresponding



values of vi = v0(wi(b,m), b) such that at (wi, vi) there is either astable or unstable equilibrium (the stability of the equilibria will bestudied in the next section). A physically equivalent viewpoint, moreconvenient for the further analysis, is the following: for each set of(w, b), where w belongs to the allowed ranges, Eq. (3.13) determinesan equilibrium value of m0(w, b) in addition to the equilibriumvalue of v0(w, b) determined by Eq. (3.11). Thus, for each b we dealwith sets of equilibrium values (wi, v0i,m0i), where wi belongs to theallowed ranges and v0i v0(wi), m0i m0(wi).

Now, for some value of b, we consider a set of equilibrium val-ues (wi, v0i, m0i) and expand the EPE ue in terms of w and v,where

w w wi, v v v0i. (3.14)

The expansion has the form

ue u0 + uww(w)2/2 + uvv(v)2/2 + uwv(w)(v),u0 ue(wi, v0i,m0i). (3.15)

In the subsequent formulas we drop the sux i for brevity. Thesecond derivatives of the EPE in Eq. (3.15) are

uww (2ue/w2)0 = [1/(1 w) 3wP/Q2]/(w2 + v20)3/2,uvv (2ue/v2)0 = [1/(1 w) + 3wP/Q2]/(w2 + v20)3/2,uwv (2ue/wv)0 = 3v0w(2w 1)/[(w2 + v20)3/2Q2], (3.16)

where the sux 0 at the derivatives means that after the dieren-tiation one should set v = v0(w) and m = m0(w). In Eq. (3.16) weintroduced the following notations:

P w(1 w) + v20 ,Q (w2 + v20)1/2[(1 w)2 + v20 ]1/2. (3.17)

Since generally uwv = 0, a rotation of the reference frame isrequired in order to transform the EPE to so-called normal coordi-nates, diagonalizing the matrix of the second derivatives of the EPE



[3.40, 3.41]:

w = w cos + v sin, v = w sin+ v cos. (3.18)It is easy to nd that

tan 2 = 2uwv/(uww uvv) = (1 2w)v0/P, (3.19)so that

cos = [(1 + P/Q)/2]1/2, sin = [(1 P/Q)/2]1/2sign(1 2w).(3.20)

In the normal coordinates, the EPE takes the form

ue u0 + w22/2 + v22+/2, (3.21)where

[1/(1 w) 3w/Q]1/2/(w2 + v20)3/4. (3.22)We note that + is always real. Physically, it is a scaled (dimen-

sionless) frequency of small oscillations around the equilibrium in thedirection of the normal coordinate v. In this section, any frequencyF and its scaled (dimensionless) counterpart f are related as follows:

f (R3/Z)1/2F. (3.23)As for the quantity , it is real if

Q 3w(1 w). (3.24)Using the denition of Q from Eq. (3.17), the condition (3.24) canbe re-written as:

v0(w, b) {w(1 w) 1/2 + [9w2(1 w)2 w(1 w) + 1/4]1/2}1/2 vcrit(w). (3.25)

Physically, under the condition (3.25), the quantity is the fre-quency of small oscillations around the equilibrium in the directionof the normal coordinate w.

Thus, if v0(w, b) > vcrit(w), the EPE has a two-dimensional min-imum at the equilibrium values of w and v = v0(w, b), so that the



equilibrium is stable. Introducing a scaled (dimensionless) time

(Z/R3)1/2t, (3.26)we obtain the nal expression for the small oscillations around thestable equilibrium in the form:

w() = aw[cos( + w)] cos av[cos(+ + v)] sin,v() = aw[cos( + w)] sin + av[cos(+ + v)] cos. (3.27)

Here amplitudes aw, av and phases w, v are determined by initialconditions; sin and cos are given by Eq. (3.20).

Equation (3.4) for the -motion can be re-written in the scalednotations as

d/d = m/v2. (3.28)

Substituting in Eq. (28) v() v0 + v(), where v() is given byEq. (3.27), and integrating over , we obtain the solution for the-motion

() f 2f{1 aw[sin( + w) sinw] sin+1+ av[sin(+ + v) sinv)] cos}/v0, (3.29)

where

f (1 w)1/2[w2 + v20 ]3/4 (3.30)is a scaled (dimensionless) primary frequency of the -motion. Equa-tions (3.28) and (3.29) show that the -motion is a rotation aboutthe internuclear axis with the frequency f , slightly modulated byoscillations of the scaled radius of the orbit v at the frequencies +and . In other words, the motion of the electron occurs on a conicalsurface of the averaged radius v0(w, b).

It is interesting to note the following relation between and fvalid for any b:

[(2+ + 2)/2]

1/2 = f. (3.31)

If v0(w, b) > vcrit(w), so that the equilibrium is stable, the relation(3.31) physically means that the rms frequency of the small oscilla-tions of w and r is equal to the primary frequency of the -motion for



Fig. 3.2. Sketch of the helical motion of the electron in the ZeZ-system at the absenceof the magnetic eld. We stretched the trajectory along the internuclear axis to make itsdetails better visible.

any b. If v0(w, b) < vcrit(w), the equilibrium is unstable: the quantity takes imaginary values; its absolute value || is the increment ofthe instability developing in the direction of the normal coordinatew. However, even for the unstable equilibrium, the relation (3.31)still holds for any b.

Thus, for the stable motion, the electron trajectory is a helix onthe surface of a cone, with the axis coinciding with the internuclearaxis. In this helical state, the electron, while spiraling on the surface ofthe cone, oscillates between two end-circles which result from cuttingthe cone by two parallel planes perpendicular to its axis (Fig. 3.2).

3.3. Crossings of Classical Energy Termsof Diatomic Rydberg Quasimolecules

In this section, our goal is to nd out whether or not classical mechan-ics reproduces crossings of energy terms of the same symmetry. In thequantum TCC problem, terms of the same symmetry means termsof the same magnetic quantum number m [3.73.11]. Therefore, inour classical TCC problem, from now on we x the angular momen-tum projection M and study the behavior of the classical energy atM = const 0 (the results for M and M are physically the same).Since m M/(ZR)1/2 in accordance to Eq. (7), then using Eq. (3.13)we obtain

R(w, b,M) = M2/[Zm20(w, b)], (3.32)



so that z(w, b,M) = wR(w, b,M) and (w, b,M) = v0(w, b)R(w,b,M). For any b > 0, for any w from the allowed ranges (controlledby the value of b), and for any M 0, the internuclear distance R,the location of the orbital plane z, and the radius of the orbit haveeach its individual unique equilibrium value given by the functionsR(w, b,M), z(w, b,M), and (w, b,M), respectively.

For simplicity, we consider the limit where aw = av = 0. Inother words, here we disregard the small oscillations of the z-and -coordinates and focus at the primary motion (i.e., at the-motion), where the electron moves around the circle of the radius(w, b,M) = v0(w, b)R(w, b,M). In this situation, the total energyE of the electron coincides with the EPE given by Eq. (3.6), so that

E = M2/(22) + U(z, ). (3.33)

Similarly to Eq. (3.8), we introduce a scaled total energy e as follows:

E (Z/R)e[w, v0(w, b),m0(w, b), b],e[w, v0(w, b),m0(w, b), b]

= m20/(2v20) (w2 + v20)1/2 b[(1 w)2 + v20]1/2. (3.34)

Using Eqs. (3.10) and (3.13), the latter quantity can be re-written as

e(w, b) = [w(1 w) + v20(w, b)/2]/{(1 w)[w2 + v20(w, b)]3/2}.(3.35)

Now we substitute R(w, b,M) from Eq. (3.32) in Eq. (3.34) andnd:

E(w, b,M) = (Z/M)2m20(w, b)e(w, b). (3.36)

Thus, for any b > 0 and M 0, Eqs. (3.32) and (3.35) determinein a parametric form (via w) the dependence of the energy E on theinternuclear distance R, i.e., the classical energy terms.

Figure 3.3 shows the dependence of the scaled energy (M/Z)2Eon the scaled internuclear distance (Z/M2)R (both quantities aredimensionless) for b = 3. The results are astonishing. There is morethan one classical energy term namely, there are three terms ofthe same symmetry. Just this is already counterintuitive. Moreover,



Fig. 3.3. Classical energy terms: The dependence of the scaled classical energy (M/Z)2Eon the scaled internuclear distance (Z/M2)R for Z = 3Z.

two of these classical energy terms cross. (We call it V-type crossingsince it resembles the inclined letter V.)

We emphasize that the above example of Z /Z = 3 representsa typical situation. In fact, for any pair of Z and Z = Z there arethree classical energy terms of the same symmetry and the upper termalways crosses the middle term. (For Z = Z there is only one termin the corresponding plot and no crossing as should be expected.)

For arbitrary Z /Z = 1, these three terms have the followingorigin. At R , the lower term corresponds to the energy of thehydrogen-like ion (HI) of the nuclear charge Zmax max(Z , Z),(E (Zmax/M)2/2), slightly perturbed by the charge Zmin min(Z , Z). This ion is in a Stark state, corresponding classicallyto the zero projection A0 of the RungeLenz vector [3.40] on theaxis Oz and corresponding quantum-mechanically to the zero electricquantum number q n1 n2 (n1, n2 are the parabolic quantumnumbers [3.42]) as should be expected for the circular states. AtR 0, the lower term corresponds to the energy of the HI of thenuclear charge Z +Z, (E [(Z+Z )/M ]2/2), i.e., to the so-calledunited atom [3.73.11].



At R , the middle term corresponds to the energy of the HI ofthe nuclear charge Zmin (E (Zmin/M)2/2), slightly perturbed bythe charge Zmax. This ion is in a Stark state, corresponding classicallyto A0 = 0 and corresponding quantum-mechanically to q = 0 asshould be expected for the circular states.

At R , the upper term corresponds to a near-zero-energystate (where the electron is almost free). In terms of the parameterw, this term at R corresponds to w 1/(1 + b1/2) w0. Ifthe ratio Z /Z is of the order of (but not equal to) unity, the upperterm at R can be described only in the terminology of ellipticalcoordinates (rather than parabolic or spherical coordinates), meaningthat even at R the electron is shared between the Z- and Z -centers. However, in the case of Z Z, the potential at w = w0is very close to the potential due to an isolated charge Z , so thatthe upper term can be asymptotically considered as the Z -term. Itcrosses the middle term, which asymptotically is the Z-term (sinceZmin = Z for Z > Z). Likewise, in the case of Z Z, the potentialat w = w0 is very close to the potential due to an isolated charge Z,so that the upper term can be asymptotically considered as the Z-term. It crosses the middle term, which asymptotically is the Z -term(since Zmin = Z for Z < Z).

Thus, when Z and Zdier signicantly from each other, the cross-ing occurs between two classical energy terms which can be asymp-totically labeled as Z- and Z -terms. This situation classically depictscharge exchange. Indeed, say, initially at R , the electron was apart of the HI of the nuclear charge Zmin. As the chargesZ and Z

come relatively close to each other, the middle and upper classicalterms cross and the electron is shared between the Z- and Z -centers.Finally, as the charges Z and Z go away from each other, the electronends up as a part of the HI of the nuclear charge Zmax.

Physically, crossings and quasicrossings in the TCC problem arepossible only because the problem possesses an additional (to E andM) conserved quantity (integral of motion) [3.43]A = (1/2)(p MM p)

ez M2/R Zz/r Z (R z)/|R r|+ Z , ez = R/R,(3.37)



where p is a linear momentum vector, the sux z means a z-projection. The last term in Eq. (3.37) was added to facilitate thetransition at R to the Stark eect for the HI of the nuclearcharge Z slightly perturbed by the charge Z . The quantity Ais a z-projection of a generalized RungeLenz vector [3.43]. For our classicalcircular TCC states this quantity becomes:

A(w, b) = Za(w, b),

a(w, b) w[w2 + (1 w)2 + v20(w, b)(1 2w)/(1 w)]/[w2 + v20(w, b)]

3/2 m20(w, b) b. (3.38)

In the quantum TCC problem, two terms of the same symmetry(m = m) should dier by their angular elliptic quantum numbers fora quasicrossing (s = s+1, while k = k) or by both elliptic quantumnumbers for a crossing (s = s, k = k) [3.93.11]. So, in either caseit should be s s s = 0. To nd a classical correspondenceto these selection rules, we recall that for each quantum term,the eigenvalues E*, A*, M* of the operators E, A, M are somefunctions of the quantum numbers k, s, m (in particular, M = m).Therefore, the quantum number s should also be some function of E*,A*, M*: s = g(E*, A*, M*). The eigenvalues E*, A*, M* correspondto the classical quantities E(w), A(w), M(w) of our circular TCCstates, so that s = g[E(w), A(w), M(w)]. It turns out that at thevalue of w = wc, corresponding to the crossing of the upper andmiddle classical terms, we have:

E(wc) = M (wc) = 0, A(wc) = 0, (3.39)

where the sign stands for the derivative with respect to w. Con-sequently, in the vicinity of wc we have: s = [(g/E)E(wc) +(g/A)A(wc) + (g/M)M (wc)]w = (g/A)A(wc)w. So, ifit were not for the presence of A and for the fact that A(wc) = 0,we would have had s = 0 contrary to the above selectionrules. Thus, the classical crossings are intimately connected with thedynamical symmetry of the TCC problem, and the remarkable setof rules from Eq. (3.39) is a classical counterpart of the abovequantum selection rules.



We also note that at the crossing we have = 0, where is given by Eq. (3.22). So, among the two crossing classical energyterms, the middle term corresponds to stable equilibria of the three-dimensional motion, while the upper term corresponds to unstableequilibria of the three-dimensional motion. Thus, the value of wc cor-responding to the crossing can be determined as a root of either one ofthe following three equations: from E(wc) = 0, or from M (wc) = 0,or from

v0(wc, b) = vcrit(wc), (3.40)

where vcrit(w) is dened in Eq. (3.25). In any case, one should acceptonly the root wc within the interval (0,1), the endpoints 0 and 1being excluded.

It turns out that the form of the parametric dependence can besignicantly simplied and some of the properties of it can be foundanalytically by introducing a new parameter

=(

1w 1

)1/3. (3.41)

In this case, w = 0 will correspond to = + and w = 1 willcorrespond to = 0, thus > 0 in the allowed regions. The pointsw1 = 1/(1 + b1/2) and w3 = b/(1 + b) dening the allowed regions0 < w < w1, w3 < w < 1 (here we assume b > 1) will correspond to1 = b1/6 and 3 = 1/b1/3 (notice that 0 < w < w1 corresponds to+ > > 1 and w3 < w < 1 corresponds to 3 > > 0).

From now on we will denote the scaled internuclear distance(Z/M2)R as r:

(Z/M2)R = r = 1/m20, (3.42)

where the second equality in Eq. (3.42) is the consequence of Eq.(3.32). Also from now on we will use for the scaled energy (M/Z)2Ethe notation 1 and for the scaled Hamilton function (called in thisbook classical Hamiltonian) the notation h:

(M/Z)2E = 1, (M/Z)2H = h. (3.43)Obviously, h = 1, since classically H = E.



The scaled energy terms 1(r) will take the following form inthe -parametrization dened by Eq. (3.41):

1(,b) = (b2/3 4)2((3 2) + b2/3(23 1))

2(3 1)2(6 1) , (3.44)

r(, b) =

b2/32 1(6 1)3/2

(b2/3 4)2. (3.45)

The parametric plot of 1(, b) from Eq. (3.44) versus r(, b) from(3.45) with the parameter varied from 0 to 1/b1/3 and from b1/6 to+ for b = 3 will yield the same graph as in Fig. 3.3.

3.4. Eects of a Static Magnetic Field: Stabilizationof Diatomic Rydberg Quasimolecules

We consider here CRS of the ZeZ -system in the presence of a uniformmagnetic eld B parallel to the internuclear axis. For z R or for(Rz) R when the electron is mainly bound to the Z or the Z ionand is perturbed by the other fully stripped ion, these circular orbitsdepict Stark states which correspond classically to zero projectionof the RungeLenz vector [3.40] on the axis OZ and quantally tozero electric quantum number k = n1 n2, where n1 and n2 arethe parabolic quantum numbers [3.42]. The classical Hamiltonian isgiven by (in atomic units):

H(, z) = M2/(22) Z/(2 + z2)1/2 Z /[2 + (z R)2]1/2+M +22/2, B/(2c). (3.46)

Here M is the constant z-component of the angular momentum and is the Larmor frequency expressed in practical units as (s1) 8.794 106 B(G).

Introduce the following scaled quantities:

b Z /Z, u /R, w z/R, m M/(ZR)1/2, M3/Z2, h HM 2/Z2 = M2/Z2, (3.47)



where E is the energy. In these notations, the scaled Hamilto-nian/energy is

h(u,w, ) = m2(u,w, ),

(u,w, ) m2/(2u2) 1/(u2 + w2)1/2 b/[u2 + (1w)2]1/2+/m2 + 2u2/(2m6). (3.48)

The conditions for dynamic equilibrium are

h/w = m2{w/(u2+w2)3/2b(1w)/[u2+(1w)2]3/2} = 0 (3.49)and

h/u = m2{m2/u3 + u/(u2 + w2)3/2 + bu/[u2 + (1 w)2]3/2

+2u/m6} = 0. (3.50)Equation (3.49) shows that equilibrium along the internuclear axisdoes not depend on the scaled magnetic eld . In terms of theequilibrium value w0 of w, the equilibrium value of u can thereforebe expressed as

u(w0, b) = {[w0(1w0)2]2/3b2/3w20}1/2/{b2/3[w0/(1w0)]2/3}1/2,(3.51)

which only exists within the following allowed ranges,

0 w0 < b/(1 + b) and 1/(1 + b1/2) w0 1; b < 1;0 w0 1/(1 + b1/2) and b/(1 + b) < w0 1; b > 1;0 w0 1; b = 1

(3.52)

of w0. Equation (3.50), represents the condition for equilibrium inthe orbital plane and can be re-written in the form

m(w0, b, ) = {f/4 + (f2/4 + j)1/2/2+ [f 2/2 j + (f3/4)/(f2/4 + j)1/2]1/2/2}1/2,

(3.53)

where, in terms of u(w0, b), given by Eq. (3.51),

f(w0, b, ) u4(w0, b)/[u2(w0, b) + w20]3/2+ bu4(w0, b)/[u2(w0, b) + (1 w0)2]3/2, (3.54)



and

j(w0, b, ) [u4(w0, b)2/18]1/3g (4/g)[2u8(w0, b)4/3]1/3,(3.55)

with

g(w0, b, ) | 9f2 + [81f4 + 768u4(w0, b)2]1/2|1/3. (3.56)The plus and minus signs in Eq. (3.53) correspond, respectively to thepositive and negative projections of the angular momentum alongthe magnetic eld. For each set {b,m, } of parameters, Eq. (3.53)determines the equilibrium value w0 of the scaled z-coordinate of theorbital plane.

The internuclear distance R was considered to be frozen. Inorder to reproduce the electronic terms, i.e., the dependence ofthe electronic energy on the internuclear distance, one should nowallow R to be a slowly varying adiabatic quantity (slowly varyingwith respect to the electronic motion, as in the BornOppenheimerapproach [3.44]).

We consider energy terms of the same symmetry which, for thequantal ZeZ -problem, means terms with the same magnetic quan-tum number M [3.73.11]. Therefore, in our classical ZeZ -problem,from now on we consider xed projection of the angular momentumM and study the behavior of the classical energy keeping M constant.

We introduce the scaled internuclear distance similarly toEq. (3.43):

r(w0, b, ) = 1/m2(w0, b, ). (3.57)

On substituting w = w0 into Eq. (3.48), then,

h(w0, b, ) = m2(w0, b, )[u(w0, b), w0, ]. (3.58)

Thus, for any positive ratio of the nuclear charges b > 0 andfor any value of the scaled magnetic eld , the dependence of thescaled energy h on the scaled internuclear distance r is determined byEqs. (3.57) and (3.58) in terms of the parameter w0, which takes allvalues from the allowed ranges specied by Eq. (3.52). In other words,



Fig. 3.4. The scaled electronic energy h versus the scaled internuclear distance r for theratio of the nuclear charges b = 3/2 at the absence of the magnetic eld (h and r aredened by Eqs. (3.58) and (3.57), respectively).

Eqs. (3.57) and (3.58) determine the classical electronic energy termsfor any strength of the magnetic eld, including the strong eld case.

Figure 3.4 shows the scaled electronic energy h versus the scaledinternuclear distance r for b = 3/2 in the absence of magnetic eld.

We note that the upper and middle energy terms terminate atsome r = rmin, so that there are no CRS at r < rmin for thesetwo energy terms. The classical energy of the CRS acquires animaginary part at r < rmin, corresponding quantally to virtualstates/resonances. There may well be non-CRS at r < rmin in thesame energy range.

At this point it might be useful to clarify the relation between theclassical energy terms h(r) and the energy E. The former is a scaledquantity related to the energy as specied above: E = (Z/M)2h.The projection M of the angular momentum on the internuclear axisis a continuous variable. The energy E depends on both h and M .Therefore, while the scaled quantity h takes a discrete set of values,the energy E takes a continuous set of values (as it should be inclassical physics).

We now turn on the magnetic eld. Figure 3.5 shows the scaledelectronic energy h versus the scaled internuclear distance r for b =3/2 at = +1.1, i.e., at a moderate value of the magnetic eld. Wenote that > 0 corresponds to BM > 0, while < 0 corresponds to



BM < 0; remember B and M are the z-projections of the magneticeld and of the angular momentum, respectively, and that the Ozaxis is directed from the charge Z toward the charge Z .

Figure 3.5 shows that the magnetic eld corresponding to =+1.1 and higher values, under the condition BM > 0, lifts the entireupper and middle energy terms into the continuum. Figure 3.6 showsthe scaled electronic energy h versus the scaled internuclear distance

Fig. 3.5. Same as in Fig. 3.4, but at the scaled magnetic eld = +1.1. We note that > 0 corresponds to BM > 0, while < 0 corresponds to BM < 0; here B and M arez-projections of the magnetic eld and of the angular momentum, respectively; the Ozaxis is directed from the charge Z toward the charge Z.

Fig. 3.6. Same as in Fig. 3.5, but at = +2.7.



r for b = 3/2 at = +2.7, i.e., at a larger value of the magnetic eld.It is seen that the magnetic eld of this value (and of higher values),under the condition BM > 0, lifts all three energy terms into thecontinuum.

These CRS above the ionization threshold, shown in Figs. 3.5and 3.6, are classical molecular counterparts of the quantal atomicquasi-Landau levels or resonances. The latter were discovered exper-imentally by Garton and Tomkins [3.45] (for theoretical referenceson atomic quasi-Landau resonances, see, e.g., the book [3.46]).

Now we explore the stability of the nuclear motion in the ZeZ -system. The electronic energy E(R,B) becomes a crucial part of theeective internuclear potential

V (R,B) = ZZ /R + E(R,B) (3.59)

for the relative motion of the nuclei. The scaled internuclear potential

v VM 2/Z2, (3.60)then reduces (c.f., Eq. (3.58)) to

v(w0, b, Z , ) = m2(w0, b, ){[u(w0, b), w0, ] + Z }. (3.61)For any set {b, Z , }, Eqs. (3.57) and (3.61) therefore determine

the dependence of the scaled internuclear potential v on the scaledinternuclear distance r in terms of the parameter w0 which takes allvalues within the allowed ranges specied by Eq. (3.52). In otherwords, Eqs. (3.57) and (3.61) determine the classical eective inter-nuclear potential for any strength of the magnetic eld, including thestrong eld case.

Figure 3.7 shows the upper and middle branches of the scaledeective internuclear potential v versus the scaled internuclear dis-tance r for Z = 2 and Z = 3 in the absence of the magnetic eld. Itis seen for any starting point at the middle branch, that the systemwould nd the way to lowering its potential energy without anyobstacle and would end up at an innitely large internuclear distance,thereby resulting in dissociation.

The same is true for the lower branch (not shown in Fig. 3.7). Inother words, in the absence of the magnetic eld, the CRS-system,



Fig. 3.7. The upper and middle branches of the scaled eective internuclear potential v(dened by Eqs. (3.60) and (3.61)) versus the scaled internuclear distance r for Z = 2and Z = 3 at the absence of the magnetic eld.

Fig. 3.8. The same as in Fig. 3.7, but at the scaled magnetic eld = 0.3 (note thatBM < 0).

associated with the middle or lower branches of the EPE, is not reallya molecule, but only a quasimolecule because the molecular orbital isantibonding. The corresponding classical result was obtained previ-ously by Pauli [3.47] for the molecular hydrogen ion H+2 . The upperbranch in Fig. 3.4 displays a very shallow minimum of v = 0.0688located at r = 8.7.

We now turn on the magnetic eld. Figure 3.8 shows the upperand middle branches of the scaled eective internuclear potential vversus the scaled internuclear distance r for Z = 2 and Z = 3 ata relatively small scaled magnetic eld = 0.3 (with BM < 0).



Fig. 3.9. The same as in Fig. 3.8, but at the scaled magnetic eld = 1.

It is seen that the minimum in the upper branch became signicantlydeeper and moved towards lower r.

We note that the scaled magnetic eld || = 0.3 would corre-spond to the magnetic eld B 105 G for |M | 30. The magneticeld B 105 G would be typical for magnetic fusion devices underconstruction.

Figure 3.9 shows the same as Fig. 3.8, but for = 1. As themagnetic eld increased, it is seen that the minimum in the upperbranch becomes further deepened and moves even closer to the origin.

The cusp formed by the upper and middle branches inFigs. 3.73.9 reects the fact that the upper and middle energy termsfor the corresponding electronic terms terminate at some r = rmin as already noted above. Although present in CRS, this cusp may notappear in non-CRS.

Figures 3.73.9 reveal magnetic stabilization of the nuclear motionfor the case of BM < 0. Indeed, in the absence of the magneticeld, the potential well is very shallow. It is known that too shallowpotential wells do not have any quantal discrete energy levels (see,e.g., book [3.42]). Moreover, if this system is embedded in a plasma,then due to the known phenomenon of the continuum loweringby the plasma environment (see, e.g., books/reviews [3.483.50] andreferences therein, as well as Sec. 3.7), the minimum of this veryshallow potential well in Fig. 3.7 could be absorbed by the loweredcontinuum. The magnetic eld dramatically deepens the potential



well and therefore stabilizes the system for the case of BM < 0. Themagnetic eld can therefore transform the quasimolecule into a real,classically described molecule so that the molecular orbital becomesbonding.

The particular example of the system chosen for Figs. 3.73.9corresponds to the CRS of an electron in the vicinity of the nuclei ofHe and Li. Both nuclei are usually present in magnetic fusion plas-mas. Moreover, in such plasmas, Rydberg states of either of thesenuclei result from charge exchange with ions of higher nuclear chargethat are always present in magnetic fusion plasmas. Relatively largemagnetic-eld strengths are also present. It should be therefore pos-sible to observe magnetic stabilization of the quasimolecule HeLi4+

present in these practically important experimental devices.Our analysis has also shown that a similar magnetic stabilization

of Rydberg quasimolecules in CRS is displayed by other (though notall) ZeZ -systems characterized by the ratio of the nuclear charges inthe range: 1 < Z /Z < 3. Our results open up this phenomenon forpossible further theoretical and experimental investigation.

3.5. Eects of a Static Electric Field on Diatomic

Rydberg Quasimolecules: Enhancementof Charge Exchange and of Ionization

We consider a TCC system, where the charge Z is at the originand the Oz axis is directed to the charge Z , which is at z = R. Auniform electric eld F is applied along the internuclear axis inthe negative direction of Oz axis. We study CRS where the electronmoves around a circle in the plane perpendicular to the internuclearaxis, the circle being centered at this axis.

Two quantities, the energy E and the projection M of the angularmomentum on the internuclear axis are conserved in this congura-tion. We use cylindrical coordinates to write the equations for both.

E =12(2 + 22 + z2

) Zr Z

r Fz, (3.62)

M = 2, (3.63)



where is the distance of the electron from the internuclear axis, is its azimuthal angle, z is the projection of the radius-vector ofthe electron on the internuclear axis, r and r are the distances ofthe electron from the particle to Z and Z , respectively.

The circular motion implies that d/dt = 0; as the motion occursin the plane perpendicular to the z-axis, dz/dt = 0. Further, express-ing r and r through and z, and taking d/dt from Eq. (3.63),we have:

E =M2

22 Z

2 + z2 Z

2 + (R z)2

Fz. (3.64)

With the scaled quantities

w =z

R, v =

R, b =

Z

Z, = ER

Z,

l =MZR

, f =FR2

Z, r =

ZR

M2(3.65)

our energy equation takes the form below:

=1

w2 + v2+

b(1 w)2 + v2

+ fw l2

2v2. (3.66)

We seek the equilibrium points by nding partial derivatives of by the scaled coordinates w, v and setting them equal to zero. Thisyields the following two equations.

f +b(1 w)

((1 w)2 + v2)3/2=

w

(w2 + v2)3/2, (3.67)

l2 = v4(

1(w2 + v2)3/2

+b

((1 w)2 + v2)3/2)

. (3.68)

From the denitions of the scaled quantities (3.65), 2 = 1/r andE = (Z/R). Since r = ZR/M2, then E = (Z/M)2/r, wherer = 1/2 can be obtained by solving Eq. (3.68) for . Substituting into the energy equation, we get the three master equations for this



conguration.

1 = p2(

1(w2 + p)3/2

+b

((1 w)2 + p)3/2)

(

w2 + p/2(w2 + p)3/2

+b((1 w)2 + p/2)((1 w)2 + p)3/2 + fw

), (3.69)

r = p2(

1(w2 + p)3/2

+b

((1 w)2 + p)3/2)1

, (3.70)

f +b(1 w)

((1 w)2 + p)3/2=

w

(w2 + p)3/2, (3.71)

where E = (Z/M)21 and p = v2. Thus, 1 is the true scaledenergy, whose equation for E does not include the internuclear dis-tance R. The scaled energy 1 and the scaled internuclear distance rin Eqs. (3.69) and (3.70) now explicitly depend only on the coordi-nates w and p (besides the constants b and f). Therefore, if we solveEq. (3.71) for p and substitute it into Eqs. (3.69) and (3.70), we willhave the parametric solution 1(r) with the parameter w.

Our focus is at crossings of energy terms of the same symmetry,i.e., of the same angular momentum projection M . Therefore, wex M and study the behavior of the classical energy terms at M =const 0 (the results for M and M are physically the same).

Equation (3.71) does not allow an exact analytical solution for p.Therefore, we will use an approximate analytical method.

Figure 3.10 shows a contour plot of Eq. (3.71) for a relativelyweak eld f = 0.3 at b = 3, with w on the horizontal axis and p onthe vertical. The plot has two branches. The left branch spans fromw = 0 to w = w1. The right one actually has a small two-valuedregion between some w = w3 and 1 (w3 < 1). Indeed, at w = 1, thereare two values of p: p = 0 and p = f2/3 1. Thus, the two-valuedregion exists only for f < 1.

The right branch touches the abscissa at w = 1 and at somew = w2. Analytical expressions for w1 and w2 are bulky and wedo not reproduce them here. They can be found in our paper [3.24].



Fig. 3.10. Contour plot of Eq. (3.70) for a relatively weak eld f = FR2/Z = 0.3 atb = Z /Z = 3.

Fig. 3.11. The same as in Fig. 3.10, but for a relatively strong eld f = FR2/Z

9814619922

Documents

Transcript of 9814619922