Advances in Atomic and Molecular Physics, Volume 18

Advances in

ATOMIC A N D MOLECULAR PHYSICS

VOLUME 18

CONTRIBUTORS TO THIS VOLUME N. ANDERSEN L. A. COLLINS A. S. DICKINSON

G. W. F. DRAKE A. HIBBERT B. R . JUNKER WALTER E. KAUPPILA J . MORELLEC

S. E. NIELSEN D. W. NORCROSS D. NORMAND

G. PETITE D. RICHARDS LEONARD ROSENBERG TALBERT S. STEIN

ADVANCES I N

ATOMIC AND MOLECULAR PHYSICS Edited by

Sir David Bates DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST. NORTHERN IRELAND

Benjamin Bederson DEPARTMENT O F PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 18 1982

@) ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishen

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LIBRARY OF CONGR~SS CATALOG CARDN U M B E R : 65- 18423 ISBN 0 -12-003818--8 P R l N T t D IN T H E UNITED STATES O F AMERICA 82 83 x4 8 5

9 8 76 5 4 3 2 1

Contents

...

Vlll

CONTRIBUTORS

Theory of Electron-Atom Scattering in a Radiation Field Leonard Rosenberg

I. 11. 111. IV. V. VI .

Introduction Bremsstrahlung Asymptotic States Scattering Theory Generalized Low-Frequency Approximations Concluding Remarks References

Positron-Gas Scattering Experiments Talbert S . Stein and Walter E . Kauppila I. Introduction 11. Experimental Techniques for Total Cross-Section Measurements 111. Total Cross-Section Results IV. Differential Scattering Cross Sections V. Inelastic Scattering Investigations VI. Resonance Searches VII. Possible Future Directions for Positron Scattering Experiments References

1 6 14

23 37 49 50

53 55 64 84 86 91 92 93

Nonresonant Multiphoton Ionization of Atoms J . Morellec, D . Normand, and G . Petite

I. Introduction 11. The Theory of Multiphoton Ionization 111. Absolute Measurements of Multiphoton Ionization

Cross Sections V

98 101 119

vi

CONTENTS

IV. Experimental Results: Comparison with Theory V. Destructive Interference Effects VI. New Trends VII. Conclusion References

Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions A . S . Dickinson and D . Richards I . Introduction 11. 111. IV. V. VI.

Angle-Action Variables Rotational Excitation Uniform Approximations Semiclassical Theories Conclusions References

133 140 151 157 I 60

166 167 170 183 186 198 200

Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena B . R . Junker I. Introduction 11. Gamow-Siegert States 111. Complex-Coordinate Theorems; and Properties of the Wave Functions IV. Variational Principle V. Variational Calculations VI . Many-Body Theories VII. Nondilation Analytic Potentials VIII . Complex Stabilization Method IX. Discussion References

Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems N . Andersen and S . E . Nielsen I. Introduction Theoretical Models 111. Experimental Techniques IV. Results and Discussion V. Conclusions References 11.

208 210 214 227 229 243 244 247 256 260

266 27 1 279 287 303 305

CONTENTS

vii

Model Potentials in Atomic Structure A . Hibbert I. Introduction 11. Simple Semiempirical Model Potentials 111. Potentials Based on Hartree-Fock Formalism IV. Core Polarization V. Relativistic Model Potentials VI. Conclusions References

309 31 1 317 327 332 336 338

Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules D . W . Norcross and L . A . Collins I. Introduction 11. General Formulation 111. Approaches and Approximations IV. Applications V. Conclusion Nomenclature References

34 1 350 360 377 390 392 393

Quantum Electrodynamic Effects in Few-Electron Atomic Systems G . W . F . Drake I. Introduction 11. One-Electron Systems 111. Light Muonic Systems IV. Two-Electron Systems V. Few-Electron Systems VI. Concluding Remarks and Suggestions for Future Work References

399 401 424 426 446 454 456

INDEX CONTENTS OF PREVIOUS VOLUMES

46 1 478

This Page Intentionally Left Blank

Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

N. ANDERSEN, Physics Laboratory 11, H. C. eJrsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark (265) L. A. COLLINS, Group T4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (341) A. S. DICKINSON,* Laboratoire d’Astrophysique, Universiti de Bordeaux, Talence , France ( 165) G. W. F. DRAKE, Department of Physics, University of Windsor, Windsor, Ontario, Canada N9B 3P4 (399) A. HIBBERT, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland (309)

B. R. JUNKER, Department of the Navy, Office of Naval Research, Arlington, Virginia 22217 (207) WALTER E. KAUPPILA, Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202 (53) J. MORELLEC, Service de Physique des Atomes et des Surfaces, Centre d’Etudes Nucle’aires de Saclay, 91 191 Gif-sur-Yvette, France (97)

S. E. NIELSEN, Chemistry Laboratory 111, H. C. eJrsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark (265) D. W. NORCROSS, Joint Institute for Laboratory Astrophysics, University of Colorado, National Bureau of Standards, Boulder, Colorado 80309 (341) D. NORMAND, Service de Physique des Atomes et des Surfaces, Centre d’Etudes Nucleaires de Saclay, 91 191 Gif-sur-Yvette, France (97) G. PETITE, Service de Physique des Atornes et des Surfaces, Centre d’Etudes Nucleaires de Saclay, 91 191 Gif-sur-Yvette, France (97) D. RICHARDS, Faculty of Mathematics, Open University, Milton Keynes, MK7 6AA, England (165)

*Present address: Department of Atomic Physics, University of Newcastle upon Tyne, Newcastle upon Tyne NEI 7RU, England.

ix

X

CONTRIBUTORS

LEONARD ROSENBERG, Department of Physics, New York University, New York, New York 10003 (1) TALBERT S. STEIN, Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202 (53)

ll

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL. 18

THEORY OF ELECTRON-A TOM SCATTERING IN A RADIATION FIELD LEONARD ROSENBERG Department of Physics Nett. York University N e w Y d , Nett' York

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . A. InfraredProblem . . . . . . . . . . . . . . . . . . . . . . . B. Bloch-Nordsieck Theory: Coherent States . . . . . . . . . . . 111. Asymptotic States . . . . . . . . . . . . . . . . . . . . . . . . A. Electron in a Plane Wave Field . . . . . . . . . . . . . . . . B. Dressed-Target States . . . . . . . . . . . . . . . . . . . . . IV. Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . A. Formulation, . . . . . . . . . . . . . . . . . . . . . . . . . B. Approximation Techniques . . . . . . . . . . . . . . . . . . V. Generalized Low-Frequency Approximations . . . . . . . . . . . A. Connection with the Classical Limit . . . . . . . . . . . . . . B. Modified Perturbation Theory . . . . . . . . . . . . . . . . . C. Intermediate Coupling . . . . . . . . . . . . . . . . . . . . . D. Strong Coupling . . . . . . . . . . . . . . . . . . . . . . . . VI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. . . .

1 6 6 10 14 14 17 23 23 28 37 37 40 43 46 49 50

I. Introduction Interest in the theory of electron-atom scattering in a radiation field has increased in recent years as a result of the important role it plays in the study of plasma heating (Geltman, 1977), gas breakdown (Kroll and Watson, 1973), and laser-driven fusion (Brueckner and Jorna, 1974). Of course, the theory has long been a subject of study, particularly in connection with the bremsstrahlung process. Owing to the dominance of stimulated emission and absorption, the theory of scattering in a laser field takes on a form somewhat different from the more familiar theory of spontaneous bremsstrahlung. However, the underlying physics is the same in the two cases, and it is interesting to trace the connections. In the lowfrequency domain the two theories show a remarkable confluence, as will 1 Copyright @ 1982 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

2

Leonard Rosenberg

be described here in some detail. The problem of scattering in a lowfrequency laser field has come under close theoretical scrutiny recently for several reasons. The effect of the field on the projectile becomes more pronounced for a given field intensity at the lower frequencies. Furthermore, the theoretical analysis simplifies considerably in the low-frequency limit; the projectile-field interaction, on the one hand, and the projectile interaction with the target atom, on the other, can be accounted for by separate calculations, whereas at higher frequencies the dynamics of the various interactions are inextricably mixed. The low-frequency domain is also of interest on more general grounds since it provides an example of the applicability of the correspondence principle to the analysis of the motion of the projectile in the field. The feature of the theory of scattering in an external field which most clearly distinguishes it from the field-free version is the necessity, in the former case, of replacing the usual plane wave solutions, describing propagation of the colliding particles in asymptotic states, by solutions which properly account for the interaction of the particles with the field. The modified plane wave solution for the electron may be thought of as representing the sum of an infinite set of perturbation terms which account for the stimulated emission and absorption of laser photons; the influence of the target atom on the electron is ignored here since the colliding particles are assumed to be well separated in initial and final states. The effect of the target-field interaction on the asymptotic states must also be included. Of course, it is only during a finite time interval that the particles are influenced by the field. This physical feature may be accounted for in the formalism by requiring the field to vanish outside a certain space-time domain and by introducing localized wave packets to describe the motion of the particles (Neville and Rohrlich, 1971). In practice one usually takes these packets to be infinitely broad and allows the field to be of infinite extent, although, as emphasized by Kruger and Jung (1978), a more realistic description of the spatial and temporal properties of the field can be crucial in many cases. Application of the apparatus of time-dependent scattering theory leads to the following qualitative picture. The colliding particles propagate in a field-free region in the remote past. This is indicated as region 1 in Fig. 1, which is a schematic representation of the strength of the particle-field interaction during the various stages of the scattering process. We wish to determine the probability for a transition which takes the system from such an initial state to a final state in the distant future, where the particles are once again free of the influence of the field (region 5 ) . Penetration of the particles through the fringe region (separating regions 1 and 2) effectively switches on the field. One may think of this as an adiabatic process,

ELECTRON-ATOM SCATI’ERING IN A RADIATION FIELD

3

PRECOLLISION

I I I

I I

+ t-+a

FIG.1 . Schematic representationof the field intensity experienced by the electron-atom system in different stages of the collision process.

in which the state of the system at any instant is determined as the solution of the wave equation appropriate to the instantaneous value of the field intensity. The field is thought of as being switched off in an analogous way. Assuming that the intensity dependence of these “dressed states” is known, we can trace the evolution of the wave function as the system passes through the fringe regions by simply varying the intensity parameter appropriately. In practice we calculate the probability for transitions between dressed states (the system passing from region 2 to region 4). These probabilities are related to the observed transitions by following the development of the dressed states as the interaction is slowly switched on and then off. A discussion of the scattering problem based on this adiabatic picture has been presented by Deguchi et nl. (1979) in the context of the many-body Green’s function formalism. The necessity of introducing modified plane wave states is by no means peculiar to the problem of scattering in an external field. Level shifts associated with sporituneoirs emission and reabsorption of photons by a charged particle are a familiar feature of renormalized perturbation theory. Modified plane waves also play a role in the theory of scattering in the presence of a long-range Coulomb interaction. Here, as in the case of scattering in a radiation field, there is a residual interaction which must be incorporated in the construction of the asymptotic states. Three-body scattering theory provides still another example in which long-range interactions must be accounted for at the outset (Faddeev, 1961). The interparticle potentials may be of short range, but a pair of particles may interact with the third particle acting as a distant spectator. The effect of this interaction, which takes place over an infinite domain of configuration space, must be included in formulating the problem (setting up the bound-

4

Leonard Rosenberg

ary conditions or, in an integral equation approach, choosing the kernel) to avoid mathematical difficulties. It is sometimes helpful to keep in mind the formal analogy connecting the three-body problem with the scattering problem of present interest; the external field plays the role of the third body. With the introduction of modified plane waves the theory of scattering in an external field can be developed in parallel with the field-free version. In effect one replaces each plane wave by a superposition of infinitely many such waves, each term in the expansion corresponding to the presence of a definite number of virtual photons in the field. Such sums must be truncated in actual calculations, and convergence must be checked numerically. A review of this approach to relativistic quantum electrodynamics in the presence of an intense laser field has been given by Mitter (1975). In the present treatment of the subject we will be concerned, for the most part, with methodology; computational techniques which are capable of dealing with the intense field version of the theory and realistic atomic models are still at an early stage of development. The emphasis we shall place on the low-frequency domain is a reflection of the fact that here interesting and useful results can be extracted from the theory without detailed numerical computation. The external field has the effect not only of modifying the scattering process through an alteration of the plane wave states but also of inducing processes which would not take place in the absence of the field. As a primary example consider an electron-atom system which can form a bound (negative ion) state. The existence of this state will not appreciably affect the scattering in the absence of the field for scattering energies well above threshold. In the presence of the field, however, the incident electron can lose just the proper amount of energy through stimulated emission so that it becomes temporarily bound in the ionic state. This is, of course, an unstable state since the system can absorb energy from the field. The net effect is the appearance of an induced resonance. The second half of this process, in which energy is absorbed and the electron liberated, is of considerable interest in its own right. The observation that multiphoton ionization can be viewed as a component of a scattering process is potentially useful since approximation methods developed for scattering may be carried over to the study of ionization. This point will be illustrated below in the description of a low-frequency approximation for multiphoton ionization. The unstable nature of atomic states in the presence of the external field should in principle be accounted for in setting up a formal theory of scattering by composite targets. Of course, if the lifetime of the unstable

ELECTRON-ATOM SCA’lTERING IN A RADIATION FIELD

5

target is large compared with the collision time, the instability will have little practical effect. For sufficiently strong fields target ionization will play a major role, and very little attention has thus far been given to the treatment of this effect. In practice the dressing of the atomic target by the field has been accounted for either in first-order perturbation theory (Mittleman, 1980) or in a resonance approximation which allows for virtual transitions involving a few bound states of the target (Hahn and Hertel, 1972; Gersten and Mittleman, 1976a; Mittleman, 1976). The term “resonance” is meant here to include the strong coupling among nearly degenerate states which can occur at low frequencies; the significance of this effect in the context of the scattering problem was discussed by Perel’man and Kovarskii (1973). The dressed-atom states required in such applications can be constructed using well-known approximation techniques (Shirley, 1965). It may be mentioned here that these techniques frequently involve the use of a gauge transformation as a method for introducing the electric-dipole approximation in the description of the interaction between the bound system and the field (Gopert-Meyer, 1931; Power and Zienau, 1959). An analogous procedure can be used in the electron-atom scattering problem; such a gauge transformation turns out to be particularly useful in the development of low-frequency approximations, as discussed below. A useful review of the theory of single-photon bremsstrahlung in electron-atom scattering has been given by Johnston (1967). Geltman (1977) has compared the classical and quantum formulations of the treatment of multiphoton processes in free-free transitions and has provided a brief review of applications to the study of plasma heating by an intense laser beam. A survey of the theory of free-free transitions, along with a description of beam experiments, has been given by Gavrila and Van der Weil (1978). The status of experimental studies of the subject has been reviewed more recently by Andrick (1980). Atom-atom scattering in an external field is a subject of some interest (De Vries et al., 1980), but we shall not touch upon it here. Another aspect of the subject of free-free transitions not discussed here deals with the use of laser beams to create a population of target atoms in an excited state (Hertel and Stoll, 1977; Bhaskar et nl., 1977). Our emphasis will be on recent developments in the theory of multiphoton processes in electron-atom scattering. We shall be particularly concerned with the connections which exist between the time-dependent and time-independent versions of the theory, the relationship between the classical and quantum descriptions in the low-frequency limit, and on the unification of our understanding of stimulated and spontaneous multiphoton processes for low-frequency radiation fields.

Leonard Rosenberg

6

11. Bremsstrahlung A. INFRARED PROBLEM

We begin with a discussion of the theory of spontaneous bremsstrahlung, with emphasis on the special features associated with the infrared limit. While this is a well-studied subject, it provides us with an interesting example of a nonperturbative treatment of the particle-field interaction, a treatment which has much in common with methods discussed later in connection with the stimulated bremsstrahlung problem. Particular care must be taken in describing the electron in initial and final states; the energy of the electron-field system is well defined asymptotically so that states differing in the number of infrared photons can have closely spaced levels. This near degeneracy leads to a breakdown of ordinary perturbation theory. In the case of spontaneous emission the breakdown has a dramatic effect-it leads to the well-known infrared divergences-which was first analyzed by Bloch and Nordsieck ( 1937). We shall now sketch the Bloch-Nordsieck analysis in the context of nonrelativistic potential scattering. A generalized version, which allows for an atomic target, and which includes a higher order correction in the frequency, has been worked out recently (Rosenberg, 1980b). Since the spacing between energy levels hw is small (compared, say, with the electron kinetic energy) in the low-frequency limit, one expects that a classical description of the electron interaction with the radiation field, corresponding to the limit h + 0, will be appropriate. (One might also expect difficulties with the usual perturbation expansion since e'/hc x for h 0.) Let us then recall some basic results of classical radiation theory. Consider the scattering of an electron from a center of force; the electron momentum changes from p to p' in the course of the collision. The scattering potential is assumed to be of short range so that the concept of a finite collision time T~ is valid. Let R,., represent the energy radiated in the frequency interval 0 s w zs oy , where the maximum frequency wY is small enough so that oflc 0) or emission ( n - n ’ < 0) of a specified number of laser photons is then obtained, by suitable modification of the argument leading to Eq. (8), in the form da(n

-

n’)/dfl= (dv‘O’/dfl)J:,,,(X)

(9)

To improve on Eq. (8), one must include the effect of the electron-field interaction in intermediate states. As mentioned above, the first-order correction term has been calculated; the result can still be expressed in terms of the measurable field-free scattering cross section. The improved cross-section sum rule (Rosenberg, 1980a) differs from the lowest order version in the appearance of small kinematic corrections; these account for the energy loss in accordance with the classical mechanism of radiation reaction in which a loss of kinetic energy is accompanied by the radiation of an equal amount of energy into the field. Low (1958) obtained a low-frequency approximation of a similar nature for single-photon spontaneous bremsstrahlung. The nontrivial feature common to these lowfrequency theorems is the fact that the scattering amplitude which enters is on the energy shell. The contributions to the amplitude associated with photon emission or absorption before or after the scattering takes place are off the energy shell, but these off-shell effects are canceled when the contribution associated with radiation in an intermediate state is included. Kroll and Watson (1973) obtained an analogous result, an improvement over Eq. (9), for scattering in a laser field. A convenient way to arrive at these first-order corrections is by means of a gauge transformation. We return to this matter in Section V in the broader context of scattering by an atomic target.

Leonard Rosenberg

14

111. Asymptotic States A. ELECTRON IN A PLANE WAVEFIELD It was first shown by Volkov (1935) that the Dirac equation for an electron interacting with an external plane wave field can be solved exactly. These solutions play an essential role in the formulation of intense field quantum electrodynamics and have been studied by a number of authors. For reviews see Eberly (1967) and Mitter (1975). Here we shall derive these solutions starting with the nonrelativistic Schrodinger equation (Keldysh, 1965; Bunkin and Fedorov, 1966). The fact that the field is intense justifies the use of a classical description. It is useful to keep in mind the close connection between the classical and quantum descriptions of the field; as will be seen below, the relation is essentially one of Fourier transformation. Consider a multimode field with each mode having the same propagation direction which we take to be the z axis. The classical vector potential in the Coulomb gauge is given by the sum over modes A,

=

+ A+e-ik.ll

$&i(Aieikitl

1

0

i

with k, = oi/c and 11 = z - ct. By retaining the z dependence we include electron recoil effects, typically of order v / c , which had been neglected in the discussion of Section 11. The transverse polarization vector Ai is represented as X i = i cos xi + i!; sin x i , where the angle xr specifies the state of polarization for each mode. Note the relations &*A: = 1 and &*At = cos 2 x i . We look for a solution of the Schrodinger equation

a

ifi - V, = at

(-ihV

-

eA,/c)2 q P

2P

in the form Wp = exp -i(Epr/h)$,, with JlP(r,1) = exp i [ ( p * r / h )+ S,]. If we define E , = p 2 / 2 p + A , where A = ( e 2 / 2 p c 2 ) d fwe , obtain an equation for S, of the form

$xi

Here we have omitted terms involving (d2S,/du2)and (dS,/di02since they may be seen to give rise to corrections which are negligible in the nonrelativistic limit. Equation ( 11) is easily integrated. Rather than continue with an analysis of the multimode field we restrict the following discussion to the single

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

15

mode case for simplicity. [The need for representations of the laser field more realistic than this has been emphasized by Kruger and Jung (1978) and by Zoller (1980).] By suitable choice of normalization we have (dropping the mode index i)

S,(ku) = p sin@

+ 0) +

(Y

sin 2ku

(12a)

Here we have defined a = (A cos 2x)/2q

(12b)

with q

=

h 4 1 - PZ/PC)

(12c)

and have introduced the real parameters p and 0 as the magnitude and phase, respectively, of the complex number

b In terms of the intensity I alternative forms

= (-e/pcW*p/q = w 2 d 2 / 8 7 r c and

(124

wavelength A , we have the

with A. = 1 p m and Zo = 1 MW/cm2. Choosing the nominal values A = 10 pm and p2/2p = 10 eV we find that an intensity 2 lo2 MW/cm2 is required before p takes on values k 1, a domain which (as will become more apparent in the following) signals the onset of significant multiphoton effects. It should also be noted that for1 S lo6 MW/cm2, and with the wavelength and energy chosen as above, the term involving a in Eq. (12a) will be small compared with the first term; taking (Y = 0 leads to a considerable ) vansimplification in the analysis. For circular polarization (x = ~ / 4 (Y ishes identically. Introducing the Fourier expansion

with inverse

Leonard Rosenberg

16

we may represent the semiclassical wave function as r

qp(r,t ) =

2 ymexp[-i(Ep m=-z

-

rnhw)t/h

+ i(p - rnhk)*r/h]

(14)

One may interpret this expansion in terms of the absorption and emission of photons; these are, of course, virtual processes, not kinematically allowed. Real emission and absorption of laser photons can take place in the presence of another particle to which the electron can transfer momentum. Properties of the expansion coefficients y m have been discussed by Nikishov and Ritus (1%4), Brown and Kibble (1964), Reiss (1962, 1980b), and Leubner (1981). We take note of the following: y m ( j ,0)

(1)

(1W

= e-’moJ-m(p)

(2) Recursion relation: According to Eq. (13) we may write 2t?iym= (inI-1

IoZn

d(eimb)

exp i [ p sin(+

+ e) + a sin 241

Integrating by parts we immediately obtain 2mym + p[eieym+i+ e-’ym-ll

+ 2a[ym+, +

~m-21

= 0

(15b)

(3) Addition formula: x

Here the primes on p and a indicate the appearance of the momentum p’ rather than p in the definitions (12d) and (12b). Equation (15c) is verified by using the integral representation (13) to evaluate the left-hand side. The sum over I can then be performed using the relation 2.

(27r)-I

2

expi/(+ - 4’)

=

fi(4 - 4’)

(16)

/=--3

and the complex numbers j and 6’ combined by vector addition. [The analogy between the addition formula (15c) and the relation (7) for the coherent states is clear.] The suitability of the choice of wave functions (14) as the modified plane waves in the formulation of a theory of scattering in a laser field is confirmed by constructing a wave packet solution from a linear superposition of these states, following the prescription of Neville and Rohrlich (1971). One finds that the center of the packet follows a trajectory determined by the classical equations of motion for an electron in the field, thereby permitting a proper physical interpretation of the theory. (The classical trajectories will be discussed in more detail later on.)

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

17

In a quantum treatment of the field the vector potential appropriate to the case where only a single mode is occupied initially, and spontaneous emission into unoccupied modes is ignored, is A

=

( 2 n h ~ ' / o L ~ ) ~ ' ~ (+ a XatX*e-ikz) e'~~

The asymptotic wave function is written as exp -i(Enpf/h)+np, where Jlnp satisfies

It represents the state which evolves from the unperturbed state In ; p) (corresponding to the field containing n photons, the electron having momentum p) as the electron-field interaction is switched on adiabatically. One looks for I+,,p) in the form of a superposition of unperturbed states, all with the same total momentum p + nhk but with different numbers of virtual photons in the field; the problem then reduces to a deterrnination of the expansion coefficients. It turns out (Kelsey and Rosenberg, 1979) that the coefficients are just the ym introduced in Eq. (14), that is,

I+,rp)

X

ym(b,

=

+ m ;P

-

mhk)

(18)

lIl=-x

where b and a are defined as above using the correspondence I = w 2 d 2 / 8 n c= ( n h w / L 3 ) c .This assertion is easily verified by substituting the expansion (18) into the Schrodinger equation (17). With the energy chosen as Erp = p 2 / 2 p + A + nhw = Ep + nhw, we find that Eq. (17) will be satisfied provided that the expansion coefficients satisfy the recursion relation ( 15b). Thus, choosing the expansion coefficients in the form (13) does indeed provide us with a solution. The addition formula (1%) may be used to establish the orthonormality relation (+,7~pf~Jl,lp) = 8,,.,,8(p'- p). The connection between the quantum and semiclassical solutions is summarized by the relation

(n

+ ml+,,,>= T-'

/oTdte-imml

+P

3

T

= 2m/w

(19)

A general discussion of the relation between the external field description and the fully quantized theory has been given by Bialynicki-Birula and Bialynicka-Birula (1976).

B. DRESSED-TARGET STATES Let us temporarily ignore the presence of the projectile and study the target-field states. The basic approximation made here is to treat these

18

Leonard Rosenberg

states as if they were stable, that is, we ignore the possibility of ionization of the target by the field. To deal with the case where the ionization rate is significant would require a formulation of the scattering problem in terms of initial states containing more stable subsystems. The problem of scattering with three or more subsystems in the initial state is a difficult one, even in the absence of an external field, particularly if these subsystems carry a net charge so that long-range Coulomb interactions must be accounted for. Not surprisingly, little attention has been given to this problem. Of course, even if one ignores target ionization by the field, ionization can take place by electron impact. Such a process involves a final state containing three charged particles interacting with the field, and the wave function for such a state is not known in closed form. We shall not consider processes of this complexity here since a theoretical investigation in this area, based on approximate treatments of the final-state wave function, has only just begun (Cavaliere et a / . , 1980; Baneji and Mittleman, 1981). The eigenvalue equation for the isolated target is expressed as H,lb,) = r y J b , )v, = 0, 1, . . . . In the presence of a single-mode field, but with no target-field interaction, the state vector in the occupation number representation is of the form In )lb,,).After the interaction has been turned on, we have the eigenvalue equation (HT

+ HF + Hk)lDun) = E v n l D v n )

(20)

The energy can be written as Eyn= c,, + nhw + r,, where r, represents the level shift induced by the field. It can be shown that r, depends only on the intensity of the incident beam. That is, ignoring photon depletion effects, r,, is fixed as n varies by a finite amount from one photon state to another. For this reason we have omitted a subscript n on r,. (A similar situation was encountered earlier in connection with the projectile-field level shift A.) The time-dependent wave function associated with the dressed atom = exp --i(E,,,t/#i)~D,,). state D,,, is I*”,) A quantum description of the field has been adopted above. The connection with the semiclassical procedure can be made in a manner analogous to that described earlier for the projectile-field states. Here we base the discussion on the general theory of periodic potentials and in particular on the so-called quasi-energy method (Zeldovich, 1973). Thus, we look for a solution of ih(@Pl,/dt)= (HT + X;)*,,where Xk,the interaction Hamiltonian in the semiclassical description, is period-ic in time, with period T = 2 r / w . Introducing the quasi-energy E, we write V,,= exp(-iE,r/h)D,, where D, is periodic. It may then be expanded in the Fourier series ?j

D,

=

2

m=-m

rm(V)~imw~

(21)

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

19

In analogy with Eq. (19) we expect that the relation (m

+ nlD,,)

T

=

T-lI0

dte-‘”’’’’‘D,

(22)

provides the connection between the semiclassical and quantum solutions. (It is assumed that the solutions have the correct adiabatic limit, passing over to the unperturbed solutions as the interaction with the field is turned Off.)

To establish the above connection, we must show that the D,,can be expanded as

wiih the same expansion coefficients as appear in Eq. (21). This is accomplished by examining the coupled equations which determine the expansion coefficients; they turn out to be identical in the quantum and semiclassical cases. Thus, inserting the expansion (23) into the quantum equation (20) we obtain the equations

(HT+ (n + m)hw

-

m

E,,)rm(u) +

( n + mlH;ln

+ m ’ ) r m t ( u=) 0

m)=-m

(24)

for the expansion coefficients. (In practice this infinite set of coupled equations must be truncated in order to generate approximate solutions; the discussion is purely formal at this stage.) In a similar way we insert the expansion (2 1) into the semiclassical equation [HT

+ X; - i h ( a / d t ) ] D ,= E,D,

(25)

Since Xk is periodic, we may exhibit its time dependence in the form 22’; = (Xk)meimwl. The coupled equations for the expansion coefficients then become

zi=-m

(HT+ mhw - E,)Tm(v) +

m

m’=-.X

(%‘b)m-mflrmt(v) = 0

This is seen to agree with the quantum version (24), provided that one makes the identification E,, = E, + nhw and notes the relation (Baym, 1969) ( n + mlH;ln + m ’ ) = (X;)m-m,. In atomic radiation theory, and in problems involving intense external fields as well, one frequently represents the atom-field interaction in the dipole approximation. That is, in the semiclassical form, Xi + - e z l rl*E,(O, f), where the sum runs over target electron coordinates and EJr, t ) = (-l/c) d & / a t is the classical electric field. This repre-

Leonard Rosenberg

20

sentation is expected to be valid in the nonrelativistic domain and for fields of relatively long wavelength. The dipole approximation is conveniently introduced through a gauge transformation. (For a recent discussion see Reiss, 1980a). We review that procedure here; it will be useful in the scattering problem as well. In its semiclassical form the gauge transformation is generated by the function A = -El rl-Ac(rl,t), which introduces the transformation ql,+ exp(iAe/hc)P,, in the dressed-target wave function. This is equivalent, in the quantum treatment, to a unitary transformation represented by exp(g,), with (Sachs and Austern, 1951)

where Z is the number of target electrons. The eigenvalue equation satisfied by the transformed state 6 , ,= exp( -g,)D,, is obtained by transformation of the Hamiltonian appearing in Eq. (20). This is accomplished by first noting the relation pl - eA(rl)/c

=

es'{pl

+ (e/c)hl*E(r,)}e-g~

where A is a unit vector in the z direction and E(r,)

=

i ( ~ / c ) ( 2 n h c ~ / w L ~ ) ~-~citX*e ~(~Xe'~.~ -Iker)

represents the quantum form of the electric field. The transformation of HF is carried out using the expansion e0THFe-OT

= HF + [ g T ,

&I + (1/2 !)[gT,[gT,HF13+

*

*

.

Since the double commutator is a c number, the terms shown explicitly are actually the only ones which contribute. Note that the operators ci and N t may be thought of as representing quantities of order n On the other hand, the double commutator is independent of the external field strength; it represents a level shift arising from spontaneous emission and absorption and may be ignored here. Using the relation [ g , . H F ] = e Ey==, rJ*E(r,), we find that the transformed eigenvalue equation is (HT

with

+ HF + i7&)p1#,) = El,,pLj,)

(27)

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

21

In the nonrelativistic limit and in the long wavelength regime we have H$ = - ~ , e r J * E ( Oto) good approximation. In general all states of the isolated target are coupled through the interaction with the field, and it will not be possible to find exact solutions of Eq. (27). Approximate solutions can be found by truncating the space of states which form the basis. As a simple example suppose we have a twofold degeneracy. In the low-frequency limit, the effective strength of the coupling between these two states will be given by the ratio of the interaction energy to the photon energy (in analogy with the situation encountered earlier in the discussion of the projectile-field states). This coupling may be large even for moderate external field strengths. One would account for it exactly and attempt to include coupling to other states in ordinary perturbation theory. If the degeneracy were not exact, one could account for the splitting using an almost degenerate perturbation theory (Rosenberg, 1980~).To give some flavor of the calculational procedure, let us assume that the unperturbed states of the target Ib,) and Ib,) are degenerate, with energy E . With all other states ignored, the eigenvalue equation to be solved is of the form (27), with HT= ~ [ b ,(b,1 ) + Ib, ) ( b21]and, in the dipole approximation, assuming linear polarization,

We look for a solution in2the form (23), or rather its gauge-transformed version, with T,(v) = CKll&’(v)b,. v = 1, 2, subject to the condition In,.,,)+ lb,,)ln)as the interaction is switched off. From the Schrodinger equation (27) we obtain a set of coupled recursion relations for the expansion coefficients c k ) ; these become decoupled when they are reexpressed in terms of the combinations ck) ? c g ) .By comparison with the recursion relations for the Bessel functions we readily identify these linear combinations with Bessel functions .I% Y),,,where ,,(

and m ’ = m - (EL,,,- E - n h w ) / h w . The irregular (Neumann) solution must be excluded to ensure the normalizability of the state vector. By the same reasoning rn’ must be an integer. We take m ‘ = m since E,,, must reduce to E + nhw as the target-field coupling is switched off adiabatically. These considerations lead to the result

Leonard Rosenberg

22

with Em = e + nhw. The orthonormality property of the solutions may be verified using the integral representation.

and the closure relation (16). Having determined the expansion coefficients we may construct the semiclassical version of the above solution using the correspondence (22). This leads to

ID,)

2

=

C

t[exp(iY cos w t )

+ (-

l)VfKexp(-iY cos wr)]lb,) (31)

K=l

One easily verifies that this satisfies the version of the Schrodinger equation (25) appropriate to the dipole approximation as well as the adiabatic condition 0, + b , , for Y + 0 (Keldysh, 1965; Kovarskii and Perel'man, 1972). Once the dressed-target states have been constructed, they may be used in conjunction with the projectile-field states to arrive at a solution of the Schrodinger equation appropriate to the asymptotic domain in which the field interacts with both the projectile and the target, the latter taken to be infinitely massive. The semiclassical version of the solution is simply the = El. + Ep. We may anticipate = P,,P,, with quasi-energy product that a knowledge of the semiclassical solution allows us to construct the quantum solution = exp(-iE,npr/h)$L,np using the analog of Eq. (19) or (22). This would lead to the representation m

(n

+ ml$,,nP)=

I'm-m,YmrIp- m ' h k )

(32a)

m' n-m

with the coefficients Tm-m.and ymlobtained from the Fourier series expansion of the semiclassical dressed-target and dressed-electron solutions, respectively. Equation (32a) is equivalent to

It may be verified directly (Rosenberg, 1978) that this form satisfies the Schrodinger equation in its quantum form, namely, (He

+ HT + HF + H')IJlunp )

=

EvnpIJlmp )

(33)

provided that the r coefficients satisfy Eqs. (24), and the energy is given = Eyp+ nhw. The version (32a) has an obvious physical interpretaby Ev,,p tion as the probability amplitude for finding an additional m photons in the

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

23

field given that n photons were present initially. Since the electron and the target emit photons independently, one forms the product of the amplitude for emission of in ' photons by the electron with the amplitude for emission of the remaining nz - rn ' photons by the target, and then sums over all m ' to arrive at Eq. (32a).

IV. Scattering Theory A. FORMULATION By adopting the semiclassical picture we may take over standard timedependent scattering theory (Schiff, 1968), but with suitably modified asymptotic states. [Some of the mathematical problems which arise in this generalization of ordinary scattering theory to include an external radiation field have been discussed by PrugoveEki and Tip (1974)l. To begin with the simplest case let us consider single channel potential scattering. In the asymptotic domain the wave function satisfies Eq. (10) in which the potential V does not appear. The full Schrodinger equation is

We saw in Section 111 that in the case of a single-mode field (to which we specialize in the following) the asymptotic wave function may be expanded in the form

According to Eq. (19) the Fourier coefficient function +,(m)and the solution of the quantum equation (17) are related by ( n + ~ Z I J " , ~ ) = $,(m). It may be demonstrated (Kroll and Watson, 1973) that the periodicity property of the asymptotic wave function is preserved when the scattering potential V is included. In analogy with Eq. (34) we have

Here again we may relate the coefficient function to the solution of the corresponding quantum version of the Schrodinger equation. We have ( n + ml&J = $,(rn), where = E,,pl$np)and H = He + H ; + HF + V. This relation between the quantum and semiclassical wave functions

24

Leonard Rosenberg

may be verified by repeating the argument which led to Eq. (22) or, equivalently, by comparison of the Lippmann-Schwinger integral equations satisfied by these scattering wave functions. The transition matrix element is

which is, of course, an integration over space and time in the coordinate representation. This matrix element may be written, with the aid of the Fourier expansions (34) and ( 3 3 , as m

Rather than formulate the scattering problem in the semiclassical picture, we may, alternatively, make use of the occupation number representation (Rosenberg, 1977). The scattering matrix element for the transition In; p ) -+In'; p ' ) is S,,,p,:np

=

6,l,,l

&P'

-

P) - 27ri6(EdP,-

~,lp)T,l~p~:"p

with

The rules of correspondence between the semiclassical and quantum wavefunctions now allow us to express the semiclassical matrix element (36) as an infinite sum of terms, each corresponding to a transition involving the emission or absorption of a definite number of photons; we find

A relativistic version of this connection formula has been used by Brown and Goble (1968). An alternative expression for the T-matrix element is obtained by representing the wave function in terms of the resolvent G(E) = ( E - H ) - l as =

IhP> + G(~,lp)vl+,,p)

A positive infinitesimal contributior to the energy Enpwill be understood here, rather than explicitly indicated. Equation (37) then becomes Tn'p':np

= (+n*pflV+ VG(Enp)Vl+np)

(38)

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

25

With initial and final states assumed known the calculational problem reduces to a construction of the matrix elements (rn'IG(E)lrn)of the resolvent. In general, of course, only a finite number of photon states can be included, and one must study the convergence of the calculated amplitudes as the dimensionality of the matrix representation of G is increased. Preliminary studies of this nature have been made by Jung and Taylor (1981). As indicated above, the calculational problem may also be formulated in terms of an integral equation of the Lippmann-Schwinger type. One writes G(E) = Go(E)+ Go(E)VG(E),where Go is defined formally as [ E - ( H - V)]-I and has the eigenfunction expansion r

.

This leads to the integral equation

with the Born amplitude given by V,r'p':,1p

=

~$fl~p~l~l$,~p~

(41)

An alternative version of the integral equation is provided by the so-called space translation method (Henneberger, 1977). The above discussion may be extended to include the case of a composite target. It will be convenient to treat the scattered electron as distinguishable from the target electrons with the understanding that antisymmetrization will ultimately be imposed by taking the proper linear combination of direct and exchange amplitudes. The asymptotic states satisfy equations of the form (33), which we write more compactly as ( H - E,,)[$,,) = V,,l$,,). Here /3 is a channel index which not only represents the set of observables which defines the state but also serves to distinguish the projectile from the target electrons. V, is the net interaction between projectile and target. The transition amplitude takes the form TB'B =

with E,,, = E,, and E

=

($fi,IV,,

+

V,,,(E - H)-'Vp($p)

(42)

E,, + i0. The differential cross section is

dm/m=

(2n)4h2Cc*(p

yp)p-pfp12

It was pointed out earlier that passage to the electric-dipole approximation, which is useful in the construction of the target-field states, can be effected by introducing a unitary transformation. Such a transformation also proves to be useful in setting up low-frequency approximations, as described later on. Since we are dealing with a system containing the

Leonard Rosenberg

26

target plus an additional electron the transformation operator is taken to be exp(g), with %+ 1

g = (ie/hc)

rj*A(rj) 1=1

The sum now runs over the coordinate indices of dI the electrons. The Hamiltonian of the system may be expressed as H = exp(g)H exp(-g), with

{ [p, + (e/c)irj

=

E(rj)I2

-

2P

l=l

e r j E(rj)]

+ HF + V

(43)

Here V represents the sum of interparticle Coulomb potentials. To study the effect of this transformation on the asymptotic states, we define &, = exp( -~,)I/J~, with g , given by Eq. (26). (A channel label specifying which of the electrons is the projectile is omitted to simplify notation.) The Schrodinger equation satisfied by JIB is readily seen to be (He+ HT + HF + HL + = E p l & ) , with g& given by Eq. (28). Writing g = g , + g , , we have exp( -g)JlP = exp( -g,)&. An analogous relation is obtained for the final-state wave functions using the appropriate decomposition g = g: + g ; . Since the resolvent transforms as (E - H)-I = exp(g)(E H)-Iexp(-g), it follows that Eq. (42) is equivalent to

n;)I$,)

T~~~= (e-o;JlgllVB+

v,.(E- H)-1V,1e-geJIB)

(44)

The transformed electron-field states appearing in Eq. (44) are of the form exp( -ge)$,,p. They may be constructed by expanding the exponential and making use of the representation (18) for $,,,; this leads to ~-oeI$,,~)

1'"

= 1E-m

(d4/2r)eil@piSp(@)

0

x [I - (ie/hc)A,(r$) re

+ . . .]In + I)lp - Ihk)

where re is the electron coordinate. Ignoring photon depletion effects, we set [ u , (it] = 0, in which case the expansion may be put back in exponential form. Then, with re recognized as the generator of momentum translations, we find cgeI$,,,)

=

5 1'" (dr$/2r)e"@eisp~@)

I=-

m

x In

O

+ I>lp - (e/c)Ac(4) - Ihk)

(45)

As may be seen from this result, the correspondence (19) between the

semiclassical and quantum solutions is preserved under the gauge transformation.

27

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

While we are focusing primarily on the free-free transition, some discussion of the bound-free transition is appropriate since the two processes are closely related physically. In multiphoton ionization, for example, the electron which has been ejected from the atom, but is still within the range of the atomic potential, can absorb energy from the laser field. In experiments performed recently (Agostini et al., 1979) electrons which have absorbed more than the minimum number of photons required for the ionization to proceed have been observed. A proper description of the dynamics of the final state, in which the electron, target, and field are mutually interacting, involves all of the scattering-theoretic apparatus with which we have been concerned here. In particular, in the strong-field or low-frequency limit a perturbative treatment of the final-state electron-field interaction will be inadequate. It becomes necessary then to modify the standard perturbation approach to multiphoton ionization (Lambropoulos, 1976) in order to build in the properly modified final-state wave functions at the outset. Since multiphoton ionization can be thought of as the second half of an induced resonance (as discussed in Section I), it is natural to make use of the scattering formalism as the basis for the development of a modified ionization theory. In fact, the procedure for applying scattering theory to the treatment of unstable states is worked out in the text by Goldberger and Watson (1964). A transcription of that method to the external field problem of interest here begins with an analysis of the resolvent (E - H)-' appearing in the expression (44) for the T matrix. The resonance corresponds to a near singularity in the resolvent. To isolate this contribution, we identify the discrete intermediate state IB ) in which the electron is temporarily bound. This negative ion state satisfies H'O'IB) = 8 ) B ) ,whereH'O' = - HF - H' is the electron-atom Hamiltonian. The resonance condition 8 + n & o = p 2 / 2 p + E" + nhw expresses the fact that the intermediate state IR) = IB)lno) is nearly The resolvent may be decomposed degenerate with the initial state I ,!J~, ~). into resonant and nonresonant parts with the aid of the projection operators P = IR ) (RI and Q = 1 - P . We have the identity (Mower, 1966)

c

=

G' + ( 1 + G'F)IR)(E

-

ER)-'(R[(l + H'G")

where cp= [ Q ( E - H)Q]-' and ER

=

8

+ ~ & C+ O(RI(H' + H ' G G ' ) I R )

It follows that Tf3.,= T$, with the replacement of Tfi,R(E- ER)-ITR,. Here

+ T i l P ,where TBIpis obtained from Eq. (44),

c by cQ.The resonant contribution is T i , , - -

TB*R= (e-g&&,I(Vp' + V p l C Q H ' ) l R )

=

(46)

28

Leonard Rosenherg

may be identified with the ionization amplitude and TRBis the timereversed amplitude for the process of capture into the intermediate unstable state. Expansion of the resolvent GQin Eq. (46) in powers of the interaction H ' generates a perturbation series for the ionization amplitude corresponding to successive absorption of photons. It differs from the standard perturbation expansion in that the effect of the field on the final state of the electron-atom system is built into the wavefunction. The first term T> 1 will be negligible due to rapid oscillations of the integrand as 4 and 4' vary between 0 and 2n. The argument would fail if the phase were stationary for some value of I, but this possibility is excluded by the assumption that S; is at most of order unity. Assuming that the off-shell amplitude t is a smooth function of 5 and t',we expand it in a Taylor series about 6 = 5' = 0 and ignore second- and higher order terms. The leading term is the

44

Leonard Rosenberg

on-shell amplitude t(E - (n + I)ho; (q' - q)2). The first-order correction terms, one proportional to 5 and the other to vanish. To see this, note that in the term proportional to 5 we- may write leftmelSp(m)= - i[(d/d+)eft m ] p f S p ( m ) [ I ,

and integrate by parts. The surface term vanishes so that we have effectively replaced 1 by -SL. (An additional contribution coming from the derivative of I with respect to 4 is of higher order and may be dropped.) In a similar way we see that the first-order correction term proportional to 6' makes no contribution. Equation (68) is still fairly complicated since it involves a double integral as well as an infinite sum. Now according to Eqs. (69), and the condition k*A, = 0, we have, to first order, (4' - 9)'

= TO -

2(p'

-

p)*(e/c)[A,(4') - A J 4 ) l

where T~ =

(p' - p

+ (n'

-

n)hk)*

The t amplitude may be expanded in a Taylor series about leading term is seen to be

(72) The

T = T ~ .

2

T , , ~ , , ~ ~ ~ : ~ , , , ,y:+l-n,(fi', ,= a ' ) y l ( b ,a ) t ( E - (n + /)nu. To)

(73)

I=-=

Upon examining the correction term, one finds (Rosenberg, 1981) that in the absence of resonances it is actually of second order and hence may be neglected. Iff is resonant, this second-order correction is promoted to first order. When this correction is added to the leading term (73) one findsfor the case of linear polarization, where A is real and the calculation simplifies somewhat-that the form (73) is retained, but with T~ shifted to (Pi - Pl)*, where pi

=

P - Ihk - pl[q/(p*A)lA

pi

=

p' - (n

+ 1 - n')hk - p(n + 1 - n')[q'/(p'.A)]A

Consider now single-photon emission in the weak-coupling limit of the low-frequency approximation (73). Keeping only the terms corresponding to I = 0 and I = 1 and making use of the approximations (491, we find T,,,,,+I~,:,~,,, zs (j'/2)"r[(p2/2p) +

To] -

(j/2)"f[(p2/2p) +

EY -

hu.

TO]

(74)

This represents an external field version of the low-frequency approximation for single-photon bremsstrahlung (Low, 1958; Feshbach and Yennie,

45

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

1962; Heller, 1968). In the resonant case, we add to Eq. (74) a first-order correction term of the form

AT = ( e / c ) ( 8 ~ c l / w ~ ) ” ~( pp)*X* ’ X

(a/a7){f[(P2/2P)f

EY

- h w , TI -

l(P2/2P)

Eu, T)IJr=r,

The resulting approximation corresponds to that displayed in Eq. (21) of Feshbach and Yennie (1962), specialized to the case considered here of a neutral, infinitely massive target. Those authors suggested that a knowledge of the spontaneous bremsstrahlung amplitude in the neighborhood of a resonance could be used to obtain information on the phase change experienced by the on-shell scattering amplitude as the energy variable passes through the resonance value. With the aid of the laser, useful extensions of investigations of this type to include stimulated bremsstrahlung should be feasible. In the absence of resonances the t amplitude in Eq. (73) may be expanded to first order as

f ( E - ( n + / ) h U , To)

f -

/hw(df/de)

with t and its derivative evaluated at the energy E - nhw. The sum over I may be performed by writing

-/-ylG, a) =

1’“( d 4 / 2 ~ ) e ” ~ e * ~ ~ ( ~ ’ S ; ( + ) 0

and using the closure relation (16). This leads to the alternative version (Rosenberg, 1979a, 1981)

lo 2H

T , W , ~ ;=~ ~ , (d+/27r) exp[i(n’ - n)41 x exp{i[S,(+) - Sp~(+)lMeup(+),7 0 )

(75)

with cup(+) given by Eq. (60). Relativistic versions of Eqs. (73) and (75) have been derived (Rosenberg, 1980d) from the general formulation of the low-frequency approximation developed by Brown and Goble (1968). The total cross section, summed over all final states of the field, is given by the expression

If the low-frequency approximation (75) is used to represent the T matrix the sum over ti’ may be performed, the result then reducing to the classical expression given in Eqs. (63). Tomcarryout the sum one introduces the representation 6(x) = (27~)-’[-=ds exp(ixs) for the energyconserving 6 function in Eq. (76). The sum could now be performed were

Leonard Rosenberg

46

it not for the t i ’ dependence of order, T~

t(e,,,, T

sz (p’ -

P ) ~+

(/I‘

T~

- n)2hk*(p’ - p)

= r(elJp,(p’ - p)? +

~ )

in Eq. (75). We therefore write, to first

(11’ -

n)2hk*(p‘- p) at/th17=,p,

(77)

The replacement n ’ - n + Sb. - S ; is justified, to the requisite accuracy, by an integration by parts; Eq. (75) is then transformed to

x exp{i[S,(4) - Sp44)lMer.p(4)r ~(4))

(78)

with ~ ( 4defined ) as in Eq. (64). At this stage, all of the 12’ dependence is in exponential form, and the sum in Eq. (76) may be carried out using Eq. (16). This leads to the stated result, namely, that the quantum total cross section reduces to the classical value d a , in the low-frequency limit. We may also compare the classical and quantum expressions for the average energy transferred from the field to the electron. The classical expression, as we have seen, is given by the ratio of dU,, defined by Eq. (65), to da,. Proceeding analogously in the quantum case, we determine the average energy lost by the field, call it ( ( n - n ’ ) h w ) ,by defining

dU

( 2 r r ) ‘ ( p / p ) h 2 ~ d 3 p ’-( nn ’ ) h w 6(E,,,,,,

=

- El.np)lTl,,n,p,:r.np12 (79)

n’

Then ( ( n - n ’ ) h w )is obtained as the ratio d U / d a . The sum in Eq. (79) is readily evaluated using the approximation (78), along with an integration by parts. The result is just the classical expression dU, and, since we have already identified da with da,to first order, we have confirmed the equality, to this order, of the classical and quantum expressions for the average energy transferred to the electron. The similarity between these sum rules and those derived by Bloch and Nordsieck (recall the discussion of these sum rules in Section II,A) should be noted. We shall see that the sum rules just obtained for the case of intermediate coupling hold in the strongcoupling regime as well.

D. STRONG COUPLING

If the ratio of the electron-field interaction energy to the photon energy is large compared to unity, fluctuations in the field energy will, on the

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

47

average, greatly exceed h w . As a result, it will no longer be appropriate to treat the off-shell parameters 6 and t', given by Eqs. (70), as first-order quantities. Instead, we may apply a stationary-phase argument. The phases in Eq. (68) are stationary when the conditions I +Sb(+) = Oand ( n + I - n') + Sl,.(c$) = 0 are satisfied. Since these correspond precisely to the conditions 5 = 5' = 0, the evaluation of the r amplitude in Eq. (68) at the points of stationary phase places it on the energy shell. The energy variable is E - (11 + I)hw, but with I replaced by -Sb(c$) (resonances are assumed not to be present here), this becomes

E

-

nhw

+ hwSl,(c$) = ( P 2 ( 4 ) / 2 p )+ G = eu,,(4)

Since the energy variable, in this form, is independent of I, the sum in Eq. (68) may be performed. The result is Eq. (75) so that version of the low-frequency approximation is justified for both intermediate- and strong-coupling regimes. In evaluating the total cross section we use the approximation (75) in Eq. (76). The replacement n ' - n + S ; , - Sl, in the expression (77) for ther amplitude is now justified by the stationary phase argument. [There are values of n' for which 'the phase in Eq. (75) is not stationary for any 4 in the range of integration, but such terms make a negligible contribution to the sum over photon states.] The transition matrix may then be taken to be of the form (78) and the derivations of the sum rules for the total cross section and the average energy transferred to the electron go through as described above for the case of intermediate coupling. The results may be summarized by stating the following external field version of the BlochNordsieck theorem: (1) In the static, or extreme strong-coupling limit, where the frequency approaches zero with the electric field strength fixed, the average number of photons emitted or absorbed approaches infinity. This follows from the fact that for any finite value of n' - n the amplitude (75) vanishes due to the increasingly rapid oscillations of the integrand as the static limit is approached. (2) The total probability for scattering, independent of the number of photons emitted or absorbed, can be obtained from a knowledge of the cross section appropriate to scattering in the absence of the radiation field. The field affects the kinematics of the collision in a manner which allows for a classical interpretation of the interaction of the electron with the field. (3) While the average number of photons emitted or absorbed is infinite in the static limit, the energy transferred between the electron and the field is finite and calculable from a classical description of the electron-field

Leonard Rosenberg

48

interaction, with the collision assumed to take place instantaneously and without influence from the field.

If we ignore electron-recoil effects of order p / p c and specialize to the case of linear polarization, in which case &(4) = &A cos 4, the condition that the phase in Eq. (75) be stationary takes the form ( n’ - n ) h o

=

-(e/pc)[(p’ - p) *&A

COS(~,,)]

With cos 4s,,thus determined, we have

F’(4s,)= p + A n ’

-

n)hoA/[A*(p‘

-

p)]

and eL,p(&p) = [Pz(&,,)/2p]+ E,. . The momentum-transfer variable reduces to (p’ - p)‘ in the no-recoil approximation. Equation (75) then becomes T,,.,,.p,.wlp = -’l~(XMel,p(4sP), (P‘ - P)*) $1

(80)

with X given by Eq. (48). This is the result of Kroll and Watson (1973) generalized to the case of scattering by an atomic target. The approximation (80) can also be justified for the intermediate-coupling regime (Mittleman, 1980). An approximation of the strong-coupling type can be applied to the multiphoton ionization problem as formulated in Section IV,A. Let us examine the lowest order approximation, Eq. (47), for the bound-free transition amplitude. Since we are ignoring contributions from the modified interaction H ’ , we adopt the approximation (67) for the final state. Using the appropriate version of Eq. ( 4 9 , we find

Tj&

=

1;

( d 4 / 2 ~ )exp[-i(no -

n’)41 exp[-iS,,(4)]Fp,(q(4))

(81)

where F,,(q)

(ql (b,.,lVfi,IB)

q(4) = p‘ - (e/c)A,(4) - (no - n ’ ) h k Since the energy absorbed by the negative ion must exceed the electron affinity - 8, we must have no - 12’ >> 1 in the low-frequency limit. This suggests an approximate evaluation of the integral in Eq. @I), based on the method on steepest descents. The stationary phase condition is no - n ’ + SL, = 0. For those values of 4 thus determined (they are complex in the - E ” , . With q satisfying this present case), one finds that q 2 ( 4 ) / 2 p= condition, the function Fp’(q) is determined from a knowledge of the asymptotic normalization of the bound-state wave function of the negative ion (Goldberger and Watson, 1964). This normalization factor can be re-

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

49

lated to the residue of the bound state pole in the physical field-free scattering amplitude. A detailed analysis of the approximation (81) along the lines just indicated has been carried out by a number of authors (Keldysh, 1965; Nikishov and Ritus, 1966, Perelomov ef al., 1966). The expression for the total ionization probability obtained in this way has a very simple physical interpretation. The result is identical to that which would be obtained by calculating the probability for ionization by a field E, = ,&J sin of for t fixed, this calculation being performed using the tunneling approximation for ionization by a static field (Oppenheimer, 1928). The ionization probability thus obtained is then averaged over a period. It is interesting to note the similarity between this adiabatic picture of the ionization process and that which emerges from the sum rules for free-free transition probabilities.

VI. Concluding Remarks The emphasis placed on low-frequency approximations in this review reflects the fact that theoretical progress to date has been largely restricted to this regime. It is in some sense fortunate that our understanding of the dynamics of the scattering process is greatest in the low-frequency domain, where, for a given field strength, the effective coupling is largest and the interesting multiphoton effects are most prominent. Additional interest is attached to the low-frequency approximation since it provides insight into the study of the connection between the classical and quantum descriptions of the particle-field interaction. This connection is most apparent in the form taken by the sum rules discussed in Section V. One has also been able to clarify, in the study of low-frequency approximations, the relationship which exists between the theory of scattering in an external field and the treatment of spontaneous infrared radiation. Here again the relationship shows up most clearly in the sum rules. In all of these considerations special care must be given to the case of resonant scattering. It seems likely that in this case particularly the external field can serve as a probe, revealing details of the scattering process not readily available by other means. It is not difficult to identify deficiencies and limitations in the theoretical picture developed thus far. One of the most glaring is the lack of a practical and generally applicable procedure for dealing with targets which carry a net charge. “Coulomb problems” also arise in the study of impact ionization in the presence of the external field. Even when the initial electron energy lies below the ionization threshold, the target can be

50

Leonard Rosenberg

ionized by the field in the precollision or postcollision stages; one requires, strictly speaking, a theory of scattering by an unstable target. It should also be emphasized that much of the theoretical work has been based on an oversimplified description of the laser field which precludes detailed comparison with experimental results. Finally, there is the sheer complexity of the calculational problem which will demand a considerable amount of ingenuity in the development of reliable approximation techniques.

ACKNOWLEDGMENTS This work was supported in part by the Office of Naval Research under Contract No. N00014-76-C-0317 and by the National Science Foundation under Grant No. PHY-7910413.1

am indebted to Dr. R. Shakeshaft for reading the manuscript and for making a number of useful comments and suggestions.

REFERENCES Agostini, P., Fabre, F., Petite, G., Mainfray, G., and Rahman, N. K . (1979). Phys. Rev. Lett. 42, 1127. Andrick, D. (1980). Proc. I t i t . Conf. Pltys. Electron. A t . Collisions, I l t h . 1979 p. 697. Andrick, D., and Langhans, L. (1978). J. Pltys. B 11, 2355. Baneji, J., and Mittleman, M. H. (1981). J. Phys. B 14, 3717. Baym, G. (l%9). “Lectures on Quantum Mechanics,” Chapter 13. Benjamin, New York. Bebb, H. B., and Gold, A. (1966). Phys. Re\*. 143, 1. Bhaskar, N. D., Jadusziliwer, B., and Bederson, B. (1977). Pkys. Rev. L e f t . 38, 14. Bialynicki-Birula, I. (1977). Actu P / i y s . Austriacu. SuppI. 18, I11. Bialynicki-Birula, I., and Bialynicka-Birula, Z. (1976). Phys. Re\*. A 14, 1101. Bjorken, J. D., and Drell, S. D. (1965). “Relativistic Quantum Fields.” p. 203. McGrawHill, New York. Blanchard, P. (1969). Commrrn. Moth. Phys. 15, 156. Bloch, F., and Nordsieck, A. (1937). Phys. Re\,. 52, 54. Brandi, H. S., Koiller, B., Lins d e Barros, H. G. P., Miranda, L. C. M., and Castro, J. J. (1978). Phys. R e ) , .A 17, 1900. Brown, L . S., and Goble, R. L. (1968). Pliys. Rev. 173, 1505. Brown, L. S., and Kibble, T. W. B. (1964). Pliys. Rev. 133, A705. Brueckner, K. A., and Jorna, S. (1974). Rev. Mod. Pl?ys. 46, 325. Bunkin, S . V.,and Fedorov, M. V. (1966). Soit. P h y s . 4 E T P (Engl. Transl.) 22, 884. Cavaliere, P.,Ferrante, G., and Leone, C. (1980). J . Pltys. B 13, 4495. Combe, P., Mourre, E., and Richard, Y. L. (1975). Cornmuti. Math. Phys. 43, 161. Conneely, M. J., and Geltman, S. (1981). J. Pltys. B 14, 4847. Damburg, R. J . , and Kolosov, V. V. (1978). J . P h y s . B 11, 1921. Deguchi, K., Taylor, H. S . , and Yaris, R. (1979). J . Phys. B 12, 613.

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

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DeVries, P. L., Lam, K. S.,and George, T. F. (1980).Proc. fnr. Cod. Electron. A t . Collisions. Ilth, 1979 p. 683. Eberly, J. (1%7). Prog. Opt. 7, 361. Faddeev, L. D. (1961). Sov. Phys.-JETP (Engl. Transl.) 12, 1014. Feshbach, H. (1958). Ann. Phvs. ( N .Y . ) 5 , 357. Feshbach, H., and Yennie, D. R. (1%2). Nucl. Phys. 37, 150. Gavrila, M., and Van der Weil, M. (1978). Comments A t . Mol. Phys. 8, 1. Geltman, S . (1977). J . Res. Natl. Bur. Stand. 82, 173. Gersten, J., and Mittleman, M. H. (1976a). Phys. Rev. A 13, 123. Gersten, J., and Mittleman, M. H. (1976b). In “Electron and Photon Interactions with Atoms” (H. Kleinpoppen and M. R. C. McDowell, eds.), p. 553. Plenum, New York. Glauber, R. J. (1963). Phys. Rev. 131, 2766. Goldberger, M. L., and Watson, K. M. (1964). “Collision Theory.” Wiley, New York. Gopert-Mayer, M. (1931).Ann. Phys. (Leipzig) [51 9, 273. Hahn, L., and Hertel, I. V. (1972).J . Phys. B 5, 1995. Heller, L. (1968). Phys. Rev. 174, 1580. Henneberger, W. C. (1977). Phys. Rev. A 16, 1379. Hertel, I. V., and Stoll, W. (1977). Adv. At. Mol. Phys. 13, 113. Jackson, J. D. (1975). “Classical Electrodynamics,” 2nd ed. Wiley, New York. Jain, M., and Tzoar, N. (1978). Phys. Rev. A 18,538. Jauch, J. M., and Rohrlich, F. (1976). “The Theory of Photons and Electrons,” Suppls. S3 and S4. Springer-Verlag, Berlin and New York. Johnston, R. R. (1%7). J . Quant. Spectrosc. Radiat. Transfer 7, 815. Jung, C. (1979). Phys. Rev. A 20, 1585. Jung, C. (1980). Phys. Rev. A 21, 408. Jung, C., and Kriiger, H. (1978).Z . Phys. A 287, 7. Jung, C., and Taylor, H. S . (1981). Phys. Rev. A 23, 1115. Keldysh, L. V. (1%5). Sov. Phys.-JETP (Engl. Transl.) 20, 1307. Kelsey, E. J., and Rosenberg, L. (1979). Phys. Rev. A 19, 756. Kovarskii, V. A., and Perel’man, N. F. (1972). Sov. Phys.-JETP (Engl. Trans/.) 34, 738. Kroll, N. M., and Watson, K. M. (1973). Phys. Rev. A 8, 804. Kriiger, H., and Jung, C. (1978).Phys. Rev. A 17, 1706. Kriiger, H., and Schulz, M. (1976). J . Phys. B 9, 1899. Lambropoulos, P.(1976). Adv. A t . Mol. Phys. 12, 87. Landau, L. D., and Lifshitz, E. M. (1%2). “The Classical Theory of Fields,” 2nd ed. Addison-Wesley, Reading, Massachusetts. Langendam, P. J. K., and Van der Weil, M. J. (1978). J . Phys. B 11, 3603. Langhans, L. (1978).J . Phys. B 11, 2361. Leubner, C. (1981). Phys. Rev. A 23, 2877. Lompre, L. A., Mainfray, G., Manus, G., and Farkas, Gy. (1979). Phys. Rev. Lett. 43, 1243. Low, F. E. (1958).Phys. Rev. 110, 974. Mitter, H. (1975).Acta Phys. Austriaca, Suppl. 14, 397. Mittleman, M. H. (1976). Phys. Rev. A 14, 1338. Mittleman, M. H. (1978).Phys. Rev. A 18, 685. Mittleman, M. H. (1979a).Phys. Rev. A 19, 99. Mittleman, M. H. (1979b). Phys. Rev. A 19, 134. Mittleman, M. H. (1979~).Phys. Rev. A 20, 1%5. Mittleman, M. H. (1979d).J . Phys. B 12, 1781. Mittleman, M. H. (1980). Phys. Rev. A 21, 79. Mower, L. (1%6). Phys. Rev. 142, 799.

52

Leonard Rosenberg

Neville, R. A., and Rohrlich, F. (1971). P h y s . Re\*. D 3, 1692. Nikishov, A. I., and Ritus, V. I. (1964). Sort. Phys.-JETP ( B i g / . Trunsl.) 19, 529. Nikishov, A. I., and Ritus, V. I. (1%6). Sov. Phys.-JETP ( E n p l . T r a n s / . )23, 168. Nordsieck, A. (1937). Phys. Rev. 52, 59. Oppenheimer, J. R. (1928). Phys. Rev. 31, 66. Perel’man, N. F., and Kovarskii, V. A. (1973). Sov. Phvs.-J€TP (Engl. Trans/.) 36, 436. Perelomov, A. M., Popov, V. S., and Terent’ev, M. V. (1966). Sov. Phys.-JETP (EnpI. Trans/.) 23, 924. Power, E. A., and Zienau, S. (1959). Philos. Trcrns. R . Soc. London, Ser. A 251, 427. PrugoveEki, E., and Tip, A. (1974). J . Phys. A 7, 586. Reiss, H. R. (1%2). J . Math. Phys. 3, 59. Reiss, H. R. (1980a). Phys. Rev. A 22, 770. Reiss, H. R. (1980b). Phys. Rev. A 22, 1786. Rosenberg, L. (1977). Phys. Re\,. A 16, 1941. Rosenberg, L. (1978). Phys. Rev. A 18, 2557. Rosenberg, L. (1979a). Phys. Rev. A 20, 275. Rosenberg, L. (1979b). Phys. Rev. A 20, 457. Rosenberg, L. (1979~).Phys. Re,,. A 20, 1352. Rosenberg, L. (1980a). Phys. Rcw. A 21, 157. Rosenberg, L. (1980b). Phys. Reif. A 21, 1939. Rosenberg, L. (1980~).Phys. Rev. A 22, 2485. Rosenberg, L. (198Od). Phys. Rev. D 22, 527. Rosenberg, L. (1981). Phys. Rev. A 23, 2283. Sachs, R. G., and Austern, N. (1951). Phys. Rev. 81, 705. Sargent, M., 111, Scully, M. O., and Lamb, W. E.. Jr. (1974). “Laser Physics,” App. H . Addison-Wesley, Reading, Massachusetts. Schiff, L. I. (I%@. “Quantum Mechanics,” 3rd ed. McGraw-Hill, New York. Schrodinger, E. (1926). NNtrtrw,isuenschafteri 14, 664. Shirley, J. H. (1965). Phys. Rev. B 138, 979. Volkov, D. M. (1935). Z. Phys. 94, 250. Weingartshofer, A., Holmes, J. K., Caudle, G., Clarke, E. M., and Kriiger, H. (1977).Phys. Rev. LCW. 39, 269. Weingartshofer, A., Clarke, E. M., Holmes, J. K., and Jung, C. (1979). Phys. Re\*. A 19, 237 I. Zeldovich, B. (1973). So\*. Phys.-Vsp. (EnpI. Trans/.) 16, 427. Zoller, P. (1980). J. Phys. B 13, L49.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,VOL. 18

POSITRON-GAS SCATTERING EXPERIMENTS TALBERT S . STEIN and WALTER E. KAUPPILA Department of Physics and Astronomy Wuyrie State University Detroit, Michigan

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Positron-Beam Production . . . . . . . . . . . . . . . . . . . B. Total Cross-Section Experiments . . . . . . . . . . . . . . . . 111. Total Cross-Section Results . . . . . . . . . . . . . . . . . . . . A. Inert Gases at Low Energies . . . . . . . . . . . . . . . . . . B. Inert Gases at Intermediate Energies . . . . . . . . . . . . . . I Introduction

11 Experimental Techniques for Total Cross-Section Measurements

C. Positron and Electron Comparisons for the Inert Gases . . . . . . D. Tests of the Sum Rule . . . . . . . . . . . . . . . . . . . . . E. Molecular Gases . . . . . . . . . . . . . . . . . . . . . . . IV. Differential Scattering Cross Sections . . . . . . . . . . . . . . . . V. Inelastic Scattering Investigations . . . . . . . . . . . . . . . . . A. Positronium Formation Cross Sections . . . . . . . . . . . . . B. Excitation and Ionization Cross Sections . . . . . . . . . . . . VI . Resonance Searches . . . . . . . . . . . . . . . . . . . . . . . VII. Possible Future Directions for Positron Scattering Experiments . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 55

60 64 64 76 79 80 82 84 86 86 89 91 92 93

I. Introduction Although the first electron-atom total cross section (QT) measurements were reported in the early 1920s (Ramsauer, 1921a), and the positron has been known to exist since the early 1930s (Anderson, 1933),there were no direct measurements of positron-atom total cross sections until the early 1970s (Costello et a f . , 1972a). One of the main obstacles to performing such measurements earlier was the difficulty of producing a sufficiently intense e+ beam of well-defined low energy. Positron-atom scattering experiments are of interest because they involve interactions of antimatter with matter, and also because they can help provide a better understanding of the scattering of electrons by atoms and molecules, a subject of great importance to many different fields of 53 Copyright @ 1982 by Academic Press, Lnc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

54

Talberr S. Stein and Walter E. Kauppila

science and technology such as plasma physics, laser development, gaseous electronics, astrophysics, and studies of the earth’s upper atmosphere. During the past decade, additional interest in positron-atom (molecule) collisions has been stimulated by the discoveries of 0.51 MeV e + 1973) and annihilation gamma rays coming from solar flares (Chupper d., from the direction of the center of our galaxy (Leventhal et nl., 1978). Such annihilation gamma rays can provide considerable information on the type of environment which exists at the site of their origin if sufficient information can be obtained on the ways in which positrons interact with H, H2, and other atoms and molecules of astrophysical interest (Crannell et d., 1976; Bussard et d.. 1979). Comparisons between e+-atom (molecule) and e--atom (molecule) scattering reveal some interesting differences and similarities. The static interaction (associated with the Coulomb field of the undistorted atom) is attractive for electrons and repulsive for positrons, while the polarization interaction (resulting from the distortion of the atom by the passing charged projectile) is attractive for both projectiles. The exchange interaction contributes to e- scattering (due to the indistinguishability of the projectile and electrons in the target atoms) but does not play a role in e+ scattering. The combined effect of the static and polarization interactions is that they add to each other in e- scattering, whereas there is a tendency toward cancellation in e + scattering. This results in smaller total scattering cross sections, in general, for positrons than for electrons at low energies. As the projectile energy is increased, the polarization and exchange interactions eventually become negligible compared with the static interaction (which has the same magnitude for positrons and electrons). This results in a merging of the corresponding e+ and e- scattering cross sections at sufficiently high projectile energies. Two scattering processes which occur for positrons (but not for electrons) are annihilation and positronium (Ps) formation (real and virtual). Annihilation is not expected (Massey, 1976) to be a significant effect at the energies which have been used in e + scattering experiments (>0.2 eV). On the other hand, Ps formation has been found to be an important factor in e+-gas collision studies. A compact survey of the e+-gas scattering measurements reported during the first decade of activity in this area is provided in Tables IA and IB. Total scattering cross sections have been measured for the inert gases and a variety of molecules by several different groups. However, Table IA indicates that experimental areas beyond QT measurements, such as differential cross sections, elastic, excitation, ionization, and Ps-formation cross sections, and searches for resonances (temporary bound states) are in very early stages of exploration. Positron scattering experiments have been discussed in two recent review articles (Griffith and Heyland, 1978;

POSITRON-GAS SCATTERING EXPERIMENTS

55

Griffith, 1979) and in recent progress reports (Kauppila and Stein, 1982; Stein and Kauppila, 1982),but the field is evolving sufficiently rapidly that there are several different experiments which have been reported since the review articles were written. Our main goals in writing this article are to ( I ) point out what we feel are the most significant developments in the first decade of e+-gas scattering experiments, (2) search for some consistent patterns in the experimental results in cases where several different groups have investigated the same collision processes, (3) present some puzzling questions raised by the new generation of experiments which go beyond QT measurements, and (4) indicate some experimental areas of e+-gas scattering which we feel would be interesting and feasible to investigate in the near future.

11. Experimental Techniques for Total Cross-Section Measurements A. POSITRON-BEAM PRODUCTION

The difficulty and expense of producing intense, monochromatic e+ beams stands in sharp contrast to the relative ease with which one can produce such beams of electrons. Table I1 summarizes the characteristics of the e+ beams used by various groups that have reported QT measurements. The experiment of Costello et d.(1972a) utilized a %-MeV electron linear accelerator to produce positrons by pair production (PP). All of the other laboratories listed in Table I1 use commercially available 22Naas e+ sources, except for the Detroit group, which uses the proton beam of a 4.75-MeV Van de G r a d accelerator to produce an llC e+ source by the reaction “B(p, n)”C (Stein et d.,1974). A variety of moderators used in either a backscattering ( B ) or transmission ( T ) mode, have been found to yield low-energy positrons with relatively narrow energy distributions when exposed to the high-energy, broad energy-width fluxes resulting from p+ decay. Two properties of moderators which are of particular interest in e+ scattering experiments are ( I ) the energy width of the emitted slow e+ energy distribution, AEFWH”(“full width at half-maximum”), and (2) the “conversion efficiency” ( 6 ) defined as the ratio of the slow e+ emission rate of the moderator to the total rate of positron production by the radioactive source. For the moderators which have been used in e+-gas scattering experiments, E ranges from less than lo-’, in the case of the Au-plated mica transmission moderator used by Jaduszliwer et al. (1972) to lop5 in the

TABLE IA A

Gas

SURVEY OF THE

Energy range (eV) ~

He

0.3-1000

Ne

0.25-1000

Ar

0.4-1000

Kr

REPORTED POSITRON-GAS SCATTERING MEASUREMENTS

Gas

Groupsa

~

~

Groups"

~

G I , T I , LI, L2, T2, T4, L3, T5,SI, L7, S2, Al, D3, BI, A3, L9, B2, D5 L2, L3, T3, T6, L7, D3, S3, A3, L9, B2, D5 L1, L2, L3, T3, T6, DI, L7, S3, L9, A4, B2, D5 L2, L3, L7, D4, B2, D6

0.35-960

Energy range (eV)

Xe H2 D2

Nz 0,

co

COP CHI

0.35-800 1-600 4-400 0.5-1000 2-600 2-400 0.5-600 19-600

L3, D4, A4, 8 2 , D6 L4, D2, L8, LIO, D8 L4 L4, D2, LIO, S4, D8 L6, L10 L4 L6, D2, L10, D8 L10

I

Differentid cross sections

Gas

Energy range (eV)

Range of angles

Group

Ar

2.2-8.7

20-60"

A2

Gas

Energy range (eV)

He Ar

14-34 7- I8

Energy range (eV)

Gas

4.5-20 4-18

Gas

Energy range (eV)

Group

Total inelastic cross sections

50-170

He

L5

Partictl excitation plus ionization cross sections

23-50 20-40 15-40

He Ne Ar

A5, A6 A6 A6

Resonctnce searches ~~

~~

Gas

Energy ranges (eV)

Ar

8.5-9.5, 11.0-12.0 ~~~

~~

Energy increments (eV)

Beam-energy width (eV)

Group

0.025

0

(5)

Then the classical probability distribution function is P ( I i , If)

=

I

(2n)rAVd+iS(If - IZ(+i, 11))

(6a)

where P,(Ii, If) = ( 2 n ) - ' ~ [ ~ ( I ~ ) / ~ ( ~ i ) l - '

and where the Jacobian (7) is evaluated at $1" (1 = 1, . the real roots of

(7)

. . , m),which are

Mdi, 1,) = If

(8)

If there are no real roots, 9 ' = 0; if two or more roots coincide, P may be infinite. Note that P i s not a probability, but 9(1,, K ) t K is the probability that the value of the final action variables lies in the range (K, K + dK). The evaluation of the integral (6a) is usually nontrivial and is discussed further in Section V. The distribution €unction P is related to the classical probability of a transition from quanta1 states n, to nr by the density-of-states correspondence principle (see Clark et d.,1977):

P cl(ni + nr) = h " 9 ( I i , If) Jb

=

(n,

+ y)h

(9) (s

=

i, f )

Some approximations to Pcl(ni+ nr) are discussed in Section V.

(10)

A . S . Dickinson and D . Richurds

170

111. Rotational Excitation A. INTRODUCTION We now consider rotational excitation in collisions between atoms and rigid linear molecules as the simplest realistic inelastic collision. The Hamiltonian of the system is

H

=

If/2A

+

P 2 / 2 p + V ( R , x),

cos x = R * i

(11)

where Z, = IJ , I , J, being the molecular angular momentum vector relative to the molecule center of mass 0, A is the moment of inertia, P is the relative momentum, p the reduced mass, R the position vector of the atom relative to 0, and i. lies along the molecular axis. Coordinates for the rotor motion are the action variables 1, and I , = J,-iand the conjugate angle variables 8, and O m , respectively. Here 0, measures the rotation of the rotor in its plane of rotation, and 8, is the angle variable conjugate to ,Z (see Fig. 1). We shall also use /3 = cos-*(Zm/Zl),0 s /3 4 7 ~ For . free motion we have 0, = wt + 4,, w = I l / A , and 8, = 4,, 4, and 4, being constants (see Section 11,B).For a rotor initially at rest these coordinates are undefined, and instead we use the two polar coordinates i = (8, 4), which are now sufficient to specify the initial state of the rotor.

B. DESCRIPTION OF A COLLISION We consider an atom incident with impact-parameter vector b = ( h , &) and velocity ui, along the quantization axis; the initial values of the molecule action variables are If,Zl, = 1; cos pi, and those of the angle variable phases are 4; and 4h. After the collision the atom moves in the direction (€3, 01, leaving the rotor with action variables 1, and I,; all final variables depend upon (b, v, I j , I f , +;, +h). The fully differential cross section is

171

INELASTIC HEAVY-PARTICLE COLLISIONS

FIG. 1. Diagram showing the angle-action variables of the free rotor. IfO-N is the line in which the rotor disk and O-.v-y plane intersect, then On is the angle between 0 - N and i and 0, is the angle between 0 - N and the rotor axis. The angle between i and J, is 8.

Because of azimuthal symmetry

+ +(+b - 6;~b , 4;)

@ = d'b

and the other final variables depend only on the combination (&, and so we have

=

sin

( 4 ~ 2

x 6(I, -

(13) - +&),

lm lzT b db

r:, S(Z,

-

dr$j l z n d Q b S ( 0 - 0') (14)

1;)

where only + b = 0 need be considered (see Gislason, 1976). Note that the cross section is now independent of @'. However, for other quantization axes in general the cross section would depend upon Qf (for a discussion 3 Zh see Dickinson and Richards, 1980). Clearly the orbits that give cross sections for one choice of z axis suffice for any other choice, but we are unaware of any simple connection between cross sections referred to different axes comparable to that obtained in quantum mechanics (Alexander et NI., 1977). For a rotor initially at rest, the integrations over and d m are replaced by integrations over and cos 8 , and the integral is multiplied by T. Because of the symmetry of the Hamiltonian under panty inversion @(+j,+k, pi) = €I(+; + T , - +k, T - pi), and similarly for Z,, while

+

+,

172

A . S . Dickinson and D . Richards

Q(&;, ,#,A, pi) = -@(&; + r, - &A, r - pi), and similarly for I , . Thus, we have the fundamental symmetry

For a stationary target the corresponding symmetry is where 6 + - 6. Pattengill ( I979a) discusses the practical details of a classical calculation. An alternative choice of coordinates, corresponding to that employed in quanta1 calculations, leads to six instead of ten first-order equations of motion, but at the price of more complicated equations (Miller, 1971; Kreek and Marcus, 1974).

C. ROTATIONAL RAINBOWS

I . Uti ( r 1wrrgrtl Rtr it1hou.s The experiments of Bergmann et (11. (1980, 1981) on scattering of Na, by He have stimulated considerable interest in rotational rainbows. The observations have been interpreted largely using the quantum infiniteorder-sudden approximation (Parker and Pack, 1978; Schinkeer d., 1981). Here we discuss the basic classical behavior underlying rainbows in inelastic scattering; like rainbows in potential scattering, these are fundamentally classical in origin. Rainbow scattering has been reviewed by Thomas (1981) and Schinke and Bowman (1982). We begin by considering the most detailed cross section a30/(an afJ t l f m ) the ; effects of averaging are considered below. Experiments at this level of detail are not yet available. For clarity we consider only the case for which d @ / d b # 0 so that the equation 8' = 8 ( b , & m ) may be inverted to give h as a periodic function of 4 = (&j, & I , and from now on O f i s treated as a parameter. Then the cross section (14) may be written as a double integral: +J,

where Fk are continuous periodic functions of the equation I:,

=

4. If we now fix I k ,

F3(4)

defines a set of lines in 4 space on each of which odically. The cross section (16) may be written

(17)

F 2 ( 4 ) varies peri-

173

INELASTIC HEAVY-PARTICLE COLLISIONS

where .Pparameterizes each line of the set defined by Eqs. (17). Since F2(,P”)is periodic, it has stationary points, so whenlf has these stationary values, the integral (18) has a square-root singularity, similar to that obtained in the rainbow region of atom-atom scattering. Similar behavior is obtained in the other two variables so that the rainbow can be observed in any of the final variables. This feature is discussed extensively by Thomas (1981). Points at which d 8 / d b = 0, that is, in the angular-rainbow region, may give rise to stronger singularities. This problem is being investigated. While this discussion was for the case of three observables depending on three initial variables, the same square-root behavior in all observables will be obtained near a rainbow in any cross section measured with maximum resolution. 2 . A\wriged Rciinbon,s Observations of the fully resolved differential cross section are very difficult so we consider the effect of averaging over some final variables. First consider the level-to-level cross sections azu/aO aZ,, which are currently experimentally accessible (Bergmann et d.,1980, 1981). Other averages may be treated similarly. For simplicity we take the target to be initially at rest. Then

= ( 4 n sin

8f)-110m

b dh

I

d i 6(ef- 8)6(Zf

-

Z,)

(19)

As before, we consider the case where a e / d b # 0, which is typical of conditions in most current experiments. Then Eq. (19) becomes

I 1M

aza = ( 4 n sin ef)-l d i b - 6(Zf - Zj(i))

aa az,

As I,@) is defined on a spherical domain, for all @‘it has stationary points at which the cross section shows some structure. Korsch and Richards (1981) have discussed the behavior of a2u/(dR aZ,) in detail and have shown that at a maximum or minimum of Zj(i) the cross section has a finite step, while at a saddle point there is a logarithmic singularity but no “bright” and “dark” side. This behavior is shown in Fig. 2, strictly for the transition probability

P

=

( 4 n ) - ’ / di.6(Zf - Zj(i))

which displays identical structure.

A . S . Dickinson and D . Richards

174

We emphasize that these types of singularity will occur anywhere the cross section of interest can be reduced to a double integral over a single delta function. Cross sections displaying this characteristic shape were obtained earlier by Gentry (1974) and very recently by Beck et al. (1981). Additional averaging, for example, calculating d d d 0 summed over all I j, leads to still weaker structure at angles where 8 is stationary as a function o f h and i. Each type of stationary point has its characteristic structure, typically a continuous cross section with a discontinuous derivative (Dickinson and Richards, 1979). When allowance is made for the blurring due to quanta1 effects, such structure appears to be at the limit of unambiguous experimental detection. D. RIGID-SHELLSCATTERING A fairly simple model yielding considerable insight is scattering by a rigid body, normally an ellipsoid of revolution. Investigated in 1958 by 1

1

I

I

1

I

I

1

,

I

I

,

I

020-

I

I I

I

I h c h -

/

H v

a /

/

/

/

I

/

0’

I

0

4

8

I

I

12

FIG. 2. Graph showing the structure of the rotational rainbow in the classical cross section. The full curve shows the angular momentum distributionP(1:) [Eq. (21)l for a model system. The broken curve shows the 10s approximation toP(fj) for the same system. (From Korsch and Richards, 1981.)

INELASTIC HEAVY-PARTICLE COLLISIONS

175

Muckenfuss and Curtiss, and later by La Budde and Bernstein (1971), it has proved especially fruitful in the hands of Beck ef nl. (1979a,b, 1981) for interpreting rainbow phenomena, particularly in K + N2 and K + CO backward scattering. The model is simple because all quantities of interest are determined from conservation laws and depend only upon the vector of the point of contact. This vector is specified by three quantities, one less than is needed to determine the result of a soft collision. For an initially stationary shell, 8‘ and ZJ’depend only upon two variables. Then it is easily seen (Korsch and Richards, 1981) that the cross section dzcr/(dR dZ,) has a square-root singularity at the rainbow value of Z,, which depends solely on the shell shape. For off-center rotation there is usually a double rainbow (Beck et al., 1979a). For initially rotating molecules the cross section shows a logarithmic singularity and a finite step (Beck et d . , 1981) consistent with the discussion in Section III,C,2. A very useful feature of this rigid-shell model is the simple relationship between the rainbow values of Wand I,, particularly for large moments of inertia. Because of the lack of an attractive potential this model is most relevant to collisions involving large changes in I, and backward scattering (Korsch and Schinke, 1981). The cross section dcrldl, has been calculated by Bhattacharyya and Dickinson (1982) for highly eccentric ellipsoidal surfaces to model H2 collisions with HC,N and HCJV, which are of astrophysical interest.

E. N U M E R I C CALCULATIONS AL Apart from calculations concerning rainbows, discussed above, and direct comparisons with quanta1 calculations (Pattengill, 1979a, and references therein), classical methods have been used for some time (La Budde and Bernstein, 1971; Loesch, 1980; and references therein) to investigate translational to rotational energy transfer. Briefly, introducing 0,, the the “rainbow” angle for da/dR, as a characteristic boundary, for 8 > 8,, repulsive anisotropy causes substantial energy transfer, while for 8 0), of the complex k plane. Those on the positive imaginary axis correspond to bound states and, for a pure outgoing wave, exp(ikr), lead to square-integrable functions. On the other hand, those lying in the lower half of the complex k plane lead to asymptotically exponentially diverging functions, exp(ik,r + k l r ) . The corresponding states are called Gamow (1931) or Siegert (1939) states or resonances, particularly those whose energy lies in the lower right-hand quadrant of the higher sheets of the complex energy plane. A. BIORTHOGONAL SETS The concept of a biorthogonal basis (Hokkyo, 1965) is a convenient formalism for a discussion of these states and complex scaling. Consider J,, , which satisfies

ih(aJ,,,/at) = HJ,, = En$,

(2)

where

H

=

-iV2

+U

(3)

and I&, which satisfies ih(dt/&/af) = HtJ,& = EL+&

where

H: =

-gVt)2

+ I/*

HS is the adjoint of H relative to J, and $IT if

(4)

RESONANCE CALCULATIONS USING COMPLEX SCALING

2 11

From Eqs. (3) and (9,Eq. (6) becomes 0

=

![+'*(TJI) - (Tt4')*JI]d7

=

(h2/2m)

1[(VP

-

v + ~ ) ~ +d7 I *+J Iin

I

nos d~

(7)

where Green's theorem has been used. The probability current density is S = (h/2irn)[4'*V$ -

(V4')*JIl

(8)

and n is the normal to the closed-surface s. If the boundary conditions on 4T are chosen such that

s=o

(9)

=

(10)

v2

vt2

that is, V2 is self-adjoint with respect to

H + = H*

=

c$t

and JI. Consequently,

-4V2 + I/*

(1 la)

Ek = EZ

(1 1b)

Then Eq. (6) yields

(E,

-

E ~ ) ! J I ~ * J(Em I ~~ En)(+;I+rn) ~ = = 0

(12)

I f n # rn, (JIhlJIm) =

0

and the functions JI; form a biorthogonal set with

(13)

JIn.

B. INCOMING-OUTGOING WAVES First consider a function satisfying the time-independent Schrodinger equation (Berggren, 1968) HJIn = En&

(14)

and the boundary conditions lim rJIn= 0 r-+O

V(r@n)

with

- ikn(rJIn)

( 15a)

(15b)

212

B . R . Junker

More generally, the boundary condition (l5b) is determined by the physics of the problem. Equation (15b) denoted an outgoing wave with wave vector k,. In addition, we consider a second solution $; satisfying the adjoint equations

H't,!~h= Eg$i

(17)

and the boundary conditions lim r@,

=

0

r-4

with

E;E

=

i(kg

k;E) = Jlk,12 exp(fip,)

(19)

Here, again, in the general case, the form of Eq. (18b) is determined by the boundary condition on $, and Eq. (9). Equation (18b) denotes an incoming wave with wave vector, kg, or equivalently an outgoing wave with wive vector, -k:. Thus, for and ;$I satisfying Eqs. (15b) and (18b), Eq. (9) is satisfied. That is, they form a biorthogonal set with respect to each other and are the right and left eigenvectors of H as discussed by Lovelace (1964). Since the eigenfunction corresponding to - k g is related to the one corresponding to k, by complex conjugation (Humblet and Rosenfeld, 1961), the resonance can actually be defined either in terms of an outgoing wave with wave number k, or in terms of an incoming wave with wave number k;. More precisely, 4' of Eq. (6) is the complex conjugate of $(-rn), where ~n is the component of angular momentum along some space-fixed axis. Otherwise the angular integrations later would yield zero for all matrix elements. This choice is possible since E is independent of tn (Mahaux, 1965). If, in addition to Eqs. (9) and (1 la) being satisfied, H* = H

(20)

we find that 4, is just the complex conjugate (time reversal) of JI(-rn). On the other hand, if Eq. (20) is not true, but there exists a unitary transformation W such that WH"W-1

=

H

(21)

the time reversal of $ is $1'

=

WT,$

(22)

RESONANCE CALCULATIONS USING COMPLEX SCALING

(Goldberger and Watson, 1964), where WH*W-'W JI* = HJIf' = E*JI'r

213

(23)

However, we see from Eq. (6) that $+

=

W-1JIty-m) = $ *( - m )

(24)

This then gives the relationship between 4' and the time reversal of JI, While the discussion above has been quite general, we shall now, for simplicity, take the surface s to be a sphere so that n * S = (h/2im)[~$~*(a$/dr) - (aC$'/ar)*+l

(25)

We are thus interested in functions JIn , JI; satisfying boundary conditions

(a@,,/ar>

- ikn(rJIn)

ma)

(dr+',/ar)

- -ikz(r$L)

(26b)

As noted above, the imaginary part of k, is less than zero and from Eq. (26b) so is the imaginary part of -k*,. Thus, both JI, and JIL are asymptotically divergent and some means of defining the integrals in Eqs. (6) and (12) must be developed. Zel'dovich (1961) suggested defining them by multiplying the integrand by exp(-ar*) and taking the limit of the result as (Y -+ 0. [A rigorous discussion of this approach is given by Berggren (1968)l. Romo (1968) suggested defining them in the upper half of the complex k plane, where JI,, and JI; are well behaved, and then continuing the result to the lower half of the complex k plane. These two approaches have been shown to be equivalent (Gyarmati and Vertse, 1971). Finally, a third completely equivalent approach was suggested by Dykhne and Chaplik (1961) in which the integral is evaluated along a ray in the upper complex-coordinate plane such that the angle 8 of the ray is greater than arctan(-E,/E,). [Note that 8 should be the absolute value of arctan (-EI/ER)and that 8 need only be greater than one-half of the angle according to Eq. (16).] The use of biorthogonal sets and the bracket ($+I$) is a generalization of the normal inner product on a Hilbert space. For bound states

WIJI) = (+I+)

(27)

This earlier formalism eliminates the need for and the ambiguities in the C-product formalism (Moiseyev ei al., 1978a) as noted by Mishra et al. (1981a) and the r-product formalism (Weinhold, 1978). The results that we have discussed up to this point are rigorous for potential scattering and, under certain conditions, for the N-particle prob-

B. R . Junker

214

lem, as discussed in Section I. We shall assume they are valid for the Coulomb potential also.

111. Complex-Coordinate Theorems and Properties of the Wave Functions As discussed in Section 11, complex scalings, and complex coordinates in particular, have been used in short-range scattering as appropriate to nuclear interactions. For example, Lovelace (1964) employed these techniques to evaluate matrix elements for three-particle scattering with short-range potentials. On the other hand, the studies of Aguilar and Combes (1971), Balslev and Combes (1971), and Simon (1972, 1973) have used the analycity of the operators to determine the spectral properties of the operators and the nature of the wave functions. Their results are particularly significant to atomic and molecular physics since they apply to potentials of the form r-a (0 < (Y < 2). A. COMPLEX-COORDINATE THEOREMS Basically the complex-coordinate theorems define a “rotated” Hamiltonian, We), which is obtained from the original Hamiltonian X ( r ) , by means of the transformation T(r) = qr

(28)

where q = a exp(i8)

(29)

The spectrum of X(0)contains the following elements. (1) Energies corresponding to bound states, which are identical to the bound states of X ( r ) , (2) Real energy thresholds for the cuts of X(8), which are the same as the thresholds for the cuts of % ( r ) , but the cuts of X(8) are rotated down onto the lower sheets by an angle 28 [note that the energies associated with a given cut of X(8) are given by (ET+ E exp(-2i8)], where E is times the energy on the cut for X ( r ) above the threshold energy E T . (3) Discrete complex eigenvalues E K + iE, , which are uncovered when

0 > +(arg(& + iEJJ= p

(30)

RESONANCE CALCULATIONS USING COMPLEX SCALING

215

where ER is relative to the threshold with respect to which 8 is defined. (Note that the eigenfunctions of these complex poles of the resolvent are square integrable .) (4) Complex thresholds for complex cuts, which will not be of interest to us here. The relationship between the spectrum of %(r) and %(@)is illustrated in Fig. 2 for some of the ' S states of H-. In Fig. 2a each of the cuts for X ( r ) beginning at a threshold for a state of hydrogen atom lies along the real energy axis. However, each of these cuts is rotated down by an angle 28 for X ( 8 ) and an uncovered resonance is indicated. In their original proof, Balslev and Combes (1971) assumed that the potential V of the Hamiltonian contained symmetric two-body interactions which were compact with respect to the kinetic energy operator; V was required to admit an analytic continuation into a region in the complex-coordinate plane. Such potentials include gr-P (0 < p < 4) and superpositions of Yukawa potentials. Then, for 0 < 8 < ~ / 4 they , proved the above result. Simon (1972) extended this result to include both interactions of the form gr-0 (4 s p < 2) and admit values of 0 up to some maximum Om,,, where Om,, defines the region of analytic continuation of the potential, after which it is no longer a bounded operator-valued function (Reed and Simon, 1978). For example, Om,, is ~ / for 2 a Yukawa potential and infinity for a Coulomb potential. ial

-

E=-0.5 f X

-- --

E =-0.125 E = -0.055

BOUND STATE

FIG.2 . lS spectrum of H-:(a) for X ( r ) ;(b) for X ( 0 ) .

B . R . Junker

216

While the above theorems apply to sums of Coulomb potentials, as formulated, they are not directly applicable to the Born-Oppenheimer (BO) approximation for molecules. The difficulty arises when one considers dilating the electron-nuclear attraction potentials (Simon, 1978). Consider a system with M nuclei and N electrons. Then the electron-nuclear attraction potential is

Dilation about a given nuclear center, say B , causes no problem for terms of the form -ZB ELl (r, - rj1-l but converts the singularities at all other nuclear centers, r A, into circles of square-root branch points. While there is no theoretical difficulty in using the total molecular Hamiltonian and scaling all coordinates including nuclear coordinates, the BO picture offers a number of practical advantages (McCurdy and Rescigno, 1978; Simon, 1979). For example, many important processes such as dissociative attachment arise from a repulsive potential surface for nuclear coordinates, but an electronic resonance state. Such a state would be a continuum state for the entire system if the total Hamiltonian were used. Another difficulty which arises when the total Hamiltonian is used is that each vibrational and rotational state would correspond to a threshold for another cut. Thus, from a practical point, there would be so many cuts rotated down that determining which eigenvalues approximated cuts and which approximated resonances in an actual calculation would be difficult. To circumvent these difficulties Simon (1979) defined a method called “exterior complex scaling.” In place of the transformation given by Eq. (28), he defined a transformation TRo(r) = r . =

Ro + a exp(ie)(r - Ro),

O s r s R ,

(324

Ro 4 r

(32b)

with boundary conditions +(R,)

=

+ ’ ( R i )=

$(Rot) a-l exp( -ie)+’(R:)

(334 (33b)

Here R,(R,+) corresponds to approaching from below (above), and the prime in Eq. (33b) corresponds to differentiation with respect to r. Simon then showed that if V is local, central, and dilation,analytic under scaling about some center, it and the Coulomb potential between any two electrons are analytic with respect to the transformation defined by Eq. (32). Finally the spectrum of the Hamiltonian, x R , , ( O ) , obtained from X ( r ) by the transformation (32), was shown to be independent of Ro and to have the same properties as that obtained with H(0).Thus, XRo( 0 ) would only

RESONANCE CALCULATIONS USING COMPLEX SCALING

2 17

have thresholds and cuts associated with electronic channels, and the resonances would be associated with electrons. An approach along these lines was actually heuristically discussed earlier by Nicolaides and Beck (1978a) for atomic systems.

B. WAVE-FUNCTION PROPERTIES Thus far, we have confined our discussion in this section to the spectral properties of the Hamiltonian operators. As for the wave functions, if U(p)@(r)= exp(np/2)@(exp(p)r)

(34) where U ( p )is an element of a one-parameter family of unitary dilations ( p E R ) on L z ( R n ) ,has an analytical continuation to complex p, @(r) is said to be dilation analytic. Similarly for the transformation (32), we have

uRt~(p)@(~) = J (r)@(TRe(r))

(35)

where (I’) = [ ~ ~ ‘ ~ R ~ ~ ( ~ l ) l [ ~ ~ R i , ( ~ l ’ I ) / ~ ~ l ~ ’ ’ ~

(36)

[ r ; 1 ~ R ~ , ( r 2 ) l [ d T R ~ , ( ~ Z ) / d r 2 ‘1 ”‘ 2’

Here exp(np/2) and J ( r ) are necessary to make U and URounitary, that is, in terms of integrals they convert the volume element for @(r) to the appropriate volume element for @(exp(p)r)and @(TRo(r)),respectively. Aguilar and Combes (1971) and Balslev and Combes (1971) showed that the domain of bound states of X ( 8 ) is obtained by transforming the domain of %(r) according to Eq. (34). That is, the bound states of X(8) are simply analytical continuations of the bound states of X ( r ) so that their only dependence of r , a, and 8 is of the form ar exp(iO), apart from the trivial phase factor in Eq. (34). Junker (1978a) (see also Simon, 1978) illustrated how the resonance wave functions also depended on r , a,and 8 only as ar exp(i8) and were thus analytic continuations of some function of X ( r ) . Junker and Huang (1977, 1978) suggested that these functions were just the Siegert functions associated with resonances of X ( r ) . Later, Junker (1982) showed explicitly that analytical continuation of the Schrodinger equation and the boundary conditions for the wave function for a Gamow-Siegert state yields functions with properties which are compatible with the complexcoordinate theorems. That is, since the long-range behavior of these functions is given by Eq. (26a), one obtains, under the transformation (28), the long-range form of the transformed Siegert function: $s

- r i l l exp[ilk,)ar,+,

exp(i(8 -

p))l+drl exp(ie), . .

.)

(37)

B. R. Junker

218

Consequently for 8 > p, $3 is square integrable with respect to the bracket defined in Section I1 and is thus a proper eigenfunction. This is in agreement with the complex-coordinate theorems above. This kinematic dependence of the bound- and resonance-state wave functions o n r , a, and 8 only as ar exp(i8) is similar to other kinematic properties of the wave functions such as parity, permutation, and spatial symmetries. The similarities between the bound and resonance wave functions enables one to understand many of the properties of resonance wave functions in terms of bound-state wave functions. For example, the effect of the cut rotating through the bound states, that is, the wave function being analytically continued from a value of 8 which is infinitismally greater , to discontinuously than 57/2 to a value infinitismally less than ~ / 2 is convert the bound-state wave function from an asymptotically divergent function to an asymptotically convergent function. Similarly, the effect (Junker, 1981) of the cut rotating through a resonance pole, that is, the wave function being analytically continued from a value of 0 which is infinitesimally less than p to a value infinitesimally greater than p, is to discontinuously convert the Siegert function from an asymptotically divergent function to an asymptotically convergent function. On the other hand, the difference between the bound-state and resonance wave functions is that the former are innately real functions of r or ar exp(i8), while the latter are innately complex functions of r or ar exp(i8) for X ( r ) and X ( 8 ) , respectively. This arises from the complex boundary conditions, Eqs. (15b) and (161, which the latter satisfy, as opposed to the real boundary conditions, which the former satisfy. This leads, of course, to the real energies of the former and the complex energies of the latter. The analyticity of the bound and resonance wave functions implies that if these functions are known at some value o f a exp(iO), say, aIexp(iO,), a completely equivalent (in the sense that they yield the same eigenvalue) set of functions can be obtained for a exp(i8) equal to a2exp(i8,). The transformations which transform 9 ( a l , 8,) to 9 ( a z ,8,) are

T(r) = ( a Z / a 1exp(i(& ) - 8d)r

(38)

T,t,,(r) = r =

Ro + (a,/aJ exp(i(8,

(394 -

e,))(r - Ro)

(39b)

depending on whether the radial coordinate is being scaled over its entire range or only over some external region. This result will be used in Section VIII to show that the resonance functions can be computed directly without rotating the Hamiltonian. On the other hand, the functions associated with the energies along the cut cannot depend on r , a, and 0 in the form ar exp(i8) except for the

RESONANCE CALCULATIONS USING COMPLEX SCALING

2 19

threshold (Junker and Huang, 1978; Junker, 1978, 1982). For the natural choice for the cut (Newton, 1966), which in the case of X ( 8 ) corresponds to the rotation of the cut down by 28, the functions defining the scattering states go asymptotically as $E

- A(k, 8)Xdri

9

9

r ~ exP(ik*rN+i) )

+ B(k, 8)X,r(ri, . . . , rN) eXp(-ik*I'N+i)

(40)

This results from the fact that the natural choice for the cut corresponds to the asymptotic form of the wave function for the scattered particle being a linear combination of eigenfunctions of the momentum operator. Transformation (28) implies p + a-lp exp(-iO)

(41)

Then exp( -iO)V, exp(+ik*r) =

?exp(-i8)a-'k

exp(2ik.r)

=

ET + $(k12a-2 exp(-2i8)

(42)

so that

E

(43)

where E,r is the threshold energy and a-l only scales the energies along the cut. Equation (43) is consistent with the cuts being rotated down into the complex energy plane by angle 28. Another point concerning the scattering functions is that solving the single-particle Schrodinger equation

[T exp(-2i8)

+ V(r exp(i0)) - E exp(-2i8)]x(r,

8) = 0

(44)

where we have set a = 1, subject to Eq. (42) yields the exact same solution as solving the Schrodinger equation

[T

+ exp(2ie)V(rexp(iO))

-

E ] x ( r , 0) = 0

(45)

subject to the boundary condition

-iV, exp(?ik*r) = k k exp(Lik*r)

(46)

where k is real. Eq. (45) is simply the Schrodinger equation for real Y but a complex potential. The resulting phase shifts will thus be complex and the magnitude of the S-matrix elements will no longer be unity. Finally, below we shall require the wave functions for the adjoint of Z(0) which form the biorthogonal set with the wave functions of X(0). The results of Section II,B must be modified to incorporate the fact that

~ ( e =) v ( e ) +

we)

(47)

220

B . R . Junker

although for the special case of real X ( r ) P ( 8 ) = X(-8)

We then choose 2 (H)$n(O)

=

x-(O)$X(8) = E;$n(o)

(49)

where $,(8) = T;$;(r) = T+$;(I’) = T - v T x $ n ( ~-wz) .

(50)

Since from Eq. (37) $ n ( 8 ) is square integrable when 0 satisfies Eq. (30), $ , ( H ) is also square integrable and the sets of functions $,,(8) and $ , ( H ) form a biorthogonal set. Certainly, an immediate consequence of the complex-coordinate theorems is a clear justification of the technique of Dykhne and Chaplik (1961) for defining otherwise divergent integrals over Siegert functions. A more interesting implication is the possibility of representing the resonance wave function in terms of a square-integrable basis. This latter possibility is the motivating force behind the concepts in Sections IV-IX.

C. T I M EREVERSAL Analogous to the discussion after Eq. (20), one can define a transformation W for real V ( r )such that Wx-(H)W-’ = X(H)

( 5 1)

within the space spanned by the bound and resonance states, since their energies are independent of 8. Then

X@)(W+,”(O))= En(W + ; W

(52)

Where W corresponds to the transformation W(r.1 =

I’

W ( r )=

I’

exp(2iH)

(53)

or

=

R,

+ exp(2i8)(1’- R,)

(54b)

depending on whether W e ) resulted from transformation (28) or (32). Thus, analogous to Eq. ( 2 2 ) , one can define the time reversal of $,#as

+:(e)

=

w$;(e) = wT,+,(e)

(55)

RESONANCE CALCULATIONS USING COMPLEX SCALING

22 1

which is not square integrable for 0 satisfying Eq.(30). On the other hand, Eq. (49) requires

we)+;(@

=

E;i+ltn(O)

(56)

Thus,

+W =

W-l+K(O, - m ) = T,&,(e, - m ) =

TOTH+&)

(57)

where the reason for G n ( - m ) has been discussed in Section II,B.

D. H Y P E R V I R ITHEOREMS AL The definition of a complex virial theorem for the analytically continued bound and resonance states has been extensively discussed by Froelich et ( I / . (1977), Brandas and Froelich (1977), Yaris and Winkler (1978), Moiseyev et ( I / . ( 1978a,b), and Brandas et cd. (1978). Following Hirschfelder (1960), one can, in fact, consider a class of hypervirial relationships. For a general operator W, which is a function of the coordinates and conjugate momenta, a matrix element of the Heisenberg equation of motion with respect to a resonance state can be defined using the methods of Section I1 as

4(+hllw(~)l ) ) / d l = -i(+hl[W , +ln

=

-i&

-

~ n ) ( ( T ~ + L ) l W ( e ) l T " + ~ ~ ) (58)

If m = 1 1 , Eq. (58) is zero as long as (T;+LIW(B)lTH+,)is finite. Here THis the operator defining the transformation (28) or (32). Then 0

=

(+kl[W(r), x(r)Il+n)

=

(Ti+;I[w(e), x(e)IJT"+n)

(59)

Note that this defines a set of complex &independent constraints for a whole class of operators. If W is chosen to have the form W,. =

2 (ri*pi + pieri) i

(60)

B. R. Junker

222

Again this complex virial theorem is 0 independent for wave functions with the correct kinematic 8 dependence and the transformations (28) and (32) are used only to define &independent matrix elements. Finally, all of the expressions discussed in this section reduce to their usual forms for bound-state functions.

E.

ANALYTICAL

MODELS

A number of analytical models have now been reported which clarify the theorems and comments in Sections II1,A-II1,D. We shall simply give the results here surpressing the real scaling factor a. I . E.\-potietititrlWell Poteritid (Junker, 1981)

The rotated Hamiltonian for S waves is X ( 0 ) = - ( e x p ( - 2 i ~ ) / 2 ~ ) ( d ~ / d-r ~Vo ) exp[-exp(it))r/tr]

which yields the Schrodinger equation [ ( d ’ / d q 2 ) -k ( I / q ) ( d / d q )+ 4b’((x’2/q2) f

K’2)]X

=

0

where b

= N

exp(-i0)

q = exp(-r/b)

k exp(i6) = (2pE)1’2exp(i8)

A’

=

K’

= K

exp(i8) = (2pV0)1/2exp(i0)

The solutions are of the form X = J-2ad2N

K ) J ~ a / c i ( 2 UK f ) )

-

Jzaki(2U K)J-paki(b K q )

which asymptotically yields

x where

C[S exp(ikr exp(it)))- exp(-iAr e x p ( i ~ ) ) ]

(63)

RESONANCE CALCULATIONS USING COMPLEX SCALING

223

Note that the only dependence on 8 in Eqs. (65)-(67), other than the explicit dependence in Eq. (66), can arise through and k . The poles o f S arise from =

J-zakl(2LIK)

0

(68)

since values of k satisfying 1--1(2[1ki

+

I)

=

o

(69)

yield only the trivial solution (ter Haar, 1964). Since all quantities in Eq. (68), with the possible exception of k , are independent of 8 and the location of the poles must be 8 independent, k must also be independent of 8. Thus, for bound states the solutions for X ( 0 ) are clearly analytical continuations of the solutions for X ( r ) , and Eqs. (59) and (61) yield &independent values. While there exist real zeroes of J,(Z) (Watson, 1966) for real v (bound states), we know of no theorem proving the existance of real zeroes for complex v (resonances). On the other hand, the scattering states are required to behave asymptotically as linear combinations of momentum eigenfunctions with eigenvalues k p exp(-i8), with p real. Thus, -i exp(-iO)(d(exp(&ikr exp(i8)))ldr) = ?k exp(+ikr exp(i8)) (70)

implies k is proportional to exp( - i 8 ) . Thus, S depends explicitly on 8 and for 8 # 0. IS1 # 1

(71)

as discussed in Section II1,B.

2. Coirlomhic Potentials Coulombic potentials have been discussed by Junker and Huang ( 1977, 1978), Junker (1982), and Nicolaides and Beck (1978b). The bound-state wave functions can be shown to simply be analytical continuations of the normal hydrogenic functions, that is,

$,(,(e)

= N,(r

exp(i8))' exp[-Zr exp(i@)/n

x L:'+l[2Zr exp(ie)/n] y;"

(72)

where N , is the radial normalization factor, Z is the charge and L;'+' is an associated Laguerre polynomial. Obviously, Eqs. (59) and (61) are again 8 independent. Junker (1982) has also discussed the continuum solutions. Fork real, the solution for a rotation angle 6 and energy, Sk2 exp(-2iO), is = Clr'

exp(ikr),F,(I + I

+ iZZ' exp(i8)k-1; 21 + 2 ; -2ikr)

(73)

B. R. Junker

224

Note that the only dependence on 6' is in one of the indices of the confluent hypergeometric function. Then SI(6') = exp(2icrJ -

I'(/ + I + iZZ'k-' exp(i6')) r(1 + 1 - iZZ'k-' exp(i8))

(74)

Again for H # 0. (S,(H)I+ 1 3 . Anci1ytic.d Models

ltith

(75)

Resonances

Analytical examples which yield resonances are more difficult to develop. Doolen (1978) considered a potential of the form V(r) = -yr-2

+ I'

(76)

for s-wave scattering. While this potential technically is not dilation analytic (see, however, Reed and Simon, 1978, p. 116), the results are instructive. Junker (1982) gave the simple extension of this model to arbitrary partial waves. That is, one seeks solutions of

+ &[I(/ +

1

(77)

((21 + 1)' - 8y)1'2]

(78)

1) - 2y] ex~(-2iO)r-~ - E Fl(r, 8 ) = 0

Defining

I'

=

-+[I

-

so that the coefficient of r - 2 becomes +Ir(/' the form Fdr, 6')

= Cp

+ I),

one obtains solutions of

exp[ikr exp(iO)][kr e x p ( i ~ ) ] * '

x lFl(I' + 1

+

21'

+ 2 ; -2ikr

exp(i6'))

(79)

with

(a) For 8 y < (21 +

k

=

-i/[n

+

+(I

the poles of S,(k) occur for

+ [(21 +

- SY]'/~)],

n

=

0, 1, 2, 3, . . . (81)

RESONANCE CALCULATIONS USING COMPLEX SCALING

225

(b) For 8 y > (21 + 1 1 2 , the poles of S,(k) occur for

2{ ~ [ 8 y (21 + 1 )21112- i [ 2 n + 13) ' [211 + 1]* + [ 8 y - (21 + 1)2]

n = 0. 1 , 2 . .

..

(82)

Note that [8y - (21 + I)']], [2n + 1 I, and the denominator in Eq. (82) are all positive. Equation (82) thus illustrates the complimentary nature of the poles of the S matrix in the lower half of the complex4 plane. The corresponding wave functions are F J ~ H, )

=

c,,exp[ilklr exp(i(8 P))][Jklr.exp(i(8 p))],' x ,F1[-n; 21' + 2 ; -2ilkl1' exp(i(8 - p))] -

-

(83)

where here 1'

= -+[I

p

=

+ i(8y

-tan{ - [ 2 n

- (21

+

+ 1 ]/ [8y

1)2)1/21 -

(84a)

(21 + 1)2]1'2}

(84b)

Equation ( 8 3 ) illustrates the dependence of F,,, on onlyr exp(i8), as well as the square integrability of FnI for 0>

P

=

(85)

31arg(E,I)I

Simon's ( 1979) external complex scaling transformation ( 3 2 ) can be clearly illustrated with this model also. For the n = 0. 1 = 0 resonance the explicit wave functions take the form (Junker, 1982) Foo(kr)= C0[r e ~ p ( - i p ) ] - [ ~ +exp[ir ' ~ ] / ~exp(-iB)],

OsrsR,

Foo(k[Ro(l - exp(i8))+ =

I'

(86a)

exp(iH)])

Co[exp(-iP)Ro(l - exp(i8)) + r exp(i(8 - p))]-[1+iv31'2 x exp[i exp(-iP)Ro(l - exp(il)))] x exp[ir exp(i(8 -

with

P))],

Ro

I'

(86b)

226

B. R. Junker e-iBF~o(k[Ro(l - exp(i8)) + r exp(i8)l =

-+[1

+ i(3)1/2]Coexp(-ip)

x [exp(-ip)R,(l

+r

exp(i(8 -

-

exp(i8))

p))]-[3+fV31/2

x exp[i exp(-ip)Ro(l - exp(i8))I x exp[ir exp(i(8 -

p ) ) ] + iCo exp(-ip)

x [exp(-ip)R,(I - exp(i8))

+ r exp(i(8

-

p))]-[l+ifi51’z

x exp[i exp( -ip)Ro( 1 - exp(iO))]

x exp[ir exp(i(8 -

p))],

Ro 6

Y

(87b)

At r = R,,, Eq. (33) is clearly satisfied. Yaris et ( I / . (1978) discuss a separable model potential which can be solved analytically to illustrate the functional dependence of the boundand resonance-state wave functions on r exp(i0) (Junker, 1982). It also demonstrates that analytically continuing the wave function back to 8 = 0 yields a wave function which satisfies a Siegert boundary condition asymptotically. These examples illustrate a very important point which was noted in Section II1,B and will be reiterated throughout the rest of this contribution. That is, Eqs. (65)-(67) and (72) show that the complex nature of the bound-state wave functions arises only through r exp(iO), so that the bound-state wave functions analytically continue back to real functions of r for 8 = 0. On the other hand, Eqs. (83)-(84) and the separable model potential of Yaris et a/. (1978) (see Junker, 1982) vividly illustrate that the resonance wave functions are innately complex functions over and above their dependence on r exp(i+) so that they analytically continue to complex functions of r for 8 = 0. By writing q for r exp(i8) in the wave functions and Hamiltonians, one observes very clearly that the complex energies of the resonance states arise from this innate complex nature of the resonance functions and not from the complexification of the coordinates themselves. As noted at the end of Section 11, the points illustrated by these model potentials are rigorously backed by potential or two-body scattering theory. We shall assume they are also valid for the Coulomb N-particle problem.

RESONANCE CALCULATIONS USING COMPLEX SCALING

227

IV. Variational Principle Few resonance phenomena of interest to atomic and molecular physics admit analytical solutions or can be solved by direct numerical integration. As a result, powerful variational and many-body techniques have been developed as computational tools for bound-state and scattering problems. The similarity between the bound and resonance states, which is demonstrated by the complex-coordinate theorems, suggests that such techniques could also be applicable to the determination of approximate resonance wave functions and eigenvalues. We shall discuss a variational principle and some related concepts in this section and configuration interaction (CI) expansions, self-consistent field (SCF) technique, and many-body methods in the following sections. A. VARIATIONAL FUNCTIONAL

Consider the functional

where xi is defined below. Note that xi of Junker (1982) corresponds to in Eq. (88). Let an( 6 ) be an eigenfunction of X(0). Then F( an) is just the energy, E n , associated with an. If anis changed by an infinitesmal amount, a@,,, and second-order terms in F(an+ 8Qn) are neglected, one obtains

x'*

6F(@n(o))J@L*( e ) @ n ( O ) d r

+ E n [ J 6 @ F ( 8 ) Q n ( 8d)r + J@L*(8)8 1, it will have a nonzero angle cusp. While not being of particular computational value, this result does give some insight into the types of behavior of the 8 trajectories which have been observed in various calculations. 3. ModiJed Doolen-Tvpe Basis

The difficulties encountered using a Doolen-type basis arise from the lack of properly incorporating the nonlinear kinematic &induced oscillations into the basis functions used to represent bound and resonance

RESONANCE CALCULATIONS USING COMPLEX SCALING

239

states and the “bound” part of scattering states of X ( 0 ) .Recognizing this, Junker and Huang ( 1977, 1978) and Junker (1978a) suggested an alternate basis. They represented the bound and the quasi-bound part of the resonance wave function in terms of STOs with the proper argument r exp(i0) and the part describing the scattering particle in terms of functions of the form

x?

=

exp(5ipr) exp(- vr)( 1 - exp( -r))/r

(1 18)

where v ranged over a number of values. The effect of this was to restore an independent-particle picture so that physical and chemical insight could be used in constructing the variational wave function as is the case for bound states. They suggested viewing the wave function as the sum of two parts \I’

=

*Q+ 9

p

where

d is the antisymmetrizer, 4 is a target function, Vv4could be viewed as the quasi-bound part (correlation configurations for shape resonances and closed channels plus correlation for Feshhach resonances) and qpcould be considered the open channel or “continuum” part of the total wave function. Unlike in the normal Feshbach formalism, W p and VQ are not required to be orthogonal since here one always uses the total wave function q. The results they obtained for the three-electron ( 1 ~ 2 s ~He) ~ Sresonance were very significantly more stable with respect to variations in 8 than any previous two-particle calculation. As Table I (Junker, 1978b) illustrates, the real part is stable to six significant figures over a variation of three orders of magnitude in 8 while the imaginary part is stable to four signifiTABLE I

p : , ( ~FOR ) (ls2s2)*SHe- RESONANCE H

- E K (a.u.) - E ,

0.001

2.19075

0.005 0.05 0.08 0.10

6 6 6 6

x

103

(a.u.)

0.01039 0.05 108 0.22000 0.22290 85

0

0.20 0.40 0.60 0.80 I .oo

-ER (a.u.1 -E, 6 6 6 6 6

x 103 (a.u.)

82 82 82 82 87

240

B . R . Junker

cant figures over more than a variation of an order of magnitude in 8. This result is particularly significant since the wave function used contained only 5 1 configurations, whereas far less stability was obtained in previous calculations on two-particle systems with much larger wave functions. Since the nature of various configurations is clear in this construction, the effect of different types of correlation could be studied. In particular, Junker ( 1978b) investigated the effected of improved target-state wave functions on the resonance parameters. Rescigno et 01. (1978) also suggested a technique to circumvent the difficulty with trying to represent the &induced oscillations and applied it to the lowest ( 1 ~ ~ 2 s ~ kBe--shape p)~P resonance. They constructed a wave function of the form

where 4 is a SCF target function for Be in this case. As above + T is considered to be a function of I' exp(ie), while the xi's are taken to be functions of I' only. They obtained a very well-defined approximate position of 0.75 eV and width of 1.11 eV. McCurdy and Rescigno (1979) noted that this technique could be extended to include correlation like the method above by using a multiconfiguration wave function for T. This extension is equivalent as long as the linear coefficients in the multiconfigurational 4 'r are determined in the resonance calculation and not in the neutral atom calculation. The former approach corresponds to treating the full ( N + I)-particle correlation problem, while the latter would correspond to using a sort of N-particle correlated frozen core for the ( N + 1) -particle calculation.

B . SELF-CONSISTENT FIELD CALCULATIONS Complex self-consistent field (SCF) techniques were first discussed by McCurdy et 01. (1980) and later by Froelich (1981). While the derivations of the complex SCF equations emphasized slightly different formalisms, the results and subsequent applications to actual calculations (McCurdy et d . , 1980, 1981; Rescigno et ~ 1 1981; . ~ Mishra et d., 1981b) are identical. Froelich (1981) derives the Hartree-Fock operator, R'. from the standpoint of a biorthogonal variational scheme. That is, for two determinants, Di(&r(sl). . .) and D(rLi(x,). . . ) with

(4114,)

=

a*,

(122)

RESONANCE CALCULATIONS USING COMPLEX SCALING

241

Froelich requires stationarity of the Lagrange functional L ( D ' ,D ) = ( D ' I X ( H ) J D ) / ( D ' I D-)

2

Aij((+il$Jj)

- S,)

(123)

i.j

where Ail are Lagrangian multipliers. This yields a set of equations similar to the ordinary SCF equations, that is,

w e , 4, , . . . ,

,.

.

.)+, =

A*]+,

( 124a)

hj4t

( 124b)

1

R'(H, 41

9

.

*

9

$19

*

*

=

.Mi

j

where

R7= Hi(H) +

c (2Jf(0)

-

KS(H))

( 125b)

j

and J and K are the normal Coulomb and exchange integrals except they are defined with respect to the biorthogonal basis. One then requires a solution of Eq. (124). While we have formally used the notation and approach of Froelish (1981), we do not restrict +i and $Ji to be real functions of I' as he did, which led him to several erroneous conclusions. Before noting some of the results of calculations using Eq. (124), several comments concerning the properties of these equations are in order. First, one must recall that the operators R and R' are state-dependent operators. Froelich (1981) requires that

a:(+)= R(H*)

( 126)

a(e*)= R y e )

(127)

so that R(H) and a'(@merge to the same Hermitian operator, a(,-), as H + 0. Equations (126) and (127) imply that all of the complex nature of the operators a(@) and R'(0) is due to exp(i0). While this is valid for bound states for which f U H ) and W ( H ) should merge to a single operator CUr) in the limit H -,O, this is not valid for resonances. The necessarily complex nature of the resonance functions, even for h, = 0, invalidates Eqs. (126) and (127) in this case. Note that the inclusion of complex basis functions either as complex linear combinations of real basis functions or as single basis functions with complex nonlinear parameters produces an R(0) and R'(0) which do not satisfy Eqs. (126) and (127) and which do not merge to a single i2(1')as

B. R.Junker 8 + 0. In fact, at 8 = 0, the inclusion of Siegert functions could yield operators, O ( r ) and O + ( r ) which , would have the correct properties for determining approximate SCF resonance energies and resonances (an alternate approach is discussed in Section V111,C). While Froelich notes that O ( r ) would contain a potential which is not compact relative to Xo if Siegert functions are used, Section I11 shows that the Hamiltonian, resonance wave functions, and boundary conditions have analytical continuations so that the methods of Section I1 can be used to define well-behaved potentials just as they were used to define well-behaved integrals for Siegert CI calculations. The restriction of constructing Q(8) and a’(@ from real basis functions also led Froelich to erroneously conclude that, if basis functions which depended on r exp(i8) were used, only real energies would result. To the contrary, the SCF equations describing a resonance are complex over and above any dependence on exp(iO), and it is this complex nature of the SCF equations which produces the complex resonance energies. The nonsymmetric transformation of the Hamiltonian, but not the basis functions, in effect substitutes for not explicitly including the necessarily nonlinear complex nature of the exact resonance function, while it also slows down the convergence. The resulting complex linear coefficients must account €or both the lack of explicit inclusion of exp(i8) in the basis functions and lack of explicit inclusion of the innate complex nature of the exact-resonance wave function in the basis functions. (This can also be said of the complex linear coefficients in the CI calculations above.) In particular, McCurdy et al. (1981) have recently reported the convergence of the SCF solution for the 2DCa- resonance using the real Hamiltonian and real basis functions. This is, in fact, equivalent to using Z(0) with real basis functions which depend functionally on r exp(i0). In this calculation the complex linear coefficients only had to account for the necessarily nonlinear complex nature of the exact resonance function. The first application to the computation of resonance parameters was by McCurdy et a / . (1980), who computed the position (0.70 eV) and width (0.51 eV) for the lowest 2PBe--shape resonance. While the 8 trajectories did not have a clear region of stability and some convergence problems were encountered in these initial calculations, these problems have been overcome in more recent calculations (private communication from McCurdy). Subsequently, Rescigno et al. (1981) have applied this SCF technique to the resonance in N;, McCurdy et al. (1981) have computed the resonance parameters of several resonances in Mg- and Ca-, and Mishra rt (11. (1981b) have computed the ground state of Be using the complex SCF equations.

RESONANCE CALCULATIONS USING COMPLEX SCALING

243

VI. Many-Body Theories Winkler (1979) and Mishra et a / . (1981a) discussed a many-body approach, starting with the solutions of the rotated SCF equations. After partitioning the total Hamiltonian into N

N

N

where q t ( 0 ) is the dilated Hartree-Fock operator and Ji(e)and Ki(0) are the dilated Coulomb and exchange operators, respectively, they performed the normal analysis for second-order many-body perturbation theory. This yielded an expression for the self-energy as in bound-state, second-order, many-body theory except that the various operators are dilated and the basis functions used are the solutions of the dilated SCF equation. Mischra er a / . (1981a) note that their result reduces to the normal bound-state expression when @ +. 0. Just as in the discussion of the dilated SCF equations, Dyson's equation is state dependent, and the analytic properties of their operators result from their particular set of basis functions and their procedure for performing the calculation, which necessarily excludes the resonance at 6 = 0. On the other hand, Dyson's equation for the resonance will be necessarily complex at 8 = 0. One way to obtain an appropriate complex Dyson's equation at 19 = 0 is to explicitly impose a Siegert boundary condition (Winkler et a / . , 1981). Palmquist et a / . (1981) have applied this approach to the calculation of several Auger resonances in Be+. An alternate scheme for developing a complex Dyson's equation which does not require the direct imposition of a Siegert boundary condition even at 8 = 0 is suggested in Section XII1,D. Donnelly and Simons (1980) suggested an alternate partitioning scheme. They write the rotated Hamiltonian as

=

XO(r.) +

xye)

( 129)

The advantage of this approach is that one needs to solve the HartreeFock equations only once as opposed to solving them for each value of 0. These basis functions, however, will be a very poor starting point since they do not possess the &induced oscillations which the solutions of the rotated Hartree-Fock equations do include. Consequently, one-electron operators contribute to the many-body scheme. There have not been enough calculations to determine whether the work saved by not having to

B. R. Junker

244

solve the Hartree-Fock equations for each value of 8 is more than the added effort in the rest of the calculation.

VII. Nondilation Analytic Potentials The potentials that have been considered in the previous sections (except for the model potential of Doolen, 1978) have been dilation analytic, that is, the complex-coordinate theorems in Section I11 have been applicable. A number of useful potentials, however, are not dilation analytic since they are not compact with respect to 2’. A. STARK A N D ZEEMANEFFECTS

The Hamiltonian describing an atom in a constant electric field is given by

Since V starkgoesto minus infinity as r i goes to infinity for aiequal to T , this potential is not dilation analytic. Nevertheless, Reinhardt (1976) investigated the application of Doolen’s method for resonances in atoms to the hydrogen atom Stark problem. He used a wave function of the form I.

N,

Where the x f ’ s were orthonormal Laguerre-type functions and transformed the Hamiltonian of Eq. (130) only by Eq. (28). The resultant 0 trajectories had the same behavior as in atomic resonance calculations, and the results were in excellent agreement with earlier calculations using other techniques (e.g., Hehenberger, 1974). In fact, the calculations for different values of L and Nl in Eq. (131) suggest that Reinhardt’s results are probably more accurate. Subsequently, Wendoloski and Reinhardt (1978) used the same techniques to determine the effect of an electric field on the width of the 2s2p H--shape resonance. The fact that one need not explicitly impose an asymptotic boundary

RESONANCE CALCULATIONS USING COMPLEX SCALING

245

condition is a considerable advantage here. In “square” parabolic coordinates x = pu cos

c#l

(132a)

y = pu sin

c#l

( 132b)

z = i(p2

-

u2)

(132c)

Damburg and Kolosov (1976) give the asymptotic form of the solution of Eq. ( 130) for the hydrogen atom as (133a)

M Y )- ( B / v ) ~ i n [ ( F ’ / ~ / 3 +) v(Er/F1/’) ~ v + x]

(133b)

where A , B, and x are constants depending on E,and F. For the general N-particle problem in spherical coordinates this would imply an asympto, a i , and c0s3l2ai. tic form with exponentials containing r:I2, r % / 2 COS’/~ Even if one attempted to formulate a Siegert calculation in terms of ( p , u ) coordinates, very complicated integrals requiring numerical integration would result. Cerjan rt (11. (1978a,b) have given a heuristic argument for using complex coordinates in the Stark problem, while Herbst (1979), Graffi and Grecchi (1980), and, particularly, Yajima (1982) have provided a rigorous foundation for such an approach. In the former two articles the concept of the numerical range of an operator is used to show that under the transformation (28) with 0 s 8 s 7r/3 the continuous spectrum of the hydrogenic Hamiltonian with a constant electric field is a subset of the lower half of the complex plane bordered by a line passing through the origin and rotated up from the positive energy axis by 8. Thus, resonances associated with the field-free bound states may be expected to be exposed. Herbst (1979) has, in fact, shown that the rotated Stark hydrogenic Hamiltonian has no continuous spectrum. Herbst and Simon (1981) have generalized these results to N-particle systems. The inclusion of magnetic fields in Schrodinger operators and the resultant dilation analyticity is discussed by Avron et nl. (1978, 1981). Then Chu (1978a) calculated the level shifts and field ionization widths as a function of the ratio of the field strengths for a hydrogen atom in crossed electric and magnetic fields using a Doolen-type basis.

B. MULTIPHOTON IONIZATION An atomic system in a laser field experiences a temporally periodic electric field. The Floquet theorem (Shirley, 1965) asserts that quasi-

246

B. R. Junker

energy-state solutions, q E ,of the form qE= exp(-iEt)@,(r, 1 ) exist, where (PE(r.t ) is periodic in time. For a discussion of the existence of such solutions see Chu (1978b) and Salzman (1974). Nevertheless, assuming such a solution exists and transforming to a coordinate system that rotates 1 ) as (Chu, with the field (Manakov et d.,1976), one can write aE(r, 1978b)

aE(r, t ) = exp(iwtL,)c$,(r)

( 134)

where w is the frequency of the field,

and Q is the quasi-energy operator whose form depends on the radiation field. Chu and Reinhardt (1977), and Chu (l978b, 1979), have applied the transformation (28) to the quasi-energy operator Q to determine the multiphoton ionization of hydrogen atoms including hydrogen atoms in an additional constant magnetic field. They used a Doolen-type basis and obtained very well-converged results. More recently, Chu (1981) has applied this formalism to molecular photodissociation of a model molecular system.

C. C U B I CA N H A R M O N OSCILLATOR IC MODEL Spectroscopic data on diatomic molecules is referenced to anharmonic corrections to a harmonic oscillator (Herzberg, 1950)of which the simplist such correction is a cubic anharmonic oscillator. A cubic term with a negative coefficient, however, produces a potential which is not dilation analytic (see, however, Caliceti et nl.. 1980). Nevertheless, Yaris et d. (1978) used a Doolen-type basis to compute the shape resonances that would be expected for such a potential. Their results were in very good agreement with a WKB calculation of the same resonance states. Note, however, that while this calculation with the cubic potential illustrated the power of these techniques to compute resonance parameters, the cubic potential itself has a number of undesirable properties, such as the fact that a particle goes to infinity in a finite time, which make this potential physically objectionable (Simon, 1982). Yaris et d.(1978) strongly suggested that these results along with those of Reinhardt (1976) imply that these numerical techniques have a much wider range of applicability than to just dilation analytic potentials.

RESONANCE CALCULATIONS USING COMPLEX SCALING

247

VIII. Complex Stabilization Method The dependence of the wave function for bound and resonance states on exp(i8) is a symmetry constraint imposed on the wave function by the analytic structure of the Hamiltonian and boundry conditions just like other symmetry constraints such as permutational and point or continuous group symmetries (Junker, 1982). This kinematic dependence on CYI’ exp(i8) is, in fact, responsible for the independence of the energies of these states on CY and 8. As noted in Section I, the complex energy of the resonances results from the innate nonlinear complex structure of the corresponding wave function and not from the transformations (28) and (32). In fact, since the Hamiltonians X ( r )and X ( 8 ) , along with the boundary conditions for the bound and resonance states, are related by a similarity transformation, one should expect that X ( 8 ) should have many of the properties of the “Hermitian” Hamiltonian X ( r ) as opposed to the general properties of a complex non-Hermitian Hamiltonian, which are often given for the reasons for the occurrence of these complex energies. The complex boundary conditions on the exact improper eigenfunctions describing resonances for X ( r ) in effect make X ( r ) non-Hermitian. The effect of the transformations (28) and (32) is to transform the asymptotically divergent improper eigenfunction into an asymptotically convergent proper eigenfunction just as the bound states are proper eigenfunctions for n/2 > H > -7712. but improper eigenfunctions for 3n/2 > 8 > n/2 or -3n/2 < 8 < - 7 1 2 . As noted in Section V, the computation of 8 trajectories and stabilization of the resonance eigenvalues with respect to variations in 8 by means of a variational principle is simply determining the value of 0 for which the &independent basis best approximates the exact &dependent wave function. That is, 8 is, in effect, playing the role of a nonlinear variational parameter. Two recent calculations illustrate this clearly. Donnelly and Simons (1980) computed the complex energy for the lowest 2P-shape resonance in Be-. The value of 8 at which the complex eigenvalue, E ( 8 ) , stabilized was less t l m t Arg(E,) (Bardsley, 1980). In addition, McCurdy ct NI. ( 198 1) were able to converge the complex SCF equations to a stable complex root for the 2D Ca--shape resonance at 8 = 0 using the real Hamiltonian with real basis functions with complex linear coefficients. All of these calculations used square-integrable basis functions without an explicit imposition of any complex boundary conditions. As a third illustration of the fact that an explicit boundary condition need not be imposed, regardless of the value of 0 (Junker, 1980a,b), consider C Y ~

B. R. Junker

248

the calculation of Doolen (1975) for the ( 2 ~ )H-~Feshbach ~s resonance. He used a wave function of the form

YO=

c. c l m , ( ~ ) ( r : r+r ryr;)ry2 exp[-u(r, +

r2)1

( 136)

1,m.n

where the c,,,(O) were determined variationally. However, as we noted in Section III,B the bound-state and resonance wave functions are analytical functions of 8 so that an equivalent wave function at some value H2 can be obtained from the wave function at some value O1 by the transformation (38) or (39). Consequently, if the resonance energy stabilizes at 8, for the wave function in Eq. (136), a completely equivalent wave function (in that it yields exactly the same eigenvalue) at O equal to zero is

Y O=

C

c l m n ( ~ , ) [exp(-iel))Yr2 (rl exp(-iO,))m

1,m.n

+ ( r 2 exp(-i~,))"(r2exp(-iO,))'](r,, x exp[-tr exp(-iO,)(r,

exp(-iO,))"

+ r,)]

(137)

Note that Y nis square integrable even though we have indicated above that the exact resonance wave function at O = 0 diverges asymptotically exponentially. This result is true of all previous complex-coordinate calculations. This implies that as long as the expansion (136) converges for X(O), the expansion (137) should converge for X ( r ) . Thus, variational approximations to the resonance energy can be computed using X ( r ) with a square-integrable basis and without explicitly imposing any boundary conditions such as a Gamow-Siegert boundary condition. In fact, the Siegert calculations in Section V,A,l are simply special cases of such calculations with X ( r ) in which a boundary condition is explicitly imposed. However, the model potential calculations of Junker (1981, 1982) illustrate that the exact asymptotic form of the exact wave function is not needed. Morgan and Simon (1981) considered the Weierstrass transform of V(r), that is,

I

V c ( r )= ( 2 7 r ~ ) - ~ 'exp(-lr12/2c)V(x ~

-

r)dr,

E

>0

(138)

They showed that eigenvalues, E f , of XYO) converge to eigenvalues E of X ( O ) since the potential VYO) converges uniformly to V(0) for the Coulomb potentials in atomic and molecular problems. In addition, the properties of VTr) imply that the matrix element ($( - O)l Xcl JI( - 0 ) ) exists for any finite N-matrix approximation and the uniform convergence of V q r ) to V(r) implies that E j -+ EN as E goes to zero. Then assuming, as

RESONANCE CALCULATIONS USING COMPLEX SCALING

249

above, that Eq. (136) converges, one has that scaling the parameters as opposed to the Hamiltonian should yield the same spectrum. This is, in effect, a realization of Simon’s (1979) exterior complex scaling.

STABILIZATION PROCEDURE A. COMPLEX As a result of the above arguments, Junker (1980a,b, 1982) suggested the following procedure for computing the complex energies for resonances using the unrotated Hamiltonian with a square-integrable basis without explicitly imposing any boundary condition such as given by Eq. (ISb). Instead of computing 6, trajectories, one would compute nonlinear parameter trajectories and stabilize E ( y ) with respect to variations in the nonlinear parameters y i . The stabilization could be performed by scaling each parameter individually, scaling groups of parameters, or globally scaling all parameters simultaneously. This scaling can either be a real scaling or a complex scaling. Before discussing applications of such a procedure, we should make several comments and observations. Since we are using X ( r ) , either some of the basis functions must be complex or some of the nonlinear parameters must be complex scaled. This is to be expected since, as we ernphasize again, the exact resonance wave function is necessarily nonlinearly complex. Interpreted in this manner, Doolen-type calculations are seen to correspond to a global complex scaling of all parameters simultaneously, while the modified Doolen-type calculations correspond to complex scaling only those basis functions describing the unbound electron. That is, in the latter cases all other basis functions are considered functions of ar exp(i0). Consequently, scaling an individual nonlinear parameter y i is equivalent to considering all other basis functions to be functions of a i y j exp(iOj). Siegert calculations correspond to including specific complex basis functions. Certainly, the most general procedure would be to stabilize E ( y ) with respect to a complex scaling of each nonlinear parameter individually. Except for certain simple cases, such a procedure is too time consuming and, fortunately, is generally not necessary from a practical point of view. Although the N-particle atomic and molecular Hamiltonians are not separable, the complex part of orbitals representing inner shells, for example, can be expected to be very small and probably accountable through the complex linear variational coefficients. In addition, one could use the approximate wave function q,.from the variational calculation to define a final global complex scale factor 7) such that the virial theorem

250

B. R . Junker

Eq. ( I 10) is satisfied. For the Coulomb potentials in atomic calculations, the virially optimized resonance energy is then E,v = - (+';.IVpPtP)2/(4 (q2pr))

(139)

just as for bound states (Lowdin, 1959). An iterative application of the virial expression as discussed in Section V,A,2 is not expected to be necessary since qrshould already be a good approximate wave function. Since the earlier calculations discussed in Section V are just special cases of this more general complex stabilization method and the modifications of Doolen's method appear to have solved the convergence problem for general N-electron atoms, one might ask the need for considering a more general scheme. There are, in fact, a number of computational advantages (Junker, 1981, 1982). First, since 0 in these earlier calculations is just a nonlinear scale factor, values of 8 other than just /3 < 0 7r/4 are meaningful. Second, a number of different complex scale factors can be used to describe, for example, different (s, p, d, etc.) polarization contributions or coupling to several lower continua. Third, BO Hamiltonian offers no difficulty since the Hamiltonian is not scaled directly. Fourth, the difficulty in describing the unnecessary kinematic &induced oscillations in the exact bound- and resonance-state wave functions of X ( 0 ) is eliminated when X ( r ) is used. Fifth, since X ( r ) is used, configurations for a variational wave function can be constructed which incorporate physical and chemical information such as spatial and angular correlations, polarization, etc. Sixth, basis sets with only a small number of complex functions or caleulations in which only a small number of basis functions are complex scaled are possible. While Doolen-type bases do not require any complex integrals be evaluated for Coulomb potentials, the calculations are restricted to two-particle systems, and even then very large basis sets are required. While these calculations with more general scalings require the computation of integrals with complex basis functions (actually all integrals except certain two-electron integrals and certain two-center integrals can be transformed to real integrals with a complex multiplicative factor), these complex integrals present no problem since they can be evaluated by the same techniques as the corresponding real integrals. The ideas presented here in no way are aimed at negating previous complex-coordinate calculations. On the contrary, they are simply aimed at showing that much more flexibility is permitted in the construction of variational wave functions and in the performance of variational calculations than was used or thought to be allowed in traditional complexcoordinate calculations. In the following sections we shall discuss how these considerations can be incorporated into CI, SCF, and many-body calculations.

RESONANCE CALCULATIONS USING COMPLEX SCALING

25 1

B. CONFIGURATION INTERACTIONCALCULATIONS Configuration interaction calculations using these ideas have been performed on a model potential (Junker, 1981, 1982) and the ( l s 2 ~ ~ He)~S Feshbach resonance (Junker, 1980b), while preliminary calculations have been reported on the lowest 2P Be--shape resonance (Junker, 1981) and field ionization of the ground state of hydrogen (Junker, 1981). In discussing these calculations we shall give some of the calculational strategies and reasons for them. For the model potential (Junker, 1982) ~ ~ V(r) = 7 . 5 exp(-r)

(140)

wave functions of the form (141a) i=1

* z ( N ) = q l ( N )+

CN+~Y-'(1

-

exp(-ar)) exp(ik,r) exp(k,r) (141b)

V 3 ( N )= Vl(N) + cN+lexp(ik,r) exp(-k,r)

(141c)

N

v,(N) =

2

exp(-y,r)

~ ~ r ~ i - 1

i= 1

+ exp(ikr)[cN+lexp(-ar) + c

~ exp(-(a + ~

+ E)T)]

(141d)

were used with the real Hamiltonian. Values of N = 10 and 14 were considered for the first three and N = 10 and 12 for the last one. The first corresponds to a Doolen-type calculation; the second contains in addition a Siegert function; the third contains a function which, unlike the Siegert function, is square integrable and does not contain r-l; the last wave function is just the sum of real functions augmented by two squareintegrable complex functions. Although y was optimized, the first variational wave function yielded very poor results in all cases until 0 = 0.5 radians. On the other hand, the second and third wave functions produced virtually identical results, at all values of 8,which were very good even down to 8 = 0. This illustrates the unimportance of the exact asymptotic form of the wave function. In the case of the latter wave function, all of the parameters y i k . a,and E were taken to be real, and only real scalings were used even though complex scalings would have given more flexibility. Thus, the only complex integrals required were those involving the last two basis functions. All N + 2 linear coefficients were determined by a variational calculation. The form of the variational wave function (141d) was based on the follow-

.

252

B . R . Junker

ing ideas. The oscillations in the exact resonance wave function in the region of the potential will not be regular and will depend on both the energy and the potential. Consequently, real functions might be more appropriate in representing this portion of the wave function than complex functions with regular oscillations. Some number of complex functions must be added to represent the complex nature of the exact wave function. By choosing E small, the variational calculation can reduce the radial distance over which the regularly oscillating functions are nonzero if cN+, is approximately - c ~ + ~In. all calculations that have been performed with variational wave functions of the form of Eq. (141d), this has been the result. The approximate resonance eigenvalue E,. ,was stabilized with respect to the individual parameters y i , X , and the pair [a. ( a + E)] individually but with respect to only real scalings. Since €r is a hypersurface with respect to the nonlinear parameter space, all of the problems associated with the optimization of nonlinear parameters which occvr in normal bound-state variational calculations are, of course, present in these calculations. In fact, the only difference between the manner in which parameters are optimized in bound-state calculations and these calculations is that in the former nonlinear parameter trajectories are computed and points of minimization of the energy are located, while in the latter nonlinear parameter trajectories are computed and points of stabilization of the energy are located. Also in the former only real scalings are required, while in the latter complex scalings are, in general, applicable. Since the imaginary part of Er is strongly dependent on E, a , and k , €, is stabilized (both the real and imaginary parts simultaneously) with respect to these parameters first. Then the structure projections Pi for each basis function xi is computed, where

P, =

1 (x*l%)l

(149,)

The nonlinear parameters y i are then optimized in order of decreasing Pi so that the most important parameters are optimized first. Stabilization with respect to nonlinear parameters with very small Pi’s sometimes cannot be obtained because they are too unimportant to have much of an effect of E,.. Typical nonlinear parameter trajectories fork and one of the yi’s are given in Tables I1 and I11 (Junker, 1982). If desired this procedure can be iterated. Just as in bound-state calculations, one must avoid pathological behavior in the nonlinear parameter trajectories which can result from “numerical” linear dependence during the variation of the nonlinear parameters. A final global scaling of the nonlinear parameters for the real functions can also be performed. Again, as in bound-state calculations,

RESONANCE CALCULATIONS USING COMPLEX SCALING

253

TABLE 11 E,(k )

~~~~

~

I S4000 1.63625 1.73250 1.82875 I .92500 2.02125 2. I1750 2.21375 2.3 1000

-7.39 -5.86 -3.74 - 1.52 0.69 2.93 5.32 8 .OO

0.55 1.91 1.62 0.64 -0.32 -0.85 -0.70 0.19

‘I A E , and AkI are the differences between a.u. The successive values of %, in units of value of 6, ( X = 1.925) is 3.42639 - 0.012758i a.u.

the larger the basis set and number of configurations is, the less sensitive Er is to any given value of any given nonlinear parameter. For the above model potential the wave function (142d) with N = 10 yielded 3.426397 - 0.0127641 a.u., while the one with N = 12 produced 3.426395 - 0.0127751 a.u. The resonance energy from a numerical integration of the Schrodinger equation produced 3.42639 - 0.0327751 a.u. (Bain ct LII., 1974).

3.0912 3.1584 3.2256 3.2928 3.3600 3.4272 3.4944 3.5616 3.6288

2.07 I .75 1.38 1.11

I .02 1.19 1.70 2.60

-3.98 -2.29 - 1.46 - 1.21 - 1.37 - 1.83 -2.51 -3.29

‘I A E I , and A,!?, are the same as in Table 11. E , ( y , = 3.36) is 3.42638 - 0.012757i a.u.

B. R. Junker

254

For the ( l s 2 ~ ~ He) ~ Sresonance calculations, the part of the wave function representing GQwas taken from a previous calculation (Junker, 1978b). Two different representations were used for q p:

(143b) As noted above, if

E is small, the complex functions only contribute in a small radial region around a-l for qb. On the other hand, the complex function in @! contributes over a much larger region around a-l. Thus, while both wave functions gave identical results for Er, the results from the first one were far more stable. In order for these calculations to be useful, one must be able to distinguish the eigenvalues approximating resonance energies from those approximating cuts. In these calculations the latter eigenvalues do not lie approximately on straight lines rotated down by 28, but do lie on well-defined curves (see Figures 1 and 2 of Junker, 1980b). In addition, although the real parts of the approximate continuum eigenvalues near the thresholds are very similar for @b and @,; the imaginary parts are entirely different. This is in contrast to the Er'sfor the two wave functions. The *P Be- calculations illustrate the usefulness of multiple complexscale factors. In this case polarization correlation is critically important. Consequently, a variational wave function of the form (Junker, 1981)

+

2 P[cfk'cp(1s22s"3d;,D)xLP[r exp(-it),)] k=l

+ ~.k!++.~~q( ls22s"3d; ID)xLp[rexp( -&)]I}

(144) was used where P denotes Clebsch-Gordon coupling of the products of the q ' s and x's to produce a function with overall 2Psymmetry. Here we have used a combination of modified Doolen-type bases for representing the various correlations for the unbound particle permitting the complexscale factor to be different for the different components. The optimum value for HI was found to be about 0.65 radians, while both O2 and 0, had

RESONANCE CALCULATIONS USING COMPLEX SCALING

255

optimum values of 0.005 radians. The small values for O2 and O3 probably reflect the fact that these contributions correspond to closed channels. E,. was also stabilized with respect to the nonlinear parameter of the 2s basis function. As to be expected, the 2s functions expanded radially relative to the neutral atom to accommodate the additional electron. This same behavior is evident in a comparison of the calculations of Rescigno et d . (1978) with those of McCurdy L't NI. (1980). The addition of polarization reduced the position and width to 0.27 and 0.22eV, respectively, for the wave functions used, which is in reasonable agreement with the empirical polarization calculations of Kurtz and Orhn (1979). The wave function used for the calculation of the field ionization of the ground state of hydrogen was

+ r' exp(iklr)[d: exp(-arr) + dk exp(-al + e J r ) ] }

(145)

E r was stabilized with respect to each group of parameters k , , [ a r ,(ar+ e l ) ] , and [ y f , i = I , N r ] for each I separately. Two wave functions were used-one with L = . 3 and one with L = 6. The former produced -0.517598 - 0.0023361 a.u., while the latter yielded -0.517561 0.002270i a.u. This can be compared with the results of Hehenbergeret nl. (1974), which gave -0.51756 - 0.00227i a.u. These examples should illustrate the potential flexibility in constructing variational wave functions and performing the variational calculations.

C. SELF-CONSISTENT F I E L DCALCULATIONS As with all calculations discussed in Section V, an equivalent formulation of the SCF calculations using X ( r ) instead of X ( 8 ) is straightforward. For example, the Be- calculations of McCurdy et al. (1980) represent one such formulation. Alternately, they could have constructed a single Slater determinant of the form in Eq. (143) and computed a k trajectory as opposed to a 8 trajectory. J. McNutt and C. W. McCurdy (personal communication) computed the resonance parameters for the IP Be- resonance using X ( r ) with a variety of basis sets for the unbound particle. The basis sets contained from five to ninep functions and were performed at 8 = 30,40, 50, 60,70, and 80". In one case, denoted I, all nonlinear parameters for thep functions were complex scaled, while in the other case, denoted 11, the four tightest functions were not scaled, that is, they were kept real. Some of the results are given in Table IV, which contains the nonlinear parameter trajectories

B . R . Junker

256

TABLE IV

COMPLEX SCF CALCULATIONS F O R (ls22s2kp)'P Be- RESONANCE 0 = 30"

Basis set"

-El
0 and E > 0 is also rigorous. While the variational principle discussed in Section IV is, of course, well founded, the same cannot be said of the simple statements relating the stabilization points of 0 and nonlinear parameter trajectories to regions of maximum overlap of the approximate solutions with the exact solutions (Junker, 1980b, 1981; Atabek et NI., 1981). Along the same lines, the convergence of an expansion such as given in Eq. (137) to the exact function is an assumption. The demonstrations of the relation between using a complex-scaled X ( r ) with a real basis and a complexscaled basis with X ( r ) by Junker (1980a) and Morgan and Simon (1981) rest upon this assumption. If it is valid, one can use X ( r ) with complex basis functions; if it is not, neither X ( r ) or X ( 8 ) are rigorously useful for determining approximate resonance energies and wave functions. Since the approach taken in this article has basically been from a heuristic point of view, we shall here briefly note the results which eventually led to the general calculational procedures discussed in Section VIII. Certainly the elegant results of Aguilar and Combes (1971), Balslev and Combes (1971), and Simon (1972, 1973) initiated interest along this direction. The suggestion by Doolen (1975) that one compute 8 trajectories and associate points of stability on these 0 trajectories with approximations to the resonance energy was a significant computational development which motivated much of the work which followed. The subsequent fieldionization calculations of Reinhardt (1976) and cubic anharmonic oscillator model calculations of Yaris el al. (1978) strongly suggested that these calculational techniques were applicable to a far more general class of potentials than dilation analytic potentials, and indeed a number of theoretical results have addressed various classes of more general potentials, for example, the Stark problem (Herbst, 1979). The analysis and

258

B . R . Junker

calculations of Junker and Huang (1977, 1978), Junker (1978a), and Rescigno et a/. (1978) provided the insight necessary to extend these calculational techniques to systems with arbitrary numbers of particles. The calculations of McCurdy and Rescigno (1978) and Moiseyev and Corcoran (1979) in which they performed the calculations complex scaling the basis functions instead of the Hamiltonian were significant not so much for their potential application to molecules as for the implied generalization of the computational techniques. Again, the external complex scaling discussed by Simon (1979), while aimed at facilitating molecular calculations, was most significant in its generalization of the original complex-scaling theorems. These earlier developments motivated the development of the complex stabilization method and the use of nonlinear parameter trajectories suggested by Junker (1980a) and the proof of the validity of the complex basis-function calculations by Morgan and Simon (1981) and Junker (1980a). Finally the calculations of Donnelly and Simons (1980) and particularly those of McCurdy et al. (1981) clearly strengthen the argument that 8 is merely playing the role of a nonlinear parameter in traditional complex coordinate calculations. The above discussion is not intended to be an exclusive list of the significant developments in complex scaling, but is instead aimed at those developments which were most significant in the development of the techniques for computing resonance wave functions. Indeed, a number of significant theoretical results have been based on the complex-scaling theorems. These include, for example, Reinhardt’s (1977) use of the dilation transformation to put rigorous constraints on the possibility of the existence on bound states in the electronic continua and the demonstration by Simon (1973) and Hunziker (1977) of the absence of bound states and resonances above the threshold for complete breakup [note, however, that several experiments have been interpreted in terms of resonances above the threshold for complete breakup (Walton et d., 1971 ; Taylor and Thomas, 1972; Peart and Dolder, 1973)l. In addition, we have not discussed the spectral representation of the Green function nor those calculations which employ such a representation. The computation of photoabsorption and photoionization cross sections (Rescigno and McKoy, 1975; Rescigno et d.,1976; Sukumar and Kulander, 1978; McCurdy and Rescigno, 1980)are examples of such calculations. Another area, which we did not discuss and which also makes use of the wave functions determined by the methods we considered in the previous sections, is the determination of partial widths (Yaris and Taylor, 1979; McCurdy and Rescigno, 1979; Noro and Taylor, 1980). Finally, we have tended to emphasize calculations which do not explicitly impose a Siegert boundary condition since calculations tend to strongly indicate that it is unnecessary to

RESONANCE CALCULATIONS USING COMPLEX SCALING

259

explicitly impose such a boundary condition and it is certainly undesirable to do so in many calculations such as field ionization and molecular resonances. It may, however, prove useful to extract the Siegert components from the approximate wave function after the calculation. Finally, we should note that the complex-scaling transformations (28) and (32) which we have discussed are but one of the types of transformations which have been investigated theoretically (Simon, 1978). Complex boosts (Combes and Thomas, 1973) correspond to the transformation ( 146a)

x+x p j p - a ,

a E C

(146b)

The real part of a shifts the thresholds for the cuts while the imaginary part converts a cut into a parabola. A generalization of Eq. (146) to (147a)

x+x p

+

p-

(I

Vf

(147b)

has been discussed by Simon (1975), Deift et a / . (1978), and Herbst and Simon (1981). Coordinate translations, X--,X+U,

UEC

(148)

have been discussed by Avron and Herbst (1977) and Herbst (1980) and applied to a model problem by Cerjan et (11. (1978b). Finally, we have also not considered the large amount of literature on complex angular momentum and Regge poles (Regge and De Alfano, 1965, and references therein; Sukumar and Bardsley, 1975; Sukumaret ul., 1975). One could, of course, then consider combinations of these various transformations. In conclusion, we have attempted to present the properties of Hamiltonians and wave functions under a dilation transformation and techniques, whose validity rest on these properties, for determining approximate complex resonance energies and wave functions. These methods closely resemble bound-state calculations, but are directed to approximating a necessarily complex function. This leads to a loss of the upper bounding property for an analogous variational principle and of rigorous convergence properties of the basis-set expansions.

ACKNOWLEDGMENTS I am especially indebted to Joe McNutt and Bill McCurdy (Ohio State University) who performed the SCF calculations discussed in Section VIII and to Barry Simon (California

260

B. R . Junker

Institute of Technology) for critical reading the manuscript, making many useful comments, and supplying many references. I would also like t o thank Bill McCurdy and Norman Bardsley (University of Pittsburgh) for reading the manuscript and giving useful comments on the manuscript.

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B . R . Junker McCurdy, C. W., Rescigno, T. N., Davidson, E. R., and Lauderdale, J. G. (1980). J . Chem. Phys. 73, 3268. McCurdy, C. W., Lauderdale, J. G., and Mowrey, R. C. (1981). J. Chem. Phys. 75, 1835. Mahaux, C. (1965). N r d . Phys. 68, 481. Manakov, N. L., Ovsyannikov, V. D., and Rappaport, L. P. (1976). Sorr. Phyc. JETP 43, 885. Mishra, M., Froelich, P., and Ohm. Y.(1981a). C h c m . P h y s . L e f t . 81, 339. Mishra, M., Ohm, Y.,and Froelich, P. (1981b). Phys. Lett. 84A, 4. Moiseyev, N. (1981a). M o l . Phys. 42, 129. Moiseyev, N. (1981b). Preprint. Moiseyev, N., and Corcoran, C. (1979). Phys. R e v . A 20, 814. Moiseyev, N., and Weinhold, F. (1980). I n r . J. Qrtcrutrrm Chem. 17, 1201. Moiseyev, N., Certain, P. R., and Weinhold, F. (1978a). M u / . Phys. 36, 1613. Moiseyev, N., Certain, P. R., and Weinhold, F. (1978b). Int. J . Qrtantrtm Chem. 14, 727. Moiseyev, N., Friedland, S., and Certain, P. R. (1981). J. Chem. Phys. 74, 4739. Moler, C. G., and Stewart, G. W. (1973). S l A M J. Nrrmer. Ancrl. 10, 241. Morgan, J. D., 111, and Simon, B. (1981). J. Phys. B 14, L167. Newton, R. G. (1960). J. Math. Phys. I, 319. Newton, R. G. (1966). “Scattering Theory of Waves and Particles.” McGraw-Hill, New York. Nicolaides, C. A., and Beck, D. R. (1978a). Phvs. Lett. 65A, 11. Nicolaides, C. A., and Beck, D. R. (1978b). I t i t . J. Qrtatitrtrn Chem. 14, 457. Nicolaides, C. A., and Beck, D. R. (1978~).In “Excited States in Quantum Chemistry” (C. A. Nicolaides and D. R. Beck, eds.), p. 383. Reidel, Dordrecht. Noro, T., and Taylor, H. S. (1980). J . P h y . B 13, L377. Palmquist, M., Altick, P. L., Richter, J., Winkler, P., and Yaris, R. (1981). Phys. Rev. A 23, 1795. Peart, B., and Dolder, K. T. (1973). J . Phys. B 6, 1497. Reed, M., and Simon, B. (1978). “Methods of Modem Mathematical Physics: Vol. IV, Analysis of Operators.” Academic Press, New York. Regge, T., and DeAlfaro, V. (1965). “Potential Scattering.” North-Holland Publ., Amsterdam. Reinhardt, W. P. (1976). I t i t . J . Qirantrrm Chem. S 10, 359. Reinhardt, W. P. (1977). Phys. Rev. A 15, 802. Reinhardt, W. P. (1982). Ann. R e v . Phys. Chem. (in preparation). Rescigno, T. N., and McKoy, V. (1975). Phys. R e v . A 12, 522. Rescigno, T. N., McCurdy, C. W., and McKoy, V. (1976). J. Chem. Phys. 64, 477. Rescigno, T. N., McCurdy, C. W., and Orel, A. E. (1978). Phjls. R e v . A 17, 1931. Rescigno, T. N., Orel, A. E., and McCurdy, C. W. (1981). J . Chem. Phys. 73, 6347. Romo, W. J. (1968). Nrtcl. Phys. A 116, 617. Saltzman, W. R. (1974). Phys. R e v . A 10, 461. Shirley, J. H. (1965). Phys. Rev. B 138, 979. Siegert, A. F. J. (1939). Phys. R e v . 56, 750. Sigal, I. (1978a). Bull. A m . Moth. Soc. 84, 152. Sigal, I. (1978b). Metn. A m . Math. SOC. 209. Simon, B. (1972). Commrtn. Mirth. Phys. 27, I. Simon, B. (1973). Ann. Math. 97, 247. Simon, B. (1975). Trans. Am. Math. Soc. 268, 317. Simon, B. (1978). Itit. J. Quuntum Chem. 14, 529. Simon, B. (1979). Phys. L e u . 71A, 211.

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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS, VOL. 18

DIRECT EXCITATION IN ATOMIC COLLISIONS: STUDIES OF QUASI-ONE-ELECTRON S YSTEMS N . ANDERSEN Physics Laboratory I1 H . C . 0 r s t e d Institute University of Copenlingen Copenhogen Denmark

and

S . E. NIELSEN Chemistry Labomtory 111 H . C . 0 r s t e d Institute V n i l w s i t y of Copenhagen Copenhagen. Denmark

I. Introduction . . , . . . . . . . . A. Background and Early Ideas . . B. Collision Systems . . . . . . . C. Qualitative Considerations . . . 11. Theoretical Models . . . . . . . . A. Atomic Basis , . . . . . . . . B. Molecular Basis , , . . . . . . 111. Experimental Techniques . . . . . A. First-Generation Experiments . . B. Second-Generation Experiments C. Third-Generation Experiments . IV. Results and Discussion . , . . . . A . Optical Spectra . , . . . . . . B. Total Cross Sections . . . . . . C. Polarizations . . . . , . . . . D. Excitation Probabilities . . . . E. Coherence Analysis . . . . . . V. Conclusions . . . . . , , , . , . References . . . . . . . . . . . .

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266 266 267 268 271 272 278 279 280 281 282 287 288 289 293 2% 298 303 305

265 Copyright @ 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

ISBN 0-12-00381&8

266

N . Andersen and S . E. Nielsen

I. Introduction A. B A C K G R O U NADN D E A R L Y IDEAS

The study of inelastic processes in binary atomic collisions has a long history. More than 50 years ago elastic scattering experiments of alkali ions on rare gases (Ramsauer and Beeck, 1928) were extended to ionization studies in Ramsauer’s (Beeck, 1930) and Millikan’s (Sutton and Mouzon, 193 1) laboratories, and the results interpreted in terms of potential curve crossings in the transiently formed quasi-molecule (Weizel and Beeck, 1932). Stueckelberg’s ( 1932) theoretical analysis of the curvecrossing phenomenon gave results in good agreement with experiment. Optical spectrometric studies of the collision-induced radiation were also taken up (Hanle and Larche, 1932; Maurer, 1936). The early developments are summarized in interesting review papers by Beeck (1934) and Maurer (1939). In 1949 Massey extracted experiences from many previous inelastic collision studies (e.g., London, 1932; Massey and Smith, 1933) in formulating a general adiabatic criterion for an inelastic process with energy defect AE. The process is unlikely to occur if the collision time, estimated as the ratio between an effective interaction length N and the internuclear velocity I * , is much longer than the time corresponding to the natural frequency M/h of the process, or

PEalhv

>> 1

(1)

Based on this idea Hasted (1951, 1952, 1960) analyzed the velocity dependence of a large number of cross sections, notably for charge exchange processes, and found that the maxima occurred when the left-hand side of Eq. ( 1 ) is unity, estimating ( I , the effective length, by 7-8 A. The application of electron spectroscopy (Blauth, 1957; Moe and Petsch, 1958) and collision spectroscopy, i.e., energy-loss analysis of the particles scattered at a fixed angle (Afrosimov and Fedorenko, 1957; see also Ramsauer and Kollath, 1933), together with the availability of large Hartree-Fock computer codes for quantum chemical calculations of molecular properties led to further advances. The MO (molecular orbital) model (Fano and Lichten, 1965; Lichten, 1967; Barat and Lichten, 1972) became the common framework for a detailed understanding of many inelastic processes in the “molecular” region, where the internuclear velocity u is smaller than the velocity u, of the electron(s) responsible for the inelastic processes which occur as one or several transitions between molecular potential curves at characteristic internuclear distances.

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In the high-energy limit ( u >> v , ) most cross sections decrease rapidly, and Born-approximation calculations normally fit the experimental data well (Bell and Kingston; 1974). In the intermediate-energy region ( u = u,) the experimental cross sections may have large values (de Heer, 1966). Theoretical progress in this region, where many channels often interact strongly, has been relatively slow and the gain in understanding accordingly rather modest, often not much beyond the Massey criterion (1) and the maximum rule. In the monumental work Electronic and tonic Impact Phenomenci, Massey and Gilbody (1974, Vol. IV, p. 3122) summarized the situation as follows, “. . . there is need for further experimental study, particularly of simple cases, with much improved precision, particularly for excitation, in order to determine the validity of different theoretical approximations.” SYSTEMS B. COLLISION A reasonable starting point for a deeper insight into the mechanisms responsible for the inelastic events, including the interesting mediumenergy range, would thus be to study collision systems with few active electrons-preferably just one-and a small number of reaction channels. One may here distinguish between (a) genuine one-electron systems, as H+-H or HeZ+-H, where the interactions are accurately known, and (b) “quasi”-one-electron systems, as alkali atom-closed-shell systems, where one may model the interactions in suitable ways. The latter category can be divided further into systems (bl) with a (near) symmetry of the two closed-shell cores, for example, Li+-Li or Li+-Na, where the valence electron interacts with two (singly) charged cores, and systems (b2), where one of the cores is neutral, for example, H-He, Li-He, and Na-He. All three groups of systems have been subject to numerous invest igations. Following the quest for simplicity stated above, one should avoid systems where charge exchange is important (a, bl), and instead consider one-center problems with the active electron located on the same center before and after the collision. Second, we shall exclude systems involving hydrogen or hydrogen-like ions, where the 1s electron is strongly bound compared to the excited discrete states, which are situated in a relatively narrow band below the continuum, and which furthermore are degenerate with respect to L , the orbital angular momentum quantum number. This level structure may be contrasted to alkali-like atoms (see Fig. l), where the first excited level, np, is well separated from both the n s ground state and the higher excited levels, in particular, the continuum. Inspection of

N . Andersrn and S . E . Nielsen

268

35,P,d

-

2S,P

-

-)

n=4

,3d ‘3P

‘3s

w w

2P

H

Li

FIG.1. Binding energies En, of some excited levels for H and for a typical alkali atom, Li. in units of the ionization energy E,.

Fig. I suggests that the most prominent inelastic process is the 17s + tip resonance transition of the alkali atom. Theoretically, one might hope that just a few states will form a basis sufficient for a fair description of the process. Experimentally, the prominent alkali spectral lines are situated in or near the visible range, where the techniques for determination of absolute cross sections, polarizations, etc., are well established.

C. Q U A L I T A T ICONSIDERATIONS VE Before discussing specific results it is useful to consider some further qualitative aspects which may guide the selection of proper theoretical methods and experimental techniques. The systems are characterized by the presence of a single, loosely bound valence electron outside two closed-shell cores, Fig. 2a. Inspection of the figure suggests that the collisions may be divided in two groups of different complexity:

(i) Violent (small impact parameter) collisions, where the two cores interpenetrate significantly. Excitation takes place at well-localized molecular curve crossings. In this case, rare gas electrons play an active role during the formation and breakup of the quasi-molecule. The events depend crucially on the internuclear distance R (see Fig. 2b) and thereby on impact parameter. (ii) Soft (large impact parameter) collisions with insignificant core-core interaction, and accordingly small deflection angles. Here the rare-gas

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(a)

269

(bl

Fib. 2. (a) Schematic picture of a quasi-one-electron system. A is an alkali-like atom consisting of a small, positively charged, closed-shell core surrounded by a single, loosely bound valence electron in the ground state. B is a rare-gas atom. (b) Diagram showing the three main interactions of the problem. The interaction between the two heavy cores A - B determines the scattering angle.

electrons remain relatively unperturbed during the collision and excitation is mainly induced by the direct interaction between the valence electron and the rare gas (e-B, Fig. 2b). In this case the inelastic transition of the alkali atom is thus induced by a nonlocalized interaction, caused by the passage of the rare-gas atom, acting effectively as a structureless, spinless particle. Here we expect the excitation probability to depend mainly on the duration and strength of the interaction, and only weakly on impact parameter. Below we shall refer to these two cases as medianisms (i) und (ii), respectively, or more descriptive, but less precise, as “molecular” and “direct” excitation. It will be seen how the nature of these processes are reflected in the degree of success of various experimental and theoretical approaches. For mechanism (i), we expect a rather large variability in the inelastic processes from system to system, depending on the intrinsic characteristics of the specific quasi-molecule, as previously seen in numerous investigations of closed-shell interactions (Barat, 1973, 1979; Olsen ct ( I / . , 1979a). For the simpler mechanism (ii) less is known, but a more uniform picture is expected with the processes depending primarily on suitably scaled variables. We end this section by a simple argument which isolates the key parameter from which scaling properties for mechanism (ii) may be predicted, an argument which also underlines the connection between this excitation mode and phenomena in neighboring subfields of atomic physics. Figure 3 illustrates various ways of inducing a transition between two atomic levels I and 2 by external forces. AE = / i U o is the 1-2 energy difference. Figure 3a illustrates the well-known technique of using a laser with tunable frequency u . The 1-2 transition probability can be estimated from the Fourier transform (FT) of the field. The transition probability peaks sharply when the laser frequency u equals the natural frequency vo .

N . Andersen and S . E . Nielsen

270

'f1

I

l--2 AE= hv0 Transition Probability (FT)

J

I T-

1

Resonance if : T = x

I

Resonance i f :

1 -dv = vo

A rtirne VO

Resononce i f :

0 = v 2AE '

Or

-

FIG.3 . Three different ways of inducing a 1 2 transition by an external force: (a) laser; (b) crystal lattice, channeling; ( c ) single collision. See the text for a discussion.

Figure 3b shows schematically the passage of an ion with velocity u in a channeling direction of a single crystal with periodicity d. Again, when the frequency of the field from the crystal lattice u/d matches v o , the 1-2 transition probability increases. This phenomenon, the Okorokov effect (Okorokov, 1965), has been thoroughly investigated by the group at Oak Ridge (Moak et al., 1979). Figure 3c finally concerns the case of a single binary collision with an effective interaction length u. In analogy with Fig. 3a and b, the 1 + 2 excitation probability peaks when the effective collision time N / U matches

DIRECT EXCITATION IN ATOMIC COLLISIONS

27 I

(half) a period I / v o of the natural frequency, or AEo/hv = 7~ (2) This result suggests that the Massey parameter AErr/hu (a reduced collision time) is a key quantity in the analysis and comparison of results for different collision systems. Because the perturbation in Fig. 3c has a much broader frequency spectrum, the excitation probability peak is much broader than in Fig. 3a and b. The following sections will review the present level of understanding of excitation in quasi-one-electron systems like alkali atom-rare gas (Li-He, Li-Ne, Na-He, etc.) and alkaline earth ion-rare gas (Be+-He, Be+-Ne, Mg+-He, etc.), illustrated by recent experimental and theoretical results. The whole energy region, from low through medium to high energies, will be covered, with the main emphasis on mechanism (ii). It will be seen how the recent advances substantiate the early ideas of the 1930s and 1940s and provide a deeper, quantitative insight into the physical processes qualitatively outlined above.

11. Theoretical Models In this section we shall consider theoretical approaches to collisional excitation in quasi-one-electron systems with particular emphasis on the asymmetric alkali atom (alkaline earth ion)-rare-gas collisions discussed in the introduction, i.e., inelastic events (A

+ e)41,,,, + B + (A + e)nlm+ B‘”’

(3)

where A is the closed-shell core of the quasi-one-electron projectile (valence electron ground state n,OO) and B is the closed-shell target. Thus far, practically all investigations of many-electron diatomic systems have used the classical trajectory concept of heavy-particle.motion. The description of the electronic scattering state, however, has been based upon a variety of possible expansions from the fully relaxed adiabatic molecular states via various so-called diabatic molecular states to completely unrelaxed atomic states of the separated atoms. A particular and vexing problem is connected with either approach, the necessity for including so-called electron translational factors (ETF) to ensure Galilean invariance of the scattering state. An excellent discussion of these subjects, with particular emphasis on slow atomic collisions, may be found in the recent review article by Delos (1981). A general review of the theory of fast heavyparticle collisions has been given by Bransden (1979). The choice of rep-

272

N . Andersen and S . E . Nielsen

resentation for the electronic state basically depends on the collision energy of interest. The representation preferred is the one for which a small subspace most closely accounts for the actual development of the electronic charge distribution during the collision. Hence, at very low energy presumably fully relaxed adiabatic (Born-Oppenheimer) molecular states and in the high-energy range undistorted separated atom states may be the natural choice. Quite often, however, in heavy-particle collisions it is the intermediate energy range that is of major interest, and either approach may prove unsatisfactory. Molecular expansions have been modified to meet the physical requirements at higher impact energies, e.g., by imposing constraints on the basis set in the form of frozen, separatedatom orbitals (Courbin-Gaussorgues et d.,1979). Likewise, atomic basis expansions may be modified to account for relaxation effects, e.g., by inclusion of united-atom orbitals into the basis (Fritsch and Lin, 1982). A very different approach to overcome the inherent deficiencies of a n y basis-set choice is currently being investigated (Horbatsch and Dreizler, 1981; Horbatsch et d.,1981) in calculations of the time development of the many-electron density with aid of time-dependent Thomas-Fermi theory. Likewise, Terlecki or t i / . (1982) have proposed for H-H+-like systems a solution based on a classical hydrodynamic interpretation of the Schrodinger equation, thus obtaining equations for the quantum mechanical probability density during the collision. A. ATOMICBASIS The genuine one-electron systems H-H+, H-He'+, H-Li3+, etc. are inherently charge transfer systems and require two-center (ETF) expansions of the electronic state. They have been investigated with respect to direct and exchange excitation since the pioneering work of Bates (1958) and Bates and McCarroll(l958). The atomic states are known exactly and the interactions are simple. Hence, these systems have been obvious test cases for choice of method and expansion and for convergence (e.g., Wilets and Gallaher, 1966; Rapp and Dinwiddie, 1972; Rapp, 1973, 1974; Shakeshaft, 1975; Ryufuku and Watanabe, 1978, 1979; Theodosiou, 1980; Janev and Presnyakov, 1980). However, in spite of their simple structure these collision systems are not trivial, as pointed out in the introduction. The hydrogen-rare-gas systems might be considered as a natural first choice of quasi-one-electron systems without the complications of charge transfer channels. Atomic basis close-coupling calculations have been performed for H-He by Flannery (1969) and Bell et 01. (1973, 1974) on the basis of the electrostatic interaction of the electron.and the ground-state

DIRECT EXCITATION IN ATOMIC COLLISIONS

273

He atom. Good agreement with experiment for H(ls-2s, 2p) excitation was obtained for medium- to high-energy collisions, essentially in the energy range where Born predictions are fair (Bell and Kingston, 1974). Analogous atomic basis calculations for the H-Ne system, however, predicted H( 1 s-2s, 2p) cross sections up to an order of magnitude above the experimental results (Levy, 1970). I . AlXrili Atom-AIXrili l o t i Systcvns

The quasi-one-electron systems Li-Li+, Li-Na+, etc., have the advantage of well-separated lower excited atomic states (cf. Fig. l), and it is tempting to model these systems as one-electron systems assuming an effective electron-alkali core interaction. Storm and Rapp (1972) made a careful analysis of this approach and its problems. One-electron valence states were based upon an effective-charge Coulomb potential for the electron-atom core (A) interaction (Rapp and Chang, 1972, 1973a): V,(r) = -Z,,,/r.

=

-[(Z,

-

Z , ) / r ] ~ - ~ ’+~ r( /l 2 a ) - Z , / r

(4)

of the alkali atom Hamiltonian H,

=

-4v: + V , ( r )

(5 )

where Z , is the charge of A and Z, the charge of the A nucleus. The value of the fit parameter LY is set by requiring that the eigenvalue problem

(H,

-

EiOXi,,

=

0

(6)

corresponding to the alkali ground state x ~ reproduces , ~ the spectroscopic ionization potential. Solving Eq. (6) with that samea now generates excited states ( d i n ) with an energy spectrum in good agreement with experiment. Impact-parameter close-coupling calculations (Chang and Rapp, 1973; Rapp and Chang, 1973b) based upon the Hamiltonian H = -tV: + V , ( r , ) + VII(rH),V , and VHbeing the model interactions [Eq. (4)], and a two-center (ETF) expansion in terms of the atomic states xj’ and xy of Eq. (6), did lead to reasonable predictions for magnitude and energy dependence of the total cross sections for Li+, Na+-Li, Na collisions. The experimental oscillatory structure, however, was not obtained, and the disagreement was attributed to failure of the atomic representation at smaller impact parameters. 2. Alkrili-Like Atotn-Riire-Gas Systems As pointed out in the introduction, these systems appear as sensible test systems for mechanism (ii). For the alkali atom or alkaline earth ion the

N . Andersen and S.E . Nielsen

274

simple one-electron model may be retained with valence states obtained from Eq. (6), and the rare-gas ground state may be the only target state in the product channels. The price paid, however, is the complication due to the electron-rare-gas interaction, which is much more difficult to model than the electron-ion interaction. One-electron model potentials. The Hamiltonian for the valence electron in the field of the two closed-shell cores A (alkali or alkaline earth ion) and B (ground-state rare-gas atom) may be expressed as (Fig. 4)

(I.

H,

=

H A + VI = -4V;

v, =

+

VA(rA)+ V,(rB, R)

VR@B, R) + W*,(R)

(7) (8)

where we have separated from the total projectile-target interaction V, the part W A , ( R ) which is independent of the electron coordinate; VA is the potential of Eq. (4). Various model potentials have been proposed for VB, the simplest being the electrostatic interaction of the electron and the undistorted charge distribution of the ground-state rare-gas atom, used in most studies of hydrogen-rare-gas collisions (Bell et a/., 1973, 1974):

where a0is the Hartree-Fock ground state of the N-electron rare-gas atom. Semiempirical modifications of VHFwere developed (Bottcher, 197 1; Bottcher rt d., 1973) to account for the long-range distortion of the raregas atom: vl3l(rl3) VB&,

=

R) =

+ Vsr(rH) + vd(rB) V d r d + Vcp(rH,R)

VHP

'

)('I3

+ vq(rB) ' l l ' 8

*

FICj. 4. Scattering geometry for a quasi-one-electron system.

(10) (1 1)

275

DIRECT EXCITATION IN ATOMIC COLLISIONS

where V,, and V , are the induced dipole and quadrupole interactions of the electron and the rare-gas atom ( w 4 , w gand w8 are cutoff functions at short range) and V , , is the cross-polarization interaction due to the projectile ion core. The added short-range term V,, includes parameters determined empirically for VB, to reproduce the experimental momentum transfer cross section for electron-rare-gas collisions as function of energy. The potential energy curves obtained from H , and the Bottcher model potentials have been used to evaluate collision-induced absorption in the Li, Na-He systems (Bottcher et al., 1973) and collisional broadening of Mg+ lines in He collisions (Bottcheret al., 1975). Valironet al. (1979) and Philippeet ul. (1979) have critically examined the validity of the model potential approach for the interaction of a valence electron with a rare-gas atom. They point out that the Bottcher potentials have been made too attractive (spurious bound states) in order to represent the low-energy electron scattering data. Instead, they propose to include a nonlocal potential term to account for the Pauli principle. Actually, Baylis (1969) and later Pascale and Vandeplanque (1974) have introduced a local Gombas potential to represent the Pauli exclusion forces in a rather different semiempirical approach to the electron-raregas interaction: V B P V ( ~R) ~ ,= 4 [ 3 ~ ~ p B ( r B ) + I ~ V&d ’~

+ V,,(ro)* d r o

- TB)

*

v(rB - ro)

+ vcp(rB R) . d 9

r B -

ro)

(12)

where pB is the electron density of the rare-gas atom, V, and V,, are the same as above [Eqs. (lO)-(ll)], and q ( x ) is the Heaviside unit step function. The range parameter ro is determined by a fit of the resulting ground-state well depth for the alkali-rare-gas system to experiment. Again, the potential energy curves obtained from the resulting Hamiltonian HI have been used to evaluate collision broadening, e.g., that of Na lines by He (Wilson and Shimoni, 1975a,b), and for calculations of finestructure transitions, e.g., those for K in He (Pascale and Stone, 1976; Pascale and Perrin, 1980) and for Na in He, Kr (Gaussorgues and Masnou-Seeuws, 1977). h. Impuct excitation. The one-electron model potentials discussed above have been used to obtain excitation amplitudes in the medium-to-highenergy range for alkali atom or alkaline earth ion-rare-gas collisions, solving the time-dependent Schrodinger equation in the straight-line impactparameter approximation based upon H1 [Eq. (711

[;(a/&)

-

Hl]9(r, 1 ) = 0

(13)

N . Andersen and S . E. Nielsen

276

‘in a one-center atomic expansion (no ETF complications) based upon the eigenstates x p [Eq. (611

W, t ) =

2 a,(t)xp(rA)e-*4f= 2 ije j

(14)

j

with the initial conditions u , ( - x ) = GJ,,,,M). The amplitudes ij satisfy the coupled equations

From the close-coupling solution obtained for the excitation amplitudes q ( h , E ) = u,(r -+ x ) all information about the inelastic collisions may be derived, such as excitation probabilities P,(b, E ) = (tr,(h, Ell2, coherence parameters, total excitation cross sections a j ( E )= 27-r dhhPj(b,E ) , and polarization of the subsequent emission. The core-core interaction W,,,(R) of H I may be discarded from Eq. (IS)for convenience, contributing a trivial common phase factor exp(-i4(h, E ) ) ,4 = W ( R ( t ’ ) ) t l t to ’ , all excitation amplitudes uj(b,E ) . The direct excitation mechanism was first investigated in close-coupling calculations of tios-n0p excitation in 1-100 keV Be+, Mg+-He, Ne collisions based on the Bottcher potential V,,, (Nielsen and Dahler, 19761, followed by a comparison of model potential calculations [V,, , VH,, and Vllz of Eqs. (9)-( 11)l for these same systems (Nielsen and Dahler, 1977). Excitation predictions have been obtained for Li, Na-He, Ne collisions in a comparative study of the VHF,V,,], V,, ,and Baylis [ VHIJ,of Eq. ( I2)1 model potentials (Maniquert d.,1977) and for Na-Ne based upon V , , (Gross Pt l i l . , 1978). The following conclusions may be drawn from the Li-He results in Fig. 5 . The Bottcher models VH, and Vl,z grossly overestimate the excitation cross sections and will not be considered further. The Baylis potential V,,,, and the electrostatic potential V,,,.., although based upon very different interaction models, give reasonable predictions of crosssection shapes and magnitudes. The simplest potential V , , has been studied in more detail for selected systems in calculations of excitation probabilities, polarization of emission, and coherence properties (Nielsen and Dahler, 1980; Andersen Pt d . , 1979c, 1982), and the results are discussed and compared with experiments in Section IV.

I

1

r~otlels. For Na(3p) excitation in Na-Ne collisions Courbin-Gaussorgues rf ( I / . ( 1979) have modified a many-electron molecular basis study to explain the direct excitation process at higher energies. A perturbed atomic basis was obtained using the frozen SCF orbitals of the Na+-Nel state at internuclear distance R = 20 a.u. Energies and

c. Mtrny-rlctm)ti

x+

DIRECT EXCITATION IN ATOMIC COLLISIONS

277

E IkeVl FIG. 5 . LiUs -, 2p) total cross section as a function of impact energy E for direct excitation in Li-He collisions. Results obtained from three-state (2s, 2p0,2p,) close-coupling calculations based upon the model potentials V , , , V,,, V,,,., and V,, are compared with experiment.

potential couplings are calculated from single-configuration states built from Na and Ne orbitals, and effects of dynamic coupling and ETF were estimated. The scattering predictions obtained represent a considerable improvement over the Na-Ne results based on VHFand VHP\.. In an attempt to improve the simple electrostatic model potential V,, and to bypass the semiempirical approaches, a classical trajectory theory of a quasi-one-electron projectile (A + e) and a (quasi-) two-electron target (B + 2e), e.g., He, has been proposed (Nielsen and Dahler, 1981). The rationale for this approach is that a complete treatment of the projectile valence electron nrid the two electrons of the target may allow identification of the Pauli exchange and exclusion interactions neglected in the electrostatic potential VHF.A case of particular interest for the present discussion is neglect of target excitation channels. Assuming a “frozen” ground-state target, equations for the valence-electron-state amplitudes result which may be brought into a form analogous to the Eqs. (15) of the one-electron model, Ij,) defining an ETF corrected valence orbital xj%-it’z

I

with an effective one-electron Hamiltonian [cf. Eqs. (7)-(8)]

278

N . Andersen and S . E. Nielsen

a projection operator Qo = 14;) ($$I onto the target ground state, and the usual exchange operator K Oof the target orbital 4:. The effect of discarding both of the operators Qo and K O (i,e., neglecting electron indistinguishability) is to generate Eqs. (15) of the one-electron model based upon the electrostatic interaction V H F .The additional terms of Eq. (16) all arise due to the use of proper antisymmetrized states, and they may be interpreted as Pauli exchange and exclusion interactions. One may notice that the V,,F model appears as the exact high-energy limit ( u + x ) of Eq. (16) since these terms vanish due to the ETFs of their integrands. This approach offers a systematic nonempirical way of introducing part of the many-electron effects missing in the simple one-electron model calculations based upon V,,,.. Preliminary calculations for Be+-He collisions show very large improvements over the VHFpredictions at lower energies. B. MOLECULARBASIS Most of the investigations performed thus far with molecular state expansions have been for genuine one-electron systems, following the initiating work of Bates and McCarroll (1958), who first introduced ETFs to account for electron translation in the so-called perturbed-stationary-state (pss) theory. Charge exchange studies have been performed for H-He2+ by Piacentini and Salin (1974, 1977), Winter and Lane (1978), Hatton rr rrl. (1979), and Vaaben (1979), and for other one-electron systems (e.g., H-Be4+) by Hare1 and Salin (1977) and Vaaben (1979). The alkali atomalkali ion quasi-one-electron systems have been studied in molecular expansions by Melius and Goddard (1974), Shimakura et t i / . (1981) and Okamoto et ril. (198 I ) . In all of these investigations charge transfer channels are important. A major point has been the proper handling of the electron translation problem and the evaluation of its importance (see also Riera and Salin, 1976; Ponce, 1979; Vaaben and Taulbjerg, 1981). The interpretation of couplings in slow many-electron atomic collisions in a molecular orbital basis was greatly stimulated by the work of Lichten (1967) and Barat and Lichten (1972). Their electron promotion model has been used for qualitative as well as quantitative predictions of numerous atomic scattering processes, in particular, for closed-shell systems and innershell excitation (see, e.g., review articles by Sidis, 1975; Fastrup, 1975; Eichler and Wille, 1978; Meyerhof and Taulbjerg, 1977). In the case of excitation of quasi-one-electron atoms by rare gases, however, very few applications exist. Bell cr trl. (1976) and Benoit and Gauyacq ( 1976) have performed limited two-state calculations for H-He, mainly to prove that a molecular mechanism (i) is required to account for

DIRECT EXCITATION IN ATOMIC COLLISIONS

279

t \ 3p,,

r >r

C 0

3SNa

I

1

7 2 R,

-A

”

c

R,,

Internuclear distance (o.u.) FIG.6. Simplified representation of orbital energies as functions of internuclear distance for Na-Ne.

the magnitudes of H(2p) excitation observed at lower energies ( E < 1 keV). The only thorough investigation until now is the study of Na(3p) excitation in Na-Ne collisions by Courbin-Gaussorgues et al. (1979). These authors analyzed the process in terms of a two-step curve-crossing molecular mechanism, first suggested by Anderson et 01. (1969) for K-rare-gas systems and later by Fayeton et al. (1976) for Mg+-rare-gas systems. For the case of Na-Ne Courbin-Gaussorgues et ~ J I (1979) . obtained the diabatic MO-correlation diagram, shown schematically in Fig. 6, with the Na valence electron initially placed in the 3su orbital. Violent collisions with distance of closest approach less than about 1.5 a.u. allow 3s + 3p excitation via electron transfer from the 3su to the promoted 4fu orbital at the curve crossing C1with subsequent transfer at C2to the 4pu orbital. Single and double excitation of Ne may occur in diabatic passages of the C , crossing. Ab initio potential energy curves and couplings were calculated, and seven-state quantum close-coupled scattering equations were solved. Calculations of Na(3p) excitation in Na-He collisions are in progress (Courbin-Gaussorgues et al., 1981).

111. Experimental Techniques The optimal experimental results needed to assess the validity of the various theoretical models described above are the quantum mechanical amplitudes (inlrn = anlm(b,E) for excitation of a given state Inlrn), as

N . Andersen and S . E . Nielsen

280

functions of impact parameter h and energy E. This would allow a direct comparison between experiment and the immediate output of the closecoupling impact-parameter calculations. The series of experimental techniques used to study excitation processes approach this prrfcw t..\-perirnerit (Bederson, 1969, 1970) to various degrees. In the present context we are concerned with excitation of levels which decay by optical emission. The "classical" experiment is to determine the production rate of photons of a certain wavelength as function of impact energy E , expressed as a total emission cross section creup(E). Comparison with theory then involves a series of summations over discrete and continuous undetermined observables. This may be schematically summarized in the formula vtheOr,(E) =

2 c c lom 2 7 ~ b l ~ , db l~l~ E

A

m

(18)

(1)

( 5 ) (4) ( 3 ) (2)

The averaging procedure consists of the following steps: (1) The amplitudes are squared in order to get the excitation prob-

abilities. ( 2 ) Integration over all impact parameters b. (3) Summation over all magnetic quantum numbers 111. (4) Summation over all states which directly, or by cascade processes, decay by emission of the photon observed. ( 5 ) Eventually, a summation over channels with simultaneous target excitation. To which degree an experiment is able to discriminate against these summations depends, of course, on its level of sophistication. In the present context it is convenient to divide the experimental techniques into three classes according to the degree to which they take advantage of the collisional symmetry: jirll 4n sirmtntrtiori, or use of cylindricd and pltririr symtnett?. With the historical development in mind, one can roughly speak of three corresponding generations of experiments. A. FIRST-GEN E R A T ION EXPERIMENTS

Experiments in this group determine total cross sections, irrespective of the direction of the scattered particle or emitted photon. lbtd Cross Sc.ctions

A typical experimental setup is sketched in Fig.. 7. A monoenergetic, isotopically pure beam from an accelerator is passed through a cell con-

DIRECT EXCITATION IN ATOMIC COLLISIONS

28 1

To Monochromator Window Polarizer Beam from Accelerator

To Beam Integrator

Gas Cell FIG.7. Schematic diagram of an experiment for determination of total cross sections and polarizations.

taining the target gas at a pressure sufficiently low to assure singlecollision conditions. The beam intensity is measured by a Faraday cup or a neutral beam detector. The light emitted from collisions in the cell is analyzed and detected by a monochromator. The numerous things that have to be taken into account in this type of experiment are discussed extensively in the literature (de Heer, 1966; Thomas, 1972; Andersen Pt al., 1974, 1976; Olsen et d., 1977) and are omitted here. We shall mention only the anisotropy of the radiation pattern, or equivalently, the light polarization. In the geometry of Fig. 7 with observation direction perpendicular to the beam axis, the total cross section is proportional to Zll + 24, where I,, and 4 are the intensities of the light polarized parallel and perpendicular to the beam axis, respectively. Counting statistics is usually better than 296, but due to calibration problems the absolute values often have an uncertainty of 20-50%, depending on wavelength. With reference to the excitation mechanisms (i) and (ii), one would expect that first-generation experiments may be able to reveal information about the direct mechanism (ii), which mainly depends on collision time and only weakly on impact parameter. Little is, however, expected to be learned about mechanism (i), which depends crucially on impact parameter, a quantity not determined by a first-generation experiment.

B. SECON D-GENERATION EXPERIMENTS This class of experiments takes advantage of the cylindrical symmetry of the collision geometry. Two techniques have been used. I . Light Polmizotion Stirdies

A by-product of the total emission cross-section measurements described in Section III,A, is the light polarization II = (Zll - lJ/(Zl, + 4).This ratio is nonzero when the magnetic sublevels of the upper level of the optical transition studied are nonstatistically populated (Percival and Seaton, 1958). The beam axis is here the natural axis of quantization. Thus, II is a measure of the relative, average deformation of the excited electron

282

N . Andersen and S . E. Nielsen

cloud with respect to the beam axis (II > 0: cigarlike shape; n < 0: disklike shape). Analysis of the polarization gives information on the relative distribution of excitation over the magnetic sublevels, step (3) in Eq. (18). For a p level in particular, the ratio ul/aObetween the urncross sections can be determined (Andersen et al., 1979e).

2. Differential Energy-Loss Analysis A powerful technique used for the study of low-energy inelastic processes has been energy-loss analysis of particles scattered through a fixed angle 8 (see Fig. 8). The impact parameter h may then be calculated if an estimate of the core-core potential is available (Sondergaard and Mason, 1975). For charged particles the energy loss has mainly been determined electrostatically, while the time-of-flight technique (Morgenstern et cil., 1973) has been used for neutrals. Differential energy-loss analysis delivered the key to the detailed interpretation of many ion-atom and atomatom collisions in terms of mechanism (i) (Barat, 1973), where the impact-parameter dependence reveals the underlying physical processes, triggered at relatively small impact parameters. Evidently, since the impact-parameter dependence of the two mechanisms (i) and (ii) are expected to be quite different (cf. Section I,C), this technique should enable clean studies of the two mechanisms separately. For valence-electron excitation induced by mechanism (ii), two experimental problems arise. The energy loss of just a few electron volts may in several cases be difficult to resolve. This problem may be solved by the photon-scattered ion coincidence technique. Second, excitation may take place with appreciable probability even at rather large impact parameters, where scattering angles are extremely small (cf. Fig. 2).

C. THIRD-GENERATION EXPERIMENTS A third-generation experiment takes advantage of the planar symmetry of the collision geometry. In favorable cases one may then approach the perfect e.~perivirnentmentioned above. The technique was introduced into atomic collision physics by the groups in Lincoln (McKnight and Jaecks, 1971; Macek and Jaecks, 1971; Jaecks er id., 1975), Paris (Vassilev et d . , 1975), and Stirling (Eminyan er id., 1973; Standage and Kleinpoppen, 1976). The same kind of information may be derived from the inverse process, that is, collisions with a laser-excited beam (Hertel and Stoll, 1977; Schmidt er nl., 1982).

DIRECT EXCITATION IN ATOMIC COLLISIONS

283

Gas Cell FIG.8. Schematic diagram of an experiment for differential energy-loss analysis. If the scattered particles are neutral, the incident beam is chopped and the time of flight from the collision cell to the detector is determined. For ions electrostatic energy-loss analysis may be performed.

I . Coherence Anulysis Figure 9 shows very schematically the experimental geometry used. It combines the techniques illustrated in Figs. 7 and 8. Polarized photons and scattered projectiles are measured in coincidence. For p-state excitation all averages (1)-(5) in Eq. (18) may be decomposed if the energy loss of the scattered particle is accurately determined. Why this is sufficient for a complete determination of the amplitudes anlmmay be seen from the following argument. The orbital part of the p state is represented by the ket

144 = 4 P o ) + allPd + a-IlP-1)

= aolPx)

- fi4lPZ)

(19)

where use has been made of the reflexion symmetry in the scattering plane, implying that a_, = -al. The direction of quantization is along the incident beam as before. In a semiclassical picture the radiation pattern of the subsequent p s decay may thus be visualized as arising from two coherently excited dipoles Ip,) and Ipt) of strengths laOIZand 21a11* oscillating along the z andx axis, respectively. The coherence parameters A and x ,

Q

I

Coincidence

I

Photomultiplier

epoiarizer O

h

1

4 - Plate

FIG.9. Schematic diagram of a polarized photon-scattered-particle coincidence experiment. The 1 p,) and 1 p,) orbitals are shown.

N . Andersen and S . E. Nielsen

284

(Standage and Kleinpoppen, 1976) are defined by A = l ~ ~ ~xl ~ = / 9 , arg(cil/no), where 9 = Icio12 + 21a112. Thus, A determines the relative strength of the two dipoles, while x is their phase difference. Determination of the polarization ellipse by Stokes parameter analysis, conveniently measured in the direction + y perpendicular to the scattering plane, will allow evaluation of A and x. Together with the light intensity, (A, x) determine uo and N , apart from a common, arbitrary phase. An important advantage of coherence parameters is that they are dimensionless quantities derived from relative measurements, and thus not influenced by the normalization problems encountered in total and differential cross-section measurements. Various experimental effects, .due, for example, to insufficient energy resolution, may complicate the analysis. If so, or if states with angular momentum larger than unity are analyzed, a full determination of the radiation pattern is desirable, and one has to perform additional measurements such as photon analysis in the.r direction. The general analysis will be the subject of the following section. 2. Tlicory of the Erptvirnerit Fano and Macek (1973) have treated angular distributions and polarization properties of collision-induced radiation. We parameterize the experimental geometry of Fig. 9 as shown in Fig. 10. C N S ~ . The angular distribution of electric dipole radiation is given by an expression of the form

o . T/ie g c n c 4

I

=

+

S ( l - +h(2)A,det $/PA,d:t cos 2p

+ +lP0,detsin 2p)

q

Polorizer Axis

FIG.10. The geometry of the experiment of Fig. 9. The angular momenta of right-hand circular (RHC) and left-hand circular (LHC) photons propagating along the +y direction are indicated.

DIRECT EXCITATION IN ATOMIC COLLISIONS

285

where

A?;'

= ci

(L,L,

+ L,L,)

0;?= a ( L , ) with l/a = L(L + 1)9,are the nonzero components of the alignment and are geometorientation tensor characterizing the excited level; h") and rical factors determined by the angular momentum quantum numbers of the upper and lower level of the transition. The angle p describes the photon polarizaticln: p = 0 corresponds to linearly polarized light, p = 71-/4 to photons with helicity + 1, or left-hand circularly polarized light (LHC), and p = -n/4 to helicity - 1, or right-hand circularly polarized light (RHC); (0, 4, $) are the Euler angles of the photon detector system. The relative Stokes parameters in the +y direction are

P =

-

f(RHC) - I(LHC) I(RHC) + f(LHC)

-

-6h"'OCn1 1I U

where f y = 4 + / P ( A p 1+ 3A;$I). In the +s direction, one obtains anal, = 0 , and Ps = 0, whereI" = 4 + ogously P4 = ~ / I ' ~ ) ( A+; 'A) ~g 1 ) / I X Ps /7(2)(Ap1 - 3A;:I). A measurement of the four polarizations ( P 1 ,P 2 ,P a , P 4 ) allows evaluation of the four alignment and orientation (or shcrpe) parameters Eqs. (20a)-(20d), which represent the maximum amount of information that can be extracted from the radiation pattern. h. Specific case: P + S transitions. In this case the number of parameters can be reduced even further. In fast collisions where spin-dependent

N . Andersen and S . E . Nielsen

286

forces can be neglected a fully coherent excitation of a p state can be described by Eq. (19). In terms of the (A, x) parameters defined above, one obtains Am1 0 = 3(1 - 3 A ) , A'"' = [X(l 1+ cos x A Cz+O I =

HA

-

I),

OF'!

=

-[A(l -

x + 3ACo1=

sin

Notice that these parameters are not independent: AS,' 2+ - 1. For a lP -+ IS transition with no fine structure (fs) or hypefine structure = -2. Then (hfs) / I " ) = 2 and

P1 = 2 X - 1,

Pz

P3 = 2[X(l - A)]"* sin x.

P, = 1

=

-2[X(l - A)]"'

cos x

The degree of polarization P = [PI. where = ( P I ,Pz,P3) is seen to be unity. In this simple case analysis in the + Y direction only is sufficient for a complete determination of the coherence parameters. However, several complications occur in practice. Internal forces which give rise to fs and hfs will reduce the polarizations, but can be accounted for by appropriate correction factors (Andersen et al., 1980). More problematic to handle are cascade effects which may show up when the experimental setup is unable to discriminate photons arising from collisions where the upper level of the transition is populated directly from photons where the level is populated by a cascade process from a higher excited level, thereby reducing the coherence of the light. Models which describe and correct for these effects have been developed and shown to work reasonably well when cascade effects do not exceed 10-20% (Andersen et ul., 1980). The degree of polarization P, corrected for fs and hfs effects when necessary, is a convenient measure of the degree of coherence (Andersen et d.,1979b). Most serious problems arise if the experimental setup sums photons from several incoherently excited channels of comparable importance. This is typically the case when simultaneous excitation of target and projectile is important. Then the degree of polarization may become very small, and evaluation of coherence parameters is meaningless. c. Specific case: D + P transitions. Generalization of the treatment above to coherent excitation of states with orbital angular momentum L > 1 poses the question of whether the radiation pattern contains sufficient information for unambiguous determination of the expansion coefficients for the 2L + 1 magnetic substates of the excited state. We discuss the case

DIRECT EXCITATION IN ATOMIC COLLISIONS

287

of a d state in analogy with the p-state considerations above. The state is represented by the ket

I$)

=

aoldo) + 4 d l ) + a-lld-1) + azldz) + 0-zld-2)

Again, reflexion symmetry requires a, = - a 1 , = a2. Nienhuis (1980) introduced the coherence parameters A = Ino12/9, p = 2(a1I2/9,x = arg(a,/u,), $ = arg(az/al),where B = la01* + 2)a1I2+ 21aZl2.Then AT'= 1 AS$

-

= -3p

ACoI 1+ -

2 A - 3 CLl2

+ 2[A(1

[ h p / 3 ] 1 ' cos z

(214

-A >(

O c o l = - [Ap/3]1/z sin

p)/3]1'2 COS(>(

+ [p(l - A

x

- p)]"'

- +[p(l - A

+ JI) cos JI

- p)]1'2sin $

(21b) (21c)

(214

As seen above, the shape parameters (AS"',A;$, A;$, OF)can be determined from a Stokes-parameter analysis. However, the next step, inversion of Eqs. (21a)-(21d) cannot be done: Two solutions, (A,, p l , xl, JI1) and (A2, p 2 ,x 2 , ~,b~),are obtained. The coherence parameters of a d state can accordingly not be determined from an experiment of the type outlined in Fig. 9. The experiment has to be sophisticated even further in order to obtain an additional constraint (Andersen et al., 1982).

IV. Results and Discussion In this section we discuss theoretical and experimental results, ordered according to the classification given above. The first study of quasi-oneelectron systems was performed by Anderson et al. (1969). They measured total cross sections for the K-rare-gas collisions in an energy region (r a/.. 1979e). The agreement with experiment is very satisfactory in going from light to heavy projectiles, from ions to neutrals and through the He, Ne, . . . target sequence. Excitation cross sections resulting from three-state ( n s , n p O , np,) atomic basis close-coupling calculations with the one-electron model potentials VHIand VHI,\ (cf. Section II,A,2,a) are shown in Figs. 13 and 14. The direct excitation mechanism (ii) with the electrostatic potential VIIF can account for the overall shape and magnitude of the cross sections for the He target systems at medium to high energy. The Baylis model VHIs\ does predict the observed shape but the magnitude only within a factor of two, in excess of the experimental uncertainty of about 30%. For Ne target collisions the VFiFmodel grossly overestimates the cross-section

-

DIRECT EXCITATION IN ATOMIC COLLISIONS

01 01

29 1

YLp

'

1

'

1

10 E IkeV) F I G . 13. ?2S-22P Be(I1) emission cross sections in (a) Be+-He and (b) Be+-Ne collisions: experimental data ( 0 )of Andersen 1'1 nl. (1976); theoretical atomic basis results using V:,k (---), and the many-electron model (-.-I, the one-electron model potentials VIIb(-), Eq. (16).

I

1

,.,

-I

E (keV) F ~ L14. . 3*S-3'P Na(1) emission cross sections in (a) Na-He and (b) Na-Ne collisions: experimental data ( 0 )of Olsen el t r l . (1977) and (. . .) of Mecklenbrauck ct crl. (1977); theoretical atomic basis results using the one-electron model potentials V , , (-), VRP, (---), and t r h i / r i r ; ( ~molecular basis results (-.-).

292

N . Andersen and S . E . Nielsen

magnitudes and their widths, and the maximum positions are shifted toward higher energies. In an attempt to modify the VHbmodel for Ne target systems, a scaling of the overemphasized electrostatic interaction has been investigated in calculations of excitation in Be+-Ne collisions (Andersen cf d., 1979~).Scaling the interaction by a factor 1/3 leads to the cross-section predictions Vp,k of Fig. 13, now in fair agreement with experiment but for a shift of the predicted E,, below the experimental results. The many-electron modification of the VH, model [Eqs. (16)], taking into account the exchange interactions with the target atom, has thus far been probed only for the Be+-He system in the low-energy range with neglect of ETF in three-state close-coupling calculations (Nielsen and Dahler, 1982). The resulting 2s + 2p excitation cross sections shown in Fig. 13 represent a significant improvement over the V , , , results, and the agreement with experiment is quite satisfactory. The predictions of the molecular basis calculations for Na(3s 3p) excitation in Na-Ne collisions (Pedersen ef d.,1978: CourbinGaussorgues et u/., 1979) are shown in Fig. 14. It is seen that the lowenergy cross sections are well explained in shape and magnitude by the molecular mechanism (i). In the energy range near the Massey maximum the frozen atomic orbital calculations by the same authors, modeling mechanism (ii) (cf. Section II,A,2,c) predict and the cross-section shape quite well. The improvement over the one-electron models Vl,l. and V,,,,,is significant, although the cross-section magnitudes predicted are still too large by a factor of about three. The energy range in between, 1-5 keV, is poorly described by either mechanism, partly explainable by the omission of two-electron target excitation in the molecular calculations. The Na-Ne system offers at present the most striking example of the roles of mechanisms (i) and (ii) for excitation in the quasi-one-electron systems, to be explored further in the second-generation experiments. The effects of higher excited states have been investigated in nine-state (2s, 2 ~ 0 ,2p,, 3s, 3 ~ 0 3p,, , 3d0, 3d1, 3d2) atomic basis close-coupling calculations for Li-He collisions ( V , , model) by Nielsen et ti/. ( 1978) in a comparison with experimental emission cross sections. Figure 15 shows the resulting cross sections for 2p, 3s, 3p, and 3d excitation. The slight increase in 2p excitation over the earlier three-state results supports the validity of the simple three-state description of 11s + np excitation. From the excitation cross sections and the knowledge of lifetimes and observation geometry one can derive the estimates of the emission cross sections (within the nine-state basis), shown in Fig. 15, together with the experimental results. The resonance line emission is well accounted for over most of the energy range. The n = 3-level emissions shown are predicted within a factor of two or better at energies about and above the Massey --f

293

DIRECT EXCITATION IN ATOMIC COLLISIONS

10-9

'I

-

10 E (keV) (0)

100

1

10

100

10-l~

E (keV) (b)

FIG. 15. Li(2s / I / ) excitation cross sections (a) in Li-He collisions and the resulting atomic emission cross sections (b) predicted from three-state (---) and nine-state (-) basis calculations using the one-electron model potential V H F ;experimental results ( 0 )of Nielsen c'f t r l . (1978) and (0) of Staudenmayer and Kernpter (1980).

maxima, but fall well below the experimental results at lower energies. Notice the importance of the 3p .+ 3s cascade for 3s + 2p emission. We thus conclude that the prominent peak of the total cross section conforms to the simple direct excitation picture and is often well accounted for by a one-electron model.

C. POLARIZATIONS The total cross-section results supported the concept of the Massey criterion as a key quantity for understanding the excitation processes. This line of thought will be further explored in the analysis of the light polarization measured as described in Section III,B,l. Use is made of the fact that the collision time is short compared to the characteristic times of fs and possible hfs couplings. The orbital angular momentum may then be treated as completely uncoupled from the spins of the electron and the nucleus (Percival and Seaton, 1958), and the polarization n of the / I %/ I 2P multiplet, or the 11 zS,,z-/~ zP3,2component, is given by a relation of the type

n

=

krr(ao - a,)/(bao + ca,)

(22)

N . Andersen and S . E . Nielsen

294

Appropriate values for k , [ I , h . and may be calculated or found in the literature (Andersen rt o/., 1979e). Furthermore, the cross-section ratio of the I I zSl/2-~i 2P,,, and 17 zSl,z-n 2P,,, multiplet components is 2 : 1. For heavier projectiles and longer collision times this is no longer the case (Kempter cJt d.,1974a; Speller Pt d.,19791, illustrating the breakdown of the Percival-Seaton hypothesis. Equation ( 2 2 ) is used to convert measured polarizations into relative cross sections, a o / a ,, which then are plotted versus the Massey parameter, o r the reduced collision time. This requires an estimate of an effective interaction length (1. For the direct interaction between the valence electron and the rare gas, the sum of the orbital radii of the valence electron (large) and the rare-gas atom (small) is a reasonable choice. One might include a common scale factor, but when accounting for ro/(iiii-o changes from system to system, this factor is immaterial. The results are displayed in Fig. 16, which shows ao/alversus Massey parameter for the Li, Be'. Na, Mg+-He, Ne, Ar systems. Isoelectronic systems exhibit similar behaviors when plotted in these units, a strong evidence for the importance of the (reduced) collision time. Another strik-

0

5

0

5

0

5

FIG. 16. Cross-section ratios for the magnetic sublevel populations (r,,I for the resonance transitions, plotted versus the Massey parameter. Experimental results for the Li, Be', Na, Mg+-He, Ne, Ar combinations ( 0 ,neutrals: 0, ions) have been taken from Andersen ct t r l . (1976, 1979e). Olsen c't rrl. (1977). Mecklenbrauck ei t i / . (1977). and Staudenmayer and Kempter (1980). Theoretical curves: ( - - - ) H F potentials, xk, for Re+, Mg+-He (Nielsen and Dahler. 1977): ( . " ) Baylis potentials, x h (Manique ('1 d.,1977): (-. .-) Na-Ne (Courbin-Gaussorgues ('I r r / . , 1979); (-. -) Be+-He, the many-electron model [Eq. (16)J.

DIRECT EXCITATION IN ATOMIC COLLISIONS

295

ing feature is the oscillatory structure, also found for the heavier rare-gas target Ar, Kr, and Xe, and for the K, Zn+-rare-gas systems (Andersen r f d.,1979e). The agreement in magnitude between theory and experiment gets gradually better with increasing sophistication of the interaction model. Models for structure in polarization curves at low energies in terms of molecular curve crossings have been proposed by Alber rt cd. (1975) and Bobashev and Kharchenko (1978). The oscillatory structure, common to all the theoretical estimates, is an inherent property of the direct excitation mechanism and can be ascribed to the oppo.~itctinw . ~ . w n ~ mofy the 11s-lip, and I I s-np, matrix elements (Nielsen and Dahler, 1980). How this structure arises is clear from a look on the development of the probabilities dirring the collision, as they are obtained in the solution of the time-dependent Schrodinger equation. For later use, we select the Be+-Ne collision as an example. Figure 17 shows the development of the pr and ps excitation probabilities for collision energies 14.5 keV (near the maximum of the total cross section; cf. Fig. 13), 3.6 keV (a doubling of the collision time), and 2.6 keV, at an impact parameterh = 1.6 a.u. As the collision time successively increases beyond the characteristic time of the system, defined through the Massey criterion (Fig. 3), an oscillatory behavior of the charge cloud develops during the collision, which causes an alternating formation of nearly pure Ip,) and Ip,) states at the end of the collision.

z (a.u.) FIG.17. Calculated development of the excitation probability for pz (-) and pr (---I during Be+-Ne collisions with fixed impact parameter b = 1.6 a.u. at various energies.

N . Andersen and S . E . Nielsen

296

Although this picture depends somewhat on h , the oscillatory time dependence survives to some extent integration over the impact parameter, giving rise to the a o / u l structures of Fig. 16. Further manifestations of these oscillations will be presented below.

D. EXCITATION PROBABILITIES Differential energy-loss analysis allows determination of the excitation probability versus impact parameter (cf. Section III,B,?). Excitation due to mechanism (i) should display a singular dependence upon impact parameter, in contrast to the direct excitation which may occur over a wide range of distances (Riiof Fig. 6). A striking confirmation of this is found for the Na-Ne system (Fig. 18). The experimental results show at low energy a sharp rise in excitation at h = 1.5 a.u., the curve-crossing distance predicted for the Na-Ne system. Significant target excitation occurs a t the same distance, also in agreement with Fig. 6. With increasing energy excitation sets in at impact parameters outside the curve crossing. The similarity in h dependence between theory and experiment is evident. The impact-parameter dependence of the excitation probability due to direct excitation need not necessarily be monotonic as for the Na-Ne system. Figure 19 shows theoretical Bef(2s + ?p) excitation probabilities for Be+-Ne collisions, based upon the VS,, potential (cf. Section IV,B and Fig. 13). We observe that a smooth oscillatory variation of 9 withh and E develops with increasing collision time. This behavior is typical for the

-

1

F v

'

1

'

1

Theory

h

.-L

f

30

Mechonism (i) .2

g

--0 C

0

15

'G 15

w

Mechonism (ii)

n m 0

z

0

1

2 b(o.u.1

3

O

-e-&= I- -2 3 b(0.u.)

FIG.18. Na(3s + 3p) excitation probabilities in Na-Ne collisions as functions of impact parameter for selected impact energies (Pedersenef d.,1978). Compare with the total cross sections in Fig. 14. The theoretical curve for mechanism (i) corresponds to I keV.

297

DIRECT EXCITATION IN ATOMIC COLLISIONS

b (a.u.)

Be+(?s+ 2p) theoretical excitation probability in Be+-Ne collisions as function of impact parameter and energy, predicted from atomic basis calculations and the one-electron model potential V b l . Compare with the total cross sections i n Fig. 13. Fit,. 19.

predictions of the one-electron model calculations (Nielsen and Dahler, 1980). It is a result of the probability flow in and out of the 2p,, 2p, states along the projectile trajectory, for fixed h dependent upon the collision time relative to a characteristic time as expressed in the Massey parameter. When integrating b 9 over b to obtain total cross sections local minima or shoulders may survive well below the Massey maximum, as noted for Mg+-He collisions (Andersen pt d., 1976; Nielsen and Dahler, 1976). Often, however, the oscillatory structures are washed out, and a smooth monotonic low-energy dependence results (e.g., Be+-Ne, Fig. 13). Figure 20 shows the measured b, E dependence of Be+(2s 2p) excitation in Be+-Ne collisions (Olsen r f (11.. 1979b), covering only the low-energy part of the theoretical study for h = 1.1-2.1 a.u. Although discrepancies may be found in the details, it is apparent that the predicted oscillatory pattern is confirmed experimentally. Contrary to this the exci-

-

A 0 '52, c

sno l o n

2 0 05a 000b la.u.1 FIG.20. Be+(2s + 2p) experimental excitation probability in Be+-Ne collisions as function of impact parameter and energy (Olsen et ( I / , , 1979b). Notice the change of ordinate scale from Fig. 19.

N . Andersen and S . E . Nielsen

298

tation probability for Be+-He shows a monotonic variation with E at low energies (Andersen et d.,1980).

E. COHERENCE ANALYSIS We now proceed to results obtained in third-generation experiments by the technique described in Section III,C. Examples have been selected to illustrate to which degree the physics is understood and where challenges re main.

I . Conditiotis .for Coherence Figure 21 shows the excitation probability and the corresponding degree of polarization for the Mg+-He, Ne, Ar collisions versus impact parameter (Andersen et al., 1979b). The probability curves for the quasisymmetric Mg+-Ne system are very similar to the results obtained for the isoelectronic Na-Ne system (Fig. 18): a molecular curve crossing mechanism near h = 1.3 a.u. with direct excitation at larger distances. No energy-loss analysis of the scattered particles are performed, and so for this system the degree of polarization decreases from almost unity at large impact parameters to a very small value inside the curve crossing, where simultaneous target excitation is important (Fayeton et d., 1976). This illustrates how the light coherence is destroyed when several excitation channels contribute to the emitted resonance line radiation. The asymmetNe

Ar

-

ZI

.-

d

3n 0.5 e

0

0.0

i":l 1 2 3 Impact parameter (a.u.1

7keV *15keV

0

1

2

3

FIG.2 I . Excitation probability and fs-corrected degree of polarization versus impact parameter for the Mg(1I) 3*S-3*Presonance lines in Mg+-He, Ne, Ar collisions.

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ric Mg+-He, Ar systems show a different behavior. The transition between the regimes where mechanisms (i) and (ii) are active proceeds in a smoother way. Theoretical ab initio calculations have not yet been done for these systems, but the observations probably reflect a less diabatic behavior at the molecular curve crossings analogous to C1and C2of Fig. 6. The coherence of the light is maintained inside these crossings in agreement with the observation of weak target excitation (Fayeton et d., 1976). The deviation from unity of the degree of polarization is ascribed primarily to cascade processes (Andersen et al., 1979b). 2 . Coherence Studies r,f S

+

P Excitation

The oscillatory pattern that develops for mechanism (ii) at long collision times, and illustrated in Figs. 17, 19, and 20, has been studied in further detail for Be+-Ne. Figure 22 shows experimental values of (9,A, x) for fixed impact parameterh = 1.6 a.u. versus collision energy (compare with Fig. 20), and for fixed energies 4 and 6.25 keV versus impact parameter 0.4 P

0.2 0 1.0

x 0.5

0 TZ

m

jE r-if

X

0

c

ll

1

2

5 10 E (keV)

b (a.u.)

2

b (a.u.)

9 and coherence parameters A and x for the Be(I1) Fiti. 22. Excitation probability ' 2 2S-22P resonance line in Be+-Ne collisions. The measurements have been performed for fixed impact parameter h = 1.6 a.u. (column I ) as function of energy, and for E = 4 keV (column 1) and 6 . 3 keV (column 3) versus impact parameter.

300

N . Andersen und S . E . Nielsen

(Andersen et d., 1980). For h = 1.6 a.u., a low-energy minimum of 9 is associated with a maximum close to unit of the A parameter, and a fast change [ A x [ = 7~ of the phase angle with energy. The change in phase is easily understood from a plot in the complex plane of the corresponding ratio c i l / ~ i o (Fig. 23). The ratio passes close to zero at an energy near 4 keV. The insert illustrates how the phase angle x = arg(n,/n,) changes rapidly by -rr (Case I ) , rr (Case 2), or jumps abruptly (Case 3) in this region. The passage is so close that the experiment is not able to discriminate among these possibilities. A similar phenomenon is seen near b = I .5 a.u. for E = 4 keV. At higher energies, 6.25 keV, the curves are more smooth: the collision time is not sufficiently long for the amplitudes to develop structures. Corresponding theoretical results with the V i F potential (Nielsen and Dahler, 1980) are shown in Fig. 24. Despite the shortcomings of this potential, especially at low energies, many trends are reproduced. Furthermore, the calculations reveal the time development of the wavefunction tltrt-ing the collision. How the A structure at low energies emerges can thus be visualized by a comparison of Figs. 17 and 24. Contrary to the Be+-Ne case the A and x parameters for the Be+-He system vary in a much smoother way with h and E , as illustrated in Fig. 25 (Andersen of d.,1980). The origin of the difference between He and Ne results is not yet understood.

FIG.23. Plot of the r r , / ~ i Oamplitude ratio for fixed impact parameter h = 1.6 a.u. as function of energy in kiloelectron volts, corresponding to the first column of Fig. 22. The = r phase change near 4 keV. inset shows the origin of the

301

DIRECT EXCITATION IN ATOMIC COLLISIONS

It

X

I

I :

0

:

;:

1 : I ;

;

.,\

'

I

/-'

2

5 1 0 1 2 1 2 E (keV) b (a. u.) b (a.u.) FIG.24. Theoretical calculations corresponding to the measurements in Fig. 22. First column: ( . ' ) l.Sa.u.;(-) 1.6a.u.;(---) 1.7a.u. Secondcolumn:(-)3.6keV;(---)4 keV. Third column: 6.2 keV.

3. Colierencr Study of S + D Ercitrition

Coherent S

+

D excitation has only been studied for the process Li(?s) + He

+

Li(3d)

+

He

by subsequent analysis of the Li(1) 22P-32D decay photons (Andersen c>t

d.,1982). Experimental conditions were chosen so that the excitation is

U)

1

2

5 1 0

E (keV)

1

2

5 1 0

E (keV) (b) (0) FIG.3 . Experimental coherence parameters (a)A and (b)X as functions of energyE and impact parameter h in Be+-He collisions.

N . Andersen and S . E . Nielsen

302

-1

-1

2

10

5

2 E(keV)

5

10

-

FIG.26. Experimental and theoretical Stokes parameters P , . P , , P,. P, for the Li(1) 22P-3zD transition induced by the process Li(2s) + He Li(3d) + He at an impact parameter h = 0.95 a.u.

coherent. The Stokes parameters show a characteristic structure with collision time, while the impact parameter dependence is fairly weak. Measured Stokes parameter components (PI, P2, PI, P4) as defined in Section III,C,2,a are shown in Fig. 26, together with theoretical curves obtained in the same nine-state calculations as the cross section results of Fig. 15. The energy is varied for fixed impact parameter h = 0.95 a.u. At an energy near 3 keV, P, approaches the maximum value (for a 2P-2D transition) and P3 exhibits a corresponding minimum; P2 and P4 vary little with energy in the whole region. As expected at low energies, the agreement between theory and experiment is not quantitative, but the calculations nevertheless reproduce the major experimental features.

I

u 3.3 keV

I

I -5

I

I I I 5 10 15 L (a.u.) FIG.27. Impact-parameter calculation illustrating the development of the P3 minimum in Fig. 26. The theoretical quantity plotted is the circular polarization that would be measured if the excited state decays at the corresponding point on the trajectory.

-1

I

0

DIRECT EXCITATION IN ATOMIC COLLISIONS

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Although d-state coherence parameters cannot be unambiguously determined from the experiment, theory may still indicate how the P3 minimum develops. Figure 27 shows the variation during the collision of the theoretical parameter that is equal to P3 at infinity for energies selected around the minimum. Also, in this case, characteristic oscillations gradually develop when the collision time increases. The phenomenon may again be associated with an opposite time symmetry of the ns-nd,( -ndz) and ns-nd, matrix elements. Coherence analysis, supported by a theoretical display of what happens during the collision, thus provides very detailed insight into the physics of the collision and emphasizes the Massey parameter as the quantity governing the time evolution of the valence-electron cloud.

V. Conclusions For the m o l e c h r tneckrrnism ( i ) the primary excitation is qualitatively understood in terms of the molecular orbital MO model, and fair quantitative agreement between.experimenta1 and theoretical results based on rib inirio calculations has been obtained for the Na-Ne system up to the level of excitation probabilities. Concerning shape parameters, no theoretical results have yet been obtained from first principles to confront with the experimental results. Although this situation will probably improve, it appears questionable whether the class of quasi-one-electron systems is the optimal starting point for an understanding of shape parameters for the molecular excitation mechanism. The main purpose of the joint experimental and theoretical venture outlined above has been to gain a deeper insight into the direct excitation mechrrnistn ( i i ) . The studies have revealed that the quasi-one-electron systems are nearly ideal for this purpose, with the collision time as the essential parameter governing the time evolution of the shape and dynamics of the charge cloud of the valence electron. This evolution occurs with a characteristic frequency or time constant, roughly equal to the collision time corresponding to the maximum in the total cross section. The physical observables exhibit features with a simple, monotonic behavior at higher collision energies. With increase of collision time beyond the characteristic time, oscillatory phenomena gradually develop, causing rich structures in the probabilities and shape parameters. The physical processes are most clearly displayed when the data are scaled by means of the Massey parameter, AErrlhu, a convenient reduced collision time. Theoretically, close-coupling impact-parameter calculations are very suitable for a description of these processes. For collisions with short

304

N . Andersen and S . E. Nielsen

collision times, that is, medium and high energies, a model with just one active electron, a small basis set, and simple model potentials is often sufficient for a satisfactory description. At longer collision times, where the probability has ample time to flow around among the various states, the theoretical predictions become much more sensitive to the details of the model used. The possibility of following this development rirrring the collision much elucidates the physical processes. Simple one-electron models have enabled systematic excitation calculations for a series of quasi-one-electron systems, covering the range from low through medium to high energies. The semiempirical potentials of Bottcher, Baylis, Pascale, and Vandeplanque were designed to model the alkali-rare-gas systems at very low energies. The resulting parameterization may not be optimal, however, for the interactions which are important for collisions at higher energies. Indeed, we have found that the simple electrostatic model potential for the electron-rare-gas interaction generally performs better. More elaborate many-electron models have so far been applied only to a few quasi-one-electron systems. The results have been encouraging, and it will be of considerable interest to explore the predictions of these models at medium to low energies also with respect to shape parameters. The comparison of theory with results from all three generations of experiments constitutes an extremely sensitive test of model Hamiltonian and choice of representation and represents a major challenge for theoreticians. Further progress is therefore expected for the description of collisions of long duration. Experimentally, extension of the studies to a larger impact-parameter range (i.e., extremely small scattering angles) is desirable, together with refined energy-loss analysis of the scattered particles in order to discriminate against cascade processes and channels with simultaneous target excitation. Studies of higher angular momentum states may prove interesting, though the inherent simplicity of the (ns, np,, npZ)concept is necessarily lost. In the spirit of Massey and Gilbody (Section 1,A above), we believe that these efforts should still be concentrated on the simplest systems, where the physical processes are expected to be most clearly manifested, and where accurate results are within reach of both theory and experiment.

ACKNOWLEDGMENTS We are much indebted toT. Andersen, M. Barat, C. L. Cocke, Ch. Courbin. J. S. Dahler, J . Fayeton, J . Qstgaard Olsen, E. Horsdal Pedersen, J. Pommier, V. Sidis and P. Wahnon for

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fruitful collaboration. This research was supported by travel grants from the Danish Natural Science Research Council (NA), Institut FranCais in Copenhagen (NA), and NATO (SEN).

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 18

I I ATOMIC STRUCTURE A . HIBBERT Depurtment of Applied Mathemutics and Theoretical Physics The Qiieen'~Uni\*ersity of Belfii~r Belfast, Northern Irelond

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Simple Semiempirical Model Potentials

111. IV. V. VI.

. . . . . . . . . . . . . . .

Potentials Based on Hartree-Fock Formalism, Core Polarization . . . . . . . . . . . . . . Relativistic Model Potentials . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

309 3 11 3 17 327 332 336 338

I. Introduction While impressively accurate atomic energy-level calculations for twoelectron systems have been available for a considerable time (Pekeris, 1958; Midtdal, 1965; Accad et al., 1971), beam-foil spectroscopy has recently achieved such high resolution (e.g., Berry, 1982) that even with the most accurate calculation of the small QED effects (Drake, 1982) the theoretical values lie firmly outside the experimental error bars. Although there remains much interest in two-electron systems, many theoreticians have in recent years turned their attention to larger systems, in part because of the need for atomic data in fusion research and in astrophysics. The differences between theory and experiment in two-electron systems are minute compared with uncertainties in calculations on larger systems. As we have remarked elsewhere (Hibbert, 1975a, 1979), the degree of accuracy drops considerably in moving from two- to three-electron calculations, while for atoms or ions with more than four electrons, it is virtually impossible (from a computational point of view) to give a proper description of the correlated motion of all the electrons. Customarily, either the correlation of the outer electrons only is treated accurately, or calculations are limited to the treatment of pair correlations. Some atomic properties (including many transition probabilities and 309 Copyright @ 1982 by Academic Press, lnc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

310

A . Hihhert

collision strengths) are governed largely by the arrangement of the outer electrons of an atom or ion (those with the largest mean radius). A rather simple analysis (which may be considered as the zero-order state in a perturbation expansion) ignores the influence of the inner electrons (or core), which manifests itself in higher orders of the perturbation series. The simple analysis is clearly inadequate. For example, if we were to ignore the electronic core potential in neutral calcium, the two valence electrons would be described by a simple two-electron potential, and the wave functions and energies would be those of.Ca’8+,or of some other two-electron ion, depending on how the screening of the nucleus by the core electrons was handled. This is an unsatisfactory description of neutral calcium, but it does lead to a set of mathematical equations which can be solved to a high degree of accuracy. It is a question of balancebetween the adequacy of the mathematical model of the physical system and the computational tractability of the model. The full Schrodinger (or Dirac plus Breit) equation of a large atom is generally a good model, but computationally too complicated. To neglect the core completely is to allow the pendulum to swing too far in the opposite sense. A desirable “middle” aim would be a set of equations which describe both the correlation of the outer electrons and the important influences of the core, while being open to reasonably accurate solution within the limitations of the model. It is the purpose of this article to discuss methods of steering this middle course, through the construction of either model potentials or pseudopotentials. Some workers distinguish between these two terms. One means of doing so (see, e.g., Kahn et a/., 1976) is by labeling as model potentials those in parametric form whose parameters are chosen to give the best possible fit to some observed feature such as the low-lying energy levels. In contrast, one could reorganize the Hamiltonian of the problem so that it could be recognized as a Hamiltonian for the valence electrons with the core influence absorbed into a modified potential (pseudopotential). Since our major concern is to consider methods which are effective in treating large atoms, we shall not place too much emphasis on the distinction between these two approaches. In one sense, the potentials of the Schrodinger and Dirac Hamiltonians are model potentials. Rather more tractable are the potentials of their respective single-particle approximations-the Hartree-Fock and Dirac-Fock methods. In some ways, these methods satisfy our criteria for the middle course fairly well. They are simple in concept, being independent-particle models. They allow a core to be defined since they allow a shell description of the electrons, and the core effect is preserved by means of an orthogonality barrier through which the valence electrons have a low probability of passing. Their energy-level description is (with

MODEL POTENTIALS IN ATOMIC STRUCTURE

31 1

some exceptions) in good qualitative and moderately good quantitative agreement with observation. But their prediction of other atomic properties is more frequently poor (see Hibbert, 1975a), so that more accurate treatment of electron correlation is needed. But just as they form a good basis for such methods, so they form a good starting point for deriving model potentials also. Just as one can modify the full Hamiltonian to obtain a pseudopotential, so one can modify the Hartree-Fock or DiracFock potentials to form others-others which may be as readily soluble as are the Hartree-Fock equations themselves. We shall pursue the details of this scheme in later sections.

11. Simple Semiempirical Model Potentials Alkali atoms and ions provide an obvious testing ground for model potentials, since with only one valence electron, the analysis leads to a single equation for the one-electron wave function. The ultimate aim is to be able to treat systems with several valence electrons. But there are certain features of model potentials common to both monovalent and multivalent atoms which are worth observing. A more detailed discussion of early work is given by Weeks et a / . (1969) and by Kahn et a / . (1976), who draw comparisons and contrasts with their own work. For a monovalent ion, one would expect the behavior of the potential for large r to approach the Coulomb approximation potential -(Z - N,)/r

where Z is the atomic number and N , the number of core electrons. The solution of the Schrodinger equation based on the potential (l), with eigenvalue set equal to the ionization energy from the particular level under consideration, has been discussed by Bates and Damgaard (1949), who applied the method to the evaluation of oscillator strengths of atoms with one or two valence electrons outside closed-shell cores. In spite of the simplicity of this potential, it gives oscillator strengths which in many cases agree well with results obtained from much more sophisticated calculations, particularly for transitions from Rydberg series, and it is still widely used. But the method is not always reliable (Bromander et a/., 1978). Another very simple form of model potential arises from assuming the form (1) for r > R, for some suitable R , and for r s R , the potential could be infinite, or be equal to the value at the boundary, - ( Z - N , ) / R , or even zero.

312

A . Hibbert

A somewhat more flexible form was used by Hellmann (1935) and more recently by Szasz and McGinn (1965): V,(I')

=

- [ ( Z - N,)/I']+ ( A / r ) c Z K r

(2)

in which A and K are adjustable parameters. Following Hellmann, Szasz and McGinn (1965) replace the Schrodinger equation by the one-electron wave equation (given here in atomic units) [-$Vz

+

V,(r)]@(r)

=

&(r)

(3)

for the ground and first excited s and p states of several alkali atoms and ions, although in doing so, they assume particular analytic forms for the wave functions 0,which also contain adjustable parameters. This combined set of parameters is then chosen to give the exact experimental energy of the ground state and the best possible approximation to the excited-state energies. We wish to make two observations. First, the Values ofA are greater than the corresponding values of ( Z - ~ , ) so , that for small I ' , the potential (2) displays a repulsive barrier characteristic of potentials modeling the orthogonality effects of electronic cores. Second, the optimum values of A and K will be dependent on the flexibility of the form of @-that is, on the flexibility of the basis set used to represent @. Adequate though Eq. ( 2 ) was found to be in some cases (Hellmann, 1935), in others-such as the metals Cu, Ag, and Au-it was less satisfactory. This led Ladanyi (1956) to propose a more flexible form V,(I') = - [ ( Z - N,)/I'] + ( A / I ' ) E - ' ~ '+ ( B / I ' ) F ' ~ ~

(4)

Having obtained values for A , B , K , and A as described above for the positive ions Mg+, Ca+, Sr+, Ba+, Ra+, Zn+, Cd+, and Hg+, Szasz and McGinn (1965) use this data to obtain model Hamiltonians for the corresponding two-electron neutral atoms in the form

-IVY

-

+Vg

+

V,,(rl)

+

V , 2 ( ~+z )I ' ; .

(5)

This reduction of the Schrodinger equation for a large atom to an effective two-electron problem allowed Szasz and McGinn to represent the wave function either explicitly in terms of the interelectronic distance r I 2 or as a configuration interaction (CI) expansion. The effect of the core, particularly the core-valence orthogonality requirement is modeled by the potential. There are, in the literature, many other variants on the form of the one-electron potential, mostly sharing the basic characteristics of Eq. (2) or (4). It is worth discussing one more of these, which has been applied to a variety of atomic systems, and also two computer programs which have

313

MODEL POTENTIALS IN ATOMIC STRUCTURE

been widely used. Following a detailed analysis of earlier work, Green ef a/. (1969) proposed the form

V,(r)

=

-r-'[(Z - N,)

+ N,R(r)]

(6)

+

(7)

where R(r)

=

[H(er'd- I )

1I-l

and the values of the parameters H and d were to be chosen to give the best fit to a specific set of experimental energy levels. For large r , Eq. (6) reduces to Eq. (2), with A replaced by - N , / H and K by 1/2d, while for smallr, Vti(r)- - Z / r . This particular form has been used to evaluate oscillator strengths for transitions from the ground states of members of the lithium (Ganas, 1979a, 1980a,c), beryllium (Ganas and Green, 1979), boron (Ganas, 1979b,d), carbon (Ganas, 1979c,e,f), and oxygen sequences (Ganas, 1980b),and for neon (Ganas, 1978b)and argon (Ganas, 1978a). The results of energy levels and oscillator strengths for lithium compare very well with those from highly accurate calculations and experiment (see Tables I and 11). This is to be expected, since the lithium sequence is well suited to TABLE I ENERGY LEVELS (a.u.) OF LITHIUM RELATIVE TO IONIZATIONL I M I T Experimental Szasz energy and (Johansson, Ganas" McGinn State 1959) (1980~) (1965) -0.19660 -0.07457 -0.03883 -0.02376 -0 .O I602

Goddard (1968)

Kahn and Goddard (1972)

Victor and Laughlin (1972)

-0.19633 -0.07475 -0.03893 -0.02381 - 0.0 I605

-0.1%16 -0.07479 -0.03895 -0.02382 -0.01605

-0.198 I5 -0.074 I9 -0.03862 -0.02364 -0.01595

-0. I9809 -0.074 17 -0.03861 -0.02364

Moore ef ril. (1981)

2s 3s 4s 5s 6s

-0.19816 -0.074 I9 -0.03862 -0.02364 -0.0 I595

-0. I985 -0.0745 -0.0385 -0.0235 -0.016

2P 3P 4P 5P 6P

-0.13025 - 0.05724 -0.03 I98 -0.02037 -0.0141 I

-0. I305 -0.0575 -0.032 -0.0205 -0.014

-0.1 1233 -0.05 107

-0.12875 -0.05685 -0.03 I82 -0.01 405

-0.13020 -0.05723 -0.03198 -0.02038 -0.0141 I

-0.13005 -0.05717 -0.03 195 -0.02036

3d 4d 5d 6d

- 0.0556I

-0.0555 -0.0315 -0.020 -0.014

-0.05533

-0.05558 -0.03126 -0.02000 -0.01388

-0.05562 -0.03 I28 -0.02002

-0.05561 -0.03 I28 -0.02001

a

-0.03 128 -0.0200 1 -0.01390

Ganas' results were given in Rydbergs to three decimal places.

314

A . Hibberr TABLE 11 Li

OSCILLATOR STRENGTHS FOR

Moore Ganas (1980~)

/I

IS2 2 S * s - l S 2

Caves and Dalgarno (1972)

llp*P

Weiss (1963)

('1 ( I / .

(1981)

A"

B

HF

CI

0.749 0.0044 0.0041 0.0025

0.753 0.00450 0.00414 0.00249 0.00153 0.00099

0.746 0.00477 0.00430 0.00258 0.00158 0.00103

0.768 0.0027

0.753

McGinn ( 1969a)

~

2 3 4 5 6 7

0.749 0.00449 0.00415 0.00249 0.00154 0.00099 'I

0.768 0.0032 0.0034 0.0021 0.0013 0.0009

A , no core polarization; B . with core polarization.

such a treatment. It is interesting, however, to see how well the method compares with, for example, CI calculations for systems with more than one valence electron. In Table 111, we give the oscillator strengths obtained by Ganas and Green (1979) for the 2sZ1S+ 2s2p 'P transition in the beryllium sequence. It can be seen that the model potential values are much closer to the accurate CI calculations than are those obtained from the Hartree-Fock (HF) method. A similar situation obtains for the oxygen s ~for S ,the 2p43P sequence (Ganas, 1980b) transition 2p43P+ 2 ~ ~ ( ~ S ) nbut -+ 2 ~ ~ ( ~ S ) r transition, id~D the model potential results are around 40%' higher than the CI values obtained by Pradhan and Saraph (1977). For the latter transition, configuration mixing of the 4S core with the 2p3*Pand 2D TABLE 111 OSCILL ~ I

O R STRENGTHS (LENGTH VALUES) OF I N THE

2s2'S-2s2p'P TRANSITION

B E R Y L L I USEQUENCE M

Model potential calculations

Ion

Ganas and Green ( 1979)

Laughlin e l rrl. ( 1978)

Be I B 11 c 111 N IV OV

1.124 1.017 0.843 0.703 0.593

1.372 1.012 0.764 0.614 0.513

HF, Nicolaides McGinn (1969b) 1.766

Y I (11.

(1973) 1.68 I .45 1.12

0.887 0.743

CI. Hibbert ( 1974) 1.371 I .02 1 0.776 0.619 0.518

315

MODEL POTENTIALS IN ATOMIC STRUCTURE

is rather more important than for the former transition. Essentially, this version of the model potential scheme gives reasonably good values for oscillator strengths whenever correlation effects or configuration mixing are not too strong. The first of the two computer programs we shall discuss was developed by Klapisch (197 I). The central potential is expressed in terms of a number of variable parameters {ai}. If we assume that the radial charge density due to a closed subshell of4 electrons with orbital angular momentum I and spherical symmetry takes the form --YX1.2'+2e --(Ir

where X is the normalization factor ( Y ~ ~ + ~ + / ( 2) ~ I!, then the potential energy of another electron at radial distance r in the field of this charge density and a point nucleus of charge Z at the origin is -r--"cLf(I,

a.

1.)

+ ( Z - 411

(8)

where

x

21+1

f ( / ,a , r ) = e-ar

[I - j / ( 2 /

+ 2)](ar)'/j!

(9)

j=O

Klapisch defines an average function for completely closed I? shells: g ( L , a , 1.)

=

[2(L+

L

l)2]-1

2 (41 + 2 ) f ( I , a ,

1.)

/=0

where n

=

L

+

I . Then the form of potential used is N,

V(q, a , 1.) = -r-I

(Z

-

N,)

+ s=1 ..

y,g(L,, as,1.)

where the first summation is over completely closed shells, the second summation is over open shells, and as before, N , is the number of electrons in the core. The form of Eq. ( 1 1) clearly expresses the physical principles on which it is based. But if instead we express Eq. ( 1 1) in exponentials and powers of I ' , then the relationship with other model potentials is more noticeable. One such form, used by Klapisch (1967) for alkali atoms V(r)= -r-'[(Z - N,)

+ &4+-anr]

+ N,e-"lr + clzre-":'"+

*

*

. (12)

316

A . Hibbert

has characteristics similar to Eq. (4), but contains more variable parameters, and satisfies the boundary conditions

I n his program, Klapisch (1971) treats all the a's and (1's as variable parameters, normally chosen either to minimize the energy average of several states or to give the best least squares fit to experimental energy levels (or ionization potentials). This program has been widely used. Perhaps its power can be seen in the relatively simple calculation of the hyperfine splitting of the ground 'S states of alkali atoms (Klapisch, 1967) using Eq. (12). It is difficult to achieve accurate theoretical results for this quantity, and the ratio of the Hartree-Fock to the experimental value ranges from 0.7 for 6Li to 0.4for I T S . The values obtained from Eqs. (3) and (12) differ from the experimental values only by about 10%. The other program, also widely used, and also employing a potential satisfying Eqs. ( 13), is described by Eissner and Nussbaumer (1969). Their potential is a modification of the Thomas-Fermi potential and may be written in the form

for some suitable radius r o , wherer

v,

=

= p

~p ,=

0.8853Z-""[N/(N - I)]?

[ ( Z - ivc)/r,l - (sK:/12K,)

K , and K k are measures of the exchange and kinetic energies of the electron gas, such that K % / K , = 0. 19/tro (Gombas, 1956) and the function +(x) satisfies +"(.I-) = .r-1/2+3'2 and the boundary conditions

4(0)

=

[dJ'(.Y)],=,,

4(xo)

0, =

= (sp/I2Z)(K:/K,)x,

{4(x,)

-

[ ( Z - Nr)/Z]}.r;1

One further variable parameter, A , is introduced by redefining x = r / @ . In practice a different value of A is chosen for each value of I so as to make IEF"" - E;""I a the sum of the energy deviations of appropriate states minimum. These two programs, though very different, maintain the conceptually simple local form required by Eq. (3). Both are more sophisticated than the earlier ones we have discussed. But they differ from the other potentials in one other, very important aspect: they depend on the angular momentum of the valence electron(s).

xi

MODEL POTENTIALS IN ATOMIC STRUCTURE

317

111. Potentials Based on Hartree-Fock Formalism The model potentials described in the previous section all contain parameters which may be varied to reproduce as closely as possible the observed energy levels (or a subset of them) of the atomic system. Model potentials are normally constructed in order to evaluate some other properties of the atom or of the atom within a molecular system or a solid. For an accurate evaluation of such properties, it is necessary to have a satisfactory representation of the iiuw frrnctiotis of the appropriate atomic states. This will not necessarily be achieved when the energy spectrum is correct, particularly for atoms with more than one valence electron. The model potential parameters would normally be chosen within a single configuration approximation. The inclusion of CI could modify the wave functions and energy levels considerably, as happens for the all-electron Hartree-Fock (HF) approximation. It is desirable to achieve a satisfactory representation of hot/? the wave functions and the energy levels. This may require the use of CI for the valence electrons with a model potential for the core, which, in a single configuration approximation, could actually give a fairly poor representation of the energy spectrum, as sometimes occurs in the HF approximation. In this section, we shall discuss a number of model potentials (or pseudopotentials) which arise from the rewriting of the H F equations or those of one of its variants. For the most part, these potentials will not depend on adjustable parameters, but will model the H F approximation (and therefore its inherent inaccuracies) itself. But then the inclusion of correlation in the valence shells (e.g., by CI) will more closely match what is normally done for all-electron calculations and which in principle leads to an exact solution. In this way, the wave functions should be a better representation of the all-electron wave functions, and can be used with greater confidence in the calculation of other properties. Independent-particle models have their own local potential in the form of the Hartree potential

where the sum extends over the core orbitals. If we integrate out the angular dependence of r, we obtain a radial potential which could be used in Eq. (3). This potential ignores electron exchange, although when the valence electron is, in a probabilistic sense, far from the core, this may not be a serious problem. Otherwise, the Hartree potential may be modified by the addition of an effective exchange term (Slater, 1951; Cowan, 1967;

318

A . Hibbert

Migdalek, 1976a), which, like the Thomas-Fermi model, is based on the electron gas model. This potential (Slater, 1951) takes the form

-[(8 I /32&")

[f'nl(r)]2]"3

(16)

nl

the last term representing exchange between the valence and core electrons. Herman and Skillman (1963) further refine this process by requiring V ( r ) to be given by Eq. (16) for I' < I ' ~ while , for I' > I'o the correct asymptotic form of - ( Z - N , ) / I 'is assumed, being chosen as the radial distance at which these two forms are equal. Cowan (1967) and Migdalek ( I 976a) use variants of this scheme for which the sums in Eq. (16) are over the core electrons only and (Cowan, 1967; Rosen and Lindgren, 1968) for which the specific analytic form of the exchange term is more flexible. This process therefore has the same roots as the scheme of Eissner and Nussbaumer (1969) discussed above. But the HF method cannot be treated directly in this way for although it takes account of exchange, the resulting potential is nonlocal. Moreover, the requirement of orthogonality between the valence and core orbitals (to obtain unique HF orbitals) leads to considerable computational difficulties, particularly when the orbitals are represented in terms of analytic basis functions; the two-electron integrals are very time consuming to compute . To some extent this effort can be reduced for atomic systems by using the frozen-core HF approximation; the core orbitals are determined by a full HF calculation of the ionic core, and are then kept fixed in the HF calculation of the valence orbitals. This scheme has been discussed by McEachran st ((1. (1968), and applied to the calculation of energy levels and oscillator strengths of various atomic systems (see, e.g., Cohen and McEachran, 1978, and the references therein) for which the ( N - I ) electron core is a unique LS state. Thus, in carbon, the wave function can be written in the form

9

=

A[@(Is'2s22p2P)~(111)]

(17)

where A is an antisymmetrizing operator which also couples @ and C#I to give the required angular momenta of the state. Application of the variational principle leads to a single equation for 4(nl).We give in Table IV a selection of their oscillator strengths for sodium and compare them with other calculations. The frozen-core values lie below those of the full H F calculation or of the model potential of Weisheit and Dalgarno (19711, but

319

MODEL POTENTIALS IN ATOMIC STRUCTURE TABLE IV OSCILLATOR

STRENGTHS FOR

THE

TRANSITIONS 3SgS-np'P

OF SODIUM

Source

Type of calculation

3P

4P

5P

6P

McEachran P I d.(1969) Biemont (1975) Weisheit and Dalgarno (1971)

Frozen-core HF HF Model potential Without core polarization With core polarization MCHF with core polarization

0.988 1.05

0.0127 0.0136

0.00183 0.00199

0.00055

0.991

0.0153

0.00249

0.00081

0.969 0.965

0.0138 0.0127

0.00210

0.00065

Froese Fischer (1976)

they lie closer to values obtained when correlation and core polarization (Froese Fischer, 1976) are included. This scheme has been extended by Seaton and Wilson (1972), Seaton (1972), Saraph (1976), and Radhan and Saraph (1977) tb allow several core functions to be coupled to one-electron functions as in Eq. (17). For example, Saraph (1976) first approximated the 1sg2sz2p6 Mg2+ core HF functions, using the code of Eissner et 01. (1974)SUPERSTRUCTURE-which minimizes the total energy by varying the parameters in a statistical model potential of the form (14). The valence orbitals 3s, 3p, 3d, 4s, and 4p of Mg+ were obtained in an effective frozen-core scheme (17), not in fact by solving the usual bound-state problem, but by solving the integrodifferential equations of the closely related Mg2++ e- scattering problem. The Mg wave functions are written as a sum of terms in each of which a Mg+ core function is coupled to a one-electron function, and these new functions are determined by solving the corresponding equations of the Mg+ + e- problem (see also Burke and Seaton, 197 1 ) . The wave function is therefore of CI type, and although the results of energy differences and oscillator strengths compare well with the best available, we note that we have moved far from our search for a local potential representing the core, in which the problems of corevalence orthogonality can be overcome. Moreover, while frozen-core schemes may be tractable for atomic systems, they are still very time consuming for large molecules. If we consider an atom with a single valence electron, then the H F equation for the valence orbital 4, is

HHF4,. = [-fV- ( Z / r ) + VHF14" =

€"$"

(18)

320

A . Hibbert

where V H F is the sum of the usual direct and exchange terms and is nonlocal. A local operator UHFcan be derived from V H Fin a formal manner by writing UHF

=(

v HF4)/ 4\

(19)

\

so that Eq. (18) becomes [-$V2 -

(Z/I') +

UHF]+\

=

€\4\

(20)

This substitution does not help in the determination of , but if @\ is the valence orbital of the lowest lying state, and if U H Fand V H Fresult in the same (or at least very similar) spectrum of states, then U H Fcould be used to generate the valence orbitals {4i} of higher lying states: [-$V2

-

( Z / I ' ) + U"F14, =

(21)

Since 4, is a HF orbital, it will be orthogonal to core orbitals. In particular, the part PI of & will be orthogonal to core radial functions of orbitals with the same I value as 4,. For these cases, P, will have zeros (nodes) which in general will not coincide with the zeros of V H F . Consequently, U H Fwill have singularities for certain values of I ' , which will artificially of { &} to have zeros at these same radial force the radial functions {Pi} distances. An alternative treatment of orthogonality is provided by the pseudopotential scheme set out by Phillips and Kleinmann (1959) and discussed by Szasz and McGinn (1965) and by Weeks et a/. (1969). If we write

where is a core orbital and ( I , is a constant, the substitution of Eq. (22) into Eq. (18) gives

Then if HHF&=

E&,

Eq. (23) can be written as

where

since from Eq. (221, a, = Cx,l&). In this way, the orthogonality of x, to the core orbitals is removed and its effect replaced. by the pseudopotential Vp. It is different in character from the model potentials described earlier

MODEL POTENTIALS IN ATOMIC STRUCTURE

32 1

in that it does not depend on parameters which are to be determined semiempirically, and the resultant potential is nonlocal. Since each core orbital 4, is a solution of Eq. (24) with the same eigenvalue E , , then the values of { ( I ( - } in Eq. (18) are arbitrary, and in general can be chosen so that x\. is nodeless except at I’ = 0 (Szasz and McGinn, 1967). In this case it is possible to define a local potential

up= K V H F+

(26)

VP,X\I/X,

without singularities. Szasz and McGinn (1967) and McGinn (1969a) determine pseudopotentials for monovalent atomic systems by first performing a full HF calculation on the core, and by determining V‘ in the frozen-core approximation. Their results for energy levels and oscillator strengths are comparable with other schemes (see Table 11). The freezing of the core orbitals is a satisfactory approximation in such cases. This approach has been extended to systems with two valence electrons by Szasz (1968). There are further pseudopotential contributions, in addition to Vp, giving a combined effective one-electron pseudopotential, which we shall write as Vr”. The Hamiltonian takes the form 2

HI2

=

C [-$Vf - ( Z / I ’ ~+) V y F + Vf“] +

~/1’12

(27)

i= 1

where c‘ =

Ni’

=

NO(2 - N o ) (XolXo)

where X , is the two-electron valence eigenfunction of Eq. (27): H E X , = Edr,

(30)

Since X , is a two-electron function, it may be represented either as a product of one-electron functions (McGinn, 1969b),for which the energies and oscillator strengths should be comparable with (frozen-core) HF (see Table III), or, in terms of interelectronic coordinates (Szasz and McGinn, 1972), allowing a proper treatment of valence correlation. This version of the pseudopotential formalism is applicable to any electron, or pair of electrons, not merely to valence electrons, but we shall not pursue this possibility, since our principal interest is in considering physical effects in which the valence electrons play the most important role. One of the unsatisfactory features of the use of Eq. (26) is the nonuniqueness of x\., associated with the arbitrariness of the coefficients { o c } . Kahn and Goddard (1972) demonstrate that in the core region the pseudopotential U p is strongly dependent on the choice of { a c } .The requirement of a nodeless xv is not sufficiently restrictive. Possible schemes

322

A . Hibbert

for specifying x, more precisely have been discussed by Cohen and Heine (1961). An alternative procedure for determining x, has been proposed by Goddard (1968). In his GI method, the wave function is a spin-projected HF determinant in which each orbital is allowed to be spatially different. For the case of the ground state of lithium, this "different orbitals for different spins" scheme is equivalent in CI or MCHF terminology to allowing configuration interaction between ls22s and lsf22s.The GI radial functions are not required to be orthogonal. The GI orbitals satisfy HFtype equations

H"'4,

(3 1)

= c,$,

where the G1 potential is again nonlocal and, as in the H F method, can be considered as the average potential due to the other ( N - 1) electrons. The orthogonality requirement for the H F Is and 2s orbitals results in a node in the 2s function. The removal of the orthogonality constraint in the GI method causes the G I 2s function to be nodeless, and therefore allowable in Eq. (26) to determine a local effective potential. Specifically if we rewrite Eq. (31) as

(-W - ( Z / r ) + V")42,

=

€2\42$

(32)

Ep,&,

(33)

then

(-fV* - ( Z / r ) +

u(;')&, =

where U'rl

=

cZ,

+ (Z/r) + 4Vv242,/42,

(34)

is a local potential which may be obtained once Eq. (32) has been solved for &, and e2,. One assessment of the usefulness of such a potential is the comparison of its spectrum with that of HF or of experiment. Goddard (1968) has computed the energies of various states of Li with Eq. (34) and we display his results in Table I. It is clear that, while the energies of the *S states are in reasonable agreement with experiment, those of other angular momentum are less well described. Kahn and Goddard (1972) found it necessary to modify the above method, by defining a different potential U f l for each I value, defined by

WVr)

=

+ (z/r) + 4V24nr/4n/

(35)

where ( & , , r , e n / )form the solution of the GI equation with the lowest state for each I: 2s, 2p, 3d, etc. The explicit I dependence of Eq. (35) is seen by noting that hV24nr/+nl

=

- [ I ( / + 1)/2r21 + (f'Y2pnr)

(36)

MODEL POTENTIALS IN ATOMIC STRUCTURE

323

where r-lPn1(r)is the radial part of &[. The results of energy values of excited states obtained in this way are also shown in Table I. It can be seen that, while the 2S states have energies, as expected, essentially the same as those of Goddard (I%@, the energies of the 2Pand 2Dstates are much improved. This establishes the need for /-dependent models or pseudopotentials, which may be written in the form

where llrn ) (Iml is an angular momentum projection operator and U,(r)is given by Eq. (35). [Equation (35) depends on angles only through the angular momentum I, but, from Eq. (36), is a radial operator.] Kahn and Goddard (1972) found that while Us and Up differed considerably, Up and Uc,were quite similar. In general, it is a reasonably good approximation to assume that Ul can be replaced by U,,, where / > L and ( L - 1) is the largest angular momentum of the core orbitals. The effective local potentials { U / , / < L } represent the direct Coulomb and nonlocal exchange potentials of HF together with the core-valence orthogonality effect, the last being comparable with the potential V p of Eq. (25). For / 3 L , the valence functions are automatically orthogonal to the core, so that I// models only the direct and exchange potentials. While it is true that the G1 orbitals used in the above analysis are unique (unlike the HF scheme where the orbitals are determined only up to a unitary transformation), it is not the only possible scheme for obtaining a suitable function x, for use in Eq. (26). Thus, for example, O'Keefe and Goddard (1969a,b), following Cohen and Heine (1961), in calculations on solids, use effective potentials (26) with x\ written in the form (22) and the coefficients { r r , } chosen to minimize the kinetic energy of x\. They find that the x\ determined in this way is very close to the G1 orbital. This suggests that it might be worthwhile considering other ways of choosing the coefficients {uc}. Kahn et trl. (1976) and Melius and Goddard (1974) have examined this procedure from essentially the same point of view. Their aim is to choose the {cI,} so that xl should have no radial nodes (except, for I > 0.at I' = 01, should have the smallest possible number of spatial undulations, and should follow the H F valence orbital as closely as possible. Melius and Goddard (1974) simply determine (0,)so that x, is the best least squares fit to the Slater function n', -cr

(38)

= (-241'2

(39)

where

5

324

A . Hibbert

The rather more sophisticated scheme of Kahn rt crl. (1976) requires the minimization of the functional

subject to

The first term in (40) involves the expectation of the core projection operator and is included to emphasize the requirement that x\ should approximate the H F valence orbital. The other requirements for x\ , listed above, are emphasized by the second term in (40). The parameter A allows the weighting of these two terms to be varied. The additional constraint (41) is included to reduce the number and amplitude of oscillations of x\ in the core region. Both methods lead to functions x\ which are small and nodeless in the core region. These are important properties, especially if the functions are represented in terms of an analytic basis set. For example, Melius and Goddard (1974), in representing x4% of potassium in terms of Gaussian basis functions, find that only a few such functions are needed to give an adequate representation, and those essentially describe the valence region. The large-exponent core-dominated basis functions, important in the HF representation to reproduce the cusp condition at the origin, are no longer necessary because of the flat nature of x\ close to r = 0. Since the time for calculating the two-electron radial integrals is essentially proportional to the fourth power of the number of basis functions, this reduction in the number of basis functions represents a large saving in computational time in molecular calculations to which Gaussian functions are well suited. For atomic calculations and those molecular calculations which are more suited to Slater functions, there are also substantial savings in time, though not so dramatic as with Gaussian functions, since the number of Slater functions needed for H F is generally not so great. An analytic expression for the effective potential determined by direct inversion of x\ is still rather complicated even when x\ is represented by only a few basis functions. A numerical representation of this potential is more easily achieved, and it is possible to fit this numerical function to an assumed analytic form if so desired. But it is interesting to look at an even simpler version of this process. As we discussed above, Melius and Goddard (1974) determine the coefficients { a c } in Eq. ( 2 2 ) by fitting to a single Slater function (38). The resulting x\ is then a linear combination of HF orbitals or of the basis

MODEL POTENTIALS IN ATOMIC STRUCTURE

325

functions (Slater or Gaussian) used to represent them. But it is not equal to a single basis function. If, however, we do simply replace xv by the single Slater function (38), we obtain a quite simple form of I/, , using Eqs. (35) and (36):

The first right-hand term in Eq. (42) adds a repulsive barrier of t n ( n + I)/r2 to the usual centrifugal term, thus preventing collapse of the solutions of the model potential equation into core orbitals. The second term simulates the screening of the nucleus by core electrons and the final term is zero if the choice (39) is made. We (Harte and Hibbert, unpublished) have performed some calculations with a model potential of the form U1(r) =

n(n + 1) - /(/ Zr'

+

1) -~ Z

-

N,

I'

+ -Ae-"' r

(43)

For / 3 L , we have chosen n = / so that the first term vanishes, leaving us essentially with a Hellmann potential (2). For 1 < L, we choose the value ofrr for which the Slater function (38) gives the best fit to the outer loop of the lowest valence orbital P n l .The parametersA and K are then chosen so that the lowest solution of

[-tv2+ ul(r)l4nd1')= E n l 4 n d Y ) is the Slater function (38) and cnl is the experimental orbital energy for the appropriate monovalent system. In Tables V and VI we present some values of oscillator strengths for magnesium and silicon, using these potentials. Velocity values are not given for model potential calculations, since they emphasise the form of the orbitals in the core region, which is generally different from all-electron orbitals. For the case of an atom with N, valence electrons, we have written the valence Hamiltonian as

x N,

H, =

N,

h(vi)

+

I: r;'

(44)

{ L. The N,-electron wave function was of CI form. The radial functions were obtained as sums of Slater orbitals, the exponents of which were optimized on the energies of the two states involved in the appropriate transition, using the computer code CIV3 (Hibbert, 1975b) modified to

326

A . Hibbert TABLE V OSCILLATOR STRENGTHS IN MAGNESIUM All-electron with valence correlation

Transition 3S*'S

J;

f,

-

Hibbert"

Froese Fischer ( 1975)

Harte and Hibbert (unpublished)

Victor and Laug hl in ( 1973)

1.743 1.711 0.157

I .757 1.736 0.160

I .732 -

1.717

0.155

0.619 0.636 0.048

0.670 0.049

3s3plP:

-

S E (a.u.) 3s3p'pU 3s3dlD:

J; h

Model potentials

S E (a.u.)

Using wave functions of Thompson

ci

ti/.

(1974). The same configurations were used

by Harte and Hibbert to represent the valence wave function in their model potential

calculation.

include the model potential (43). The results we obtain in this way are quite close both to the all-electron CI calculations which take correlation into account in the valence shells and, for Mg, to the more sophisticated model potential calculations of Victor and Laughlin (1973). The use in a multivalent Hamiltonian (41) of the model potentials which have been derived for the corresponding monovalent ion follows the scheme of many other workers, including Szasz and McGinn (1965) and Victor and Laughlin (1973). It is interesting to note how well such a scheme performs with four valence electrons (see Table VI). Kahn et (11. (1976) note that this procedure can give poor results when applied to molecules, and they propose instead a pseudopotential that is based on TABLE VI O S C I L L A T O R S T R E N G T H S OF T H E

3S*3pp3P+ 3S'3p4S3P0 S T A T E

OF

si I

Calculation

J;

A € (a.u.)

Model potential (Harte and Hibbert, unpublished) CI with valence correlation (Hibbert, 1978)

0.225

-

0.197

0.207

0.192

0.189

MODEL POTENTIALS IN ATOMIC STRUCTURE

327

neutral atom HF wave functions. For atoms with several valence orbitals, the HF equations take the form

where J , , K,. are the usual core direct and exchange operators and W,. is the sum of direct and exchange interactions among the valence orbitals { I $ , } . As with monovalent systems, these equations may be solved for the lowest valence orbital for each I value, and, for each I < L, the pseudoorbital can be determined as before [e.g., by choosing the coefficients in Eq. (26)l. The pseudoorbital equation

I -3v2 - (274+ U(r) +

W,({XV,

(where { 4; } are the HF orbitals with I pseudopotential U(r)=

E\

3

4~.l~lx,. =

f V X \

L ) can be inverted to give the

+ ( Z / r ) + [(R2+ W,)x,l/x,.

(47)

which is again / dependent. This pseudopotential takes explicit account of the valence-valence interactions, whereas the potential based on a monovalent ion does not. The use of ionic orbitals in model potentials for bivalent atoms has already been seen to be moderately successful, but the effective potential experienced by the valence electrons of the-atom can be expected to differ increasingly from the corresponding monovalent ions as the number of valence electrons increases. Equation (47)provides a possible means of taking this change into account. A similar procedure to Eq. (47)has been explored by Melius et a/. (1974)to obtain model potentials for Fe and Ni. They find that the use of Eq. (47) reproduces energy separations much more closely for Fe than a model potential such as Eq. (35) based on the monovalent ion Fe7+.

IV. Core Polarization The pseudopotential (model potential) schemes discussed in the previous sections assume that the core can be treated as static, that interactions between the core and valence electrons can be ignored. In general, this cannot be valid. For example, in the case of 2p43P+ 2p%d3Dtransitions in oxygen considered by Ganas (1980b), it is not sufficient to include only

A . Hibbert the 4S term of the 2p3 “core” of the excited state. There should be configs config)ii uration interaction with 2p3(*P)nd,2p3(*D)nd,and also 2 ~ ~ ( ~ D urations. The implied frozen-core assumption of the model potentials could be modified either by treating only Is* as the core, or by using several different cores, as adopted by Seaton and Wilson (1972). The question of what actually constitutes the core is also raised by the *D states of copper, in which the Rydberg series 3dI0tid is perturbed by 3d94s2.Indeed, the existence of such perturbers generally turns a system with few valence electrons into one with many. But even if one does peel back to a ‘S core, which is common to all states and/or configurations, the assumption of a static core is still not justified and may not be a satisfactory approximation. The electric field of the valence electrons will modify or polarize the core, and this will affect the results obtained from either an all-electron frozen-core scheme or a pseudopotential treatment of the valence electrons based upon it. For a single valence electron, first-order perturbation theory produces a modification to the potential by the addition of the term V,,(I‘)= -ac1/2r4

(48)

for large I ’ , where atIis the static polarizability of the core. Clearly this additional potential cannot be allowed to extend within the core region without modification. One possibility (Beigman et d., 1970) is to modify Eq. (48) to the form VL(I’)=

-ad/[2(Y?

+ r.3’1

(49)

which has the same asymptotic form as Eq. (48) but which remains finite for small I ’ , with Vi,(O) = V,,(r0), so that ro represents an effective core radius. Alternatively, the values of acland ro can be treated as parameters adjusted to fit alkali spectra (Beigman et 01.. 1970). The same asymptotic conditions are satisfied by the potential VL(r) = -a,,r””2(r~ + I”0) 31

derived by Bayliss (1977). A different type of cutoff process is used by Dalgarno and co-workers (Dalgarno et a / . , 1970; Weisheit and Dalgarno, 1971; Caves and Dalgarno, 1972): where

W,(x) = I

-

exp(-.rn)

(52)

and the second term in Eq. (51) represents the quadrupole polarizability of the core. For alkali systems, Dalgarno ef a/. (1970) set up a model poten-

MODEL POTENTIALS IN ATOMIC STRUCTURE

329

tial consisting of the Hartree potential, V:, given by Eq. (51) and additional terms containing variable parameters: ~ ( r=) 2

2 j[l+c(r~)l~/lr- r'lldr' C

+ VY, + Ar-Kr + BrePKr

(53)

the factor 2 arising because of doubly occupied orbitals &(r). The parameters a;, A and B are chosen to give the best least squares fit to the alkali spectrum, while K = r i l . This potential therefore combines the physically based Hartree and polarization potentials with the model potential characteristic of parameter optimization. Since even less sophisticated procedures obtain reasonably good energy levels and oscillator strengths for alkalis, it is not surprising that the very accurate results listed in Table I for the Li energy levels were obtained (Victor and Laughlin, 1972). An analysis of the effect of core polarization on oscillator strengths has been carried out by Hameed of cil. (1968) using perturbation theory. If Hc denotes the Hamiltonian of the core, and QC the lowest eigenstate, and if V(R, r) denotes the Coulomb interaction between the core and valence electrons (coordinates R and r, respectively), then a possible model potential for the valence electron of a monovalent atom is

H,. =

-tV2

- (Z/r)

+ V(r)

(54)

where is the expectation of V with respect to the core function aC. It is possible to set up a perturbation scheme based on ( H , + H,.) as the zero-order Hamiltonian, and

A = V(R, r)

-

v(r)

(55)

as the perturbation. To first-order in A, the dipole matrix element ("flrlYi)for transitions between states described by qi and Qf is modified to

(w -

(Y(r/r3)~w

(56)

where a is the polarizability of the core at the transition frequency. In most applications, the transition frequencies of the valence electron will be small compared with the excitation frequencies of the core, and so that core polarization may be assumed to follow the valence electron adiabatically, and the sturtic dipole polarizability of the core cqI, may be used in place of (Y in Eq. (56). The correction in Eq. (56) differs from the original operator by the factor a d / r 3 . A similar factor occurs (apart from constants) in comparing Eq. (48) with the electron-nucleus potential - Z / r . Hameed rt ( I / . (1968) find that the core polarization effect is small for the

330

A . Hibbert

resonant transition in lithium, reducing the oscillator strength by less than 1%. But for the resonant transition in cesium the size of this reduction increases to 16%. An even more striking effect is noted by Weisheit and Dalgarno (1971) in the 4s np transitions in potassium. We give a subset of their results in Table VII. It may be seen that while the resonant transition is reduced by only a few percentage points, the oscillator strengths for n > 12 are reduced by a factor of more than three. For two valence electrons, Victor and Laughlin (1972) use a Hamiltonian of the form

-

where the parameters of LI(ri)are those determined for the corresponding monovalent ion. The valence two-electron wave function is then expressed in CI form. In Tables I11 and V, we present some of their results for oscillator strengths of transitions in the beryllium sequence (Laughlin c v d., 1978) and in magnesium, respectively. It may be seen that the model potential results compare well with.other CI results. The use of Eq. (57) is therefore seen to be a satisfactory alternative to the all-electron Hamiltonian for these effective two-electron systems. For more than two valence electrons, the difficulties experienced by Melius et a/. (1974) in employing model potentials derived from monovalent systems are likely to be met in the use of Eq. (53). As for monovalent systems, the magnitude of the core polarization correction increases with the number of core electrons, although not monotonically. Hameed ( 1972) has analyzed the effect of core polarization TABLE VII E F F E 1CO

F C O R E POLARIZATION ON

f II

4 6 8 10 "

4S-!lp

f'

'

(no correction)

(with correction)

1.03 (O)* 1.39 (-3) 1.48 (-4) 4.09 ( - 5 )

9.80 ( - I ) 9.14 (-4) 7.14 ( - 5 ) 1.60 ( - 5 )

Weisheit and Dalgarno (1971) lo".

* p(q) implies p x

O S C l l I ATOR S T R E N G I H b IN POT4SSIUhl"

11

12 14 16

.I'

f'

(no correction)

(with correction)

1.69 (-5) 8.61 (-6) S.06 (-6)

5.75 ( - 6 ) 2.66 ( - 6 ) 1.47 ( - 6 )

MODEL POTENTIALS IN ATOMIC STRUCTURE

33 1

on energy differences and oscillator strengths for two-electron systems. In particular, he considered the correction to the nsnp 'P + 3Penergy difference, which is an example of the modification to the valence-valence interaction, and which the last two terms in (57) model. Using a perturbation treatment similar to that of Hameed et al. (1968), Hameed (1972) showed that the dominant correction to the 'P + 3Penergy difference is approximately AECP

= -3(y

while the proportional correction to the n s2'S strength is approximately

4fCP/f= -2adL/Z1)

-

(58) n sn p 'P oscillator

(59)

In Eqs. (58) and (59)

12 =

lr: Pn.s(r)pnp(r)r+ d ~ '

(61)

and r 1 P n S ( r )r-*Pn,,(r) , are the radial functions of the orbitals. An alternative to the use of a polarization potential such as Eqs. (49)(5 1) is the modification of calculations which have not included core polarization, by means of Eqs. (59)-(61). This process has been applied (Hibbert, 1982) to some all-electron CI calculations of the 5sz ' S + 5s5p 'P transition in the cadmium sequence, but with correlation only included among the valence electrons. The oscillator strengths are reduced by 20% for Cd I increasing to 30% for Xe VII. Substantial corrections also occur in the alkaline earth atoms (Hameed, 1972). If such corrections are to be determined with some degree of accuracy, it is essential to have corresponding accuracy for a d ,and to have a reasonable means of determining r,, the effective core radius. Several schemes for obtaining Y, have been proposed. Hameed (1972) uses the value of I' for which the probability density of the core falls to 10% of its maximum value. Hafner and Schwarz (1978b) use twice the mean radius of the outermost core functions. More refined calculations of ro would appear to be unnecessary. It is rather less straightforward to obtain values for ad.For neutral atoms, the literature does contain a range of values, although the highest and lowest quoted for a given atom can differ considerably. For heavy ions, the available data are rather sparse. If core polarization corrections are going to be made generally, particularly for the heavier atoms and ions, then it is necessary to direct research effort toward the evaluation

332

A . Hibbert

(or experimental determination) of dipole (and perhaps quadrupole) can polarizabilities of such atoms. An estimate of the effect of errors in aCI be made by eliminating 1, from Eqs. (58) and (59), so that the proportional change in the oscillator strength (59) depends on An error of 25% in for a 30% core polarization correction (59) results in an error of only 3-4% for the oscillator strength, which is well within the uncertainties of most calculations. On the other hand, an error of 25%) in a d ,for a 70% core polarization correction (59), leads to an error of 20-30% in the[ value. Polarization potentials V,’, have been added to other model potentials. Friedrich and Trefftz (1969) add Eq. (49) to a model potential similar to that of Klapisch (1971). Moore and Liu (1979) derive a polarization potential rather more general than Eq. (51) and add this to a statistical exchange model of the HF potential (Herman and Skillman, 1963). The process is applied to lithium (Moore ct (11.. 1979) and to other alkali atoms (Moore et NI., 1981). We present some data for lithium in Tables I and 11, and it can be seen that the results are in general agreement with other methods.

V. Relativistic Model Potentials The potentials we have discussed so far have been applied in a nonrelativistic approximation, and we have considered oscillator strengths only of allowed transitions. If we wish to consider forbidden transitions, it is necessary to use .&j’ (or at least intermediate) coupling of angular momenta, and this requires some treatment of relativistic effects. For heavy atoms and ions, a relativistic treatment is necessary, even for allowed transitions. Since it is for atoms with many electrons that a model potential formalism is especially useful, we must now turn to a discussion of relativistic model potentials. We shall see that many of the main schemes developed for a nonrelativistic treatment have been modified to incorporate relativistic effects. There are two ways of treating relativistic effects: either by the addition of the Breit-Pauli operators to nonrelativistic equations, or by the fully relativistic Dirac method. In each case, we shall be seeking to derive a local model potential or pseudopotential which models the core and relativistic effects sufficiently to allow the accurate calculation of atomic properties. Laughlin and Victor (1974) have extended the use of the model potential of Weisheit and Dalgarno (19711, which allows for corrections due to core polarization, to the calculation of transition probabilities of intercombination lines. They first determine one-electron eigenfunctions of a

333

MODEL POTENTIALS IN ATOMIC STRUCTURE

Schrodinger equation with pseudopotential (53). These are then used in a CI expansion of the two-electron wave function with Hamiltonian (54). The relativistic effects are treated by a perturbation analysis, with this two-electron wave function as the zero-order function and the Breit-Pauli operators: spin-orbit, spin-other orbit, spin-spin, as the perturbation. For the spin-orbit interaction, Laughlin and Victor use

where the sum runs over the valence orbitals, and Z , is chosen to give the best possible fit to the fine structure of the monovalent positive ion. For the n'L states of the ion, Z , is approximately independent ofn,for fixed L. The method has been applied to intercombination lines in beryllium and magnesium by Laughlin and Victor (1974), and to the 2sz1SO+ 2s2p3P1 transition in the beryllium sequence by Laughliner a / . (1978). In Table VIII we compare the latter results with those of other methods which include all the electrons in the calculation, but in fact only treat correlation for the valence electrons. The model potential transition probabilities are somewhat larger than other recent work, but set against the H F values of Garstang and Shamey (1967), all the correlated methods are in moderate agreement. The difficulty with this transition is in obtaining the correct magnitude of the components of the 'PI configurations in the 3P1 wave function. For fairly small atoms, the inclusion of the spin-dependent Breit-Pauli operators is usually sufficient to determine energy levels and oscillator strengths. But as the number of electrons increases, the other non-finestructure operators of the Breit-Pauli Hamiltonian, particularly the TABLE VllI TR A N S I TI O PROHAHIL N I T I E S(in sec-') FOR T H E 2sz1S0+ 2s2p3Pl T R A N S I T I O N I N T H E B E R YLII U M SEQUENCE

I on Be I B 11

c 111 N IV

ov

Ne V11

Model potential, Laughlin et a / ( 1978)

Miihlethaler and Nussbaurner (1976)

Glass and Hibbert (1976)

HF, Garstang and Sharney (1967)

0.27 10.65 110.4 604.2 2374 20,500

0. I7 96.0 ? 5.0" 2180 17,100

0.24 8.1 86.0 495 I990 17,400

0.71 20.0 I90 920 3600 29,000

CI

" Nussbaumer and Storey (1977).

9

CI,

334

A . Hibbert

Darwin and mass correction terms, begin to play an important role. Hafner and Schwarz ( 1978a) have incorporated these operators into the model potential, arguing that the felativistic behavior of the core electrons has an influence on the outer electrons. Accordingly, for a monovalent atom, their one-electron Hamiltonian for the valence electron is written as

+ V(r) + 2

Herr= - 3 V z

Vu(r)l!jmj)(~m,l

(63)

1.l.ml core

The functions VLj(r) incorporate the effects of the pseudopotential which models the core-valence orthogonality, together with the spin-orbit, mass correction, and Darwin terms. The sum over angular momentum in Eq. (63) is similar to that in Eq. (37), with the additional option ofj-dependent radial core functions and therefore potentials V N ( r ) Hafner . and Schwarz (1978a) choose V ( r ) in the form of the Hellmann potential (21, while V l j ( r ) are represented as sums of exponentials, the parameters being chosen to reproduce the spectrum of the monovalent system, including the finestructure splitting. Denoting the complete Hamiltonian in Eq. (63) by HeR, Hafner and Schwarz (1978a) consider atoms with two valence electrons using the Hamiltonian H ( I , 2) = H;ff + Hprf+ r F . , writing the two-electron wave function as a CI expansion. The wave functions were used by Hafner and Schwarz (1978b) to compute transition probabilities of transitions in a number of heavy atoms and ions. They corrected the results for the effects of core polarization, as described in the previous section. It can be seen from Table IX, where we present results for transitions in copper, silver, and gold, that these effects can be substantial. Although the spread TABLE IX O S C I L ~ A TSTRENGTHS OR FOR

rHE

PRINCIPA RESONANCE L L I N E SOF Cu, Ag, Au Hafner and Schwarz (1978b)

Ion cu I

Transition 4s + 4p

Ai3 I

5s + 5p

Au I

6s

‘I

Lvov (1970).

-

6p

j

+J

Without With polarization polarization

‘

4-4

Moise (1966).

0.280 0.560 0.196 0.410 0. I49 0.350

0.336 0.677 0.361 0.739 0.364 0.802

1-1 1-1 1-1 1-1 1-1 L‘

Curtis P I

ti/.

(1976).

Migdalek and Bayliss (1978)

Experiment

0.214 0.432 0.198 0.413 0. I48 0.339

0.22,“ 0.153’’ 0.41,’ 0.322” 0.247,“0.196” 0.506,“0.459b 0.19,” 0.076b 0.41,“ 0.18”

‘‘ Penkin

and Slavenas (1963).

MODEL POTENTIALS IN ATOMIC STRUCTURE

335

of experimental results is rather wide (in Table IX we give only a small selection of them), the oscillator strengths evaluated with core polarization corrections are much closer in line with experiment than are thef values obtained without such corrections. An alternative local one-electron potential is given by Cowan and Griffin (1976). They incorporate the mass correction and Darwin terms into a local form of the nonrelativistic HF equations: 2r2

=

ciPi(r)

+

V i ( r )- Aa2{q - Vi(r)}2

(64)

In Eqs. (64), (Y is the fine structure constant, and V,(r)is a local version of the HF potential (Cowan, 1967) in which the nonlocal terms are replaced by a local statistical exchange potential. The spin operator is not included in Eqs. (64) so that the radial functions {Pi } do not dependbn j. The set of equations (64)are solved self-consistently, in a manner similar to the basic HF equations. Cowan and Griffin applied this process to the 5f36d7s2state of uranium, and compared their values of a number of atomic properties with those from Dirac-Fock (DF) calculations. The major relativistic effects observed by comparing DF and HF orbitals are seen to be included in the solution of Eqs. (64): for example, the relativistic contraction of the s and p orbitals and the relativistic expansion of the outer d and f orbitals. Moreover, the binding energies of closed-shell atoms obtained from Eqs. (64) are much closer to the DF values than are either HF, or nonrelativistic HF to which are added the mass correction and Darwin contributions obtained by first-order perturbation theory. The modeling of the DF Hamiltonian thus requires these relativistic operators to be incorporated into the potential. Nevertheless, the omission of the spin operator does mean that spin-orbit parameters do differ, substantially in some cases, from DF values. The use of Eqs. (64)leads to a set of orbitals {Pi}, which, apart from their j independence, are quite similar to the DF orbitals. Kahn et a/. (1978) have generated pseudopotentials with these orbitals in an analogous manner to the nonrelativistic case (Kahn et a / . , 1976). The relativistic effects are contained in the potential, so that the equations for the valence orbitals are nonrelativistic, and spin-orbit terms are added by means of perturbation theory. The pseudoorbitals {Qv}were written as linear combina-

336

A. Hibbert

tions of core and valence orbitals

[compare Eq. (22)], and the coefficients {(I,} were chosen to satisfy conditions equivalent to Eqs. (40) and (41), with A infinite. The resulting (nodeless) pseudo-orbitals {&} (where v = nl, I < L, and n is the lowest value not found in the core orbitals) were then used in Eq. (47) to obtain /-dependent local pseudopotentials. For / > L, the valence orbitals { P , } were used directly in Eq. (47). Model potentials or pseudopotentials can also be derived on the basis of the Dirac or DF equations. The derivation of a pseudopotential which is dependent both o n j and / has been considered by Lee et t i l . (1977). Their approach is a generalization of the nonrelativistic treatment developed by Kahn et a/. (1976) and therefore parallels the work of Kahn et ( I / . (1978), but with the basis of the DF method. Since in practice Lee et d.(1977) do not include the small component part of the relativistic wave functions, and since the valence orbitals are treated nonrelativistically (the relativistic effects being included in the pseudopotential), the major difference from the work of Kahn et d.(1978) is the j dependence of the potentials. Migdalek (1976a) has solved an approximation to Dirac-Fock equations in which the nonlocal potential is replaced by a local statistical exchange potential similar to that of Cowan (1967). Both large and small components were used in the evaluation of oscillator strengths of effectively one-electron systems (Migdalek, 1976a,b,c), and the influence of core polarization is added through the potential (50), (Migdalek, 1980; Migdalek and Bayliss, 1978, 1979a,b). In Table IX, we present values obtained in this way for the resonance transitions of copper, silver and gold. For the last two atoms, there is close agreement between Migdalek and Bayliss (1978) and Hafner and Schwarz (1978b). Their results differ for copper, but we note that the sum of the 4 - 4 and 4 - Q oscillator strengths of Migdalek and Bayliss is much closer to the accurate nonrelativistic result of 0.624 obtained by Froese Fischer ( I 977) than is the sum of the values obtained by Hafner and Schwarz.

VI. Conclusions In this article we have described and compared a number of different methods of treating the effects of core electrons on the valence electrons by means of a potential or potentials, so that the equations for the valence

MODEL POTENTIALS IN ATOMIC STRUCTURE

337

electron functions contain no explicit reference to core functions. The alkalis, with a single valence electron outside ' S cores, are natural testing grounds for such methods. As far as their application to lithium is concerned, a glance at Tables I and I1 reveals that the methods we have described are of comparable accuracy. For two-electron systems, the results obtained by the various methods differ from each other largely because of the differences in the degree of correlation included among the valence electrons (by configuration interaction or the use of interelectronic coordinates) and, for the heavier systems, by whether or not core polarization effects are included in the calculation. Relatively simple model potentials with a few variable parameters and a core polarization term seem adequate in these systems. It is in the case of atoms with several valence electrons, or of heavy atoms where there are relativistic changes in the spatial character of the orbitals, that the more elaborate methods, based on the construction of pseudopotentials from nodeless pseudoorbitals (Kahn et al., 1976), become important. For atoms with several valence electrons the use of the pseudopotentid (47), which incorporates the valence-valence interaction, to determine the radial functions is preferable to the use of a model potential based solely on the core orbitals. Heavy atoms can be treated in a similar manner (Kahn et ({I., 1978) if the core orbitals, from which the pseudopotential is constructed, are obtained as solutions of equations such as (64), in which the principal relativistic effects of the core are introduced into the Hamiltonian. Some care must be exercised in deciding on which electrons form the core, particularly in the case of heavy atoms. The most simple possibility is to treat all closed subshells as in the core. If the model potential includes the effects of core polarization, then this definition of the core will probably be sufficient to allow a reasonably accurate evaluation of orbital energies of the valence electrons and of oscillator strengths of transitions involving them. In practice, most all-electron calculations would be of frozen-core type, with the core defined in a similar manner. However, Lee et d.(1977) pointed out that such a simple procedure for determining the core may not be satisfactory in molecular applications. They discussed in detail the case of gold, and suggested that the choice of the core should be made on the basis of the values of the orbital energies. They found that 5d5/2,,/2have orbital energies closer to that of the 6sIl2than to those of the core, and concluded that the 5d10 subshell should be treated along with the 6s electron rather than being included in the core. One great advantage of the use of model potentials is the saving in computational time, especially for atoms with many electrons. Although we have limited our discussion to the case of atomic structure calcula-

A . Hibbert tions, the savings in time are even more substantial when model potentials are used in molecular calculations or in scattering calculations.

REFERENCES Accad, Y.,Pekeris, C. L., and Schiff, B. (1971). Pliys. Rcv. A4, 516. Bates, D. R., and Damgaard, A. (1949). Philos. Trcrns. R . Soc. 242, 101. Bayliss, W. E. (1977). J. P h y ~ B. 10, L583. Beigman, I . L., Vainshtein, L. A., and Shevelko, V. P. (1970). Opt. S/JPCI,O.\C.28, 229. Berry, H. G. (1982). Nrrcl. Irirrrrrrn. Mrthotls 202, in press. Biemont, E. (1975). J . Qcrnnt. Spc,ctrosc. Radicrt. Trari.sfi.r 15, 53 I . Bromander, J., Duric, N., Erman, P., and Larsson, M. (1978). Plrys. Scr. 17, 119. Burke, P. G., and Seaton, M. J. (1971). Methods Comprrt. Phys. 10, 1. Caves, T. C., and Dalgarno, A. (1972). J. Q I I N ~Spectrosc. II. R d i a t . Trlrmfi’r 12, 1539. Cohen, M. H., and Heine, V. (1961). Pliys. Reis. 122, 1821. Cohen, M . , and McEachran, R. P. (1978). J . Qrcanr. Sperfrosc. R d i r r . TrnnJ;fiv 20, 295. Cowan, R. D. (1967). Phys. Rev. 163, 54. Cowan, R. D., and Griffin, D. C. (1976). J. Opt. Soc. A m . 66, 1010. Curtis, L. J., Engman, B., and Martinson, 1. (1976). Phys. S u . 13, 109. Dalgarno, A., Bottcher, C., and Victor, G. A. (1970). Clicvn. Phys. L P I I .7, 265. Drake, G. W. F. (1982). N d . Itistrrrm. Methods 202, in press. Eissner, W., and Nussbaumer, H. (1969).J. Plcys. B 2, 1028. Eissner. W., Jones, M., and Nussbaumer, H. (1974). Comprct. P h y s . Commrrii. 8, 271. Friedrich, H., and Trefftz, E . (1969). J. Q r r ~ i i r t . Speci~.o.sc.Rrrt/irrt. Trtrri.sfi~9, 245. Froese Fischer, C. (1975). Crrn. J. Pliys. 53, 338. Froese Fischer, C. (1976). C m . J. Phys. 54, 1465. Froese Fischer, C. (1977). J . P h y s . B 10, 1241. Ganas, P. S. (1978a). J . Clicvri. P/ry.s. 69, 1019. Ganas. P. S. (1978b). J . Qcrniit. Spectrosc. Rrrdirrt. Trrrrisfer 20, 461. Ganas, P. S. (1979a). Z. Phys. Abr. A 292, 107. Ganas, P. S. (1979b). Pliys. Lett. 73A, 161. J . Opt. Sot. A m . 69, 1140. Ganas, P. S. (1979~). Ganas, P. S. (1979d). J. Chem. Phys. 71, 1981. Ganas, P. S. (1979e). Astron. Asrrophys. Srcppl. Ser. 38, 313. Ganas, P. S. (19790. Plr.v.\icir 98B+C, 140. Ganas, P. S. (1980a). J. Opt. Soc. Anz. 70, 251. Ganas, P. S. (1980b). Mol. Plry.5. 39, 1513. Ganas, P. S. (1980~).f n t . J. Qrrtrrrtrrrn Chem. 17, 1179. Ganas, P. S . . and Green, A. E. S. (1979). Phys. R n . . A 19, 2197. Garstang, R . H., and Shamey, L. J. (1967). A.strophys. J. 148, 665. Glass, R., and Hibbert, A. (1976).J. Phys. B 11, 2413. Goddard, W. A. (1968). Phys. Rev. 174, 659. Gombas, P. (1956). Hwidh. P h y s . 36, 109. Green, A. E. S., Sellin. D. L., and Zachor, A. S. (1%9). Pliys. R1.r.. 184, I . Hafner, P.. and Schwarz, W. H. E. (1978a). J. Pkys. B 11, 217. Hafner, P., and Schwarz, W. H. E . (1978b). J . Phys. B 11, 2975. Hameed, S. (1972). J . Plrys. B 5, 746.

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Hameed, S., Herzenberg, A., and James, M. G. (1968). J. P/iy.s. B 1, 822. Hellmann, H. (1935). J. Chem. Phys. 3, 61. Herman, F., and Skillman, S. (1963). “Atomic Structure Calculations.” Rentice-Hall, New York. Hibbert, A. (1974). J. Phys. B 7, 1417. Hibbert, A. (1975a). Rep. f r o g . Phys. 38, 1217. Hibbert, A. (1975b). Compur. Phys. Commrrn. 9, 141. Hibbert, A. (1978). J. Phys. 40, C1-122. Hibbert, A. (1979). III “Progress in Atomic Spectroscopy” (W. Hanle and H. Kleinpoppen, eds.), p. I . Plenum, New York. Hibbert, A . (1982). Nricl. Insrrirni. Mc,/hotb 202, in press. Johansson, I. (1959). Ark. Fys. 15, 169. Kahn, L. R., and Goddard, W. A. (1972). J. Chmi. Plrys. 56, 2685. Kahn, L. R., Baybutt, P., and Truhlar, D. G . (1976). J. Chem. Phys. 65, 3826. Kahn, L. R., Hay, P. G., and Cowan, R. D. (1978). J . Cliem. Phys. 68, 2386. Klapisch, M. (1967). C. R. Acod. Sci. (Poris) Ser. B 265, 914. Klapisch, M. (1971). Comprrt. Phys. Commtrn. 2, 239. Ladanyi, K. (1956). Acro Plryc. Hirng. V, 361. Laughlin, C., and Victor, G. A. (1974). Astrophys. J. 192, 551. Laughlin, C., Constantinides, E. A., and Victor, G . A. (1978). J . Phys. B 11, 2243. Lee, Y. S., Ermler, W. C., and Pitzer, K. S. (1977). J. Chem. Phys. 67, 5861. Lvov, B. V. (1970). Opr. Spwrrosc. 28, 8. McEachran, R. P., Tull, C. E., and Cohen, M. (1968). Cun. J. Phys. 46, 2675. McEachran, R. P., Tull, C. E., and Cohen, M. (1969). Con. J. Phvs. 47, 835. McGinn, G. (1969a). J . Chem. Phps. 50, 1404. McGinn, G . (1969b). J. Chern. Phys. 51, 5090. Melius, C. F., and Goddard, W. A. (1974). Pliys. R e v . A 10, 1528. Melius, C. F., Olafson, B. D., and Goddard, W. A. (1974). Chcm. Phys. L e t / . 28, 457. Midtdal, J. (1965). Phyx. Ri.1.. 138, A1010. Migdalek, J. (1976a). Con. J. Phys. 54, 118. Migdalek, J. (1976b). Crrn. J. Phys. 54, 130. Migdalek, J. (1976~).C m . J. Phgs. 54, 2272. Migdalek, J. (1980). J . Phys. B 13, L169. Migdalek. J., and Bayliss, W. E. (1978). J. Phys. B 11, L497. Migdalek, J., and Bayliss, W. E. (1979a). J . Phvs. B 12, 1113. Migdalek, J., and Bayliss, W. E. (1979b). J. Phys. B 12, 2595. Moise, N. L. (1966). Astrophys. J. 144, 774. Moore, R. A., and Liu, C. F. (1979). J . Phys. B 12, 1091. Moore, R. A., Reid, J. D., Hyde, W. T., and Liu, C. F. (1979). J . Phys. B 12, 1103. Moore, R. A., Reid, J. D., Hyde, W. T., and Liu, C. F. (1981). J. Phys. B 14, 9. Miihlethaler, H. P., and Nussbaumer, H. (1976). Asrrori. Asrrophys. 48, 109. Nicolaides, C. A., Beck, D. R., and Sinanoglu, 0. (1973). J . Phys. B 6, 62. Nussbaumer, H., and Storey, P. (1977). Astron. Asrropplijls. 64, 139. O’Keefe. P. M., and Goddard, W. A. (1969a). Phys. Rev. 180, 747. O’Keefe, P. M., and Goddard, W. A. (1969b). Pliys. Reif.Lctr. 23, 300. Pekeris, C. L. (1958). Phys. Rcr. 112, 1649. Penkin, N. P.. and Slavenas, I. Y. (1963). Opt. Specrrosc. 15, 3. Phillips, J. C., and Kleinmann, L. (1959). Plips. R e v . 116, 287. Pradhan, A. D., and Saraph, H. E. (1977). J . Phys. B 10, 3365. Rosen. A., and Lindgren, I. (1968). Phys. Rev. 176, 114.

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Saraph, H. E. (1976). J . Phys. B 9, 2379. Seaton, M. J. (1972). J . Phys. B 5, L91. Seaton, M. J., and Wilson, P. M . H. (1972).J. P h y s . B 5, LI. Slater, J. C. (1951). P h y s . Rei,. 81, 385. Szasz, L. (1968). J . Cliem. P h g s . 49, 679. Szasz, L., and McGinn, G. (1965). J . Cliem. Phys. 42, 2363. Cliem. . Phys. 47, 3495. Szasz, L., and McGinn, G. (1967). .I Szasz, L., and McGinn, G . (1972). J . Chem. Phys. 56, 1019. Thompson, D. G., Hibbert, A., and Chandra, N. (1974). J . Phys. B 7, 1298. Victor, G . A . , and Laughlin, C. (1972). Chcin. Pliys. L e f t . 14, 74. Victor, G. A., and Laughlin, C. (1973). Niccl. fti.strio?i. Methods 110, 89. Weeks, J . D., Hazi, A., and Rice, S. A. (1%9). A d v . Cllem. Phvs. 16, 283. Weisheit, J . C., and Dalgarno, A. (1971). Chcm. Phss. L e f t . 9, 517. Weiss, A. W. (1963). Asfrophys. J . 138, 1262.

II

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 18

RECENT DEVELOPMENTS IN THE THEORY OF ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES D. W. NORCROSS Joint Institiire f o r Lnborutory Astrophysics Unikvrsity of’ Colorado Nuiionol Birreau c!f Standards Boiiltier. Colorado

L. A . COLLINS Lo.? Alnmos Nritionril Luborritory Lo.? Alutnos. Neb$,Mexico

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Outline of the Review . . . . . . . . . . . . . . . . . . . . . B. Background and Overview . . . . . . . . . . . . . . . . . . . 11. General Formulation . . . . . . . . . . . . . . . . . . . . . . . A. The Scattering Equations . . . . . . . . . . . . . . . . . . . . B. Fixed-Nuclei Approximations . . . . . . . . . . . . . . . . . . 111. Approaches and Approximations . . . . . . . . . . . . . . . . . . A. Solution of the Scattering Equations . . . . . . . . . . . . . . . B. Specification of Cross Sections . . . . . . . . . . . . . . . . . IV. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A Comparative Study for LiF, and Other Molecules . . . . . . . . B. Vibrational Excitation . . . . . . . . . . . . . . . . . . . . . V. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 343 344 350 351 355 360 361 372 377 378 388 390 392 393

“No experimental result is to be taken as certain until confirmed by theory” Eddington (apocryphal)

I. Introduction Much of the recent progress in the theory of electron scattering by polar molecules has been stimulated by practical considerations, such as the 34 1 Copyright @ 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

342

D . W . Norcross and L . A . Collins

effort to develop a more efficient source of electrical power from fossil fuels by using magnetohydrodynamics (MHD). The primary determinant of the electrical resistivity of the bulk plasma is thought to be electron collisions with molecules, both preexisting and formed in the combustion process (Spencer and Phelps, 1976). The cross section of interest is the momentum-transfer cross section vl,,and it is known to be in general much larger for polar than for nonpolar molecules. This is illustrated in Fig. 1, which shows a selection of experimental and theoretical results for a wide range of dipole moments. Without dwelling here on the merits of the various results shown, we wish only to emphasize the sheer magnitude

14C

700

120

600

-I00

500

N I 0

-

0,

v)

t

N O

E 80

-

100

0

aJ

A

v

H

b

7

>

I

6(,

300 W

200

I00

2

4

6

8

10

12

D (Debye) F I G . I . Total momentum-transfer cross sections as a function of dipole moment. The ) are from thermal-energy swarm measurements. (Reprinted from experimental data (u,, Collins and Norcross, 1978.)

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

343

of the cross sections. By comparison, Ev\,for H2is roughly 100 eV-rr$ in the energy range from thermal to -10 eV. Of the many other areas of science and technology in which electron collisions with polar molecules are important, two stand out. One is the field of laser development and modeling. Polar molecules (e.g.. CN, HCI) are and will continue to be important constituents of existing and new laser systems (e.g., Quick Pt d., 1976; Kliglerer d.,1981). The other is in the area of the physics of the interstellar medium, in which many polar molecules and polar molecular ions (e.g., HCN, CH+) play an important role (e.g., Johnston, 1967; Dickinson er d..1977; Dickinson and Flower, 1981). In both of these areas the relevant cross sections are often those for rotational excitation. Just as for momentum transfer, the cross sections for these processes are strong functions of the dipole moment and can be several orders of magnitude larger for polar than for nonpolar molecules. Polar molecules are also intrinsically interesting for a variety of reasons. The sheer magnitude of the cross sections makes them ideal candidates for measurements in both swarms (e.g., Christophorou and Pittman, 1970) and beams (e.g., Vuskovicci d., 1978; Rohr, 1979). The fact that the scattering is dominated by a long-range interaction makes them attractive theoretically, since perturbation methods would appear to be applicable. Theoretical interest thus traces back to some of the earliest work in scattering theory (e.g., Massey, 1932). Polar molecules are also different from nonpolar molecules in much the same way that positive ions differ from neutral species, i.e., in some approximation they can bind an infinite number of states (Turner, 1966). Much of the recent work, in both theory and experiment, on the interaction of electrons and polar molecules has been stimulated by the exciting discovery of very pronounced resonances in near-threshold vibrational excitation (Rohr and Linder, 1975). This subfield of physics is thus both very important and quite active, but much yet remains to be accomplished. A. O U T L I NOF E

THE

REVIEW

It is not our purpose here to delve further into the area of applications, but rather to summarize recent work in the theory of electron collisions as applied to highly polar molecules (loosely defined for the purpose of this review article to be those for which the dipole moment is 20.5 a.u. = 1.27 D). In particular, we hope to leave the reader with an impression of the commonality, rather than the diversity, of the wide variety of approaches that have been used or suggested. In our discussion of both theory and calculations the emphasis is on

344

D . W . Norcross and L . A . Collins

elastic scattering and rotational excitation, but vibrational excitation is also briefly discussed. Experimental results are mentioned where useful for illustration and comparison, but not extensively reviewed. We are happily relieved of the burden of providing an exhaustive bibliography, due to the recent reviews of Itikawa (1978), Burke (1980), and Lane (1980), and thus we shall concentrate on the literature that has appeared over the last few years. We wish to acknowledge at the outset the contribution that these reviews have made to our own: that by Itikawa reviews the state of both experimental and theoretical work specifically for polar molecules; those by Burke and Lane are more general reviews, and we follow the notation and nomenclature of the latter as closely as possible. Since we have tried to minimize overlap with these reviews, the reader is encouraged to study them. In spite of the existence of these reviews, there is more than a little of concern to us here. The last few years have seen several significant advances in both formal theory and numerical techniques. Of particular interest are continuing efforts to study the applicability and utility of simple approximations in the scattering formalism and advances in our ability to accurately describe the interaction potential beyond the firstorder effect of the dipole. In terms of efforts to carry out practical computations, these two directions are obviously quite complementary. In Section I,B, we present a general overview of the unique features of scattering by polar molecules. with a view to motivating and providing direction for the discussion to follow. Section I1 is devoted to a synthesis of the formal theoretical hardware and Section 111 to numerical techniques that are essential to scattering calculations and to an adequate description of the interaction potential. The emphasis here is on approaches that employ the most powerful and general techniques, but it should be understood that one of their cardinal virtues is the provision of results against which simpler approximations can be tested. In Section IV we discuss and compare some of the recent applications of this machinery, the various approximations made thereto, and the general physical picture thus developed. Section V is a summary and contains suggestions for future work. B. BACKGROUND A N D OVERVIEW

In his review, Lane (1980) emphasized the importance of taking “every advantage of reliable approximations, and occasionally a few unreliable ones,” in recognition of the fact that high rigor is usually unaffordable in

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

345

any realistic computation. Owing to the complexities associated with the multicenter nature of the interaction, not only is the potential itself usually difficult to generate, but it is perforce highly anisotropic. In both regards the problem is at the outset more difficult even to set up than for an atomic system. The anisotropy introduces many long-range multipole moments to first order, as opposed to none or at most a few in atomic systems, thereby coupling many more angular momenta in a partial-wave treatment. In polar molecules the most important of these is associated with the permanent dipole moment, yielding an interaction that goes as r 2(for r the coordinate of the scattered electron). The only analogous situation in atomic collisions is the r P 2 coupling between degenerate states of hydrogenic systems. As Itikawa (1978) pointed out, polar molecules are not only typical (most molecules are polar), but also in some respects the easiest to treat computationally. Dipole moments are among the best known of molecular constants and, if large enough, the long-range dipole interaction can dominate the collision process. To first order, then, it may be possible to obtain reasonably accurate results by ignoring m y other details of the electron-molecule interaction, particularly those at short range which are exceedingly complex. The validity of this statement is obviously a function of the size of the dipole moment, and it requires further qualification. It will break down ( 1 ) at low projectile energies and/or small scattering angles, when the energy and/or momentum transfer, respectively, are comparable to the spacing between states involved in the transition; and (2) at high or resonant energies and/or large scattering angles, when penetration of the molecular-charge cloud is relatively more important. It should also be obvious that these qualifications will be more important for one type of cross section than for another. The cross section for momentum transfer, for example, with its attendant reduced weighting of forward scattering, might be expected to be relatively more sensitive to the interaction at short range. This point deserves emphasis: Tlie pcrrticrrlrr prowss o f ititerrst is ( I S irnportrrrit it1 cietermining the rcrngc. of rdidity c?f pcirticrrlrir (ippro.-rit?i~rtioti.S ( i s (ire tliP physicril pcrrlimeters of the scattering process.

I. Tlic First Bortr Appi.ci.-rit?irrtioti The fact that the dipole interaction occurs at long range, and is perhaps even dominant, leads to the conclusion that the simplest forms of perturbation theory may be useful. Even if not highly accurate, they can be helpful in assessing the validity of various approximations. This partially

346

D. W . Norcross and L . A . Collins

accounts for the importance and enduring value of two of the earliest papers in the field: those of Massey (1932) and Altschuler (1957). These, along with the later extensions by Takayanagi (1966), Crawford rt d. (1967), and Crawford ( 1967) provide very simple forms for cross sections in the first Born approximation (FBA). The Coulomb-Born approximation for polar molecular ions has been developed by Chu and Dalgarno (1974) and Chu (1975, 1976). It will enlighten our discussion to have several of these results set down at the outset. We consider a molecule initially in rotor state j with dipole moment D in atomic units (1 a.u. = 2.5418 D)and an incident electron with kinetic energy k 2 in rydbergs ( R , = 13.606 eV). The differential, momentum transfer, and integrated cross sections for a (vibrationally elastic) transition fromj to,j‘ (=.j 5 1) are, in the FBA for a point dipole rotor,

is the kinetic energy of the where j, is the larger of j and j’,and outgoing electron. Here and throughout this review, atomic units are used unless otherwise noted. Thus cross sections are in square Bohr ( ( I : = 0.2800 x m2). The simple forms of Eqs. (1)-(3) illustrate the major qualitative features of scattering by polar molecules. The process is in general dominated by transitions for which IAjjJ = 1 and by forward scattering. The cross sections are in general quite large. The greatest sensitivity to the rotational energy spacing occurs at small scattering angles and in the integrated cross section. The history of developments in theory have centered around attempts both to establish the degree of accuracy of Eqs. (1)-(3), and to overcome their many limitations. Chief among them are: (1) that no account is taken of interactions other than the dipole, either at short range or of higher multipolarity at long range; (2) that no transitions other than for IAjjl = I are allowed; and (3) that for very large dipole moments, one of the elementary requirements of perturbation theory breaks down; i.e., quantum mechanical unitarity is violated (Crawford ~f d.,1967). All of these problems except ( I ) have been addressed in a large number of calculations in which the scattering equations have been solved more or less exactly for the dipole potential, usually through the use of closek I 2

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

347

coupling techniques. The earliest calculations of this type for polar molecules were performed by Itikawa and Takayanagi (1969) and Crawford et al. (1969). Unitarity is automatically enforced in such an approach, and the effect was found to be most significant for scattering out of the forward direction. This result is consistent with the classical picture of forward scattering being dominated by large impact parameters, for which perturbation theory is most likely valid. In consequence, the integrated cross section is much less seriously affected than the momentum transfer cross section, typically reduced by 10% and a factor of 3, respectively. One general conclusion that may be drawn is that for transitions with 1 Ajl = 1, first-order perturbation theory is correct for highly polar molecules only to the extent that large impact parameters (or the quantum mechanical analog of large partial-wave angular momenta) dominate the collision process. The other side of this coin is the success of a wide variety of relatively simple methods that employ perturbation theory, but which go beyond first order for the dipole interaction. A second feature of calculations in the close-coupling class is the automatic inclusion of contributions from the dipole interaction to transitions with JAjI # 1 by what is often referred to as ladder coupling, e.g.,j = 0 + 1 + 0 a n d j = 0 + 1 + 2 . Another feature is the fact that solutions of this kind cannot usually be obtained unless the r-2 singularity of the dipole potential at the origin, which is harmless in the FBA, is somehow removed, e.g., by the ad hoc introduction of a smooth cutoff or hard sphere. This might be viewed, however, as a crude model for short-range effects. There have also been several attempts to model long-range interactions other than the dipole, e.g., quadrupole and polarizability effects. All of these (ladder coupling by the dipole interaction, short-range, and longrange nondipolar, interactions) were found to be in general significant for transitions with IAjI # I . The latter two also contribute noticeably to scattering out of the forward direction, and with decreasing significance, to the momentum transfer and integrated cross section, for transitions with IAjI = I . A second general statement that can safely be made, then, is that it is also necessary to consider more than just the dipole potential if accurate results are to be obtained for any but a few selected processes. This is less true for total (summed over all final rotor states) than partial cross sections, owing to the fact that enforcement of unitarity and higher order effects usually affect the magnitude of the cross section in opposite directions. One point of view might be that this is misleading in that it obscures important effects, but it might also be held fortunate in that total cross sections are often of the greatest practical importance.

348

D . W . Norcross and L . A . Collins

2. The Fi.red-Nirclei Apprauimtrtion Another topic that is pervasive in the literature of electron-molecule collision theory is the utility of the fixed-nuclei (FN) approximation. In this approximation the collision process is assumed insensitive to the instantaneous momenta of the nuclei and is treated initially as elrrstic scattering by a molecule held fixed in space. This kind of impulse approximation is particularly appropriate to electron-molecule collisions as the classical collision time t ( = L / u . where u is the electron velocity and L is the effective range of the interaction) is usually much less than the molecular rotational (L sec) or vibrational (L sec) periods. For polar molecules the extent of the long-range dipole potential clearly strains the basic assumption of this approximation, and the validity of the FN approximation for polar molecules has generally been viewed as rather limited [see, e.g., comments in the reviews of Itikawa (1978) and Lane ( 198011. That the difficulty with the FN approximation is nontrivial can be seen by taking the limits of Eqs. (1)-(3) as the moment of inertia of the k ) , yielding molecule tends to infinity ( k '

-

d<Jj,y/tiR = (2/3k')D2[j>/(2j+ 1 ) ] ( 1 a],'y =

-

cos O)-'

( 8 x / 3 k 2 ) D ' U > / ( 2+ j I)]

(4)

(5)

and infinite ~ 1owing , ~ to ~the divergence of Eq. (4) at H = 0. Summed over all final rotor states, Eqs. (4) and ( 5 ) are identical to the results originally obtained by averaging the FN scattering amplitude in the FBA over all molecular orientations (Altschuler, 1957). This essential breakdown of the FN approximation is neither an artifact of the FBA, nor the fact that Eqs. ( I ) - ( 5 ) are obtained for a potential that behaves incorrectly as r 2 at the origin. It would pertain even for an extrct FN treatment of the scattering problem for any real polar molecule (Garrett, 1971a). This follows from the facts that the breakdown is associated with the interaction for large angular momenta and that the FBA eventually becomes exact in this regime. Thus the FBA can be used to assess the reliability of the FN approximation in spite of the fact that it does not account correctly for short-range or higher order interactions. The regime of small angular momenta presents a different set of difficulties, all of the essential physics of which can be illustrated by considering an interaction that behaves as 0 / r 2for I' 2 r0 and is constant for I' s r O . Residual interactions, including those at short range, are incorporated in the parameter r O . We must consider, then, the behavior of the solutions of the equation {(d'ldr.')

-

"(N

+

I)//.? 1

+ k")rr(I')

=

0

(6)

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

349

where N(N

+

I)

for I’ 3 I ’ ~Defining . v = [N(N behaves as (O’Malley, 1965) 771

A-0

for D

5

q

=

+

l(1 + I ) - D

1)

(1 +

+ t11’2, the

(7) scattering phase shift

1-Vh/2

( 8)

D, = 1(1 + 1)

1-0

+ 1, and as { ( l + 4 h / 2 + tan-I[-tan()v) In k + 6) + tanh()v(r/2)]}

(9)

for D 2 D,. The constant 6 in Eq. (9) is characteristic of the residual interaction. Equation (6) admits of no bound solutions for D 5 D, , but an infinite number for D > D, (Turner, 1966). In contrast to the situation for a short-range potential where q1 Ak2‘+‘, r~

A-0

the scattering cross section associated with Eqs. (8) or (9) diverges at threshold, going through an infinite number of oscillations in the process for D > D,. Threshold behavior is, for D < D,, independent of the residual interaction. These general properties remain valid for scattering by a real polar molecule, in spite of the necessary generalization of Eqs. (7)-(9). In this case, N ( N + I ) becomes an eigenvalue of a simple tridiagonal matrix determined by the coupling of scattering channels with I’ = I 2 1 by the potential D cos 8 / r 2 (e.g., Clark, 1979).The critical values D, of the dipole moment are still defined as those for which N ( N + I ) = - 1/4, but their values change, e.g., 0.639 and 3.792 a.u. for scattering symmetries with lowest angular momentum 1 = 0 and 1, respectively, compared with 0.25 and 2.25. The generalized forms of Eqs. (8) and (9) become the sum of the eigenphases of the multichannel scattering matrix, and the total scattering cross section at threshold will be similarly ill behaved. This pathological behavior can be eliminated only by introducing the effects of nuclear motion in the scattering equations. Then the analogs of Eqs. (8) and (9) will become well behaved at threshold; the values of D, will become dependent on, for example, the rotational constant and short-range interaction; and the potential will only support a finite number of bound states for D > D, (Garrett, 1971b, 1980). We might expect, however, that the qualitative behavior of low-energy scattering suggested by Eqs. (8) and (9) will persist for scattering energies down to those characteristic of the rotational spacing. The preceeding discussion requires only slight modification for vibrationally inelastic scattering, in which case D 2 in Eqs. (1)-(5) is simply

350

D. W . Norcross and L . A . Collins

replaced by I (t'[DIt")l2.If the FN approximation is applied oi11.v to rotation, then Eqs. (1)-(3) apply with k and k ' the momenta relative to the initial and final vibrational states. All cross sections are well behaved in this case. If, however, the FN approximation is applied to vibration as well, then Eqs. (4) and (5) apply. The FBA can also be used to illustrate other general properties of collision cross sections within the FN approximation. Consider Eqs. (1)-(3) generalized to rovibrational transitions, with X and k ' taken to be independent o f j andj'. Summing over allj', the result is clearly independent o f j . We can also make use of the completeness of vibrational wave functions to sum over all r ' , obtaining cross sections that depend on t' only very weakly, primarily through the term ( t'lD21t') . This can often be approximated quite accurately by the square of the value of D at the equilibrium internuclear separation. Thus when the FN approximation i.7 valid, very great simplification can be expected. Thus while the breakdown of the FN approximation is absolute, it is limited in its impact, and this approximation has found much useful application for electron collisions with polar molecules. The absence, however, of any simple prescription for exploiting the computational simplifications inherent in the dominance of the dipole interaction, in perturbation theoretic approaches, and in the FN approximation, and for incorporating rovibrational dynamics at a level adequate for accurate specification of rill cross sections of interest, has been one of the most challenging problems in the field. This should be done without sacrificing the ability to treat the m t i w interaction at a level of accuracy adequate for almost any practical purpose and in a way that can readily be generalized from the simple molecules of the test bench to the complicated molecules of the real world.

11. General Formulation In the following summary of general formulas we rely heavily on the development presented by Lane (1980), but will emphasize those points peculiar to polar molecules. Details of derivation will be avoided except where necessary. For our purposes it will be adequate to restrict the initial and final states to singlet states of a linear molecule, but the range of possible molecular states involved in the collision process will not be otherwise limited. The first section is an unavoidably dry summary of some of the essential equations of molecular scattering theory. This sets the stage for introducing the F N approximation in the next section, where some novel concepts pertinent to polar molecules are discussed.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

351

A. THESCAT-TERING EQUATIONS The most general representation for the total wave function for the collision system with the molecular target initially in state ti is

q,@,X )

=

Fn,nr(r)+nW

A

(10)

n'

where r is the radial coordinate of the scattered electron, X refers to the totality of molecular coordinates, and A implies antisymmetrization. The is related to the scattering amplitude .fn,n, for asymptotic form of Fn,nt(r) the transition t? to ti' by

-

Ffl,,4r) r-7. exp(ik,. r) 8s;

+ ~ . - lexp(ikn~rlf,,n~(kn9 b#) ( I 1)

where k, and k,, are the initial and final momenta of the electrons. Their magnitudes are related by

A:, - A:

=

2(En - En*)

.

(12)

where En and En, are the total energies of the initial and final molecular states. The differential cross section for the transition n to n' is

Other quantities of interest are the integral and momentum transfer cross sections, given by

The state indices 17 and ti' include all relevant quantum numbers. These are the electronic state a and the projection A, of its angular momentum on the symmetry axis of the molecule, the vibrational state u , and the rotational statej and its projection m,. In most applications the degenerate magnetic substates are summed andor averaged over, and in many cases sums over all final rotational and vibrational states are taken as well. A conventional treatment of the coupled-states expansion for V n(r, X), and the resulting scattering equations, is adopted in the balance of this section for pedagogical reasons. Other treatments are discussed in Section II1,A.

D . W. Norcross and L . A . Collins

352

I . The Liihorirto)~~-Fi.\-rci Coordincrtc Fmme

For a coordinate frame fixed in the laboratory (LF) with its origin at the molecular center of mass, Eq. (10) can be expanded as W F ( r , r \ l , R) = A

C r-l!l;F(r)@kF(F,

r t I ,R)

(16)

P

where r\I and R refer, respectively, to the molecular electronic and nuclear coordinates, andp is a channel index. The channel functions are the following (ignoring spin coupling):

C( j l J ; ~ ~ ~ Z ~ ) Y ~ ~ / ( ~ ) R ~ , \R, ), ( ~(17) ) ~ ~ , ( ~ ) +

r t l , R) = m/

lnJ

where is a spherical harmonic, CGIJ; mlm3) is a Clebsch-Gordan coefficient, and R?,,,,,,(k), &(R) and +ly(r,l;R) are rotational, vibrational, and electronic molecular-state wave functions, respectively. Here, for purposes of illustration we have specialized to the coupling scheme of Hund's case (b) (Herzberg, 1950) and ignored subtleties such as the dependence of +,, o n j (Temkin and Sullivan, 1974; Choi and Poe, 1977a,b). . The channel functions are eigenfunctions o f J 2 and J , , with eigenvalues J ( J + I ) and mj + m l ,respectively, and the channel index p = ( m $ ; J ) . They could also be constructed as eigenfunctions of parity, but this would add nothing essential to our present purpose. The molecular state wave functions satisfy

[Hmt

- E m n , I 1, full account of rovibrational dynamics must be taken. In actual calculations one chooses the particular coordinate representation that is most appropriate, i.e., the B F frame for I s I, and the L F frame for I > I , . It is also helpful if I, can be chosen large enough that exchange, perhaps even short-range interactions, can be neglected or crudely approximated in the LF frame. This method therefore encounters minimal difficulties with convergence of the expansion of Eq. (10) over 111, o r j and u . in the two frames. The AFT method is clearly just the RFT method with r, = x for I 6 I, and r, = 0 for I > I , , and involves the additional assumption that the accumulation of phase due to H , , , )and/or H,,, is negligible for I d I, and r > r , . Recent applications of the AFT method include the work of Collins Pt t i / . (1980b) and Siege1 et a/. (1980, 1981a,b). The transformation Eq. (35) is used for I G I,, and thus becomes a part of the method, not just a pedagogical device. The argument suggesting a preference for the alternative form Eq. (38) might still be made now in the interest of minimizing dependence on the choice of I, near threshold, but the problems attending its application to polar molecules persist. Any difference between the results of two calculations using the RFT and AFT methods should not be significant except near threshold, where the effect of the additional approximation involved in the latter must be balanced against their common limitation. The /,-conserving modification remains an attractive extension for vibrationally elastic scattering, but by introducing dependence on J does imply more work. This would still leave the problem of near-threshold vibrational excitation. Before concluding that all the labor thus implied is necessary, it is important to recall that the FN approximation is employed, in both methods, for only a limited range of collision parameters, and thus that errors in energy dependence, even magnitude, of particular elements of LFK may be inconsequential in the ultimately calculated cross sections. With this in mind, a very simple combination of Eqs. (35) and (38) might be used with the AFT method: Eq. (35) for all elements of l,FK with I ' = I ? 1 , and Eq. (38) otherwise.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

359

A more general approach has been suggested by Nesbet (1979). In this energy-modified adiabatic (EMA) approximation, the BF-frame energy k: is treated as a parameter. The elements of K1-Ffor a p(rrticu1cirj and j ’ might then be obtained using the transformation of Eq. (35) with elements of K R F obtained at the BF-frame energy k: = kpkpr. This preserves the unitarity of the S matrix. Another consequence of such a choice is that there is a well-defined relationship between threshold in the BF frame (k,. .+ 0) and in the L F frame ( k p r-+ 0). Thus, for example, any near-threshold electronic structure in the BF-frame calculations will be reflected at ewry rovibrational threshold in the final results. This basic idea was developed almost simultaneously by Domcke ef crl. (1979), who argued that this nonadiabatic modification to essentially adiabatic theory (see also Varracchio, 1981) might be the key to understanding threshold structure observed in vibrational excitation for several molecules (e.g., Rohr, 1979). This analysis was later extended (Domcke and Cederbaurn, 1981) to incorporate the special problems associated with near-threshold scattering by polar molecules. 3. The Adicibrrtic.-Niic.lci Approsinitrtioti

-

The RFT method applied to all partial waves in the limit r, x and the AFT method in the limit I, x reduce to what is referred to as the adiabatic-nuclei (AN) approximation for cross sections. Without modification the AN approximation has serious limitations for polar molecules, as discussed in Section I. Nevertheless, we shall summarize the essential formulas of this approximation here for reasons that will become apparent. In discussing this approximation it is more convenient, and conventional, to work directly in terms of the scattering amplitude [the lack of any more compelling reason for doing so was emphasized by Chandra (1975b) and Choi and Poe (1977b)l. In this approximation the analog of Eq. (30) is

x

(i)l-/’+l

--j

BFT$(R)YIm(R,)* Y, Fmp(iq,) (40)

where it is understood that RFT(R)and BFK(R) are related by Eq. (32). Integrating over all R for particular initial and final vibrational-state wave functions yields the so-called adiabatic-nuclear vibration amplitude

.fE,L,(kq,kq,;k > = Cvlf~~,(k,, kq*;R)lo’)

(41)

Integrating over all R for particular initial and final rotational wave functions yields the general AN amplitude (Faisal and Temkin, 1972; Henry

D . W . Norcross and L . A . Collins

360

and Chang, 1972) .f,4",,(kp,k,,)

=

(jmjA,lf%(k,,

k , ; k ) b f ~ ~ ~ , A l V , ) (42)

The angular integration is accomplished by a rotation of coordinates from the BF to LF frames. Comparing the result with Eq. (30), it follows that Eq. (42) can be obtained from Eq. (30) with kpkpt+ k,k,, and ',FTreplaced by the result of Eq. (35) applied to HFT. With Eq. (42), Eq. (13) becomes k L 7 * l n j ,

=

l~'t,'J'mj,/dQ (k,,/k,) I (jm,Aulf:,?, (k, , k,, ; fi )I j

1

' M I , A,Y ) ''

(43)

The kinematic ratio k,,/k, is often neglected, but its retention ensures conservation of energy and current and that cross sections satisfy the principle of detailed balance (Chang and Temkin, 1969). This latter point is clearly most relevant near threshold. We also note in this regard that the essential ambiguity of the choice of k , and kqt is resolved (Norcross and Padial, 1982) by the choice k,k,. = k,kp, suggested by the identification Eq. (42) with Eq. (30). This is a simple variant of the more general EMA approximation. By adopting the geometric mean for the energy at which the scattering amplitude in Eq. (43) is defined, instead of for the transformation Eq. (35), this variant of the EMA approximation is much simpler to apply. It does, however, sacrifice preservation of exact quantum mechanical unitarity for the implicit many-channel LF-frame S matrix. The EMA approximation does not, unfortunately, solve any of the essential near-threshold problems for rotational excitation of polar molecules. Consider Eq. (43) evaluated for only the dipole interaction in the FBA. The geometric mean merely permits an unambiguous identification ofk2 in Eqs. (4) and (5) as the electron kinetic energy relative to the initid rotor state. The differential cross section at 8 = 0 and the integral cross section remain infinite.

111. Approaches and Approximations We now discuss in a little more detail methods employed to solve the infinite sets of coupled integrodifferential Eqs. (23) and (27). The solutions [Eqs. (16) and (25)] contain all of the basic scattering information needed to construct any measurable quantity, e.g., Eqs. (13)-(15). As a practical matter, however, approximations must be made to both the scattering and cross-section equations and to the interaction potential. It is these approx-

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

361

imations, and the numerical techniques designed to best exploit the simplifications thereby introduced, that presently concern us. The number of techniques and approximations applied to polar molecular systems is legion, and the remainder of this review could be occupied with merely a list of them. In keeping with our earlier caveat, we shall confine ourselves to discussing newer methods and to reporting developments in the older methods that have arisen since the appearance of reviews of Lane (1980) and Itikawa (1978) and of our own paper (Collins and Norcross, 1978). A. SOLUTION

OF T H E

SCATTERING EQUATIONS

We consider first the exact static-exchange (ESE) approximation, which presently represents the highest level of sophistication to which coupled-channel techniques have been applied to polar systems. In this approximation the sum over electronic states of the target molecule is truncated at the ground state. The nonlocal exchange terms in Eqs. (23) and (27) are treated exactly, but electronic excitation and correlation effects, e.g., polarization, are neglected since the target molecule remains frozen in its ground electronic state. We then consider approximations and improvements commonly made to the interaction potential, and then approximations to the solutions of the equations themselves.

I. Evrrct

Sttitic

Eschtitige

Here we describe briefly the various methods used to solve Eq. (27) in the BF frame with an essentially exact treatment of exchange. In all such and H,,, have been completely neglected. The methods work to date, HVib of solution divide into three areas: L 2 ,close-coupling (CC), and variational techniques. In the L 2 techniques, the radial space is usually divided into two regions: an inner region in which all the short-range electrostatic and exchange interactions must be treated and an outer region where the potential is local (usually in its multipolar form). In the inner region, the system wave function Eq. ( 2 5 ) is expanded in a basis of square-integrable functions (usually Slaters or Gaussians) and the solution treated very much as in a bound-state problem. This expansion allows full advantage to be taken of powerful quantum chemistry bound-state codes that have been developed to utilize such bases. The solution then proceeds by matching the wave function to the more conventional asymptotic form on the boundary of the inner and outer regions. The two most successful L z methods

362

D . W . Norcross and L. A . Collins

applied to electron-molecule collisions have been the R-matrix (Schneider, 1975; Schneider et al., 1979, and references therein), and the T-matrix (Rescigno et d . . 1975; Fliflet, 1979) methods. The T-matrix method suffers from the limitation that strong long-range forces are difficult to include. The L 2 techniques have also been used to search for resonances in electron-molecule scattering using variants of the stabilization technique (Taylor et a / . , 1966). Recent work of this type includes the studies of LiF by Stevens (1980) and Hazi (1981), HCI by Goldstein et ( I / . (1978), and H F by Segal and Wolf (1981). While not capable of yielding scattering data directly, such studies seek to discover resonance structure that may significantly affect a variety of processes involving the collision of electrons and molecules. The CC approaches (Arthurs and Dalgarno, 1960) directly address the solution of the coupled integrodifferential equations represented by Eq. (27). The methods divide into iterative and noniterative techniques. Since exchange is nonlocal, the differential equations cannot be propagated directly outward as is standard practice for a local potential. The iterative procedure (Collins et d.,1980a) circumvents this problem by initially making a simple local approximation to the exchange term. The resulting coupled equations are solved, and the solution is used to obtain a better approximation to W t $ . The procedure is continued until a stable form for W:: is obtained. Three noniterative approaches have been pursued. One involves converting the exchange integrals in Wt$, to differential equations (DE) and solving the resulting set of coupled equations for the solution 11:; and the integrals simultaneously. This technique, originally applied to homonuclear systems (Burke and Sinfailam, 1970; Buckley and Burke, 1977), has also been applied to electron scattering from polar molecules (Raseev ef N / . , 1978). The second approach is based on the integral equations (IE) formulation of Sams and Kouri (1969), as generalized to multichannel electron-molecule collisions (Morrison ef a / ., 1977; Collins and Norcross, 1978). Rescigno and Ore1 (1981) have extended this method to encompass an exact treatment of exchange and found significant improvements in computational time by representing the exchange kernel in separable form. The third form of the noniterative method is the linear algebraic (LA) approach. In this method, the coupled differential (or integral) equations are converted to a set of LA equations by imposing a discrete quadrature on the equations. The method has long been used in electron-atom collisions (Seaton, 1974) and was first applied to molecular scattering by Crees and Moores (1977). A recent revival of the method (Collins and

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

363

Schneider, 1981), which also exploits the separable exchange kernel, has been successfully applied to several polar molecules. The variational approaches have evolved along the lines of two wellknown prescriptions: the Schwinger and the Kohn. A Kohn procedure using a trial wave function composed of free waves and the static solution has been developed by Collins and Robb (1980). McKoy and co-workers (Lucchese et d.,1980; Watson et ul., 1981) have developed an iterative-variational scheme based on the Schwinger principle. Both L 2 and continuum wave functions are used in the basis. The Schwinger variational prescription is used to start and correct the solution, and the Lippman-Schwinger equations are also solved at each iteration to further improve the results. An approach that might be usefully incorporated into any of these, and one which draws its inspiration from heavy-particle collision theory, involves the expansion of the continuum wave functions in terms of an adiabatic basis in which the nuclear coupling is weak (Mullaney and Truhlar, 1978; Clark, 1979, 1980; Vo Ky Lan et ul., 1981). A candidate for such a basis would be the set of functions which diagonalize the dipole interaction, or the complete static potential. The extension of this method to nonlocal interactions remains untested. Table I provides a list of recent work using ESE techniques for polar molecules. The existence of such a large variety of ESE techniques should indicate that no one method is particularly superior for all applications. TABLE I POLAR MOLECULES FOR WHICH EXACT-STATIC-EXCHANGE CALCULATIONS HAVEBEENPERFORMED Method T matrix Iterative CC

Noniterative CC DE IE LA Kohn variational Schwinger variational

Molecule

co

Reference

LiH, LiF CH+ LiH, CO HCI

Levin ei a / . (1979) Collins ei a/. (1979a) Robb and Collins (1980) Collins ei a / . (1980a) Collins ei d.(1980b)

CH+ LiH LIH LiF, HCI LiH, CO LiH

Raseev e: d.(1978) Rescigno and Orel (1981) Collins and Schneider (1981) Collins and Schneider (unpublished) Collins and Robb (1980) Watson er a / . (1981)

D . W . Norcross and L . A . Collins The CC techniques have generally been confined to a single-center expansion for the bound and continuum wave functions. The strong coupling provided by the nuclear singularities causes the single-center expansion to converge rather slowly in the vicinity of the nuclei. This in turn requires that large sets of coupled equations be solved. One procedure for circumventing this problem is to use a prolate spheroidal coordinate system (Crees and Moores, 1977) in which the nuclear potential no longer provides any coupling. However, the use of such a coordinate system is restricted to diatomics and linear polyatomics. In the LA scheme, a great simplification can be introduced into the solution of the set of coupled equations by selecting a different quadrature for each scattering channel. Since high partial waves are concentrated near the nuclei, they can be accurately represented over a small region of space by a quadrature of only a few points. Thus, adding more partial waves in order to better represent the solution around the nuclear singularities does not significantly increase the order of the matrices. Complications with the L2 techniques arise from an entirely different direction. The square integrable, two-center basis functions are quite effective in representing the solution in the vicinity of the nuclei. They are, however, not as well suited to represent the oscillatory channel wave functions associated with the low partial waves. Thus, a large basis must be employed in order to properly mock the oscillatory behavior of the continuum solutions. This seems to be a particularly acute problem for polar molecules. A procedure to circumvent this difficulty for scattering calculations has been devised (Watson and McKoy, 1979) within the Schwinger variational method by introducing both L2 and continuum basis functions. 2 . Ai.’pro.rit?itrtiotIs t o tlio Ititertrc~tiotiPotentid Despite the success of these ESE methods in treating such polar systems as CO, HCl, LiH, and LiF, they are fairly restricted at present to small molecular systems. However, the modeling of many physical processes requires a knowledge of electron collisions with much larger polar systems (e.g., KOH, CsF), or results of greater accuracy than the ESE approaches can yield. We now turn our attention to ways of improving and/or simplifying the complicated electron-molecule interaction potential. Tlio polurizcrtioti potetitiol. The neglect of the closed electronic channels may be a serious omission for low-energy elastic scattering. The

(I.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

365

closed channels represent the polarization effects which arise from the distortion of the target molecule by the incident electron. Such effects are not included in the ESE approximation because the charge cloud remains frozen in the ground state and cannot relax in response to the incident electron. Polarization effects are known to be important at very low and near-resonant energies for nonpolar systems (Chandra and Temkin, 1976; 1977; Schneideret al., 1979) and for weakly polar systems Morrison et d., like CO (Chandra, 1975a, 1977). For strongly polar systems, the longrange dipole potential (-I--*) .may completely dominate the polarization potential (-r4). As has been shown for the strongly polar system LiF (Collins and Norcross, 1978; see also Fabrikant, 1979), polarizatip is probably a negligible effect for energies above a few tenths of electron volts. However, for a more weakly polar system like HCl (Gianturco and Thompson, 1977), the effects of polarization may be important. This problem can be remedied by including the closed electronic channels either directly by a many-state CC scheme, or by an optical potential approach. Unfortunately, such calculations have so far been confined to small homonuclear systems (Chung and Lin, 1978; Klonover and Kaldor, 1978; Weatherford, 1980). Effective, local-model polarization potentials with one or more empirically chosen parameters have, however, proven of value in describing the electron collisions with several nonpolar systems (see Lane, 1980). The semiempirical approach to polarization may permit, through the choice of parameters, some account to be taken of other effects such as exchange that are also treated approximately (e.g., Collins et d.,1980b). h. Tlir r.vc.licitigr potential. As discussed in the previous section, the nonlocal nature of the exchange term causes the main complications in solving the coupled equations. Most attempts at modeling exchange 'have centered on replacing the exchange terms with a local form. Such a replacement drastically simplifies most coupled-channel calculations. Three main approaches have been devised to accomplish this simplification. One approach involves replacing the nonlocal exchange term W:$ with a local, energy-dependent form based either on a free-electron-gas description (Riley and Truhlar, 1975; Morrison and Collins, 1978) or a semiclassical treatment (Furness and McCarthy, 1973). Of the plethora of model-exdhange potentials that have been devised, a form of the freeelectron-gas potential (HFEGE) suggested by Hara (1967) seems to give consistently the best results for low-energy electron collisions with nonpolar (Morrison and Collins, 1978) and weakly polar (Morrison and Collins, 1981) molecules. However, a semiclassical treatment of exchange, which

366

D. W. Norcross and L . A . Collins

has been shown to give reasonable results for CO at intermediate energies (Onda and Truhlar, 1980), has not yet been applied to any highly polar system. A note of caution is in order: The first application of the HFEGE potential to a strongly polar system resulted in the appearance of broad resonances (Collins and Norcross, 19781, the existence of which was at first confirmed by molecular structure calculations (Stevens, 1980). These were later proven to be spurious in molecular structure calculations (Hazi, 1981) employing more diffuse functions, and in scattering calculations (Collins ct d., 1979a). An alternative approach has been to introduce exchange by imposing orthogonality of the bound and continuum solutions (Burke and Chandra, 1972). As originally devised, the exchange term in Eq. (27) was dropped, and a solution was calculated for the static potential V&$. The resulting continuum solution was forced by the method of Lagrange undetermined multipliers to be orthogonal to all bound orbitals of the same symmetry. Since it is used in conjunction with the static potential, this method is usually designated as orthogonalized static (0s).This method is in effect a prescription for enforcing the Pauli principle, which for closed-shell molecules demands the exclusion of the scattering electron from the occupied molecular orbitals. This procedure has been applied to HCI, HF, H,O, and H,S by Gianturco and Thompson (1977, 1980). One drawback of the method is that exchange effects can only be included in scattering solutions for symmetries in which there are occupied bound orbitals. This can be quite a severe limitation, as is evident in the resonant II, symmetry in c-N, collisions. Studies of the orthogonality procedure (Collins el ( I / . , 1980b; Morrison and Collins, 1981; Salvini and Thompson, 1981) have shown that it generally produces too weak an exchange potential. The weakness of the above two procedures can be overcome to some extent by combining them. Imposing orthogonality on the solutions of the model potential removed the spurious resonances mentioned above in the case of LiH and LiF (Collins et d., 1979a) and produced results in good agreement with ESE calculations for these and for HCI (Collins et ( I / . , 1980b). In fact, in all cases for polar and nonpolar systems so far considered, the orthogonalized-static-model-exchange (OSME) procedure gives the best agreement with ESE results (Morrison and Collins, 1981). Conversely, there is some evidence that for polar molecules, in the absence of spurious resonance structure, orthogonalization does little to improve the HFEGE potential (Collins et a/., 1980b; Salvini and Thompson, 1981). The next level of approximation used in many polar calculations involves ignoring the exchange interaction altogether

c. The stcitic porential.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

367

and using only the static (S) potential in Eqs. (23) and (27). The static potential is determined by averaging the electrostatic interaction of the incident electron with the molecular charge cloud and nuclei over the ground-state wave function of the molecule. A Legendre expansion of the molecular charge density is usually made so that the angular part of the matrix elements V,(r) can be evaluated analytically, i.e.,

For small values of i’, the radial expansion coefficients u A ( r )represent the strongly attractive near-molecular field. For large values of I-, the terms have the form u A ( r ) - y A / r A + ’ , where y h is the Ath multipole moment of the static potential. For example, the dipole moment D is given asymptotically by --u1(i’)r2. In CC calculations, the static solution is found by systematically increasing the number of channels and of terms in Eq. (44) until convergence is reached. The charge density (or its Legendre expansion) used to generate the static potential is also an essential ingredient in several of the model-exchange potentials. New computer codes to generate these expansions have appeared (Morrison, 1980; Schmid r f (11.. 1 980). Another set of models is formed by making approximations to the static potential. The one most thoroughly studied is the truncated static [S(A,)l, in which the sum in Eq. (44)is artificially truncated at a value of A(=A,). This procedure gives an approximate representation of the short-range part of the static potential in addition to retaining the correct asymptotic form for the dominant long-range interactions. d. The dipole cirtoff pottwticil. Another popular approximation to the electron-polar molecule potential is the dipole-cutoff form [DCO(r,)], first introduced by Itikawa and Takayanagi (1969). The long-range dipole potential is cut off at small radii to avoid the r-2 singularity at the origin as, for example, by

u 1 ( 4 = - ( D / r 2 ) { I - e~p[-(r/r,)~]}

(45)

The correct long-range behavior of the dipole term is maintained, and a crude approximation to the short-range potential is achieved through choice of the cutoff radius r c . Rudge (1978a,b) has employed a modified form of the cutoff-dipole potential for calculating cross sections for a number of polar systems. Instead of Eq. (43,a spherically symmetric component ( U J is added to the dipole term. This component is represented by a hard sphere with the radius tuned so as to produce the correct electron affinity for the molecule

368

D. W . Norcross and L . A . Collins

(Rudge, 1978~).The spherically symmetric component mocks to some extent the short-range potential. The resulting set of LF-frame coupled equations is solved numerically within the CC approximation. We refer to this as the close-coupled hard-sphere (CCHS) method. This kind of approach can be generalized to include higher multipolar interactions and/or polarization effects, as in work (Saha rr d.,1981) on CO and HCN. The crudest model of the electron-polar molecule interaction is the point dipole form u l ( r ) = - Dr-*. This form is used predominantly in the Born approximation, but it has also been used in approaches based on the Glauber approximation (see Allan and Dickinson, 1980; Dickinson and Flower, 1981; and references therein) and semiclassical perturbation theory (see Allan and Dickinson, 198 1 , and references therein). Its high-order singularity at the origin considerably reduces its effectiveness for more elaborate collisional calculations (Garrett, 1981), but the exact solution (Clark and Siegel, 1980) for the point dipole potential has been found to be useful for generating T-matrix elements for intermediate angular momenta (Siegel et NI., 1980, 1981a,b).

c. Tlic point dipole potcriticil.

3 . Collisionril Appro.\.iniritioris

In the previous section we discussed various forms employed to represent the electron-molecule interaction. We now consider in more detail the techniques and approximations employed to solve the sets of coupled equations given by either Eqs. (23) or (27). Closr cwirplitig. Of the various coupled-channel techniques described in Section III,A, 1, the CC approximation has been the most widely used. It has been used to obtain exact solutions of the scattering equations for all the model potentials described in Section III,A,2 except the point dipole. In this approximation the infinite sums over quantum numbers in Eqs. (23) or (27) are truncated at finite values. The solutions to the finite-order CC equations are approximate representations of irk;, and i t : $ . The accuracy of these representations are tested by systematically increasing the number of terms retained in the expansions until a particular quantity, say the cross section, converges to within a given tolerance. A detailed description of the procedures used to guarantee accurately convergent scattering quantities is given by Morrison and Collins (1978j. Once the coupled, finite-order equations are established they could, of course, be solved by any of the techniques described in Section III,A,I. In any case, for a given potential, the CC prescription guarantees a system-

(I.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

369

atic procedure for obtaining an exact solution to the coupled equations. This is important for comparisons of model potentials, as differences can be ascribed solely to the potentials and not to the collisional algorithm. (Calculations employing this approach are listed in Table 11.) One aspect of the numerical solution of the CC equations that is unique to polar molecules is the extended radial range of the interaction and the large number of partial waves required to converge the scattering equations. If, for example, outward propagation into the asymptotic region is employed, these two effects can conspire to demand a very large radius for matching to the asymptotic boundary conditions, i.e., rmax a lOI,,,,,/k. Thus the calculations of Gianturco and Rahman (1978) for HCI, who used I',,,;,, = lOOu,, agree with those of Itikawa and Takayanagi (1969), who used I',,,,, as large as 1500 n o , only at the highest energy considered. Another aspect peculiar to polar molecules relates to a standard measure of numerical convergence, the sum of the eigenphases of the manychannel S matrix. While this quantity is strictly convergent, even in the BF frame with the F N approximation, it may appear to converge quite slowly. I t is easily seen, however, that the bidiagonal character of the S matrix for high partial waves (in the limit of large I , only elements with A/ = tl are significant) leads to contributions to the sum that enter in pairs with comparable magnitude and opposite sign. Thus convergence of the eigenphase sum may be a misleading measure of the accuracy with which the short-range interaction has been treated. h. Tho i i ~ r r l t i ~ ~ I r - s c ~ t i t t ~mrtliod. ~'iiz,~ In the multiple-scattering method (MSM) of Dill and Dehmer (1974), space is divided into three regions and various approximations are enforced within these distinct spatial areas. The first region consists of spheres around each atom, in which the molecular potential field is approximated by a spherically symmetric local TABLE I1 R E C Er ~CALc U L ~ T I O N SF O R POIA R MOLECULES EMPLOYING T H E CLOSE-COUPLING APPROACH AND V A R I O MODEL U~ POTENTIALS Molecule

Reference

LiF, K I , CsF HF, HCI LiF LiH, LiCI, CsF, NaCI, CsCI. KI KOH, CsOH

Collins and Norcross (1978) Gianturco and Rahman (1978) Rudge (1978a) Rudge (1978b) Collins r t ol. (l979b)

T-hlotecule HCI H,O, H,S HF CO, HCN HCI H2O

Reference Collins e t N I . (1980b) Gianturco and Thompson (1980) Rudge (1980) Saha et r i l . (1981) Norcross and Padial (1982) Salvini and Thompson ( I 98 I )

370

D . W . Norcross and L . A . Collins

form that includes the static potential and a model exchange potential. In the third region, outside a sphere constructed around the atomic spheres, the long-range multipolar form of the interaction, perhaps including polarization effects, is used (Siegel et d . , 1976), and the equations are solved by standard CC techniques. The potential is assumed constant in the interstitial region. The total scattering wave function is constructed from the solutions of the simplified scattering equations in each region using the boundary conditions at the zone interfaces. The choice of spherically symmetrical zones greatly reduces the complexity of the scattering equations in each zone. The validity of this particular set of approximations to the molecular potential field can be determined by careful comparisons with ESE calculations. The method has been applied to a large number of polar systems, including OCS (Lynch c1 d . , 1979); LiF (Siegel et d . , 1980, 1981a); CsCl (Siegel ct d . , 1981b); and it seems to provide an efficient prescription for tracing the general features of the scattering properties over wide energy ranges (see Dehmer and Dill, 1979). c. The distor?ed-~~ri~ metlind. In the distorted-wave (DW) method, the channels are treated in pairs and back coupling is ignored. This is a weak-coupling approximation. Single-channel scattering equations are solved to introduce distortion effects into the elastic channels, and the transition probability is calculated by perturbation theory. The method, along with its unitarized form, has been applied to rotational excitation of several strongly polar molecules (Rudge rt d . , 1976), but has been shown to give rather poor results for momentum transfer cross sections and for differential cross sections for H > 15” (Collins and Norcross, 1978; Rudge, 1978a). The method seems much more suited to calculating cross sections for electronic transitions (Fliflet et a/., 1979, 1980). d. Cltrssicul mid setriiclri.ssic.trl tnrthods.

Numerous classical and semiclassical approaches to the motion of the incident electron and rotor have been developed. Dickinson and Richards (1975) treated the motion of the electron classically and determined the transition probability from firstorder time-dependent perturbation theory. Dickinson ( 1977) has considered the exact classical solution of an electron scattering from a dipole potential and derived a set of simple, analytical formulas for the total integrated and momentum transfer cross sections by employing classical perturbation theory (CPT). Two approaches have employed classical S-matrix theory to represent the scattering of an electron from a strongly polar molecule. Smith and

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

371

co-workers (Mukerjee and Smith, 1978; Hickman and Smith, 1978) have applied semiclassical perturbative scattering (SPS) techniques (Miller and Smith, 1978) to scattering in the LF frame. This approach treats both the electron and rotor motion by CPT and is thus restricted to high angular momentum states of the rotating molecule. This restriction has been eased by Allan and Dickinson (1981) who still treat the electron motion by CPT, but use a quanta1 representation for the rotor. The S-matrix elements are calculated in the BF frame and transformed to the LF frame using Eq. (35). This semiclassical suddenS-matrix method (SSSM) combines the best aspects of a number of earlier semiclassical approaches and yields for the point-dipole potential a set of simple, analytical expressions for the partial and total integrated cross sections. While these expressions have been derived for the point-dipole potential, the procedure can be extended to include other moments of the potential. Several other approaches warrant mention. The Glauber approximation has been applied to electron-polar molecule collisions by Ashihara et trl. (1973, while Fabrikant (1976, 1977a,b, 1978, 1979, 1980) has developed an effective range theory (ERT) which includes a representation of the long-range multipolar effects as well as a parametric representation of the short-range potential. Band (1979) has combined CC and classical techniques to produce a quasi-classical CC procedure. The method seems capable of producing differential cross sections in reasonable agreement with more sophisticated models to about 90". Botw crppr.o.~imcrriotis. We consider three forms of the Born approximation: the first Born approximation and two unitarization prescriptions. The T matrix in the FBA, which was discussed in Section I,B,l, depends on an integral over the incoming and outgoing plane waves and the interaction potential which induces the transition. For example, in the BF frame, FN approximation, the FBA T-matrix element is given by

CJ.

where j,(kr.) is the Bessel function of order I and Vll, is the interaction potential. For highly polar systems, unitarity is strongly violated in the FBA (Itikawa, 1969; Clark, 1977). Two prescriptions have been devised to impose unitarity on the T matrix. The BII approximation (Seaton, 1961, 1966; Levine, 1969) takes the form

372

D. W . Norcross and L . A . Collins

while the BIII approximation (Seaton, 1966; Levine, 1971) is given by

T,,,,,

= 1

-

exp(2iB)

(48)

The unitarized Born approximations applied to the dipole interaction alone not only satisfy the requirement of unitarity but also account to some extent for second-order interactions (lAj1, 1A/1 f 1 ) not allowed for in the FBA (Itikawa, 1969). This, along with their relative simplicity and the ease with which they are generalized to arbitrary long-range interactions, makes them very useful in generating T-matrix elements for intermediate angular momenta (Padial et l i / . , 1981). A modified form of the FBA has been suggested by Rudge (1974) in which the integral is truncated at a value of the radius characteristic of the molecular size. Numerical techniques for evaluating the Born partial wave integrals have been discussed by lnokuti (1980).

B.

S P E C I F I C A T I O N O t CROSS S E C T I O N S

Since one of our goals is to exploit as fully as possible all of the simplifications attending BF-frame calculations with the FN approximation, it will be helpful to first recall several of the general theorems of the AN approximation as applied to cross sections. We will then consider how these theorems can be made most useful in the case of polar molecules. I. Tlio AN Apl',o.uit,ititioii

We sum Eq. (43) over allj' and tnj,, assuming k,. to be insensitive to j ' and using the completeness of rotational wave functions: then nveraging over i n j and using the addition theorem for the rotational wave functions, we obtain

1

d ~ < , , . j . , , ~ , .= ~ / (Xpz/X,,)(4~)-' dfl If,".z,Ck,, , k,. :

dR

(49)

This is independent of j and is conveniently viewed as a vibrational cross section obtained in the BF frame, averaged over all orientations of the molecule. Another cross section often of interest is the energy-loss, or stopping, cross section. This is defined by

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

373

It has recently been shown (Shimamura, 1981) that if both initial and final states are eigenfunctions of the same H,,, ,then Eq. (50) is independent of .i. Next, consider Eq. (49) summed over all u ' . Using Eq. (41) and the completeness of vibrational wave functions yields c/a,,,,,,,,/dR = ( k P , / k , ) ( 4 r - '

1

(ullf%(kq,

kqg; R)I21u) dR

(51)

This is not independent of u . If, however, the initial vibrational wave function is well localized in R about the equilibrium separation R,, , then Eq. (51) can be approximated by

1

da,,,,J,,,/dn = (kp,/kP)(4n)-' If,".F,f(k,,k,, ; R,)12 d f i

(52)

It is obvious that to the extent the AN approximation is valid, a great deal of information can be obtained from a quite limited calculation (or from a few measurements), and that the interpretation of physical systems is greatly simplified. With the identification of Eqs. (42) and (30), the actual evaluation of cross sections is conventionally accomplished using the transformation Eqs. (35) or (38) and the LF-frame formulas presented by Arthurs and Dalgarno ( 1960). An extremely useful alternative follows by using a standard contraction formula (Rose, 1957) for the product of two rotational wave functions in Eq. (43), which when then summed over mj, and averaged over t ~ leads > to

2 C(j l e j ' ;AaA,.-A3'(doZ,,,,,/dR)

d ( ~ , , , . ~ , ~ ~ ,= , , ,( ~k z, ,/, d/ kOp )

(53)

1,.

where dcr:;t,,ct,vr/dfl is independent of .j and j ' . This kind of factorization has been a general feature of the development of the AN approximation (Oksyuk, 1966; Chang and Temkin, 1969, 1970; Temkin and Faisal, 1971). Equation (53) applies to symmetric-top as well as linear molecules, and there is an analogous result for asymmetric-top molecules (Norcross, 1982). Such factorization is not, however, unique to the AN approximation. It also appears in perturbative treatments not in any way employing the FN approximation (e.g., Dalgarno and Moffett, 1963; Crawford, 1967). The use of a factorization based on I, has also been shown to be of great value in full LF-frame calculations (Chandra, 197.5~).The physical implications of associating I , with the angular momentum exchanged during the collision have been thoroughly explored by Fano and Dill (1972). The advantages of this kind of factorization are apparent. Having once obtained the partial cross sections in Eq. (53) for a (usually small) range of values of I , , cross sections for any rotor state are easily obtained. Conversely, if the physical conditions for validity of the AN approximation

374

D . W . Norcross und L . A . Collins

are deemed to hold, then the calculation or measurement of cross sections for a few rotor transitions may suffice to determine them all. Consider also Eq. (53) summed over aI1.j'. This yields

d ~ r ~ , , . j , ~ , ~=t . ~(A,,,/L,,) ldR

2 dc&.,,,,,,/dfl

(54)

1,.

For linear molecules, Eq. ( S O ) reduces (Norcross, 1982) to

Equation ( 5 5 ) has also been generalized to symmetric-top and asymmetrictop molecular (Norcross, 1982).This kind of factorization into dynamical and geometric parts has also been extended to characterize scattering by state-selected (m, as well a s j ) polar molecules (Allan and Dickinson, 1980) as a function of azimuthal as well as polar angle. Thus Eqs. (49) and (SO) can be simply represented as a sum over cross sections partial in I,.. The partial cross sections may also be used as parameters in a least-squares fit to analyze rotationally unresolved energyloss spectra (Shimamura, 1980). We see that even if only rotationally summed cross sections are of direct interest, there is the potential for obtaining a great deal of additional information by carrying out calculations in such a way as to extract the individual partial cross sections in Eq. (54) as a byproduct. It also turns out to be vastly more economical (Norcross and Padial, 19821, a happy coincidence! We also note that these conclusions are not dependent on the FN approximation for vibration. If H\i,,were to be retained in Eq. (27) and the alternative representation used, then Eq. (40) would depend on /?,not R, and (1 would carry vibrational indices. The integral in Eq. (41) would then be superfluous, but Eqs. (42), (43), and ( 5 3 ) - ( 5 5 ) remain valid. These would then describe the "hybrid" theory of Chandra and Temkin (1976: see also Choi and Poe, 1977b). 2. Applic~itionof C1oslrr.r

Now let us consider the evaluation of Eq. (49) in a little more detail. It can be expressed as a Legendre expansion, the coefficients of which are written in terms of BF-frame T-matrix elements (Burke and Chandra, 1972). Consider the fact that the FBA is ultimately correct for the elements HFT$in the limit I , I ' -+ =. It is easily shown (e.g., Crawford ct (11..

ELECTRON SCATTERlNG BY HIGHLY POLAR MOLECULES

375

1967) that for the dipole interaction treated in the FBA, for NII I and I' only the value I, = 1 contributes in Eq. (53), leading to Eq. (4). It follows that the essential failure of the AN approximation for polar molecules manifested in Eq. (4) only occurs for I, = 1 , and that for any other value of I, the AN approximation will be perhaps no worse an approximation for polar than nonpolar molecules. In addition, referring to arguments made in the Section I for the case X , A,,, we note that it may be quite adequate even for I , = I for the differential cross section for scattering out of the forward direction, and for the momentum transfer cross section. Thus the A N approximation has found much useful application both for partial and total differential and momentum transfer cross sections (Gianturco and Thompson, 1977, 1980). Near threshold the AN approximation will certainly be inferior to the frame-transformation methods, particularly for I, = 1 . For I, # 1 , the range of energy over which the AN approximation is valid may not differ significantly for polar and nonpolar molecules. Use of the modified transformation from RFK to LFK suggested in Section II,B,2 may lead to improved results for I, # 1 , but to the extent this is necessary the simple theorems summarized above no longer hold. A word of caution is in order, however. It has been shown (Norcross and Padial, 1982) that etw-y coefficient in the Lengendre expansion of Eq. (53) is strictly divergent for I, = I . Thus without very careful attention to the evaluation of these coefficients, i.e., carrying out the sums in such a way as to achieve exact cancellation of the divergent terms (e.g., Chandra, 1975a), very serious errors could result, particularly in the differential cross section. The use of simple closure formulas is a solution to this problem. Consider Eq. (13), now expressed in terms of Eq. (30). In actual computations the partial wave sums involved in the evaluation of Eq. (13) can be very slowly convergent, particularly away from thresholds, this being an artifact of the essential divergence of the FN approximation. Therefore Eq. (13) is usually replaced (Crawford and Dalgarno, 1971) by

-

da,,,*/dR

=

(dtF,,,*/dfl) + A(dU,,,,/dfl)

A ( d a , , , , / d f l )= ( A , ~ / k , ) { V f ; ~ n , ( k p,)Iz

(56) '}

Ifi,Fni(kpi)l

(57)

The first term in Eq. (56) is a simple approximation for the cross section, dominated by long-range multipole interactions, and the second term in Eq. (57) is the .s(itne cross section in partial wave form. It is assumed that contributions from the two terms in Eq. (57) cancel out for large I and I'. Thus, while Eq. (56) and Eq. (13) are formally identical, numerical convergence is greatly speeded up. This technique, applied using .f,9N,,(kq,)

376

D. W. Not.c.ros.v r i n d L . A . Collins

from Eq. (42) instead ofj'k$(kp,), and Eq. (4) and its partial wave representation in Eqs. (56) and (57), will permit efficient and problem-free evaluation of Eq. (53). 3 . Tlir M E A N Approsirnrr f i o i i

Another method has been suggested that combines the best features of frame-transformation methods and the AN approximation. These are the suitability of the former to calculations for polar molecules, particularly near threshold, and the simplicity of the latter. Since Eq. (57) is dominated by a finite range of angular momenta, it is reasonable to consider the AN approximation (for vibration andlor rotation) for this term otily. This multipole-extracted adiabatic-nuclei (MEAN) approximation (Norcross and Padial, 1982) is therefore based on the same working hypothesis as the AFT method, but its mechanics are considerably simpler. With the A N approximation, i.e., Eq. (43), for both terms in Eq. (57),the second term serves to effect cancellation of contributions to the first for large angular momenta, for which the AN approximation breaks down. This kind of approach was first suggested in the context of total cross sections for vibrationally elastic scattering (Fabrikant, 1976; Dickinson, 1977; Collins and Norcross, 1978), i.e., using Eq. (49) to evaluate Eq. (57). An additional feature of the MEAN approximation is the use of the geometric mean kpkg,= k,kp. for the evaluation of Eq. (43). The closure formulas, as usually applied, adopt the FBA for the dipole potential for the quantities dF,,,./di2 and? Using the FBA in the MEAN approximation would correspond roughly to using the F B A for / > I , in the 1980), or I' > I-, in the RFT method (Clark, AFT method (Siege1 of d., 1979). The MEAN approximation is not, however, subject to any such restriction, since dZ,,,,/dR and? could be obtained including interactions other than the dipole in the FBA, or even by going beyond the FBA. For highly polar molecules, rapid cancellation in the evaluation of Eq. (57) using the FBA for ,/ = 1 may be hampered by violation of unitarity, and for I , = 0 and 2 by ladder coupling of the dipole interaction. This suggests use of, for example, a unitarized Born approximation for the dipole potential. The essential breakdown of the FN approximation near threshold discussed in Section I,B,2 can be dealt with only by going to a coupledchannel treatment for dt?,,n,/dflandf. For D s D, the choice of shortrange potential is immaterial, but for D > D,. it might be possible, if difficult, to mock the effect of the term involving 6 in Eq. (9). In any event, by reducing the contribution from Eq. (57) to the final result, errors associated with the use of the AN approximation are reduced. This is analogous to choosing r , ( / , ) closer to zero in the frame-

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

377

transformation methods, i.e., additional accuracy is obtained at the expense of additional work. There is nothing to preclude use of the /,-conserving modification, or the inclusion of H,.ib,in BF-frame calculations. The only requirements are that the second term in Eq. (57) differ from the first term in Eq. (56) only in the use of the AN approximation at some level, and that contributions to the two terms in Eq. (57) cancel identically in the limit of large / and 1 ' . To the extent that the effect of long-range interactions is confined to the first term in Eq. (561, problems attending its application near threshold are mitigated. Use of the geometric mean variant of the EMA approximation for the scattering amplitudes in Eq. (57) thus becomes particularly attractive. All of the implications and theorems of the AN approximation can now be applied to Eq. (571, even if invalid for the first term in Eq. (56). It is clearly desirable to retain the FBA for this term and its analog in Eq. (57) as far as possible, as sums over final states are thereby greatly simplified.

IV. Applications Most recent applications of the theoretical tools described in Section 111 are summarized in Tables I and 11, or referred to in the text. It is noteworthy that we seem to be rapidly approaching an era when highly sophisticated calculations become routine. This is not to suggest that simple models will lose their value. They will continue to provide the majority of needed data and to help elucidate the basic physical processes that very sophisticated calculations may obscure. It is becoming possible, however, to put these models to more demanding theoretical tests. We subject several of these models to such a test in this section by choosing representative cases from each general type and performing a detailed, systematic comparison for a single molecule, LiF. Significant progress has also been made recently in experimental work. Data are now available on differentialcross sections for vibrationally elastic and inelastic scattering by HF and HCI (Rohr and Linder, 1976): H,O (Seng and Linder, 1976); KI (Rudge el d . , 1976); HBr (Rohr, 1977a, 1978a): LiF (Vuskovic er d . , 1978); HCN (Srivastava cr id., 1978); H,S (Rohr, 1978b); and CsCl (Vuskovic and Srivastava, 1981). Comparison of these data with the results of calculations serves not only as a test of theory, but also, in most cases, as the only way to put the measurements on an absolute scale. In this section we will also consider conclusions which have, and may, be drawn from such comparisons.

A. A COMPARATIVE STUDY

FOR

LiF,

AND

OTHER MOLECULES

A meaningful comparison of the various models can be achieved only in terms of a standard. For this purpose, we select the ESE approximation. This choice is justified along two lines of argument. First, if we exclude the effective polarization potential, all the models presented in the previous section are basically approximations to the static-exchange equations. These models, from the OSME to the FBA, contain no more (and usually a great deal less) information than the ESE case. Thus, the use of the ESE results as a standard against which to judge the other theoretical models is justified. Second, in a more general sense, the ESE results should provide a reasonable representation of the elastic scattering process away from resonances, above rotational thresholds, and below electronic thresholds. This expectation is borne out to some extent by the comparison of the theoretical and experimental results. Therefore, the ESE calculations serve not only as a theoretical standard, but also as a moderately good representation of low-energy elastic scattering for strongly polar molecules. Before embarking on a detailed comparison of the various models, we review some general features of the cross sections for electron-polar molecule collisions. A typical total differential cross section is strongly peaked in the forward direction, drops precipitously for the intermediate scattering angles (220"),then rises at the larger scattering angles (2loOD). The main contribution to the small-angle scattering comes from distant encounters. For such encounters, only the long-range dipole potential can have any influence since the electron never approaches near enough to be affected by the local molecular field. Such distant encounters, which have large impact parameters, correspond quantum mechanically to high partial waves (or large values of /). These partial waves are associated with channel wave functions which encounter large centrifugal barriers which in turn confine the waves to large radii and prevent them from penetrating into the short-range field of the molecule. Thus, the scattering of these high partial waves is due primarily to the long-range dipole potential. This implies that a simple scattering approximation such as the Born should describe the collisional process in this regime quite well. As the scattering angle is increased, the effect of close encounters and therefore of low partial waves becomes more pronounced. The low partial waves can penetrate their associated weak centrifugal barriers and can be greatly affected by the short-range field. Therefore, we must also increase the sophistication of our models of the short-range interaction. We shall see in the following discussion more quantitative demonstrations of these basic observations.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

379

Although ESE calculations are not available for most strongly polar systems that have been studied recently (e.g., CsCI, KI, CsOH, and KOH), similar comparisons (Collins and Norcross, 1978; Collins et a/., 1979b), with simpler models (e.g., DCO, BII, CPT, and other semiclassical methods), support the basic conclusions reached below concerning the reinfive accuracies of these models. In addition, studies on the weaker polar system HCl (Padial ef d . , 1982) have led to similar conclusions, although polarizations effects can be of more importance. Where possible, we compare calculations employing the same target parameters, such as bound molecular wave functions and multipole moments, and the same collisional code. This procedure, coupled with a sedulous attention to convergence, serves to highlight the basic physical differences among the models and to eliminate spurious differences due to the use of numerical techniques of varying degrees of accuracy. We choose as a representative system the LiF molecule since it possesses the requisite large dipole moment ( D = 2.54) and a sufficient number of electrons and strong enough nuclear charges to produce a significant shortrange potential. We have selected the following set of models for detailed comparison: exact static-exchange (ESE), orthogonalized-static-model exchange (OSME), static-model exchange (SME), static (S); truncated static [%A)], dipole cutoff [DCO(r,)], semiclassical sudden S matrix (SSSM), classical perturbation theory (CPT),unitarized Born (BII), and first Born. All models requiring molecular wave functions used the near-Hartree-Fock result of McLean and Yoshimine (1967) with the fol!owing properties: El,,, = - 106.9916 hartree, Re, = 2.9877a0, and D = 2.54eci0. In addition, we compare these results with those of the multiple-scattering (MSM) method (Siege1 c v d.,1980), and the close-coupling hard-sphere (CCHS) method (Rudge, 1978c), and the Glauber approximation (I. Shimamura, private communication). The ESE calculations were performed with the CC linear algebraic method (Collins and Schneider, 1981), which takes full account of the nonlocal nature of the exchange interaction. The CC integral-equations propagation method (Collins and Norcross, 1978; Collins ef CJI. 1980a; Morrison and Collins, 1978) was employed for the OSME, SME, S , S(2), and DCO models. The static potential was augmented by the HFEGE local exchange potential for the OSME and SME cases. For a given local potential, the eigenphase sums and T-matrix elements calculated by the linear algebraic and integral equation codes agreed to within better than 1%. Thus, a direct comparison of the results of these two methods is valid. The CC codes were used to calculate the BF-frame FN T-matrix elements for the lowest few symmetries (typically, m s 6) for which the

3 80

0. W. Norc~ossrititl L . A . Collins

short-range effects were not negligible. These T-matrix elements were combined with BII and FBA elements for the higher symmetries and partial waves in the MEAN approximation (Norcross and Padial, 1982)to produce integrated, momentum transfer, and differential rotational excitation cross sections. Finally, simple analytical expressions were used for the SSSM (Allan and Dickinson, 19811, CPT (Dickinson, 1977), and FBA results. We are at last ready to present the comparison of the models. We break the discussion into three parts: integrated, momentum transfer, and differential, according to the types of cross sections most typically calculated. A few minor points should be noted for clarification. First, in the following tables we present results at 3.0 eV, which coincide with entries in our earlier paper (Tables XI and XIII, Collins and Norcross, 1978). The results for a particular model show slight differences between these two sets of calculations. This difference arises primarily from the use of slightly different convergence criteria and dipole moments (2.54 a.u. in this study as compared to the 2.59 a.u. in the older article). The use of a different dipole moment (and sensitivity to the choice of the rotational constant) is most apparent in the v(0 + 1) cross section. This should be taken into account in the comparison with other independent calculations for the integrated cross sections. Second, in the tables of our earlier paper the designation SE was used for SME.

I . It t t egrri t d Cross S rctiot I S In Table 111 we present the partial and total integrated rotational excitation cross sections at three energies. We recall that the integrated cross section is simply the differential cross section integrated over all angles with each point being weighted equally. As we observed, the total differential cross section for an electron scattered by a highly polar molecule is strongly peaked in the forward direction. Therefore, the dominant contribution to the total integrated cross section originates in this small-angle region. Since the main contribution in this region comes from high partial waves whose scattering is mainly determined by the long-range dipole potential, we expect any model that simply takes into account the asymptotic dipole field to provide a fairly good approximation to the total integrated cross section. Indeed, the FBA results are only in error by less than l5%, while if the simplest account is taken of distortion (e.g., the SSSM, CPT, or BII approach), the agreement with the ESE cross sections is significantly improved. In fact, all models beyond the FBA give results within a few percent of each other. Therefore, for the total integrated cross section, the very simple, analytical formulas provided by SSSM or CPT seem adequate to determine the total cross section to within 5% or better.

TABLE 111 A N D TOTAL( S U M M EPARTIAL) D INTEGRATED CROSSSECTIONS" FOR LiF FROM PARTIAL SEVERAL MODELS

Model

0-0

0-1

0-2

0-3

0-4

I

6225.3 6221.1 6239.8 6230.6 6213.5 6438.8 5895.5 6200.5 6162.2 6050.9 6925.9

For transitions from the ground rotor state at 1.O eV ESE OSME S S(2) DCO(0.5) MSMb CCHS' SSSM CPT BII FB A

373.3 370.5 397.4 386.4 381.9 364.3 373.9 325.9

5641.9 5647.8 5612. I 5640.0 5613.7 5856.7 5350.1 5660.1

155.6 150.8 158.6 156.9 157.7 165.7 128.8 153.2

45.5 43.4 59.1 38.5 49.4 92.5 34.6 42.9

9.1 8.7 12.5 8.7 10.9 9.6 8.2 18.4

242.7

5666.4 6925.9

110.0

25.8

6.1

For transitions from the ground rotor state at 3.0 eV ESE OSME S S(2) DCO(0.5) MSMb CCHSC** SSSM CFT BII FBA

105.5 106.0

113.0 157.0 116.9 90.4 107.3 108.6 83.7

2159.4 2159.3 2147.5 2150.2 2145.6 2371.3 2014.8 2156.0

56.5 58.1 54.8 42.5 54.4 53.6 49.2

2156.6 2578.2

13.5 14.1

51.1

14.5 12.4 14.0 14.3

2.6 2.6 3.2 2.8 3.1 2.3 2.7 6.1

36.6

9.6

2.4

15.1 11.1

2337.5 2340.0 2333.6 2363.7 2334.4 2529.9 2188.0 2336.1 2323.6 2288.9 2578.2

For transitions from the ground rotor state at 10.0 eV ESE OSME S S(2) DCO(0.5) MSMb CCHS'.* SSSM CPT BII FBA

" In 06.

*

2.59 a.u. 2.47 a.u. Interpolated.

D D

=

=

37.3 37.2 25.6 48.4 37.4 26.1

737.9 739.1 737.7 737.2 734.3 767.8

15.6 16.2 11.3 15.4 14.2 12.4

5.3 4.6 3.2 2.2 3.O 4.4

I .8 I .6 0.9 3.8 0.8 0.4

32.6

735.3

15.3

4.3

1.8

27.7

733.9 862.0

11.1

3.9

1.1

798.0 798.7 778.7 807.0 790.3 811.2 738.3 789.3 785.6 777.6 862.0

D. W . Norcross

and L . A . Collins

The partial integrated cross sections provide a more severe test of the models. The transition for j = 0 to j = 1 is dominated by direct dipole coupling and is therefore similar in its behavior to the total cross section. The cross sections for transitions to final states other thanj = 1 are influenced only indirectly by the dipole potential through ladder coupling, and are more sensitive to the other molecular moments and the short-range potential. The OSME model gives cross sections for these transitions consistently to within 10% or better of the ESE standard. While the other models can approach this accuracy for a given transition and energy, they do not do so in a consistent fashion and give somewhat erratic results. For example, at 3.0 eV the S cross sections o(0 +. j ’ ) are in error by no more than 12%, while at 10.0 eV this error rises to over 30%. The S and DCO models yield partial integrated cross sections accurate to within 10-5057, while the S(2) and BII cases provide results in the accuracy range 2050%. The SSSM model is remarkably accurate, given its simplicity, but to assure an accuracy in the partial integrated cross sections of better than 1596, we must resort to a model of at least the sophistication of the OSME. 2. Momoitirnr-~crrisfcr. Cross Sections In Table IV we present the total and partial momentum-transfer cross sections. The momentum-transfer cross section provides an even more stringent test of the models. Since the momentum-transfer cross section is a weighted integral of the differential cross section which deemphasizes small-angle scattering, we expect this cross section to be much more sensitive to the short-range interaction. We note that the cross sections in the FBA approximation are in error by factors of 2 or 3, confirming our expectations. The OSME model consistently produces total cross sections within 5% of the ESE results. As with the partial integrated cross sections, the other models yield rather erratic results. The S and DCO models appear capable of yielding results accurate to 20% or better, while the S(2) and BII models can vary in accuracy from 10% to 50%. Both the MSM and the CCHS procedures give total momentum transfer cross sections to within an accuracy of about 15% at low energies. As the energy increases, they appear to give less satisfactory results (errors, 230%). This trend may arise from the greater importance of high partial waves and potential moments, which are more approximately handled by these techniques at these elevated energies. The CPT approach yields a cross section that behaves as 395(eV-a i ) / E and is uniformly too large. The Glauber approximation, on the other hand, yields (I. Shimamura, private communication) a cross section that behaves as 169(eV-ui)/E and is uniformly too small. In view of the fact that the SSSM model yields

TABLE IV PARTIAL. A N D TOTAL(SUMMED PARTIAL) MOMENTUM-TRANSFER CROSS SECTIONS' LiF FROM SEVERAL MODELS Model

0-0

0- 1

0-2

0-3

FOR

0-4

1

12.6 12.1 17.8 11.8

256.I 265.8 215.2 275.7 244.4 225.8 284.3 395.2 209.1 735.2

For transitions from the ground rotor state at 1.0 eV ESE OSME S

S(2) DCO(0.5) MSMh CCHS' CPT BII FBA

54.1 58.7 59.0 92.9 53.4 46.8 88.0 25.5

79.2 91.7 64.4 74.5 66.2 77.5 110.0

78.6 735.2

60.9 56.8 66.0 58.8 44.I 49.3 40.8

49.2 46.5 68.0 37.2 51.8 39.6 34.6

64.1

31.6

15.0

12.6 11.0

9.3

For transitions from the ground rotor state at 3.0eV ESE OSME S

S(2) DCO(0.5) MSMh CCHSr.d CPT BII FBA

14.5 12.6 23.3 46.6 22.8 11.4 15.1

9.0

21.8 20.3 22.8 41.3 20.2 20.4 22.5

24.3 25.8 24.8 11.9 20.9 17.1 18.5

14.4 15.3 16.6 11.3 14.5 11.4 14.8

3.7 3.8 4.5 4.2 3.O 3.7

26.1 245.1

20.4

11.6

3.7

3.7

78.8 77.8 92.1 114.8 82.4 63.3 74.6 131.7 70.7 245.1

For transitions from the ground rotor state at 10.0 e V ESE OSME S

5.0

S(2) DCO(0.5) MSMh CCHS'.' CFT BII FBA fl

In ( 1 ; . D = D =

2.59a.u. 2.41a.u.

Interpolated.

4.9 3.9 7.6 10.6 5.9

3.2

16.3 15.4 21.6 10.5 11.0 11.1

7.7 73.5

6.0 6.7 4.2 9.3 4.6 4.1

6.2

2.8

5.0

2.5

4.2 I .7 2.5 5.3

5.0 I .o

5.3

4.6

1.6

0.9 0.6

36.3 34.5 34.8 34.I 29.8 27.0 20.4 39.5 22.3 73.5

I). W. Norcros.v t i t i d L . A . Collim

384

differential cross sections in good agreement with the Glauber approximation, it might be expected to yield similar results for the momentumtransfer cross section. Thus, even at the level of the t o t d momentumtransfer cross section, we observe quite a spread in the results from the various models. The effective-range theory (ERT) of Fabrikant (1976, 1977b) provides a useful framework for rationalizing these various results. The total momentum-transfer cross section in this theory, which is intimately related to the threshold behavior of the eigenphase sum mentioned in Section I,B,2, takes quite simple forms, e.g., for D >>D, (see Collins c't t i / . , 1 979b), EvxI = C[l

-

A sin()vllnX. -

4)]

(58)

where C, A , and v are constants that depend only on D , and 4 depends on 1 IOD both D and the short-range interaction. The ERT predicts that C = 280 eV-ai, A 0.2 and v 1.5 for LiF. The form of Eq. ( 5 8 ) , with these values, is in remarkably good agreement with the CCHS results and with more extensive DCO(r-,) calculations (Fig. 2). The BII, CPT, and Glauber models yield values of C well outside the range C( 1 k A ) and fail to reveal the sinusoidal behavior of Eq. ( 5 8 ) , not because of inadequate treatment of the short-range interaction, but because of inadequate treatment of the most penetrating s-wave contribution for which D >>D,. = 0.639 a.u. For energies above a few electron volts, of course, the basic assumptions of the ERT will break down at the same time that short-range interactions become relatively more important. It has, however, been

-

I

10-1

-

-

I

I

I

1

1

1

1

1

1

I

1

I

I

I

I

I

I

10

E(eV)

FIG.2. Total momentum-transfer cross sections for LiF from several dipole cutoff models and from the CCHS model ( 0 )of Rudge (1978): --, DCO ( 0 . 3 ) : ---,DCO (0.50); , DCO (0.75); ----, DCO (1.00).

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

385

remarkably successful in analyzing swarm data (Fabrikant, 1977b), where the average over an electron-velocity distribution greatly reduces the amplitude of the oscillations (Collins et NI., 1979b),and thus the sensitivity to the one adjustable parameter 4 . The partial momentum cross sections show even a greater sensitivity to the models. The OSME model can only be relied on to about 15% accuracy for the various partial cross sections. The other models do not show any degree of systematic behavior and can be in error from 10 to loo%. An investigation of the partial cross sections reveals an interesting point-that the accuracy to which various models are able to reproduce the total momentum-transfer cross section may be somewhat accidental. For example, the CCHS method gives crLIto within I I% of the ESE result at I .O eV. However, this effect is produced by this model overestimating u \ ~ ( O + 0) and a x I ( O 4 1) by 63 and 395%,respectively, and underestimating (+\,(O ---* 2) and a\,(O+ 3) by 33 and 3096, respectively. Therefore, for accurate partial momentum-transfer cross sections, we are forced to use a method at least as sophisticated as the OSME. 3. D i f i r c v h l Cross Sections

In Figs. 3 and 4 we present total differential cross sections for r-LiF collisions. The basic differences in the models described for the integrated and momentum-transfer cross sections manifest themselves in the differential cross section to varying degrees according to the scattering angle. For small scattering angles ( s15"), the cross section is dominated by far encounters, and all of the models are in better than 5% agreement with the FBA values. For intermediate angles (15-60"), the various model cross sections remain within 30% of each other but begin to significantly depart from the FBA curve. In this region some account of the distortion of the continuum wave function must be taken into account. Such simple methods as the Glauber (Shimamura, 1979), SSSM, CFT, and BII seem adequate. For example, the simple analytical form of the CPT model gives cross sections within 25% of the ESE standard over this angular range. For larger angle scattering, a more sophisticated representation of the short-range interaction is required. We observe that in the region of 120180", the various models can have differences of factors of 2 or more, but that the MSM results are within 30% of the ESE standard for angles up to 120". The partial differential cross sections can show much more pronounced differences among the models. Also in Figs. 3 and 4, we compare the results of the ESE, SME, DC0(0.5), and FBA models with those of the experiment performed by

386

8 81

Fici. 3. Total differential cross section for LiF at 5.44 eV from several models:-, FBA; , MSM; - - -, DCO (0.5);- -, ESE. The measured data (at 5.4 eV) of Vuskovic ti/. (1978) are normalized to the ESE results at 40".

Vuskovic ct a / . (1978) at 5.4 and 20.0 eV. We have normalized the experimental results to our cross section at 40". The agreement is reasonably good at both energies. The SME results at 20.0 e V are in excellent agreement with ESE results at all angles. We have also investigated the effect of adding an effective polarization potential to the ESE case; however, this does not appear to significantly alter the cross section, even in the backward-scattering region.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES IO2C

lo-!

1

I

1

1

387

I

I

I

I

I

1

30

60

90

120

150

1

180

8 F I G .4. As in Fig. 3 at 20.0 e V -, FBA: - - - -, MSM; - - -, DCO (0.5); - -,

SME.

We note that Vuskovic et nl. originally normalized their results to the CPT cross section at 40", but that the present normalization to the ESE results raises the experimental values by 29% at 5.4 eV and 17% at 20 eV. A similar revision of the normalization of experimental data for KI (Rudge o r d., 1976) has been suggested (Collins and Norcross, 1978). This illustrates a fundamental difficulty in the normalization of experimental data using calculations. In the angular region lo" where theoretical differences are small, experimental uncertainty may be large, whereas in the region

-

388

I). W. N o i ~ r o s soritl L. A . Co1liri.s

-40" where its uncertainty is smaller, calculations may differ by as much as 30%. Normalization to calculations also facilitates the extraction of integrated and momentum-transfer cross sections by extrapolation of the experimental data beyond the angular range of the measurements. In view of the importance of small-angle scattering, particularly to the integrated cross section, it may be most accurate to normalize at the largest angle for which several models agree, and to adopt the theoretical cross section for smaller angles. This was the approach used in recent work on CsCl (Vuskovic and Srivastava, 1981). A contrary tack was taken by Srivastava ct t r l . (1978) in the analysis of their measurements on HCN. They took their absolute normalization of the differential cross section from simultaneous measurements on helium, and estimated the contribution to the total cross section from angles less than 20" by normalizing the differential cross section in the FBA to their results. This involves changes in the small-angle contribution by factors of as much as 2.5, which are much larger than credible on the basis of any theory. Renormalization of these data as suggested above would result in changes of as much as a factor of 2 in the integrated cross sections obtained, but much less in the momentumtransfer cross sections. B . V I B R A T I O NEXCITATION AL

Most attention paid to date to electron-polar molecule collisions has been focused on rotational excitation. However, one area of vibrational excitation has received considerable theoretical treatment. This involves the question of the mechanism(s) which produce the sharp threshold peaks and broader high-energy resonances observed for electron scattering from several hydrogen halides. Models have been proposed based on threshold resonances (Taylor c>t ul., 1977; Segal and Wolf, 19811, virtual states (Nesbet, 1977; Dube and Herzenberg, 1977), final-state interactions (Gianturco and Rahman, 1977), and broad shape resonances away from 1979; Domcke and Cederbaum, 1981). The qualthreshold (Domckeot d., itative shape of the threshold feature in HCI was reproduced in the models of Dube and Herzenberg, and Domcke and Cederbaum by a suitable choice of parameters. Rudge (1980) carried out a more elaborate vibrational CC calculation for H F and was also able to reproduce the qualitative features of the measurements with suitably chosen parameters in the model potential employed, although great sensitivity to their choice was noted.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

389

More recently, the observation of similar features at vibrational excitation thresholds in several nonpolar molecules (see Rohr, 1979) upset the developing consensus that the polar nature of the molecule was an essential element. Herzenberg ( 1979) subsequently argued that “the deviation from spherical symmetry does not seem to enter the calculations in any essential way,” and showed that the feature could be reproduced using a virtual-state model without any long-range potential or low-energy resonance. The apparent correlation of the magnitudes of the resonance features in HF, HCI, and HBr with their polarizability and polarizability derivatives was noted by Rohr (1978a) and Gianturco and Lamanna (1979). Measurements reported by Azria et a / . (1980a) have added a new twist. It was shown that the previously reported (Rohr, 1977a, 1978a) peaks in vibrational excitation of the u = 3 to u = 5 states of HBr were due to negative ion formation, and it was suggested that this effect was completely responsible for the observed (Rohr, 1977b) peaks in (nonpolar) SF,. This has not only revived interest in an interpretation stressing the importance of the dipole field, but also presents difficulties for the virtualstate picture. This is because threshold resonances associated with a virtual state might be expected (Nesbet, 1977) to appear at every vibrational threshold. The resonance interpretation of Taylor et a / . (1977) remains quite controversial. The stability of the lowest lying roots in their stabilization calculation has been challenged (Krauss and Stevens, 1981). Nesbet (1977) interprets them as artifacts of a virtual state, but Domcke and Cederbaum (1981) exclude both resonances and virtual states, arguing instead that they are due to an “extremely steep decrease of the fixed-nuclei phase shift near threshold.” They predict that this should be observed in FK scattering calculations for vibrationally elastic scattering. In the most elaborate calculations to date on scattering by HCI (Padial ef d., 1982), the eigenphase sum was found instead to be effectively constant below 0.3 eV and to approach the limit suggested by Eq. (8). There is obviously no clear evidence favoring any particular interpretation of the threshold features. Final determinations may have to await more detailed calculations. In addition to our own work on HCI, very sophisticated calculations are in progress (Rescigno ef a/., 1982) on vibrational excitation of HF, the preliminary results of which suggest that the observed threshold feature is reproduced without any obvious resonant behavior in the eigenphase sum as a function of energy for FN scattering. The feature emerges, rather, from the finite behavior of the eigenphase sum near threshold [see Eqs. (8) and (9)l and from rapid variation with internuclear separation at low energies.

390

D. W . Norcross and L . A . Collins

V. Conclusion Given the length and the detail of the previous discussion, we are clearly faced with a nearly impossible task in any attempt to write a summary and are forced to side with Winston Churchill when he remarked, “One never finishes a writing project, just abandons it.” However, we should state briefly the main areas of recent progress in the theory of electron-polar molecule collisions discussed in this review, and indicate the directions in which future research should aim. Three areas in particular have witnessed an extensive amount of development over the past few years: (1) Classical and semiclassical methods have been extended and refined such that certain scattering cross sections, such as the total and partial integrated (for small j ‘ ) and small-angle (560”) differential, can be obtained from fairly simple analytical expressions. This development frees, to some extent, the more sophisticated scattering methods to concentrate on the large-angle scattering regime. (2) Recent years have seen significant developments of sophisticated mathematical and numerical techniques for treating the nonlocal character of the electron-molecule exchange interaction. While these techniques have been applied at present to only a few small molecular targets at the static-exchange level, their extension to more complex systems and coupling schemes seems likely in the near future given the performance of the present codes and the developments in computer systems. These methods not only provide benchmark results to which more approximate procedures can be compared, but also provide at present the only reliable means of obtaining large-angle ( 21000) differential cross sections. (3) An extensive endeavor has been mounted in order to extract LFframe cross sections from BF-frame FN scattering calculations. Most of these schemes employ the frame transformation either explicitly or implicitly. Reliable rotational excitation cross sections can be extracted using these techniques for scattering energies several times greater than threshold (and possibly smaller).

In the final part of this concluding section, we wish to list briefly some areas that require further study: (1) The problem of electronic excitation, both real and virtual, demands attention. Several of the studies presented in Section IV span an energy range which includes open electronic channels. While a distorted-wave calculation can give an estimate of the electronic-excitation cross section, we must await the advent of an electronic CC procedure in order to

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accurately assess the effects of the excited states on the elastic scattering. In addition, at very low-energies where all electronic channels are closed, a more systematic study of the correlation or polarization effects must be made and the theory extended beyond the currently used semiempirical form. (2) More extensive studies are warranted of the range of validity of the adiabatic approximation and the various “hybrid” schemes such as the MEAN approximation for calculating LF-frame cross sections. From our first point it seems evident that the BF-frame FN approximation will become even more important in scattering calculations as they progress to treat vibrational and electronic excitation; since it is only within this formulation that high-precision calculations at the static-exchange level (or higher) are feasible. Convincing tests of such schemes are formidable tasks, since highly converged calculations in both the L F and BF frames, e.g., recent work on H2(Feldt and Morrison, 1982), are required in order to distinguish actual differences due to the approximations from those arising from different levels of convergence. Such studies must also treat several polar systems with a range of dipole moments in order to derive trends and exclude the possibility of a pathological syste;. (3) More effort is needed in an attempt to reconcile theoretical and experimental results. Unfortunately, at present, their respective “regimes of greatest confidence” do not overlap well for the differential cross section. More accurate experimental results at small angles (520% and theoretical models routinely capable of a precision of -10% in the angular range 20-60” would permit much less ambiguous normalizittion of experimental results. Once this normalization question is resolved, then the experimental results at large angles (2120”) could be employed to distinguish among the various theoretical models in the regime where they are most sensitive to the details of the short-range potential. Measurements which resolve individual rotational transitions would also be of very great value. (4) The various models (OSME-BII) must be improved and extended to much larger polar systems. While the OSME model can be used to produce results in quite good agreement with ESE, this model is almost as expensive to implement as a full static-exchange procedure. The results from the other models for partial momentum-transfer cross sections and differential cross sections beyond -90” are not encouraging. While testing these refinements on small molecules, we look to their eventual value in applications for molecular systems too large for more sophisticated procedures. ( 5 ) We wish to encourage more study by our vitally important counterparts in the molecular structure field of potential energy curves and prop-

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erties of polar molecules. Not only are wave functions needed for scattering calculations, but information is required on the internuclear bond dependence of such properties as polarizabilities. Contributions of this type include the work on the polarizabilities and dipole moments of LiF and Be0 (Yoshioka and Jordan, 1980), the structure of KOH (England, 1978), and properties of the alkali halides (Williams and Amos, 1980; Gianturco and Lamanna, 1979). (6) Finally, the focus here on rotational and vibrational excitation should not obscure the importance of other processes in contributing to our understanding, and challenging our models, of the electron-molecule interaction. The observations (Novick rt d . , 1979) of resonances in photodetachment of several sodium halide negative ions, suggesting autodetaching 211 states of the negative ions, provide one such interesting problem. Other very fruitful ways of studying this interaction include dissociative attachment and associative detachment. There has been considerable recent work devoted to the hydrogen halides (Zwier ef ( I / . , 1980; Azria rt d . , 1980a,b; Allan and Wong, 1981), and results for other highly polar molecules are beginning to appear (Teillet-Billy et a / . , 1981), but detailed theoretical models are still being developed (e.g., Bardsley, 1979; Gauyacq, 1982). One of the greatest challenges for future work is to exploit the results, including wave functions, of some of the sophisticated scattering models discussed here to study these processes. Rigorous numerical techniques have been developed (Schneider et n / . , 1979) only for the R-matrix approach, but this has not yet been successfully applied to vibrationally elastic scattering by any highly polar molecules.

NOMENCLATURE AFT Angular frame transformation AN Adiabatic nuclei BII, BIII Unitarized Born approximations BF Body fixed cc Close coupling CCHS Close-coupling hard sphere CPT Classical perturbation theory DC Critical dipole DCO(r,) Dipole cutoff model DE Differential equations DW Distorted wave EMA Energy-modified adiabatic ESE Exact static exchange ERT Effective range theory

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES FBA FN HFEGE IE LA LF MEAN MSM 0s OSME RFT S S(h)

SME SPS SSSM

393

First Born approximation Fixed nuclei Hara free-electron gas exchange Integral equations Linear algebraic Laboratory fixed Multipole-extracted adiabatic nuclei Multiple-scattering method Orthogonalized static Orthogonalized static model exchange Radial frame transformation Static Truncated static Static model exchange Semiclassical perturbative scattering Semiclassical sudden S-matrix method

ACKNOWLEDGMENTS One of us (D. W. N.) would like to thank the Harvard-Smithsonian Center for Astrophysics for its hospitality, and assistance in the preparation of this article, while resident as an Associate (1980-1981), and Lorraine Volsky and her staffat JILA for suffering through countless revisions.

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QUANTUM ELECTRODYNAMIC EFFECTS IN FEW-ELECTRON I I ATOMIC SYSTEMS G . W. F. DRAKE Dqxirttnent of Physics University of Windsor Windsor. Canadrr

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 401 . . . . . . 401 . . . . . . . 405 . . . . . . . 412 . . . . . . 424 . . . . . . 426 . . . . . . . 429 . . . . . . 436 . . . . . . 446 . . . . . . . 448 . . . . . . . 450 . . . . . . . 454 . . . . . . 456

11. One-Electron Systems . . . . . . . . . . . . . . . . . . . . . . .

111. IV.

V. VI.

A. Lamb-Shift Theory . . . . . . . . . . . . . . . B. Precise Lamb-Shift Calculations . . . . . . . . . C. Comparison with Experiment . . . . . . . . . . Light Muonic Systems . . . . . . . . . . . . . . . Two-Electron S.ystems . . . . . . . . . . . . . . . . A. Calculations and Results for He and Li+ . . . . . B. High-Z Extrapolations . . . . . . . . . . . . . . Few-Electron Systems . . . . . . . . . . . . . . . . A. Relativistic Hartree-Fock Calculations . . . . . . B . Z - ' Expansion Calculations . . . . . . . . . . . Concluding Remarks and Suggestions for Future Work References . . . . . . . . . . . . . . . . . . . . .

. . . . .

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I. Introduction The development of modem quantum electrodynamics began in 1947 with the discovery of the Lamb shift in atomic hydrogen (Lamb and Retherford, 1947). According to one-electron Dirac theory, the 2s,, and 2p,,, states should be exactly degenerate, although an upward shift of about 0.03 cm-' in the 2 ~ , state , ~ had been suspected for several years from spectroscopic observations (Pasternack, 1938). The dramatic wartime advances in microwave techniques made it possible for Lamb and Retherford to observe directly the 2 ~ ~ ~ ~ - 2transition p,, frequency and to perform the first accurate measurement of the upward shift in the 2s,,, energy relative to the 2p,/, state. This was rapidly followed by Bethe's (1947) estimate of the electron self-energy, with a result in rough agreement with Lamb and Retherford's preliminary measurement. This work has initiated a long sequence of Lamb-shift measurements of ever399 Copyright @ 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

400

G. W . F . Drake

increasing precision in hydrogen and the one-electron hydrogenic ions. The experimental progress has been paralleled by a corresponding sequence of increasingly accurate theoretical calculations. The generally good agreement between theory and experiment has provided an important check on the computational techniques of quantum electrodynamics in the presence of strong fields (i.e., the Coulomb field of the nucleus). Equally important confirmations of quantum electrodynamics in the lowenergy regime provided by atomic physics have been obtained from studies of the anomalous magnetic moments of the electron and muon (Gidley and Rich, 1981: Van Dyck et ( I / . , 1981) and of the properties of positronium and muonium (Rich, 1981). The above-mentioned high-precision work at low energies complements high-energy scattering experiments, which test the lowest order quantum electrodynamic predictions down to interaction distances as small as R 5 cm (Hofstadter, 1975). The older high-energy experiments are reviewed by Lautrup et (11. (1972) and will not be further discussed here. Studies of quantum electrodynamic effects in atomic systems have been extended in the past several years beyond the original Lamb-shift measurements in hydrogen. Some of the new directions are the following: high-precision Lamb-shift measurements in highly ionized oneelectron ions and muonic systems: (2) theoretical and experimental developments for Lamb shifts and higher order relativistic effects in two-electron ions; (3) extensions of the above techniques to many-electron systems; (4) special effects, such as positron emission, which occur during the high-energy collisions of heavy ions. ( 1)

Item (1) has recently received thorough reviews from both the experimental (Kugel and Murnick, 1977) and theoretical (Brodsky and Mohr. 1978) points of view. The present review summarizes the older work and then gives more extensive coverage of developments since 1977. Items (2) and (3) are also reviewed in considerable detail. Developments concerning item (4),which is covered in the review of Brodsky and Mohr (1978), is beyond the scope of the present work. In addition, the energy levels of muonic atoms have been reviewed by Borie and Rinker (1982). There are a number of motivations for the continuing interest in the Lamb shift and other quantum electrodynamic effects in atomic systems. First, the Lamb shift, particularly in high-2 hydrogenic systems, remains one of the best ways of testing the predictions of quantum electrodynamics in a situation where the electron propagator is far off the mass shell, and, consequently, perturbation expansions in powers of Z a are of limited usefulness. The region with Z a > 1 has not been probed at

40 1

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

all until recently. Second, the properties of high-2 two-electron ions have come into prominence because of their importance in plasma diagnostics and fusion research. Lamb-shift-type and other higher order relativistic effects must be included in order to obtain transition frequencies of spectroscopic accuracy. The calculation of these contributions remains a difficult computational problem. The difficulties become even more severe for many-electron atoms.

11. One-Electron Systems The study of quantum electrodynamic effects in one-electron systems has now reached a high state of refinement, both theoretically and experimentally. This section briefly summarizes the material contained in the older reviews of Lautrup et a/. (1972), Kugel and Murnick (1977), and Brodsky and Mohr (1978), and then discusses in greater detail the progress that has been made since 1977. A. LAMB-SHIFT THEORY

The Lamb shift predominantly affects the nsl,, states of hydrogenic ions since the wave function remains finite at the nucleus. As shown in Fig. 1 , the dominant effect is to shift the n s , / , states upward relative to the np,,, states by about one-tenth of the np,n-np,l, fine-structure splitting, although the np states are also affected by a smaller amount. The physical origin of the Lamb shift can be understood as follows. In lowest order, the two contributions are the electron self-energy and vacuum polarization, as illustrated by the Feynman diagrams in Fig. 2. The electron self-energy, which arises from the emission and reabsorption of virtual photons, can be treated qualitatively as the interaction of the electron with the zero-point oscillations of the electromagnetic field (Welton, 1948). This tends to smear the electron charge over a mean square radius of (Bjorken and Drell, 1964) ( ( S r ) 2 ) = (2a3u2/.rr)In(Za)-l =

(5.838 x lo-’, cm),

for Z

=

1 (1)

Here ( I = h’/rne’ is the Bohr radius, a = e2/hc is the fine-structure constant, and Z is the nuclear charge. The corresponding correction to the interaction energy with the Coulomb field of the nucleus is ( 6 ~ =) (V(r + 6r). - V ( r ) ) = 4((sr)*(V2V>

(2)

402

G. W.F. Drake

I

I I

I

I

I I I

I I I I

2

I I

%

2

//s=i057.8 MHz

1-L-

FIG.I . Energy-level diagram for the n = 2 states of hydrogen showing the progressive splittings produced first by relativistic corrections in one-electron Dirac theory and then by QED corrections. The diagram is drawn roughly to scale.

Using V2V

=

4nZp2ti3(r)

(3)

I ~l,,,d0)1*= Z36d7m3a3

(4)

for the electron probability density at the nucleus, the energy shift for s-states is AE,

=

(8Z4a3/3m3) ln(Za)-'R,

(5)

where R, = e2/2a is the Rydberg unit of energy for infinite nuclear mass. It is usual to express Lamb shifts in Rydberg frequency units defined by Ry = R J h . Equation ( 5 ) for n = 2 and Z = 1 then gives a frequency shift of -1000 MHz.

(a)

( b)

FIG.2. Feynman diagrams for (a) the electron self-energy and (b) the vacuum polarization. Double solid lines represent a bound electron in the Coulomb field of the nucleus.

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

403

The vacuum-polarization correction can be thought of as arising from a polarization of the virtual electron-positron pairs surrounding a bare charge. The physically measured charge is therefore smaller than the bare charge since it is screened by the polarized vacuum. The screening distance is of the order of the Compton wavelength X = w. The resulting correction to Coulomb’s law was first investigated by Uehling (1935). To lowest order in (Y, the interaction energy between an electron and a point nucleus of chargeZ is (Akhiezer and Berestetskii, 1965; Schwinger, 1949)

+ AVVp(r)

(6) and the first-order perturbation correction to the 2s1,,-2p,,, energy splitting is =

Vo(r)

The Lamb shift has been measured in the muonic system p--HeZ+ (Bertin d.,1975; Bone and Rinker, 1982) as well as in ordinary hydrogenic ions. If p is the reduced mass m,rn,/(rn, + rnN) of the muon-nucleon system and me is the electron mass, then the vacuum-polarization term scales roughly as p 3 c 2 / m $ ,while the self-energy scales only as pc2. Thus, vacuum polarization dominates the Lamb shift of muonic systems, while self-energy terms dominate the Lamb shift of electronic systems. For a particle of reduced mass p and Bohr radius ug = h2/pe2,the nonrelativistic wave functions in Eq. (7) are

et

+(2s) =

(~/~,)3/2(2fi)-~(2 -

p)e-P/2

+(2p) = (z/a,)3’2(2~)-ipe-~’2

(8) (9)

where p = Zr/u,. Substituting Eqs. (8) and (9) into Eq. (71, integrating overr, and changing the remaining variable of integration in Eq. (6) from 6 to z = I / ( yields

where 5 =

2

I,

(1

+ ZZ/2)(1 - Z 2 P 2 ( 1 + pz14

dz

R, = e2/2rr,, is the Rydberg for a particle of reduced mass p and p = ZX/(2crU)measures the extent to which the orbital radii lie within the

G . W.F. Drake

404

vacuum-polarization region. The expression I ( p ) can be expanded in powers of p as n=o

with To = 1, TI = - 2 5 ~ 1 3 2 ,and the remaining Tnare given by the recurrence relation Tn

= {[(n

+ 3)(n + 4)1/[(n

-

l)(n + 5)lITn-z

(13)

For an electron, p = aZ/2, A/a, = a (neglecting reduced-mass corrections) and I @ )= 1. Then Eq. (10) reduces to AEvp= -(a3Z4/15~)R, 2 -27 MHz

for Z = 1

(14)

However, for a muon (m,/m, = 206.769), X/a, = am,/m, and p = 1. In this case, the higher terms in Eq. (12) make an important contribution. The lowest order vacuum-polarization correction is given in Table I for several muonic systems. For p < 1, the power series in Eq. (12) is convergent, but for the heavier ions, p > 1 and the integral in Eq. (1 1) must be calculated numerically. An accurate calculation requires also relativistic self-energy and finite nuclear-size corrections (see Section 111). The point of Table I is that for increasing Z , the small values of I @ ) cause AEvpto increase much more slowly than the Z4 dependence indi-

Zam,/(2me)

-

TABLE I

LOWEST ORDER

System e--'H, p--1H1 p--2D,

p--3He2 c(--~H~, p--BLi3 p--'Li3 p--lBe, pL--11B5 p--l*Ce

p--I4N, p--'60,

/L--l@F, p--ZoNe,,

V A C U U M - P O L A R I Z A T I O N C O R R E C T I O N S FOR S E V E R A L

plm,

0.999456 185.841 195.739 199.271 201.068 202.940 203.478 204.198 204.659 204.832 205.107 205.312 205.541 205.602

P 3.64669 x 0.678076 0.7 14189 1.45415 1.46726 2.22 139 2.22727 2.98020 3.73367 4.484 18 5.23856 5.99294 6.74960 7.50176

43)

0.991 106 0.284695 0.270532 0.115589 0. I14154 0.0614161 0.061 1584 0.0378325 0.0255789 0 .0184046 0.0138300 0.0107579 0.0085935 0.0070260

M U O N I CS Y S T E M S =\P

-26.8435 MHz -0.20502 eV - 0.22763 - 1.61189 - 1.66577 -4.66497 - 4.68242 -9.25583 - 15.3756 - 22.9986 -32.1464 - 42.7867 -54.9310 - 68.5 I24

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

405

cated by Eq. (10). In fact, for Z > 4, the upward correction due to finite nuclear size exceeds the downward vacuum-polarization correction, causing the overall Lamb shift to reverse sign. The results in Table I for the muonic systems up to P - - ~ H ~are , in good agreement with the calculations by DiGiacomo (1969, 1970), Campani (1970), and Borie (1975). B. PRECISE LAMB-SHIFT CALCULATIONS

The traditional method of calculating the dominant self-energy and vacuum-polarization corrections has been to evaluate successively higher order terms in powers of Z a . This approach is discussed in detail by Erickson and Yennie (1965) and reviewed by Taylor er al. (1969). Since then, a number of important advances have been made. We start by summarizing the low-order terms which are known exactly. The expansion of the radiative corrections in powers of Z a for an electron with quantum numbers rdj has the form AE

=

8m c2a (Za1 4 {A40 + ,441 In(Za)-, 6nn3

+ ASo(Za)

G . W . F . Drake

406

71112

-

31nn

+

32

601 240

4-1- - -

q= 1

BMSI

=

2 (197 + 4

40

-

7T2

144

In 2

77 -)60nZ

81,0

+ -43 ((3)

where 1/(1

+

for j = / + 4 for j = / - $

I)

[(x) is the Riemann zeta function and In cnl is the Bethe logarithm of

the average excitation energy calculated by Schwartz and Tiemann (1959) for ti = 2 and by Harriman (1956) for n = 3, 4. The more recent highprecision results of Klarsfeld and Maquet (1973) are listed in Table 11. Poquerusse (1981) has obtained asymptotic expressions for In En[ with the result In (A) Z 2 R , = 2.723 -

+

(ti

5 0.0003 +0S15 0.405)2

-0.3

(21 + I)(P + 1

+ 0.04)

+

for I

(21 + I)(n

=

0

0.56

+ 0.751 + 0.3912

4 3 * 0.00007 (m)

for / > O

(29)

The above formulas are accurate to within the stated error limits for all n . The vacuum-polarization parts A:: and A:: not listed in Eqs. (17-28) are zero. The terms AE,, and A E R in Eq. (IS) are finite nuclear mass and nuclear size corrections given by (Erickson and Yennie, 1965; Brodsky

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

407

TABLE I1 VALUES OF THE BETHELOGARITHM CALCULATED B Y KLARSFELD A N D MAQUET (1973) State IS

2s 2P 3s 3P

In(E . ~ / Z * R , ) 2.984 1285558 2.8 I 17698931 -0.0300167086 2.7676636125 -0.0381902294

State

In(€.(/Z2R,)

3d 4s 4P 4d 4f

-0.oO5232 148 I 2.74981 18405 -0.0419548946 -0.0067409 -0.0017337

and Mohr, 1978)

where

PER

7

(+)

2(Zc~)~rnc~ 3n3

where R is the root-mean-square radius of the nuclear charge distribution and X = 0.386159 pm. The term Ci, in Eq. (30) was originally calculated exactly only for n = 2, but Erickson (1969) has shown it to be correct for all n. Also, the term B,St has been recalculated and earlier errors corrected, as discussed by Lautrup at al. (1972). The contribution from higher order binding-energy corrections, represented by G(Za) in Eq. (16), is still a matter of controversy. The difficulty is that in an exact relativistic calculation, the exact DiracCoulomb-Green's function must be used. Although a form involving an infinite sum over partial waves is known (Wichmann and Kroll, 1956; Zon et NI., 1972; Zapryagaev and Manakov, 1976), no closed-form expression exists. The problem is severe because the uncalculated terms in the expansion (16) become of the order of magnitude of the calculated terms for

G . W.F. Drake

408

TABLE 111 VALUESOF C s E ( Z aA) N D G U ( Z a ) TABULATED B Y MOHR(1976)

1 10 20 30 40

- 20.13(34)' - 17.674(28)

- l5.776( 1 I ) - 14.1376(62) - 12.6650(28)

50

-0.5587 - 0.5015 -0.471 1 -0.4576

The notation -20.13(34) means -20.13

2

0.34.

Z z 10, and the expansion seems to diverge for olz 2 1 . Erickson (197 1) and Mohr (1976) have done independent calculations of the self-energy part of G ( Z a )by performing a numerical summation over the partial wave expansion of the exact Dirac-Coulomb-Green's function and subtracting off the known lower order contributions, thereby avoiding an expansion in powers ofZa. The results obtained by Mohr (1976) forZ = 10, 20, . . . , 50 are shown in Table 111. Numerical difficulties prevented explicit calculations for Z < 10. but Mohr estimated the smallZ behavior by fitting the results for Z = 10, 20, 30 to a series of the form GSE(Za)= A::

+ b(Za) In(Zcr)-* + c(Za)

(32)

in analogy with the corresponding high-order terms in the vacuum polarization. The results of the fitting procedure are A:: = -24.064 A 1.2, b = 7.3071, and c = 15.6609 for the 2 ~ ~ , ~ - 2 energy p ~ , , difference. The result GSE(0)= -24.064 2 1.2 is within the error limits of the earlier estimate - 19.08 2 5 by Erickson and Yennie (1963, but disagrees with Erickson's (1977) value - 17.246 2 0.5. The difference corresponds to a Lambshift difference of 0.049 MHz in hydrogen, which is much greater than the accuracy of recent experiments. A recent recalculation by Sapirstein (1981), which treats as perturbations certain small terms in the equation for the Dirac-Coulomb-Green's function, yields GSE(0)= -24.9 & 0.9 in agreement with Mohr's value. Although Sapirstein's (1981) calculation is for the Isll2 ground state, the n dependence is expected to be small. ) probHis result therefore indicates that Mohr's values for G S E ( Z ~are ably substantially correct. Mohr ( 1976) has similarly calculated the vacuum-polarization part GVP(Za)of G by writing it in the form GVP(Za) = G"(Za)

+ GWK(Za)

(33)

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

409

where C"(Za) is the contribution from the integral over the Uehling potential as shown in Eq. (7) after subtracting the known lower order terms contained in Eq. (IS), and G W K ( Z ais) the contribution of third order in the external potential calculated by Wichmann and Kroll (1956). The latter is given by

=

0.04251 - 0.10305(Z~~) + ...

(34)

and Mohr's numerical values for GL'(Za)at Z = 10, 20, and 30 are shown in Table 111. Fitting a functional form analogous to Eq. (32) yields G c ( Z a )= -0.557

?

0.003

+ 0.221(Za) In(Za)-2 - 0.128(Za)

(35)

The sum of Eqs. ( 3 2 ) , (34), and (35) is G(Za) = -24.598 t 1.2

+

+ 7.528(Za) In(Za)-2

15.429(Za) I+_ O.l(Za)

(36)

for the 2s,,-2pI,, energy splitting. The last term in Eq. (36) is an approximation for the uncertainty due to higher order omitted terms. A further contribution to the theory of Lamb shifts is an argument by Borie (1981) that finite nuclear size effects should be included directly in the calculation of the low-order level shift ofris,,, states. Referring back to the arguments leading to Eq. ( 3 ,this means that ( V V ) should be replaced by (4mp(r)), where p(r) is the nuclear charge density. For a point nucleus, p(r) = Zes3(r),and the usual results are recovered. Using a calculation analogous to that of Zemach (1956) for the size correction to hyperfine structure, and of Friar (1979) for the size correction in muonic atoms, Borie finds that in lowest order (p(r))ns= Z~l+ns(0)1~[1 - (2az/M(r)d

(37)

where

Assuming an exponential form for the charge distribution, then ( r ) ( 2 )= 3 5 R / ( 1 6 4 )

(39)

Thus, the s-state Lamb-shift terms A40 + A41 ln(Za)-, in Eq. (15) should be multiplied by the factor [l - 35ZaR/(16flX)] to give an additional additive correction of

G . W. F . Drake

410

For hydrogen and deuterium, the additional shifts are -0.042 and -0.104 MHz, respectively, assuming R, = 0.86 fm and RD = 2.10 fm. The comparison of these results with experiment is discussed in Section II,A,3. The validity of the Borie correction has been questioned by Erickson and Yennie (quoted by Borie, 1981) on the grounds that it may be canceled by higher order and relativistic effects. In summary, the 2~,,~-2p,,,Lamb shift for hydrogenic ions is obtained from Eq. (15) by calculating S = AE(2s,,,) - AE(2p,,,) with the result S = [mc.za(Za)4/6i7]{ln(Za)-2 + ln(e2,,/~2,0) + (91/ 120)

+ 0.32208a/r + 2.2%22rcuZ

+ (Za)2[-QInZ(Za)-2+ 3.91842 In(Za)-2 + G(Za)]J + s\, + SH + SB

(41)

where

with s

=

[I

-

( Z C X ) ~ ]and '/~,

and G ( Z a ) is given by Eq. (36). Equation (43) contains some higher order relativistic corrections obtained by Mohr (1976) in addition to the lowest order term from (31). He also estimates that the uncertainty resulting from uncertainties in the nuclear radius is ASR/SR

2:

O.~(ZCX)~ + (2AR/R)

(45)

The calculated values of the Lamb shift are given in Table I V f o r Z from I to 30. The tabulated numbers are essentially the same as those given by Mohr (1976), with the following exceptions. For hydrogen, the revised proton-charge radius of R = 0.862 2 0.012 fm (Simonet a / . , 1980) has been used in place of the value 0.81 t 0.02 fm used by Mohr. This change increases S by 0.016 MHz. Also for He+, the revised nuclear radius R = 1.674 k 0.012 fm (Sicker a/., 1976; Borie and Rinker, 1978) has been used in place of 1.644 fm. The other nuclear radii are representative values derived from electron-scattering data (Elton et ( I / . , 1967; de Jager et d.,

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

41 1

TABLE IV V A L U E S OF THE

Z 1 1

2 3 4 5

6 7 8 9 10

II 12 13 14 IS 16 17 18 19

20 21 22 23 24 25 26 27 28 29 30

A 1

2 4 6 9 II 12 14 16 19 20 23 24 27 28 31 32 35 40 39 40 45 48 51 52 55 56 59 58 63

64

2S1,2-2p1,2LAMB-SHIFT s,EXCLUDING THE CONTRIBUTION, A N D OF SB

sg

R (fm)

S"

-Stl

0.862(12) 2.10(2) 1.674(5) 2.56(5) 2.52(2) 2.4(1) 2.43 I ) 2.54(2) 2.72(3) 2.90(2) 3.02(4) 2.94(4) 3.01(3) 3.03(3) 3.09(2) 3.19(2) 3.24(2) 3.34(3) 3.45(5) 3.41(3) 3.48(3) 3.54(8) 3.60(1) 3.60(5) 3.66(5) 3.72(7) 3.73(6) 3.80(5) 3.78(3) 3.93(3) 3.934)

1057.883(13) 1059.241(27) 14042.36(55) 62737.5(6.6) 179.791(25) 404.57( 10) 78 1.99(21) 1361.37(47) 2 196.2l(92) 3343.1( 1.6) 4861.1(2.7) 6809.q4 .O) 9256.0(5.8) 12264.7(8.O) 15907.(1I) 202544 13) 25373417) 31347420) 38250.(25) 46133.(29) 551 16.(37) 65259.(55) 76651.(56) 89345.(78) 103.482(98) 119.12(13) l36.32( IS) 155.25(18) 175.85(21) 198.54(26) 223.03(32)

0.043 MHz 0.104 2.18 22.0 0.081 GHz 0.22 0.50

1.04 2.02 3.6 6.0 8.8 13.1 18.4 26 35 47 61 79

97 120 147 178 209 0.246 THz 0.29 0.33 0.38 0.42 0.48 0.53

a Calculated with Mohr's (1976) values for G S E ( Z agiven ) by Eq. (32). To obtain the corresponding Lamb shifts with Erickson's (1977) values, add Z6[0.04925 - O.O5278(Za) In(Zd-* + O.O2%I(Za)] MHz.

1974), and from muonic-atom transition energies (Engfer et al., 1974). The additional Borie correction term SBis not included, but is tabulated separately since its validity is still open to question. Its effect (relative to the experimental precision) is particularly great in the case of deuterium. The values of the fundamental constants used are a-* = 137.03604, K = 386.159 fm, and mcza5/6r = 135.643665 MHz.

G . W . F. Drake

412

c. COMPARISON WITH

EXPERIMENT

The uncertainties in the calculated Lamb shifts shown in Table IV arise principally from two sources: (1) higher order binding-energy corrections, represented by G ( Z a ) ,which increase in proportion toZs; and (2) nuclear radius corrections, which increase in proportion to Z4. Since the Lamb shift itself increases roughly in proportion to Z4, the experimental precision required for a significant test of the G ( Z a ) term is roughly k 10Z2 ppm. Thus, less precision is required at high Z than low Z for the same theoretical significance. Also, uncertainties arising from the nuclear radius correction, which is predominantly a non-QED effect, become relatively less important with increasing Z . It is therefore of importance to perform both high-precision measurements at low Z and lower precision measurements at high Z. Important advances have been made in both regimes since 1977, as discussed below. The overall comparison between theory and experiment is summarized in Table V and Fig. 3. In Fig. 3, ( S - s ) / Z 6is plotted for the various hydrogenic ions, where S is a Lambis the Lamb shift calculated as in Table IV, shift measurement and except that Gt[,hr(Z) given by Eq. (32) is replaced by the average value

s

C~E(= Z S~[ G): ; ’ : , ~ , ( Z ~ ) + ~ E F i c k s o n ( Z c ~ ) I

(46)

The lower solid bars are then Mohr’s values for the Lamb shift as tabulated in Table IV, and the upper solid bars are Erickson’s values as calculated by adding the expression at the bottom of Table IV. These are not quite the same as those tabulated directly by Erickson (1977) since he uses slightly different values for the nuclear radii. The same remarks apply to the entries in Table V. The dashed lines in Fig. 3 are the corresponding theoretical values when the Borie correction S , is added. The experimental results and their comparison with theory are discussed in Sections II,C,l and II,C,2.

I . Lo

M e a s rr rcm iw ts

~9-z

For the systems H, D, He+, and Liz+,most r l f i i e s o n n t i c ~rec~hriiyrres. ~ high-precision measurements have been based on the rf resonance technique originally employed by Lamb and Retherford (1947, 1950, 1951, 1952). The basic idea is to subject a beam of atoms or ions in the metastable 2s,,, state to an rffield which is tuned through the 2s1/,-2p,/, transition frequency. (Alternatively, the rf field can be held constant and the tuning done by Zeeman shifting the transition through resonance.) The resonance is detected by monitoring either the disappearance of ions in the 2s1,, state

(1.

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

413

TABLE V COMPARISON OF THEORETICAL A N D EXPERIMENTAL LAMB SHIFTS I N HYDROGENIC IONS Theory*

Ion

Technique"

Reference Lundeen and Pipkin (1981) Newton c't ( I / . (1979) Robiscoe et ( I / . (1970) Triebwasser et a / . (1953) Cosens (1968) Tnebwasser ct ( I / . (1953) van Wijngaarden and Drake ( 1978) Narasimham and Strornbotne (1971) Lipworth and Novick (1957) Drake ('I r r / . (1979) Dietrich et ( I / . (1976) Leventhal (1975) Kugel et ( I / . (1972)

SOF rf rf rf

rf rf A

Value 1057.8439) 1057.862(20) 1057.90(6) 1057.77(6) 1059.24(6) 1059.00(6) 1059.36(16)

Mohr (1976)

Erickson (1977)'

1057.883(13) 1057.840(13)

1057.929(13) MHz 1057.886(13)

1059.241(27) 1059.137(27)

1059.287(27) 1059.183(27)

rf

14046.2(1.2)

14042.36(55)

14045.12(55)

rf

14040.2(1.8) 14040.2(2,9) 62790.(70) 62765.(21) 780.1(8.0)

14040.18(55)

14042.94(55)

62737.5(6.6) 627 15.5(6.6) 781.99(21) 781.49(21) 2 196.2l(92) 2194.19(92)

62767.4(6.6) 62745.4(6.6) 783.67(21) GHz 783.17(21) 2204.98(92) 2202.96(92)

A

rf rf

QR

Curnutte ct t i / . (1981) Lawrence P I ( I / . (1972) Leventhal P I d . (1972) Kugel i>t t i / . (1975) Murnick d ( I / . (1972) Wood ct ( I / . (1982)

A

Gould and Marrus (1978)

QR

QR QR LR QR LR

2192.(15) 22 15.6(73) 2202.7(11.0) 3343.1(1.6) 3339(35) 3339.5( 1.6) 3405(75) 3 1 190(220) 31347(20) 3 1286(20) 38250(25) 38 IOO(6OO) 3817l(25)

3360.3( 1.6) 3356.7(1.6) 3 196x20) 31904(20) 39100(25) 39021(25) ~

" SOF, separate oscillating fields;

rf, rf resonance; A, anisotropy; QR, quench rate; LR, laser

resonance. The second entry of each pair includes the Bone correction term Se. Calculated from Mohr's values by adding the expression at the bottom of Table IV.

or the appearance of Ly-a photons produced by the rapid process 2p,,, + IS,,^ + hv. Since the resonance width of 99.7Z4 MHz is about a tenth of the Lamb shift being measured, a major source of error arises from locating the resonance center with sufficient precision. To reduce the linewidth problem, Lundeen and Pipkin (1975, 1981) have used Ramsey's (1956) separated oscillatory-fields technique, in which the single rfregion described above is replaced by two rfregions separated by a distance L along the beam as shown schematically in Fig. 4. The fractional quenching of the metastable 2s,/, state is then measured for relative

G. W . F. Drake

414

"r t

I d

I b

T

I

T

I I

'----'

F I( i . 3. Comparison between scaled theoretical and experimental one-electron Lamb shifts, expressed as deviations from the theoretical mean. The upper horizontal line for each ion is Erickson's (1977) theory, the lower horizontal line is Mohr's (1976) theory, a n d 3 is the average of the two. The experimental data (see Table V) are labeled by method of measurement according to: (0) microwave resonance: (A) anisotropy measurement: (+) quench-rate measurement; (0) laser resonance. The dashed lines are the Borie (1981) corrections to the theoretical values.

phases of 0" and 180" between the two rf fields. The difference signal I ( w ) = Q(w, 0) - Q(w, 1800) resembles an interference pattern with a central peak at h w = E(2sIl2)- E(2p,,,), whose width decreases as L increases. The theory is described in detail by Fabjan and Pipkin (1972). In effect, only those 2pIl, states are detected which are formed in the first rf field and live long enough to traverse the distance L to the second rf field. The amount of narrowing that can be achieved is limited by the progressively larger amount of signal relative to noise that is lost in the subtraction procedure asL is increased. As shown in Table V and Fig. 3 , Lundeen and Pipkin's high-precision result of S = 1057.845(9) MHz for hydrogen lies below the theoretical values when the Borie correction S,, is excluded, but agrees with Mohr's value of 1057.840( 13) MHz when S , is included. Lundeen and Pipkin estimate that their experimental precision could be improved to c0.001 MHz. A measurement of similar precision for deuterium as Lundeen and Pipkin's result for hydrogen would provide a definitive test of the SBcorrection, since the effect is particularly large. It is evident from Fig. 3 that the present level of experimental accuracy is not sufficient to draw firm conclusions. Only in the case of deuterium is the SBeffect well separated from

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

415

Resonance Cavity

**

Frequency

Frequency FIG.

4.

Schematic diagram comparing (a) the traditional single-resonance method with

(b) the Ramsey separated-oscillatory-fields method of measuring the Lamb shift. The inserted graphs show the line narrowing that is obtained with method (b).

the G S E ( Z a )and other uncertainties. A high-precision measurement would therefore be particularly valuable as a test of nuclear size corrections. The separated oscillatory-field method has also been applied to the determination of the 32S,,,-32D,,, interval in hydrogen (van Baak et al., 1980, 1981) with the result AE = 4013.155(48) MHz, in good agreement with the theoretical value 4013.195(7) MHz. A complete listing of earlier high-n measurements is given by Erickson (1977). The accuracy relative to uncertainties in the theoretical values is generally lower than for the n = 2 results. A precision measurement of the lsl/, Lamb shift in hydrogen has been reported by Wieman and Hansch (1980). Their experiment starts with a powerful, highly chromatic dye laser with a wavelength near 4860 A. After frequency doubling, this is used to excite the 1s1,2-2s1/2transition by Doppler-free two-photon spectroscopy (Bloembergen and Levenson, 1976). A component of the Balmer-p n = 2 + 4 transition is simultaneously recorded with the fundamental laser frequency. Since the two

416

G. W. F . Drake

absorption frequencies would be exactly the same in the absence of relativistic and QED corrections, the small frequency shift between them of about 4760 MHz can be used to determine the ground-state Lamb shift. After correcting for systematic effects and subtracting the known QED shifts for the excited states, they obtainS(ls,,,) = 8151 2 30 MHz, in good agreement with the theoretical value of 8149.43 2 0.08 MHz (Erickson, 1977). Their value for deuterium is 8177 -+ 30 MHz, in good agreement with the theoretical value 8172.23 2 0.12 MHz. The largest source of error is the frequency shift introduced by the pulsed dye-laser amplifiers. Wieman and Hansch (1980) also obtained a higher precision value for the hydrogen-deuterium isotope shift in the lsl,z-2sl~ztransition frequency which is sensitive to the term

AERR = ( m / M ) ( a 4 / 8 n 4 ) t n c z arising from relativistic recoil corrections (Bethe and Salpeter, 1957; Barker and Glover, 1955). It contributes -23.81 MHz for hydrogen Is,,, and - 11.92 MHz for deuterium Is,,*.The measured total isotope shift is 670992.3 k 6.3 MHz, in good agreement with the theoretical value 670994.96 ? 0.81 MHz. This is the first experimental confirmation of the AERR term.

h. The qurnching ciriisotropy method. Standard microwave techniques cannot be extended to ions heavier than Liz+ because the Lamb-shift frequency becomes too great for microwave sources. Several alternative methods have been devised for the higherZ ions, as further discussed in the next section. One of these, the quenching anisotropy method, has been tested in He+ and found to yield a value for the Lamb shift comparable in precision to the microwave resonance measurements (Drake et al., 1979). The basis of the method is that when a beam of hydrogenic atoms in the metastable 2s,,, state is quenched by a dc electric field, the induced Ly-a radiation intensity possesses an anisotropy in its angular distribution which is proportional to the Lamb shift and roughly independent ofZ. The ratio Zl,/L of the intensities emitted parallel and perpendicular to the applied electric field direction determines the anisotropy R = (Zll - &)/(ZIl + 4);R can be measured to high precision by simultaneous photon counting in the two directions as shown in Fig. 5. Rotating the field by 90" allows the relative sensitivities of the two counting systems to be canceled out. This also reduces most other systematic errors to at most quadratic effects. A further advantage is that the final result depends only weakly on the level width of the 2p1,, state. The theory of the anisotropy method is discussed in detail by van

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

PREWENCHING

COCLIMATOR

FARAMY

CYLINDERS

417

GAS CELL

CUP

S, with THIN FILM

DETECTOR

F I G .5 . Schematic diagram of the apparatus for the etectrostatic-quenching-anisotropy method of measuring the Lamb shift. (From Drake f'c c d . . 1979.)

Wijngaarden and Drake (1978) and by Drake et al. (1979). Although many small corrections must be taken into account, the basic physics can be understood as follows. Taking the z axis as the dc field direction, the adiabatic field-perturbed initial 2sll, state is +i

=

~-"~[+(2~1/2) + a+(2~1,2)+ /"~P~/JI

(47)

where a and /3d are mixing coefficients determined by diagonalizing the complete Hamiltonian, including the external field, in the 2s1/,, 2pl12, 2p,/, basis set. In the electric dipole approximation, the emitted radiation intensity with polarization vector & is

I ( @ )a 1 (++* r19(1s1,*)>l2

(48)

The total radiation intensity emitted in a direction 8 to the dc field is obtained by summing Eq. (48) over two perpendicular 0 vectors, both perpendicular to the direction of propagation. The result is an isotropic contribution from the 12p,,, - l ~ , , , )term ~ and anisotropic contributions from the 12p,/, - ls1/2~2 term and the 2pll, - Is,,, - 2p3/, cross term. The sum of all three is I(8) a 1

+ Re(p)(l - 3 cos2 8) + lpI24(5 - 3 cos2 6 )

(49)

G . W.F. Drake

418

with p

= P/a.

In the limit of weak fields,

where r,, = 99.71Z4 MHz is the 2p-level width. The “uncorrected” anisotropy is therefore Ro

= - [3

Re(p) - +IIp211/[2 - R d p ) + ~ ~ P I ‘ I

( 5 1)

with p given by the leading term in (50). The above formula emphasizes that Ro = -(3/2) Re(p) is determined primarily by the ratio of the Lamb shift to the fine-structure splitting. In the absence of hyperfine structure, the small corrections to Ro can be written in the form (Drake et d.,1979)

R = Roil + (6R/Ro)n +

(6R/Ro)re1

+ (~R/RJMZI + R21EI‘ + R41E14 (52)

where ( 6 R / R 0 ) ,accounts for perturbations of the Is,,, final state by all np states, and of the 2s,,, state by np states with n > 2, (6R/R0),, arises from relativistic corrections to the transition-matrix elements, and (6R/Ro)>l, accounts for an interference term between the 2p,,,- Is,,, electric dipole magnetic quadrupole (M2) decay mode decay mode and the (Hillery and Mohr, 1980; van Wijngaarden and Drake, 1982). The three terms together contribute a correction of about -202‘ ppm. The terms R2 and R, arise from higher order external field perturbation corrections. Since these corrections are typically quite small, the external field strength I El need be known only approximately. In the limit of weak fields, R becomes independent of I El, Numerical values of the parameters in Eq: (52) are listed in Table VI for several ions with zero nuclear spin. With the TABLE VI

DATAFOR DERIVING THE LAMB S H I F T FROM T H E Q U E N C H I N G ANISOTROPY OF IONS WITH ZERO NUCLEAR SPIN

14.04205 0.1179656 5.822 x lo-‘ -0.37 x 10-5 -2.37 x 10-5 0.64 x -6.54 x 1 0 - 5 I .0352

781.99 0.08 17 182 1.577 x lo-” -2.23 x 1 0 - 4 0.61 x -5.71 x 1 0 - 4 1.0150

21%.21 0.0726620 1.000 x

-4.02 x 10-4 1.09 x - 10.09 X 1.01 14

4861.1 0.0658649 1.194 x -6.33 x 1 0 - 4 1.72 x 10-4 -15.70 X 1 .OO91

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

419

help of these numbers, Ro can be determined from the measured value of R , and then Eq. (5 1 ) can be solved for the Lamb shift S. This is most easily done by calculating R,T at the theoretical value ST.Then the measured Ro corresponds to a Lamb shift of

ST{1 + [b(R, - R ~ ) / R ~ ] } with h --- 1 . Values of RB, ST,and b are given in Table VI. S

5

(53)

The accuracy of the anisotropy measurements for deuterium (k 150 ppm, van Wijngaarden and Drake, 1978) and He+ (2200 ppm, Drakeet a / . , 1979) listed in Table V was primarily limited by photon-counting statistics. An improved experiment is in progress for He+ which is designed to reduce the error to about 240 ppm. A further anisotropy measurement for 07+ is discussed in the following section. 2 . High-Z Mcwsirremrnts Measurements of Lamb shifts in high-Z hydrogenic ions have all been based on one of the three experimental techniques: ( 1 ) quench-rate measurements (2) laser resonance measurements (3) quenching anisotropy measurements

For ions heavier than Liz+,the experimental difficulties are compounded by the necessity of using a high-energy accelerator such as a van de Graaff to strip the ion beam to predominantly one-electron ions, followed by excitation to the 2sllZmetastable state. Higher beam velocities are also required as the decay rate of the 2s1, state increases. In the nonrelativistic limit, the 2sllZ-1s,,, spontaneous decay rate is given by (Klarsfeld, 1969) yz:, = 8.2294Z6

+ 2.496 x

sec-'

(54)

where the first term is the 2E1 contribution and the second is the MI contribution. Relativistic corrections to the 2E 1 part are accurately represented by the formula (Goldman and Drake, 1981) yzs(2E1 )

=

8.22943Z6

I

+

1.39448(OrZ)' - 2 . 0 4 0 ( ( ~ 2sec-l )~ 1 + 4.6019(cuZ)2

(55)

with an error of ?0.05% in the range 1 c Z s 92. More detailed tabulations are given in Goldman and Drake (1981) and by Johnson (1972). Qirench-rritc. measirrc~mmts. The first quench-rate measurement of the Lamb shift was done by Fan et c d . (1967) for Liz+. In experiments of this

(1.

G . W . F . Drake

420

type, the Lamb shift is determined indirectly from the rate at which an ion beam in the 2s1,, state is quenched by an electric field. The external field mixing with the 2p,/, and 2p,,, states causes the 2s1,, state to decay more rapidly than the field-free rate given by Eq. (54). The intensity of Ly-a photons is measured as a function of position along the ion beam and a decay curve plotted to find the field-induced decay rate y . This is related to the Lamb shift in lowest order perturbation theory by the Bethe-Lamb equation (Lamb and Retherford, 1950)

+ O(JE(4) where

T

=

E(2s,/,)

-

E(2p3/,)

v = (2Sl/,~Z~2PI,,)/L/ = (fi/Z,[I w = (2s,/,Jz(2p3,,)/N = (&/Z)(I

-

(5/12)a2Z21

-

Qa2Z2)

where [El is the electric field strength in volts per centimeter and S , T , and expressed in megahertz. The numerical factor in Eq. (56) is (2 Ry/E,J2 where Eo = 5.14225 x lo9 V/cm is the atomic unit of field strength. Fan rt (11. (1967) discuss higher order perturbation corrections to Eq. (56) and hyperfine-structure effects. Holt and Sellin (1972) present a nonperturbative time-dependent theory which reduces to the above result in the limit of weak fields. Recently, Kelsey and Macek (1977) and Hillery and Mohr (1980) have shown that Eq. (56) has a rigorous QED foundation to lowest order in a / r . A complete discussion of the rotational and polarization-dependent asymmetries exhibited by the quenching process is given by van Wijngaarden and Drake (1982). Of particular interest are effects produced by interference terms with spontaneous MI and fieldinduced M2 transitions. Using the above quenching technique, Fan et (11. (1967) obtained a Lamb shift of 6303 1 2 327 MHz. This result has since been supplanted by the more accurate rfresonance results shown in Table V, but at the time, it was the only available measurement for an ion heavier than He+. By 1972, and Fn+using quench-rate measurements had been extended to C5+,07+, the Bell-Rutgers and Los Alamos van de Graaff accelerators, with the results shown in Table V and Fig. 3. Many systematic effects such as low signal-to-noise ratios, electric field calibration, beam-velocity calibration, and beam bending in the quenching field combined to limit the accuracy of these experiments to 0.5-1%. As shown by Fig. 3, this was not sufficient to provide a definitive test of theory.

rpI,are

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

42 I

In order to provide a better test of theory with the same level of experimental accuracy, Gould and Marrus (1978) have attempted the most ambitious quench-rate experiment to date to determine the Lamb shift of Ar17+. They used the Lawrence Berkeley Laboratory Super HILAC to accelerate bare argon nuclei to approximately 340 MeV (4 x log cmlsec). The beam was then passed through a carbon foil located in a homogeneous magnetic field of B = 16 kG. A fraction of the bare ArIB+nuclei picked up an electron in the 2sl12state and were quenched to the ground state by the (e/c)v x B motional electric field of 6.5 x 105 V/cm. This was sufficient to produce a field-induced decay rate about comparable to the natural decay rate y s = 2.859 x lo8 sec-l. The exponential decay of the 2sII2state could therefore be observed over a distance of about 20 cm. An average of four runs taken at different field strengths yielded a Lamb shift of 38.1 k 0.6 THz (revised from 38.0 2 0.6 THz following a recalculation of y Z s ;see Goldman and Drake, 1981). As shown in Fig. 5 , this tends to support Mohr's theoretical value of 38.25 k 0.025 THz over Erickson's value of 39.10 k 0.025 THz. The S,, correction reduces these figures by only 0.08 THz. The primary source of error in the above experiment was the uncertain background contribution from the A P + 1s2s1S0+ ls2 'So two-photon continuum. The initial ls2s 'So population was estimated to be 0.03 & 0.01 of the 2s 2S,12 population. Uncertainties from other systematic effects, especially severe beam bending in the magnetic field and associated geometrical corrections, were all smaller. This experiment appears to be about the limit of what can be achieved with quench-rate measurements without major improvements in technology. 6. Lasc~r resotwtlce tiie(i.siII'PtnPtlfs. Laser resonance experiments in high-Z ions are conceptually similar to the microwave resonance techniques discussed in Section II,C, I ,a. The primary difference is that a laser frequency is chosen which closely matches the Lamb-shift [or E(2p3/,) E(2sl12)= AE - S ] transition frequency of a high-Z ion. Several possible candidates for such matches identified by Murnick (1981) are listed in Table VII. Tuning can be achieved either by changing the frequency of the laser or by taking advantage of changes in the Doppler shift as the intersection angle between the laser beam and the high-velocity ion beam is altered (Doppler-shift tuning). The first laser resonance experiment (Kugel et id., 1975) used an HBr gas laser beam crossed with a 50-64-MeV Fa+ ion beam, together with Doppler-shift tuning to measure the AE - S transition frequency. The accuracy of the result S = 3339 35 GHz was limited by major problems associated with the rf noise generated by the pulsed-laser discharge, the

*

G . W.F . Drake

422

TABLE VII C A N D I D A T E S FOR

Transition (GHz)

2

2 5 6 7 9 10 16 17 20

HY DROGENIC

(p-He2+) (B'+) (C5+) (N"+) (FR+) (Nes+) (SlS+)

(CP+) (Cat@+)

AE

-S S

368473 405.3 783.7 25030

AE - S AE - S AE - S S S

68832 105201 699883 31930 56990

S S

AE

-

ION

LASER-RESONANCE SPECTROSCOPY"

Laser

Tuning"

Dye CH,Br CH,F Spin-flip, CO, pumped HBr HF Dye CO, CO

C DS DS C DS DS C D DS

Average power (mW) 75 2 6 10 10

400 100 1 0 x 103 1 x 108

From Murnick (1981). DS, Doppler-shift tuning; C, continuous: D, discrete.

nonreproducibility of the laser-mode structure from pulse to pulse, and subtle geometrical effects arising from the Doppler-tuning method. The low-Ly-a photon-count rates observed at resonance necessitated a sacrifice in laser-beam quality for maximum count rate. Recently, a very significant laser resonake measurement of the Lamb shift in C P + has been reported (Wood et d.,1982). In this work, a highpower COz laser was used to induce directly the 2s1,,-2p1~,transition, and the resulting 2p1,z-1s1~p X rays at 2.96 keV were detected. Tuning was done by a combination of Doppler-shift tuning and discrete laserfrequency changes. A schematic diagram of the apparatus is shown in Fig. 6 and a typical resonance curve is shown in Fig. 7. The control of systematic errors in this experiment, especially those arising from variations in the intersection region of the two beams as the laser frequency was changed, required particular care in the design of the laser and associated optics. The experimental result, 31.19 k 0.22 THz for the Lamb shift of 35C116+is in good agreement with Mohr's value 31.35 k 0.02 THz, but lies three standard deviations below Erickson's value 31.97 f 0.02 THz. The SBcorrection lowers the theoretical values by only 0.06 THz and therefore has little effect on the comparison between theory and experiment (see Fig. 3 and Table V). The quenching anisotropy method described in Section II,C, 1 ,b was originally proposed as a method for measuring Lamb shifts in high-2 ions. Preliminary results have re-

c. Quencking onisotropy measirrements.

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

423

I R POWER DETECTOR

J

ALIGNMENT COLLIMATOR

SPECTROMETER

DAY P /

CARBON ADDER

FOILS

\

LASER , BEAM

-‘\$Or

ANTICHARGE SCATTERING C OLL I MATOR

F I G .6. Schematic diagram of the apparatus for the laser-resonance measurement of the Lamb shift in CIi6+.(From Wood of o/., 1987.)

cently been reported for 07+ (Curnutte et a/., 1981). In this experiment, a ions from the Kansas State University tandim van de Graaff beam of 07+ is quenched by an (e/c)v x B motional electric field as shown schematically in Fig. 8. The quenching magnet has four pole pieces arranged as in a standard magnetic quadrupole lens; but the magnetic coils are wired so that the magnet is a dipole whose magnetic field direction can be changed from horizontal to vertical and can be reversed. This allows the anisotropy to be measured independently of the detector efficiencies and related

-

10

0

1

I

I

1

I

I

I

I

-

-

27

30 33 36 REST FRAME FREQUENCY ( T H d F I G 7. . Typical laser resonance data for the Lamb shift of CP+.The solid curve is the best fit to the data. (From Wood ef d . , 1982.)

424

G. W. F. Drake

F I G .8. Schematic diagram of the apparatus for the quenching anisotropy measurement of the Lamb shift in O'+. (From Curnutte cr 01.. 1981.)

systematic effects. Direct Stark quenching with a static electric field was found to be impractical because of the large field emission and ionization background produced. Magnetic field quenching gives a signal-to-noise ratio of about 10 : 1. The primary factor currently limiting the accuracy of this experiment is an uncertainty in subtracting the background from the signal. There is an apparent 20% increase in the background when the quenching field is turned on whose origin is not clearly understood. It may be an unjustified assumption to take the spectrum of the additional background to be the same as the spectrum of the zero-field background. Taking this uncertainty into account, the present experimental value for the Lamb shift is S = 2192 k 15 GHz. Since the statistical uncertainty is only -t0.2% and can be further reduced, it will probably be possible to improve the precision to the + 3 MHz region or better when the background problem is properly understood. In summary, there are now measurements of sufficiently high precision to provide a significant test of theory at both low and high Z. As can be seen from Fig. 3, the overall best agreement with the high-precision experimental data appears to be Mohr's theoretical values with the additional finite nuclear size correction SBincluded (lower dashed line). However, a further high-precision measurement for deuterium is needed to verify that the finite nuclear size effects are being adequately treated in the calculations.

111. Light Muonic Systems Interest in light muonic Lamb shifts arises primarily from a measureet d.(1975). Their experiment in the exotic system K - - ~ H ~by~ Bertin +

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

425

ment is basically of the laser-resonance type, using a ruby-pumped IR dye laser to stimulate the 2~,,~-2p,~, transition at X = 8120 A. Subsequent higher precision measurements by Carboni et NI. (1977, 1978) yielded the values E(2p3,,) - E(2s,,,) = 1527.5 _t 0.3 meV and E(2pl12)- E(2sllz)= 1381.3 2 0.5 meV. As discussed in Section ILA, the Lamb shift of light muonic atoms is dominated by the vacuum-polarization term, and approximate numerical values (using nonrelativistic wave functions) are given in Table I. The other important terms to be considered are the 2p3,,-2pll2 fine-structure splitting AEFs = [ ( a Z ) 4 p c 2 ] / 3 21 [+ Q(5a2Z2)] =

0.045283(p/me)[I

+ i(5a2Z2)] (meV)

(57)

the finite nuclear-size correction to the 2s1,, state

AE,>V12+ B in the second term. AsZ varies, T generates a continuous range of transformations which tend to the correct limit in both extremes. Results using the above techniques have been published for the Is2 'S0-ls2p 3P1and ls2 'S0-ls2p 'P,transitions for ions up t o 2 = 100 by Drake (1979), and for the ls2s 3S,-ls2p 3P, and ls2s 3S,-ls2p 3P2transitions for ions up t o 2 = 50 by DeSerioer al. (1981). The calculations have been extended to the ls3d 'D and 3D states by Drake (1981). Results for

-

'

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

44 1

the fine-structure splittings of the ls3d 3D,,,, states are shown in Fig. 11. At Z = 10, sin H = 0.209, indicating that strong singlet-triplet mixing is present in the 3D, configuration. Values up t o Z = 25 are given in the above reference. Although a unique separation of relativistic and QED terms is not well defined, energy differences as calculated above can be taken as representing the relativistic but non-QED contribution to the transition frequency. After adding the mass-polarization correction, given by the expectation value of the operator

Hu = M-'p,-p2

(104)

and subtracting from the experimental transition frequency, what remains is the Lamb shift of relative order a3Z4ln(aZ), together with terms not included in the expansion (91), the leading one being a4Z4Et.One can therefore expect about 1% accuracy in the Lamb shift at low-to-moderate Z if the latter correction is not taken into account. Since accurate two-electron Bethe logarithms for excited states are not available, one is forced to make estimates. The simplest approximation is to use the sum of the individual one-electron terms discussed in Section II,B. [Note that if anomalous magnetic-moment corrections of relative order a / r are included in B p , then they should be excluded from the Lamb shift. This means for example, omitting the term EI'I, from Eq. (77).] Since (a3(r)) = Z 3 / r n 3 for an ti/ configuration, DeSerio et [ I / . (1981) sug-

-12

c

3D-'

FIL.1 1 . Theoretical fine-structure splittings of the Is3d 3Dstate of helium-like ions;E is the center of gravity of the configuration and 2 is the nuclear charge.

G. W. F. Drake

442

gest defining the two-electron Bethe logarithm to be

for a lsrd configuration so that the hydrogenic Lamb-shift difference is recovered in the limit of large Z. Using the data in Table 11, this leads to 1 . ~ 2= ~ )28.095Z2R,

(106)

~ ( l ~ 2 =p )19.695Z2R,

( 107)

E(

However, for Z = 2, this gives In[€(Is2s)/Rm]= 4.72 in poor agreement with the value 4.38 ? 0.02 calculated by Suh and Zaidi (1966). An alternative approach involving hydrogenic wave functions with adjustable effective nuclear charges is discussed by Ermolaev (1973, but there is clearly a need for more accurate calculations at higherZ. Results for the ls2s 3S,ls2p 3P,,and 3P2transitions are compared with experiment in the following sect ion. 2 . Comparison \i-ith Experiment

Transitions of the type ls2s 3S,-ls2p 3 P J ( J = 0, 1, 2) are particularly sensitive to the Lamb shift. Since the Lamb shift increases approximately in proportion to Z4, while the nonrelativistic energy difference increases only as Z, the ratio is given approximately by AE,,/AE= ~ ~1.4 x

10-623

Thus, for FeZ4+,the Lamb shift contributes about 2% of the total energy difference. Several measurements have recently been made of the wavelengths of the ls2s 3S-ls2p 3P transitions in the rangeZ = 8-26 (Davis and Marrus, 1977; Berry rt d.,1978, 1980; O’Brien et d . , 1979; Armour rt d., 1979; DeSerio et d.1981; Buchet rt a/., 1981; Stampet a/., 1981). Most of these measurements use the method of beam-foil spectroscopy with a fast-ion beam source to excite the ls2p 3P state and a scanning-vacuum monochrometer to record transitions to the ls2s 3Sstate. The main sources of error are systematic effects resulting from the large Doppler shifts associated with fast-ion beams ( u / c 0.1). The work of Stamp et rd. (1981) uses a tokamak plasma in place of a fast-ion beam as an excitation source. In addition to avoiding problems with large Doppler shifts, tokamak plasmas have several advantages over the theta-pinch sources used in earlier work on helium-like ions (Baker, 1973; Elton, 1967; Engelhardt and Sommer, 1971), as discussed by Stamp rt NI. (1981). They report measurements for 06+ and F7+. The experimental transition frequencies are summarized in Table XIV.

-

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

443

Both DeSerio et ( I / . (1981) and Stamp ef al. (1981) have made detailed comparisons with theory, including relativistic and one-electron Lambshift corrections. They both find that there are systematic discrepancies which increase in proportion to Z 3 along the isoelectronic sequence. This comes predominantly from higher order terms in the expansion (S3(rl) + S3(r2)) =

(2Z3/n)[b

+

(Sl,o/2n3) + dlZ-'

+ d2Z+ +

*

*

-1

(108)

multiplying the other terms in Eq. (78). Values of the first few dr obtained from variational perturbation theory with 50-term correlated variational basis sets (e.g., Drake and Dalgarno, 1970) are listed for several low-lying states in Table XV. DeSerio et [ I / . have estimated the two-electron corrections, using Eqs. (106) and (107) to approximate the two-electron Lamb shifts and found reasonably good agreement as is shown by the dashed curve in Fig. 12. It is also of interest to compare theory and experiment for the finestructure splittings of the ls2p 3P0,3P1 and 3P2states in helium-like ions. These are particularly sensitive to higher order one- and two-electron relativistic corrections. In addition, the 2 3P1 state becomes strongly per-

TABLE XIV E Y P ~ R I M E NVT4 A L ULE S

Ion

He I Li I1 Be 111 B IV

cv

N VI 0 VI1 F VIll Ne IX Al XI1 Si XI11

s xv

CI XVI Ar XVII Fe XXIV

tOR THE

lS2S 3s,-lS2p 3PJ TRANSITION FREQUENCIES"

3P2

3p,

3p0

Reference

9230.795 18228.198(I ) 26867.9(.2) 26871 3 . 7 ) 354293 ) 4402 1.6( I ) 527193.6) 61588.3( I . 5 ) 70700.4(3.0) 8 0 1 2 0 3 1.3) I I 1 I lO(100) 122746(3) 148493(5) 162923(6) 178500(300) 368960( 125)

9230.87 1 18226.108(1) 26853.1 (.2) 26856.3( .7) 35377 43886.U I ) 52429.0( .6) 61036.6( I . I ) 69743A(3.0) 78566.3( 1.3)

923 1.859 I823 1.303 26867.4(.7) 35393.2 43899.0(1) 52413.9( 1.4) 60978.2( 1.5) 69586.0(4.0) 78266.9(2.5) 104930(100) 1138134) 132 l98( 10) I4 1643(40) I5 l350(250) 232558(550)

Meggers (1935) Holt er d.(1980) Lofstrand (1973) Eidelsberg (1972) Edlen (1934) Edlen and Lofstrand (1970) Baker (1973) Stamp er d.(1981) Engelhardt and Sommer (1971) Engelhardt and Sommer (1971) Denne p r a/. (1980) DeSerio er cd. (1981) DeSerio er d.(1981) DeSerio er cd. (1981) Davis and Marrus (1977) Buchet er t i / . (1981)

444

G . W . F . Drake

rl -2

-.j

-6

(b)

L

F I G .12. Comparison of theory and experiment for the QED corrections to the WS, ?p3P2(a) and 2s3SS, 2p3P0(b) transitions of helium-like ions: Ethincludes only one-electron QED terms, and the dashed curve represents an estimate of the two-electron corrections. (From DeSerio ot d.,1981.) ~

turbed by the 17 'P, states. Myers et rrl. (1981) have recently exploited the near coincidence of the C02 laser frequency with the 2 3P,-2 3P2 M 1 transition of 11F7+ to measure directly the J = I + 2 transition frequency. Tuning was of the Doppler-shift type, but in this case the laser angle was held fixed and the beam velocity ( - 0 . 4 8 ~ )from the University of Oxford

445

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS TABLE XV EXPANSION COEFFICIENTS FOR T H E CALCULATION OF (83(rl) + 63(r,))0 State Is2 1s

ls2s 1s ls2s 3s ls2p 'P ls2p 3P a

do

di

4

4

4

1 9/16 9/16 1/2 112

-0.66841 -0.14546 -0.10583 0.01083 -0.04304

0.17805 0.1 1145 0.03124 -0.02422

0.00354 -0.02398 0.00733 - 0.00469 0.01310

-0.00012 -0.01712 0.00154 0,02370 -0.00352

0.06070

See Eq. (108).

TABLE XVI A N D EXPERIMENTAL DATAFOR THE FINE-STRUCTURE S U M M A ORFYTHEORETICAL INTERVALS OF T H E ls2p 3P0,1,2STATES' ~~

Ion

Transition

Theory

Bez+ B3+

2- 1

c4+

2-0 1-0 2-0 1-0 2- 1 2-0 1-0 2-0 1-0 2- I 2-0

14.89 36.35 - 16.29 123.04 - 12.72 299.18 8.26 290.92 609.48 58.07 1107.56 150.05 957.51 1856.05 298.94 6366.00 8924.38 16261.5 21286.2 27395.5

2-0 1-0

N5+

06+

F7+

F7+

1-0

AI11+

sin+ SM+

c~5+ AP+

2-0 2-0 2-0 2-0 2-0

Experimentb 14.8(3) 36.3(8) - 16.0(8) 122.6(1.4) - 12.9(1.4) 305.6(1.4) 15. I ( 1.4) 290.5 610.1 58.4( 1.5) 11 14.4(3.0) 157.8(3.O) 957.88(3)' 1853.6(2.4) 299.4(2.4) 6180.( 150) 8931.(5) 16295.(11) 21280440) 27 150.(400)

In cm-l. From data tabulated in Table XIII. High-precision measurement by Myers et a / . (1981).

Difference 0.09(30) 0.05(80) 0.29(80) 0.4( 1.4) -0.2(1.4) -6.4( 1.4) -6.8(1.4) 0.4(0.6) -0.q 1.5) -0.3(1.5)

-6.8(3.0) - 7.8(3 -0)

-0.37(3) 2.4(2.4) -0.5(2.4) 1864150) -6.q5.0) -33.(11) 6.(40) 245 .(400)

G. W. F . Drake

446

van de Graaff was varied by stepping the current of the momentumanalyzing magnet. After subtracting hyperfine-structure effects, they obtained the result v21 = 957.88 ? 0.03 cm-'. This direct measurement provides a severe test of theory, as discussed below. In addition, several other less accurate fine-structure splittings can be obtained by subtracting the transition frequencies to the 2 3 S , state shown in Table XIV. The comparison of the above experimental data with theory is summarized in Table XVI. The theoretical values were calculated as described in Section IV,B, 1 and Drake (1981). They therefore contain the following higher order relativistic corrections: (1) from one-electron Dirac theory, a6Z6n7c2(1 + a 2 Z 2 + . . .) (2) from Breit-interaction terms, a6Z5mc2(l+ a2Z2 + + . . )

not contained in the calculations of Accad rt t i / . (1973). For example, these extra terms increase the value for v21of Nee+from 955.26 to 957.51 cm-l. Although the latter value is closer to the high-precision measurement 957.88 0.03 cm-' of Myersrt d.(l98l), the difference is more than ten standard deviations. The discrepancy could be accounted for if the term a4Z4E,4in Eq. (91), which was not included in the calculation, contributed approximately 0.2a4Z4Ry. The less accurate results for the other ions are in reasonably good agreement, except for N5+,F7+,and S4+. For all of these, the differences arise primarily from the location of the J = 0 level relative to the other two.

*

V. Few-Electron Systems The analysis of QED effects in atomic systems containing more than two electrons is obscured by the lack of high-precision nonrelativistic eigenvalues and relativistic corrections available for two-electron systems. It therefore becomes more difficult to make a reliable subtraction of these contributions from the observed transition frequencies in order to reveal the specifically QED effects such as the Lamb shift. As will become evident below, the conclusions that can be drawn about the Lamb shift from the experimental data depend rather strongly on the approximations used in other parts of the analysis. Consider the ls22s 2S,,,-1s22p 2P1,2 and 2P3,2transitions of the Li isoelectronic sequence as an extensively studied example. Large-scale Hylleraas-type (HT) variational calculations, combined with con-

447

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

figuration-interaction (CI)terms, have been done for the nonrelativistic energies of the 2S and 2Pstates of neutral Li (Larsson, 1968; Sims and Hagstrom, 1975). Unfortunately, the most accurate 150-term result for the 2S state is uncertain by -10 cm-l, which is large compared to even the unscreened hydrogenic 2 ~ , ~ ~ - 2 pLamb , ~ , shift of 2.1 cm-I. Less accurate variational calculations have been extended up to Z = 8 (Perkins, 1976). Even here, the -200 cm-l uncertainty in the calculation is large compared to the hydrogenic Lamb shift of 73.2 cm-l. In the absence of more accurate calculations for low Z, the only hope is to investigate the region 2 k 12, where QED effects may rise above the uncertainty in the high-2 approximation methods described below. A large body of experimental data is available in this region, either from laboratory plasma observations (Fawcett, 1970) or from solar observations (Widing and Purcell, 1976; Behring rt a/., 1976; Sandlin et al., 1976; Dere, 1978). Edlen (1979) has obtained an accurate semiempirical fit to the observational data. His results, which are listed in Table XVII, provide a convenient comparison with the ci priori calculations summarized in Table XVIII and discussed in more detail in the following two sections. TABLE XVIl COMPARISON OF CALCULATED VALUES OF THE 1Sz2p 'plp-lS'2S 's112 TRANSITION FREQUENCY" WITH A S E M I E M P I R IFIT C ATO L T H E EXPERIMENTAL DATA* (lo00 cm-l) ~~

Cheng er Z 10 12 14 16 18 20

22 24 26 28

a / . (1979)

Shestakov (1979)

Berry et ol. (1980)

Edlen (1979)'

EL(Z)d

129.06 161.00 193.09 225.48 258. I 291.2 324.8 358.8 393.4 428.4

127.978 159.831 191.870 224.170 256.783 289.786 323.209 357.124 391.582 426.635

128.068 159.953 192.031 224.373 257.030 290.068 323.539 357.494 391.988 427.076

128.152 160.012 192.063 224.377 257.020 290.046 323.516 357.486 392.019 427.182

-0.162 -0.309 -0.531 -0.846 - 1.276 - 1.838 -2.556 -3.451 -4.546 -5.864

EJZ

-

sF

-0.085 -0.180 -0.334 -0.561 -0.880 - 1.304 - 1.850 -2.529 -3.354 -4.329

First three columns. Edlen (1979). The entries in this column represent the experimental data with an uncertainty of about -t0.040 x lo3 cm-'. E,, is the unscreened hydrogenic Lamb shift used by Berry et ul. (1980) and Shestakov (1979). Screened hydrogenic Lamb shifts withs = 1.60 used by Edlen (1979).

G . W .F. Drake

448

TABLE XVIII SUMMARY

OF

COMPUTATIONAL METHODSFOR T H E lS22p 2P11,-lS22S FREQUENCIES OF HIGH-Z L I T H I U M - L IIONS KE

Method Dirac-Foc k

Z-' expansion With empirical corrections

Electron correlation correction

Relativistic correction

None

( h l Bl h = )

AE,O

($"FIBIJI"F)

AE,O

Screened Dirac total energies Screened Dirac outer electron energies

AE,O

TRANSITION

2s,12

Lamb-shift correction Screened hydrogenic Unscreened hydrogenic Unscreened hydrogenic Screened hydrogenic

Reference Cheng er d.(1979) Shestakov (1979) Berry i>f ( I / . (1980) Edlen (1979)

A. R E L A T I V I S THARTREE-FOCK IC CALCULATIONS The generalization of the usual Hartree-Fock method to include the single-particle Dirac Hamiltonian together with electron-electron interaction terms is discussed in detail by Grant (1970). The resulting Dirac-Fock (DF) equations have been programmed by Desclaux (1975) and employed by Cheng er a / . (1 979) in an extensive survey of transition wavelengths and rates in the isoelectronic sequences of lithium-like ions through fluorinelike ions up to Z = 92. They estimated the vacuum-polarization term by calculating the expectation value of the Uehling potential in Eq. (6) and the self-energy term by using the screened hydrogenic value for each orbital (Desclaux et a / . , 1979). The screened nuclear change Z - s was determined by requiring that the hydrogenic orbital radius ( r ) have the same value as for the DF orbital. In addition, the Breit-interaction term Eq. (65) was included as a first-order perturbation. The results of these calculations show in Table XVII are systematically larger than Edlen's semiempirical fit to the experimental data and the other two calculations for two basic reasons. The first comes from the nonrelativistic correlation energy. The total nonrelativistic energy extracted from Eq. (91) is

449

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

where

as obtained from Eqs. (8 l)-(88). Hartree-Fock calculations contain the exact E! and EP, but they do not include the electron-correlation contributions to E;, E!, . . . . However, E: and E! can be expressed as sums of one- and two-particle matrix elements which have been calculated to high accuracy (Horak et d.,1969; Ivanova and Safranova, 1975, and earlier references therein). The exact results (in atomic units) are

E0(22S)= -9Z2/8

+

E0(2'P) = -9Z2/8

+ 1.0935262 - 0.5285756 +

1.0228052 - 0.4081652

-

0.0230/2 *

*

+-

a

*

whereas the Hartree-Fock expansions are

EtF(22S) = -9Z2/8 EiF(22P) = -9Z2/8

+ +

+ . 1.0935262 - 0.469462 - 0.10758/2 + . . . 1.0228052 - 0.354549

-

0.04135/2

*

*

The corresponding expressions for the transition energies are AE0(2P- 'S) = 0.070721Z - 0.1204104

AEtF('P

- 2S) =

0.0707212

-

+ O(Z-l)

0.114913 - 0.06623/2

( 1 13)

(114)

The difference of -0.005497 + O W 1 ) is the error in the Hartree-Fock transition energy due to nonrelativistic electron-correlation effects. For sufficiently large Z the error, therefore, tends to the constant value 0.005497 a.u. = 1207 cm-'. Subtracting this amount from the results of Cheng c't crl. (1979) in Table XVII brings their values into better agreement for low Z , but discrepancies remain at high Z. The second possible source of error comes from the screening approximation used to calculate the Lamb shift. Their use of the value of ( 1 . ) as a criterion for choosings may lead to an overestimate of s and therefore an underestimate of IEJ. A larger Lamb shift would further decrease their transition frequencies at high 2 as required. The DF calculations of Shestakov (1979) include the above correlation correction. However, he uses unscreened hydrogenic Lamb shifts for the

G. W .F. Drake

450

QED correction which, as discussed further in the next section, is probably an overestimate. His results therefore came out consistently lower than Edlen's. Screened Lamb shifts with s = 0.7 would bring his results into close agreement in the range 18 S Z S 28. B.

z-' EXPANSION

CALCULATIONS

Calculations based entirely upon Z-' expansion techniques have been studied for many years (Layzer and Bahcall, 1962; Dalgarno and Stewart, 1960; Doyle, 1969; McKibbon and Stewart, 1969; Snyder, 1971, 1974). The expansion of the nonrelativistic energy Eo has already been discussed in Section V,A. The above work also makes use of the corresponding expansion E2

=

aZZ4[E% + EfZ-'

+ EEZ-2 +

. . .]

(115)

for the leading Breit-interaction correction given by the second column of terms in Eq. (91). For an N-electron atom containing 4, electrons of type t , Ez is trivially calculated from the Sommerfeld formula

with E m

=

-(2r?f)-1{[4/(j, +

t)l - a>

(117)

and E: can also be calculated exactly for an N-electron atom by taking linear combinations of two-electron values (Doyle, 1969). Since the Breitinteraction BIZalso connects states of the same total angular momentum and parity within the basis set of hydrogenic states which are degenerate in zero order, a diagonalization step as in Eq. (92) may also be necessary (Layzer and Bahcall, 1962). No direct calculations of E: or higher order terms have been attempted for systems containing more than two electrons. The truncation of Eq. (115) after the first two terms is not sufficiently accurate for most applications. Again, consider as an example the lsz2p zP-lsz2s 2S transitions of lithium-like ions. Doyle (1969) obtained a substantial improvement in the 2P,,z-2P3,2fine-structure splitting by writing the relativistic energy difference in the form

45 1

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

The expansion of Eq. ( 1 18) then correctly reproduces the leading two terms of Eq. ( 1 IS), together with an allowance for higher order terms. The expansion coefficients listed in Table XIX yield the value s = 1.7449. Snyder (197 1 , 1974) has suggested a more elaborate procedure in which the relativistic energy of eachji- coupled orbital is screened separately. He writes

E2 =

a2

2 q,Eg(t)(Z -

(1 19)

t

where the orbital-screening parameters ut are determined from a twoelectron screening matrix via at

=

2 (41,- & t , ) d t l t ’ )

( 120)

1’

The a(tlt’)are determined by solving the pair of equations

+ q r E g ( t ’ ) a ( t ’ ( t=)

-iEq(r, 1 ’ ) q , E t ( r ) b ( r l t ’ ) l *+ q t G ( t ’ ) [ ~ ( ~ ’ l t = ) l 2iG(t,1’) qtEg(t)u(tlt‘)

(121) (122)

where ET(r, t ’ ) and E,2(r, t’) are the first- and second-order two-electron expansion coefficients. Snyder extracted the E f ( r , t’) from the variational calculations of Accad rt 01. (1971) to obtain the screening parameters listed in Table XIX. Although this method incorporates the two-electron E,2 terms, there is no guarantee that the N-electron E,2 obtained by expanding Eq. ( 1 19) and collecting coefficients of Z2 will even be close to the right value. However, his calculated fine-structure splittings reproduce the experimental data to better than 1% f o r 2 z 7. Berry er (11. (1980) have used Snyder’s screening parameters to estimate the total relativistic energy of lithium-like systems by writing Ere’=

2 {4tE:lraC(Z

-

at)

+ [(Z - ~ , ) ~ / 2 n , ] }

(123)

t

TABLE XIX R E L ~ T I V I S TENERGY IC EXPANSION COEFFICIENTS A N D SNYDER PARAMETERS FOR L I T H I U M - L IIONS“ KE

zs,,2 2Pl,* 1S22P3l* *P,,* IS22Pl,, Is2

‘So “

In atomic units.

-371128 -371128 -331128 - 114

0.68029 0.78896 0.57084 0.48014

0.47366 0.50858 0.49748

-

SCREENING

1.3224 I .7944 2.3476 -

G. W . F. Drake

452

where the first term contains the exact Dirac energy for nuclear charge Z - (+!.Thus, all higher order one-electron relativistic corrections are also included. The second term subtracts out the leading nonrelativistic energy contribution. Using Eq. ( I 13) for the nonrelativistic energy (augmented by the Hartree-Fock term -0.06623/2) and adding unscreenrd hydrogenic Lamb shifts, they obtained the results for the 2 2P112-22S,12 transition frequency shown in Table XVII. The values agree with Edlen's experimental fit to within the experimental accuracy for Z 2 14. Berry et d. therefore concluded that the QED corrections are close to the unscreened hydrogenic values. However, this conclusion does not agree with the parameters obtained from Edlen's (1979) semiempirical fit to the experimental data. His expression for the 2 2P,,,-2 2S,,, transition frequency can be written in the form AE(2P,/z-2Sl/z)= AENR

-

Ar('S)

+ Ar('P,)

-

3 Ar('P)

- Ai.(2S) + AL('P,)

(124)

where AEy,