IC/90/290

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

DYNAMIC TOROID POLARIZABILITY OF ATOMIC HYDROGEN

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION

A. Costescu

and

E.E. Radescu

1990 MIRAMARE -TRIESTE

IC/9Q/290

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS-

DYNAHIC TOROID POLARIZABILITY. OF ATOMIC HYDROGEN

A. Costescu

Department-of Physics, University of Bucharest,P.O. Box Hg-11, Bucharest, Romania

and

E.E. Radescu

International Centre for Theoretical Physics, Trieste, Italy,and

Institute of Atomic Physics, P.O. Box Mg~6Bucharest, Romania.

ABSTRACT:

The concept of toroid polarizability introduced in previous work is examined

within the familiar context of nonrelativistic quantum Coulomb systems, in a wa>

unbiased by approximations. The dynamic (i.e. frequency (<*>) dependent) toroid dipole

polarizability Y W ) of a (nonrelativistic, spinless, ground-state) hydrogenlike atom is

calculated analytically in terms of (essentially) one Gauss hypergeometric function.

The static result take* on the simple formy( tO = 0) = (23/60X2Z~*a* (<*=fine

structure constant, Z = nucleus charge number, aQ = Bohr radius). Y W ) characterizes

the linear response of the Byttem to a conduction and/or displacement (time

dependent) external current. Th« method of calculation (based on the use of the

integral representation for tha natirelativistic Coulomb Green's function) is presented

in detail. The imaginary part of y((J) above ionization threshold is also computed in a

simple closed form. Comparing ^ (<•»)= 0) with the already known (exact) results for

the electric multipole polari I abilities (for which, as a byproduct, we present in an

Appendix a considerably simplified expression, to our knowledge the simplest reported

as yet), one sees that although for H-like atoms the toroid effects appear as very small

indeed, they are however increasing with Z. A comparison with analogous (but, this

time, only order of magnitude) evaluations for (charged) pions, indicates that the role

of the induced toroid moments (as against that of the usual electric ones) increases

drastically when passing from atomic to hadron physics; it is argued that this trend

might continue further, at the sub-hadronic level.

MIRAMARE - TRIESTE

September 1990

* To be submitted for publication.

r

L INTRODUCTION

There is now increasing interest in toroid multipole momentB [ H [ 2 l Although

their importance does not seem to have been fully recognized until recently, they are

presently being studied in a variety of contexts (see, for instance, refs. [3M11] and the

literature cited therein). Since the subject of toroid multipole moments may look

somewhat unusual to some readers, before starting with the genuine presentation of

our own results, in order to make this paper selfcontained, we have firstly to recall

shortly some generalities about multipole decompositions, with particular accent on the

less familiar ctasi of toroid multipole moments; for clarity purposes, we shall proceed

in a pedestrian way, l im i t i ng a ' s o the preliminary considerations only to the extent

really needed for the further understanding of this article.

A general configuration of charges and currents is described by the charge and

current densities *(?,t), 0,t) which satisfy the continuity relation

By Helmholz-Neuman's theorem (see, for instance, ref./12/), one may write j(x,t) in

terms of three scalar functions LM,

The four scalars P , *W , <df, ~)C are related through the condition eq.(l.l) and so, as

shown in textbooks (for us, the best sti l l remains that of Blatt and We&kopf /13/)J

three families of independent multipoles are generated, by means of which one can

describe the most general type of source, for the description of the radiation emitted

by the source, there are left, of course, only two families of multipoles (the usual £l£

and Mi. waves), because of the transversality of the electromagnetic waves (which, on

account of gauge invariance, amounts to one condition more for the scalar ("f6"1) and

vector (A ) potentials of the external fields, in comparison with the case in which

one deals with the sources P , j).

-2 -

The history of toroid moments, in our view, begins with Zeldovich's work / I / .

He was the first to note that a closed toroidal current (which cannot be reduced to a

usual charge or magnetic multipole moment) represents, in fact, a certain new kind of

dipole. He did this when, in connection with the discovery of parity violation in weak

interactions, considered a new type of electromagnetic interaction (invariant under

time reversal but odd under parity). He observed that, if one allows for violations of

the discrete space-time symmetries, a spin 1/2 particle might possess,besides the usual

electric and magnetic dipole moments, a third kind of dipole characteristic, which was

named (with Kompaneyets's help) "anapole" to distinguish i t from the usual electric

and magnetic dipoles; he also gave the first rough estimate of this third dipole for an

elementary particle and found its classical interpretation in terms of a toroidal

current. Thirty years later, there is as yet no experimental measurement of the

anapole (toroid dipole) for an elementary particle, although b> the Glashow, Salarn,

Weinberg electroweak theory, every lepton and quark must possess one.

If the history of the toroid moments begins with Zeldovich's pioneering paper

/ l / t there is too, as always, a pre-history. Who did read well the book of Blatt and

Weisskopf, for instance, might recall some deep remarks about muttipole contributions

coming from induction currents (the curl of magnetization) in a case when multipole

expansions are carried out in connection with a magnetically polarizable medium. So, in

a sense, one may say that the toroid structures were (indirectly) singled out already

(e.g.) in ref. / I } / .

In the work by Yu.M.Shirokov, A.A.Tcheshkov and V.M.Dubovik (summarized in

the reviews 111) it has been shown that the toroid dipole (Zeldovich's anapole, in

essence) represents, in fact, only the first element of a whole independent class of

("toroid") multipole moments. A complete parametrization for the most general

configuration of charges and currents (in both classical and quantum electrodynamics)

has been obtained in terms of three families of electric, magnetic and toroid multipole

moments and distributions, generated by the three independent scalar functions "J , y ,

- 3 -

X- existing in the problem. Mean radii of various order can also be unambiguously

defined for any fixed multipalarity order (2 - pole order, •£. = 1,2,...) and type

(electric, magnetic, toroid) which ,together with the multipole moments^achieve a full

characterization of an arbitrary kind of source in terms of a convenient set of

parameters (in general, this set is infinite in classical electrodynamics but finite in the

quantum case, because of angular momentum quantization).

We start now presenting the multipole . decomposition 111 of a classical current

density j(>?,t) up to terms including the toroid dipole moment in which we are primarily

interested. Using the identity

.ne other multipole momenta and radii are defined as customarily:

and formally expanding the J -function in a series in -f , organizing then the products

).•?., j . f f k , etc., according to irreducible rotation group tensors and taking into

account the continuity equation eq.(l.l), one arrives at

+ [terms with higher derivatives of § {$)] -

a similar expansion for f (x*,t) does not contain anything unusual;

+ [terms with higher derivatives of b (i?)];

the dot means differentiation with respect to time (t); summation over repeated

indices will be understood throughout this paper; Q., M-, T. are, respectively, the

jdipole e lec t r ic , magnetic and toroid moments of the system:

(1.5)

(1.6)

We shall concentrate on the piece containing the toroid dipole moment in the

identity eq. (1.4); it does not seem to have been always paid due attention. When a

system of charges and currents [specified by ? ()?,t), j(x\t)] interacts with the external

electromagnetic f ield"£ex t , He X t described by the potentials 'JeX\\,l),'Aext(1i,t) the

contribution of the toroid dipole moment T- to the interaction energy

appears as /1/./2/

- -Tit), [v,TTkt)]^ -Tw [ f jwhere 3 and (4T) D are the external conduction and displacement currents.

This is a contact interaction, i.e. the toroid dipole T(t) interacts with the external

fields only if it overlaps with the source of the magnetic field (which may be a

conduction and/or a displacement current).

The multipole projections of P , j (and of the interaction energy W) may be, of

course, continued further; as usually, for simplicity reasons, one has then to work in a

< than^spherical basis ratherYln a cartesian one (see ref./2/). So higher turuid multipoles will

appear (toroid quadrupole, octupole, toroid radii, etc.) alongside with the usual higher

electric and magnetic multipoles and their various radii. Since in this paper we shell

deal only with the first element of the torpid class of multipoles, we shall not bother

about higher toroid moments any !onge?(from noting presently the important fact that

- 5 --A-

- e l e c t r i £ /

all toroid multipoles distinguish themselves definitely from the usuafTfand magnetic

ones in what concerns their properties under parity and time reversal operations.

In classical electrodynamics, a typical example of a current configuration

which possesses only a toroid dipole moment and no other multipole characteristics, is

a closed toroidal current (circulating along the meridians of the toroid) with an even

number N of turns of winding (an odd N would amount to a current flowing on the

parallels of the toroid and then a magnetic moment would also be present). If the

current intensity is I and the volume of the torus is \ B straightforward calculation

with eq.(1.6) (in toroidal coordinates, for instance, or, easier, by considering pairs of

circular currents, equal, but circulating opposite to each other) shows /Z/,/7/) that the

(classical) toroid dipole moment of such a current configuration is

_, fv 1 V4TTC

( 1 - 1 0 )

where77is a unit vector along the toroid axis.

The (intrinsic) toroid moments of elementary quantum systems are ruled out

either by invariance under parity P or invariance under time reversal T (depending on

whether the multipolarity order is odd or even); this happens because, in the quantum

case, the toroid moments must, of course, "lie" on the system's spin direction and the

spin has opposite transformation properties under the discrete symmetries P, T. So,

the intrinsic toroid moments of elementary quantum objects should actually be

extremely small, being determined by (very small) parity or time reversal violating

interactions. Toroid dipole (anapole) moments of leptons, quarks, nuclei, etc., have

nonetheless been to some extent, theoretically investigated (see, for instance, refs.

/14/). We mention also the strange happening that there is a whole class of nowadays

topical (but as yet undiscovered) particles, the Majorana fermions (i.e. truly neutral

(selfconjugate) fermions, for whom particle end antiparticle are one and the same

thing) whose possible electromagnetic structure is required (by CPT-invariance alone)

to consist only of toroid moments and distributions; for them, all other usual electric

- f i -

end magnetic moments and distributions are forbidden (see ref. /€/). A spin 1/2

Majorana fermion would appear so as the cleanest elementary carrier of the toroid

dipole.

In refs. /7/,/8/ it has been, however, emphasized that while intrinsic toroid

moments and distributions of elementary quantum systems are forbidden by parity or

time reversal invariance, in general there is nothing to prevent the appearance in such

systems of induced toroid moments and distributions when (time varying and/or

nonhomogeneous) external fields are present, because, in this case, a new direction

(that of the external field or current) is available and therefore the selection rules

may be eluded. Indeed, in the quantum case, the linear response to the particular

interaction from eq.(1.9), according to well known nonstationary perturbation rules

(see, for instance, ref. 715/), is described by the following dynamic (i.e. frequency (CO)

dependent) toroid dipole palarizability /7/,/8/

em Jtci.ii)

E , E denote the energies of the ground and excited states of the unperturbed

Hemiltonian; as usual the ground state contribution (if automatically nonvanishing)

is to be taken off from eqs.Q.ll), The toroid dipole moment induced in the system

(irrespective of whether or not the system does have a nonzero intrinsic one) has

~"̂ extthe following Fourier components (linear in the inducing current ^ x H ):

X (1.12)

The ("toroid") polarizability V..(ii*) which so emerges (/7/,/B/) is obviously different

from the usual electric and magnetic ones (whose definitions differ from the expression

eq. (1.11) by the replacements T. —»Q., T.-> M.); it has dimension of length to the

-7-

fifth power, as the usual electric and magnetic quadrupole polarizabilities, but cannot

be reduced to the latter As emphasized in refs. /7/,/B/, the toroid polarizabilities, be

they frequency dependent or static (u)= 0), dipole-type (as in eqs.(l.ll)) or of a higher

muttipolarity order (when in definitions analogous to eqs.(l.ll) will enter the operator

for the corrsponding higher toroid multipole moment instead of that for the toroid

dipole T.), cannot be re-expressed in terms of the usual (frequency dependent or

static) electric and magnetic multipole polarizabilities. The toroid polarizabilities are

so new, independent characteristics of the body; they appear in connection with a third

class of (toroid) multipote moments and their consideration is required for a complete

description of the linear response of the Bystem to electromagnetic perturbations. As

shown in refs./7/,/8/, they are characterizing a new type of polarization phenomenon.

To understand intuitively what means inducing a toroid dipole moment, one may note

that in the presence of an external conduction or displacement current, some of the

system's constituent charges may well begin to "move" (speaking classically) on very

complicated orbits, for instance, on "eight-like" ones. While inducing a magnetic dipole

means, say/ inducing a circular current, inducing a toroid dipole would mean

analogously inducing an "eight-like" (closed) current, or a (coplanar) pair of circular

currents (equal, but circulating oppositely to each other), or a coaxial collection of

such pairs of currents (a toroidal current). Such current configurations, topologically

nontriviai, have no resultant magnetic moment but still represent a certain (induced)

dipole characteristic, an (induced) toroid dipole.

The toroid polarizabilities (and the induced toroid moments) being not fobidden

by the discrete symmetries, give rise to effects which, in the real world, are not a

priori^ compelled to be very small. In refs./7/./B/ some of such possible effects were

analyzed. In particular, in connection with the elastic scattering of light on the

considered system (i.e. real Compton scattering), it has been shown that unlike the

(static) electric and magnetic dipole polarizabilities ot . ^£0 = °\ ftZ-\( a ) =

subscript <£. indicates the multipole (2 -pole) order) which are fixing the angular

- 8 -

structure of the low energy Compton amplitude in the second (photon) energy order,

the (static) toroid dipole polarizability V t ,(u) = 0) enters only in the fourth energy

oder, together with the usual (static) electric and magnetic quadrupole polarizabilities

_2(W= 0), pjr _2(cJ= 0), and some derivatives of the usual (dynamic) dipole

polarizabilities, like, for instance, that of the electric one

(1.13)

-0

The present paper is mainly devoted to the examination of the toraid

polarizability concept (previously introduced in refs./7/./8/) in the framework of a

perfectly known situation, that of the familiar nonrelativisitc quantum Coulomb

systems. Specifically, we are going to present in full detail a method which enabled us

to calculate analytically the (frequency depedent) toroid dipole polarizability v" (cJ)K M

of a nonrelativisttc, spinless hydrogenlike atom in its ground state. The method is

essentially based on the use of the integral representation for the nonrelativistic

Coulomb Green's function /16/ in the form obtained by Schwinger and on the fact that

a certain "basic" momentum space integral (which is at the root of many exact

calculations in problems concerning the interaction of hydrogenlike atoms with the

electromagnetic field) can be taken exactly /17/. The (exact) result for the static

toroid dipole polarizability of a H-like atom takes on the simple form

(1.14)

where <dL is the fins structure constant, Z is the nucleus charge number and a is the

Bohr radius. This is the "toroid" analog of the well known static electric dipole

polarizability ,,,, , — H

found in 1926 by Epstein and Waller /18/.

- 9 -

r TI—n

Some of the main results of our investigation were already shortly

communicated without proof in ref . / lU/ and with parts of the proof merely outlined in

r e f . / H / ; here we are going to show concretely and with all necessary details how these

results were obtained; also, we shall present and discuss some byproducts (found by

means of enough complicated technical procedures), like, e.g., the calculation (in a

(H)compact form) of the imaginary part of V . W ) (above the ionization threshold),

important in applications. Working so in a usual context (that of nonrelativistic atomic

physics) and in a way unbiased by approximations, we can clarify completely all

aspects regarding the origin, properties and role of this less known type of (toroid)

polarizabilitv. We find out that for H-like atoms (and in atomic physics, in general) the

ef fects of the induced to ro id moments appear as very small indeed wi th

respect to those of the corresponding usual e l ec t r i c ones ( t h i s may be one

of the reasons why the (induced) toroid moments have not been paid proper attention

in atomic physics so far). However, albeit small, these toroid effects turn out to

increase with the nucleus charge number Z and this allows for prossible practicalproblemsJ

applications in certain atomic physics/Oike plasma physics). In this paper, such possible

applications to atomic physics Btudies will only be briefly touched upon. Instead, we

shall assess the relative importance of the induced toroid moments (measured by the

toroid polarizability) as against that of the usual induced electric and magnetic ones

(measured by the usual electric and magnetic multipole polarizabilities o ^ ((0),

&t (O), first for atoms (by taking the hydrogenlike atom as a typical example), and

then for hadrons (by considering the case of the charged pion), with the aim of getting

some hints on what might happen at even smaller distances (or, otherwise, larger

characteristic excitation energies), at the subhadrontc level. In the H-like atom case

we shall use for comparison the available (also exact) results of ref./19/ regarding the

frequency dependent electric multipole polarizabilities oC (ti>) [for ground-state

hydrogen, because of the spherical symmetry, apart from the Langevin-type (contact)

magnetic polarizabilities which will be discussed separately, there is no Q. (£J)J, In

the case of the (charged) pion we shall use some numerical estimates tentatively

-10-

obtained in reC/7/ for the pion's dipole toroid polarizability in conjunction with (also

order of magnitude) estimates of the pion's electric and magnetic quacirupole

polarizabilities found previously in refs./10/. Looking then at what the situation is in

the two cases, at the atomic length scale of 10~ cm, on one side, and at the hadronic

length scale of ID" cm, on the other aide, we reach the important conclusion that

the role of the (induced) toroid moments increases considerably when going from

atomic to particle physics problems. Indeed, while generally very small for atoms, the

toroid dipole polarizability appears, for hadrons, to be of the same order of magnitude

as the usual electric and magnetic quadrupole polarizabilities (as already noted above,

it is with the latter that the comparison has to be made). We give then arguments

(based on some features of such topical theories as supersymmetry, strings,

superstrings) that this trend might continue further, so that at the sub-hadronic level

1 ft(at length scales of 10~ cm, for instance, i.e. at such distances which are expected

to be explored e.g. by the HERA electron-proton collider at DESY), the (induced)

toroid moments (and effects related to them) may become as predominant over the

usual (induced) electric and magnetic ones, as the latter were dominating over the

(induced) toroid moments in atomic physics. So the main lesson which seems to come

out as a result of the present investigation may be formulated as fallows; the more

"elementary" the object is, the better might it respond to an external current

(VXH 6 X t ) rather than to the external f ie lds"?e x t , lHE x t themselves (in other words, the

better might it respond to a contact interaction with the source of the external fields,

rather than to the field direct!^. This conclusion might have far reaching consequences

concerning our understading of the hadron structure. The hadron's "ultimate"

constituents would then have to look as self-protected (self-screened) electromagnetic

toroidal structure, almost insensible to external fields but dramatically reacting to

external currents when the latter succeed to penetrate them (recall that a classical

toroidal solenoid with a constant current intensity in the wire is almost dead for

constant, homogeneous external fields Ee , \-fxti but would orient its axis on the

-11-

direction of an external current density (YXrV x t ) , if the latter penetrates the

solenoid).

This paper is organized as follows: in Section II we study the hydrogenlike

atom in an external field configuration with CyxH)6" homogeneous in space but, in

general, time dependent, and show concretely, on this particular example how the

concept of toroid poilarizability (previously introduced in refs./7,B/ on genera!

grounds) appears naturally in connection with problems in which a (conduction and/or

displacement) current is flowing through the system. The starting formula for the

(H)frequency dependent toroid dipole polarizability of the H-like atom Y . (cj) is derived

using first order nonstationary perturbation theory. A toroid analog of the Langevin

diamagnetism (previousiy found in ref./21/) is discussed here in connection with the

contact term of the nonrelativistic interaction Hafniltonian (i.e. the term quadratic in

the vector potential describing the specific external field configuration Dne is

interested in). The straightforward calculation of the corresponding contact Langevin-

type contribution to the toroid dipole polarizability^f ^ is presented. In Section 111

(H)the nonstardard, long and tedious calculation of V (<*>) is carried out. It involves

some less known techniques (based on the use of Gauss hypergeometric functions); we

have chosen to present them in due detail since their knowledge may be helpful in

(H)other similar applications as well. Some alternative (equivalent) forms for V as a

function of the frequency CO are given, because for various further purposes, one or

other of them may prove particularly convenient. In Section IV we calculate directly

the first excited state contribution to V L t [ j ) ; besides providing powerful checks for

the cumbersome evaluations performed in Section II, this allows us to evaluate how far

the exact result fo rV is with respect to the approximation in which only the firstXU\

excited state contribution is retained; this is important since in most practical

applications, when the calculations cannot be carried out exactly, all one can usually

do is to consider contributions coming from few relevant excited states; the results of

Section IV may be thus helpful, e.g. in situations regarding many electron atoms. In

-12-

(H)Section V we calculate in closed form the imaginary part of V . (W) above ionization"•1

threshold; so one can get exact formulas for the matrix elements of the toroid dipole

operator between the ground state and a final Coulomb scattering state. In Section VI,

starting from a known exact resu l t obtained in the second of refs./17/, we obtain a

low energy expansion of the (forward) amplitude for elastic scattering of a photon on a

(nonrelativistic, spinless, ground state) hydrogenlike atom, valid to the fourth

frequency order inclusively. Such an expansion help3 seeing within the framework of an

exactly soluble problem (that regarding the hydrogenoid atom and its interaction with

the radiation) how the various polahzabitities of the system do appear in the low

energy Compton scattering amplitude. Section VII contains a comparative analysis of

the effects related to the toroid polarizability as against those related to the usual

electric (and magnetic) ones, first in atomic physics and then in hadron physics.

Thereafter, on the basis of what is presently known for H-like atoms, on one side, and

for (charged) pions, on the other side, some speculative remarks on a possible dramatic

increase of the role played by the toroid moments at the subhadronic level are put

forward. Among some aspects of nowadays topical theories which are invoked to

support such speculations, one particularly notes supersymmetry, in which Majorana

fermions ere currently occurring. The boson-fermion symmetry is very intriguing,

indeed, and leads often to surprises (see, e .g., ref.722/).

Finally, in an Appendix we show how a previous exact result (obtained in

(H)ref./19/) for the frequency dependent electric multipole polarizabilities oi (tO) of a

£

(nonrelativistic, spinless, ground state) hydrogenlike atom (valid for any multipolarity

order t ) can be put in a much simpler form, involving only one Gauss hypergeometric

function rather than three such functions, as in ref./19/j to the best of our knowledge,CH)the expansion for <<, (uJ) found in this Appendix is the simplest hitherto reported.t-

-13-

IL THE HYDROGENLIKE ATOM IN AN EXTERNAL FIELD

CONFIGURATION WITH V X H 6 * 1 HOMOGENEOUS

The (nonrelativistic) quantum mechanical Hamiltonian describing the

interaction of a hydrogenlike atom with an external field configuration for which

V x f i e x t is homogeneous in space (but, in general, time dependent) may be taken as

TO-(2.1)

where P =-itiV, e = - | e | and m denote the charge and mass of the electron, the scalar

potential U> for a nucleus of charge -Ze = Z | ef is

..-it .

while the vector potential~A'T\?,t) ia chosen / 2 1 / to be

C2.2)

(2.3)

with the vector l(t) 7- independent but known as a given vector function of time. One

sees that"A*TV,t) by its definition eq.(2.3) satisfies

The Lorenz condition

(2.4)

(2.5)

is obviously satisfied, too, in view of the time-independence of the scalar potential lfl.

As it is well known, since the gauge satisfies eq.(2.4), one may not bother

about symmetrization in the third term of the r.h.s. of eq.(2.1) and hence one may use

instead the following equivalent form of the Hamiltonian

From eq.(2.2) and eq.(2.3) which specify the form of the scalar and vector

potentials, it is seen that we are dealing with the movement of a charge e in the

following external fields / 2 1 /

fc (lL,t)=-J.y-.±-3L.:_-fc_

(we recall that summation over repeated indices is understood throughout while the

dot means differentiations with respect to time). The fields are such that the external

displacement current is / 21 /

(2.8)

while ^7xHe x is homogeneous in space (but, in general, time dependent) / 2 1 / :

The sources of the external f ields"Eext, H e x t from eqs.(2.6),(2.7),"jext and 0 e X t , are

(2.10)

(2.11)

From the Maxwell equation

-IS-

(2.12)

(verified, obviously, by eqs.(2.B), (2.9), (2.1Q))j one sees that for a general time

dependence of the vector I(t) (constant in space), V x H e x receives contributions from

both the external conduction (3BXt) and the displacement [(4T )" ~Eext] currents. If I(t)

is constant in time as well,

Ttti-r , (2.13)

only the external conduction current "3BX wi l l be present in eq.(2.12) which simplifies

then to

3 ~ J . (2.14)

So, in the static case when V * H e x is taken as being not only homogeneous in space,

but time independent as well, our problem reduces to that of a hydrogenlike atom

immersed in an external constant current density.

With H' denoting the system's wave function, the (system's) average current

density in the presence of the external field A given by eq.(2.3) is known to be

despite the explicit presence of the vector potential, one also knows that j / •. is gauge

invariant. For transitions between different quantum states, one may analogously

consider (see ref./13/, chapter XII) nondiagonal transition current densities (using, this

time, different eigenstates of the (perturbed) Hamiltonian):

Returning now to the expression of the Hamiltonain eq.(2.1') we note that the

third term (the one linear inA^ ) may be alternatively written, using eq.(2.3), as / 2 1 / :

(2.15)

-16-

where

(2.16)

is, by eq.Q.6), just the one particle operator for the toroid dipole moment (in the

absence of the external fields).

Equation (2.15) expresses thereby the "toroid dipole interaction" we shall be

dealing with in this paper; it is almost entirely analogous to the usual electric and

magnetic dipole interactions - t !e x t( t )J? and ^f? x t ( t ) .M ((? and T^ are the electric end

magnetic dipole moment operators). Eqs. (2.15), (2.16) show concretely, within the

familiar nonrelativistic quantum mechanics framework, how a new ("toroid" dipole

type) interaction, which, seemingly, had not yet been paid due attention in the

literature, arises naturally in connection with a particular external field configuration.

The same toroid dipole interaction (aa shown in detail in Section I) appears on general

grounds when correctly performing a multipole analysis of the interaction Hamiltonian

in the presence of an arbitrary external electromagnetic field. The toroid dipole

interactions and the toroid dipole moment come alongside with the usual quadrupole-

type interactions and moments. However, the former and the latter are independent

entities, since the toroid dipole cannot be reduced to the usual quadrupoles. Equations

(2.15). (2.16) appear then as a particular realization (relevant for quantum mechanical

appl icat ions)of the general formalism presented in r e f s . / 2 / .

The first two termB in ^ f t ) as given by eq,(2,l') wi l l be taken as the

unperturbed HamiJtonianJ/jj, while the last two terms wil l be viewed SB perturbations:

(2.1B)

(2.1?)

-17-

T~T

*W (t) as given by eq.(2.19) (which is quadratic in A , or alternatively, in I(t))

represents a "toroid" analog I'lll of the usual part of the Hamilton)an responsible far

the Langevin diamagnetism and will be discussed in detail later on in this Section. For

the time being we shall concentrate on that part of the perturbation [W oid-diDole'1^

which is linear in A (orT(t)) and recall first, for completeness, some basic formulas

of the usual nonstationary perturbation theory, necessary below in the subsequent

derivations.

Let us take Vx£?xt(?,t)s W*7c)"?(t) (which, aa said before, is chosen to beT-

independent) as harmonic in time with the period &>:

ioiL I

-" > (2.20)

(The superscript (c) indicates time-independence; (VxHe x ) is now not only "r-

independent, but time independent as well). I f the perturbation acts on the system

(switched on. adiabaticolly at t = -••) ,

V - F + F (2.21)

with the operator F time-independent then (see, for instance, ref./15/), the first order

correction superscript (1) below) to the diagonal matrix elements of any observable f is

given by the formula

r- Cr to r /" ' r I .

U(2.22)

where the sum extends over the entire (unperturbed) spectrum (discrete and

continuous),f r- \ I

(2.23)

-18-

{E are the eigenvalues of the unperturbed Hamiltonian), whereas f denote the

matrix elements of the operator f attached to the observable f .calculated v i t h the

etgenfunctions Y^° of the unperturbed Hamiltonian

(2.24)

In the specific case under consideration here, that of the toroid dipole interaction

(eqs.(2.15)), with ( f x H ) homogeneous and harmonic in time as in eq.(2.20), the

operator F in eq.(2.21) has the concrete form

(2.25)

with the toroid dipole moment-operator T given by eq.(2.16). So, taking for the

observable f of interest the system's toroid dipole moment itself and confining

ourselves to the consideration of the ground state (label a = 0) only, eq.(2.22) givet us

the first order correction to the ground state expectation value of the toroid dipole

moment , , . • • * i -i

. I " • 1 (u>) , £ J , (2-26)

coming from that part of the perturbation Qh^ Q j d rf. . (t)] which is linear in ~A .

According to eq.(2.22), the (frequency dependent) vector amplitude of the oscillating

toroid dipole moment induced in the (ground state) atom, T l n u c e («), will be given by

(U) j (2-27)lL M Yt. () { j

with the (frequency ((0) dependent) toroid dipole potarizability of the (ground-state,

(H)nonrelativistic, spinless) hydrogenlike atom y .. {«) having the expression (of the same

type, of course, as the general formula displayed i n e q . ( l . l l ) :

Y.(HJM 7

(2.28)

. ~Ee

-19-

The toroid operator T. figurating here is the specific differential operator given by

eq.(2.16); the sum extends over all excited states in the entire (discrete and continuous)

spectrum (due to its spherical symmetry the ground state does not contribute anyway);

because of spherical symmetry arguments one may anticipate that the tensor y .. (U))

must be of the form

Y.(H)a> J- Y ^ <"9 )

To summarize, until now, in this Section, we have shown how for a specific

physical system, the hydrogenoid atom, the new concept of "toroid" polarizability

(previously introduced in a general context in refs./7/,/B/) appears naturally when one

starts studying the atom's interaction with a field configuration (specified, e.g., by the

vector potential /V̂ from eq.(2.3)) in which tyxfiF*1 (generally time dependent) is(H)homogeneous in space. The toroid polarizability V - (W) from eq.(2.2B) expresses the

linear response of the (ground state) H-like atom to such Bn interaction and, according

to eq.(2.27) (explicitly derived before by means of the usual nonstationary perturbation

rules) gives the contribution to the toroid dipole moment induced in the s-state coming

from that part of the perturbation which is linear in the inducing agent (VxHe ). So,

although (because of the parity invariance of the Hamiltonian) the intrinsic toroid

dipole moment of the s-state H-atom is zero, the atom does acquire an induced toroid

dipole moment in the particular external field configuration mentioned above. The

(H)quantity Y (CJ) will be (exactly) computed in the next Section. Before doing this, we

have to return, however, to the detailed analysis of the Langevin-type part of the

interaction Hamiltonian H. (t), eq.(2.19), which is quadratic in the inducing agent

(VxH ). This time we shall restrict our considerations to the case in which VXH

is not only homogeneous, but also constant in time, i.e.

v*T{Rt) = (v xff^f' = Jo.1 * * c

(2.30)

1(0 in eq.(2.3) will be taken therefore now as a true constant threevector I(t)g I .

The whole Hamiltonian ̂ from eq.(2.17) becomes then time-independent and by first

-20-

order (stationary) perturbation theory there will be the following energy shift ill! in

the ground state energy coming from the contact piece J t (eq.(2.19)) (the piece

id dioole in the stationary case does not contribute in the first

perturbative order since, as mentioned before, there is no intrinsic toroid dipole

moment in the ground state |0>, i.e. <0 I Tj 0> = 0^:

2-inc1 V tot

Writing (KFT = rZ$ c ' ) 2 cos2 0, we have set cosZ B = 1 - sinM) = i/3 in view of the

spherical symmetry of the ground state. Now, using the concrete expression of the

ground state wave function (displayed for convenience in eq.(3.B) below), one has

(2.32)

(a^ i8 the Bohr radius, aQ =1iZ/,\ =ti/«<.mc, ot = fine structure constant). Writing the

energy shift as

(2.33)

one then finds the following Langevin-type contribution to the toroid dipole

polarizability

(2.34)4001**£- fa ~-?A

the derivatives

- 2 1 -

(2.35)

- • r ' - -

can be viewed as yielding the "contact" (Langevin-type) part of the toroid dipole

moment acquired by the atom, T . The full toroid dipole induced in the atom's ground

state by (V xAeJ<t)'c \ however, will be given by

with• ( H ) (H) C2.37)

because one has to add to the energy shift £i,EQ (related by eq.(2.33) to Y L o t > t a i n e d

from rl in the first order stationary perturbation theory Iff. itself is quadratic in

another contribution, quadratic in iffxJPPxtyc' too, which comes in the

second perturbative order from the piece JYT Q o i d d oJ (linear in (VxH e x t ) ). The

(H)latter piece, which gives rise to the first term in the r.h.s. of eq.(2.37), Y ( ^ = °)>

will be completed as a zero frequency limit of the dynamic (i.e. frequency dependent)

H (H^

toroid dipole polarizability V (<*)). In turn, Y(jjj w ' " b e evaluated analytically for any

U) , in the next Section, in the framework of a first order nonstationary perturbation

calculus for interactions with time-periodic fields. The static limit Y ( ( k > = 0)

obviously could have been equivalently obtained from the energy shift caused by the

time independent inter action tij^.^^ j - a i B ('•*• that given by eq.(2.15) in which

one has now to replace V xfiext(?,t) and"1(t) by (y x?|ext)(c) a n d i ^ in second order

stationary perturbation theory.

In connection with eq.(2.36) it is worth recalling that while the moments

induced in the system depend both on the inducing fields and the system, the corres-

ponding polarizabilities (associated with the respective type of induced moments) are

quantities which depend only on the system and not on the inducing fields. With

eq.(1.14) (which anticipates the result of the cumbersome calculation presented in

Section III) and with eq.(2.34), eq.(2.37) leads to

(2.38)

-22-

ir m s

and, therefore, by eq.(2.36), the total toroid dipole moment induced in a (ground-state)

hydrogenlike atom by an external (time independent) magnetic field of constant

(2.39)1

By analogy with diamaqnetism, we shall call this way of inducing a toroid dipolet effects

moment a dia-toroicHtaccording to the minus sign, the induced toroid dipole lies

opposite to the direction of the inducing agent (V xHe > t trc ' ) . As it is seen fron

eq.(2.36), the result expressed by eq.(2.39) appears as a consequence of large

compensations between ^ %0i= 0) (the "normal" static toroid dipole polarizability,

(H)which is "para-toroidal") and the contact (Langevin) termV^ which is "dia-toroidal"

and prevails. Consequently, the (ground-state} hydrogenoid atom should behave as a

"dia-toroid" substance in a field configuration with ^ xH6" uniform in space and

constant in time, whereas in the case of sufficiently small homogeneous but periodic in

time ^7XH (small, so that the Langev in type contribution to the (now time-

xHexdependent) interaction '(-lamiltonian Jj-/. , quadratical in \JxHex , may not count too

much as against the piece ff-j- - , .. , , linear in ^ xH ), its behaviour will be

(H)controlled by the frequency dependent toroid dipole polarizability Y (<*?).

Concerning the Langevin-type "die-toroid" effect / 21 / put in evidence for H-

like atoms by eqs (2.33), (2.34) above on account of the contact interaction M , we(that ,

mentiomfa straightforward formal generalization to more complicated atoms (with

more than one electron; here we continue to ignore spins) may be easily achieved just

as one does in the analogous case of the well known Langevin dia-magnetism. The

vector potential is still given by eq.(2.3) with Kt) taken as constant in time I(t)=~1 c =

= (C/4TT) (yxH 6 * ) . The (time-independent) Hamiltonian for the many electron atom

immersed in a constant external current density J =1 is then

(2.40)

-23-

with j t denoting the Hamiltonian of the atom in the absence of the external current,

the sum aver a, extends over all electrons of the atom. For a state

((2.41)

of the

unperturbed Hamiltonian K j , one would have the following contribution to the energy

shift £^E^,, caused byj>|! in the first perturbative order,

and hence the following Langevin-type contribution to the atom's toroid dipole

polarizability:

(2.45)Y =-

Equations (2.44), (2.45) represent the toroid analog of the corresponding usual formulas

expressing the Langevin dia-magnetism in the case of the atom's interaction with a

constant homogeneous magnetic field H (see e.g. ref./15/):

(2.46)

(Z.47)

However, there is a certain obstacle in pushing too far the analogy with the usual

Langevin diamagnetism. Whereas for a many electron atom in a constant external

magnetic field, if one takes V to be a state of zero total angular momentum, eqs

-24-

46) and (2.47) will automatically give the whole effect (i.e., the whole induced

magnetic moment), in the toroid case (i.e. when one considers the atom immersed in a

constant external current density}, specifying ty to be an s-stBte would not be in

general sufficient to get rid of contributions to the energy shift coming (in the second

perturbative order) from the piece Wroroja'.djpole) ^ ' n e a r '" t n e e * t e r n a l current

density) and therefore Y , as given by eq.(2.45) will give in general only a part of the

atom's total toroid dipole polarizability (otherwise, this has already been explicitely

shown above for Hydrogen). The practical value of eqs (2.44), (2.45) appears to much

reduced in comparison with that of the analogous eqs (2.46), (2.47), unless one may find

some specific systems (or quantum configurations) for which the contribution of

may be made if not zero, at least small enough with respect to that ofToroid-dipole

j-f . Even so, because of the possible presence of usual quadripole Langevin

diamagnetic contributions (which are also expressible in terms of Z.<<f'|r l^^), the

toroid and the usual effects get mixed and to disentangle them one would have to work

with ideally prepared external field configurations. Irrespective, however, of practical

possibilities,inclusion of the toroid contributions remains anyway a matter of complete-

ness whenever multipole decompositions of the interaction Hamiltonian are employed.

For the ground state H-like atoms the Langevin magnetic (dipole) polari-

zability ^ takes on the known form

(2.48)

(H)ft I with its higher multipole analogs are the only existing magnetic polarizabilities

for this particular system (The "normal" (i.e. coming from the Ap term of the

Hamiltonian) dipole and higher multipole frequency dependent magnetic polarizabilites

for (ground state, nonrelativistic, spinless) H-atoms are obviously identically zero

because of spherical symmetry). The case of ground state Hydrogen (for which all

possible induced magnetism is only of a contact, Langevin type) appears then

particularly instructive in revealing the previously introduced (/7/,/8/) concept of

-25-

toroid polarizability, since now one can compute everything exactly and therefore

make perfectly clear that a new type of polarizabitity, which is different from the

usual electric and magnetic ones, must be introduced if one wishes to achieve a

complete description of the atom's interaction with a most general configuration of

(H)external fields. While ->̂ ( i j ) clearly has nothing to do with the usual frequency

dependent electric multipole polarizabilities (because the latter involve, by definition,

only matrix elements of operators representing moments of the charge density and do

not contain any"P* = - i t i V operators, related to the current density), it has nothing to

do either with the usual frequency dependent magnetic multipole polarizabilities,

simply because for s-state Hydrogen there is none of the kind (the contact Langevin

polarizabilities like the dipole one p ^ are of a distinct origin, as stressed out

(Hi ?

before). By its definition, "jj- J(£j) involves operators of the type x.x.P. and r P.

arising in the particular combination specified by eq.(2.16), which is not reducible to

that for a usual magnetic quadrupole moment. Although the appearance of a third

class of multipole palarizabilities (the toroid ones) alongside with the usual electric

and magnetic ones may in fact be well understood on general grounds alone (see

refs./7/,/8/), the present concrete analysis within the familiar framework of the

Hydrogen-atom renders this subject particularly transparent.

Before facing, in the next section, the difficult task of computing the dynamic(H)

toroid dipole polarizability V (^)t some comments on the chosen gauge may be in

order. In this paper we shall consistently work with the vector potential as given by

eq.(2.3). While obviously not at all compelling, this has the advantage of facilitating

from the beginning the introduction of objects with a well defined physical meaning.

We have in mind in the first place the type and order of multipolarity of the response

function under investigation, the toroid dipole polarizability, which appears in

connection with Hxaroid-dlpotc^'^ ' a e e ot1s-<2-15^- w w e n a d worked (as obviously we

perfectly could, afterall), for instance, in the new gauge (for simplicity, we refer now

to the static case only):

- 2 6 -

with X taken for exemplification purposes as

(2.50)

when

(2.51)

(i.e. y x H e x t and A, or, alternatively, V x V « A1 and A1 are colinear), then the direct

appearance of quantities with well defined multipolarity would have been last and

subsequent rearrangement of terms would have consequently been needed to ensure

again the appropriate physical interpretation of the results. Such artifacts would have

been by no means unexpected, because in the gauge from eqs.(2.51) a new direction

(specified by the unit vector TD is in fact put in by hands, thus compromising the

original equivalent role played in the problem by the space coordinates x,y,z; with the

expression of A- from eq.(2.3) no shortcomings of this type are appearing. This, of

course, does not mean that in other similar applications other gauges, like the one of

eqs.(2.51) could not turn out to be preferable.

We note that the practical question of preparing concretely external field

configurations corresponding to A is secondary to the issues discussed in this paper;

what is really of primary importance to us is that in a power series expansion in x,y,z

of the potentials A(x,y,z,t), AQ(x,y,z,t), apart from the usual type of pieces leading to

the various usual electric and magnetic multipales, simply for reasons of completness,

one must also include contributions, like A , corresponding to the less familiar toroid

multipotes. Disregarding the latter would simply mean loss of generality; while for

some physical systems and some situations such neglections may be of l i t t le

importance, for others they are certainly misleading.

-27-

10. CALCULATION OF ThE DYNAMIC TOROID POLARIZABIL1TYf jJJ

We want to calculate exactly the frequency dependent toroid dipole

polarizability for a (nonrelativistic, spinless) hydrogenlike atom (nucleus charge =

= Z\e\) in its ground state \ H,s-state> (energy EQ), i.e. the quatity>( •- (O) as given

by eq.(2,28) of the previous section:

-t-

(the ground state \H,s-state> does not contribute anyway to the sum over the

intermediate states |n> which belong to the whole spectrum, discrete and continuous).

The matrix elements of the toroid transitions can be evaluated by means of the one

particle operator for the system's torpid dipole moment III, eq.(2,16):

(we recall that P . = i r i J / 3 x k is the momentum operator and summation over

repeated indices is understood throughout; e = - |e[ end m denote the charge snd the

mass of the electron). Strictlyspeaking one should use for <a|T.|n> the expression

following from the classical formula eq.(1.6) with the "transition" current densities

<a[ j.(r)Jn> written in the symmetrical form

0.2)

-28-

\u are the wave functions in the coordinate representation). As known from textbooks

(see, e.g., ref./13/, Chapter XII), the symmetrical expression of eq.0.2) ensures the

fulfillment of the current conservation condition by <alj.(?)( n> and the corresponding

"transition" charge densities

(3.3)

In our case, however, one may use directly the toroid operator T. from eq.(2.16) in

which P acts to the right; indeed, integrating by parts in eq.(3.1) and noting that

(3.4)

one sees immediately that both terms from the r,h.s. of eq.(3.2) give the same

contribution to <a I'T. I n>.

Denoting

r- J- , r , + (3.5)

one has

with

the ground state wave function is

(3.6)

(3.8)

(3.9)

(3.10)

-29-

r - r r "T-

the last equality sign in eq.(3.6) follows from invariance under rotations, the ground

state being spherically symmetric,

(a) We shall use first the explicit simple form of the ground state wave function

ujQ(j(r) in order to reduce the calculation of T,-CJlJ as given by eq.(3.7) to that of a

more simple object. We shall show that

To this aim we use eq.(2.16) to write

Zk^)-7~ <•*-> ̂ ^ ) ̂ Sy +J^ikwith

(3.12)

(3.15)

(3.16)

Noting that

(3.17)

(3.18)

one has immediately

-30-

l']

(3.19)

which implies

Now, because

(3.20)

one may perform the following replacement under the integral over r, in eq.(3.13)

(3.21)

By the same arguments, under the integral over *?, in eq.(3.1£) one may do the

replacementtil ^

(3.22)

Therefore one hasM

and

(3.23)

.••-D&•)

-31-

7T I ttic- J £-

(3.24)

Equations (3.23), (3.24), (3,20) and (3.17) then lead to ei, .(3.11) which we wanted to

prove.

(b) In the next step of the derivation we shall get rid of the sum over the

intermediate states | n> by putting in evidence the known (nonrelativistic) Coulomb

Green's function. Using

(3.25)

Equation (3.11) may be written as

(3.26)

By Fourier transforming, the curl bracket in eq.(3.26) may be expressed as

where

• *

and

(3.Z7)

(3.28)

-32-

The calculation of i(j?,)iiR) is elementary ;

\

(3.30)

So, eq.(3.26), after relabelling for convenience (j^) by (-^^t becomes

(3.31)

where

(3.32)&T2-

(c) New we shall make use of the integral representation for the nonrelativistic

Coulomb Green's function in the form obtained by Schwinger /16/

fat)a.33)

(3.34)

-33-

T—r" -ri

(the integration contour starts at the point 1 on the real axis, encircles the origin in

the counter clockwise sense and returns to 1) in order to express eq.(3.31) as

V ' 2^ L(3.35)

1) in order to express eq.(3.31

in terms of the basic integral 111/

(3.36)

previously introduced and calculated in refs./17A By means of repeated application of

the residuum theorem for complex functions, there it has been found that 111 I

(3.37)

where

r

(3.38)

(3.39)

So, at this stage not only have we got rid of the sum over the intermediate states, but

also of the six integrations over~p^,*p^, at the expense of being left with one contour

integration and with some derivatives which can be straightforwardly performed:

-34-

itp^f^)rf(3.40)

(d) The derivatives can be safely taken under the integral sign. One has first

with

this brings eq.(3.40) to the form

* ( " 1 . c

(3.42)

X A" A

where

(3.43)

Furthermore,

(3.44)

end hence

-35-

'+ XT(Ort (3.46)

where

(3.47)

(e) At this stage, before taking the last two derivati ves with reapect to \ \ we

find it convenient to carry out the integration firstly, in order to bring juat now into

the game the Gauss hypergeometric function F(a,b,c;z) whose known properties (see

ref,/23/) shall be next used to simplify the result. Noting that /23/

?

one has

and so one can write eq.(3.46) as

(3.50)

with

- 3 6 -

i To avoid taking two derivatives from a combination (J ) of three

hypergeometric functions F, we shall now express the quantity Q, (eq.C3.5D) in terms

of only one such F. The result reads

( 3-5 2 )

To get it, one has to use (see ref./23/)

(3.53)

for a = 3 -T , b = 6 and the formula

/ T i t

(3.54)

(which is nothing else but eq.(3,53) twice iterated) for a = 2 -X , b = 6. This leads to

32 (A't-jM5"

(3.55)

and

(3.56)

(A'+Xt.)

- 3 7 -

T r - - • - - r - - T ~ T "

After a l i t t le algebra, using also the concrete expression of T ( \ ' ) eq.(3.45), one finds

eq.(3.52).

(g) Now comes the most boring part of the work; i t consists in taking the second

derivative with respect to X' >n eq.(3.50) (with g ( \') as given by eq.(3,52)). There is

nothing tricky about that except for one thing: one should exercise some core in order

to maintain during such operations the same hypergeometric function F(4-Z (6,5-"E ;T)

which we have just introduced to simplify eq.(3.51). "Maintain" means having it with

the parameters unshifted. To achieve this, we shall use the formula

(3.57)

which can be worked out using ref./23/. All the rest is pure algebra, rather awful, by

the way. We give only the results; the intermediate one, expressing the first derivative

with respect to \', is:

o

end the last one of this paragraph, which provides the desired second derivative with

respect to V1 at \ " = \ :

X6 (^(vo

(3.59)

now "? stands for

(3.60)

(h) Equation (3.59) may be further simplified by using the relation jlil

which gives

(3.61)

(3.62)

From eqs.(3.50),(3.59),(3.62),(3.47) one has then

2.0

(3.63)

where aQ denotes the Bohr radius

(i) CXir calculation has been so completed. By eqs.(3.6) and (3.63), the dynamic

(i.e. frequency dependent) toroid dipole polarizability of a (nonrelativistic, spinless)

hydrogenlike atom (of nucleus charge number Z) in its ground state is

(3.65)

-39-

where F(e,b,c;z) is the usual Gauss hjpergeometric function with the series expansion

L (3.66)

we also recall the notations

(A*. i t (3.67)

Y (U) as given by eq.(3.65) is an even analytic function of CO having (in the complex

U) -plane) the right singularities at the right place. The singularities on the positive

real axis are given by the term involving the first Gauss hypergeometric function (i.e.

that corresponding to i = 1). They are: 1°. Simple poles at the (J-valuesftcJ = E - EQ,

2n = 2,3,4,... (E = Ep/n represents the discrete spectrum of the H-like atoms)

introduced by the poles at 1 . = n, n - 2,3,4,...; the poles at "£•• = 2, T = 3, " t , = 4

appear explicitly factorized in eq.0,65), while all those at ~C, - n , n = 5,6,... are

contained in the hypergeometric function F( l , -1-Tj j h ^ 2°. A branch cut along the

real axis in the complex CO -plane above ionizetion threshold (i.e. for Cj >*J th,

th = 1^0^' t h e ^ u m p o f Y ^ ^ across it, given by J*n-Y(tJ), will be computed (in

closed form too) in Section V.

(j) Equation (3.65) may be alternatively written as

(3.68)

Equation (3.68) will prove itself much more convenient than eq.(3.65) later on, when

-40-

taking the imaginary part of V (to) f o r b) above ionization thresholdHcJih = / EQ | .

Equations (3.6a) and (3.65) ere equivalent because

(3.69)

which follows from

c-i A_ I", . c-2.(3.70)

which, in turn, follows from

^c-,* ) •(3.71)

(which should be known ! see ref./23/), applied first for a = 1, then for c ->c - 1 and

a = 1 to eliminate F(l,b,c-ljz).

If one wishes to incorporate further the pole (2 - "C •)" into the

hypergeometric function, so as to have contained in the letter the whole discrete

spectrum of the H-like atom, one may put ;;q.(3.60) in yet another form :

Y'H>

(3.72)

I n d e e d , i f o n e w r i t e s e q . ( 3 . 7 1 ) f o r B = 1 , c = 3 - t , b = l - t , c - b = 4 , o n e h a s

- 4 1 -

T T

(J'73>

and eq.(3.6B) goes into eq.(3.72).

One sees that unlike eq.(3.65), the alternative expressions of N / H \ U » given in

eq.(3.68) and eq.(3.72) apparently suffer from the presence of a (spurious) singularity

at "Z. = 1. It cancels out, of course, as it will be soon shown explicitly.

(k) We write down finally the very simple (exact) result for the static (i.e. CO = Q)

toroid dipole polarizability of a (nonrelativistic, spinless) hydrogenlike atom in its

ground state. Using eq.(3.65) at (J=D (when T j = T^ = 1, " ^ =}2 ' ° B n d h e n c e t n e

hypergeometric function reduces to F = 1), one finds immediately

Numerically, one has

,tH)X {0^.0) ~ O.&& x \0~ 2 x-iO (3.75)

This is the toroid analog of the static electric dipole polarizability

2. &(3.76)

found in 1926 by F.Epstein and LWaller / IB/.

(D Finding the static limit of the toroid polarizability >f (cJ = 0) as displayed in

eq.(3.74) above is not so immediate if instead of eq.(3.6S) one uses the alternative

forms of• i\

( given in eq.(3.68) or eq.(3.72), because of the spurious multiple pole

factors (absent in eq.(3.65)) ( £ . - 1)" , respectively (T j - D" , whose cancellation

must be carefully carried out. This task, albeit tedious, is not formidable after all.

Next, we shall show for illustrative purposes how the static limit expressed by

eq.C3.7G) may be obtained from eq.(3.68) as well. Knowledge of such procedures is in

fact quite important in avoiding despair when faced with this kind of problems without

having at hand an alternative way to circumvent the difficulty.

When taking the desired limit

one has to keep the first three terms in the series expression of the hypergeometric

function eq.(3.66) (higher terms will not contribute in the limit). One has then

successively

loo 2 * X^A

f *S"-

5o, one recovers (as it must) the expression of V ( cJ= 0) already obtained

eq.O.74).

-43-

IV. FfftST EXCITED STATE CONTRIBUTION TO

In most applications the calculation of polarizahilities can not be done exactly

and in order to get order of magnitude estimates one usually has to exploit the

supposed good convergence of the sum over the system's excited states by retaining

only few of them, either the first ones or those expected to be more relevant; often

one is taking in fact as a prime indication the contribution of the first excited state.

Having been able, in the present work, to compute exactly the toroid dipole

polarizability V (ta» of (ground' state) H-like atoms, we may take advantage of the

virtues of an exact calculation by assessing in a precise way how close the first excited

{Histate contribution to V (bi) would turn out to be in this case with respect to the

already known exact result. Such a comparison may prove instructive when studying

atomic systems more complicated than hydrogenoid ones, since as a rule no exact

(H)solutions are available then. Computing the first excited state contribution to V (cJ)

will moreover provide powerful additional checks of the exact result obtained for(H)V ((*)), because, for instance, we shall achieve in this way an independent direct

calculation of the residuum of Y CcJ) at the poleW= E(n_2) ~ Em-1) = E(n-2)~ECT

Such checks are obviously quite important if one recalls the intricacy of the

calculations performed in the previous Section.

We want therefore to find

(4.1)

" W> " f l t J - i

where the supplementary superscript (n=2) on V means that only the first excited

state (with the principal quantum number n=2) has been retained in the defining

relation eq.{2.28)(l' denotes here the usual orbital quantum number). Using the concrete

-44-

expression of the toroid dipole operator eq.(2.16) in conjunction with eqs.0.17), (3.18),

and noting that ^V^QQ I x; | ^ Q Q ? = °> o n e n a s ^(n=2) = 1 ' * E 0 = " 1 ' 4 I E o l ' :

. ± A — n(6, L

r, A!,. 1-uJ) 1

with

•IBut

T) -srr J V t ;X*.

and after some little work one gets

5o the desired result is

(4.2)

( 4 3 )

(4.4)

(4.5)

For it) - 0 it leads to the following (first excited state) contribution to the static

»'H)/,.i_n\toroid dipole polarizability V

(4.7)

-45-

r T

(H)Comparing with the exact value of Y (u)=0)i eq.(3.74), one sees that in the specific

case of nonrelativistic H-like atoms considered in this paper, the approximation of

retaining only the first excited state contribution to Y (U)=0) would lead to an

underestimation of about 15% only (the factor (3/4)(8/9) ac 0.33 in eq.(4,7) compares

with 23/60 2*0.38 in eq.(3.74)). This conclusion may prove helpful for orientative

purposes in other applications regarding more complex atomic or molecular systems.

(H)With the aid of eq.(4,6) one may check the residuum of the exact Y (CO) (as

given, for instance, by eq.(3.6B)), at the poleTicO=E. .— E_ = 3/4 j EjJ • The Po le

singularity a tnW = 3/4 |EQI appears through the factor ( Z - t , ) " in front of the first

hypergeometric function from eq.(3.68) (recall that by eq.(3.67),"C, = ( l^ i tJ/ \EQ\)" )

and therefore

- t>ffU. b<uX) -

So, on one side, one has from eq,{4,6)

(4.B)

while, on the other side, from eq.{3.68) results

But with the defining series expansion of F(a,b,c;z), eq.{3.66), one gets easily

3 2-1 21-11 3d (4.11)

-46-

and hence

4(4.12)

yihich coincides with the right hand side of eq.(4.9). So, the residua of

V- Qji) and yr ~ id) at• the pole singularity corresponding to the first

excited state come out all right indeed and this represents a comforting, check

of the final formulas obtained in Section III .

Having at hand the exact result f o r ^ ' (to), one may find immediately its

residue at any other pole (i.e. at t i tO= E ^ E Q for any n = 2,5,4,...) just as we did it for

the first excited state (i.e. in the case n=2). The explicit knowledge of these residua

provides closed form expressions for the hydrogenic toroid dipole matrix elements.

Indeed, from eq.(2,28) we have

(4.13)

All one has to do now in order to compute the r.h.B. of eq.(4.13) is to use any of. (H)

eq.(3.65), (3.68) or eq.(3.7Z) expressing ' y (63) in conjunction with the relation

4.14)

which generalizes eq.(4.B) for any n=2,3,4,.... This procedure is similar to the one

(H)followed in ref./19/ in connection with the usual electric polarizabilities (>l * (id).

To substantiate these comments, we note thst, for example, eq.(4.12) may be

viewed as a formula for the matrix element of the toroid dipole operator between the

initial (ground, n=l) state and the excited state (n=2,1 =1, (i=0):

(4.15)

0 « *\-—— ts.C ot !l-

-47-

As indicated in refs./2/ the knowledge of such toroid matrix elements is important

because they contribute to the transition rates W, . (the toroid multipoies of the

source establish together with the usual electric ones, the electric type radiation (L£. )

emitted by the system); in those cases in which the usual long wave approximation

kr « 1 is no longer satisfactory (as e.g. by Roentgen transitions in heavy atoms), the

toroid contributions may be substantial. For instance, for the 2P-»1S transition in H-

like atoms, one has /2/

2?-MS

and one sees that the toroid correction (the term with l/3(o( Z) above) may indeed

became quite large (for heavy ions) with respect to the usual part of radiation given by

the electric dipole moment (--|£Q^\ , Q = electric dipole operator). The same

conclusion holds also (see refs./2/) for higher discrete transition as well as for

transitions into continuum (ionization processes). The toroid matrix elements of the

continuum to ground-state transitions wil l be computed exactly In the next Section;

they are given by the imaginary part of (C/J5 (above ion izat ion threshold) for

which we ere going to find a closed form expression.

-48-

V. IMAGINARY PART OF Y (H)(<<3) ABOVE IONIZAT10N THRESHOLD

When the intermediate states |n> in eq.(2.2B) belong to the continuous

spectrum (i.e. when ]n> is a Coulomb scattering state | u - , L= l , L > specified, say,

(H)by its energy £ , angular momentum L and its z-projection L ) Y (U) acquires an

imaginary part expressing its discontinuity across the branch cut"K|i^\ > | Egl • One can

then obtain closed form expressions for the matrix elements of the toroid dipole

operator between the initial (n=l) state and the final Coulomb scattering state

[u ,L=1,L > just in the same manner as we have found the toroid matrix elements for

the bound state - bound state transitions in Section IV. Indeed, from eq.(2.28) one has

(H)In this Section we shall compute the imaginary part of V (W) above

ionization threshold lib) h = |EQI by means of a procedure analogous to the one

(H)employed in refs./17/; to this aim we shall use the form of V (0>) as given by

eq.(3.68) which is by far more convenient for the present purposes than eq.(3.65) or

eq.(3.72). Only the first term (the one with i=l) in eq.(3.68) contributes. We note that

now one has

(5.2)

When (JQ varies in the real interval (1,"°), ~J, satisfies |*"T, I = l (i.e. ~j. runs along

the unit circle in the complex }, -plane), while ~J , satisfies D < Jf, < 1. From eq.(3,6S)

we thus have

i x i -, r ,-,\•(5.3)jrt*.'

-49-

T"

But

(5.4)

21

| W f X 4 • * + t < J

and the square bracket from the r.h.s. of eq.(5.4) may be transformed with the aid of

the following formula for analytic continuation of a hypergeometric function F of

variable z to F-functions of variable z~ (see ref./23/):

Specifying the above equation to 8=1, b=l-"Zp

relation

one gets

end using the known

(5.6)

To satisfy the condition I arg(- |.) \ < TT one must take

(5.7)

hence

-v-- e T. ¥11,1. (5.8)

-50-

Furthermore, because (see ref./2J/)

one has

(5.9)

In conclusion, one finds so from eqs.(5.3), (5.4), (5.6), (5.9) the desired result:

We recall that here

CJ. > ^ >

(5.11)

-51-

VL LOW ENERGY EXPANSION OF THE AMPLITUDE FOR ELASTIC

SCATTERING OF PHOTONS BY A HYDROGENLIKE ATOM

At this stage, we find it useful to report below a low energy expansion of the

quantum mechanical amplitude for elastic scattering of a photon on a spin less,

nonrelativistic, ground state hydrogenlike atom valid to the fourth order in the photon

frequency CJ inclusively. This amplitude (retardation included) is given by the Kramers-

He isenberg-WBller matrix element

In this Section we set for simplicity n = c = 1, XpSj, are here, respectively, the initial

and final momenta of the scattered photon of energy tO = \"X, \ = VK-A I~S*I and"?- the

corresponding initial and final polarizations, P denotes as before the electron

momentum operator - i y . The first term in eq.(6.1) represents the contribution coming

from the ~K term in the interaction Hamiltonian, treated in the first order

perturbation theory, whereas the sum expresses the (second order) contribution of the

~KVtixm. In refs./17/ the matrix element |YLhas been computed in closed form; below

we shall display the result:

' * ' l ' ' I A M / \ L , j ' (6.2)

(6.3)

(6.3')

A and other notations in the following formulas are those already used in Section

-52-

III ((*) now denotes the photon energy whereas in Section III *O stood far a general

frequency characterizing the periodicity of the external perturbation). Cf represents

the contribution of the first (contact) term in eq.(6.1)j it is

. - 2 -

(6.4)

The functions P{$L), Q(Sl) depend also on the scattering angle 0 which satisfies the

relation

• (6.5)

P(5L), P(5l2) ^ d QCSlj), QiSl^) represent the contributions of the sum in eq.(6.1);

P(SJ, Q(il) have been expressed tin the second of refs./17/) in terms of Appell's

generalized hypergeometric function F, (of four parameters and two variables; see

ref./23/)

as follows

(6.6)

(6.7)

(6.70

The variables x,,x2 of the Appell function F. are given by

(6.8)

-53-

•- r T

To achieve a series expansion of P(j l ) , Q(j l) in Cti powers it is convenient to use the

formulas (also obtained in the second of refs./17/)

(6.9)

which express P(J1), Q(JD as infinite sums over usual Gauss hypergeometric functions

F multiplied by factors containing {J -powers. The argument u in F is given by

(6.10)CJ1

(For p = 0 in the sums of eqs.(6.9),(6.9'), the factors in front of the square brackets

should be taken as 1).

4Obtaining the desired series expansions of M, N in powers of CO (to (0

inclusively) becomes then only a matter of tedious algebra. Next, we outline the

concrete derivation only for the amplitude M which survives in the forward direction

and in which we are mainly interested in this paper'(the derivation for N is similar and

for it we shall give only the final result).

Recalling that

f

we write P(JJ.) as

J

T (

) (6.13)

(6.14)

where (...) means "terms containing (^-powers higher than (0 ". Noting that

(6.15)

^ ' X1-

-55--54-

we get^ N , 3

J^>

(6.14)

Now we shall exploit that last of eqs.(6.15) (which shows that u is of order fOn for

small COQ) in order to obtain the needed expansions for the hypergeometric functions

sti l l remaining in eqs.(6.16), (6.17). We use /23/

(6.1B)

which for c = a+1 gives

Together with the formula

already used in Section III, eq.[6.19) leads to

(6.19)

0.54)

(6.20)

-56-

Setting a = 2-X. and b = 4 in eq.(6.20) one finds

Vx-(6.21)

which, with eqs.(6.15), leads to

Noting that

(6.23)

one has then

(6.24)

and hence eq.(6.16) yields the desired expansion for P :

(6.25)

- 5 7 -

•• r * r T T"

Therefore we find

+• —

H\ V

I * W -

It remains now to get an analogous expansion for P TZ.) (for P l ' ( t ) one already has

the expression eq.(6.14)). Due to the (J factor in front of the hypergeometric function

in eq.(6.17) this is much simpler as before, since one can use

(6.27)

and by means of only elementary calculations one gets

and hence

-f- -r

Equation (6.14) leads directly to& * * *<•-•>

We—)

The expansion of the contact term O^n eq.(6.3) is

(6.29)

(6.30)

(6.31)

Collecting the results we finallu. get the desired expansion for the scalar4

- 5 8 -

amplitude M enterring the matrix element liL in eq.(6.2):

]S*tf (6.32)

< JFor the Becond amplitude N in eq.(6.2) one finds analogously

In the forward direction (X^ = ^ , 0 = 0°) without change of polarization

finds, in particular,

^42. (Jif

The factor (- o^/m) in front of M, N andYV[,above was formally included to ensure the

usual normalizations; the optical theorem is given by

(6.34)

where <T, is the nonrelativistic total cross-section of the photoeffect, retardation

included.

Next, we shell briefly comment on eqs.(6.32), (6.32'), (6.33). They offer us an

excellent opportunity to see within an exact context how some very general theorems

-59-

on polarizabilities (oftenly applied and discussed in connection with cases in which

approximations can not be avoided) are operating.

From the works of Gell-Menu, Goldberger and Thirring /24/, Baldin /25/,

Lapidus /26/ and Petrunkin /Z7/ one knows that on the basis of very general

assumptions a low energy theorem can be derived which says that the coefficient of2

CO in the (properly normalized) forward Compton scattering amplitude on any target

should coincide with the sum of target's static electric and magnetic (dipole)

polarizabilities e t j ^u^O) + &t _j(iJ =0). From eq.(6.33), looking at the coefficient of

tO , we see that indeed, for hydrogenlike atoms, this theorem is exactly verified (the

Thomson (i.e. (J=0) term in the scattering amplitude (containing the target's mass in

the denominator) is absent in our case since we work in the limit in which the nucleus

(and hence the atom) is infinitely heavy):

w-to = x2.

2* (6.35)

There is one important point in connection with eq.(6.35) which needs to be

emphasized. Since for ground state hydrogen, because of spherical symmetry, there is

no magnetic polarizability except for the Langevin contact pieces like thB dipole one

(H)A L already mentioned in Section II,

(2.48)

it is interesting that the latter arises in the forward amplitude eq.(6.33) from the term

P ^ t i j ) + P®'(X J (as given by eq.(6.26)); it should not be forgotten that the quantity

P ( t ) comes from the infinite sum over the whole (continuous and discrete) spectrum

(see the second term in eq.(6.D) and is unrelated to the first (contact) term in eq.(6.1).

In the forward direction, the contribution of the contact term to I IL (O,0 = 0") comes

only through the quantity Q, and from eq.(6.4) it is seen that dW,9 = 0") = 1; the only

-60-

role of y in //|_ (u),Q = 0°) is therefore to cancel the unity in the expression of

P (T,) + P (Zjh eq.(6.26). The intricate way in which the Langevin magnetic

dipole polarizability ji . combines with the Epstein-Waller static electric dipole

polarizability /18/

(1.15)

to ensure the exact validity of the very general low energy theorem /24/-/27/ (as

applied to the particular system under consideration here, the hydrogeoid atom in its

ground state) is quite remarkable and illustrates once again the virtues of exact

calculations. In this respect we may say that we have rederived in this Section the

exact results for oC^(cJ=0) eq.(1.15) and B *H ' eq.(2.48) starting from an exact

evaluation of the "Compton" amplitude on hydrogen eq.(6.1). Strictly speaking, we

t (iJ =0) + S 5 ;actually did this here only for the >funf| ot (ii> =0) + fi . ; to obtain separately

oL: (cJ=0) and & ^ one should go out of the forward direction and exploit also the

expansion of the N-amplitude eq.(6.32'), comparing thereafter the whole, nonforward,

result with the general low energy expansion of the Compton amplitude as given e.g. in

refs./20/,/27/. in connection with the first term in the coefficient of cj in eq.(6.33),

the same may be said about the derivative with respect to (0 at uJ =0 of the dynamic

(frequency dependent) electric dipole polarizability(HI

M. (6.36)

This assertion will be proved in the course of the discussion from the last Section VII;

(H)( asthere we shall get eq.(6.36) starting dirsctly from the exact result for c<

given, for instance, in rpf./19/ or in the Appendix of the present paper.

Concerning the second and the third terms in the coefficient of h) from the

r.h.s. of eq.(6.33), to establish their relationship with the (electric quadrupole,

-61-

• r

Langevin-magnetic quadrupote, toroid dipole, etc.) polarizabiliUesj is by far a more

elaborated matter; the rtght in terpre ta t ion ^~O) a n / j and <J rf an/j[ terms may

be found in a separate paper /28/ devoted to a full and exact analysis of the role

played by the electric, magnetic and toroid polarizabilities in low energy Compton

scattering. Here we confine overselves onl)* to the following quali tat ive remarks.

(H)The static electric quadrupole polarizability txt (4J =0), which is known to be1-3-

of the type S / 2 (see eq.(7.3) below^will enter (but not alone) the second term in/ (H)

the a) - contribution from eq.(6.33). The static toroid dipole polarizability Y, (**=£))

together with the contact (Langevin-type) piece YU<H) G-e. Y ^ o j g , ^ ^ ^ ^ ) ^ ^

as given by eq.(2.38), wil l contribute to the third term in the CU piece of eq.(6.33).

However, there wilt be also a contribution of exactly the same type (i.e., <* a.Qtj. )

coming from a Langevin (contact) usual quadru polar magnetic polarizability

/^LQuadruDole' t h e l a t t e r originates (in complete analogy with the familiar (dipole

type) Langevin magnetism) from the*^ part of the Hamiltonian, in connection with a

magnetic quadrupole piece in the vector potential

With such a vector potential, the t e r m ^ p1 in the Hamiltonian will represent just

the part linear in the external field of an usual magnetic quadrupole-type interaction;

for the present discussion only the part quadratic in the external field [A J , is, of

course, of interest. To disentangle the toroid (and electric) effects from the purely

magnetic ones (which all get mixed in the forward «mplitude) one must work at 0 / D°

and include in the analysis both amplitudes (M and N) from eq.(6.2). For more precise

clarifications on this subject, we send the reader to the forthcoming paper /2B/.

-62-

• •UULM. I! » j i Ji is rffc-

VIL DISCUSSION OF THE RESULTS AND SOME COMMENTS

In connection with the elastic scattering of tight on an arbitrary system (here

referred to as Compton scattering) in refs./7/,/8/ it has been shown that unlike the

usual electric and magnetic dipole (£ =1) polarizabilities oC^ _1(u)=Q) and &,_AO=0)

which establish the low energy behaviour of the amplitude to the second order in the

light frequency 63 , the static toroid dipole polsrizability Yt ,(lJ=0) enters beginning

only with the next relevant (fourth) CO order. V » _,((J=0) appears therefore on the

same footing with the system's (static) quadrupole (electric and magnetic)

polarizabilities ot » _2(iJ=0) and fi i^J^*®' b u t r e l a t e d t 0 different angular

structures in the low energy expansion of the Compton scattering differential cross-

section. In "real" (i.e. both photons massless and transverse) Compton scattering

V" , _,(u)=0) enters in general accompanied by a " ta i l " coming from the derivative with

respect to CO (at(0=0) of the usual electric dipole dynamic polarizability oL / ,(<t)=0).

)~0(7.1)

In the particular case of real Compton scattering V" . ,(C0=0) and oi^'^i('J=0) appear

jointly in the amplitude through the combination V , ,(<J=0) + e£'_A<J=0) and so

the toroid effects as a rule get mixed together with those related to the usual

(frequency dependent) induced dipole electric moments. This reflects the already

known situation (see refs./2/) that while there are three types of multipolarity for

sources (electric, magnetic, toroid), there are, of course, only two types of

multipolarity for radiation (the usual El and IM1 waves); toroid type sources emit

electric-type radiation. In general, in order to disentangle the effects pertaining to

>Y . ,(uJ) from those related to oi / . I ^ ) D n B should look into processes in which a

spectral (O) analysis rather than a purely angular one can be carried out, such as

inelastic electron scattering, far instance, or virtual Compton scattering, etc. Despite

-63-

this sort of degeneracy (with respect to the type of the source; different types of

sources may yield the same type of radiation), which is present also in real Compton

scattering, the toroid polarizabMities ^ (CJ) and the usual electric and magnetic ones

o(. (U), Pt('d) (as emphasized in refs./7/,/B/) are independent characteristics of the

body, can not be reduced to each other and reflect different material properties of the

system. The case considered in the present paper, that of the hydrogenoid atoms,

presents the particular advantage of allowing a clear separation between the toroid

and the usual electric and magnetic effects on account of the additional parameter Z

(characterizing the strength of the bounding electric Coulomb field) existing in the

problem. Indeed, oi,'£ =1(c0=0) Bnd Yj=1((d=0) have a different Z dependence and the

"degeneracy" we have spoken about above may, in principle, be overcome in this case

by comparing measurements on ions with different nuclear charge numbers Z.

To get an intuitive picture of what we mean by inducing a toroid dipole

moment into a system, we note that (speaking in a classical language) whereas a

charge moving on a closed circular orbit gives tise to a magnetic dipole moment, the

toroid dipole moment is related to charges moving on "tight-ltke" closed orbits. When

an atom is placed in a magnetic field with V xH e x t t 0 (say, if a constant external

conduction or displacement current is being flowed through the atom), some of the

electrons may well begin to "move" on such eight-like (or other sort of topologically

complicated) orbits and the toroid polarizability V" is just measuring the extent to

which that should actually happen. Once the Hamiltonian describing the system's

text "~*ext, H does contain the independent dipole

pieces ~Q? 8 X t , -~S!i3e>lt, -T.'7x"Hex t, then each of the corresponding response

functions (i.e. the electric, magnetic and taroid (dipole) polarizabilities) can, in

principle, be measured.

After these comments we start looking now into the relative importance of

the effects coming from the induced taroid moments (specified by Y (u>)) as against

that of the usual (induced) electric and magnetic ones (specified by the familiar

-64-

electric and magnetic polarizabilites). Specifically, here we shall first compare

V , ,(U)=G) with tt' - Au)=Q) and oC,__(£j=0) for (nonrelativistic, spinless) H-like

at Dins (as a typical case far atomic physics scales) and afterwards do the same

comparison for a case characteristic to hadronic scales, that of the (charged) pion.

Looking then at what the situation is at distances of the order of the Bohr radius

S -13

( 10* cm) on one side, and at distances of the order of 10" cm on the other side,

some speculative ideas will be put forward about what might happen at even smaller

distances.

The dynamic (i.e. frequency dependent) electric multipole polarizabilities

°t>i(tO) of the ground state hydrogenlike atom have been calculated in closed form

(for any multipolarity order L ) in ref./19/ by a method which is less direct than the

one used by us to get ^A, _1('*>) (one first expresses of. (U) in terms of a nonphysical

scattering amplitude and only afterwards uses the results of refs./17/). In our notations

and conventions the result of ref./19/ reads

(In ref./19/ a factor of e (electron's charge) in the definition of the multipole operator

is left aside and only the first of the two terms in equations analogous to our eq.(2.28)

is retained).

-65-

•~r r

ft A

Actually, the exact result (eq.{7,2) above) for c<* '(u)) may be further

considerably simplified and, as it is shown in the Appendix, may be condensed to a

form in which only (essentially) one hypergeometric function F appears instead of

(essentially) three such functions F as in eq.(7.2). However, we prefer to work here

with eq.(7.Z) which is probably more familiar to the reader.

For further comparison we shall need the static electric quadrupolepolarizability ck (tJ=O); from eq.(7.2) it follows immediately that

t l

. \0'h° (7.3)

For Z = 1 this result is quite close to the old approximate result noted in Dalgarno's

basic paper /29/, which, in our definitions (different from those in ref./29/ by a factor

of d\ = e ) looks

0.1,9 MO.ho

(7.31)

Finding^ <(H\w=Q) = (<J)/dUZ] n is not so immediate; one has, for

instance, to perform a series development in U) for the dynamic electric dipole

(H*lpolarizability ot (U) as given by eq.(7.2):

. 2'

(7.4)

Using the first required terms in the developments of the appearing Gauss

hypergeometric functions

-66-

• "

after some algebra one finds

uJ + higher (even) powers of <t). (7.5)

The first term in eq.(7.5) represents the well known Epstein-Waller result / IB / for the2

static electric dipole polarizability whereas the coefficient of (0 in the second term is

just the quantity (eq.(7.1.)) we are now interested in:

_ 349 £ ^

On the basis of the result found in this paper for the static toroid dipote

polarizability of (ground-state) H-like atoms

2 lCo o .

D.74)

end taking into account eq.(7.3) and eq.(7.6), the following conclusion emerges: The

effects of the induced tore id moments for H-like atoms (and in the atomic physics, in

general) are very small with respect tc those of the corresponding usual electric

moments, the dominant ones being as a rule expressed by «(• (ft?=Q). However, because

(7.7)

-67-

one sees that the effects of the induced toroid moments, as against that of the usual

electric ones, are increasing with Z. This allows for possible applications involving, the

induced moments even in atomic physics problems, e.g. in what concerns certain

plasma components, but this subject will not be pursued here any longer.

It is perhaps because of their smallness that the toroid effects have not been

so far identified and investigated in atomic physics. In a larger perspective, what is

important to us is that they are there, however small. In this respect, we recall that

in re fa./7/,/B/ it has been shown that the toroid components of the van der Weals

forces (i.e. those van der Waals forces which arise on account of the toroid

polarizabilities, or, in other words, on account of fluctuating toroid dipoles), may prove

themselves particularly important in biomolecular physics in view of the complicated

topological structure of the macromolecules. In subsequent applications relativistic

and spin effects (not considered in this paper) should also be carefully analyzed, by

including in the expression of the toroid dipole operator the appropriate spin dependent

part (see refs./2/).

At this point, we would like to mention that there is now renewed interest on

other topics related to both atomic and elementary particle polarizabilities, such as

electric polarizability of hydrogenlike atoms with Z \ 137, polarizabilities of strongly

bound systems, etc. (see refs./3Q/) and references therein). Exact relativistic versions

of the results obtained in this paper may be looked for on the same lines as in ref./31/,

where a closed relativistic formula for ot,i was found.

We turn now to see what a comparative analysis of the induced toroid

moments as against the usual electric and magnetic ones will give when instead of a H-

like atom one is considering a typical hadron, the (charged) pion, for instance. The

example of the (charged) pion is by no means of purely academic interest. The

Compton effect on the charged pion has already been experimentally observed and

studied /32/-/3S/; the electric dipole of the charged pion has been experimentally

extracted in two different experiments /33/,/34/.

-68-

In refs./7/,/8/ an order of magnitude estimate of the static toroid dipole{IT1}

polarizabtlitv of the charged pion "V'. ,((0=0) has been obtained (by evaluating, as a1 <r*J

first indication, the A.(127O MeV)-meson resonance contribution to V» ,({*J=0) in

terms of the experimentally known radiative width I (Aj-

)C£0.6 MeV) with the

result

(IT*) (7.8)

Under the same approximations, in refs./7/,/8/ it has been found that the derivetive at

(jj (at U)-Q) of the pion's usual frequency dependent electric dipole polarizability of

interest here is

From previous work (the first of refs./20/) it is known that the (charged) pion's static

electric quadrupole polarizability is expected to be of the order

fir*) (IT*)

P • A D " 5 " AVv. (7.10)

The picture which so emerges for (charged) pions looks then as follows:

(7.11)

It Ut(sharp contrast with the corresponding one for H-like atoms (see eqs.(7.7)). While

for atoms (length scale 10" cm) the toroid polarizabilities are in general negligible

quantities, for hadrons (length scale 10" cm) they can no longer be neglected, being

expected to be of the same order of magnitude as the usual ones (of one order of

multipolarity higher, of course, because it is with them that the comparison has to be

- 6 9 -

r

made). So, eqs.(7.7),(7.11) seem to tell us that the more "elementary" the object is (or,

alternatively, the higher are the characteristic excitation energies of the system), the

better might it respond to an external (conduction end/or displacement) current

(7xT^ x t ) rather than to the external fields~£ext, Tfxt directly. What eqs.{7.7) and

eqs.{7.11) are saying must be given serious consideration ; eqs.(7.7) come from the old

goad quantum mechanics while eqs.(7.11) come from quantum electrodynamics plus

some firmly established phenomenoiogical information on pions which circumvents

quite reliably possible uncertainties related to the strong interaction part of the

dynamics; at least in as far as orders of magnitude are concerned, these results

therefore are bound to remain unshaken. Eqs.(7.7),(7.11) illustrate a considerable

increase of the induced toroid effects as compared with the usual electric and

magnetic ones when going from atomic to hadronic energy scales. The speculative idea

we would like to put forward now, on the basis of the above discussion, consists in the

following: At even smaller distances (higher energies), say at •—• 10 cm (the HERA

electron-proton collider at DESY will begin to probe the structure of matter down to

such distances), the role of the induced toroid moments might increase further so

much that the situation took just opposite to the one in atomic physics (in the sense

that the usual (induced) electric and magnetic moments might become just as small

with respect to the toroid ones as the latter were in comparison with the former in

atomic physics). Some features of such topical subjects like supersymmetric, string,

superstring theories, seem also to support this conclusion:

When dealing with induced toroid moments, we have some kind of current

flowing through the system which induces in it certain closed (toroidal) currents; this

is, in a sense, an usual transformer effect and the toroid polarizability measures in fact

its strength. We recall (see refs./2/./7/,/8/) that for a classical toroidal current (of

large and Binalt radii of the toroidal solenoid R, r) with N turns of winding (N = even)

and a current intensity I, the classical toroid dipole moment calculated with eq.(1.6) is

-70-

WM

JI V I (7.12)

(f? is the unit vector directed along the symmetry axis of the toroid and pointing in the

(common) direction of two opposite circular currents (on the meridians of the torus) at

their closest position). On account of large numbers of turns of winding, (closed)

filiform structure (Strings) with non trivial topological properties, could provide ideal

candidates for systems having large toroid polarizabilities but relatively small electric

and magnetic ones. A string picture at the aubhadronic level would therefore come well

in line with the speculative idee put forward above.

There is yet another strange coincidence which provides further indirect

support to our inferences. More than half a century after being Invented, Majorana

fermions are nowadays again under renewed scrutiny. They currently occur in grand

unified and supersymmetric theories as well as in connection with various modern

aspects of neutrino physics (double 3 -decay, neutrino oscillations, etc.). Now, the

Majorana particles single them out among the other fermions by the happening that, as

shown in ref./6/, the only electromagnetic structure they can possess, is represented

just by toroid multipole moments and distributions; any other usual electric or

magnetic multipole characteristics (like charge, charge radius, electric or magnetic

dipole moment or radii of such dipole distributions, or electric and magnetic quadrupole

moments and distributions and so on) are for them rigorously forbidden by TCP

invariance. The discussion is now going about intrinsic multipole moments and

distributions, but bound states of Majorana particles with nonvanishing intrinsic

toroidal electromagnetic structure would rather have large toroid polarizabilities just

as macroscopic substances composed of polar molecules (i.e. molecules with an intrin-

sic electric dipole moment) would have in general a large electric polarizability.

We end this paper with the remark that it may be worth considering toroid

configurations and toroid response functions in connection with non-abelian gauge

fields, more or less on the same lines as done for electramagnetism.

- 7 1 -

ACKNOWLEDGEMENTS

Parts of this work have been discussed at various times with V.M.Dubovik,

S.B.Gerasimov, B.N.Valuev and others. Boris Nikolayevich Valuev died on February

19th, 1987, at the age of f i f ty eight. Apart from his important research contributions

to various areas of ph>sics, B.N.Valuev was one of those who translated in Russian the

Schweber's book on quantum field theory. In our student days in Bucharest we have used

it to learn something on quantum fields (no English copy was then available to us),

without, of course, particularly noting at the time the names of the translators.

This paper is humbly devoted to the memory of Boris Nikolayevich Valuev. He

has never been allowed to travel to the West.

-72-

APPENDIX

COMPACT EXPRESSION FOR THE FREQUENCY DEPENDENT

ELECTRIC MULTIPOLE POLARIZABILITY OF HYDROGEN o<(,H)(*

In Section VII we have made use of the exact formula eq.(7.2) for the dynamic

(H)electric multipole polarizability of hydrogen o( (<J) (valid for any multipolarity order

I) obtained in re f . / l9 / which we rewrite below in a slightly modified version:

(A.1)

Our aim is to express the curl bracket figurating in eq.(A.l) Cthe subscript i on f . , T.

is left aside below since i t is irrelevant for the following considerations)

t

in terms of only one hypergeometric function F, To that purpose, with the aid of

-73-

ref.23, we shall first derive the following general relation:

Proof: eq. Z.B(4) of ref./23/, p.109 gives

F(a,b,a;z) = (1 - z),-b (A.3)

With eq.2.804) of ref./23/, p . l l l

c[a-(c-b)zF(a,b,c;z) = ac(l-z)F(a+l,b,c;z) - {c-a)(c-b)zF(8,b,c+l;z) (A.4)

written for c = a+1, and eq.(A.J), one gets

- (a+l-b)zF(a,b,a+2;z). (A.5)(a+l)[a-(atl-b)z)r(a,b,a+ljz) =

But eq.2.8(31) of ref./23/, p . l l l written for a a+1,

(c-a-l)F(a,b,c,z) = [c-2(a+l)-(b-a-l)z]F(8*l,b,c;z) + (a+lXl-z)F(a+2,b,c;z), (A.6)

in which one sets c = a+z, together with eq.(A.3), leads to

F(a,b,a+2;z) = [-B+(a*l-b)zK(a+l,b,a+2;z) + (s+lX

Eliminating now F(a,b,a+2;z) from eq.CA.5) and eq,(A.7) one finds

_ _ 2 r

A-t-CA.7)

(A.8)

-74-

Eq.(A.B) written for a-ja+1 reads

(A.9)

Dividing now eq.(A.S) by a and aq.(A.9) by (a+2-b) and subtracting the results, one gets

A.+2.

(A.

When multiplied by (b-2), eq.(A.lO) leads to eq.(A.2) we wanted to prove.

Returning to eq.(A.l'), we note that it may be written in a form suitable for

subsequent use of eq.(A.2). Indeed, setting

eq.(A.l') we are interested in becomes

(A.1Z)

and, therefore, by means of eq.(A.2), we have succeeded so to remain with only one

Y(<L+i, tr, CK^; h)

-75-

hypergeometric function instead of three:

^ (A-2rf

(A.13)

With l i t t le algebra eq.(A.13) may be further simplified:

fI * +

(A.14)

An even more convenient form may be obtained by noting that (eq.2.9(l) of ref.23,

p.113)

F(a,b,c;z) = ( l -z)C " 8 " b F(c-a,c-b,c}z) (A.15)

wherefrom, for c = a+1, one gets

(A.16)

and hence, with a-+a+l,

F(a+l,b,a+2;z) = ( l -z ) 1 " 6 F(l,a+2-b,a+2;z) (A.17)

With eq.(A.17), eq.(A.I4) finally takes on the form

This is the result we were essentially looking for. Al l that remains to be done is to

replace a,b,z appearing in eq.(A.lH) by the physical variables 1 , t , T , according to

-76-

eqs.(A.ll). So one finds

I j(A.19)

In conclusion, coming back to eqs.(A.l'), (A.I), (7.2), one seea that the frequency

dependent electric multipole polarizability of a (nonrelativistic, spinless) hydragenlike

Btonn, c^ C^), can be expressed (for any multipolarity order CO as

^ _ v1 L-

. &

We recall that (restoring now factors of %, c):

IE4

•tic

we also recall for convenience that F(a,b,c;z) is the usual Gauss hypergeometric

function

- 7 7 -

As it is seen from eq.(A,20), oC. (a)) is an even analytic function of b) in the complex

<*>- plane with simple poles at + <J= (En-EQ)/h, n=2,3,4,... (En = EQ/nZ is the discrete

spectrum of the H-like atom), and branch cuts along the real 4?-axis for MWl > |E I

(i.e. above tonization threshold ^ E l ) .

From eq.(A.2O) one finds immediately the following simple formula for the

static (i.e. Ci)-0) electric multipole polarizabilities of nonrelativistic spinless ground

state hydrogenlike atoms, valid for any multipolarity order / :

(A.21)

In particular, for t=l and t =2 one reobtains from eq.(A.21) the known expression of

the static electric polarizabilities in the dipole and quadrupole cases already mentioned

above in this paper:

- 7 8 -

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- 8 1 -

Stampato in proprio nella tipografia

del Centro Internazionale di Fisica Teorica