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 << 1. Writing R,., = Z(o) do, we have (Jackson, 1975) -+

-+

1;"

Here ro(r) is the position vector of the electron, lp = ( d r o / d t ) / c ,and p = d p / d r . In the nonrelativistic limit we have p << 1, and since ro/c f l r , the exponential may be replaced by exp i d (dipole approximation). Similarly, the contribution coming from lp x p may be neglected, and the retardation

-

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

7

factor (1 - p-i)-* may be replaced by unity. Finally, since << 1 and the integrand is nonvanishing only for times during which the collision takes place, expiwt may be set equal to unity. With these approximations we have 2

where v = p/p, v’ = p’/p, and p is the electron mass. In this long-wavelength approximation the radiation is independent of the details of the orbit; the actual collision may be replaced by one which changes the velocity instantaneously from v to v’. Since I ( w ) is nonzero for w --., 0, an attempt to interpret the bremsstrahlung spectrum in terms of a photon picture runs into difficulty. Thus, let us define the average number of photons emitted during the collision as fi = /:M d i , with hw(dii,) = I ( w ) dw. The approximation (1) then leads to

which diverges logarithmically; an infinite number of photons are emitted with a finite energy loss. This indicates that the usual perturbation analysis in terms of individual photons is based on an inappropriate physical model. In a correct quantum theory, divergences of the above type still arise, but are properly interpreted as divergences in photon number, not transition probabilities. We shall first go through a heuristic treatment of the quantum analysis; a more formal discussion appears in Section I1,B. We begin with the auxiliary device of placing the system in a box of volumeL3, which makes the photon spectrum discrete. The radiation energy can then be expressed as a sum over modes (which reproduces the classical expression in the limit L 3 + x ) of the form M

Rpjp=

hw*(Pi,,- prp)2 f=O

where pfp= ( h ~ ~ ) - ~ ( 2 ~ h e * / ~ * w ~ L ~ ) * ’ ~ p * X ~

Here wf and X i represent the angular frequency and polarization vector, respectively, for the ith mode. It is then natural to identify

Hi

=

(pip,

-

pip)*

(3)

8

Leonard Rosenberg

as the average number of photons radiated into theith mode. Since we are ultimately interested in calculating cross sections, we would really like to know the distribution function P(ni.)giving the probability of emission of ni photons into theith mode. It should satisfy P ( n 0 = 1 and En* niP(ni)= f i t . Then

En,

would represent the joint probability for emitting n , photons in mode 1, n2 photons in mode 2, etc. This could then be used to calculate the bremsstrahlung cross section for scattering accompanied by the emission of ni. photons into the ith mode. The condition W T ~<< 1 suggests the physical picture in which the collision takes place essentially instantaneously, with a probability determined by the cross section d d O for ) scattering in the absence of the interaction with the radiation field. We have seen, furthermore, that the dipole approximation adopted for the classical calculation of the radiation energy, which is equivalent to the neglect of electron recoil on absorption and emission of photons, leads to a radiation formula which is independent of the detailed nature of the collision. These considerations suggest the product form n2, .

d&,

. . , nM) = P(n,, n2, . . . , n M ) d d 0 )

(5)

for the bremsstrahlung cross section. Actually, this result is not very useful in itself. The probability of emitting ajnite number of photons must vanish for L 3+ since in that limit the average photon number is infinite. It follows that da(n,, n2, . . .) vanishes for L3+ =. This is not disturbing since the quantity of experimental interest is not d v ( n , , n 2 , . . .), but rather its sum over final states containing unobservable photons; these are “soft” photons whose frequencies lie below some limiting value w,, say. The summed cross section will be finite and measurable. The above discussion can be summarized in the form of two sum rules. First, the total scattering cross section, obtained from Eq. ( 5 ) by summing over all n,, n2, . . . , n M , is given by the cross section d d O )for scattering without any influence from the field. Second, the average energy radiated, defined by weighting the photon energy X i hoIntwith the collision probability, reduces to the classical radiation energy Rptpassociated with the instantaneous collision picture. These sum rules are correct to lowest order in 0 ~ 7 It~ is . remarkable that with the first-order correction term included the results can still be expressed in terms ofthe field-free scattering cross section (Rosenberg, 1980a). Analogous sum rules for scattering in a low-frequency laser field are discussed in Section V. ( ~ 1

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

9

The distribution P(n,) remains to be determined. Since we ignore recoil of the electron, the source is unchanged by photon emission-each photon is emitted under identical circumstances. One might then expect a Poisson distribution

P h i ) = (nr!)-le-Ei(at)ni

(6)

It is instructive to consider a more detailed, but still heuristic, justification of this formula. Let us recall the analogy between a radiation field containing n photons of frequency o and a harmonic oscillator of natural frequency w in then th quantum state (Sargent er al., 1974). Actually, we have a radiation field which, for a given mode, may be described classically in terms of an electric field of a specified frequency o and an amplitude corresponding to the specified energy h on. The harmonic oscillator analog would be a time-dependent state corresponding to a wave packet whose center oscillates about the equilibrium point in simple harmonic motion with frequency w and amplitude d. One takes the initial wavefunction to be a minimum packet centered at x = d and allows it to evolve according to the Schrodinger equation. The solution was obtained by Schrodinger (1926) and is reproduced in the text by Schiff (1968). From the solution one can determine the probability that the oscillator will be found in the stationary state with quantum number n . It is a Poisson distribution with Ti determined so that the average energy hwii = j K d 2 , where K is the stiffness constant of the oscillator. This argument lends credence to the claim that the low-frequency bremsstrahlung photon distribution follows a Poisson law. In any event, we have a simple mechanical model for a so-called coherent state of the radiation field. A proper derivation of the Poisson distribution will be given later. As an example consider the emission of a single observable photon during the collision; its frequency is w,, with w, < w,,, < w M .Summing the distribution (4) over soft-photon occupation numbers, with n, = ,a, for j > s, we obtain

2..

11, .111..

~ ( n , n,, , .!I,

. .. , nM) =

n M

e-%irn

e-Dfim

f=&l

Here exp( - D ) is a radiation-damping factor representing the probability that no hard photons are omitted. Observe that had we not summed over M soft photons the damping factor would have been exp(-xi-o nil, which becomes exp(-A), with Ti given by Eq. (2) in the limit L3 + m. Since i diverges, one finds zero probability for emission of a single hard photon and any finite number of soft photons. This is consistent with the earlier observation that, in fact, an infinite number of photons are emitted. However, if we expand the exponential-this corresponds to treating the

Leonard Rosenberg

10

electron-field coupling in perturbation theory-the divergence in ii at the low-frequency end of the integration range gives rise to a divergence in the transition probability. This unphysical result is the infrared catastrophe. Returning now to the case where we do sum over unobservable photons, we find M

Here w^ represents the region in k space satisfying w, S kc 5 wy. The result of the integration is

The differential cross section for scattering with an electron observed in a solid angle d o and a photon of polarization A observed in a solid angle don with wave number between k and k + dk is

This is of the form w ( d 3 a / d Rdw d R i ) = A. where w > 0 and A . is frequency independent and is obtained from a knowledge of the field-free scattering cross section. Low (1958) was able to calculate a correction term of order w , again in terms of the physical field-free cross section. Kroll and Watson (1973) obtained an analogous result for stimulated bremsstrahlung. These results can be derived as special cases of a more general formulation of the scattering problem discussed below.

B. BLOCH-NORDSIECK THEORY: COHERENT STATES Infrared difficulties in scattering theory are avoided by making an appropriate choice of modified plane waves. The necessity of including the effects of the electron-field interaction in the asymptotic states is clear physically. Furthermore, failure to do so leads to mathematical problems when one attempts to construct a formal proof of the existence of the scattering matrix (Blanchard, 1969). A comprehensive review of modern treatments of the infrared problem in quantum electrodynamics has been given by Jauch and Rohrlich (1976). Since the problem arises from soft photon interactions, we shall ignore, for the purpose of the present discus-

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

11

sion, hard photon interactions in constructing the modified plane waves. Their effects can be included later on by doing perturbation theory. Furthermore, we retain the essential features of the problem if we consider here the potential scattering case, work in the dipole approximation, and drop the A term in the electron-field interaction. (These limitations can be relaxed.) We look for asymptotic states satisfying (He + HF+ H’)I+) = El+) where He is the electron kinetic energy operator. With the sum over modes understood in the following to be cut off so that only soft photons (wt < w,) are included, we have, in the occupation number representation, HF = hwtaiat and H ’ = -(e/pc)p*A. Here af and a; are the annihilation and creation operators, respectively, for photons in the ith mode and A

( 2 5 r h ~ ~ / o ~ L ~ )+” a:) ~A~(a~

= i

is the vector potential in the dipole approximation. We work in a basis in which the polarization vector hi is real. Since the electron momentum p is conserved in the dipole approximation, the solution will be of the form IJlop) = Ip)WpIO). We are assuming that the only photons present are in the photon cloud associated with the electron. More generally, we could have, in addition, a beam of incident photons, in which case the vacuum state 10) would be replaced by the appropriate occupation number state. With the energy eigenvalue associated with the above state expressed as Eop= ( p 2 / 2 p ) + Ap, the Schrodinger equation is easily seen to be equivalent to the commutation relation r

1

which is satisfied by W, = exp[X.ipf,(af - a,)]; the level shift is given by Ap = hofpf,. This solution may be verified with the aid of the identities (Bjorken and Drell, 1965)

-xi

eA+B

= eAeBe-[A.Bl/2

,

[A, a B ]= [A, B ] e B

which hold when [A, B ] commutes withA andB. Note that the relation Eop h o f p j pwhich has been obtained for the energy eigenvalue is easily understood in classical terms by recognizing that the work done in accelerating the electron from rest to the momentum p includes a contribution arising from the radiation reaction force. This contribution, which accounts for the work done by the field, is the negative of the energy radiated during the acceleration process (Landau and Lifshitz, 1962). If one includes only the soft photon modes and assumes that the = ( p 2 / 2 p )-

cf

12

Leonard Rosenberg

acceleration time is short compared with the period 2n-/w,, the energy radiated reduces to El h w i p f p ;the negative of this is just the expression obtained above for the level shift. The quantum origin of the level shift lies, of course, in the emission and reabsorption of photons. It is a simple matter to establish the property, well known in the theory of coherent states (Glauber, 1963),

wp.wp=

Wp-,,

(7)

Using this result we can calculate the amplitude for scattering with the emission of n, photons in the ith mode. In the Born approximation it is

.

( n l . n 2 , . . . n,lW~~(p'(VJp)W,IO) =

V(p' - p ) ( n l , n2, . . . , n,lW,-,,10)

The matrix element of WP-,, can be calculated. Its square gives the probability of finding photons with the occupation numbers n , , n 2 . . . . , ris in the final state. This probability turns out to be precisely the Poisson distribution mentioned earlier: n2, .

. . , n , l ~ , - , , 1 0 ) 1= ~ P(n1)P(n2) *

P(n,)

with P(n,) given by Eqs. (3) and (6). In this way much of the earlier heuristic discussion is justified. The cross section is da/dR

=

(du'"'/dR)HornP(nl,n2,

. . . , n,)

More generally, that is, to all orders in the scattering potential V , we expect that the Born cross section should be replaced by the exact fieldfree cross section to good approximation (Nordsieck, 1937). The following physical argument may be offered to support this expectation. Nonperturbative methods are required to construct the asymptotic states because of near degeneracies. (Two states with the same electron energy but different photon number are very closely spaced.) The p,, functions play the role of effective coupling parameters-they are matrix elements of H ' / h w , . This operator may be taken to be of zero order, with H' and hwi each of first order. As for the intermediate states the collision time is so short compared with the period 2r/ol(here we assume that there are no resonances in the field-free scattering cross section-that requires special attention) that a well-defined energy cannot be ascribed to the electron during the collision. The energy denominator in the Green's function is not of the order of the photon energy (as it is in the asymptotic propagator) but some average excitation energy-large compared with the electron-field interaction energy (Bebb and Gold, 1966). This indicates that ordinary perturbation theory is sufficient to account for the effect of the field in

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

13

intermediate states. To lowest order we may ignore the field. The potential V may be replaced, to this order, by the field-free scattering operator, yielding the improved approximation dvldfl

= (da‘O’/dfl)P(n,, n2, . . , n,) *

(8)

Suppose that the initial state contains a large number of photons in a single mode (the ith mode, say) corresponding to the presence of a monochromatic laser beam of intensity Z = (nihwc)/L3.The photon number can change in the scattering process through absorption or stimulated emission. With effects of spontaneous emission ignored-they introduce corrections of order l/ni compared with the leading term-one finds (dropping the mode indexi), (n’lWp-pJJn) = J,!-,,,(X), whereJ is the Bessel function of the first kind and

X

=

(ho)-’(e/~~)(8rr~Z/o*)”*(p’ - p)*X

The cross section d v ( n - n‘)/dfl for the absorption (n - n ’ > 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,

=

0

+ A+e-ik.ll

$&i(Aieikitl

1

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 simplification in the analysis. For circular polarization (x = ~ / 4 (Y ) vanishes 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 = p 2 / 2 p + A + nhw = Ep + nhw, we find that Eq. (17) will chosen as Erp 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 = 8,,.,,8(p'- p). used to establish the orthonormality relation (+,7~pf~Jl,lp) 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 state D,,, is I*”,) = exp --i(E,,,t/#i)~D,,). 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 absorprJ*E(r,), tion and may be ignored here. Using the relation [ g , . H F ] = e Ey==, 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,)

C 2

=

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 = exp(-iE,npr/h)$L,np using the analog of Eq. (19) quantum solution or (22). This would lead to the representation (n

+ ml$,,nP)=

m

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 by Ev,,p = Eyp+ nhw. The version (32a) has an obvious physical interpretation 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 $,,,; this leads to ~-oeI$,,~)

= 1E-m

1'"

(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
can be related to the tunneling contribution to the total ionization rate; we return to this point in Section V,D. It should be noted that a scattering-theoretic approach has been used by Damburg and Kolosov (1978) to study the problem of ionization of hydrogen in a static electric field.

B. APPROXIMATION TECHNIQUES I . Potetitid S c u t t u i t i g

Reliable approximations to the transition matrix can be obtained only under special circumstances, some of which are discussed below. To begin with, we consider scattering by a potential which is weak enough so that the iterative solution of Eq. (40) can be replaced by its first term, Eq. (41). Using the expansion (18) along with the addition formula ( I ~ c ) , we find

TI+.)IP

($,I

P

IVIJl,,p) = y l l , - , , ( b-

b ' , CY

-

a')V(p' + n ' h k

-

p

-

nh k)

If we ignore the photon momentum in comparison with that of the electron-this is equivalent to assuming a spatially homogeneous semiclassical field A,. = RedA exp(--id)-we have a - a' = 0. According to Eq. (ISa), the y coefficient may be replaced by the product of an (irrelevant) phase factor and a Bessel function of argument X =

Ifi

-

6'1 = I(hW)-l(e/pc)d A * ( p ' -

p)I

(48)

The differential cross section then becomes

~ i t h p ' ~ / 2=p (p2/2p)+ ( n - n ' ) h w . In the limit of vanishing field strength we find, of course, that T,,,p..,rp +

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

29

anfnt[p 2 / 2 p , (p' - P)~]], where t(e,T ) is the physical amplitude for scattering in the absence of the field; it is obtained from the off-shell amplitude

t ( e ; q', 9) = (q'l[V + V(e - H e - V)-'Vl1q)

by setting q2/2p = q"/2p = e, with (q' - q)* = T . A perturbative treatment valid in the weak-field regime may be set up by expanding the resolvent appearing in Eq. (38) in powers of the interaction. The weakfield behavior of the y coefficients is obtained by expanding the exponential in Eq. (13). The first few terms obtained in this way are, to first order, yo = 1,

y1 zz

-+p 9

y-1

= +b

(49)

Ordinary perturbation theory is recovered in this way. However, if the frequency is low enough, the parameters p and a , which measure the effective strength of the coupling with the field in asymptotic states, need not be small. As discussed earlier in connection with the infrared problem, the coupling is enhanced in the low-frequency limit by virtue of the near degeneracies in the spectrum of the electron-field states; adjacent energy levels EllPare separated by h w . To account for this effect we retain the exact form (18) for the asymptotic states. Then, with electron-field interactions ignored in intermediate states we have G(E)= (E - He - V HF)-l, and the approximate T matrix becomes

If the t matrix is a slowly varying function of the energy, the energy variable in Eq. (50) may be replaced by p 2 / 2 p as a first approximation. The sum can then be performed using Eq. (15c), giving T,I,P,.IIP

= y'l,-,,(b - b', 0)t(p2/2p; P', P)

(51)

where again we have set a = a'. The differential cross section obtained in this way agrees with that derived earlier [see Eq. (9)] using the coherent state representation of the field. The addition formula El J f ( X ) = 1 allows us to equate the total cross section, obtained by summing the approximation (9) over all final states of the field, to the field-free cross section. This is analogous to the Bloch-Nordsieck sum rule for scattering accompanied by the radiation of unobservable soft photons, as discussed in Section I1 (see also Jung, 1979). An improved version of this sum rule is presented in Section V. Equation (51) has been frequently used in the description of plasma heating through the inverse bremsstrahlung process (Geltman, 1977). In such applications the target is a positive ion so that the (unscreened) potential contains a Coulomb tail. Since the scattering theory from which

30

Leonard Rosenberg

the approximation ( 5 1) was derived was based on the use of asymptotic states in which the effect of the Coulomb tail was ignored, the validity of this approximation requires further justification. A useful extension to include the Coulomb case is made difficult by the unavailability of exact solutions of the Schrodinger equation for an electron in the presence of both a laser field crud a Coulomb potential. An attempt to bypass this problem has been proposed (Rosenberg, 1979b). While the practicality of that calculational procedure has not been investigated, it can be employed to derive a low-frequency approximation the leading term of which is of the form (51). The derivation involves the use of asymptotic states obtained by replacing plane waves by Coulomb waves. Approximations of this type have been suggested by Jain and Tzoar (1978) and, in connection with the problem of ionizing collisions, by Cavaliere et c i l . (1980). Some mathematical work dealing with the existence of the wave operator for 1975). Coulomb scattering in a laser field has been done (Combe et d., Further work on this important problem is called for. In order to compare approximate representations of the free-free cross section with results obtained from experiments involving pulsed laser fields, the temporal shape of the laser pulse must be accounted for. Kriiger and Jung (1 978) have suggested an approximate description of the classical vector potential in the form A, = Re d ( t ) X exp(-iwr). With the assumption that the amplitude function d(r)is slowly varying in time intervals of order m-', Kriiger and Jung (see also Gersten and Mittleman, 1976b) were able to derive a time-averaged version of Eq. (9) of the form

with X ( t ) = ( e / p c h o )sP(t)X*(p' - p). The integration goes over the duration of the pulse, or that part of it which may be selected electronically in an actual experiment. The cross-section sum rule is unchanged by this averaging procedure. Spatial averaging may also be introduced (Jung, 1980). Multiphoton free-free transitions have been detected by Weingartshofer et t i / . (1977, 1979). In these experiments electrons are scattered from argon atoms and hydrogen molecules in the presence of a CO, laser field. Since the energy spread of the incoming electron beam was much greater than the widths of the resonances in the e-Ar and e-H, systems, resonance effects are effectively averaged over and may be ignored. It also appears safe to ignore the effect of the field on the target since in these experiments the photon energy hw = 0.117 eV is small compared with the target excitation energies, and the maximum laser flux (5lo8 W/cm') corresponds to an electric field whose magnitude is small compared with typical atomic field strengths. It was not possible to check for quantitative

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

31

agreement between theory and experiment since the spatial and temporal laser flux distribution was not known with sufficient accuracy. However, qualitative agreement with Eq. (52) was observed, and the results were consistent with the cross-section sum rule. It has been suggested (Jung, 1980) that one might work backward and use the free-free transition measurements to determine the average power density of the laser field. The multiple absorption of photons by electrons in free-free transitions has been observed by Lompre et (11. (1979). In that experiment the electrons have a thermal energy of lo-' eV, which is small compared with the energy 1.17 eV of the photon. The validity of the low-frequency approximation is therefore questionable in this case. In fact, the experiments were interpreted not in terms of Eq. ( 52) but rather by assuming that the N-photon absorption process depends on the laser intensity as I N . The lowest nonvanishing order of perturbation theory in the electron-field coupling implies such a power law behavior. The absorption of as many as eight photons has been detected in this experiment. 2. E&t

of Scrittrr'itig Resotwnces

As pointed out by Kriiger and Jung (1978), Eq. (50) represents a convenient starting point for the study of resonant scattering in a laser field. If the field-free scattering amplitude is resonant at an energy e,, the amplitude (50) will show resonances for initial electron kinetic energies p 2 / 2 p = e o , eo & ho,eo ? 2hw, etc., corresponding to the fact that the electron can exchange energy with the field before or after the resonant interaction. To be more explicit, we suppose that an isolated resonance exists whose width is small compared to hw. Let I, be the integer closest to [ ( p 2 / 2 p )e o ] / h w .Then, ignoring the nonresonant background in Eq. (50), we have Consider now single-photon absorption in the weak-field limit. The relevant y coefficients are well represented by the expressions given in Eq. (49). With p 2 / 2 p = eo - ho,we have lo = - 1 and T,l,p,;,lp= tbr(e,; p', p), corresponding to a single photon absorbed in the initial state. There will be a second resonance at p 2 / 2 p = e, (1, = 01, with amplitude ( - b ' / 2 ) * t ( e 0 ; p', p) corresponding to a single photon absorbed in the final state. The validity of this weak-field soft-photon approximation has been checked by Jung and Kriiger (1978) in a two-state square-well potential model. They compared it with the exact (to first order in the electron-field coupling) numerically determined free-free matrix element for absorption (see also Kriiger and Schulz, 1976). Experiments on e-Ar scattering in a weak COz laser field near the argon resonances at 1 1 eV (Andrick and Langhans,

Leonard Rosenberg

32

1978; Langhans, 1978) are in full agreement with this simple theory (or rather its generalization to account for two closely spaced resonances). The appearance of resonances leads to a modification of the crosssection sum rule (Mittleman, 1979~).In the no-recoil approximation, with the A' contribution to the interaction ignored, and with the field assumed Suppose ). that t has a to be linearly polarized, we have y l ( p ) -+ .I& resonance at energy eo with a width sufficiently small compared to h w so that for 1 # l o , the amplitude t in Eq. (50) may be treated as a smooth function of the energy. We may then write t = t R + t N R , where in the energy domain Ie - eol > h w , f R is negligible, while t N R is a smooth function for all energies. The low-frequency approximation (50) then takes the form T".p.:npJn,-n-/,(P')J-/,(P)tR[(P2/211.) - Iohw; p', PI

+

Jti

d(p -

p')fNR(p2/2/J-;

p', p)

(53)

The addition formula ( 1 5 3 , which in this case reduces to the Graf addition formula for the Bessel function, allows us to evaluate the sum

where

With the laser absent only d u ' " ' / d R is measurable; the laser, through the parameter p , allows for an additional measurement. The second (nonresonant) term in Eq. (53)vanishes for scattering in the forward direction for n' # n because of the vanishing of the Bessel function. (More generally it will vanish if the momentum-transfer vector is perpendicular to the direction of polarization.) This suggests that the laser field may be used to suppress the contribution to the scattering cross section which is only weakly dependent on energy, thereby allowing for an observation of the pure resonance scattering, unobscured by interference with the background. The method may also be useful in separating threshold cusps from the background. Model calculations, involving a single spatial dimension and a finite-dimensional photon Fock space, performed by Jung and Taylor (1981) confirm this background suppression effect. Comparison of the one-dimensional version of Eq. (50) with the essentially exact numerical results shows that the low-frequency approximation explains all of the qualitative features of the energy dependence of the cross section for free-free transitions with the exception of the fielddependent shift of the resonance position.

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

33

In the absence of an external field electron-atom scattering resonances of the “closed-channel” type can be understood in terms of composite bound states of a modified Hamiltonian from which one or more of the low-lying target states have been projected out. These composite bound states give rise to pole singularities in the effective potential, which in turn generate the resonance contributions to the scattering amplitude. This effective potential formalism (Feshbach, 1958)can easily be generalized to account for the presence of the external laser field (Gersten and Mittleman, 1976a; Rosenberg, 1977). In addition to providing the basis for a description of the effect of the field on closed-channel resonances (the dynamic Stark shift of the resonance position is one such effect), this formalism can be used to generate approximation techniques of the close-coupling type which have been widely used in field-free atomic scattering problems. Resonance theory has been applied by Mittleman (1978, 1979d) to the analysis of a “two-state’’ resonance process seen in an experiment performed by Langendam and Van der Weil (1978). They observed transitions, induced by the absorption of a single laser photon, between two resonant states of the e-Ne system. The interest in this process lies in the fact that the resolution is limited not by the energy spread of the incident electron beam but rather by the width of the resonances. This dramatically improved energy resolution allows for detailed examination of the fine structure of the resonant states. A detailed calculation of cross sections for single-photon absorption in multichannel e-H scattering in a laser field has been carried out by Conneely and Geltman (1981), who applied the close-coupling procedure to calculate the required continuum wave functions. They find that the presence of closed channel electron scattering resonances can enhance the background free-free absorption in the forward direction by several orders of magnitude. 3.

Mil Itichn11 n el

S critt ering

We now consider scattering from a model target atom whose spectrum consists of a finite number of discrete bound states, the effect of the continuum being assumed to be negligible. This example provides an illustration of how the response of the target to the field can be accounted for within the effective potential formalism. Referring to Eq. (44) for the transition matrix, we identify the Born amplitude

v ~= (e-oeJIpllVle-QeJIo) , ~

(54)

as the effective potential. (Since we ignore exchange here, we are able to simplify notation by dropping the channel index p on the projectile-field interaction operator V , .) The transition matrix satisfies a multichannel

34

Leonard Rosenberg

version of the Lippmann-Schwinger integral equation (401, which takes the form Tflyj =

Vflcfl+

V8,&fl

-

E@,)-’T@,,

(55)

8”

where the integration over projectile momenta is understood to be included in the sum over intermediate states. The asymptotic states appearing in Eq. (54) are constructed according to the procedure described in Section III,B, the overbar reminding us to make use of the gaugetransformed version (28) of the target-field interaction. Ignoring recoil effects in the representation (45) of the projectile-field state, one finds

x exp{i[S,(4) - S P W l ) (P‘(4J)IV”d4)lP(4))

Here we define P(4J) = p - eA&)/c, representing the momentum of an electron moving in a classical field A c ( 4 ) ;4 is interpreted as the phase - w f of the field in the dipole approximation and p is the time-averaged momentum. P’(4) is similarly defined. We also have V,,,,,(4) = ( ~ , , ( d ) ~ V ~ ~where , ( d ) El(&) ), = T,(v) exp(-irn&) is the gaugetransformed version of Eq. (2 1) for the semiclcrssiccrl target-field wave function. If the field is not too strong, this Fourier expansion may be approximated by a finite sum, and the sum over intermediate photon states in the integral equation ( 5 5 ) may be truncated. The resultant set of coupled equations may be solved numerically and the convergence of the procedure checked by repeating the calculation with a larger set of photon states included (Jung and Taylor, 1981). Numerical calculations for realistic systems are required to determine the range of applicability of this approach. We may, however, anticipate a simplification at low frequencies. In an adiabatic picture of the scattering process (Mittleman, 1979a), which is reasonable physically if the period of the field is large compared with the collision time, one imagines the phase of the field to be held fixed during the collision. Furthermore, one ignores the fluctuation of the field energy in intermediate states since its contribution will be small in the low-frequency limit. In this picture the scattering operator T l r C L ,corresponding (b), to the instantaneous phase 4, satisfies the coupled operator equations

Em

Note that while the effect of the field on the target is contained in the potential V,,,,,, thefin’rn of Eq. (56) is identical to that appropriate to the

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

35

field-free case; it is much simpler to deal with than the original version ( 5 5 ) . The physical transition matrix is obtained from the scattering operator by constructing a weighted average over the phase of the field of the form

x ( P’( 411T l d ($)IP( 4))

(57)

To justify this representation one goes back to Eqs. (551, lets o + 0 in the energy denominator, and then verifies that Eq. (57) provides a solution if the operators T,,,(+) satisfy Eqs. (56). The differential cross section is du(n - n ’ ) / d ~= (27T)4F2)15(P’/p)~~Y1nIP);YnP~2 With the approximation (57) for the T matrix, and with p t 2 / 2 k = ( p 2 / 2 p ) + E,. - El,, in the low-frequency limit, we have the sum rule

d u ( n - n’)/dR n-p+--m

= Jozn

~

(d4/2n){(2.rr)4p-2tl * ( P ‘ / ~ ) I

~

~~ ( 4 I)z,)I

~

The quantity enclosed in braces is the cross section for an instantaneous collision corresponding to the phase of the field. The total cross section is represented above as the incoherent average over the phase. The electron with initial momentum p is thought of as following a classical trajectory once it enters the field, arriving at the target with momentum p e A J + ) / c ; it then follows a classical trajectory until the field is switched off, leaving the electron with momentum p‘. If the field is not strong enough to appreciably distort the atomic wave functions it may still play an important role in the dynamics if the energy of the laser photon closely matches the level separation of a pair of target states. Such a situation has been analyzed by Gersten and Mittleman (1976a) for e-H scattering in a field which can cause resonant transitions between the Is and 2p states of hydrogen. The target then has an induced dipole moment even in the absence of the projectile so that the effective potential has long range components quite distinct from those which appear in the field-free case. However, some of the novel effects predicted in this theory will actually not be observed due to the loss of coherence of the dressed atomic states arising from spontaneous emission (Mittleman, 1976). The effect of the field on the target can be significant, even for nonresonant fields of moderate strength, if the target spectrum contains near

+

~

Leonard Rosenberg

36

degeneracies and the laser frequency is sufficiently low (Perel'man and Kovarskii, 1973; Brandi et ul., 1978). This can be seen clearly in the example worked out in Section III,B, in which two degenerate states of an atom are coupled by interaction with the field. The &v.ti\,e coupling strength is the parameter Y , defined in Eq. (29), which increases like w - I for fixed electric field strength. This near singularity does not appear in intermediate states of the scattering process where integration over a continuum of energies smooths out the effect of small energy denominators. To see the effect of target degeneracies more explicitly, let us place the problem in the context of the effective potential formalism described above and assume that the field is weak enough so that the target-field interaction may be ignored except for the coupling it induces between degenerate states. Solving the target-field eigenvalue problem in the subspace of degenerate states we construct semiclassical solutions which satisfy, at each instant, the closure relation I'

1'

the sum running over states in the degenerate subspace. [This relation may be verified directly for the case of a twofold degeneracy using the representation (311.1 The closure property may be used to replace the sum over dressed intermediate states in Eq. (56) by a sum over bare states. Formally, one looks for a solution of Eq. (56) in the form Tvtu= ). Ignoring target-level shifts, one finds that Eq. (56) will be satisfied, pro) 6,. - He - H T ] - l t . This identifies vided thatt satisfies f = V + V [ ( p 2 / 2 p + t as the field-free scattering operator, the physical t matrix being given by tY.P.:YP = (p'l (b$lb,,>lp).In evaluating the T matrix (57), we suppose the field to be weak enough so that P(4) = p and P'(4) = p'. Suppose further, for definiteness, that the initial target state 16,) is nondegenerate and that transitions to two degenerate states 16,) and 16,) are considered. Using the explicit solutions (3 1) for the dressed target in the final state (corresponding to a linearly polarized field) along with the representation (30) of the Bessel function, we find, for v' = 1 ,

(o,,,lrl&

-itip,:,lp sin[(n -

n')e]},

(58)

where X is defined in Eq. (48) and tan 8 = Y / X . For v' = 2 , we must interchange the indices 1 and 2 in Eq. ( 5 8 ) . If we adopt this low-frequency approximation for the T matrix, we are led to the sum rule

ELECTRON-ATOM SCAlTERING IN A RADIATION FIELD

37

In this approximation, then, the differential cross section for scattering into one or the other of the two degenerate states of the target reduces, when summed over final states of the field, to the corresponding differential cross section for scattering in the absence of the field. Let us note, finally, that the above analysis may be used as the basis for an almost degenerate perturbation theory (Rosenberg, 1980c), the level separation being treated as a small parameter.

V. Generalized Low-Frequency Approximations A. CONNECTION W I T H T H E CLASSICAL LIMIT In this section we review recent developments in the theory leading to improved and extended low-frequency approximations. As we have seen, such approximations are useful since they allow one to express the transition probability in terms of the measurable cross section for scattering in the absence of the field. They also play a role in providing analytic expressions for the amplitude against which numerical approximation procedures can be tested. Aside from these practical considerations, the development of nonperturbative techniques is of interest in the study of the relationship between quantum and classical formulations of electrodynamics (Bialynicki-Birula, 1977); such a study can lead to useful insights. The relevance of the classical limit shows up clearly in the Bloch-Nordsieck analysis of the spontaneous infrared radiation problem reviewed in Section 11. One sees that the use of classical currents to describe the source of the radiation field is justified if the field consists of a large number of very soft photons. Further insight into the nature of low-frequency approximations came from the demonstration (Low, 1958) that soft-photon amplitudes (single-photon bremsstrahlung was the particular process studied by Low) are determined, up to the first two orders in an expansion in powers of the frequency, by the charge of the projectile and the on-shell amplitude for scattering in the absence of the field. Low’s derivation was based on general considerations of current conservation, Lorentz invariance, and simple analytic properties of the scattering matrix. The particle-field interaction was treated to lowest order of (renormalized) perturbation theory. The work of Brown and Goble (1968) on relativistic scattering in an external low-frequency radiation field may be thought of as representing an extension of Low’s theorem to the problem of stimulated (or inverse) bremsstrahlung, with the particle-field interac-

38

Leonard Rosenberg

tion treated to all orders. There is some merit in reconsidering this problem with the aid of the nonrelativistic Schrodinger equation since the dynamics can then be specified completely and the argument can be carried through in a much more explicit fashion. The low-frequency approximation described in Section IV,B represents a first step in this direction. One expects, in view of the results of Low and of Brown and Goble, that the approximation can be extended to include a correction term of higher order in the frequency without requiring off-shell information. One is also led to expect that the Bloch-Nordsieck analysis could be extended to higher order in a fairly simple way. With the above considerations in mind, let us begin with a review of the classical treatment of scattering in a low-frequency external field (Brown and Goble, 1968; Kroll and Watson, 1973). Since the collision time is assumed to be short compared with the period of the field, we may determine the cross section by assuming that the collision occurs instantaneously. The effect of the field on the target, and on the electron-atom system during the collision, will be ignored. The electron with initial momentum p enters the field adiabatically (in the sense discussed in Section I). The subsequent motion is most easily visualized in a reference frame in which the initial momentum is zero, with the field assumed for the moment to be plane polarized. The electron is forced to oscillate along the direction of the electric field with frequency w . As a result of this motion there is a magnetic force which induces an oscillatory motion of frequency 2 0 in the direction of propagation of the beam. This results in a trajectory in the form of a figure-eight with axis along the direction of polarization. The work done by the field during the adiabatic passage of the electron into the field may be calculated by assuming the vector potential to be of the form A, = dX exp( - ylfl) cos( k *r - wt), with y + 0 : then, 0 with v = - e A , / p c , the work done is determined as e E , - v dt = e Z d 2 / 4 p c 2This . level shift can be expressed (for arbitrary polarization) as

A

JozT

(d4/27r)r2A36)/2pc2

and corresponds to the quantum mechanical level shift determined in Section II1,A. A transformation back to the original reference frame, taking into account the alteration of the magnetic force along k which is caused by this transformation, leads to the classical momentum

P(4) = p

-

( e / c ) A , ( 4 )-

(R/c)

x [ p * ( e / p c . ) A , ( 4) ( e ' / 2 ~ ~ c . ~ ) A+ %4 $)

(59)

The momentum is expressed here as a function of the phase 4 rather than the time (to which it is proportional to first order in v / c ) . This variable

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

39

change just compensates for the effect of the magnetic force in the direction of A,, as may be verified by solving the equations of motion directly (Rosenberg, 1979~).One sees in this way that the solution (59) holds for arbitrary polarization. [The exact relativistic solution has been given by Landau and Lifshitz (1962) and by Brown and Goble (1968).] The kinetic energy of the electron in the field is

+ (1 - p * R / p ~ ) - ' [(-e/pc)p*A, + (e2A!/2pc2)- A]

P 2 ( 4 ) / 2 p= (p2/2p)+ A x

with average value (p2/2p) + A. If there were no collision, this average energy would remain fixed. The instantaneous collision changes the momentum to P'(+), defined as in Eq. (59) but with p + p', and evaluated at the same scattering phase 4. Energy conservation requires that eJ4) = P , , , ~ . ( $ ) ,where

ey,(4) = ( P 2 ( 4 ) / 2 p +)

E"

(60)

The average energy does change as a result of the collision since the dirrction of the instantaneous momentum is changed. The energy transferred from the radiation field to the electron for a particular scattering phase 4 is @'2/2p) +

€,,,

- ( p 2 / 2 p )- E , =

hods;.(+) - SL(+)]

(61)

where, recalling Eqs. (11) and (12), we have

Since the scattering phase is not observed, the classical cross section is obtained by averaging over this phase; it takes the form duC =

lozT (d4/27r) d d 0 ) ( + )

(634

Here ddo'(4) is the cross section for scattering in the absence of the field and may be expressed (assuming a qlranrum description of the electronatom interaction) in terms of the on-shell scattering amplitude t ( r , 7);we have

du"'(4)

=

( 2 ~ ) ~ ( p / p )/ hd 3' p ' 6(e,fp44) - e.p(4))(t(evp(4), r(4))I2 (63b)

with e,,J4) defined by Eq. (60) and

T(4)

(p'(4) -

p(+))*

(64)

Note that the phase space factor has been taken to be d3p' rather than d T ' ( 4 ) . This is justified since the Jacobian of the transformation from P'

40

Leonard Rosenberg

to p‘ is unity (to first order in u / c ) , and it is the more convenient choice since p’ rather than P’ is the observable momentum. To determine the average energy transferred from the field to the electron, call it ( ( p f 2 / 2 p )+ E,, - ( p2/2p) - E, ), we multiply the energy transfer corresponding to a given scattering phase 4, shown in Eq. (61), by the collision probability, average over the phase, and divide by the average collision probability. That is, we write (( p ’2/2p) + cup- ( p2/2p) - E , , ) = dU,/da,, with

dUc = (27T)“p/p)h’

y=(d4/27T) p3Pf[(P‘2/2P) + 0

x Ne,,*p4b)- e&))lr(evp(+),

~(4))l’

lE”,

-

( p 2 / 2 p ) - GI

(65)

These classical formulations have been used as the basis for numerical analyses of free-free transition processes (Geltman, 1977). In the following sections we adopt the quantum mechanical description and point out the correspondence between the classical and quantum pictures which exists at low frequencies. B. MODIFIEDPERTURBATION THEORY

An incident laser beam of intensity I and frequency w corresponds to a classical electric field of root mean square amplitude (4571/c)lI2. For I = 10l6W/cm2, this amplitude will be of the order log V/cm, characteristic of the field strength inside the atom. We suppose in the following that I << 10l6 W/cm2, in which case it will be reasonable to treat the particle-field interaction appearing in the modified Hamiltonian H , Eq. (43), in perturbation theory. The starting point is the expression (44) for the transition matrix, which was obtained from the version (42) by means of a gauge transformation. In the process one has effectively replaced the “ p * A ” form of interaction by the “r*E” form. The appearance of the additional factor w/c in passing from the vector potential to the electric field amplitude suggests that this replacement will be particularly useful in the lowfrequency domain. One must be careful, however, if there are near degeneracies in the target spectrum, or if scattering resonances are present, since the appearance of small energy denominators may invalidate the conclusion that the r * Einteraction is of lower order than the p - A term. To deal with the degeneracy problem, one must take into account the effectively strong coupling of the nearly degenerate target states at the outset and perturb about the residual interaction (Rosenberg, 1980~).This complication will be ignored in the following. The effect of a scattering reso-

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

41

nance has been discussed by Feshbach and Yennie (1962) in the context of the low-frequency , single-photon bremsstrahlung process. These authors point out that a correction, which in the present formulation arises from the second term in the expansion ( E - H)-1 = ( E - H(0)- H F )-1 + (E

-

-

H(U)- H,)-lH’(E - H(0)- HF)-l +

...

(66)

and which is treated as a second-order term and dropped in the derivation of the Low theorem, is actually of first order in the resonant case. Unfortunately, this term is rather more difficult to estimate than the remaining first and zero-order terms given explicitly by Feshbach and Yennie. An external-field version of the Feshbach-Yennie low-frequency approximation is presented in Section V,C. Putting aside the effects of resonances as well as target degeneracies we may develop the perturbation theory by expanding the resolvent, as shown in Eq. (66). An iterative procedure for solving Eq. (27) for the dressed target states can be set up in a similar way. However, the projectile-field asymptotic states which appear in Eq. (44), and are represented explicitly in Eq. (45), must be treated nonperturbatively since here near degeneracies play an essential role. The effective coupling strength is determined by the function SJd) in Eq. (45). Its derivative, shown in Eq. (62), is measured by the ratio of the electron-field interaction energy to the photon energy. Let 9 represent the maximum value of this ratio considered as a function of the phase variable 4. [For moderate field strengths the parameter a in Eq. (12a) will be negligible in which case 92 = p ] . It is convenient to distinguish the following three regimes (Rosenberg, 1981). (1) 9 << 1, weak coupling. Here ordinary perturbation theory suffices for the determination of the electron-field asymptotic states. If, for the sake of numerical orientation, we choose hw = 0.1 eV,p2/2p = 10 eV, and I = lo6 W/cm2 we find 92 = 0.1, indicating that the weak-field regime encompasses a sizable range of laser intensities. (2) 92 1, intermediate coupling. (3) 9 2 >> 1, strong coupling.

-

With the photon and electron energies fixed at 0.1 eV and 10 eV, respectively, an intensity of lo1*W/cm2 gives 9 = lo2. Note that even at this intensity the laser field strength is still two orders of magnitude below the nominal atomic field strength of lo0 V/cm. As will be seen explicitly below, the strong-coupling regime is characterized physically by the existence of fluctuations in the field energy which are, on the average, large

42

Leonard Rosenberg

compared with ho;one must account for the effect of a large number of soft photons acting coherently. In the following we confine our discussion to the case where the interaction R’ is ignored completely; this is the lowest order in the modified perturbation expansion discussed above. [It should at least be mentioned that a determination of higher order terms in the expansion, while difficult in general, is of some interest since such corrections involve off-shell scattering information which is not otherwise easily obtainable; see, e.g., Mittleman (1979b).l Thus, we retain only the first term in the expansion (66). Similarly, with H ; set equal to zero, we have

IJI,,,)Jb”), I S D 4 l#dp,)l~lJ,) (67) and the field-induced level shift of the target is ignored. Introducing the field-free scattering amplitude

ISd

+

+

r(e;q’, q) = ( h , I (q’l[v,

+

Vde - ~(o))-lv~llq)lhJ)

(channel indices on the r amplitude are suppressed for simplicity) and making use of the representation (45). we find

with E

=

El,‘,,and q’ = p‘ - (e/c)A&’) - (I q =

P

-

+n

-

n’)hk

(e/c)A,(4) - Ihk

Electron recoil effects lead to corrections of order v/c;these are small but not always negligible in the nonrelativistic domain and are included here for completeness. The extent to which the t amplitude is off the energy shell is measured by the scalar variables

6=E 6’ = E

-

( n + I)ho - ( q 2 / 2 p )-

-

( n + I)hw - ( 4 ’ 2 / 2 p )- el,#

el,

Using Eqs. (69) and the definition (62) we find

6

+

= - ~ ( l S;(&))*

6‘ =

-q’((n

+ 1 - n ’ ) + S;,(+’))

(70)

To continue the analysis of the approximation (68), it will be convenient to discuss the intermediate- and strong-coupling regimes separately. Before doing so, we interpose a comment on the derivation of Eq. (68). The

ELECTRON-ATOM SCATTERING IN A RADIATION FIELD

43

assumption that the electric-dipole interaction is a small perturbation on the intermediate-state propagator might be questioned since the projectilefield interaction operator -eE*r, is unbounded. One could argue, on the other hand, that only the region within the range of the scattering potential contributes appreciably to the spatial integrations so that re is effectively bounded by the range of the potential (Feshbach and Yennie, 1%2). To give a more direct argument, we express the amplitude (it will be sufficient to consider the case of potential scattering) as T,,fp8:,,p = (&,p,IT(E)I&p), where = exp( -ge)l+np)and satisfies the operator integral equation

T(E) =

v + VG,(E)T(E)

-

Here Gocan be represented in the form (39) but with t,hnLnp + +,lp. Now the approximation (68) is equivalent to and this version can be verified by showing that the right-hand side provides an approximate solution to the representation of the integral equation (71) in the basis formed by the states {In;p)}. We shall not go through the demonstration here but merely remark that the problem of the unbounded nature of the dipole operator is avoided in this approach; rather, it is a low-frequency argument of the type which will reappear in the following two subsections. The argument explicitly requires that t be nonresonant so that, as indicated earlier, one should anticipate that an approximation based on Eq. (68) will omit certain first-order correction terms in the case of resonant scattering. C. INTERMEDIATECOUPLING The photon energy and the electron-field interaction energy will be taken here to be quantities of first order and their ratio assumed to be of order unity. We now show that the amplitude (68), evaluated to first order, can be expressed in terms of the on-shell amplitude t ( e ,7 ) for scattering in the absence of the field. Let us first observe that the parameters $, and t' are effectively of first order. It is true that the index I ranges over all integers, but contributions to the sum in Eq. (68) corresponding to 111 >> 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 vanish. To see this, note terms, one proportional to 5 and the other to 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 present case), one finds that q 2 ( 4 ) / 2 p= - E ” , . With q satisfying this 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.

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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 + annihilation gamma rays coming from solar flares (Chupper d., 1973) and 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

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

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

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


D7

~~

Groups are listed in chronological order. See Table IB for an explanation of researchgroup codes and reference codes.

POSITRON-GAS SCATTERING EXPERIMENTS

57

TABLE IB KEYT O RESEARCH GROUPS A N D REFERENCES LISTED I N T A B L E1A

Code

Research group

A

University of Texas at Arlington

B

University of Bielefeld Wayne State University (Detroit)

D

G

L

Gulf Energy and Environment, Inc. University College London

Reference code A1 A2 A3 A4 A5 A6

B1

B2 DI D2 D3 D4 D5 D6 D7 D8 GI

LI L2 L3 L4

L5 L6 L7 L8 L9 L10 LI 1 S

T

University College of Swansea University of Toronto

S1

s2 s3 s4 TI T2 T3 T4 T5 T6

Reference Burciaga et [ I / . (1977) Coleman and McNutt (1979) Coleman et cil. (1979) Coleman et n / . (1980a) Coleman and Hutton (1980) Coleman er ril. (1981) Wilson (1978) Sinapius el id. (1980) Kauppila et a / . (1976a) Kauppila et cil. ( I977a) Stein et a / . (1978) Dababneh el c d . (1980) Kauppila et ril. (1981a) Kauppila et ril. ( I98 1b) Stein ef [ I / . (1981) Hoffman er ril. (1982) CosteUo et cil. (1972a)

Canter et etl. (1972) Canter et ril. (1973) Canter et N I . (1974a) Coleman er rr/. (1974) Coleman et a / . (l975b) Coleman et a/. ( 1 9 7 5 ~ ) Coleman er e l / . (1976a) Griffith and Heyland (1978) Griffith er ( I / . (1979a) Charlton et ril. (1980b) Charlton er ril. (1980~) Dutton et cil. (1975) Brenton et ril. (1977) Brenton er 01. (1978) Dutton et c r / . (1981) Jaduszliwer et a / . (1972) Jaduszliwer and Paul (1973) Jaduszliwer and Paul (1974a) Jaduszliwer and Paul (1974b) Jaduszliwer et a / . (1975) Tsai et a/. (1976)

TABLE I1 POSITRON-BEAM CHARACTERISTICS FOR TOTALCROSS-SECTION EXPERIMENTS

Laboratory"

Source

Moderator*

Gulf London Toronto Swansea Detroit Texas Bielefeld

PP 22Na '%a 22Na "C 22Na 22Na

T: AulAI B: MgO/Au T Admica' B: MgOIAu T Boron B: MgOIss B: Cu(0FHC)

(I

Energy analysis

AEFWHM

TOF TOF !WE

1-2

4

0.14E

ER

TOF TOF

(eV)

-1

-1 co.1

1.5 0.4 at 6 eV

References can be found in Table IB. T refers t o transmission, B t o backscattering, Au/Al to gold over aluminum, etc. Other moderators used include T: Ni, B: MgO/brass, B: MgO/Au.

Energy range (eV) 1-26 2-1000 4-300

13-1000 0.3-800 2-50 1-6

I0

Scattering region Ell

4 II

4 B ,I

4, -FF

Detector

(no./ sec)

2Y CEM 2Y 2Y CEM

Few 100 Few
y.

Y CEM

POSITRON-GAS SCATTERING EXPERIMENTS

59

case of the MgO-coated Au “venetian blind” backscattering moderator was used by Canter et al. (1972). A conversion efficiency of 3 x reported for a MgO-coated Au backscattering moderator by Canter et al. (1974b) in an experiment which detected efficient Ps formation on various solid surfaces. Although the MgO-coated Au moderator has a relatively high conversion efficiency, it has the disadvantage of emitting a rather broad e+ energy distribution (AEFWHM 2= 1-2 eV) possibly due to the charging of the insulating powder grains. The existence of a narrow, low-energy peak in the e+ energy spectrum emitted by a successful moderator was attributed in the early stages of moderator development (Costello et al., 1972b; Tong, 1972) to the thermalization of high-energy positrons in the moderator, and their subsequent ejection from the surface due to a “negative” work function for positrons in certain metals. However, typical energy widths of the lowenergy e+ peaks observed for the first few successful moderators were relatively broad (-1 eV), until Stein et al. (1975) observed a very narrow energy width (AEFWHM < 0.1 eV), low-energy peak in the energy spectrum of positrons emitted from their boron moderator which did, in fact, appear to be consistent with a thermal energy distribution at room temperature. Throughout the development of the moderators listed in Table 11, investigators have recognized that the conversion efficiencies of various moderators were sensitive to their surface conditions, but none of the experiments referred to in Table I1 was done with well-characterized surfaces. In the first experiment using atomically clean (submonolayer contaminated surfaces) in an ultrahigh vacuum (-1O-lO Torr), Mills et al. (1978) demonstrated that positrons implanted in a clean single-crystal target can diffuse back to the surface and be emitted as Ps or as free positrons. The positrons are emitted predominantly in a forward lobe (Murray and Mills, 1980), with a maximum energy interpreted as the positron negative work function of the surface (Murray et al., 1980). These observations by Mills and co-workers have led to the fabrication of new e+ moderators with very high conversion efficiencies and narrow energy widths. Mills (1980) has found that a clean single-crystal Cu( 11 1) moderator which has been exposed to H2S in situ, and cooled to 100 K, combined with a low self~ source has an efficiency of 1.5 x the highest absorption 5 s Ce+ reported efficiency for an e+ moderator. The narrowest energy width ( < O . 1 eV) achieved by Mills et al. (1978) with their single-crystal Al( 100) moderator is about the same as that achieved by Stein et al. (1975) with their boron-moderated “C source. The single-crystal moderators referred to above may be somewhat difficult to adapt to e+-gas scattering experiments, since the conversion efficiencies of such moderators may be affected by their exposure to thc

60

Talbert S . Stein and Walter E. Kauppila

various target gases admitted to the scattering region. There have been some recent advances in moderator technology in moderate vacuums (-lo-’ Torr) which could also be useful in future e+-gas scattering experiments. As an example, Dale et a / . (1980) find that a tungsten vane moderator (prepared by heating to 2200°C in a vacuum) is stable in air, and if used with a better geometry (a larger solid angle for stopping fast positrons), the estimated conversion efficiency would be 0.7 x However, its energy width is relatively broad (1-2 eV). As a result of the relatively low intensities of the e + beams used in scattering experiments, most groups use a rather long scattering region and an axial magnetic field (sometimes curved) to transport the e+ beam through that region. The laboratories (listed in Table 11) which do not use an axial magnetic field in the scattering region are Swansea, where the positrons move along a circular path in a transverse magnetic field (a Ramsauer type of system), and Bielefeld, where the positrons move in a field-free (FF) scattering region (except for a single axial magnetic focusing lens). For energy-analysis of the e+ beams, the Gulf, London, Texas, and Bielefeld groups use time of flight (TOF), Toronto uses a 90” electrostatic analyzer ( W E ) , Swansea uses a transverse magnetic field with beamdefining apertures (BJ, and Detroit uses a retarding electrostatic field (I&). In all cases except for Toronto, the method of energy analysis also provides some discrimination against the detection of scattered positrons. The e+ energies used in Q T measurements range from 0.3 eV at Detroit to of the slow 1000 eV at Swansea and London. The energy widths, AEFWHM, e + beams used in scattering experiments are typically about 1 eV or more, except for the Detroit energy width of less than 0.1 eV. Three methods used for detecting positrons are to ( I ) observe the two coincident annihilation gamma rays (2y) with two NaI scintillation counters, (2) observe one or both annihilation gamma rays (y) with a single NaI well counter, and (3) use a Channeltron electron multiplier (CEM). The detected primary beam currents (Z,) range from less than l/sec to more than 1OO/sec.

B. TOTALCROSS-SECTION EXPERIMENTS The basic experimental method used by all of the groups which have measured QT is to study the attenuation of the e+ beam as it passes through a gas scattering region. Under “ideal” experimental conditions, QT can be obtained from the expression

I =

~ o ~ ~ - ’ ~ L Q ~

(1)

61

POSITRON-GAS SCATTERING EXPERIMENTS

where lo is the detected beam intensity with no gas in the scattering region,l is the detected beam intensity with gas of number densityn in the scattering region, and L is the path length of the e+ beam through the scattering region. The Bielefeld and Detroit systems (shown in Figs. 1 and 2, respectively) represent two significantly different experimental approaches to measuring QT. However, they share a feature that distinguishes them from the other experiments listed in Table 11; namely, in each case, absolute total cross sections for positrons and electrons have been measured in the same apparatus, using the same technique. In the Bielefeld experiment (Sinapius et a / . , 1980) the fast positrons from a 22Na source strike an OFHC Cu tube moderator used in a backscattering mode, yielding slow positrons and secondary electrons, which are used for the respective QT measurements. When a fast positron passes through a thin scintillator foil on its way to the Cu moderator, light is emitted which, when detected by a photomultiplier tube (PMT), provides one of the timing marks needed for the TOF measurement. A second timing mark is provided by the detection of the projectiles (with a CEM) which have traveled through the straight, approximately field-free scattering region. The use of a weak magnetic lens helps to focus divergent projectiles through the exit aperture without increasing their path length in the scattering region by more than 3%. The pressure in the scattering region is measured by an ionization gauge. Sinapius et 01. (1980) have recognized the need to apply an average 18% downward correction to the e+-He measurements of Wilson (1978) due to scattering in the vicinities of the moderator and the accelerating exit lens. In the Detroit experiments (Kauppila et al., 1977b), the electrons for the e- QT measurements are produced by a type-B Philips cathode, which

,Scattering reglon

Extraction grldr 1 IDeflectlon Dhtes

-

I

\ Moderator

Source

/

/

Magnetic rhleld

Exlt

Ion gauge -c

I

lenr

FIG. I . Bielefeld experimental setup for measuring total scattering cross sections. (From Wilson, 1978.)

62

Talbert S . Stein and Walter E. Kauppila

Magnetic Shielding

limating Apertures

Retarding Element Channeltron

Detector

FIG. 2. Detroit experimental setup for measuring total scattenng cross sections. (From Kauppila el crl , 1976a.)

replaces the boron target used to generate the positrons for the e+ QT measurements. The gas number density in the scattering region is determined from pressure and temperature measurements using a capacitance manometer (MKS Baratron) and thermocouples, respectively. The increase in the effective path length of projectiles due to spiraling in the axial magnetic field of the scattering region has been estimated to be a maximum of 1% (Kauppila et nl., 1977b). The Detroit group has also checked for the possibility that the target gases being studied could directly affect e+ or e- emission from the respective sources and have found no measurable direct effects of this type with the gases for which Detroit has reported QT values.

POSITRON-GAS SCATTERING EXPERIMENTS

63

In total cross-section experiments, care must be taken to minimize the number of scattered positrons which reach the detector, since inadequate discrimination against scattered positrons during the measurement of I [in Eq. ( l ) ] will result in measured cross sections being too low. In TOF experiments, some discrimination against scattered positrons is provided by their longer flight times compared with unscattered positrons. The angular discrimination achieved by the TOF method depends on the geometry of the experiment (e.g., the length of the scattering region and the distance between the scattering region and the detector), the beam energy and energy width, and the timing resolution of the electronics. Griffith et ul. (1978) in a reassessment of the angular discrimination capabilities of the original London total scattering system, have determined that an additional source of angular discrimination arises from the fact that appreciably stronger axial magnetic fields exist in their positron-detection region than in the major portion of their scattering region. This arrangement of magnetic fields results in a “magnetic reflection” of positrons scattered at sufficiently.large angles in the forward direction. Griffith et al. (1978) estimate that at 2 eV, the experiment of Canter et al. (1972) was able to discriminate against all positrons elastically scattered at angles greater than lo”, rather than the estimate by Canter et al. (1973) of 40-55”. We suspect that the recent reassessment by Griffith et al. (1978) is incomplete due to their neglect of the effect of positrons which scatter in their 150mm-long detector magnetic field region (refer to Fig. 1 of Coleman et al., 1973), which accounts for more than 15% of the total length of their scattering region. Since positrons scattered in the detector region are very close to the detector, the ability of their TOF approach to discriminate against such scattered positrons would be significantly worse, and could appreciably degrade their overall angular discrimination. The Texas group (Coleman et d.,1980a) has estimated an upper limit of the angular discrimination of their TOF system to be about 20”. In non-TOF experiments, there are several different techniques which have been used to discriminate against scattered positrons. One method is to use a retarding electrostatic field between the scattering region and the detector which serves as a potential “hill” for the axial energy of the positrons. If the retarding potential is set sufficiently close to the actual positron beam energy, it is possible to have 100% discrimination against inelastic scattering for many target atoms (e.g., the inert gases) and partial discrimination against small-angle elastic scattering, depending on the retarding field and the e+ beam energy distribution. Another method of discriminating against scattered positrons is to use a well-collimated e+ beam that must pass through a small aperture between the scattering region and the detector. The geometry of the beam and exit aperture will

64

Talbert S . Stein and Walter E . Kauppila

then provide discrimination against small-angle scattering. The Toronto group (Jaduszliwer and Paul, 1973) relied on the use of collimators in their beam scattering region, variations in their axial magnetic field strength, and Monte Carlo calculations to compensate for small-angle elastic scattering, and the use of a retarding field to discriminate against inelastically scattered positrons. The group at Swansea (Brenton et NI., 1977) used a retarding field and beam-collimating apertures for discrimination in their Ramsauer-type experiment. Although the Bielefeld group (Sinapius ef d., 1980) used TOF for analyzing their beam energy, they employed the geometrical approach (i.e., a collimated beam with a small exit aperture) to obtain an angular discrimination estimated to be 7". The ability of the Detroit group to discriminate against scattered positrons by using a retarding field and well-defined e+ beam with a small exit aperture from the scattering region has recently been analyzed by Kauppila et ( I / . (1981a). Since their retarding potentials are generally within a few tenths of a volt of their e + beam energy for energies less than 100 eV and within a few volts for energies up to 800 eV, the Detroit group should have 100% discrimination against inelastic scattering for the inert gases where the minimum energy lost by inelastically scattered positrons is more than 5 eV. For e+energies from 1 to 20 eV, Dababneher 01. (1980) have estimated angular discriminations of 15-20" for their Kr and Xe measurements, while for energies above 100 eV, Kauppilaer al. (1981a) estimate angular discriminations of less than lo". The Detroit group (Kauppilaet a/., 1981a) obtains better angular discriminations (typically 5" for all energies) for their recent e- measurements than for their e+ measurements at the same energy in the same apparatus because the more intense (by a factor of 10 or more), and nondecaying e- beams permit better overall tuning conditions to be achieved. However, the earlier Detroit low-energy e- measurements (for He and Ar by Kauppila et d . , 1976a, 1977b) were made with tuning conditions similar to those used for the low-energy positron work of Dababneh et al. (1980) and should have similar angular discriminations.

111. Total Cross-Section Results A. INERT GASESAT Low ENERGIES

The QT measurements for e+-He collisions are shown in Fig. 3, along with the results of several theoretical calculations. All of these measurements are absolute except the normalized ( n ) values of Sinapius el d.

65

POSITRON-GAS SCATTERING EXPERIMENTS

o ~ ~ ' " 5 ' ' " ' ' 10" ' " ' ' 15" "

20

Positron Energy (eV)

F I G .3. Low-energy e+-He total cross-section measurements (symbols) and theoretical calculations (curves). The thresholds for inelastic processes are indicated by the labeled arrows: "n" refers to normalized results; only the name of the lead author for each set of reported results is shown in the figure [e.g., Coleman (1979) refers to Coleman t'i NI. (1979).1 (Unless otherwise specified, the same conventions will be used in all of the following figures.) (From Stein and Kauppila, 1982.)

(1980). The values reported by Sinapius et al., are the results of Wilson (1978) corrected for scattering in the moderator and accelerator regions of their apparatus, which lowered Wilson's results by an average of 18%. Most of the measurements lie in a fairly narrow band in Fig. 3. One of the interesting qualitative features is the observation of a RamsauerTownsend effect (a minimum in QT) by Stein et al. (1978) and Sinapius et d.(1980) near 2 eV, with a steeply rising cross section at the lowest energies. All of the theoretical results in Fig. 3 indicate the existence of a cross-section minimum. Cross-section minima such as this were first observed by Ramsauer (1921b, 1923), Townsend and Bailey (1922), and Ramsauer and Kollath (1929) for low-energy e--Ar, Kr, and Xe collisions. These minima were so deep that it appeared that the target gases were nearly transparent to the projectile electrons. The RamsauerTownsend minima arise from quantum mechanical effects associated with a net attractive interaction between the projectile and the target atoms. Another interesting feature of most of the experimental data in Fig. 3 is the noticeable increase in Q T as the energy is increased above the Ps formation threshold, which shows up most clearly in the narrow energywidth measurements of Stein ef a/. (1978). The e+ experimental QTresults for Ne, Ar, Kr, and Xe are displayed in

66

Talbert S . Stein and Walter E. Kauppila

Positron Energy (eV) F I G . 4. Low-energy e+-Ne total cross section results. UN refers to the (2p-d) unnormalized results and N refers to the (2s-p) + (2p-d) normalized results of Montgomery and LaBahn (1970). (From Stein and Kauppila, 1982.)

Figs. 4-7, along with the results of several theories. In a qualitative sense, the shapes of the theoretical curves at low energies are quite similar to some of the experimental results. Stein et al. (1978) observe a rather deep Ramsauer-Townsend effect in Ne near 0.6 eV. Kauppila et al. (1976a) observe a shallow cross-section minimum in Ar near 2 eV. There are

McEachran (1979)

_--- Schrader (1979) _- Montgomery (1970) -__Massey (1966) Positron Energy (eV)

FIG.5 . Low-energy e+-Ar total cross-section results. (From Stein and Kauppila, 1982.)

POSITRON-GAS SCA'ITERING EXPERIMENTS

67

. . V

0

Dobobnrh (1980) Sinapiu, (1980)n Canter (1973) Mc Eochron (1980) Schrader (1979) Mmey (1966)

-_0

"

.

"

*

"

"

.

.

"

t

10 20 Positron Energy (eV)

30

FIG.6. Low-energy e+-Kr total cross-section results. (FromStein and Kauppila, 1982.)

dramatic increases in the measured QT values above the Ps formation thresholds for all the inert gases, illustrated most clearly in the narrow energy-width measurements made in Detroit (Kauppilaer ul., 1976a; Stein et d.,1978; Dababneh et ul., 1980). The theoretical cross section curves which extend above the Ps formation thresholds in Figs. 3-7 do not account for inelastic scattering, and thus are not expected to show the

Dobabneh (1980) (1980)n o Coleman (198Oa) v Canter (19740) McEochran (1980) ---- Schrodrr (1979)

+ Sinopiur

-

0

0 0

"

.

.

t

"

.

10

.

"

'

*

"

20

30

Positron Energy (eV)

FIG.7. Low-energy e+-Xe total cross-section results. (From Stein and Kauppila, 1982.)

Talbert S . Stein and Walter E. Kauppila

68

abrupt increases near the Ps formation thresholds which show up in the experimental results. The qualitative features of the low-energy e+ QT curves for the inert gases are summarized in Fig. 8 with the measurements of Stein rt (11. (1978) for He and Ne, Kauppilaet al. (1976a) for Ar, and Dababneh rt d. (1980) for Kr and Xe. For comparison, the corresponding e- Q.,.curves are also provided in Fig. 8, where the results of Ramsauer (1921b, 1923) and Ramsauer and Kollath (1929) for Ne, Ar, Kr, and Xe, and Milloy and Crompton (1977) for He, are used for energies below a few electron volts, and the results of Kauppilart crl. (1976a, 1977b) for Ar and He, Stein et a/. (1978) for Ne, and Dababneh et al. (1980) for Kr and Xe are displayed at the higher energies. It is interesting to note that the situation regarding the existence of Ramsauer-Townsend effects in the inert gases is nearly reversed for positrons compared with electrons in the sense that positrons exhibit Ramsauer-Townsend minima only for the lighter inert gases (He, Ne, and possibly a shallow minimum for Ar), whereas electrons exhibit Ramsauer-Townsend minima only for the heavier inert gases (Ar, Kr, and Xe). Another interesting observation that is apparent when comparing the general nature of the e + and e- curves shown in Fig. 8 is the dramatic

OO

5

I0

15

20

Porllron Energy (eV1

25

10

20

3

Electron Energy (eV1

FIG.8. Total cross-section curves for low-energy e+-inert gas and e--inert gas scattering. The arrows (in the order of increasing energy) refer to the thresholds for positronium formation, atomic excitation and ionization for e + scattering, and atomic excitation and ionization f o r e - scattering. (From Stein and Kauppila, 1982.)

POSITRON-GAS SCATTERING EXPERIMENTS

69

increase in Q T at each of the lowest energy inelastic thresholds (due to Ps formation) for e+ scattering and the lack of any noticeable changes in QTat the lowest inelastic thresholds (due to atomic excitation) for electrons. The e+-He curve also indicates a noticeable increase in the slope at the threshold energy for atomic excitation by e+ impact which was first pointed out by Coleman et al. (1975a) in an analysis of the data of Canter et al. (1973). When making these comparisons, it is important to realize that narrow resonances (not shown in Fig. 8) exist near several of the inelastic thresholds for e- scattering, while none have yet been observed for e+ scattering (see Section VI). The e + curves in Fig. 8 can be used to obtain estimates of the Ps formation cross sections (Qps) (crosshatched regions) for e+ energies between the thresholds for Ps formation and atomic excitation, assuming that the elastic scattering cross sections are smoothly varying as the e+ energy increases through the Ps formation thresholds. (For a further discussion of QB refer to Section V,A.) It is not a simple matter to ascertain which e+ QT measurements are the most reliable. Two recent review articles (Griffith and Heyland, 1978; Griffith, 1979) discuss errors in several of the experiments represented in Figs. 3-7. Although experimental groups will often make estimates of potential systematic errors in their experiments, the estimated magnitudes of such errors may be incorrect, and there may be systematic errors which are overlooked that are as large or larger than those which are painstakingly considered. An approach initiated by the Detroit group (Kauppila et ul., 1976a) is to make the corresponding measurements for each gas with electrons and positrons using the same experimental approach and system. An advantage of this approach is that most of the potential systematic errors should equally affect the e- and e + measurements. The Detroit e--He results (Kauppila et al., 1977b) are within a few percentage points of several other sets of absolute measurements (Milloy and Crompton, 1977; Kennerly and Bonham, 1978; Blaauw et al., 1980) and theoretical calculations (Callaway et al., 1968; Yau et al., 1978; Nesbet, 1979; Fon et al., 1981). The Detroit e- measurements for the other inert gases (Kauppila et d.,1976a; Stein et al., 1978; Dababneh et al., 1980) have been compared with other e- measurements and appear to be reliable. The absolute e--inert gas QT measurements of Sinapius et al. (1980) and some normalized e--He and e--Ar Q T measurements of Charlton et al. (1980a) are also in quite good agreement with the results discussed above. It is informative to examine the Q T results for low-energy e+-inert gas collisions, keeping in mind the e--inert gas results referred to above. For the e+-He case, Fig. 3 indicates that there are several calculations that agree quite well with the experiments in the vicinity of the RamsauerTownsend minimum. The variational calculation of Campeanu and Hum-

70

Talbert S.Stein and Walter E . Kauppila

berston ( 1977) and an exchange adiabatic approximation calculation of Massey et a/. (1966) remain in quite good agreement with several of the experiments u p to the highest energies of overlap. Since the calculation of Campeanu and Humberston is likely to be the most elaborate of all the above calculations, it is of interest to consider comparisons between this theory and the various experiments. Considerable attention (Campeanu and Humberston, 1977; Humberston, 1978; Humberston and Campeanu, 1980) has been devoted to trying to explain the discrepancy between their calculations and the measurements made in Detroit (Stein et a/., 1978; Kauppila et d.,1976b) and Texas (Burciaga et a/., 1977; Coleman et ul., 1979) for energies below 6 eV, where the experiments are lower. Humberston (1978) contends that the Detroit and Texas results are both too low below 6 eV due to the neglect of positrons scattered through small angles and that the measurements of Canter et d.(1973), which are claimed to agree with Campeanu and Humberston (1977) below 6 eV, are more accurate in the vicinity of 2 eV. Humberston (1978) deduces that at 2 eV, the discrepancy between the results of Campeanu and Humberston (1977) and those of Stein et a / . (1978) could be explained by an angular discrimination of 12", while an angular discrimination of 20" could explain the discrepancy with the results of Burciaga et a / . (1977). In order to determine if differing angular discriminations could provide a consistent explanation for the discrepancies between the results of Campeanu and Humberston (1977) and the various experimental results, we have calculated the percent error introduced into the total elastic for e+-He and e--He collisions due to variscattering cross section ous values of angular discrimination at various energies. The results for positrons, summarized in Fig. 9, were obtained using the s-wave phase shifts of Campeanu and Humberston (1977), the p-wave phase shifts of Humberston and Campeanu (1980), the lowest set of d-wave phase shifts of Drachman (1966), and the higher phase shifts (up to 1 = 20) from the formula of O'Malley e f a/. (1961):

(eel)

61

=

~k'P/((21+ 3)(21 + 1)(21 - 1))ao

(2)

whereP is the static dipole polarizability (1.3830:) of the target atom (He) and k is the wavenumber of the projectile. The corresponding percent errors for e--He collisions, also shown in Fig. 9, were obtained using the s-, p-, and d-wave phase shifts of Callaway et a / . (1968) (from their "EP" calculation) and the higher phase shifts (up to 1 = 20) from the formula of O'Malley et d. (1961) given above. Figure 9 clearly illustrates that the percent error in Qe,(and therefore in QT)for a particular angular discrimination reaches a maximum for e+-He collisions in the region of the Ramsauer-Townsend minimum (about 2 eV). An experiment with a large

POSITRON-GAS SCATTERING EXPERIMENTS

71

40 s wove IHIB-Cornpconu (1977)

Q .c

g 30

-

p wove lH5)-Hurnberston (1980) d wow-Drachman 11966) (3-201- OMolley (1961)

-

).

t,p,d wove Colloway (1968) I:(3--2O)-O'Malley (1961)

5

0 c .L

2

20

-

fi

8

10

FIG.9. Calculated percent errors (using the phase shifts listed) in measured e+-He and e--He total cross sections due to the neglect of small-angle elastic scattering for various angular discriminations. (From Stein and Kauppila, 1982.)

value for the angular discrimination could thus be expected to show a deeper Ramsauer-Townsend effect than actually exists. The angular discrimination in the experiment of Stein et al. (1978) was initially estimated to be about 13" at 2 eV. Figure 9 indicates that an angular discriminatioii of about 17"could explain the difference between the e+-He results of Stein et NI. (1978) and the theoretical results of Campeanu and Humberston (1977) near 2 eV shown in Fig. 3. The more recent and more complete analysis of the angular discrimination for low-energy positrons by Dababneh ef a / . (1980) (using the same apparatus as that used by Stein et a/., 1978), which gave an angular discrimination of 15-20' for e+ energies between 1 and 20 eV, is consistent with the 17" value required to explain the difference between the measurements of Stein et al. (1978) and the values of Campeanu and Humberston (1977) in the vicinity of 2 eV. The results of Canter et al. (1973) agree with those of Campeanu and Humberston (1977) below 3.5 eV, but are measurably higher than the curve of Campeanu and Humberston (1977) between 3.5 and 6 eV, and this is inconsistent with any explanation based on angular discrimination alone. Coleman et NI. (1980a) estimate that the angular discrimination of their apparatus is less than 20", which, taken with the percent error results shown in Fig. 9, would suggest a consistency of the e+-He results of Coleman et a/. (1979) with those of Campeanu and Humberston (1977),

72

Talbert S . Stein and Walter E . Kauppila

except for the lowest energies (<3 eV) where Coleman et d.41979) would still be too low. Of all the measurements, the corrected, normalized values reported by Sinapius et nl. (1980), who estimate an angular discrimination of 7, are in the best agreement with Campeanu and Humberston between 1 and 6 eV. A major part of the discrepancy between the e+-He measurements of Sinapius et 01. (1980) and Stein et d . (1978) could be due to the differing angular discriminations of the respective experiments. An observation which at first glance seems inconsistent with an angular discrimination argument is that the measurements of Stein et d.(1978) for e+-He above 7 eV are measurably higher than the curve of Campeanu and Humberston (1977), shown in Fig. 3. However, we found that the curve in Fig. 3 (obtained from the H14 curve of Fig. 1 of Campeanu and Humberston, 1977) was noticeably lower at energies above the RamsauerTownsend minimum than one which we obtained using the s-wave phase shifts of Campeanu and Humberston (1977), the p-wave phase shifts of Humberston and Campeanu (1980), the lowest set of d-wave phase shifts of Drachman (1966), and higher phase shifts (up to 1 = 20) from the formula of O’Malley et (11. (1961) referred to above, with only part of the discrepancy between the two theoretical curves being accounted for by the inclusion of the higher partial waves. The resulting theoretical curve is shown in Fig. 10, along with the measurements of Canter et (11. (1973) and Stein et a / . (1978). The percent error estimates shown in Fig. 9 indicate

FIG. 10. Low-energy e+-He total cross-section measurements of Canter i’f ctl. (1973) and Stein rt ( I / . (1978) compared with a theoretical curve obtained by using the phase shifts referred to in Fig. 9. (From Stein and Kauppila, 1982.)

POSITRON-GAS SCAmERING EXPERIMENTS

73

that a 15-20" angular discrimination (as estimated by Dababneh et al., 1980) would result in a remarkable consistency between the theoretical results in Fig. 10 and the measurements of Stein et al. (1978) not only at 2 eV, but also for all energies above 2 eV. The measurements of Canter et al. (1973) agree with the theoretical curve below 3.5 eV, are higher than the curve between 3.5 and 6 eV and lower than the curve between 7 and 13.6 eV. It is interesting to note from the percent error curves in Fig. 9 that a 15-20' angular discrimination for the low-energy e--He QT measurements would introduce less than a 5% error in those measurements from 0 to 13.6 eV in contrast to the e+-He case, which suggests that the excellent agreement of the Detroit e--He results with several other experiments and theories may be an indication that systematic errors in the Detroit measurements, other than that due to angular discrimination, are relatively small. (For some additional considerations of the e+-He situation, the reader is referred to Wadehra et al., 1981.) We have used calculations of phase shifts available in the literature to make percent error (in Qel) estimates (Table 111) for various angular discriminations and projectile energies for positrons colliding with the heavier inert gases (Ne, Ar, Kr, and Xe) in order to search for some consistent patterns in the measurements of QTfor these gases. Although we have only displayed the results obtained from one particular set of theoretical phase shifts for each gas in Fig. 9 and in Table 111, we computed percent errors using different sets of theoretical phase shifts and found that the general trends and magnitudes of the percent errors were surprisingly insensitive to the particular set of theoretical phase shifts which we used, provided that we used ten or more phase shifts, supplementing the phase shifts obtained from a particular theoretical calculation with the higher phase shifts up to I = 10 or more obtained from the formula of O'Malleyer al. (1961) referred to above. Thus, the conclusions that we draw in the following discussion using the percent error results in Fig. 9 and in Table 111 would not be significantly affected by our particular choice of the sets of available theoretical phase shifts. There is a very intriguing consistency between the Bielefeld and Detroit experimental results when the percent error data in Fig. 9 and Table I11 are taken into account. With the exception of Ne, the measurements of Sinapius et al. (1980) are consistently higher than those of Detroit (Kauppila et al., 1976a; Stein et al., 1978; Dababneh et al., 1980) for each of the heavier inert gases, as was the case for He. Our percent error estimates suggest that a significant part of the discrepancies between the measurements of Sinapius et d.(1980) and Detroit for Ar (Kauppila et d.,1976a), Kr, and Xe (Dababneh et al., 1980) could be related to the differing angular discriminations of the respective experiments (-7" for Bielefeld and

Talbert S . Stein and Walter E . Kauppila

74

15-20" for Detroit) as was the case for He. Furthermore, our percent error estimates for the e+-Ne case indicate that the differing angular discriminations of the Bielefeld and Detroit experiments would not result in a noticeable discrepancy between their measurements above 2 eV. Figure 4 indicates that the Bielefeld e+-Ne results are in reasonably good agreement with the Detroit results (Stein rt nl., 1978)above 2 eV but are lower at the lowest energies. If the Detroit e+-Ne QTresults should be shifted upward TABLE 111 PERCENTAGE ERRORS I N Q P , FOR QT MEASUREMENTS D U ETO F I N I T E A N G U L AD RI S C R I M I N A T I O N " Error in System et-Ne

e+-Ar

e+-Kr

QPI

(5%) with angular discrimination (degrees)

Energy (eV)

5"

10

15

20

25

30

45

60

90

0.14 0.54 1.22 2.18 3.40 4.90 6.66 8.70 11.02 13.60

0.6% 1.8 1.1 0.6 0.4 0.3 0.3 0.2 0.2 0.2

2.4 6.8 3.8 2.0 1.3 I .o 0.9 0.8 0.7 0.7

5.2 13.9 7.5 3.8 2.4 1.8

8.7 22.2 11.4 5.5 3.3 2.4 1.9 1.6 1.5 1.4

13.0 31.2 15.3 7.0 4.1 2.8 2.2 1.9 1.8 1.8

17.6 40.4 18.7 8.1 4.6 3.1 2.4 2.1 2.2 2.5

33.3 64.4 25.2 9.5 5.0 3.7 3.7 4.5 5.8 7.5

49.1 79.7 26.7 9.8 6.4 6.8 8.6 11.4 14.7 18.4

75.1 88.3 30.3 20.6 22.7 27.5 33.3 39.4 45.4 51.0

0.14 0.54 1.22 2.18 3.40 4.90 6.66 8.70 11.02 13.60

0.3 0.8 1.7 2.3 2.5 2.5 2.6 2.6 2.5 2.5

1.3 3.2 6.5 8.2 8.7 8.8 8.7

2.9 6.8 13.3 16.4 16.8 16.5

8.5

8. I 7.8

14.8 13.8 12.8

5.0 11.4 21.5 25.6 25.4 24.1 22.3 20.3 18.3 16.4

7.6 16.7 30.4 35.2 33.9 31.0 27.8 24.5 21.4 18.7

10.5 22.6 39.7 44.4 41.3 36.5 31.6 27.0 23.1 20.0

21.5 41.2 65.1 65.9 55.3 44.6 35.9 29.7 26.1 24.5

34.2 58.8 83.2 76.4 59.2 45.8 37.9 34.6 34.8 37.2

60.3 83.7 %.8 79.3 62.0 55.7 57.0 61.3 66.1 70.1

0.14 0.54 1.22 2.18 3.40 4.90 6.66 8.70 11.02 13.60

0.3 0.9 1.8 2.4 2.8 3.1 3.3 3.3 3.4 3.3

1.3 3.5 6.8 8.6 9.8 10.6 10.9 10.9 10.7 10.3

2.8 7.4 13.8 17.2 18.9 19.7 19.6 18.9 17.9 16.7

4.9 12.3 22.3 26.8 28.4 28.5 27.4 25.5 23.4 21.2

7.4 18.0 31.6 36.6 37.5 36.4 33.7 30.3 26.9 23.9

10.3 24.2 41.0 45.9 45.4 42.4 38.0 33.2 28.9 25.3

21.1 43.8 66.5 67.0 59.5

33.7 61.8 83.8 76.5 62.8 51.8 45.2 42.9 43.8 46.6

59.8 86.1 95.6 78.9 66.6 64.0 66.7 70.6 74.0 76.3

1.5

1.2 1.1 1.1

15.8

50.5

42.3 36.3 32.7 31.5

75

POSITRON-GAS SCATTERING EXPERIMENTS TABLE 111 (continued) Error in Qel (%) with angular discrimination (degrees) System et-Xe

"

Energy (eV)

5"

10

15

20

25

30

45

60

90

0.14 0.54 1.22 2. I8 3.40 4.90 6.66 8.70 11.02 13.60

0.3%) 1.0 1.9 2.6 3.3 3.8 4.2 4.5 4.6 4.7

1.2 3.9 7.0 9.4 11.6 13.2 14.1 14.5 14.6 14.4

2.6 8.2 14.3 18.5 22.0 24.2 25.0 24.8 24.1 22.9

4.6 13.6 22.9 28.6 32.9 34.7 34.5 33.1 31.0 28.5

7.0 19.9 32.4 38.9 43.0 43.7 41.9 38.8 35.2 31.6

9.8 26.6 41.9 48.5 51.5 50.4 46.6 41.9 37.3 33.3

20.1 47.7 67.2 69.3 65.7 58.5

32.3 66.4 83.5 77.8 68.4 60.0 55.2 54.3 55.9 58.6

58.2 89.4 93.1 79.7 73.8 74.6 77.5 79.9 81.0 81.1

51.0

45.5 42.4 41.6

Sources of phase shifts used in these calculations:

e+-Ne: ef-Ar: e+-Kr: e+-Xe:

/ I

= =

/ = I =

0 t o / = 3, Schrader (1979); 0 to I = 6, McEachran ct d.(1979); 0 t o / = 6, McEachran ef c d . (1980): 0 to / = 6, McEachran et e l / . (1980);

/ = 4 to / = I = 7 to / = I = 7 to/ = I = 7 to / =

20, O'Malley et crl. ( l % l ) . 20, O'Malley et c d . ( l % l ) . 20, O'Malley ct id. ( l % l ) . 20, O'Malley et e l / . (1961).

in energy by a few tenths of an electron volt (as discussed in Section V,A), this would make the agreement between Bielefeld and Detroit even better. The measurements of London (Canter et al., 1973) for the heavier inert gases do not appear to fit any consistent pattern in relation to the other experiments. The results of the Texas group (Colemanet a/., 1979, 1980a) are in rather good agreement with Detroit for He, Ne (Stein et d . , 1978), and Ar (Kauppilaet d . , 1976a), and with London (Canter et al., 1974a)for Xe . On the basis of the consistency of the low-energy Detroit and Bielefeld e+ QTresults for the inert gases when the angular discriminations of those experiments are taken into account, we feel that there is a reasonable chance that the Bielefeld results (having an estimated angular discrimination of about 7") may be the closest of all the measurements to being correct within their restricted energy range (1-6 eV). However, it should be recalled that the Bielefeld measurements can be subject to an appreciable correction (Sinapius et a/., 1980) when QT depends strongly on the positron energy, and this could affect their reliability. The Detroit experiments should yield reliable QT values if a correction is made for their angular discrimination (estimated to be 15-20" for the low-energy e+ experiments). Using Fig. 9 and Table I11 to make such corrections, one finds, for instance, that at 2.2 and 13.6 eV in He, the results of Stein ef d.(1978) could be about 19 and 5% low, respectively; at 2.2 and 13.6 eV in Ne, the

76

Talbert S . Stein and Walter E. Kauppila

results of Stein et d.(1978) could be about 5 and 1% low, respectively; at 2.2 and 8.7 eV in Ar, the results of Kauppila et a / . (1976a) could be about 21 and 18% low, respectively; at 1.2 and 6.7 eV in Kr, the results of Dababneh et d.( I 980) could be about 18and 23% low, respectively; and at 1.2 and 4.9 eV in Xe, the results of Dababneh et ti/. (1980) could be about 19 and 29% low, respectively. Similar corrections for the Bielefeld work would range from less than 1% for Ne to about 8% for Xe. B. INERT GASESAT INTERMEDIATEENERGIES Experimental results for He, Ne, and Ar at intermediate energies are shown in Figs. 11-13 along with several theoretical calculations. The measurements reported by Griffith c't d.(1979a) were obtained by normalization of their results to the measurements of Coleman et d . (1976a) between 30 and 100 eV. The other measurements in Figs. 11-13 are absolute. Above 50 eV, the measurements of Toronto (Jaduszliwer et d., 1975; Tsai et d . , 1976), Swansea (Brenton ct d.,1977, 1978), and Detroit (Kauppila rt d . . 1981a) are in good agreement (generally within 10% of each other) for these gases. The normalized measurements of Griffith ct d.(1979a) are also in good agreement with the other results for He and Ne, except from 1

2

3

4

I

I

I

k (iia,,)

5

6

7

8

I

I

I

I

9

Koupp~Ia(198101

\ \\

o Brenlon 11977) A Coleman (19760) + Jodurzlwer (19751 v Conler (1973) -.-Byron (197B)EBS Dewongen (1977)DW Byron (1977MM lnahull (19751B*G,mel. Dcrongen 11977)DW,e lnohuli (1974)EB

-

I &-A 0'

I " ' "

' " ' I

'

' '

I

'

'

I

'

'

I

400 700 1000 Positron Energy (eV) Fib. I I . Intermediate-energy e+-He total cross-section results. The results displayed for Jaduszliwer C I trl. (1975) are interpolated values. The codes for the theoretical results are identified in the text. except for the Bethe-Born (BB) theory of Inokuti and McDowell (1974). (From Stein and Kauppila, 1982.) 20 50

100

200

77

POSITRON-GAS SCATTERING EXPERIMENTS k (i/ao)

1

2

-

k l

.

3

4

5

6

7

8

I

1

I

1

I

I

Kauppila (1981a) Coleman (1979) Griffith (1979a)n 0 Brenton (1978) 4 Coleman (1976a) + Tsai (1976) v Canter (1973)

4 N -

E

y

I

2-

* g o .

0

E

2

c

2

"

? I

:o

-

9

0 A

.

. O

+

f 0

I-

2

..

- ------ Dewangen (1977)DW Byron (197730M 0

-.-.-

c 0

g

lnokuti (1975)B.G,lnel; Dewangen (1977)DW,el

.

0

A

.

'

'

'

~.

Positron Energy (eV)

I ? . Intermediate-energy e+-Ne total cross-section results. (From Stein and Kauppila, 1982.) Fit,.

k(t/a,) 1 k I

2

3

4

5

6

7

I

I

I

I

1

I

'#

01

100

1

Kauppila (1981a) Coleman (1980a) A Griffith (1979a)n 0 Brenton (1978) 4 Coleman (1976a) + Tsai (19761 v Canter (1973) Joachain ( 1 9 7 n O M I

.

' *.*'

' ' * I * n

20 50

8

___-

'

'

"

200 400 Positron Energy (eV)

'

*

I

700

.

'

IOOC

F I G . 13. Intermediate-energy e+-Ar total cross-section results. OM1 refers to the method I "optical model'' calculation of Joachain ct ol. (1977). (From Stein and Kauppila, 1982.)

78

Talbert S . Stein and Walter E. Kauppila

200-400 eV for He, where Griffith et a / . are 10-15% lower. For Ar, the measurements of Griffith et d . (1979a) are about 10% higher than Tsai vt ( I / . (1976) and Kauppilaet cil. (1981a) below 100 eV, and more than 10% lower than Tsai et nl. (1976), Kauppila et t i / . (198la), and Brenton et r i l . (1978) above 200 eV. The higher energy results of Canter et r i / . (1973, 1974a) and Coleman et cil. (1976a) are lower than the other measurements, and this has been attributed to their inadequate discrimination against small-angle scattering. On the basis of these comparisons, it seems that the most reliable measurements for energies above 50 e V are those of Toronto (Jaduszliwer ef d . , 1975; Tsai et d.,1976), Swansea (Brenton ef N/., 1977, 1978), and Detroit (Kauppila et N I . . 1981a). The Detroit group has also made the corresponding e- measurements on the same gases (using the same apparatus and technique), which are in remarkable agreement with the recent measurements of Blaauw et r i l . (1980) and Wagenaar and de Heer (1980). The only significant differences between the e- measurements of these two laboratories could be explained by their differing angular discriminations for elastic scattering estimated to be less than I" for the Amsterdam group and typically 5-6" for Detroit. For positrons at energies above 100 eV, the estimated angular discriminations for the Detroit work are generally 6-8" for He, Ne, and Ar (Kauppila et d . , 1981a). Hence, the measurements made in Detroit are subject to corrections, which would increase their values by an amount depending on the nature of the differential elastic scattering of positrons, and the contribution of elastic scattering to QT. Using the e+-He differential elastic cross sections calculated by Byron and Joachain (1977) [using an optical model (OM) formalism], Kauppilaer t i / . (1981a) estimate that their e+-He measurements may be an average of 1% too low due to the neglect of small-angle elastic scattering from 100500 eV. In surveying the various e+-He theoretical results, Kauppila et t i / . (l981a) found that by adding the inelastic cross sections that can be calculated from the Bethe theory with an additional "gamma" term (related to the number of electrons in the target atom) of Inokuti et a / . (1975) and Kim and Inokuti (1971), and the elastic scattering cross sections calculated by Dewangen and Walters (1977) using a distorted wave second Born approximation (DW), QTvalues (designated B + G, DW in Fig. 1 I ) are obtained which agree to within 2% of their measurements above 200 eV. Brenton et t i / . (1977) are in good agreement with this composite theory up to their highest energy, 1000 eV. The theoretical calculations for Ne and Ar, shown in Figs. 12 and 13 are reasonably close to the measurements, but do not merge with them at the highest energies of overlap. For Kr and Xe, the measurements of Canter et nl. (1973, 1974a) and Coleman et a/. (1976a) are

POSITRON-GAS SCATTERING EXPERIMENTS

79

significantly lower than the recent results reported by Kauppila et NI. (1981b). C. POSITRON A N D ELECTRON COMPARISONS FOR THE INERT GASES Comparisons have been made by Kauppila et d.(1981a,b) of the e+inert gas and e--inert gas total cross sections up to 800 eV. For the He comparisons (up to 650 eV) shown in Fig. 14, the Detroit measurements 1981a) have been used for all e+ ener(Stein et a / . , 1978; Kauppila er d., gies and for e- energies above 2 eV, while the measurements of Milloy and Crompton (1977) were used for e- energies below 2 eV. Since the e+-He and e--He measurements (from 2 to 600 eV) shown in Fig. 14 have been made with the same experimental apparatus and techniques, most of the potential systematic errors should equally affect the e+ and e- measurements. The partial neglect of small-angle elastic scattering, on the other Energy (eV) 0 2 10

50

100

200

400

60C

F I G . 14. Comparison of measured e+-He and e--He total cross sections. The lowest inelastic thresholds for each projectile are indicated by arrows. (From Kauppila cf c d . , 1981a.)

80

Tulbeit S . Stein and Walter E . Kuuppila

hand, is a source of error which, in general, does not affect the e + and emeasurements equally, since it depends on the angular discrimination (estimated to be roughly 5" for all e- energies and ranging from 15-20" at low energies to 6- 10" at higher energies for positrons) and on the differential elastic cross sections. The e'-He comparison in Fig. 14 provides a striking illustration of some of the differences and similarities in e+ and escattering. At low energies the e + cross section is about two orders of magnitude smaller than the e- cross section. This is consistent with the fact that the static and polarization interactions are both attractive in the e- case, whereas there is a tendency toward cancellation of these interactions in the e+ case. In sharp contrast to the vastly different cross sections at low energies, there is an observed merging (to within 2%) of the e- and e + results above 200 eV. Kauppila e f d.(1981a) estimate that the maximum amounts by which their cross-section measurements could be too low due to the neglect of small-angle elastic scattering are 2% for electrons and 1% for positrons. The merging of the QT curves was not expected to occur at such low energies. The e +and e- distorted wave second Born approximation (DW) calculations of Dewangen and Walters ( 1977) do not merge (to within 2%) until 2000 eV, while the composite (B + G, DW) calculations of Inokuti r t t i / . (1979, Kim and Inokuti (l971), and Dewangen and Walters referred to in Section II1,B merge (to within 2%) at 1000 eV. At 200 eV, the DW calculations for electrons are 21% higher than the corresponding e+ calculations, while there is a 14% difference for the B + G , DW composite theory and a 21% difference for the eikonal Born series (EBS) calculations of Byron (1978). The comparison e' measurements of the Detroit group for Ne, Ar, Kr, and Xe (Kauppila et d . , 1981a,b) do not indicate any merging of the cross sections at the highest energies studied.

D. TESTSOF

THE

S U MR U L E

An aspect of total scattering of positrons and electrons by atoms that has received considerable attention in recent years is the question concerning the validity of the sum rule, which is based on the forward dispersion relations of Gerjuoy and Krall (1960). For scattering by the inert gases, in which the projectile and the target atoms do not form bound states, the sum rule has the form (Bransden and McDowell, 1969)

POSITRON-GAS SCATTERING EXPERIMENTS

81

whereA is the scattering length,fi andfEBare the first Born elastic scattering amplitudes in the forward direction for direct and exchange scattering, I, is the projectile wave number, all in atomic units, and Q T is the total scattering cross section in units of m i ; . By using available e+-He and e+-Ne QT measurements (which at the highest energies are extrapolated to the Born approximation) to evaluate the integral in Eq. (3) and theoretical results for the other terms in Eq. (3), it has been determined by Bransden and Hutt (1979, Byron rf (11. (1979, de Heer ef al. (1976), Hutt rt d.(1976), Tsai et (11. (1976), Brenton et (11. (1977), Griffith et al. (1979a), and Kauppila et d.(1981a) that the sum rule is valid when applied to e+-He, Ne scattering and not valid for e--He, Ne scattering. The sum rule tests by Kauppila et (11. (1981a) displayed in Table IV provide a summary of the current situation for e*-He, Ne, Ar scattering. In the latter work we feel there are possible uncertainties on the order of a few percentage points in the evaluation of the integral term for each projectile-target combination, which is based on a fit to the QT measurements of Kauppila et d.(1976a, 1977b, 1981a) and Stein ef d . (1978). Similarly there are small uncertainties associated with some of the other terms in Eq. (3) which get progressively larger as one goes from He to Ar (where there is no known calculation of J'E). From Table IV it is seen that the sum rule also appears to be valid for e+-Ar scattering as was first suggested by Tsai et (11. (1976). The apparent validity of the sum rule for positron-atom scattering and its invalidity for electron-atom scattering has been studied by Byron et al. (1975), de Heer ef (11. (1976), Hutt et NI. (1976), and Tip (1977), and is understood to arise from the nature of singularities in the exchange ampliTABLE IV SUM-RULE TESTSOR e'-He, Ne,

System e--He e+-He e--Ne e+-Ne e--Ar e+-Ar

.t'E

A

+ 1.178b - 0.48*

+O.P -0.614" 1.7h -3.0 to -4.W ~

AND

+0.796c -0.796' +3.21' -3.21' +9.7' -9.7'

Ar SCATTERING"

-A - .f!

+ 3.943'

+.tE

I .969 1.276 I .891 3.824

0 +5.32If 0 Unknown 0

-

12.7-13.7 ~

2.56 I .25 5.10 3.88 14.1 13.4

~~

From Kauppila ei id. (1981a). O'Malley (1977). Ho (1977). Campeanu and Humberston (1977). Naccache and McDowell(1974). Hutt C I d.(1976). McEachraner crl. (1978). O'Malley (1963). Tsai et ( I / . (1976). Hara and Fraser (1975). a

J

Talbert S . Stein and Walter E. Kauppila

82

tude which causes the sum rule to fail for electron scattering. No such difficulties arise for positron scattering since there are no exchange effects between the incident positrons and target electrons. E. MOLECULAR GASES The low-energy (up to 40 eV) QTmeasurements for e+-Hz, N2, and CO, scattering are shown in Figs. 15-17, along with a few available theoretical results. Earlier measurements by Coleman et al. (1974) for Hzare not included because they have been remeasured and reported by Griffith and Heyland (1978). The measurements of Charlton er al. (1980b) are normalized. The measurements for H2 shown in Fig. 15 indicate the existence of a broad minimum below 9 eV and a dramatic increase near the Ps formation threshold. Another process (besides Ps formation) which could contribute to the large increase is dissociation of Hz, which becomes energetically favorable near 8.8 eV (the H2dissociation energy is 4.48 eV). A fixed-nuclei approximation calculation of the elastic scattering cross section by Hara (1974) is in quite good agreement with the measurements. An adiabatic nuclei approximation calculation by Baille et d.(1974) is somewhat lower than the measurements but similar in shape. The QT measurements of Hoffman er d.(1982) for N2 (Fig. 16) also show a broad minimum with increasing cross sections at the lowest energies and near the Ps formation threshold. The measurements of Coleman et cil. (1975~) for N, are generally lower than Hoffman et a/. and do not indicate an I

I

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Hoffman (1982) Charlton (1980b)n v Coleman 0976b) -Hara (19741 --- Baille 0974)

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FIG.15.

0

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20 24 28 32 Positron Energy (eV) 16

I

I

36

40

Low-energy e+-HZtotal cross-section results. (From Hoffman et

(I/.,

1982.)

83

POSITRON-GAS SCATTERING EXPERIMENTS 1

I

1

1

I

-6 8 l

N

.II

s

7

4

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.

/ N E * A A

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32 36 40

16 20 24 28 Positron ‘Energy (eV)

FIG.16. Low-energy e+-N, total cross-section results. (From Hoffman et a / . , 1982.)

increase at the lowest energies. For e+-CO, scattering (Fig. 17) the meaare much lower (20-40%) than the surements of Coleman et a/. (1975~) results of Hoffman et a/. (1982),with the latter measurements revealing an interesting bump just above the Ps formation threshold. Comparisons of e*-H, total cross sections by Hoffman et al. (1982) are shown in Fig. 18. The e*-H, results merge for energies above 200 eV, but the merging could be fortuitous if there is appreciable small angle elastic scattering for Hz. Similar comparisons for e2-N, by Hoffman et al.’(1982) 15

-

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Hoffman 0 9 8 2 ) Charltan (1980b)n v Coleman ( 1 9 7 5 ~ ) A

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vv vvvvv~v~vvvv

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1

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1

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I

Positron Energy (eV)

FIG.17. Low-energy e+-COP total cross-section results. (From Hoffman el d..1982.)

84

Talbert S . Stein and Walter E . Kuuppilu

-0

1

2

3

4

5

6

7

k(l/ao)

FIG.18. Comparison of measured e+-Hzand e--Hz total cross sections. (From Hoffman et 01.. 1982.)

indicate that the e--N, results remain above the e+-N2 results up to the highest energies studied (700 eV).

IV. Differential Scattering Cross Sections Coleman and McNutt (1979) have recently reported the first measurements of differential cross sections for the elastic scattering of 2-9 eV positrons by Ar for angles from 20-60”. A schematic diagram of their 1980b) is shown in Fig. 19. In their experiment, apparatus (Coleman et d., slow positrons pass through a I-cm-long gas cell and then travel approximately 25 cm through an evacuated straight flight tube in a strong axial magnetic field to a detector (Channeltron electron multiplier). The larger the angle through which a positron is scattered in the gas cell, the longer its TOF will be in the axial magnetic field. By alternately admitting gas into and then evacuating the gas cell, TOF histograms such as those shown in Fig. 20 are obtained. A “tail” can be observed on the long-time side of the “gas” TOF spectrum, which is associated with detected posi-

POSITRON-GAS SCATTERING EXPERIMENTS

c

85

Solenoid

FIG. 19. Texas time-of-flight spectrometer for measuring differential cross sections for elastic scattering. (From Coleman rr u l . , 1980b.)

trons which have undergone forward elastic scattering. The “difference” spectrum is obtained by subtracting an appropriately adjusted “vacuum” TOF spectrum from the “gas” spectrum. Differential cross sections can be obtained by correlating the “difference” TOF spectrum with various angles of forward elastic scattering. The differential cross sections measured by Coleman and McNutt (1979) for e+-Ar collisions are compared

GAS 0

VACUUM

’DIFFERENCE

0- 200

i e

40“

* \

TCF (nsec)

FIG.20. TOF histogram with and without argon in the gas cell to illustrate the increase in flight time corresponding to elastic scattering at various angles. (From Coleman and McNutt, 1979.)

86

Talbert S . Stein and Walter E. Kauppila

10

SCATTERING ANGLE FIG.21. Differential e+-Ar scattering results for different positron mean energies. The error bars represent statistical standard deviations. (From Coleman and McNutt, 1979.)

with the calculations of Schrader (1979) (solid lines) and “scaled-down’’ calculations of McEachran et 01. (1979) (broken lines) in Fig. 21. The agreement between experiment and theory is reasonable. Coleman et crl. (1980b) have also measured differential scattering cross sections for e--Ar collisions in the same apparatus and using the same technique as used for their positron studies and obtain reasonable agreement with other recent e--Ar results.

V. Inelastic Scattering Investigations A. POSITRONIUM FORMATION CROSS SECTIONS The crosshatched regions of the e+ QT curves illustrated in Fig. 8 can be used to provide estimates of the Ps formation cross sections (Qps)in the

POSITRON-GAS SCATTERING EXPERIMENTS

87

inert gases. Such estimates assume a smooth extrapolation of the elastic scattering cross sections from below the Ps formation thresholds and also depend on the e+ energy calibration. Recent resonance searches by the Detroit group in Ar (Stein et al., 1981) described in Section VI indicate that the energy calibration used by the Detroit group could be low by a few tenths of an electron volt. This appears to be supported by the locations of the sharp onsets of Ps formation in the inert gases shown in Fig. 8, since the abrupt increases in QT occur at assigned energies which are a few tenths of an electron volt below the predicted Ps formation thresholds (indicated by arrows in Fig. 8). In order to compare the Detroit estimates of Qpswith other experimental and theoretical results, we have extracted QB curves for He, Ne, and Ar from the crosshatched regions in Fig. 8, but we have shifted the energy calibrations used by the Detroit group upward (0.2-0.4 eV) in order to match the observed and predicted thresholds for Ps formation. The Detroit QB results for He, Ne, and Ar, based upon the QT measurements of Stein et al. (1978) and Kauppila et al. (1976a), are shown in Fig. 22, where they are compared with several other sets of experimental and theoretical results. Charlton et al. (1980~)have recently made the first direct measurements of the energy dependence of the orrho-positronium formation cross section (Q,,-ps)in He, Ar, Hz, and CH, by passing a slow e+ beam through a scattering chamber and counting triple coincidences from the 3y decay of 0-Ps. In order to compare their Qo-Psenergy dependence with the other results shown in Fig. 22, we have normalized the relative He and Ar results of Charlton et al. (1980~)by scaling their values to match the Detroit curves at the respective excitation thresholds of He and Ar. We have also set the positron energy by adding 1.8 and 2.0 V to the “applied voltage” quoted by Charlton et al. for He and Ar, respectively, in order to match the predicted Ps formation thresholds with the observed onsets. Implicit in this normalization scheme is the assumption that the Qo-ps energy dependence would be the same as the Q, energy dependence. It is interesting that the Qo-psvalues indicated by Charlton ef al. (1980~)increase only up to roughly 4 eV above the respective Ps formation thresholds for He and Ar and then begin to diminish. Based on their direct measurements, Charlton et al. (1980~)have estimated the ratio of Q,,.ps(Ar)/Q,,-p,(He) for the peak values to be 6.5. The normalized Q,,-ps results of Charlton et al. (1980~)shown in Fig. 22 indicate a ratio of QB(Ar)/Qps(He)for the peak values of about 25, which is much larger than the ratio based on the direct measurements of the London group. The Qpsvalues of Coleman et al. (1975a) for He (based upon e+ lifetime and QT measurements) are in good agreement with the QPsvalues based upon the QT measurements of Stein et al. (1978) (and with the normalized values of

88

Talbert S . Stein and Walter E. Kauppila

FIG.22. Positronium formation cross sections for e+-He, e+-Ne, and e+-Ar scattering. The S and K curves are Q,., values based on the Qr measurements of Stein rt a / . (1978) and Kauppila el ( I / . (1976a), respectively: the closed circles are the QePj results of Charltonrt t i / . (1980~)normalized as described in the text; the open squares are estimates from Coleman t t id. (1975c), based on the QTmeasurements of Canter er a ( . (1974a); C refers to estimates by Coleman E I ( I / . (1975a), based on e + lifetime and QT measurements; the FBA (first Born approximation) and FOEA (first-order exchange approximation) calculations (for Ps formed in the IS state) are from Mandal er a/. (1980), where the suffixes “a” and “b” refer to “postinteraction” and “prior interaction,” respectively: the DWA (distorted-wave approximation) calculation is from Mandal et a / . (1979); F refers to a calculation by Fels and Mittleman (1969) using a projection operator technique and the Temkin form of the polarization potential; B, P, and NP refer to the Born approximation, distorted-wave approximation (with polarization), and distorted-wave approximation (without polarization) calculations, respectively, by Gillespie and Thompson (1977).

Charltonrr al. (1980~)up to about 22 eV), but end up being about twice as large as the normalized values of Charlton et al. (19804 at 24.5 eV, while their results (Coleman et id., 1975a) for Ne and Ar are much lower than the QPs values based upon the QTmeasurements of Stein et al. (1978) and Kauppila et id. (1976a), respectively. In a recent TOF investigation (not represented in Fig. 22) of inelastic e+-He scattering, Griffith et (11. (1979b) made estimates of QB (based on numerous assumptions) which indicate that Qpsreaches a peak value of about 0.48 x 10-l“ cm2,roughly five times the peak value indicated by the normalized results of Charlton et a/. (1980~).The position of the peak indicated by Griffith et a/. (1979b) is 10-15 eV above the Ps formation threshold as compared with 4 e V above that threshold for the results of Charlton er al. (1980~). None of the theories shown in Fig. 22 has a shape which matches the results of Charlton et al. (1980~)for He. It is also interesting to note the

POSITRON-GAS SCATTERING EXPERIMENTS

89

poor agreement of the different theories with each other and with the experiments in the case of He. One indication of the extent of the disagreements between theories and experiments in the case of He is that none of the available theories could be placed in reasonable locations on the graph of the experimental results without the use of the indicated scaling factors ranging from A5 to 10. Results of a first Born approximation calculation of QPs for He by Massey and Moussa (1961) have not been included in Fig. 22 because Mandal et 611. (1975) have indicated that numerical errors were made in that calculation. The Born approximation results of Gillespie and Thompson (1977) for Ne and Ar remain above all of the experimental values. The Born values of Qps for Ar are still rising at the highest energies studied by Gillespie and Thompson (8-9 eV above the Ps formation threshold) in contrast to the Q,,-ps observations of Charlton et d.(1980~).The distorted wave approximation (P, NP) calculations of Gillespie and Thompson (1977) for Ne indicate a threshold behavior similar to the QPsvalues based on the QTmeasurements of Stein et a/. (1978) but begin to level off just above 15 eV whereas the values of Stein et al. continue a rapid increase up to the excitation threshold of Ne (16.6 eV). For Ar the distorted wave approximation (P)calculation of Gillespie and Thompson (1977) indicates an energy dependence roughly similar to the Q,,+? results obtained by Charlton ef a/. (1980~)but the theoretical QPs values are much lower than the normalized values of Charlton et a/. ( 1980~). B.

EXCITATION A N D IONIZATION

CROSS SECTIONS

Using the TOF apparatus shown in Fig. 19 (the same apparatus used for measurements of differential cross sections), Coleman and Hutton (1980) have obtained lower bounds on total excitation cross sections for 23-31 eV e+-He collisions. In this energy range, well-defined secondary peaks were observed in the TOF spectra corresponding to positrons which have lost 20.6 eV of energy and have been scattered in the forward direction at angles less than 70". A TOF spectrum for an incident e+ energy of 25.8 e V is shown in Fig. 23. At incident e+ energies above 30 eV, a secondary peak associated with ionization overlaps the excitation peak, making it difficult to assign excitation cross sections. The secondary peak corresponding to a 20.6 eV energy loss indicates that in the projectile energy range from 23 to 31 e V the total excitation cross section is dominated by excitation of the 2lS state, and that there is appreciable small angle scattering associated with this excitation process. This work has recently been extended to Ne and Ar (Coleman et al., 1981),

90

Talbert S . Stein and Walter E. Kauppila 25.8 eV

t

1

* - I

TOF (nrrc) FIG.23. TOF spectrum showing primary (unscattered) and secondary (excitation) peaks for positrons with a mean incident energy of 25.8 eV colliding with He. (From Coleman and Hutton, 1980.)

and lower bounds on “excitation plus ionization” cross sections have also been measured for He, Ne, and Ar. In Fig. 24 we summarize the current state of affairs regarding the partitioning of e+ total scattering cross sections for He and Ar between the elastic and inelastic scattering channels. The curves shown in Fig. 24 are portions of the QT and the extrapolated elastic cross section curves from Fig. 8 [based on the results of Stein et a / . (1978) and Kauppilaet a / . (1976a)l. The increase in the Detroit results due to inelastic scattering should be reliable as a consequence of the ability of the Detroit experiments to discriminate 100% against inelastically scattered positrons. The Qo.psresults of Charlton et a / . (19804 (normalized as discussed in Section V,A), and the partial excitation and partial “excitation plus ionization” cross sections of Coleman et a / . (1981) have been added to the respective extrapolated elastic cross section curves. It is clear from Fig. 24 that if the extrapolations of the elastic scattering cross sections above the Ps formation thresholds are valid, then a large part of the inelastic scattering cross section for He and Ar is unaccounted for by the normalized 0-Psformation cross section measurements of Charlton et al. (1980~)and the partial excitation and partial “excitation plus ionization” cross-section measurements of Coleman at a / . (1981). Griffith et a / . (1979b) have also used their TOF system with a localized scattering region and a long flight tube to investigate intermediate energy e+-He inelastic scattering. From studies of the shapes and positions of secondary peaks observed in their “gas-in’’ TOF spectra, Grfith et a / .

POSITRON-GAS SCATTERING EXPERIMENTS

91

-’.O Positron Energy (eV) FIG.24. Inelastic cross sections for e+-He and e+-Ar scattering. The solid and broken curves are the total cross section and extrapolated elastic cross section curves from Fig. 8, respectively. The open circles represent the relative ortho-positronium formation measurements of Charlton er d . (198Oc) normalized in the same manner as described for Fig. 22 and added to the broken curves. The asterisks and plus signs represent the partial “excitation” and ‘‘excitation-plus-ionization” cross-section measurements, respectively, by Coleman and Hutton (1980) and Colemaner al. (1981), added to the broken curves.

have deduced that ionization is the dominant inelastic process in e+-He scattering between 100 and 500 eV.

VI. Resonance Searches Resonances have been predicted to exist for e+-H scattering just below then = 2 atomic excitation threshold (Doolen et al., 19781, and associated with the first excited state of Ps in the e+-H system (Doolen, 1978), and some possible theoretical evidence of a resonance in the e+-He system near 20.375 eV (above the Ps formation threshold and just below the first singlet excitation of He) has been provided by Ho and Fraser (1976). Up to the present time, there have not been any experimental observations of e+ scattering resonances. Stein et a/.(1981) are currently using their narrow energy width (<0.1 eV) e+ beam in a transmission experiment to search for e+ scattering resonances. Figure 25 shows measurements of the transmitted beam current versus the voltage applied to the e+ source over 1.0-V ranges centered near the Ps formation threshold (9.0 eV) and the

Talbert S . Stein and Walter E. Kauppila

92

50K

90K Lo

c

c

0

u)

4-

V

e

0

V

(b)

(a)

801

.5

9.0 Applied Volts

9.

40t

I

.o

L

11.5 Applied Volts

12.

FIG.25. Transmitted positron beam currents in Ar versus voltage applied to the positron source. The error bars represent statistical uncertainties. (From Stein et ( I / . , 1981.)

lowest atomic excitation threshold (1 1.5 eV) for Ar. These data were taken at 25-mV intervals, with the primary beam attenuated by about 50%. In the data for Ar shown in Fig. 25 there is no convincing evidence of a resonance (which would be expected to manifest itself as a relatively narrow structure in the transmitted beam current). There is also no evidence of resonances in preliminary searches with He and H2. However, there is still useful information that is obtained from these studies. An abrupt change in the slope of the transmitted current curve appears in Fig. 25a which should correspond to the abrupt increase in the total cross section at the Ps formation threshold (see Fig. 8). The slope change in Fig. 25a occurs at 8.8 V, whereas the Ps formation threshold is known to be 9.0 eV. This implies that an applied voltage of 8.8 V corresponds to an e+ energy of 9.0 eV in the scattering region, which is consistent with the observation that the Detroit e+ QTcurves in Fig. 8 appear to be starting their abrupt increases a few tenths of an electron volt below the predicted Ps formation thresholds. The curve in Fig. 25b does not show an appreciable change in its slope as the energy is increased through the atomic excitation threshold (1 1.5 eV) and this is supported by the corresponding QT curve for Ar in Fig. 8, which is quite smooth between I 1 and 12 eV.

VII. Possible Future Directions for Positron Scattering Experiments A glance at Table I conveys the feeling that there are many feasible, interesting e+ scattering experiments still waiting to be done for the first

POSITRON-GAS SCATTERING EXPERIMENTS

93

time. Even in the area of Q T measurements, there are a large number of obvious candidates for target atoms and molecules which have not been studied at all. Atomic hydrogen, and the alkali atoms would be of particular theoretical interest as target atoms due to their relatively simple structure. We have discussed several “second generation” experiments which have gone beyond Q T measurements. Although such efforts demonstrate the feasibility of investigating e+differential cross sections, inelastic scattering cross sections, and resonances, it seems that these areas can still be regarded as essentially “wide open” for improved experimental techniques and for the first measurements on many different collision systems. As an example, in the area of differential cross-section measurements, only one gas (Ar) has been studied, and it has only been studied in a very limited energy range (2.2-8.7 eV) and over a very restricted range of angles (20-60”). The studies of inelastic scattering up to the present time may have raised more questions than they have answered, and indicate a need for additional direct measurements of Ps formation, excitation and ionization cross sections.

ACKNOWLEDGMENTS We would like to thank Mr. Kevin Hoffman, Mr. Paul Felcyn, and Mr. Diab Jerius for their helpful assistance, and Ms. Evelyn Williams for typing the manuscript. We acknowledge, with gratitude, the support of the National Science Foundation for our Research program.

REFERENCES Anderson, C. D. (1933). Phys. Reit. 43, 491. Aulenkamp, H., Heiss, P., and Wichmann, E. (1974). Z. Phys. 268, 213. Baille, P., Darewych, J. W., and Lodge, J. G. (1974). Cun. J . P h y s . 52, 667. Blaauw, H. J., Wagenaar, R. W., Barends, D. H., and de Heer, F. J. (1980).J. Phys. B 13, 359. Bransden, B. H., and Hutt, P. K. (1975). J . Phys. E 8, 603. Bransden, B. H., and McDowell, M. R. C. (1%9). J . Phys. E 2, 1187. Brenton, A. G., Dutton, J., Harris, F. M., Jones, R. A., and Lewis, D. M. (1977).J. Phys. B. 10, 2699. Brenton, A. G., Dutton, J., and Hams, F. M. (1978). J . P h y s . E 11, L15. Burciaga, J. R., Coleman, P. G., Diana, L. M., and McNutt, J. D. (1977). J. Phys. B 10, L569. Bussard, R. W., Ramaty, R., and Drachman, R. J. (1979). Astrophys. J . 228, 928. Byron, F. W., Jr. (1978).Phys. Rev. A 17, 170.

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Byron, F. W., Jr., and Joachain, C. J. (1977). Phys. Rev. A 15, 128. Byron, F. W., Jr., de Heer, F. J., and Joachain, C. J. (1975). Phys. Re\*. Lerr. 35, 1147. Callaway, J., LaBahn, R. W., Pu,R. T., and Duxler, W. M. (1968). Phys. Re\,. 168, 12. Campeanu, R. I., and Dubau, J. (1978). J. Phgs. B 11, L567. Campeanu, R. I., and Humberston, J. W. (1977). J . Phys. B 10, L153. Canter, K. F., Coleman, P. G., Griffith, T. C., and Heyland, G. R.(1972).J. Phps. B 5, L167. Canter, K. F., Coleman, P. G., Griflith, T. C., and Heyland, G. R.(1973).J. Phps. B 6, L201. Canter, K. F., Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1974a). Appl. Phys. 3, 249. Canter, K. F.,Mills, A. P., Jr., and Berko, S. (1974b). Phys. Re),. Lett. 33, 7. Charlton, M., Griffith, T. C., Heyland, G. R., and Twomey, T. R. (1980a). J. Phys. B 13, L239. Charlton, M., Griffith,T. C., Heyland, G. R., and Wright, G. L. (1980b). J . Phys. B 13, L353. Charlton, M., Griffith,T. C., Heyland, G. R., Lines, K. S . , and Wright, G. L. (1980~).J. f'hys. B 13, L757. Chupp, E. L., Forrest, D. J., Higbie, P. R.,Suri, A. N., Tsai, C., and Dunphy, P. P. (1973). Nfitiire (Loridori) 241, 333. Coleman, P. G., and Hutton, J. T. (1980). Phys. Re\-. Lett. 45, 2017. Coleman, P. G., and McNutt, J. D. (1979). Phys. ReLv. Lett. 42, 1130. Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1973).Proc. R . Soc. L O ~ I ~Ser. O N A 331, 561. Coleman, P. G., Griffth, T. C., and Heyland, G. R. (1974). Appl. Phys. 4, 89. Coleman, P. G., Griffith, T. C., Heyland, G. R.,and Killeen, T. L. (1975a). J. Phys. B 8, L185. Coleman, P. G., Griffith, T. C., Heyland, G. R.,and Killeen, T. L. (1975b). J. Phys. B 8, L454. Coleman, P. G., Griffith, T. C., Heyland, G. R.,and Killeen, T. L. (1975~). Atom. Phps. 4, 355.

Coleman, P. G., Griffith, T. C., Heyland, G. R., and Twomey, T. R. (1976a). Appl. Phys. 11, 321. Coleman, P. G., Griffith,T. C., Heyland, G. R.,and Twomey, T. R. (1976b). Private communication as reported by Griffith and Heyland (1978). Coleman, P. G., McNutt, J. D., Diana, L. M., and Burciaga, J. R. (1979). Phys. Re\,. A 20, 145. Coleman, P. G., McNutt, J. D., Diana, L. M., and Hutton, J. T. (1980a). Phys. Re\,. A 22, 2290. Coleman, P. G., McNutt, J. D., Hutton, J. T., Diana, L. M., and Fry, J. L. (1980b).Reis. Sci. 1ri.striirn. 51, 935. Coleman, P. G., Hutton, J. T., Cook, D. R., Diana, L. M., and Sharma, S. C. (1981). Proc. l i l t . C q f . Phys. Electron. Atom. Collisions, 12th Abstr., p. 426. Costello, D. G., Groce, D. E., Herring, D. F., and McGowan. J. W. (1972a). Ctrri. J. Phys. 50, 23. Costello, D. G., Groce, D. E., Herring, D. F., and McGowan, J. W. (1972b). Phys. Rev. B 5, 1433. Crannell, C. J., Joyce, G., Ramaty, R.,and Werntz, C. (1976). Astrophys. J. 210, 582. Dababneh, M. S., Kauppila, W. E., Downing, J. P., Laperriere, F., Pol, V., Smart, J. H., and Stein, T. S . (1980). Phys. Rev. A 22, 1872. Dale, J. M., Hulett, L. D., and Pendyala, S . (1980). Surf. Interface Anal. 2, 199. Darewych, J. W., and Baille, P. (1974).J . f h y s . B 7, L1.

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de Heer, F. J., Wagenaar, R. W., Blaauw, H. J., and Tip, A. (1976). J . Phys. B 9, L269. Dewangen, D. P., and Walters, H. R. J. (1977).J. Phys. B 10, 637. Doolen, G. D. (1978). I n t . J. Quotit. Chem. 14, 523. Doolen, G. D., Nuttall, J., and Wherry, C. J. (1978). Phys. Rev. Lett. 40, 313. Drachman, R. J. (1966). f h y s . Rev. 144, 25. Dutton, J., Harris, F. M., and Jones, R. A. (1975). J . Phys. B 8, L65. Dutton, J., Evans, C. J., and Mansour, H. L. (1981). Proc. I n t . ConJ Phys. Electron. Atom. Collisions. 12th Abstr., p. 428. Fels, M. F., and Mittleman, M. H. (1969). PhyA. Rev. 182, 77. Fon, W. C., Berrington, K. A., and Hibbert, A. (1981). J. Phys. B 14, 307. Gerjuoy, E., and Krall, N. A. (1960). Phys. Rev. 119, 705. Gillespie, E. S., and Thompson, D. G. (1975). J . f h y s . B 8, 2858. Gillespie, E. S., and Thompson, D. G. (1977). J. Phys. B 10, 3543. Griffith, T. C. (1979). Adv. Atom. M o l . Phys. 15, 135. Griffith, T. C., and Heyland, G. R. (1978). Phys. Rep. 39, 169. Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1978).J. Phys. B 11, L635. Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1979a). Appl. Phys. 19, 431.

Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1979b). J . f h y s . B 12, L747. Hara, S. (1974). J . Phys. B 7 , 1748. Hara, S., and Fraser, P. A. (1975). J . Pliys. B 8, 219. Ho, Y. K. (1977). J . Phys. B 10, L149. Ho, Y. K., and Fraser, P. A. (1976). J . Phys. B 9, 3213. Hoffman, K. R., Dababneh, M. S., Hsieh, Y.-F., Kauppila, W. E.,Pol, V., Smart, J. H., and Stein, T. S. (1982). Phys. Rev. A 25, 1393. Humberston, J. W. (1978). J. Pliys. B 1 1 , L343. Humberston, J. W., and Campeanu, R. I. (1980). J . Phys. B 13, 4907. Hutt, P. K., Islam, M. M., Rabheru, A., and McDoweU, M. R. C. (l976).J. Phys. B 9,2447. Inokuti, M., and McDowell, M. R. C. (1974). J. Phys. B 7 , 2382. Inokuti, M., Saxon, R. P.,and Dehmer, J. L. (1975). I n t . J. Rodiat. P h p . Chem. 7 , 109. Jaduszliwer, B., and Paul, D. A. L. (1973). Can. J. Phys. 51, 1565. Jaduszliwer, B., and Paul, D. A. L. (1974a). Can. J . Phys. 52, 272. Jaduszliwer, B., and Paul, D. A. L. (1974b). Can. J. Phys. 52, 1047. Jaduszliwer, B., Keever, W. C., and Paul, D. A. L. (1972). Can. J. Phys. 50, 1414. Jaduszliwer, B., Nakashima, A., and Paul, D. A. L. (1975). Can. J . f h y s . 53, %2. Joachain, C. J., Vanderpoorten, R., Winters, K. H., and Byron, F. W., Jr. (1977). J. Phys. B 10, 227. Kauppila, W. E., and Stein, T. S. (1982). Con. J. Phys. 60,471. Kauppila, W. E., Stein, T. S., and Jesion, G. (1976a). Phys. Rev. Lett. 36, 580. Kauppila, W. E., Stein, T. S . , Pol, V., and Jesion, G. (1976b). Proc. I n t . Conf. Positron Annihilation, 4th (Helsingor A b s t r . ) p. 25. Kauppila, W. E., Stein, T. S., Smart, J. H., and Pol, V. (1977a). Proc. I n t . Cotif. Phys. Electron. Atom. Collisions, 10th Abstr., p. 826. Kauppila, W. E., Stein, T. S., Jesion, G., Dababneh, M. S.,and Pol, V. (1977b). Rev. Sci. lnstrrim. 48, 822. Kauppila, W. E . , Stein, T. S. , Smart, J. H., Dababneh, M. S.,Ho, Y. K.,Downing, J. P., and Pol, V. (1981a).fhys. Rev. A 24, 725. Kauppila, W. E., Dababneh, M. S., Hoffman, K. R., Hsieh, Y.-F.,Pol, V., and Stein, T. S. (1981b). Proc. I n t . Conf. Phys. Electron. Atom. Collisions, 12th Abstr., p. 422.

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Kennedy, R. E., and Bonham, R. A. (1978). Phys. Rev. A 17, 1844. Kim, Y.-K., and Inokuti, M. (1971). Phys. Re\,. A 3, 665. Leventhal, M . , MacCallum, C. J., and Stang, P. D. (1978). Astrophys. J . 225, L1 I . McEachran, R. P., Morgan, D. L., Ryman, A. G., and Stauffer, A. D. (1977).J. Pliys. B 10, 663. McEachran, R. P.,Ryman, A. G., and Stauffer, A. D. (1978). J . Phys. B 11, 551. McEachran, R. P., Ryman, A. G., and Stauffer, A. D. (1979). J. Pliys. B 12, 1031. McEachran, R. P., Stauffer, A. D., and Campbell, L. E. M. (1980). J. Phys. B 13, 1281. Mandal, P., Ghosh, A. S., and Sil, N. C. (1975). J. Phys. B 8, 2377. Mandal, P., Guha, S., and Sil, N. C. (1979). J. P/iys. B 12, 2913. Mandal, P., Guha, S., and Sil, N. C. (1980). Phys. Rev. A 22, 2623. Massey, H. S. W. (1976). Phys. Todcry 29 (3), 42. Massey, H. S. W., and Moussa, A. H. (1961). Proc. Pliys. Soc. 77, 811. Massey, H. S. W., Lawson, J., and Thompson, D. G. (1966).In “Quantum Theory of Atoms, Molecules, Solid State” (P.-0. Lowdin, ed.), p. 203. Academic Press, New York. Milloy, H. B., and Crompton, R. W. (1977). Phys. Rev. A 15, 1847. Mills, A. P., Jr. (1980). Appl. Phys. L e t t . 37, 667. Mills, A. P., Jr., Platzman, P. M., and Brown, B. L. (1978). Phyh. Rev. Lett. 41, 1076. Montgomery, R. E., and LaBahn, R. W. (1970). C m . J . f l i p s . 48, 1288. Total cross section values were provided by R. E. Montgomery and R. W. LaBahn, (private communication). Murray, C. A., and Mills, A. P., Jr. (1980). Solid Stccte Cornmrrn. 34, 789. Murray, C. A., Mills, A. P., Jr., and Rowe, J. E. (1980). S i y f . Sci. 100, 647. Naccache, P. F., and McDowell, M. R. C . (1974). J . Phys. B 7, 2203. Nesbet, R. K . (1979). J. Phvs. B 12, L243. O’Malley, T. F. (1963). Phys. Re\,. 130, 1020. O’Malley, T. F. (1977). Phys. Lett. 64A, 1%. O’Malley, T. F., Spruch, L., and Rosenberg, L . (1961). J. M d i . Pliys. 2, 491. Ramsauer, C. (1921a). A n n . Phys. ( L e i p z i ~ 64, ) 513. Ramsauer, C. (1921b). Ann. Phys. ( L c i p z i g ) 66, 546. Ramsauer, C. (1923). Ann. Phys. (Leipzig) 72, 345. Ramsauer, C., and Kollath, R. (1929). Ann. Phys. ( L e i p z i g ) 3, 536. Schrader, D. M. (1979). Phys. Re\*. A 20, 918. Sinapius, G., Raith, W., and Wilson, W. G. (1980). J . Phys. B 13, 4079. Stein, T. S., and Kauppila, W. E. (1982). I n “Physics of Electronic and Atomic Collisions” (S. Datz, ed.), p. 311. North-Holland Publ., Amsterdam. Stein, T. S., Kauppila, W. E., and Roellig, L. 0. (1974). Reiq. Sci. Instrrrm. 45, 951. Stein, T. S . , Kauppila, W. E., and Roellig, L. 0. (1975). PIiys. Lett. SlA, 327. Stein, T. S . , Kauppila, W. E., Pol, V., Smart, J. H., and Jesion, G. (1978). Phys. Re\’. A 17, 1600. Stein, T. S., Laperriere, F., Dababneh, M. S. , Hsieh, Y-F., Pol, V., and Kauppila, W. E . (1981). Proc. 1nt. Coi!f: Pkys. Electron. Atom. Collisions, l2th Abstr., p. 424. Tip, A. (1977).J. Phys. B 10, LII. Tong, B. Y. (1972). Phys. Rein. B 5, 1436. Townsend, J. S., and Bailey, V. A. (1922). Philos. M n g . 43, 593. Tsai, J-S., Lebow, L., and Paul, D. A. L. (1976). Ctrn. J . PAys. 54, 1741. Wadehra, J. M.. Stein, T. S., and Kauppila, W. E. (1981). J. Pltys. B 14, L783. Wagenaar, R. W., and de Heer, F. J. (1980). J . Phys. B 13, 3855. Wilson, W. G. (1978). J. Pliys. B 11, L629. Yau, A . W., McEachran, R. P., and Stauffer, A. D. (1978). J . Phys. B 11, 2907.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS VOL 18

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS J . MORELLEC. D . NORMAND. and G. PETITE Seri3ic.e d e Pliysiyiie des Atnnies et des Siirjuces Centre d'Eriides Niiclkiires de Sacloy G[f-.sirr-Yi.ette. Frcince

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . The Theory of Multiphoton Ionization . . . . . . . . . . . . . . . A . General Formalism . . . . . . . . . . . . . . . . . . . . . . B . The Method of Bebb and Gold . . . . . . . . . . . . . . . . . C . The Truncated-Summation Method . . . . . . . . . . . . . . . D . Implicit Summation . . . . . . . . . . . . . . . . . . . . . . E . Numerical Results: Discussion . . . . . . . . . . . . . . . . . F. Effect of Light Polarization . . . . . . . . . . . . . . . . . . G. Coherence (Photon Statistics) Effects . . . . . . . . . . . . . . H . Breakdown of Lowest Order Perturbation Theory . . . . . . . . Ill . Absolute Measurements of Multiphoton Ionization Cross Sections . . . A . General Considerations . . . . . . . . . . . . . . . . . . . . B . Principle of the Experiment . . . . . . . . . . . . . . . . . . C . Direct Method . . . . . . . . . . . . . . . . . . . . . . . . D . Saturation Methods . . . . . . . . . . . . . . . . . . . . . . E . Identification of the Ionization Process Investigated: Background Effects . . . . . . . . . . . . . . . . . . . . . . F. Recent Improvements in the Measurement of the Laser Intensity Distribution . . . . . . . . . . . . . . . . . . . . . . IV. Experimental Results: Comparison with Theory . . . . . . . . . . . A . Absolute Cross Sections in Linearly Polarized Light . . . . . . . B . Effects of the Laser Light Polarization . . . . . . . . . . . . . V . Destructive Interference Effects . . . . . . . . . . . . . . . . . . A . General Considerations . . . . . . . . . . . . . . . . . . . . B . Two-Photon Ionization of Cesium Atoms . . . . . . . . . . . . C . Three-Photon Ionization of Potassium . . . . . . . . . . . . . . VI . New Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Above-Threshold Ionization . . . . . . . . . . . . . . . . . . B . Angular Distributions . . . . . . . . . . . . . . . . . . . . . C . Double Multiphoton Ionization of Atoms . . . . . . . . . . . . VII . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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97 Copyright 0 1982 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 0-12-003818-8

98

J . Morellec, D . Normand, and G . Petite

I. Introduction This is a review of nonresonant multiphoton ionization processes with emphases on the experimental aspect. Following the review by Lambropoulos (1976), our prime concern here is on works published since then. We have tried to be as exhaustive and objective as possible; however, it is not inconceivable that some important contributions to the field have passed unnoticed. Such omissions should be considered as completely involuntary. Multiphoton ionization is merely one aspect of the extensive field of multiphoton processes. Its history can be traced back to the early 1960s, with the first predictions and theoretical treatments of multiphoton ionization (Geltman, 1963; Keldysh 1964), the first experimental observation being reported by Voronov and Delone ( 1965). The continued interest it has aroused ever since may be due to several reasons: (1) its importance in technological applications such as isotope separation, laser induced fusion, and gas breakdown; (2) the continuous challenge it has represented to theory concerning a large variety of problem such as high-order perturbation theories, nonperturbative treatments of atom-light interaction, or to experiment, as attested by the only recently solved problem of absolute measurement of multiphoton ionization probabilities; (3) the fact that it becomes, as pointed out early by Bunkin and Prokhorov (1964), the dominant process when intensity is increased. A typical multiphoton ionization process is represented in Fig. 1. In such a process, photons with energy too low to ionize the atom are "piled up" to reach the ionization threshold. The atom therefore has to transit through a succession of intermediate states. In the most general case, such intermediate states are not eigenstates of the atom, and they are therefore often referred to as "virtual" states. Although the question of the nature of the lifetime of virtual states is rather ambiguous, it may generally be considered to be very short, that is, on the order of one sec). Therefore, multiphoton ionization can be deoptical cycle (= fined as the ionization of an atom through simultaneous absorption of several photons. Bounds between multiphoton ionization and the laser intensity are represented in Fig. 2. It can be shown that the probability of an N-photon process is, in the framework of the lowest order perturbation theory, proportional to the Nth power of the laser intensity, and thus is represented in log-log coordinates by a straight line of slope N . Several of these curves, corresponding to different orders of multiphoton ionization processes, are represented in Fig. 2, and they show that when the order of the process is increased, the intensity required for their observation in-

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-7-

FIG.1.

Schematic diagram of a multiphoton ionization process.

creases. Multiphoton ionization processes have been observed for intensities ranging from lo7 W/cmZ(two-photon ionization of alkalis) to a few 1015 W/cmz (21-photon ionization of helium). The latter figure is close to the atomic unit of intensity and brings up the question of the limits of the validity of perturbation theory. W (sec-') A

105

107

10'

laser intenrity ( W cm-' )

Fib. 2. Multiphoton ionization probability as a function of the laser intensity for different Nth-order processes.

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Another cause for breakdown of the lowest order perturbation theory is the possible presence of resonances in the multiphoton ionization process. Resonances are due to the coincidence between the energy of an allowed atomic transition and that of an integer number of photons, or equivalently to the fact that an allowed multiphoton transition toward a given atomic state is energy conserving. Resonant multiphoton ionization processes have caused a great upsurge of interest in recent years. However, they will be excluded from the scope of this contribution, since they were reviewed recently (Georges and Lambropoulos, 1980). Section I1 is devoted to a survey of the results obtained when lowest order perturbation theory is applied to the problem of multiphoton ionization. In fact, most of the fundamental theoretical work on nonresonant multiphoton ionization is quite old and has been extensively reviewed by Lambropoulos ( 1976). Recent improvements in the theory of multiphoton ionization deal with situations where lowest order perturbation theory breaks down, namely, the case of resonant processes and that of very high laser intensities. Resonant multiphoton ionization is excluded from the scope of this contribution, and there is no experimental evidence of a breakdown of lowest order perturbation theory for the strongest laser intensities used in nonresonant multiphoton ionization experiments to date. Therefore, to obtain a reasonable self-consistency and also avoid unnecessary overlap between this review and that of Lambropoulos (1976), we only give a brief survey of the formal theory of nonresonant multiphoton ionization (NRMPI) and put the emphasis on the points which have recently been subject to decisive improvements as concerns the comparison between theory and experiment. The experimental part of this contribution is divided in two sections. Section I11 reviews a number of experiments which have been performed with currently available solid state lasers to measure absolute values of multiphoton ionization cross sections at a few selected wavelengths. The results of the different methods of determination of the ionization cross section are compared with the results of different calculations in Section IV. Section V is devoted to a different type of experiment, those performed with dye lasers whose ability to continuously scan wide ranges of wavelength allows one to demonstrate and measure cross-section interference minima occurring between two resonances. The effects of light polarization are also reviewed in Section V because they were expected in some theoretical works to be particularly dramatic around the interference minima (Lambropoulos and Teague, 1976; Declemy er cil., 1981; Delone er d.,1981).

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This contribution ends with Section VI, which is devoted to new trends appearing in the field. The measurement of the characteristics of the outgoing electron allows one to obtain experimental informations on the angular distribution of the differential cross section which cannot be obtained with the current ion detection technique. Experiments on the energy spectrum of the outgoing electron gave rise to a new line of experiments on above threshold ionization (ATI), a process in which the atom absorbs more than the minimum number of photons necessary for ionization. Both of these fields together with the question of double ionization, represent a possible future area for experiments on multiphoton ionization of atoms. Recent developments have been reported in fields closely related to multiphoton ionization such as the ionization of small molecules and multiphoton ionization of atoms at high pressures, where collective effects can be of importance. They have not been included because they require quite different theoretical treatment, and also because, with only little anticipation, one can think that they both could be the subject of a review by themselves.

11. The Theory of Multiphoton Ionization A. GENERAL FORMALISM

It is now a common statement that NRMPI is conveniently handled by lowest order perturbation theory. Both semiclassical and full quantum treatment of the multiphoton ionization problem are possible. Since we neglect the influence of spontaneous processes throughout this article, both have to lead to identical results. However, one may be more convenient than the other for a specific problem. Semiclassical formalism will, for instance, handle the problem of the influence of laser bandwidth more easily, whereas the full quantum mechanical treatment has proved to be very useful in the case of resonant multiphoton ionization. The latter will be presented here, since a large majority of authors seem to prefer it. The Hamiltonian of the system “atom plus field” is where H,A is the atomic Hamiltonian, H R is the Hamiltonian of the free

J . Morellec, D . Normand, and G . Petite

102

radiation field, and V is the atom-field interaction Hamiltonian. The radiation Hamiltonian can be written as

where for simplicity the laser field is supposed to be reduced to a single longitudinal mode of frequency w k , wave number k , and polarization A , and ( I & and (ikA represent the usual creation and anihilation operators (Messiah, 1965). The eigenstates of H R are the usual Fock states Inkh) (Messiah, 1965) defined by the occupation number of the mode ( A , A). The eigenstates of H,, are obtained as the direct products of an atomic state ) by a field state In ). At t = 0, the atom is supposed to be in its ground state lg) and the field populated by 17 photons, so that our initial state can be written as

The interaction Hamiltonian is, in most cases, given by the electric dipole approximat ion

where Ekh is the polarization vector of the laser mode and L is the dimension of the quantization volume ( h = c = 1 is assumed throughout this article). Suppose that the atom ionizes by absorption of N photons. The final state of the system will be

If)

where is an atomic state belonging to the continuum. If we neglect the influence of the ionization processes involving absorption of more than N photons, which is discussed at the end of this section, the ionization probability at the time t can be computed from the evolution operator of the system as

U(r) can be computed in different ways. An integral equation can be derived from the Schrodinger equation (Bebb and Gold, 1966), leading to

U'"(t) = 1 - i

1; V ' " ( t ' ) U ' " ( t ' )

dr'

(7)

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

103

where the operators are written in the interaction representation, that is,

vtryt) = p,,tVe-wIlr

(8)

U ( t ) can also be computed from the resolvent operator of the system (Goldberger and Watson, 1964) defined as a function of the complex variable Z by

G(Z) = ( Z - H ) - '

(9)

U ( t ) is obtained through

U(r) = (27ri)-1

I, -~zG(z) ~IZ r

(10)

For positive times, Z can be replaced by x + i q , and the integration carried out over X from += to -= and the limit taken for q + +O. Because of its ability to incorporate resonance effects in an almost straightforward way, the resolvent operator formalism has been used in most of the recent work on multiphoton ionization (Beers and Armstrong, 1975; Gontier and Trahin, 1979; Petite et a / . , 1979). In the case of nonresonant multiphoton ionization by moderate fields, an exact solution of Eqs. (7) or (9) and (10) is not necessary. NRMPI can be accounted for by computing the lowest order, nonvanishing term of the perturbation expansion of U(r),that is, the N-order term for a N-photon process. This is often referred to as the "weak field approximation." The resolvent operator can be expanded in power series of V G(Z) = G"(Z)[l

+

2 (VGO(Z))n] n=1

where Go(Z) = (Z - HO)-'

(12)

whose matrix elements can easily be computed. The N-order term in the expansion of U F I ( t )is then calculated from G$Y'(Z) = (FIGo(Z)(VGo(Z))NIZ)

(13)

through an integral similar to Eq. (10). It is then a matter of algebraic manipulation to obtain the N-photon ionization probability. For detailed calculations, the reader is referred to Lambropoulos (1976) or to the original articles. The ionization probability then is written as

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J . Morellee, D . Normand, and G . Petite

where the energies w I , w A, and wF incorporate the energies of the photons that “dress” the different atomic states. The summation is carried over ( N - 1) intermediate states, which are allowed to span the complete atomic spectrum. p ( w F ) represents the density of final states. A similar result could be obtained from Eq. (7) through an iterative procedure up to the orderN in V“’(Bebb and Gold, 1966). It is then an easy matter to separate the field quantities from the atomic quantities in Eq. (14). As long as the ionization probability remains small compared to unity, that is, far from the saturation of the ionization process, an ionization rate can be defined whose dependence toward the photon flux F (photons/cm2 sec) is W):)

=

gNF.V

(15)

where uN is the total generalized N-photon ionization cross section given by

I

uN = [ ( 2 7 ~ ( ~ ) ~ ” / / 4 r r ~ ] t IM>,S’p n Q w ~ dR,

(16)

where CY is the fine-structure constant, Q is the wave number of the outgoing electron, and the integration is over the direction of propagation of the outgoing electron; M # ) contains the dependence of w N on the atomic structure and is defined by

) r depends on the laser polarization, and u1 . . * aN-l are where r ( A= N - 1 pure (undressed) atomic states. The calculation of Ml;) is now the dominant problem in the determination of the ionization probability. Calculation of M)? will rest upon several approximations. First, for all atoms, except hydrogen, the wave functions and thus the transition matrix elements are not known exactly, and one has to resort to approximations such as the quantum defect method (QDM) or more or less precise model potentials for their evaluation. Another difficult task is to perform the summation over an infinite set of atomic states, including the continuum. This summation cannot be performed exactly (at least explicitly), and one has to resort to approximations. Different kinds of approximations with different levels of accuracy have been proposed, and we now review the main approaches which have been used.

B . THEMETHODOF BEBBAND GOLD If, in Eq. (17), the denominator can be factorized, since the summation runs over complete sets of atomic states, closure relations can be used,

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

I05

and the numerator thus takes the form of a simple&’-order matrix element between lg) and I f ) . This is achieved by using an “average frequency” W(v)independent of the state to replace the atomic frequency ( w ~ ,-, wg) in each summation. A little algebra shows that this average frequency is determined for the vth summation by

M:i) being given by

Further simplification can be obtained by assuming that there is a single average frequency w independent of the order v, so that N-1

N-1

Although the number of matrix elements to be evaluated has been greatly reduced by this procedure, the evaluation of the average frequency by Eqs. (18) and (20) still involves infinite summations. However, if approximate average frequencies can be found, which can be done by trial and error, for instance, the calculation ofM#’has been greatly reduced. Bebb and Gold (1966) have applied this method to the case of hydrogen and rare gases, and their results are compared later to those of more sophisticated treatments. As a further simplification Morton (1967) proposed to replace the numerators in Eq. (17) by an “average matrix element” by using one “effective energy level.” By this method Morton has calculated the multiphoton ionization generalized cross section for hydrogen, noble gases, and alkalis.

C. THETRUNCATED-SUMMATION METHOD Another way of circumventing the problem of the infinite summation in Eq. (17) can be based on the following remarks. Starting with a given atomic state, the matrix element of I . ( ~ ) decreases as one goes to higher states. Meanwhile, the corresponding energy denominator starts to increase after a certain state. Therefore, a rapid drop of the contribution of higher states to each sum can be expected, and a good accuracy can be obtained by considering only a limited number of atomic states. Moreover, it is usually possible to estimate the error introduced by this trunca-

106

J . Morellec, D . Normand, and G.Petite

tion by using sum rules. The dimension of the summation being kept within reasonable limits, there is a large choice of methods available for the determination of the matrix elements. Usually quantum defect method (QDM) (Bates and Damgaard, 1949; Seaton, 1958; Burgess and Seaton, 1960) can be used when suitable, but also model potential calculations or even experimental determinations of the oscillator strengths can be used with profit. The truncated summation method has been used by different authors, mainly in the case of alkalis (Bebb, 1966, 1967; Lambropoulos and 1976). Teague, 1976; Teague and Lambropoulos, 1976a,b; Teague et d., The three techniques we have reviewed thus far deal with the evaluation of the infinite sum in Eq. (17). Attempts have been made to calculate the multiphoton ionization probability avoiding the explicit calculation of this infinite sum. The two most successful approaches have been the use of the Schwartz and Tiemann technique, and the use of the Green’s function method. D. IMPLICITSUMMATION

I . Tlir Method

of’ Sdi iiwrt:

cirid

Ticnitrriri

This method is based on the implicit summation technique introduced by Dalgarno and Lewis (1955; Dalgarno, 1963) and later reformulated by Schwartz and Tiemann (1959). The principle of this method can be presented with a great simplicity using the resolvent operator formalism. Consider a two-photon ionization process. The two-photon ionization probability is computed from the matrix element of the resolvent operator,

where, since we are interested in lowest order processes, we have retained in V the part representing photon absorption only, denoted v-. Straightforwardly, one has

The determination of G ii)(Z)now rests upon the calculation of the matrix element

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

107

We define the ket I VL2)(Z)) by

Ivp’(z))= ( Z

-

H,)-’V-lg)

(23)

1 V P ) ( Z ) is now the solution of the inhomogeneous differential equation (Z - H,)IV;Yz)) = v - J g )

(24)

This method can be generalized to any order by an iterative procedure, If I V Y - ” ( Z ) ) is defined by )VbN-I)(Z)) (GoV-)“-21g)

(25)

I V k v ) ( Z) and therefore G # ) ( Z ) is determined by the following differential equation: ( Z - H , ) ( V p ( Z ) )= V-lVb“-’)( Z ))

(26)

The determination of I V ~ + ” ( Z )and ) , that of Gil)(Z), now rests upon the solution of a set of ( N - 1) inhomogeneous differential equations. It is worth noting that these equations are not coupled. However, they have to be solved by increasing order to obtain the equation giving G # ) ( Z ) . The formal simplicity of the method as presented above must not conceal the fact that most of the algebra, particularly all the angular algebra, still has to be performed in the ‘ Y ’ representation, leading to rather complicated calculations. 1970; Rapoport et d.,1969; Klarsfeld, 1969, It was shown (Zon et d., 1970; Karule, 1971) that in the case of two-photon ionization of hydrogen, analytic solutions can be found in terms of hypergeometric functions. Comparison with earlier results obtained through numerical solutions of the differential equation (Zernik, 1964) led to a very good agreement. The multiphoton case has been thoroughly studied by Gontier and Trahin (1968, 1971) in the case of hydrogen. Results are presented for multiphoton processes of order up to eight. In the case of two-photon ionization, their numerical results are in excellent agreement with the other works quoted above. For completeness, we should add to the above-mentioned work on two-photon processes calculations by different authors (Robinson and Geltmann, 1967; Mizuno, 1973; Choudhury and Gupta, 1974; Chang and Poe, 1974) in which the summation is carried out implicitly as in the Dalgarno and Lewis (1955) technique. In a recent work on the four-photon ionization of cesium (Crance and Aymar, 1980a; Aymar and Crance, 1979), the same technique is used to compute multiphoton bound-bound and bound-free matrix elements.

J . Morellec, D . Normand, and G . Petite

108

The Green’s function method is based on the following remark. If we isolate the uth summation in Eq. (171, it contains the expression

which can be written in the I’ representation

which is nothing but the single-electron Green’s function of the atom G (r, r’; R) evaluated at the energy R = w g + m k . G(r, r’; 0)can be found as the solution of the following differential equation: (H,4(r)- R)G(r, r‘; 0) =

-

&r

-

r’)

(27)

M).:’ can be written, using the Green’s function as the following N-fold integral : M);) =

I

d3rp . . .

I

~ ~ r ~ + ~ ( r rNPl; ~ ) r wg ~+ ~ (( Nr -~ 1,) q . )

x I’!$?IG(rN-1rN-2;Wg -k ( N

- 2)Wk)

Details on the algebra involved in this method can be found in Lambropoulos (1976) or in the original papers. As for the Schwartz and Tiemann method, analytical solutions can be found in the case of hydrogen. Karule (1971) has calculated the generalized cross section for the two-photon ionization of hydrogen. However, this method leads to almost inextricable calculations for processes of higher orders. The use of the Sturmian representation of the Green’s function (Karule, 1975; Maquet, 1977) has allowed generalized cross sections for orders up to 16 to be calculated, giving, in addition, valuable information on the effect of light polarization. Comparison between these results and those obtained by the method of Schwartz and Tiemann shows a very good agreement. For atoms other than hydrogen, where the Green’s function is not known analytically, phenomenological Green’s functions have been constructed (Zon et d . , 1970, 1972: Davydkin et d.,1971) using QDM. Results have been obtained in the case of He 2’sand He 2%. Manakov r t t i / . ( 1973) have calculated the multiphoton ionization cross-section of alkalis

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

109

by neodymium and ruby laser light, fundamental and frequency doubled, linearly and circularly polarized. Other work on alkalis has been published since (Flank and Rachman, 1975; Flank et d.,1976; Laplanche et d., 1976a; Manakov et d.,1978; Rachman et (11.. 1978). In a recent article on the two-photon ionization of Cs, Declemy ct nl. (1981) include the effect of spin-orbit coupling. Before concluding this review of the different methods of calculation of the multiphoton ionization probability, and turning to the presentation of numerical results, we would like to add some comments on the precision which can be expected from the different methods presented above. The first three methods (average frequency, average matrix element, truncated summation) are subject to limitations of two kinds: (1) how the approximate summation approaches the exact infinite summation, and (2) exactly how the atomic quantities (wave functions, transition matrix elements) are known. The last two methods (Schwartz and Tiemann, Green’s function) are subject to the second limitation only. Therefore, they are expected to give more accurate results. One can even say that, in the case of hydrogen, they should give the best possible results for multiphoton ionization generalized cross sections, as far as lowest order perturbation theory holds. E. N U M E R I C ARESULTS: L DISCUSSION

The purpose of this section is twofold: to give a reasonably complete review of the numerical values calculated for different multiphoton ionization generalized cross sections and to comment on the possible causes of the discrepancies among different results in an attempt to determine the advantages and the drawbacks of the different methods. In the discussion, we shall emphasize the results which can possibly be compared with recent experimental work, whose reliability, as far as absolute measurements of multiphoton ionization cross sections are concerned, has been considerably improved. The case of hydrogen obviously does not fall in this category. There are few experimental results on multiphoton ionization of hydrogen (Lu Van at d., 1973). Its interest lies in the fact that, exact solutions being available in this case, it is a good test for more approximate treatments which have to be used for other atoms. Table I presents the comparison between results obtained by Morton (1967), Bebb and Gold (1966), and Gontier and Trahin (1971) on multiphoton ionization of hydrogen in its ground state, for four different wavelengths (the emission wavelengths of the neodymium and ruby lasers, fundamental and frequency doubled). Re-

110

. I Morellec, . D . Normand, and G . Petite TABLE I

MULTIPHOTON IONIZATION RATESFOR ATOMIC HYDROGEN"

h

(8)

10,600 6943 5300 347 I

N

Morton ( 1967)

Bebb and Gold ( 1966)

Gontier and Trahin (1971)

I2 8 6 4

3.9(154) 7.1(99) 1.5(71) 1.5(42)

3 .O( 149) 3.6(96) 3.3(70) 6.6(43)

0.9(96) 1.25(69) -

~~

~

~~

" Numbers in parentheses indicate the power of ten by which the first number has to be divided to obtain the ionization rate.

sults obtained by Karule (1975) and Maquet (1977) are identical to those of Gontier and Trahin (1971). Only the last publication of different authors has been utilized. We have tabulated the ionization rate (in sec-I) corresponding to a laser intensity of 1 W/cm2. The numbers in parenthesis indicate the power of ten by which the first number has to be divided to obtain the ionization rate. Whenever comparison is possible, Bebb and Gold results are closer than Morton's to those of Gontier and Trahin, which were obtained without approximation and whose reliability is confirmed by the agreement between different results obtained by different methods of equivalent accuracy. This leads to the conclusion that Bebb and Gold results can be relied on, at least as an indication of the order of magnitude of the ionization rate. Comparison between the results of Bebb and Gold (1966) and Gontier and Trahin (1971) is shown in Fig. 3, where the ionization rate is plotted versus the laser wavelength, for six- and eight-photon ionization of hydrogen. The discrepancies between the two sets of results increase in the region where the ionization rate is minimum. This can be easily understood from the fact that the minima of the ionization rates are due to destructive interferences between several ionization channels. It is well known that, in such a case, a precise determination of the different transition amplitudes is necessary to account for the interference minimum. Table I1 shows results obtained by different authors on two-photon ionization of atomic hydrogen in its ground state. The quantity tabulated in W-' cm4. These results are obtained for linearly is ( u l l ) x polarized light. Very complete data on multiphoton ionization of hydrogen in various excited states can be found in Maquet (1975).' Different conventions have been used in the literature, essentially theoretical, to present the results on ionization cross sections. In general, throughout this contribution we have respected the author's choice. The two usually used are ( I ) the generalized multiphoton

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

111

a (10.98 15

I

\

i

10”

i 10”

\ lo10

I ‘

ii I(

5000

, ,

I

h

I

7000

5129 h8 69L3 7180 FIG.3 . Six- and eight-photon generalized MPI cross sections of atomic hydrogen versus Gontier and Trahin (1971); (- . -) Bebb and Gold (1966); the photon wavelength (A): (-) ) I = I is a reminder that the ionization process starts from the fundamental state I s of H.

Alkalis have received considerable attention from both the experimentalists and the theoreticians. Calculations in alkalis rest upon approximations such as QDM or model potential, but their atomic properties are well known, which makes them one of the best tools for comparison between theory and experiment. ionization cross section uN= W / F Nin units of cmzNsecN-l; (2) that used in Table I1 (and also in Fig. 3, and by many other authors). Here u is the usual cross section in cmx ( W I F ) , and ( u / P 1 )is an intensity-independent quantity of the same kind as u N ,but in units of W1-N anzN. Of course, these two representations are linked by the elementary relation uN= E Z - l (u/P”+)where E,, is the photon energy (J). The subscript notation has been reserved throughout to the generalized multiphoton cross section (1).

J . Morellec, D . Normand, and G . Petite

112

TABLE I1 TWO-PHOTON IONIZATION CROSSSECTIONS FOR ATOMIC HYDROGEN"

Bebb and Gold .A

(A,

925 975 I020 I100 1200 I300 1400 1600 1700

( 1966)

10 660

I10 70 68 84

Chan and Tang ( 1969) 51.5 67.5 4.01 630 128 84.5 91.5 103

Klarsfeld ( 1969)

Gontier and Trahin (1971)

55.2 4.05 580 128 84.5 91.4 102

55.2 4.05 580 128 84.5 91.4 102

Kristenko and Vetc hinkin (1976)

Laplanche (1976b)

Karule (1978)

1.29 61.7 55.2 4.05 580 128 84.5 91.4 102

2.04 49.2 70.8 4.04 642 128 84.7 91.8 103

2.03 49. I 70.9 4.0 642 I27 84.5 91.5 103

Pi

trl.

( u l f )x 10'' for two-photon ionization of atomic hydrogen (u = W / F is the ionization cross section). Bebb and Gold, average frequency method; Chan and Tang, Gontier and Trahin: Schwartz and Tiemann's method. Others: Green's function method (units, W-' cm4).

Table I11 compares results obtained on the ionization rate of alkalis by different authors for a set of commonly used laser wavelengths. Bebb (1966, 1967) used a truncated summation method and QDM; Manakov et d.(1973) used the Green's function method and QDM, and so did Laplanche ef d.(1976a) and McGuire (1981). Results were obtained by Manakov et d.(1978) using the Green's function method and a model potential instead of QDM. Other results have been obtained by Teague et a / . (1976) for the three-photon ionization of Cs and K at 6943 and by Crance and Aymar (1980a)and Aymar and Crance (1980) for the two- and four-photon ionization of Cs at 5300 and 10,600 A. These results will be commented upon in Section I11 as concerns their comparison with experimental results. A general comment on Table I11 is that the overall dispersion of the results quoted there is about one order of magnitude. This would be rather poor in the case of hydrogen, but given the uncertainty on alkalis oscillator strengths and the nonlinearity of the processes, this must be considered as a satisfying agreement in this case. The largest discrepancies are observed close to resonances (three-photon ionization of Cs at 6943 A) or in the vicinity of deep mimina between two resonances (five-photon ionization of Na at 10,600 A, two-photon ionization of Cs at 5300.A). It has been proven since that resonant multiphoton ionization requires a more sophisticated treatment than lowest order perturbation theory, and we have already mentioned that the regions of the minima require a precise knowledge of the atomic wave functions. This is particularly emphasized by the strong

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

113

TABLE 111 MPI GENERAl.17ED CROS5 SECTIONS FOR ALKALIS" Two-photon ionization generalized cross sections

Atom Li Na K Rb

cs

A

(A)

3471 3471 3471 5300 3471 5300 3471 4300

Bebb (1966) 7.8 (50) 6.0 (53) 3.1 (50) 5.1 (50) 2.9 (49) 2.6 (49) 9.0 (50)

('I

Manakov t d . (1973)

2.42 (49) 6.25 (52) 3.08 (50) 1.61 (49) 2.53 (50) I .08 (49) 2.52 (49) 7.41 (49)

Manakov d.(1978)

McGuire (1981)

1.14 (49) 5.52 ( 5 2 ) 7.24 (50) 1.12 (49) 4.16 (49) 1.55 (48) 4.16 (49) 1.55 (48)

1 .o (49) 4.0 (51)

('1

4.0 (49) 1.5 (50) 4.0 (49) 2.0 (49)

Three-photon ionization generalized cross sections

Atom Li Na K Rb cs

A

(A)

5300 5300 6943 6943 6943 6943

Bebb (1967) 9.0 (81) 4.3 (77) 7.0 (80) 9.0 (78) 6.3 (78)

Manakov (1973)

('I t i / .

2.33 (78) 9.33 (78) 1.87 (79) I . 14 (78) 9.57 (78)

Manakov r i d . (1978)

Laplanche (1976a)

McGuire (1981)

-

2.0 (81) 2.0 (81) I .o (77) 5.0 (80) 4.0 (78) 1.0 (77)

PI t i / .

7.14 (82)

-

1.05 (77) 2.00 (79) 3.68 (78) 1.02 (77)

9.31 (78) 3.72 (79) I . 17 (77) I .oo (77)

Four- and five-photon ionization generalized cross sections Atom

A

(A)

N

Manakov (1973)

i'r t i / .

Manakov (1978)

C! ( I / .

~~

Li Na K Rb

cs

6943 10,600 10,600 10,600 10.600 10,600

4 5

5

4 4 4

3.52 (106) I .37 (137) 3.96 (138) 4.38 (107) 1.32 (107) 5.16 (107)

5.14 9.18 2.14 7.14

(142) (108) (108) (108)

'' In unitspf cmZvsecN-l. Numbers in parentheses indicate the negative power of ten by which the first figure has to be multiplied 17.8 (50) reads 7.8 x

dispersion between the results obtained by the same authors (Manakov rt ol., 1973, 1978),in such cases, by methods which differ only by the potential used (QDM and Fues potential, respectively). Concerning McCuire's results (19811, it should be mentioned that, as stated by this author, due to

114

J . Morellec, D . Normand, and G . Petite

his particular use of QDM, his dispersion curves may be subject to an overall wavelength shift. This has especially important consequences in the areas of the curve which exhibit rapid variation, that is, those areas close to resonances and interference minima. Rare gases have also received considerable attention from experimentalists, but far less than theoreticians. This is due to their rather complicated structure, which makes them difficult to handle, at least, for the higher members of the series. However, both Bebb and Gold (1966) and Morton (1%7) have given results on different rare gases for a set of different laser wavelengths, and the comparison of their results is given by Morton (1967). A special case is that of helium. Three-photon ionization probabilities of helium in both the P S and 2's states have been calculated by Zon et crl. (1972) and more recently by Olsenet d . (1978), Lompre et a/. (1980), and Aymar and Crance (1980). Figure 4 shows the comparison between these last two results (identical) and Olsen's, and the agreement is found to be excellent. The results of Zon er al. (1972) agree rather well with these in the case of the triplet state (within 15%) but differ by about a factor of three in the case of the singlet state, which may be due to the presence of a two-photon resonance nearby which cannot be satisfactorily accounted for in lowest order perturbative treatment.

F. EFFECTOF LIGHTPOLARIZATION All the results quoted so far have been obtained for linearly polarized light. Number of authors have published results on the effect of light polarization, which will be reviewed in this section. The influence of the light polarization on the ionization probability arises from two different causes. First, the evaluation of the matrix elements in the numerator of Eq. (17) requires some angular algebra, and the Clebsh-Gordon coefficients which appear in this algebra will depend on the polarization index A . Second, all the intermediate transitions appearing in the ionization process are subject to selection rules on the angular momentum. For instance, dipole-selection rules for a single-electron atom will be, for a one-photon transition, AJ = 5 1 , O (0 c* 0 forbidden), A M = 0 for light linearly polarized along the quantization axis, and A M = 2 I for light circularly polarized, and propagating along the quantization axis, the plus (minus, respectively) sign corresponding to the right (left, respectively) polarization. It is an easy matter (with the help, for instance, of Kastler diagrams) to demonstrate that the differences between the M selection rules result in a limitation of the number of channels available for ionization by circular light compared with linear light. Calculation of the effect of light polarization on multiphoton ionization

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

t

lo8'

115

11111111111 u,350 4

14,400

14,440

Laser frequency ( cm-') (-)

FIG.4. Three-photon ionization generalized cross sections of He (2's)and He (2?3): Lompre et c d . (1980); ( + + + ) Olsen et d.(1978).

of hydrogen have been published by Zon et al. (1972), Gontier and Trahin (1973), Karule (1978); and Maquet (1977). In addition, Zon er a/. (1972) present results of the same kind on the 23S, and 2's states of helium. The case of alkalis has been treated by Manakov et al. (1973, 1978), Lambropoulos and Teague (1976), Teague er al. (1976), Laplanche et al. (1976a), and Declemy e t a / . (1981). Particular interest has been paid to the case of two-photon ionization of cesium in the region of the minimum of ionization between the 6P and 7P resonances (see Section V). Finally we should mention that polarization effects can lead to very exciting application of multiphoton ionization as a source of polarized electrons. G . COHERENCE (PHOTON STATISTICS) EFFECTS When writing the ionization probability in the form WN = uNFN,we have supposed that the laser field was monochromatic, that is, time independent. This, of course, is not the case in most of the experiments, and the influence of the laser coherence on multiphoton ionization has received much attention. However, apart from the problem of resonant

I16

J . Morellec, D . Normand, and G . Petite

processes, which is still under investigation, no new work has been published since the review of Lambropoulos (1976), to which the reader is referred for precise information. Let us just recall a few basic facts. Equation (15), the ionization probability, has been derived in the case of a purely monochromatic field, a condition seldom fulfilled in experiments. Real laser fields will exhibit amplitude and phase fluctuations which have to be averaged in the calculation of the ionization probability. I n general, we must therefore write the ionization probability as WN

=

uNGN

(29)

where GN is the Nth moment of the photon distribution, which can be computed from the diagonal matrix elements of the density matrix of the initial field state P,, by (Glauber, 1963a,b)

Equation (15) can then be considered as a special case of Eq. (29) when a single longitudinal mode laser (whose field is best represented by a Glauber state, or coherent state) is considered. If we consider instead the case of an incoherent (or “chaotic”) light, we can prove (Glauber, 1963b) G 7‘ = N I G C O H (3 1) * N It follows that incoherent light is N ! more efficient than coherent light when an N-order process is considered. This is due to the “bunching effect” associated with noncoherent light (Mandel and Wolf, 1965). This can be easily understood from the fact that, in a nonresonant multiphoton ionization process, the atom has to transit through a number of intermediate states whose “lifetime” is very short (less than sec) since they are not energy conserving. Therefore, the simultaneous absorption of N photons has a better chance to occur if these photons are bunched (incoherent light) than if they are not (coherent light). The two typical cases mentioned above are closely enough realized by single longitudinal mode lasers (coherent light) and lasers operating on a large number of modes (50 or more: chaotic light). For intermediate situations (a few modes, mode-locked lasers, etc.) one has to rely on specific models for the state of the field (Debethune, 1972). Figure 5 shows the behavior of the sixth-order correlation function of a laser field as a function of the number of modes in the case of stationary independent modes. It is a particularly noticeable property of multiphoton ionization that a n N-photon process can be used to measure the Nth moment of the field, and therefore provides an excellent experimental check of laser fields statistical models (Sanchez, 1975).

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

t

L

/

117

/

FIG.5 . Sixth-order moment of a laser field as a function of the number of modes of the laser for stationary independent modes (SIM).

H. BREAKDOWN OF LOWEST ORDER PERTURBATION THEORY

There are some physical situations where the lowest order perturbation theory is no longer a satisfying tool for handling multiphoton ionization. The most common of these situations is that of resonant processes. Since they are not within the scope of this article, we shall limit ourselves to a few indications and refer the reader to a review by Georges and Lambropoulos (1980). In the derivation of the expression (14) of the ionization probability from the expansion (13) of the resolvent operator, a number of terms have been neglected. This was done because these terms were oscillating at a frequency equal to the energy detuning from intermediate states, and therefore averaged to zero (Lambropoulos, 1976). This, of course, is no longer true in the case of resonant processes, and results in a divergence of bq. (14) when one of the denominator vanishes. Different approaches have been used to solve this problem. Two of them have been particularly successful: the method of projection operators (Gontier and Trahin, 1979; Petite et d.,1979) and that of the effective Hamiltonian (Beers and Armstrong, 1975; Crance and Feneuille, 1977; Crance, 1978), both applied to the dressed-atom model (Cohen-Tannoudji, 1967). It has been demonstrated that, in the case of resonant processes, it is generally impossible to

118

J . Morellec, D . Normand, and G . Petite

define an ionization rate. Moreover, two different types of resonances can be distinguished, depending upon whether the resonant state is more strongly coupled to the continuum (crossing resonances) or to the initial state (anticrossing resonances). Energy shift and width effects can be accounted for this way in a satisfactory manner. Alternating current Stark shift of atomic states also can have important effects in some cases of nonresonant processes. This may occur in the vicinity of the interference minimum between two resonances; the energy shift of the atomic states, as well as the corresponding modifications of the wave functions, can result in a shift of the position of the minimum analogous to the resonance shifts observed in resonant processes (Morellec et d., 1976; Grinchuk et d., 1975; Petiteet d., 1979) and have similar effects. Edwards and Armstrong (1980) predict deviations of the law of variation of the ionization probability on the laser intensity from the I .v law. This result was obtained by computing terms up to the sixth order of the laser field for a two-photon ionization process. They found that this effect could be detectable for laser intensities of a few hundred megawatts per square centimeter or higher. However, this has not been observed by Normand and Morellec (19811, even for an intensity of 40 GW/cm2,and the discrepancy probably results from interaction volume effects. The problem of multiphoton ionization under very high laser intensities is another example of the possible breakdown of the lowest order perturbation theory. This can be expected from the following simple argument. There is neither mathematical nor physical sense in treating the electromagnetic field as a perturbation when it exceeds the atomic field seen by the electron. There is a need for a nonperturbative theory of multiphoton ionization, and although no completely satisfactory theory has been built thus far, a few attempts have been made in this direction, which should lead in the not too far future to the solution of this problem. Gontier et ti/. (1976), in a fully quantized formalism, have developed a method of summation of all the proper diagrams contributing to a given ionization process. In this way, they are able to determine an analytic continuation of the perturbation series in regions where it does not converge anymore. A “resummed” resolvent operator obtained by this method has been successfully applied to the problem of resonant processes (Gontier and Trahin, 1979), and the influence of higher order terms on angular distribution has been studied with a similar method (Gontier tv d., 1975). The final results of such calculations are expressed as continuous fractions which have to be truncated, but it must be clear that such a truncation does not restrict the maximum order of the diagrams accounted for. The same authors (Gontier et d.,1981) recently published a generalization of the Schwartz and Tiemann method to the case of resummed

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

119

operators which should be a solution to the problem of ultrastrong fields, and although no numerical application of their method is available yet, there is a hope that such calculations will be possible in the near future. Another approach has been proposed (Chu and Reinhardt, 1977). It is based on the analytical continuation of the Floquet Hamiltonian (Shirley, 1965) of the atom plus field system, obtained through a dilatation transformation I' refe. Although no further work on this method has been published yet, it seems to be a promising approach to the problem of strong fields.

-

111. Absolute Measurements of Multiphoton Ionization Cross Sections A. G E N E R ACONSIDERATIONS L Although a large diversity exists in experimental arrangements reported in the MPI domain, some general features can be usefully recalled in a ' schematic representation (Fig. 6). The main purposes of this basic apparatus are as follows: (1) Preparing the target of gaseous atoms in the vacuum vessel. The vacuum pump reduces the pressure of residual gases to to Torr. Then the gas is introduced at a pressure whose upper limit of about Torr (-4 x 1013atoms/cm3)preserves the strictly independent behavior of the atoms involved in the interaction. When the target is an alkali atomic beam, the density is in general limited to 10°-1010 ~ m - ~ .

C

F I G .6. Experimental setup for a MPI experiment.

120

J . Morellec, D . Normand, and G . Petite

( 2 ) Ionizing the atoms in the middle of the interelectrode system. The lenses are placed inside or outside the vacuum cell, depending on their focal length. The choice depends on the intensity required at the focus for the MPI process investigated. Typical focal sections of cm’ and a few 10+ cm2 are obtained by focusing a single-mode laser radiation with cylindrical or spherical lenses, respectively (Cervenan and Isenor, 1974; Alimov et ( I / . , 1973; Lompre et a/., 1977; Normand and Morellec, 1980). This allows an increase of the laser intensity by a factor of u p to lo6. (3) Separating the electric charges by a transversal electric field. Classically, the ions are repelled toward a time-of-flight mass analyzer, and the ion signal is then amplified by an electron multiplier.

Parasitic surface emission of ions and electrons due to light reflections inside the vessel is limited by tilting the windows or using antireflection coated windows. However, the most efficient precaution against surface effects is collection over a small angular aperture a (Fig. 6). The photodiode and the calorimeter measure, respectively, the time duration and the laser pulse energy. The photometric arrangement allows one to determine the spatial distribution of the laser intensity. When experiments are devoted to qualitative studies of multiphoton ionization (MPI) such as the dependence of the ion production on the laser intensity, the influence of light polarization, or resonances on the ionization rate, they do not need the measurement of the spatiotemporal laser intensity distribution I ( r , 1 ) . In that case, one has only to detect an ion signal which is proportional to the number of ions N t and measures the laser pulse energy eL, whose spatial and temporal distributions are assumed to be constant from pulse to pulse. Conversely, the determination of MPI cross sections require quantitative measurements of N t and Z(r, t ) . Until the advent of single-mode Iasers, the rotating-prism, Q-switched lasers delivered incoherent pulses of bad reproducibility and large spectral width. This led to bad estimates of spatiotemporal intensity distribution, resulting in large errors in g N and obviously poor agreement with the theoretical values of mN then available (three-photon ionization of singlet and triplet metastable states of helium, Bakos et a/., 1970; two- and fourphoton ionization of potassium, Held et d., 1972; two-, three-, four-, and five-photon ionization of potassium and sodium, Delone et d.,1974; three-photon ionization of cesium, Evans and Thonemann, 1972). These experiments have been reviewed (Lambropoulos, 1976), and we shall focus instead on more recent results obtained under more precise conditions, mainly as concerns the laser coherence properties and the accuracy of laser intensity determination.

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

B. PRINCIPLE OF

THE

121

EXPERIMENT

I . Gencwrl Determincition

of the Ion

field

Using Eq. (15), the ionization rate, it is possible to determine the experimental ion yield, that is, the number of ions created by a given laser pulse. We assume that the dependence of the laser intensity on space and time can be factorized in the following way: I(r,

1) =

IMf(r)g(r)

(32)

where f(r) and g(t) are the spatial and temporal normalized intensity distributions and I,,, is the maximum laser intensity derived from the photon flux F by I,,, = F / E p where E p is the photon energy (J). Then, the number of ions created by the laser pulse can be written as Ni = noUNTNFNVN

where no is the neutral density, and V , and tion volume and time, defined by VN =

1,f”(

r) dv

7,

(33) are the Nth-order interac(34)

where the volume integral is over the whole space. The expression (33) of the ionization yield is valid only in the case where this yield is weak enough so that the depletion of neutral atoms can safely be neglected. Some experiments utilize the behavior of the ionization in the saturation region in order to deduce the ionization probability, a point which we discuss later. The measurements of Niand no are different for each experiment and will be described further (Section 111,D).The quantities T ~ ,Z, and V , appear to be measured in the same way in all laboratories: the Nth-order interaction time T, [Eq. (35)] is generally derived from the measurement of the time duration of the laser pulse T by means of a fast photodiode. Some experiments have been performed using picosecond laser pulses. In this case, more sophisticated equipment (in most cases, built around a streak camera) was used to measure the pulse duration; T is obtained from the pulse as follows: T = g ( t ) dr. The maximum laser intensity I,,, and the Nth-order interaction volume V , are determined by photodensitometry. Figure 7 shows isodensitograms of photos taken in different Z positions on the laser propagation axis in the +X

122

J . Morellec, D . Normand, and G . Petite 53

SL

1m m FIG.7. Isodensitometric mapping of the laser beam section for four different positions along the propagation axis in the region of the best focus.

vicinity of the best focus and for constant shot-to-shot laser energy (Morellec rf d., 1976). The energy which crosses any plane of the Z axis during one pulse is EL

=

eZ =

I

I(x, y ,

Z,

t ) d.r dy d f

(36)

el., the laserenergy, is measured by a calorimeter. From Eqs. (32) and (36)

we deduce EL = TIM

I

f(X,

Y , Z) dx dY

(37)

f(s,y , z) is a dimensionless factor. If the origin of the coordinates is chosen at the best focus, we havef(o, 0,0 )= 1 . Let

I

f ( x , y,

0)d.r

dy

=

So

(38)

The term So is obtained from the isodensitometry of the photo located at the best focus. The maximum laser intensity is then IM

=

ELITSO

(39)

Finally, the generalized multiphoton ionization cross section u,,,can be expressed with Eqs. (33) and (39) as a function of all the interaction parameters :

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

123

The Nth-order interaction volume VNhas been defined in Eq. (34). Contrary to the measurement of I,, which requires the densitometry of only one photo located at the best focus, the measurement of VN needs the processing of a set of photos between z = 0 and zmaxvalues for which the ion production becomes negligible. The distribution function of the ion production in a z plane is SN(z) = J.f"(x, Y , 2) dx dY

The Nth-order interaction volume results from the integration of SN(z): (41) The determination of I, explicitly requires the separation of the time and space distribution functions, as in Eq. (32). This condition was not fulfilled in the case of lasers Q-switched by rotating prisms in the first multiphoton experiments. Then, the direction of emission of the laser system changed during the pulse (Barjot, 1971), which proves that the spatial distribution of the intensity in the focal volume was undoubtedly linked to instant time t of the laser pulse. This resulted in underestimation of IM, leading to experimental cross sections much higher than the calculated ones. Conversely, the sharp reduction of the cavity losses affects homogeneously the laser beam section when the switching utilizes Pockels cells or saturable absorbers. This validates the separation of the time and space distribution functions of the laser intensity even in the case of non-Gaussian laser beams. 2 . It~jiiericeof the Coherence of the Laser Radiation The influence of the laser temporal coherence on the multiphoton ionization has been experimentally studied (Lecompte et al., 1974, 1975) and later interpreted so that the experimental results on multiphoton ionization were used to derive the statistical properties of the laser radiation (Sanchez, 1975). Varying the number of modes of the laser from 1 to 100, they showed a dramatic enhancement of the ionization probability for multimode fields. This experiment is the first example of multiphoton ionization used as a test in another field of physics. The spectrum of the laser used in this experiment was composed of an adjustable number ( L ) of evenly spaced lines, each line containing 10 laser modes. Figure 8 shows the results of measurements of the 11-photon ionization cross section as a function of the number of lines L (data points). The broken lines show the behavior of the eleventh moment of the laser intensity obtained in a model derived from the SIM model (Debethune, 1972) by substituting to the number of laser modes M the prod-

124

J . Morellec, D . Normand, and G . Petite frr

I

1

5

10 L

F I G .8 . Eleventh-order moment of a laser field, measured from the 11-photon ionization of Xe, as a function of the number of IinesL present in the laser spectrum. Each line contains ten modes. The broken lines are deduced from the independent domain model for = I , 3, and 10.

uct L M of the number of lines by an “effective number” of phase indeThe result of the experiment was = 3, noticeably pendent modes smaller than the number of modes per line (10). This experiment allows one to confirm the validity of the “independent domain model” for such lasers (Sanchez, 1975). In practice, the coherence of the laser radiation is taken into account by two different methods depending on whether the temporal distribution g(1) of the laser intensity is resolved or not by the detection system. The former case generally corresponds to experiments where the laser operates on a small number of modes (10 or smaller) so that the fluctuations of the laser intensity have a time duration longer than the time resolution of the commercially available photodiodes. Therefore the instantaneous laser intensity measured takes effectively into account the overintensities seen by the atoms. The multiphoton ionization cross sections can then be directly determined with Eq. (40). This procedure has been applied, for instance, to the case of a Ruby laser oscillating on 6- I0 adjacent modes (Lompre et d., 1980).

m.

a

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

125

In the latter case, only an averaged temporal distribution can be measured and therefore the measured values for the pulse duration T~~~~ and ~ ~ ) different from T and the Nth-order interaction time T N ( ~are~ respectively T N . However, in the case of multimode lasers with a large number of independent modes, uNcan still be determined with use of Eq. (40),on condition that the ratio T ~ / T Nbe evaluated with Eq. (31):

An example of application is given in the measurement of the two-photon ionization cross section of cesium atoms by a 50-axial-mode dye laser radiation (Morellec et d . , 1980). 3 . Iri.flirence of Resoti~iices

Another condition of applicability of Eq. (40)is the nonresonant character of the interaction. In the case of resonant processes, as defined in the introduction, it is clear that the energy denominators appearing in Eq. (14) are no longer intensity independent, and the simple form of the ionization rate of Eq. (15) is no longer valid. In this case, it has been shown (Morellec et t i / . , 1976; Petite et d., 1979) that the "effective order of nonlinearity" k (measuring the slope of the experimental ion yield curve in log-log coordinates) and the number of photons absorbed in the process ( N ) are no longer equal. In that case, uNdepends locally and instantaneously on the laser intensity I, and the definitions of the Nth-order interaction volume and time are meaningless. This has been the main reason of the observation of order of nonlinearity different from N in the first experiments of multiphoton ionization. This, along with the underestimation of I discussed above, led to experimental values of MPI cross sections much larger than expected from the theory. This situation came to an end with the advent of single-mode lasers oscillating at variable frequency.

4.

Slitiiuitioii

oj" thr Iwiizrrtion Process

Thus far, we have only considered the case of a nonsaturated process. It is characterized by the following inequality: uNF"'TN

<< 1

(43)

In this case, one can safely consider that the number of atoms concerned for the process remains essentially unmodified during the interaction. However, some experiments are based on the properties of the saturation

126

J . Morellec, D . Normand, and G . Petite

of the ionization process (Chin and Isenor, 1970; Delone et d.,1971; Cervenan et al.. 1975; Arslanbekov et d.,1975; Lompre et al.. 1980). When the neutral atom depletion cannot be neglected, the number of ions N icreated during the laser pulse is Ni

=

no

I

du[l - exp(-(TNF.VfYr)TN)]

(44)

N ican also be developed in power series of [uflJN1as wl

=

no([aNTNFN]VN - &[mNTNF"]2vzN

+ ((-

f .'

'

l)D-'!I.l!)

[aN7NFV1"VpXN)

(45)

where V p x Nis the ( p x Nbh-order interaction volume as defined by Eq. (34). The effect of the terms of order higher than 1 in u N T N F I \ ' is to reduce the rate of growth of the number of ions given by the first-order term. Two methods for measuring the MPI cross section can be used, depending upon whether or not the saturation is observed.

c. DIRECT

METHOD

It utilizes the nonsaturated ionization regime (for which k = N ) and consists of measuring the absolute values of all the parameters of the interaction ( N i , n o , F , V N ,T ~ ) and , introducing them into Eq. (40) to calculate u N This . method is described by Normand and Morellec (1980) for two- and four-photon ionization of cesium atoms. Two difficulties are specific to this method: the measurements of Niand no, which we describe now. The number of ions created in the interaction N iis linked to the detected signal by a calibration constant q which accounts for the collection efficiency and eventually the gain of the electron multiplier (EM). The high collection efficiency is due to: ( I ) a collection electric field of good homogeneity; ( 2 ) low ion losses on the high-transparency grid and in the timeof-flight separator; (3) the large aperture of the electron multiplier. The boundaries of the interaction volume are tested by moving the focusing lens and measuring the ion signal on the cathode of the electron multiplier as a function of the position of the lens. In addition, the length of the interaction volume measured is found to be in good agreement with the photometric V Ndetermination. On the other hand, a laser intensity value is found for which N i is measurable either directly on the cathode of the

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

I27

electron multiplier or at the output of the EM. This allows the determination of the E M amplification factor for realistic experimental conditions (possible modifications of the gain by contaminating alkali atoms on the electrodes can thus be taken into account). As concerns the neutral density no in the atomic beam,it is measured by a surface ionization detector which can be moved through the focal volume. The ionization efficiency and the ion collection efficiency are carefully checked, and no is estimated to be known with an accuracy of ? 12%. The results of the measurements and their uncertainties are summarized in Table IV for both cases of focusing by a planocylindrical and a planospherical lenses. The cross sections found in both cases are nearly the same and the values given in Table VI and compared to theory in Fig. Ila and c are the mean values between the results of the two sets of experiments. Another application of the direct method has been reported by Bakos and co-workers (1976) in the three-photon ionization of metastable helium atoms. This experiment faced three main difficulties: (1) The absolute values of the atomic densities of He 2s' and He 2S3 in the discharge afterglow were determined with an uncertainty of a factor 10. (2) The Langmuir probe measuring the variation of the plasma potential gave only an indirect determination of Ni.Its calibration was subject to an uncertainty factor of 2. (3) The laser used was Q-switched by a rotating prism and, as we have seen before, the maximum laser intensity, in this case, cannot be convincingly determined. TABLE IV COMPARED CHARACTERISTICS OF SPHERICAL A N D CYLINDRICAL FOCUSlNGS uN (crnZS

N

A (nrn)

T

(nsec) ~

2

528

19.1

f

0.6

T~

(nsec)

Focusing"

So (ern')

V N (cm3)

sec,v-')

C

(1.36 t 0.05)

(1.05 2 0.08)

(6.2 t 1.7)

~~

13.2 t 0.4

x 10-3

S

( 1 3 5 f 0.06)

C

(3.40 t 0.14)

S

(2.07

x 10-5

4

1056

25.5 t 0.8

12.9 t 0.4

x 10-3 2

0.08)

x 10-5

" C , cylindrical: S, spherical.

x 10-3

(7.7 -t 0.6) x 10-6 (4.2 f 0.7) x 10-4

(1.86 t 0.3) x 10-6

x

1 0 -5 0

(7.1 t 2) x

10-50

(8.4 f 3) x 10-lW

(6.6 t 2.4) x

10-109

128

J . Morellec, D . Normand. and G . Petite

This experimental setup, owing to its simplicity, seems well adapted to relative measurements such as resonant ionization studies (Bakos et cd., 1976, Fig. 4) but much less convenient to absolute cross-section measurements.

D. SATURATION METHODS Several methods were developed to take advantage of the saturated ionization regime. The simplest one utilizes the saturation bend on the ion yield curve for which the relation u ~ F--~1 is~verified, T ~ and it allows the determination of m N with only the measurement ofF and T ~ When . saturation occurs in its highest intensity region, the interaction volume expands and the departure from the straight line cannot be accurately located on the ion yield curve, especially for the low Nth-order processes. In that case, uncertainty affects the value of u N ,which must then be considered as only an estimate of the ionization cross section. A more sophisticated method was introduced to overcome the difficulty of precisely locating the onset of saturation (saturation point). It consists in fitting the experimental data (only F and T~ being determined in absolute values) to a curve which takes account of the depletion of neutral atoms in the focal region, calculated from N i / n o V ~= .r - $ X ' ( v z ~ / v ~ )f

(1/3!)X3(V3N/VN)

* . *

(46)

where .Y = U ~ T ~ F which . " , is a rewritten form of Eq. (45). The calculated curve log(Ni/noVN)= f [ l o g ( x ) ] must exhibit the same shape as the experimental curve log(ion signal, arbitrary units) = f[log(F)] and, in particular, the same curvature between the unsaturated and saturated regions. Hence, the fit between the two curves identified the F values with the u N r N F N values and allows one to extract the cross section u N ,r N being measured as mentioned before. If the laser intensity distribution is purely Gaussian, Chin et a / . (1969) have shown that the ratio V,,/V, can be determined analytically. In that case, in the determination of uN only the measurement of eL and TN is needed. Such a case was first investigated by Chin and Isenor (1970) and applied later by Cervenan and Isenor (1974) to measure reliably the three-photon ionization cross section of alkali atoms by a single-mode ruby laser beam. The uncertainties in v3are found to be around 25% for Na, K , and Rb and 60% for Cs, so that these 'cross-section values are among the most accurate reported until now on multiphoton ionization processes. However, the basic assumption of pure Gaussian shape in describing

129

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

the laser field in the focal volume does not always correspond to the real experimental conditions. In powerful pulsed solid state lasers, the near Gaussian beam from the oscillator is often altered in the amplifiers, in particular, as a result of thermal inhomogeneities in the rods leading to birefringence, interference, and lens effects. Furthermore, the calculation of the ion yield curve (Cervenan and Isenor, 1974), assumes an interaction volume without any limitation (useful for a gas or a saturated vapor). This condition is not realized, however, in most crossed-beam experiments, where the atomic beam generally limits the expansion of the interaction volume. The saturation method is no longer subject to these restrictions if a maximum laser intensity (IOI)and p x Nth-order interaction volume ( V N , V Z N ,... , V u N obtained ) from photometric measurements are introduced into Eq. (46) (see Section 111,B). This procedure described by Delone ef d.(1971) allows the measurement of the ionization cross section cr4 for cesium, for sodium (Arslandbekov ef d.,1975), and crI1 for Xe (Arslandbekov and Delone, 1976). The accuracy for u4 is about ?50%, whereas it decreases dramatically for cr5 and ull as the higher N values emphasize the uncertainties on the laser intensity determination. Recently, Lompre ef d.(1980) determined by a similar method the threephoton ionization cross section of metastable He atoms (He 2's).The apparatus for the creation and detection of the metastable atoms is represented in Fig. 9. The experimental conditions in this experiment were quite different from those in the Bakos er a / . (1976) experiment described above (see Section 111,C). The use of an atomic beam led to HeYatomic ~ , in the Bakos et a/. experiment, beam densities of 3 x lo3 ~ m - whereas in the discharge afterglow. In return the this density was about 10'O BEAM CHOPPERELECTRIC FIELD METASTABLE ATOM SOURCE

~.

VELOCITY SELECTOR

n--ION

DETECTOR

'

METASTABLE

EMITTING D l O M

I [DATA

READ

OUT

]

FIG.9. Experimental setup for the measurement of the three-photon ionization cross section of metastable helium (Lompre et a / . , 1980).

J . Morellec, D . Normand, and G . Petite

130

purity was much greater in the beam than in the discharge afterglow, and secured the identification of the ions produced. This experiment demonstrates the possibility of measuring the MPI cross section of a gas at a very low atomic density (3 x lo3 cm-3) in a molecular background (residual gases) of much higher density (107-108~ m - ~ ) . Table V summarizes the main features of the experimental methods for determination of g N ,their domain of applicability, and their respective advantages. The choice of the method depends essentially on the observation of both the saturated and the nonsaturated parts of the ionization TABLE V COhlPARlSON

OF T H E

Domain of applicability Direct method Below the saturation of ionization process Any laser light distribution in the interaction volume

Saturation methods (A) Need results in saturated and unsaturated Ionization regimes Gaussian laser light distribution in the interaction volume Volume unlimited by the target dimensions Saturation methods (B) Need results in saturated and unsaturated ionization regimes Any laser light distribution in the interaction volume

EXPERIMENTAL METHODSUSED Parameters to be measured

FOR T H E

UN

DETERMINATION

Advantages

References

Neutral density n o Interaction volume V , Interaction time rN Absolute number of ions N Maximum laser intensity I ,

The ions are created under the lowest laser intensity compatible with detectivity and background effects The intensity-dependent level shifts and broadening are minimized

Normand and Morellec ( 1980)

Interaction time T , ~ Ion signal proportional to N i Laser energy eL

Eliminates: The calibration of the detection system The measurement of the interaction volume The measurement of the neutral density

Chin and lsenor ( 1970) Cervenan and Isenor (1974) Cervenan 1’1 trl. (1975) Boulassier ( 1976)

Interaction time T , Ion signal -Ni Interaction volume V , Maximum laser intensity I,w

Eliminates: The calibration of the detection system The measurement of the neutral density

Delone 1’1 t r l . (1971) Arslanbekov 1’1 t r l . (1975) Arslanbekov and Delone (1976) Lompre i ~ ttrl. ( 1980)

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

131

process and the ability of measuring the interaction parameters. Cervenan d.(1975) found that their analytically calculated ion yield curves assuming a Gaussian laser light distribution (Method A of Table V) have in highly saturated regions slopes B and 2 for spherical and cylindrical focusings, respectively. As these slopes are found to be independent of the focal lengths and of the other beam parameters, they can be considered as criteria of applicability of this method. However, in practice, few experiments seem to have satisfied these criteria. In most experiments using atomic beams, the slope of the saturated region is smaller than 1, mainly because the expansion of the interaction volume, when the laser intensity is increased, is limited by the dimensions of the target (Delone er a / ., 1971 ; Arslandbekov et d., 1975; Normand and Morellec, 1980; Lompre et nl., 1980).

Pt

E. IDENTIFICATION OF T H E IONIZATIONPROCESS INVESTIGATED: BACKGROUND EFFECTS The atomic species investigated are always found with other atomic or molecular species as impurities which can also be ionized through multiphoton ionization processes. The experiment is generally carried out so that the ionization process investigated largely dominates the background of the other ionization processes. On the other hand, the ions produced pass through a mass spectrometer, which assures, if necessary, the separation of the good from the spurious ion species. Usually, a time-of-flight spectrometer is used. This consists of a field-free space in which the ions previously accelerated are delayed for a time proportional to V% ( M is the molar weight) and then collected by a collector plate or the cathode of an electron multiplier. The mass analysis naturally fails in separating identical ions created through different ionization processes. Fortunately, the parameters of interaction can be chosen to favor a particular MPI process with respect to other competing processes. Generally, the experimentalist confirms the identification (1) by checking the equality of rC the effective order of nonlinearity measured with the Nth order of the MPI process expected; (2) by observing the linear dependence of the ion signal on the neutral density in order to eliminate spurious surface ionization effects which are insensitive to the variations of the gas pressure or of the atomic beam intensity; (3) by choosing the laser intensity range which assures the predominant detection of the MPI process investigated (a perusal of Fig. 2 makes this obvious). An application of this means of selection by the laser intensity

132

J . Morellec, D . Normand, and G . Petite

for the particular case of competing cesium dimers and monomers MPI processes will be examined in Section V,A. F. RECENTI M P R O V E M E N T S I N T H E MEASUREMENT OF INTENSITY D I S T R I B U T I O N

THE

LASER

We have described the importance of the control and the measurement of the maximum laser intensity in the determination of absolute MPI cross sections. When the laser intensity distribution is found to be reproducible from shot to shot over all of the useful intensity range, the tedious photometric measurements based on isodensitometry of photographic plates can be made for only a few laser shots and carried over for the entire experiment. But, as mentioned by Lompre et nl. ( 1 9 7 6 ~the variations of pump power in laser amplifiers can sometimes induce variations of the minimum focal section S o , which can be attributed to thermal inhomogeneities and the nonlinear index of the laser rods. So has to be constantly NI

Nl arb units)

s b . units)

1oL

1oL

lo3

lo3

102

102

10

10

1

- 1 Loser intensity I l a r b unts)

FIG.10. Effect of the shot-to-shot variations of the focal spot on the four-photon ioniza3.4 f tion of Cs: (a) So is assumed to be constant, the slope of the curve is found to be X 0.5: (b) So is measured at each laser shot, which leads to X = 4.0 ? 0.1.

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

133

checked, which is incompatible with the time-consuming photographic method. The solution to this problem has been found in the introduction of a video camera-digital memory-minicomputer analysis system, which produces the laser intensity map at the focus of the lens within a response 1979) or 60 sec (Lee Smith et id., 1978). By time of 5 sec (Lompre et d., means of four equally spaced reflective planes in the focal region, a rough t distribution of I can even be displayed (Lompre et a / . , 1982), which allows locating the best focus accurately. Figure 10 clearly demonstrates the physical implications of this new tool; the corrections of up to 30% applied on So for a maximum variation of the laser intensity by a factor 10 places the experimental points on a straight line of slope k = 4 2 0.1, as theoretically expected, whereas the uncorrected values of N , as a function of IMare spread on a line of approximate slope 3.4 0.5. Such a system was first used for measurements of time resolved optical spectra (Schmidt et l i / . , 1976)and is also applicable to instantaneous quantitative determination of the temporal laser intensity distribution.

*

IV. Experimental Results: Comparison with Theory A. ABSOLUTE CROSSSECTIONS I N LINEARLY POLARIZED LIGHT

The experimental values of &c are presented in Table VI with their uncertainty range. They have all been obtained at fixed radiation frequencies of neodymium glass and ruby lasers and their second harmonics. Recent experimental data are excluded from this table: the two-photon ionization of cesium atoms in the continuous range of frequencies from 460 to 540 nm, which is studied in detail in Section V within the framework of antiresonance effects. For sake of clarity the experimental a$’ values forN = 2, 3 , 4 , 5 are compared to calculated values in Figs. lla-c. But before comparing theory to experiment, these diagrams suggest some strictly experimental observations: (1) The experimental values of d!) for alkali atoms are accurately determined, the error bars generally extending over less than half an order of magnitude. ( 2 ) The uncertainties on uhL)in the three-photon ionization of metastable helium atoms reported by Lompre et a/. (1980) are reasonably small. On the other hand, the three-photon ionization of He (2 ‘S) in a discharge afterglow (Bakos et d . , 1976) results in an error bar of nearly three orders of magnitude, which does not allow for any interpretation.

J . Morellec, D . Normand, and G . Petite

134

TABLE V1

E X P E R I M E NVAL.UES T A L OF MPI CROSS SECTIONS WITH L I N E A R L POLARIZED Y LIGHT

Atom

N

A (nm)

min

Measured value

max

cs

2

528.0

4.8 x

6.7 x

He* 2s'

2

347.26

7

x

2.7 x

10-49

4.7 x

- 2.53

2

-

10-50

1.5 x

10-49

2.9 x

Na

3

694.5

4.3 x

10-77

5.4 x 10-77

6.7 x

K Rb cs He* 2s' - 2Sl

3 3 3 3 3

-

2.8 x 1.1 x 1.1 x

10-79

693.7 694.5

3.5 x 10-79 1.44 x 10-77 1.8 x 10-77

3

-

4

1060.0

4

Na Xe

-

cs

2.33

8.6 x 10-49

10-77

x 1.4 x

3.3 x lo-"'

4.4 x 10-79 1.97 x 10-77 2.9 x 10-77 4 x 10-77 5.2 x lo-""

8

3.0 x lo-"'

5.2

6.3 x lO-Ion

I .o x 10-1"7

1.6 x

1056.0

4.7 x

7.5 x 10-'W

1.03 x lO-Inn

5

1060.0

4

1.26 x

4

I1

1060.0

10-77 10-77

5

x 10-8z

x 10-33H

10-78

10-137

10-336

References Normand and Morellec (1980) Lompre et 01. ( 1980) Cervenan ei (1975)

(11.

Bakosei trl. (1976) Lompre er (11. ( 1980)

x

x

10-107

10-137

10-334

Arslandbekov et ( I / . (1975) Normand and Morellec (1980) Arslandbekov et nl. (1975) Arslandbekov and Delone ( 1976)

(3) a ' ; ) values for He 2lS and He 2,s (Lompre et al., 1980) are less ' ; ' is obtained by an indirect deaccurate than their cr, values because a termination. (4) It may seem surprising to find two values of a'b) for cesium differing by a factor 10 for almost the same laser wavelength. In fact, these values are obtained on both sides of the resonance on the 6F level, which would be tuned for 1059 nm. The proximity of this resonance is the most probable explanation of this difference between the two results, but because of the differences between the experimental conditions, it is difficult to derive any definite conclusion on this point. In the following comparative discussion between theory and experiment, the experimental data and error bars are used as criteria to dis-

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

i

I35

Manakov et al., 1973

-

McCL.1981 Teague et 1976 \ Crance and \

&-

Bebb, 1966 Declemy et

al.. 1981

Atom

LomPd et a1..1980

'f

cs

Hec2ls)

He (2's)

FIG.I la. Comparison of calculated (horizontal lines) and measured (vertical bars) twophoton ionization cross sections.

criminate between the different calculations. This comparison leads to several remarks: ( 1 ) The pioneer calculations of Bebb (1966, 1967) can provide, in most cases, a good estimation of the experimental results, the only exception being the three-photon ionization of potassium at 694.5 nm, where the value of Bebb is seven times smaller than the value of Cervenan et nl. (1975). (2) The calculations of Aymar and Crance (1979; Lompre et id., 1980) using a model potential method and implicit summation are found in all cases to be in good agreement with the measured cross-section values. It must be pointed out that in two different cases they have found that the weak field approximation was no longer valid. (a) The three-photon ionization of He 2's at 694.5 nm for which the 6IS state is only 40.5 cm-l away from two-photon resonance. (b) The four-photon ionization of cesium at 1056 nm for which the 6F,5/: states are only 79 cm-I away from three-photon resonance.

136

J . Morellec, D . Normand, and G. Petite

1o-'Bc

L

Teag

-

FIG. 1 Ib. Comparison of calculated (horizontal lines) and measured (vertical bars) three-photon ionization cross sections. Bebb: Lapl: McG: Man I: Man 2: Teag.:

1967 Laplanche rt t i / . (1976a) McGuire (1981) Manakov et c i l . (1973) Manakov er cil. (1978) Teague and Lambropoulos (1976b)

Bak.1: Bakos cr c i l . (1976): (QDM) Bak.2: Bakos rt t i / . (1976); (Hartree-Fock) Lom: Lompre r't t i / . (1980): a: weak field approximation; b: calculations with higher order terms 0 1 s : Olsen er d.(1978)

In both cases the light-induced level shift is not negligible compared to the detuning from resonance, so that the generalized cross section depends on the laser intensity. By taking into account the higher order terms of the perturbation series, the 2 'S-6 'S transition energy of He is shown to be shifted away from two-photon resonance so that the three-photon ionization cross section decreases when the laser field is increased. Conversely, for cesium, when the laser field is increased, the 6S1'2-6F3i transition energies are pushed toward the three-photon energy, and the four-photon ionization cross section of cesium is increased. For both ex-

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

I37

I10-l0 Manakov et aL.1973

1o-lo7

-

Manakov et al..l978-

l0-l

1-

t

10-

10-141 and

t-

-a

et al..

Crance. 1979

10-1

t nm Atom

!:zted:I of

1060

1056

I

cs Arslandbekov et a1..1975

1060

Na Normand and Morellec . 1980

I

Arslandbekov et a1..1975

FIG.1 Ic. Comparison of calculated (horizontal lines) and measured (vertical bars) fourand five-photon ionization cross sections.

periments, the calculations using the weak field approximation (a) are found outside the experimental error bars (Fig. l l b and c). On the other hand, the calculations where the higher order effects as well as the real experimental conditions are taken into account (b) provide good agreement with the experimental data. (3) The theoretical predictions of Delone er al. (1976), based on the quantum defect method (QDM), are generally found to be in poor agreement with the experimental results; for example, the calculation for cesium at 528 nm is about 12 times higher than the experimental value of Normand and Morellec (1980), and the calculated value for rubidium at 694.5 nm is 10 times smaller than the measurement of Cervenan et al. (1973, the discrepancy with experiment reaching two orders of magnitude for cesium at 694.5 nm. As for the more recent calculation of Manakov et d.( 19781, using the Green's function and a model potential theory (MFT),

I38

J . Morellec, D . Normand, and G . Petite

the agreement with experimental values is good for Rb, Cs, and K at 694.5 nm, while disagreement persists for Na. The two-photon ionization of Cs at 528 nm, as well as the five-photon ionization of Na at 1060 nm, provides the surprise in the sense that the calculations of Manakov r t id. (1978) are farther from the experimental values that the calculations of Manakov et d.(1973). (4) The calculations of Laplanche et d . (1976a), as well as those of McGuire (1981), using the QDM and the Green’s function formalism, are found mostly outside the experimental error bars for Na, Cs, Rb, and K at 694.5 nm. However, the theoretical predictions of Dedemy et d . (1981) agree perfectly with the experimental result for Cs at 528 nm. In fact, this excellent agreement is somewhat misleading since it can be seen in Fig. 14 that 528 nm is exactly the abscissa of the crossing point between the calculated (curve B) and measured dispersion curves for two-photon ionization of Cs as a function of the photon energy. This remark underlines the difficulty of comparing theory with experiment at a few fixed frequencies. In fact, a much more reliable comparison can be made when the MPI experiment is carried out over a wide range of photon energy, which requires the use of high-power dye lasers. Such experiments have been performed on Cs in the 18,500-2 1,740 cm-’ photon energy range (Morellec et a/., 1980) and on N a in the 16,300- 16,600 cm-’ photon-energy range (Delone rt l i / . , 1981). These results concerning the study of “antiresonance” will be presented in the next section.

B. EFFECTS OF

THE

LASERLIGHTPOLARIZATION

The ratio RN of the MPI cross sections for circularly and linearly polarized lights can be determined experimentally with a very good accuracy. Indeed, the only sources of error are the reproducibility of the spatiotemporal laser intensity distribution from shot to shot, the quality of the polarization of the laser, and the accuracy of the relative ion measurements. The comparison of the theoretical predictions with the experimental data shows that only few calculations are found inside the experimental error bars. Nevertheless, the disagreement between theory and experiment is always less than a factor of two for all the comparisons listed in Table VII. The only exception concerns the five-photon ionization of Na at 1060 nm for which the calculated value of Delone et d . (1976) as well as the value of Manakov rt d.(1978) are, respectively, 12 and 4 times smaller than the experimental data of Delone (1975). This important discrepancy is probably caused by the vicinity of a minimum for mk” of Na (see Fig. 2 of Manakov rt d . , 1978), where the accuracy of the calcula-

TABLE VII RATIOO F

Na

3 3 5

K

MULTIPHOTON I O N I Z A T I O N CROSS

530 694.5 1060 530 694.5 1060

0.42

5

0.08 2.3 t 0.12

cs

He* 2’s

3 4 3

694.5 1060 693.7

4.0 2.0

He* 2%

3

693.7

2.9 2 1.2

11 13

1060 1060

Xe Kr

2.49

0.5 5 0.1 1.2 2 .36

2 3 4 3 2

Rb

SECTIONS FOR CIRCULARLY TO L I N E A R LPOLARIZED Y LIGHTO

2.66

5

0.11

2.46

0.33 2.49 0.07 1.16 2.46

1.67 2 0.5

347.1 2.24 2

(1): Delone (1975); Delone et

2.18 4

2.16 t 0.11

694.5

0.12

2

0.11

2.31 1.28

2

0.2

2.15

5

0.4

2.45

0.6

2.82

2.35 1.09 1.42 I .68 I.OS(QDM) 0.35(HF) 2.5(QDM) 2WF)

1.07 1.14 2.22 2.71

(38 2 6)-’ (70 2 8)-l

( I / . (1976); Bakos et ( I / . (1976). (2): Cervenan er u / . (1975). (3): Kogan et a / . (1971); (1971). (4): Lompre et ( I / . (1977). ( 5 ) : Laplanche et a / . (1976a). (6): Delone et a / . (1976); Bakos et (I/. (1976). (7): Manakov et ( I / . (1978). (8): Teague and Lambropoulos (1976a); Teague et a / . (1976).

Fox et

(I/.

140

J . Morellec, D . Normand, and G . Petite

tions is poor due to the cancellation of the different channels leading to the ionization. In particular, the two above mentioned calculations have neglected the spin-orbit coupling. This reduces the number of possible ionization channels and thus tends to underestimate ukc)and consequently R5. As a conclusion to this section on absolute measurements of MPI cross sections, the following remarks are in order. Owing to the improvements in the experimental conditions and in the accuracy of the theoretical calculations, the agreement between theory and experiment in most cases is now better than one order of magnitude. Definite conclusions on the different calculational methods are difficult to draw from this comparison. One should point out, however, that changes in the atomic wave functions used bring noticeable changes in the results. This is emphasized by the two successive results of Manakov r t t i / . (1973, 1978), where the use of model potential calculations instead of quantum defect theory brings, in most cases, the theoretical results closer to the experimental ones. A striking exception concerns the five-photon ionization of sodium at 1060 nm. In this case, the latter result is four orders of magnitude lower than the former one, which was in reasonable agreement with the experiment. More comments from the authors on this point would have been useful. The agreement between theory and experiment is better when the real experimental conditions are precisely taken into account in the calculations. For instance, in the four-photon ionization of Cs at 1056 nm and three-photon ionization of He (2's)at 694.5 nm, the calculated values are brought inside the experimental error bars by taking into account the intensity dependent level shifts (Lompre et d., 1980; Aymar and Crance, 1979). However, the residual dispersion of the calculated values around the experimental points does not allow (for these discrete laser wavelengths) definitive conclusions regarding the superiority of one or the other calculational method. This can be done much more easily when the comparison between theory and experiment is carried out over relatively large laser frequency ranges, as will be seen in the next section.

V. Destructive Interference Effects A. GENERAL CONSIDERATIONS

A special interest has been paid in the last few years to the question of interference minima (antiresonances) occurring between resonances in multiphoton ionization. These cross-section minima for particular values

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

141

of the photon energy can be understood as the cancellation of the preponderant contributions of nearby atomic states. To illustrate this phenomenon by an example, Fig. 12 represents the minima of destructive interference with the corresponding spectral diagrams in the two-photon ionization of cesium atoms. The first minimum results principally from the cancellation of the opposite contributions of 6P states and 7P states to the ionization cross section. Interference minima are quite common phenomena in physics and appear every time one has to sum different amplitudes presenting a resonant character. In fact, this has been experimentally demonstrated in the case of bound-bound two-photon processes by Bjorkholm and Liao (1974). However, as will be detailed further, experiment failed for a long time to show such interference minima in the case of MPI. This has made this

2.0

2.4

2.8

3.2

3.6

Photon Energy (eV)

FIG. 12. Two-photon ionization generalized cross section (Bebb, 1966) showing the interference minima occuring between: (a) the 6P-7P levels: (b) the 7P-8P levels: (c) the 8P-9P levels.

J . Morellec, D . Normand, and G . Petite

142

question an important issue for both theoreticians and experimentalists in the last few years. Effects of light polarization have been mentioned in both Sections II,F and IV,B, and they are expected to be particularly important around the interference minima. Indeed, a change in the light polarization brings about considerable changes in the interference conditions, since it modifies, on account of the selection rules, the number of channels participating in the ionization process. Therefore, the effect of light polarization will be reviewed at the end of this section. The main difficulty to overcome in measuring MPI cross sections around a minimum of destructive interference is that the process investigated may have to compete with other ionization processes of comparable or even higher probabilities (see background effects in Section 111,E). In the case of alkali vapors, one must be very careful since dimers are always mixed with monomers in a proportion which depends on the thermodynamic equilibrium conditions. The first experimental investigation of destructive interference effects in the two-photon ionization of Cs atoms by Granneman and Van der Wiel(l975) (which we will refer to as GVW) failed due to the predominant contribution of dimers. The energy interval between the 6P and 7P levels was sampled with nine laser wavelengths between 454.5 and 514.5 nm; however, no minimum was found for the cross section, and the measured u!$'values were higher by four orders of magnitude than theoretical predictions. The density of dimers was assumed by the authors to be reduced so dramatically that a possible contribution of molecules to the ionization could be excluded. In fact, it has been proven since then (see Section V,B) that in the 104-105W/cm2intensity range used in this experiment, the ionization of atoms was completely negligible. On the contrary, dimers could be ionized through a resonant process due to the complex vibrational-rotational structure of the Cs, spectrum. As shown by the following reaction diagrams, atomic ions coming either from dissociation of Cs: or direct ionization of Cs, on a repulsive potential curve could be mixed with atomic ions created through direct two-photon ionization of atoms: Cs + 2 photons

+ 2 photons (one-photon resonant on the first molecular absorption band)

> C S ++ e

+e I 1+ photon = Cs' + Cs

Cs:

Cs,

\cs+

+ cs + e-

The time-of-flight analysis did not allow separation of the Cs+ ions produced through these three different ionization channels. It is important to note that the laser wavelength around 480 nm corresponds to a strong

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

143

absorption band of Cs2:CIIlu+ X5:. The strongly resonant character of the two-photon ionization of Cs, obviously explains the predominance of the molecular process over the atomic process although the neutral density for atoms is much higher than for molecules. In the puzzling situation created by the disagreement between the GVW experimental results and those of second-order perturbation theory, a few theoretical attempts were made to attribute the discrepancies to the turning on of the interaction (Theodosiou and Armstrong, 1979) or to laser bandwidth effects (Armstrong and Eberly, 1979); however, the assumptions on the laser pulse rise time or on the laser line shape involved in these models did not realistically describe the experimental conditions. Finally, observations of destructive interference effects were reported in for the two-photon ionization of Cs atoms and in 1980 by Morellec et d., 1981 by Delone et d., for the three-photon ionization of K atoms.

B . TWO-PHOTON IONIZATIONOF

CESIUM

ATOMS

I. Lineur Polurizution Measurements The experiment of Morellec et al. (1980) can be compared to the GVW experiment since both of them use a Cs atomic beam with similar intensities and deal with approximately the same photon energy range (18,500-21,500 cm-') (Fig. 12a). Likewise, in both experiments, the molecular density is reduced by thermal dissociation in the atomic beam apparatus. The principle of the technique is described in detail by Lambropoulos and Moody (1977), and we only recall here its essential features. The atomic beam apparatus is separated into two chambers, which are thermically decoupled and have two different functions: (1) the evaporation cell, whose temperature determines the intensity of the atomic beam; (2) the dissociation chamber, overheated by one to several hundred degrees, whose configuration makes particles collide against the walls before effusing through the nozzle. The transit of the vapor through the dissociation chamber thus reduces the molecular density without noticeably changing the atomic density. However, nondissociating particle collision can lead dimers to escape without wall collisions, and so the dissociation efficiency mainly depends on whether or not the mean free path is much larger than the dimensions of the chamber. Therefore, contrary to the assumptions of GVW, the molecular component in the atomic beam cannot be completely suppressed by vapor superheating. In the best cases, the ratio of the populations of dimers and atoms is about one part in lo5(Morellec and Normand, 1981),which is still noticeable because of the resonant character of the molecular process. In fact, the selection be-

144

J . Morellec, D . Normand, and G . Petite

tween the atomic and molecular processes is finally secured by the choice of the laser intensity range, as illustrated in Fig. 13. The 104-loj W/cm2 range used by GVW is the region of predominance of the molecular process, whereas the intensity range used by Morellec et d.(10"' W/cm2) is the region of saturation of the molecular process and predominance of the atomic one. The observation of the antiresonance minimum of Cs around the 7P level has thus been made possible by the use of both a superheating experimental values are system and a high laser intensity range. The dZL' plotted in Fig. 14 as a function of the photon energy. They are determined

A

lo5

10'

10'

lo8

10'

io'O

Laser intensity ( W / c m z )

FIG.13. Competition between atomic and molecular processes for the creation of Cs+ at 20,840 cm-I (A = 480 nm). (A) Laser intensity range ( I W - I W W/cm*) used in GVW (1975); all the Cs+ detected originate from ionization dissociation of Cs, . (B)Laser intensity range ( 10'O W/cm*) used in Morellec er ( I / . (1980), predominance of the atomic process. The ratio of the populations of Cs, and Cs is about lo+.

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

145

c

t

I I

FIG. 14. Two-photon ionization generalized cross section of Cs around 20,800 cm-l, showing the interference minimum: ( 0 )experiment (Morellec ef id., 1980). Curves: A, Lambropoulos and Teague (1976); B, Declemy c f rrl. (1981); C, Teague cf N I . (1976); D, Aymar and Crance (1982).

by means of the direct method described in Section I11 and Table V; miL)is found to have a minimum for E D = 20,840 5 10 cm-l; curves A, B, C, and D are theoretical calculations due to Lambropoulos and Teague (1976), Declemyer LII. (1981), Teagueer nl. (1976), and Aymar and Crance (1981), respectively. This comparison leads to the following remarks: (1) Although the position and the depth of the miL) valley somewhat vary with the calculational model of the atomic wave functions, the agreement with the measurements of Morellec er ul. (1980) is satisfactory enough to reject any unexplained breakdown of the perturbation theory. (2) The differences among curves A, B, C,and D cannot, on the other hand, be considered as negligible. To make the arguments clear, the positions of the miL) minimum, according to experiment and to calculations, are plotted on an energy axis in Fig. 15. It must be recalled that: (a) A and B use the quantum defect method (QDM), while C and D utilize the model

146

J . Morellec, D . Normand, and G. Petite experiment 20,840

Quantum Defect Method

Model Potential Theory

FIG.IS. Position in energy of the &’ minimum according to the experiment (Normand and Morellec, 1981) and to the different calculations of Fig. 14. The arrow shows the shift expected from the contribution of the P continuum.

potential theory (MPT), including the spin-orbit interaction and the effects of core polarization (Norcross, 1973); (b) A and C restrict the calcuto a finite number of discrete levels, while B and D take lations of a$*.’ into account the contribution of all In pj) intermediate levels (continuum included). As the positions of the minima of curves B and D are, respectively, found displaced by more than 100 cm-I as concerns A and C, it is obvious that the contribution of the continuum is significant. This is confirmed by a calculation of Aymar and Crance (1982), which shows that the minimum is shifted by about 100 cm-I toward low photon energy when the contribution of the continuum is added. When comparing the experimental data with curves B and D (both including the contribution of the continuum), it is obvious that M P T provides much more accurate results than QDM since: (1) the position of the minimum in curve D is found almost inside the experimental error bar, while the minimum of curve B is 540 cm-’ away; (2) the shape of the valley in D reproduces perfectly the experimental results. In fact, it must be pointed out that the first p levels give the main contribution in Eq. (17) since matrix elements (6slrlnpj) decrease very rapidly with 1 1 , while the energy denominator increases with n . Thus, QDM reveals itself to be inadequate to account for this experiment because it gives matrix element of poor accuracy for the first p levels. 2. Efects qf the Light Point-iztrtion

All the results presented thus far have been obtained for a linearly polarized laser field. In fact, the study of the influence of the laser polarization in the vicinity of an antiresonance can provide a very useful test for

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

147

theoretical predictions. In the case of two-photon ionization of Cs atoms, the ionization channels allowed for a linear polarization of the field are

@

0 0

+ (6s, 3, 4) + (np, 3,h) + (6s, 4, 3) (np, 8, 4) (6s, 3,3)

+

(np, 3, h)

(np, 4, h)

3,3) (np, B, 3) + (Ed, 8, 3) ( ~ d9,, i) +

(ES,

(47) (48)

(49) the atomic bound states being labeled (nl, j , mj)and the free states (el, j , m,). In fact, they are the three similar channels starting with rn, = - 1/2, but since the cross section is averaged over initial rn,’s, they can be conveniently omitted. On the other hand, if a circularly polarized laser is used, the selection rules imply the following ionization channels: +

+

t)

@

(6s, 3,

@ @

(6s, 3, -4) ( 6 3,~ -3) ~

+

+

+

(np, 9,9) + (Ed, Q,Q) (np, 3, h) + (np, 8,3) (np, 3,

+

(50)

(ed, B, B)

3) + (€4B, 8)

(51) (52)

Averaging over initial rn,’s gives the following two-photon ionization cross sections: =

G(M,, +

+ M32)

(53)

+ MJ2+ Ma2)

(54)

M22

a:‘) = (G/2)(M42

G is an atomic constant which depends on the normalization of the continuum states: M i is the amplitude of the transition probability of channel 0. By calculating the angular parts of the Mi’s, it can be easily shown that

u p = ifG(M22+ M 3 2 )

(55)

so that the ratio R2of the cross sections for circular and linear polarizations is

R2 = ukc)/u;L)= 8 [ ( M z Z+ M32)/(M12+ M Z 2 + M32)]

(56)

Because of the different dipole matrix elements involved in the M i ’ s , there is a slight displacement between the zeros of MI, M,, and M 3 , so that neither uLL)nor mi‘) can have a zero. Cross section uiL),as well as uF), passes through a deep minimum whose position and depth are dramatically sensitive to the position in energy of the zeros of M I , M z , and M 3 . The constant G disappears in the ratio Rz [Eq.(56)], and comparison can be conveniently carried out between theory and experiment (Fig. 16).

148

J . Morellec, D . Normand, and G . Petite

1.5

1.0 RZ

0.5

0

Fib. 16. Ratio rh"l/uhL'around the interference minimum: ( 0 )experiment (Normand and Morellec, 1981), compared with different theoretical results (see Fig. 14).

The full circles with error bars are the experimental data of Normand and Morellec (1981), while A , B , C, and D refer to the four calculations defined in Fig. 14. It clearly appears from the comparison that the agreement with the experiment is found to be excellent for curve D, and at least qualitatively acceptable for curve C, while curves A and B are pretty well removed. The following conclusions can then be drawn: ( I ) The experimental data emphasize the importance of the spin-orbit interaction. Indeed, if fine structure is neglected, there remains only one ionization channel for the circular polarization of the laser so that R2can have a zero, which is obviously not there in the experiment. Therefore, the calculations where the spin-orbit coupling is neglected (Rachman et ( I / . , 1978; Manakov et a / . , 1978; McGuire, 1981) have not been plotted in Fig. 16. (2) The flagrant disagreement between the experimental data and calculations A and B proves that QDM provides zeros for M , and M 3 , which are very close to each other so that R , nearly reaches zero. (3) It is interesting to note that the experimental data reach the limiting value 1.5 [Eq. (56)], thus indicating the photon energy (20,750 cm-') for which MI vanishes.

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

149

C. THREE-PHOTON IONIZATION OF POTASSIUM A recent experiment by Delone et al. (1981) has provided evidence of the strong influence of the light polarization on the three-photon ionization of potassium atoms in the 16,300-16,600 cm-I photon-energy range. As can be seen in Fig. 17, the energy of two photons lies in the energy interval of the 7d and 8d levels, as well as between the energies of the 9s and 10s levels. According to the selection rules, the ionization channels allowed for a three-photon process are as follows:

4s

+

np

Circular polarization: 4s

+

np

Linear polarization:

+

-t

ns

nd

Ef

From these diagrams, the interference minimum of v3 is expected to be deeper in circular polarization (one channel) than in linear polarization (three channels). The experiment from Delone et al. unambiguously demonstrates this, since the minimum of vbc)is found to be lower by one and a half orders of magnitude than the minimum of vbL)(Fig. 17). This experiment gives very interesting results on combined interference minima and light polarization effects. However, these authors do not observe, with circularly polarized light, the window of transparency predicted by Manakov et ol. (1978), which can be seen in Fig. 18. Delone et nl., attribute this discrepancy to the large spectral bandwidth of the laser (6 cm-I), whereas the radiation is assumed to be monochromatic in the calculations of Manakov et nl. However, two further reasons for this discrepancy may be suggested, one being theoretical in character, the other purely experimental: ( 1) Since c r y ) is found to pass through zero according to Manakov et al. (1978), the spin-orbit coupling, as well as the quadrupole electric interaction (the 9p level being only 30 cm-l away from the vbc)minimum), should be taken into account in these calculations, thus resulting in a finite minimum for uhc). (2) On the other hand, it is not excluded that K2dimers may contribute to the creation of K + ions through a mechanism analog to that previously

J . Morellec, D . Normand, and G . Petite

150

7d 95 (L,C

03

9P

8d 10s

II

II

10

10

10

I

1

32,600 32,700 32,800 32,900

33,000 33,100 33,200

0

I

2w [crn-'1

32,974

FIG. 17. Experimental three-photon ionization probability of K around 33,000 cm-' (Delone ('I t i / . . 1981): (0) linear polarization; ( x ) circular polarization.

described in two-photon ionization of Cs atoms. This molecular background can then impede the observation of a very deep minimum for &). As a conclusion to this section, the cross-section measurements performed in the vicinity of an antiresonance appear to be a very sensitive test for discriminating between the different calculational methods of dipole matrix elements. In particular, for the two-photon ionization of Cs, the superiority of the model potential theory (MPT) over the quantum defect method (QDM) is clearly demonstrated. Moreover, the comparison between the calculations and the experiments puts forward evidence that some approximations which are usually made everywhere else in the spectrum are no longer valid in the vicinity of an antiresonance. Thus, the spin-orbit coupling must be taken into account because it increases the

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

151

FIG. 18. Theoretical three-photon ionization generalized cross section of K (in units of crnasecP)around 33,000 crn-* (Manakov ef ( I / . , 1978): (-1 linear polarization; (---) circular polarization.

number of allowed ionization channels, which has a dramatic influence on the antiresonance profiles. Likewise, the contribution of the continuum for the intermediate states and perhaps also, in some cases, the contribution of the electric quadrupole interaction must be included in the calculations of multiphoton ionization cross sections to obtain an accurate determination of the interference profiles.

VI. New Trends A. ABOVE-THRESHOLD IONIZATION

As long as the goal of MPI experiments is to measure the ionization rate as a function of the atom and laser field characteristics, the collection of ions is easier and less problematic than electron collection. Due to the much higher electron mobilities, the MPI electrons cannot be efficiently

152

J . Morellec, D . Normand, and G . Petite

separated from parasitic electrons such as surface photoelectrons in the study of alkali vapors. A knowledge of all these parasitic phenomena occurring in MPI experiment allows one to adjust the parameters of the interaction so that the electrons really originating from the process investigated can be measured without any collection field. The possibility of measuring the physical characteristics of the outgoing electron, especially its energy spectrum and its angular distribution, has given birth to a new line of experimental and theoretical work, namely, above threshold ionization (ATI). Thus far, we have restricted the study of multiphoton ionization to processes where only the minimum number of photons necessary to ionize the atom are involved. But if an atom can be ionized through a N-photon process, it can also be ionized through the absorption of N + I , N + 2, . . . , photons of the same energy. Such processes are present in every multiphoton ionization process, but they have been neglected until recently. This was justified by the fact that these processes, being of a higher order, require higher intensities to produce a significant contribution to the ionization probability. The other side of this argument is that, being of a higher order, their contribution should increase faster than lower order processes with the laser intensity, and is expected to be significant in the case of strong fields. Above threshold ionization was predicted a long time ago. In an early article, Zernik and Klopfenstein ( 1965) calculated the two-photon ionization probability of a hydrogenic state for photon energies above the ionization threshold, using a method similar to Schwartz and Tiemann’s. In this article, AT1 results in an intensity-dependent correction to the Gaunt factor characterizing the single-photon ionization. Such processes have since been the subject of calculations by different authors using the Green’s function method and a Sturmian expansion (Klarsfeld, 1970; Karule, 1978; Klarsfeld and Maquet, 1979a,b) or using Schwartz and Tiemann’s technique (Aymar and Crance, 1980). Few results are available on multiphoton AT1 (i.e., AT1 involving more than one continuum-continuum transition). Gontier and Trahin ( 1980) calculated processes involving up to six-photon absorption above the minimum number in the six-photon ionization of hydrogen by the second harmonic of a neodymium laser. They showed that there is an intensity above which AT1 processes give contribution of the same order of magnitude as the lowest order process, and calculated this intensity to be of the order of lo’.‘ W/cm2. Last, we should mention the only nonpertubative treatment of ATI, which is due to Crance and Aymar (1980b), who used the effective Hamiltonian method. Their results agree with the predictions of perturbative theories in the weak field limit.

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153

In experiments on AT1 as well as on angular distribution, it must be realized that the detection system achieves simultaneously the best collection efficiency and the smallest possible perturbation of the electron characteristics. An important consequence is that the use of collecting fields is prohibited for reasons which are quite obvious in the case of angular distributions (which would be wiped out by the use of such fields), and applies also to the case of ATI. For these experiments the use of a collecting field should result simply in a translation of the energy spectrum of the electrons, but because of the finite size of the interaction volume, all electrons are not accelerated by the same amount, which is a cause of broadening of the apparatus function of whatever spectrometer is used in this experiment. Three major electron spectroscopy techniques have been used. The retarding potential technique has the advantage of simplicity but a lowenergy resolution. Spherical electron analyzers (Ballu, 1980) have been used by different authors and give the best possible resolution4own to a few millielectron volts-although this bandwidth is still large compared to the characteristic widths of most ionization processes (current ionization widths of atomic states are generally fractions of wave numbers only). Time-of-flight spectroscopy, although seldom used thus far, seems a powerful method because the time analysis of the electron signal provides informations on the complete electron spectrum for each laser shot, when other methods have to resort to scanning techniques. However, time-offlight spectroscopy offers a poor resolution for fast electrons, a drawback which could possibly be circumvented by a combination of the retarding potential and the time-of-flight techniques. The first observation of fast electrons originating from laser interaction with a gaseous target is due to Martin and Mandel(1976), who used a ruby laser focused in air, but the acceleration mechanism does not seem to result from ATI. The first unambiguous evidence of AT1 was given by Agostini et d.(1979), who used a frequency-doubled Nd-glass laser focused in xenon vapor. Using a retarding potential technique, these authors gave evidence of the absorption of one photon more than the six photons strictly necessary to ionize the Xe atom. More recently, Agostini ct a / . (1981),using a high repetition rate, frequency-doubled Nd-Yag laser and a 180" spherical electron analyzer, showed evidence of absorption of five photons more than the minimum. Their electron energy spectrum is shown in Fig. 19. This spectrum appears to be formed of two series of peaks evenly separated by the photon energy. The existence of two series results from the core fine structure splitting, that is, the energy difference between the two ionization limits corresponding to the two possible Xe core configurations. It must be noted that these results have been obtained for laser intensities of the order of 10" W/cm2, that is, about three orders

J . Morellec, D . Normand, and G . Petite

154

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F I G . 19. Experimental energy spectrum of the outgoing electron for the six-photon ionization of Xe at 5300 A (Agostinirt a / . , 198.l). The two series of peaks are due to the core fine-structure splitting, as shown in the bottom part of the figure.

of magnitude less than the intensity expected from the theory applied to atomic hydrogen. This could mean that these processes play a much more important role in MPI experiments than was believed until now. Another observation of AT1 on Xe atoms is due to Kruit ct a / . (19811, who studied multiphoton ionization of Xe by a Nd-Yag laser-both fundamental and frequency doubled-and by a Nd-pumped dye laser. These authors have used a time-of-flight spectrometer together with a “magnetic mirror” (Kruit and Read, 1982) allowing a collection angle of 27r steradians. In the case of 1 I-photon ionization of X e at 1.06 p m , they report the absorption of 7 photons above the ionization threshold, a result similar to the one reported by Fabre cf cd. (1982). In the five-photon ionization of Xenon with a three-photon resonance, Kruit et d.have observed two extra AT1 peaks. They have shown that the amplitude ratio between the lowest energy peak (no ATI) and the first excited peak (one-photon ATI) is linear in the laser intensity. This dependence agrees with a direct measurement of the intensity dependence of AT1 (Fabre ct a / . , 1982), which showed a good agreement with the predictions of lowest order perturbation theory: Within the experimental uncertainty the (N + S)-order AT1 was found to behave like thus implying

NONRESONANT MULTIPHOTON IONIZATION OF ATOMS

155

that the ratio between two consecutive orders depends linearly onl. (S is the number of photons absorbed above the continuum limit.) Finally, one of the most striking results of the experiment of Fabre et (11. (1982) is the observation of the MPI of xenon up to the order 21 (N = 1 1 , S = 10) for intensities of a few 10l2 W/cm2, whereas more than 1015 W/cm2were required to observe an ionization process of the same order in helium ( N = 22, S = 0). B. ANGULAR DISTRIBUTIONS The question of the angular distribution of the outgoing electron attracted the attention of theoreticians some years ago, but we are not aware of recent work on the subject other than the case of resonant processes. Therefore, the reader is referred to Lambropoulos (1976) or to the earlier articles quoted there. Few results concerning application to special experimental cases have since been published. Mention should be made of the work of Olsen et al. (1978) on helium, as well as that of Gontier et a / . ( 1980), who calculated the angular distribution relevant to different electron peaks resulting from AT1 of hydrogen by a frequency-doubled Nd-Yag laser. The principle of the angular distribution measurements is described below (Fabre et al., 1981). Given an initial state, the final state of the MPI process is a superposition of different partial waves, which determines the angular distribution of the outgoing electrons. Due to the selection rules, a given state of the continuum can be reached through different channels, and the determination of the angular distribution thus requires the knowledge of the radial matrix elements and the Clebsh-Gordan coefficients associated with these channels. The general form of the differential cross section can be written as U

where X is a factor depending on the laser intensity andN is the number of photons absorbed. In the case of linearly polarized light, 8 measures the angle between the direction of the ejected electron and the polarization direction of the incident light; a, depends on the properties of the initial state, the intermediate states, and partial waves of the emitted electrons but does not depend on the laser intensity. Therefore, in linearly polarized light, the experiment consists of keeping the position of the detector fixed, rotating the polarization axis (rotation of the h/2 plate around the propagation axis), and measuring the

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electron signal as a function of the angle 8 between the polarization direction and the collection system. Such experiments have been reported by Edelstein et crl. (l974), Duncanson et ( I / . (1976), Strand et a / . (1978), and Leuchs et crl. (1979). It should however be noted that these experiments were utilizing resonances on intermediate states and have to be considered as a means of extracting data concerning the particular intermediate states rather than pure multiphoton experiments. The first application of angular distribution measurements to AT1 is due to Fabre et rrl. (1981). They measured the angular distribution of electrons in the six- and seven-photon ionization of Xe atoms by a linearly polarized, frequencydoubled Nd-Yag laser beam ( N = 6 being the minimum number of photons necessary to ionize Xe). They obtained for both distributions ( N = 6 and N + I = 7) a maximum at 0 = 0" (i.e., along the light polarization axis) and a nonzero minimum around 6 = 90". Unfortunately the calculation of the differential cross section for the six-photon ionization of Xe atom is very difficult, and comparison between theory and experiment is still not possible. Therefore, there exists an interest in the angular distribution measurements of AT1 experiments, with atoms amenable to calculation (alkali, for instance). In conclusion, we must remark that above threshold ionization experiments are only at their beginning, and decisive improvements of the situation can be expected in the not too distant future. Thus far, the most important result seems to have been able to clearly identify AT1 as the accelerating process responsible for the fast electrons detected, among the various possibilities such as stimulated inverse bremsstrahlung or the ponderomotive force due to the gradient of laser intensity.

C . D O U B LMULTIPHOTON E IONIZATION OF ATOMS When an intense laser beam is focused on an atom, more than one electron can be detached by MPI processes. Thus, we encounter double (or even multiple) multiphoton ionization processes. Table VIII compares the first and second ionization potentials of different atomic species and the corresponding orders of MPI processes at 1.06 pm. With this comparison in mind, it seems reasonable to expect that much higher laser intensities will be required to observe double ionization than for simple ionization. However, surprises such as those met in AT1 experiments should not be excluded here. In contrast to the rare-gas atoms (Ar) and alkali atoms (Cs) whose second-ionization potentials are pretty high (many electrons on the outer shell), alkaline earth atoms (Ba, Sr) can be ionized through a reasonably low-order MPI process. This has given rke to the experimental

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157

TABLE VIII FIRSTA N D

SECOND IONIZATION POTENTIALS A N D CORRESPONDING ORDERS OF

NONLINEARITY FOR DIFFERENT GASES= First ionization potential (eV)

Atom Ar cs Ba

15

3.89 5.21 5.7

Sr ~

~~

N+

Second ionization potential (eV)

13 4 5 5

21 25.1 10 11

NZ+ 23 22 9 10

~~

Energy of one photon is equal to 1.18 eV.

study of two-electron MPI of alkaline earth atoms and some other atoms having two optical electrons on the outer shell (Aleksakhin et al., 1977, 1979). They have made the main following observations. First, the probability for production of doubly charged ions is surprisingly not much lower than the probability for production of singly charged ions. Second, the effective order of nonlinearity of the double MPI process experimentally measured ( A 2 + ) is much smaller than the number of photons absorbed in the interaction (N2+).In the case of barium, they have even found k 2 + smaller than k ( k 2 + = 2.9 versus k = 4). The interpretation given by the authors is essentially based on possible resonances in the two-electron bound state or in the autoionizing state. They conclude with the need for a quantitative theoretical description of resonance with two-electrons bound states and autoionizing states and further experimental investigations with tunable lasers. Regarding this matter, Chin et al. (1981) have reported the three-photon ionization of strontium atoms via resonance of the last photon with some autoionizing states in the excitation range of 5560-5640 A. They use a powerful dye laser ( I 1Olo W/cm2)and observe peaks of singly charged ions emerging from the detection background at resonances which broaden as function of the laser intensity. Let us note that the laser intensity used in the Aleksakhin et al. experiment to study the double MPI of strontium atoms at a fixed frequency was about lo2 times the laser intensity used by Chin et al. to observe the influence of resonance on the simple MPI of strontium atoms. +

+

-

VII. Conclusion This article has been centered around the comparison between recent reliable experimental determination of NRMPI generalized cross sections and their different calculated values. Agreements (or disagreements), as

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well as their possible causes, have been observed in the course of this contribution and discussed at length, and there is no point in repeating this discussion in our conclusion. We would prefer to address the following question: What are the fundamental results we are now able to derive from this ensemble of experiments? Figure 20 is a helpful illustration of the ensemble of results discussed here. It depicts the logarithm of the generalized multiphoton ionization cross sections (sign reversed, for convenience) versus the order of nonlinearity of the process. The points correspond to the different experimental values reported in Fig. 11, and the straight line is a least squares fit obtained with these points. The fluctuations of the points around the straight line are noticeably small so that one can say that when the order of the process is increased by one unit, the cross section is decreased by a factor This indeed can be derived from Eq. (16) of uN(with necessarily rough approximations concerning the “average” energy detuning in the denominator of M,,). It is also noteworthy that the fitted line leads to a value of cm2 for a single-photon ionization cross section, which seems quite reasonable. From this figure, we can conclude that this ensemble of experiments proves that lowest order perturbation theory is definitely, with rare exception, the convenient tool for tackling nonresonant multiphoton ionization. Concerning the small dispersion of the experimental points, they are simply due to different atomic situations such as the greater or lesser proximity of a potentially resonant state (it should be noted that a true resonance could shift an experimental point from the line by as much as ten orders of magnitude).

1 2 3 4 5 N FIG.20. Experimentally measured NRMPI generalized cross sections as a function of the order of the process.

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Another general conclusion which can be drawn from the comparison between experimental and theoretical results concerns the ability to obtain an accurate atomic potential. For instance, in the case of two-photon ionization of Cs, the potential described by Norcross (1973) is shown to give much more accurate results than the quantum defect method. Once such a potential is available, exact summation techniques can make the difference between a good and a poor agreement. This is especially true when there is a large cancellation between the discrete states and the contribution of the continuum therefore starts to be of importance. In fact, the study of the limiting cases where lowest order perturbation theory breaks down seems to be a much more exciting direction. We have already mentioned the question of resonant processes, which is now well understood, both theoretically and experimentally, even if no reliable absolute values of the resonant ionization cross section are available. Even the influence of the field statistical properties in the case of resonant processes seems to be reasonably well understood (see, e.g., Lompre ef d.,1981a,b, and references therein). The other limiting case, that of the ultrastrong fields, seems to be far less well understood. Lompre et d.(1976) published results on the 22photon ionization of helium, where no departure from the IN law was observed for laser intensities as high as several W/cm*. Nonperturbative theories of the multiphoton ionization are still unable to provide numerical values of the ionization cross section under ultrahigh laser fields, but progress can be expected in the not too distant future on this point. On the other hand, no experiment has supported the tunneling ionization theory (Keldysh, 1965), which was thought to dominate multiphoton ionization for ultrastrong fields. Concerning the future of multiphoton ionization experiments, the study of the electrons is definitely an important point. Even if no great surprise is expected from angular distribution experiments, no nonresonant experiment suitable to comparison with numerical results has yet been done. On the other hand, AT1 seems richer in potential puzzles. It should be noted, for instance, that the recent observations of AT1 have been obtained at laser intensities three orders of magnitude lower than those predicted by theory. Of course, the theory concerns hydrogen, and the experiment was done with xenon, both having the same order for the AT1 process. Does the atom make the difference, or is there here something more fundamental? Here again experimental results comparable with computation are yet to come. Extensions of multiphoton ionization studies toward small molecules or collective effects are, without a doubt, 'important, with many possible technical applications. Last, double-ionization studies should renew the

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question of rnultiphoton ionization in cases where the single-electron picture is no longer sufficient. All this leads to the cheerful conclusion that however important the work already done, rnultiphoton ionization is still quite an open field.

ACKNOWLEDGMENTS The authors are greatly indebted to their colleagues who, through many helpful discussions, made the completion of this work possible. In particular, they would like t o gratefully acknowledge the help of Drs. N. K. Rahman, P. Agostini, M. Aymar, M. Crance, F. Fabre, S. Geltman, Y.Gontier, L. A. Lompre, G. Mainfray, C. Manus, J. P. Marinier, M. Poirier, and M. Trahin. They would also like t o thank the following persons for the permission granted to quote their results, sometimes in advance of publication: P. Agostini, D. T. Alimov. M. Aymar, H. B. Bebb, M. Clement, M. Crance, J. L. Debethune, A. Declemy, N . B. Delone, F. Fabre, Y.Gontier, D. Goodmanson, M. Jaouen, P. K. Khabibulayen, P. Lambropoulos, G. Laplanche, L. A. Lompre, G. Mainfray, B. Mathieu, D. W. Norcross, T. Olsen, M. A. Preobrazhensky, A. Rachman, F. Sanchez, N . R. Teague, J. Thebault, M. Trahin, M. A. Tursunov, and G. Watel. They would also like to express their gratitude to the copyright owners of the material used in this work: the publishing division of the Institute of Physics (Bristol, G.B.), the North-Holland Publishing Company (Amsterdam, N.L.), I1 Nuovo Cimento (Bologne, Italy), the Physical Review (Ridge, N.Y., U.S.A.) and the Revue d e Physique Appliquee (Paris, France).

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Leuchs, G., Smith, S. J., Khawaja, E., and Walther, H. (1979). Opt. Committi. 31, 313. Lompre, L. A., Mainfray, G., Manus, C., Repoux, S.,and Thebault, J. (1976). Phys. Rev. Lett. 36, 949. Lompre, L. A., Mainfray, G., Manus, C., and Thebault, J. (1977). Phys. Rev. A 15, 1604. Lompre, L. A., Mainfray, G., and Thebault, J. (1978). Proc. SOC.Photo-Opt. Instrum. Eng. 189, 558. Lompre, L. A., Mainfray, G., Mathieu, B., Watel, G., Aymar, M., and Crance, M. (l980).J. Pliys. B 13, 1799. Lompre, L. A., Mainfray, G., Manus, C., and Marinier, J. P. (1981a). J. Pliys. B 14,4307. Lompre, L. A., Mainfray, G., Manus, C., and Marinier, J. P. (1981b). Phys. Lett. A 86, 17. Lompre, L . A., Mainfray, G., and Thebault, J. (1982). Rev. Phys. Appl. 17, 21. Lu Van, M., Mainfray, G., Manus, C., and Tugov, I. (1973). Phys. Rev. A 7, 91. McGuire, E. J. (1981). Phjw. Re\,. A 23, 186. Manakov, N. L., Ovsiannikov, V. D., and Rapoport, L. P. (1973).Proc. Irit. ConJ'. Phenorn. lorii:. Grises. l l t h , 1973. p. 29. Manakov, N. L., Ovsiannikov, V. D., Preobrazhenski, M. A., and Rapoport, L. P. (1978).J . Phys. B I I , 245. Mandel, L., and Wolf, E. (1965). Re\*. Mod. Phys. 37, 231. Maquet, A. (1975). Thesis, University of Paris VI. Maquet, A. (1977). Phys. Reis. A IS, 1088. Martin, E. A., and Mandel, L. (1976). Appl. Opt. 15, 2378. Messiah, A. (1965). "Quantum Mechanics," Vol. 2. Wiley, New York. Mizuno, J. (1973). J. Pltys. B 6, 314. Morellec, J., and Normand, D. (1981). J . Pliys. B 14, 3919. Morellec, J., Normand, D., and Petite, G. (1976). Phys. Re,,. A 14, 300. Morellec, J., Normand, D., Mainfray, G., and Manus, C. (1980).Phys. Rev. Lett. 44, 1394. Morton, K . M. (1967). Proc. Pliys. Soc.. London 92, 301. Norcross, D. W. (1973). Pliys. Rev. A 7, 606. Normand, D., and Morellec, J. (1980). J . Pliys. B 13, 1551. Normand, D., and Morellec, J. (1981). J . Pliys. B 14, L401. Olsen, T., Lambropoulos, P., Weatley, S., and Rountree, S. (1978). J . Phys. B 11, 4167. Petite, G., Morellec, J., and Normand, D. (1979). J . Phgs. (Orsny, F r . ) 40, 115. Rachman, A,, Laplanche, G.. and Jaouen, M. (1978). Phps. Lett. A 68, 433. Rapoport, L. P., Zon, B. A., and Manakov, N. L. (1%9). SOIF. Phys.-JETP (Engl. Trunsl.) 29, 220. Robinson, E. J., and Geltman, S. (1967). Plips. Rev. 153, 4. Sanchez, F. (1975). N i i o w Cimento Soc. Itnl. Fis. B 27B, 305. Schmidt, K. H . , Gordon, S., and Mulac, W. A. (1976). Rev. Sci. Instrum. 47, 356. Schwartz, C., and Tiemann, J. I. (1959). Ann. Pliys. ( N . Y.) 6, 178. Seaton, M. J. (1958). Mori. Nor. R . Astron. Soc. 118, 504. Shirley, J. H. (1965). P/i.vs. Re\'. B 138, 979. Strand, M. P., Hansen, J., Chien, R. L., and Berry, R. S. (1978). Cltem. Pliys. Lett. 59, 205. Teague, M. R., and Lambropoulos, P. (1976a). Phys. Lett. A 56A, 285. Teague, M. R., and Lambropoulos, P. (1976b).J . Phys. B 9, 1251. Teague, M. R., Lambropoulos, P., Goodmanson, D., and Norcross, D. W. (1976).Phys. Re\*. A 14, 1057. Theodosiou, C. E., and Armstrong, L., Jr. (1979). J . Pliys. B 12, L87. Voronov, G. S . , and Delone, N. B. (1965). Sol-. Phys.-JETP Lett. (Engl. Trciti.tl.) 1 , 42.

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Zernik, W. (1964). Pliys. Re\,. A 135. 51. Zernik, W., and Klopfenstein, R. W. (1965). J . Mtrtli. Ph.vs. ( N . Y . ) 6, 262. Zon, B . A . , Manakov. N . L., and Rapoport, L. P. (1970).SOY.Phys.-DoA/. ( G i g / . 7'nrml.) 14. 904. Zon. B . A , , Manakov, N . L., and Rapoport, L. P. (1972).SOI..P/i!.r.--JETP ( E / / , q / T. r t i ~ i s l) . 34, 515.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS. VOL . 18

I1

CLASSICAL AND SEMICLASSICAL METHODS IN INELASTIC HEAV Y-PARTICLE COLLISIONS A . S . DICKINSON" Laburritoire d'Astrophysiyiiet Uniiiersite de Bordecnrx Tii Ien ce France

D . RICHARDS Fric.iilty of Mathemutics Open University

Milton Keynes. England

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 Angle-Action Variables . . . . . . . . . . . . . . . . . . . . . . A Isolated Bound Systems . . . . . . . . . . . . . . . . . . . . B Classical Collisions . . . . . . . . . . . . . . . . . . . . . . I11. Rotational Excitation . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Description of a Collision . . . . . . . . . . . . . . . . . . . C . Rotational Rainbows . . . . . . . . . . . . . . . . . . . . . . D . Rigid-Shell Scattering . . . . . . . . . . . . . . . . . . . . . E . Numerical Calculations . . . . . . . . . . . . . . . . . . . . F. Approximate Methods . . . . . . . . . . . . . . . . . . . . . G. Atom-Vibrating-Rotor Collisions . . . . . . . . . . . . . . . ' . H . Rotor-Rotor Collisions . . . . . . . . . . . . . . . . . . . . . IV. Uniform Approximations . . . . . . . . . . . . . . . . . . . . . V. Semiclassical Theories . . . . . . . . . . . . . . . . . . . . . . . A Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Primitive Semiclassical S-Matrix Theory . . . . . . . . . . . . . C . Integral Representations of the S Matrix . . . . . . . . . . . . . D . The Strong-Coupling Correspondence Principle . . . . . . . . . . E . Time-Dependent Methods . . . . . . . . . . . . . . . . . . . F. The Sudden Approximation . . . . . . . . . . . . . . . . . . . G . Other Methods of Quantizing Classical Trajectories . . . . . . . . V1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

. .

.

166 167 167 169 170 170 170 172 174 175 176 181 182 183 186 186 187 190

193 194 195 197 198 200

*Present address: Department of Atomic Physics. University of Newcastle upon Tyne. Newcastle upon Tyne NEI 7RU. England . t Equipe de Recherche numero 137 du CNRS . 165 Copyright @ 1982 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-003818-8

I66

A . S . Dickinson and D . Richards

I. Introduction Intermolecular collisions are responsible for a wide variety of observable phenomena. Among the simplest energy-transfer processes are those involving vibrationally and rotationally inelastic collisions. In this review we concentrate on some of the theoretical methods needed to understand these processes. For these relatively simple collisions there is a wide variety of methods available, each with differing ranges of validity and accuracies. The choice of method depends upon the physical quantity needed, the required accuracy, and the system being studied. Thus, there is no universal practical tool equally useful for all processes and systems; careful thought is required in choosing a method. The available approximations fall into two classes: those based upon a purely quantal description and those involving a classical description of at least part of the system. Here we discuss only the latter methods. For more general reviews in this area see in particular Bernstein’s ( 1979) book Aror?i-Mol~wilcCollisioti T I ~ o r yand ~ also Dickinson ( 1979), Connor (1979), Child (1980), De Pristo and Rabitz (1980), Loesch (1980), and Muckerman (1981). Limitations of space have forced us to restrict our consideration mainly to atom-molecule inelastic collisions. Applications of classical methods have become increasingly widespread in atomic and molecular physics over the past 15 years. Apart from the reviews mentioned above, see Berry and Mount ( 1972) for atom-atom scattering, Percival and Richards (1975) and more recently Olson (1980) for applications to electronic processes, particularly charge transfer and ionization, Tully (1981) for the scattering of atoms by periodic surfaces, and Brumer (1981) for intramolecular energy transfer in polyatomics. There are at least three reasons for using semiclassical methods for heavy-particle collisions. First, purely quantal methods are too expensive to apply routinely without making drastic and unquantifiable approximations. Further, the accuracy of the available potential energy surfaces does not generally warrant such sophistication. Second, for heavy particles one has an intuitive feeling that a method based upon exact classical trajectories does rather less violence to the physics than do many quantal approximations. Finally, in many cases a study of the trajectories can yield insight into the collision dynamics, leading to either new approximations or a better understanding of old ones; for example, classical trajectories naturally focus attention upon the different time scales of a collision. We have attempted to look classically at some of the important approximations currently in use in quantal studies in this area.

INELASTIC HEAVY-PARTICLE COLLISIONS

167

Even within the class of semiclassical methods there is still a bewildering variety of approximations, the validity and accuracy of which is often unclear. One problem in understanding these methods is that they often work well for some systems but not for others, and so an understanding of the theory behind them is essential to their confident application. Such an understanding is still incomplete. Nevertheless, such methods are exceedingly useful, and there is now a wide variety of calculations to support their use in many different circumstances. As a general rule a purely classical description of a collision is accurate if highly averaged quantities are required, for example, mean energy transfers or rotational relaxation times. For more detailed quantities some quantal effects may need to be included. Often this is straightforward and the errors are at least qualitively understood, but in other areas we still have no practical tool for introducing significant quantal effects into classical calculations. In Section I1 of this article we introduce some of the basic classical mechanics needed to describe heavy-particle collisions. As reliable generalizations about the applicability of various methods are almost impossible, we concentrate in Section 111 on one problem-the rotational excitation of rigid linear molecules by structureless particles. We use this process to introduce and discuss various approximations. At the end of this section we review very briefly recent work on atom-vibrating-rotor and rotor-rotor collisions. Next we turn our attention to more general methods. In Section IV we discuss the approximate evaluation of a class of integrals arising in the general theories of Section V. There we describe the semiclassical theories which have been applied to heavy-particle collisions, attempting to show their interrelations and their relative advantages and disadvantages. In Section VI we present our conclusions.

11. Angle-Action Variables A.

ISOLATED BOUNDSYSTEMS

An isolated quantal system of N degrees of freedom and with Hamiltonian H o can be represented by a distribution in phase space. In order to apply classical mechanics to a collision, we need the appropriate classical approximation to this distribution. This approximation is obtained by utilizing a set of variables in which the momenta I = (II,Z2, . . . , IN) are

A. S . Dickinson and D . Richards

168

equivalent to the quantum numbers n = ( n l , n 2 , . . . , n,) of the system; these are called thecictiori rwricibles and are constants of the motion taking the values where the Maslov indices y (see, e.g., Percival, 1977, and references therein) take different values according to the nature of the motion in the k t h direction. For libration y = 1/2, and for rotation y = 0. The irrigir rwricihle Ok,k = 1, . . . , N , is conjugate to lk. Since the l k are all constants, Hamilton’s equations of motion give

e(r)= o(I)r + 4,

o =

aH,/aI

(2)

where &(I) is the Hamiltonian expressed in angle-action representation and o are the fundamental angular frequencies of the system. Often systems of interest are degenerate; then it is possible to choose the angleaction variables so that one or more of the fundamental frequencies are zero (Born, 1960). One such example is the rigid rotor treated in Section 111. The angle variables have the important property that all dynamical variables F defined on phase space have period 2ir in each 1 3 ~ : F(61,.

9

.

9

6)k f

2n,.

.

3

6,)

= F(61,

+

.

9

..

7

6,)

(3)

This can be understood geometrically as follows. Because there are N constants of the motion, a phase curve is confined to an N-dimensional surface in the 2N-dimensional phase space. For bounded motion this is almost always an N-dimensional torus; the action variables label the tori, while the angle variables label points on each torus. Because the action variables take specific values, the uncertainty principle precludes knowledge in quantum mechanics of the angle variables. , because of relaEach of these is randomly distributed over [0, 2 ~ 1 and, tion (2) between each angle variable and the time, this distribution is uniform. Thus, the distribution on the surface of each torus is uniform. For I given by Eq. ( l ) , this distribution of phase points corresponds to the quanta1 distribution for a system in state n. Usually a problem is formulated in a different representation, (9, p), say, and so we need the canonical transformation from (9, p) -+ €3, I) representation. In general this is difficult to find, but in most current applications the target Hamiltonian is separable, so that the elementary method described in standard texts may be used (see, e.g., Goldstein, 1980; Born, 1960). The simplicity of these special cases can be misleading; even for systems of one degree of freedom there can be problems, and we return to a discussion of this in the conclusion.

INELASTIC HEAVY-PARTICLE COLLISIONS

169

B. CLASSICAL COLLISIONS I) Consider the collision of a target having angle-action variables (8, with another system, and suppose that the target Hamiltonian is the same before and after the collision. The initial unperturbed target motion is

e = o(I)t + + i ,

I

=

I ~ , t

<< o

(4)

being the value of the angle variable at some appropriate time originusually taken to be the instant at which the projectile on its unperturbed trajectory is closest to the target. After the collision, the final value of the action variables I, will depend upon the initial conditions, in particular ( + i , Ii), as well as other parameters such as total energy and impact parameter. Similarly, a long time after the collision the angle variables behave as +i

Q = O(I,)f

+ &(+i,

Ii),

I

>> 0

(5)

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

=

I

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

(6a)

where

(7)

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

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

. . , 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’

0

I

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 “rainbow” angle for da/dR, as a characteristic boundary, for 8 > 8,, the repulsive anisotropy causes substantial energy transfer, while for 8 << 8,, the long-range anisotropy induces only small energy transfer. For 8 S Or, small to intermediate energy transfer predominates (Loesch, 1980). As an example of dzu/(dfl dZ,) we show in Fig. 3 a comparison for Ar-CO, between the measurements of Loesch (1976) and the calculations of Preston and Pack (1977). Other comparisons between classical theory and experiment for d2a/(dR dZ,) have been made for Li+-H2 (Barg et d., 1976) Li+-CO (Thomas, 1977; Thomasef d . , 1978, 1980). For the latter the

A . S . Dickinson and D . Richards

176

0

90

180

k

0

90

180

8

FIG.3. Graphs showing the relative angular distribution P ( 0 ) = sin 8 ao/an for the rotational excitation of CO, by Ar. The histograms represent the theoretical results and the solid curves represent the experimental results (Loesch, 1976). (From Preston and Pack, 1977.)

observations at lo" (Eastes et d.,1979) show structure absent from the classical calculations (Thomas et d., 1980). This is probably due to quantal interference effects occurring with an attractive potential. For the calculations of rotational relaxation rates see Parker and Pack (1978) and references therein. The I,,, dependence of cross sections has been investigated by Pattengill (1979b).

F. APPROXIMATE METHODS

I. Cltrssijctitioti It is useful to divide these into two groups: perturbative and decoupling. The former have a long history in mechanics, while the latter are more recent in their widespread application. These methods, such as energysudden and centrifugal-sudden approximations, depend on separating motions of different natural frequencies. In quantum mechanics this can produce dramatic savings in computational effort (see Dickinson, 1979, and Kouri, 1979, for reviews.) Classically these approximations reduce the number of equations of motion to be solved, with a much less marked

177

INELASTIC HEAVY-PARTICLE COLLISIONS

effect on the computer time (Mulloney and Schatz, 1980). However, we consider that a classical approach can give useful insight into the applicability of these approximations. 2. Pertirrhcition Theory To obtain approximate trajectories for the Hamiltonian, Eq. ( 1 l), we employ as the zero-order approximation relative motion in a spherically symmetric potential v ( R ) and free motion for the rotor. For normally the leading term in the Legendre expansion of V ( R , x) is employed, but see also Dickinson and Richards (1978). Classical perturbation theory (CPT) can be presented neatly in terms of the first-order contribution, A , to the action, the generating function of the transformation from initial to final variables (Goldstein, 1980):

v

using R(b. f ) obtained from motion in are determined from Alj = -dA/a+,,

v. Changes in the action variables AIm =

-aA/a+,

(23)

If the usual Legendre expansion of V ( R , x) is available, then the action may be expressed in a particularly simple form which separates the impact parameter and orientation dependence (Dickinson and Richards, 1974; Dickinson, 1980). Note that if CPT is used to obtain cross sections using Eqs. (14), conservation of flux is automatically achieved in contrast to quanta1 perturbation theory. Perturbation theory requires that the effect of the perturbation on both the relative motion and the rotor motion be small. For the former this condition will generally be satisfied if 8'is nearly independent of the rotor orientation. Numerical calculations (e.g., Loesch 1980) show that this is often the case. For the rotor we require Alj/lj to be small. Cohen and Marcus (1970) have tested numerically the accuracy of CPT for rotational excitation, particularly for AZ,, and generally confirm these qualitative arguments. Cross and Herschbach (1965) have examined two less drastic forms of perturbation theory. Apart from the early work of Cross (19671, nearly all calculations of the relative motion using CPT have made the additional, independent, sudden approximation and are discussed in the following section.

178

A . S.Dickinson and D . Richards

3. The Sudden Approximution

This method is applicable when the rotor moves slowly compared to the incident particle. Then the rotor can be held fixed during the collision, giving the simpler problem of scattering by an anisotropic potential. The results for each target orientation are then averaged over all orientations; that is, relative motion and target motion are dealt with successively rather than simultaneously. While the angles of deflection can be obtained directly, excitation of the target must be inferred from angular momentum conservation. For an initially stationary target15 = IL' - L'I, L being the angular momentum of the relative motion, andl', = L - L (Cross and Herschbach, 1965; see also Section V,F). The derivation of the cross section then proceeds as before (Section 111,B). For an initially rotating target, while the trajectories are not affected, the final values of I, and I, now depend upon their initial values. Following Bhattacharyya and Dickinson (1979), it can be shown that

where A = Zj + I ) , B = IIj - Zjl ,giving the usual extra integral for initially rotating targets. The quantal analogs of Eq. (24) are well known (see Dickinson, 1979). It should be stressed that these interrelations are consequences of the assumption of a sudden collision and are unaffected by any subsidiary approximations. For distant collisions, where CPT with straight-line paths may be used, the sudden approximation is valid when z = w b / v << 1. The collision is expected to become more sudden as the' value of b decreases. Quantitative criteria for the validity of the sudden approximation for close collisions are poorly understood, but see Dickinson and Richards (1977). Because full classical trajectory calculations are not very expensive, the sudden approximation alone is seldom used. An interesting application to Nz+ H2 (treated as spherical) collisions has been made by Bergeron et nl. (1978). They noted that the rotation of the molecule in a full classical calculation was at most 15" in various coplanar collisions. Generally small differences were obtained between cross sections d a l d l , calculated with and without the sudden approximation, as occurred in the analogous quantal calculations. Several applications have been made of CPT combined with the sudden

INELASTIC HEAVY-PARTICLE COLLISIONS

179

approximation, particularly for small-angle scattering where straight-line paths can be used. With asymptotic forms of the long-range potential analytic results for the deflection angle are often possible (Gislason and Sachs, 1977; McFarlane and Richards, 1981b), which makes understanding the results easier. Udseth et a/. (1974) have considered H+-HF, HCI collisions; Gentry (1974) and Dickinson (1977) have obtained d u / d i l for charged particle collisions with strong dipoles; Budenholzer ef al. (1977) have inferred quadrupole moments from ion-molecule cross sections; Dickinson and Richards (1979, 1981)have investigated rainbows in &ddR with Lennard-Jones potentials, finding the stationary values of 8 analytically (see also Budenholzer and Gislason, 1978). For a discussion of CPT with curved trajectories and the sudden approximation, see Richards (1 98 la). 4 . The Body-Fixed Approximation

This approximation was suggested by the considerable success of the quantum j,-conserving coupled-states approximation (McGuire and Kouri, 1974; see Kouri, 1979; and Dickinson, 1979, for reviews). Expressing the Hamiltonian in a reference frame S ' = Ox 'y 'z ' with Oz ' parallel to R, we obtain (see Tarr et a/., 1977). H HRF H,

+ H, = f'i/2p + i2/2$' = HBF

=

( J 2- lJ2- 21:

(25a)

+ 1,212A + V ( R , X) -

(25b)

i2)/2pR2

+ (lf - 12,)"'(J2 - lk)'', cos 0,/2pR2

(25c)

Neglecting the correction, H, to the body-fixed (BF) Hamiltonian HBF, the problem becomes two dimensional, 7 and 1, are conserved, leaving only four coupled equations instead of the usual ten. Furthermore, as the final variables depend only on I , p , and O,, one fewer average is required. Thus, the structure of the cross section at a rainbow singularity is more pronounced than in the full calculation (Section 111,C). The most extensive comparisons of this approximation with full classical calculations are by Mulloney and Schatz (1980) for d u / d l , in He + H, collisions. Apart from the Al, = 52h cross section, good agreement was obtained. Tarr et a / . (1977) have examined some exact orbits to see if J,. R is indeed conserved during the collision. Apart from small-impact parameters, less than one-third the radius of the well, R, in the potential, this was not generally the case. Dickinson and Richards (1977) have examined the BF approximation within CPT. Forb = 0, the error in the action

180

A . S . Dickinson and D . Richards

[Eq. (22)] vanished, and it increased initially linearly with b . For straight orbits the relative error could be considerable, of order l/(n - 2) for potentials decreasing as R -n. In CIT the BF error arises directly from the neglect of the angular motion of the atom-the value at the midpoint of the collision is assumed to apply throughout. A less drastic approximation within CPT, conserving I,,, , has been discussed by Dickinson and Richards ( 1978). The findings in the classical investigations are broadly consistent with those obtained quantally (Kouri, 1979), but the numerical results of the quantal calculations appear to be rather more accurate than examination of individual classical orbits would suggest. Quite possibly the averages needed to obtain cross sections reduce the BF errors; this certainly occurs in the CPT approximation. In the BF approximation apparently no information about I , changes can be obtained. However, in the quantal version Am changes are ob1980, for a recent discussion). Physically, the BF tained (see Khare et d., approximation requires that I, be conserved along some axis, but unless this coincides with the quantization axis, transitions in I,,, will be obtained. Classically the symmetry axis of the collision is clearly associated with the “middle” of the collision, but for nonspherically symmetric potentials this direction is not well defined; see also Richards (1981b, 1982a). 5. The Body-Fi.ved- Su ddet I A pproximn tion

If we combine the two preceding approximations, the body-fixed and the sudden approximations, we obtain the classical analog of the quantal infinite-order-sudden (10s)approximation (Parker and Pack, 1978). We name this the body-fixed-sudden (BFS) approximation. Having made the sudden approximation, x is held fixed while calculating the relative motion in V ( R , x), and so 8 ( b , x) may be calculated by quadrature. For an initially stationary target we have

where the integration is over the trajectory in V ( R , x). This type of integral can be performed directly by quadrature techniques similar to those used for calculating 0 ( 6 , x) (Dickinson and Shizgal, 1975). Thus, no solution of coupled differential equations is required. To obtain cross sections for an initially rotating target Eq. (24) may be used. The BFS approximation in this form has been compared with exact classical calculations by Bowman rt ul. (1980). However, Mulloney and

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181

Schatz (1980) have integrated the rotor equations of motion in the timedependent field due to the BFS relative motion. This loses much of the simplicity of the BFS approximation and is not the precise classical analog of the 10s approximation. Application of the 10s approximation to transport collision integrals, with reference to earlier work using BFS ideas, has been described by Maitland et al. (1981). Because of the reduced averaging over orientation, the BFS approximation gives incorrect behavior near singularities in 13~cl(I3R dZ,) (Korsch and Richards, 1981). An example of the possible error due to the BF assumption alone is shown in Fig. 2. Application of the BFS approximation to scattering by a rigid ellipsoid is discussed by Korsch and Schinke (1981).

G. ATOM-VIBRATING-ROTOR COLLISIONS Now consider atom-vibrating-rotor collisions. Because of the extra degree of freedom very few converged quantal calculations, without subsidiary dynamical approximations, have been performed (Schaefer and Lester, 1975; Raczkowski et ul., 1978; Flower and Kirkpatrick, 1982). However, the calculation of many observables, such as vibrational relaxation rates or mean energy transfers, requires considerable averaging so that the effects of quantal interference should be small and a classical method should be accurate. However, quantal tunneling can be relatively important for vibrational transitions in diatomics at near-thermal energies. Consequently, the usual quasi-classical methods (see Section V,G) may be inaccurate. Because of the relative cheapness of classical calculations compared to the difficulty of fully quantal calculations, a large number of classical calculations are available; see Gentry (1979) for a review. More recent calculations include Poppe (1978) for He + H i , Poppe and Bottner (1978) for Li+ + N2), Wagner et al. (1978) for Li + H2, Thomas et al. (1978) for Li+ + CO, Stace and Murrell (1978) for He, Ar, Xe + 03,Dove et al. (1980) for He + HI, Schatz (1980) for Li+ + CO,, and Thompson (1981a) for He + N2 and (1981b) for Ar + HF. Work using classical-path methods to preserve at least vibrational quantization is discussed in Section V,E. Quantization methods based on the exact quantal solution for a forced harmonic oscillator such as the ITFITS method (Heidrich et ul., 1971), the DECENT method (Giese and Gentry, 19741, and the INDECENT method (Gentry and Giese, 1975) are discussed by Gentry (1979).

182

A . S . Dickinson and D . Richards

Rainbow effects (see Section III,C) in vibrational excitation have been discussed by Drolshagen et a/. (1981). There is considerable current interest in quantal calculations of vibrational transitions using the 10s approximation (Section 111,F,5) for the rotational degrees of freedom (Secrest, 1975; Schinke and McGuire, 1978; Clary, 1981). No analogous classical calculations exist, and a trajectory exploration of the validity of the approximation would be valuable.

H. ROTOR-ROTOR COLLISIONS Relatively few accurate quantal calculations on rotor-rotor collisions have been performed because the large number of states involved makes them very expensive. At present such calculations require that either the collision energy be small, or at least one collision partner (preferably both) have a large rotational constant, for example, H, or a hydride: H2-H, (Monchick and Schaefer, 1980, and references therein; Heil and Kouri, 1976; Green et d.,1978; Ramaswamy er al., 1978), on HCI-H, (Green, 1977), on CO-H, (Brechignac et d.,1980; Flower et al.. 1979), HD-D, (Buck et a/., 1981), and HF-HF (Alexander, 1980). For systems with smaller rotational constants typically quantum numbers are high, and then there are many accessible states so that classical and semiclassical methods are more appropriate. Many early experiments measured only highly averaged quantities, such as rotational relaxation times, or total rotational energy transfer. For these quantities a purely classical calculation is usually adequate and often gives reasonable agreement with experiment, even for the simplest potentials: H,-H, (Alper and Gelb, 1975; Gelb and Alper, 1975), OCSOCS, N,, H,, CO, , and HCN-HCN (Turfa et al., 1977), NrCl, (Nyeland and Billing, 1978), HCI-HCl, HF, HI, HBr (Turfa and Marcus, 1979), and N2N, and C02-C0,, depolarized Rayleigh spectral linewidths (Turfa and Knaap, 1981). Such calculations are relatively simple and inexpensive and may be useful in eliminating some potential surfaces (Alper and Gelb, 1975). Detailed cross sections are more sensitive to the potential, and the variation of the cross section between systems is more difficult to understand since it depends on so many parameters. At present only collisions between a few individual partners have been considered, for example, the detailed classical calculations of Sathyamurthy and Raff (1980) on C0,H,, also Alper et al. (1978) on HF-HF, and Gelb and Alper (1979) on

INELASTIC HEAVY-PARTICLE COLLISIONS

183

H2-HD. A more general understanding of such collisions is not yet available and, clearly, further detailed studies are required. When there are many accessible states, a purely classical treatment should be adequate (but see below) and often the quantal sudden approximation will also be valid. Alternative methods (exploited by Bhattacharyya et al., 1977, for HCI-HCl; Ray et al., 1976, for HCN-H2, Itikawa, 1975, for N,N,) are based upon the time-dependent Schrodinger equation (see Section V,E) and approximations to it. The relative merits of this method, which uses approximate dynamics but quantizes the bound state correctly, over exact classical trajectory calculations are not clear, and as far as we are aware no systematic comparison of the methods has been made. For systems interacting through dipole potentials the dipole-allowed transitions can have very large cross sections. For these tunneling may be important (see Section V,A), and so a purely classical treatment is invalid. However, the straight-line classical path method is normally valid. For fast collisions, the sudden approximation (Cross, 1971; McFarlane and Richards, 1981a) may be valid; for slower collisions, adiabatic corrections may be used (Alexander and DePristo, 1979; McFarlane and Richards, 1981a) to obtain estimates of the cross section.

IV. Uniform Approximations In many semiclassical scattering theories (see Section V), integrals occur of the form F ( A ;c) =

I*

dxg(x) exp i ~ f ( x ;c)

(27)

where A is a large real number [typically O(llh)I, x = (xl, . . . , x N ) ,c = (c, , . . . , cM) are a set o f M “control parameters,” for example, molecular orientations or impact parameter, andf and g are sufficiently wellbehaved functions of order unity. Usually the integral F is the first term in an asymptotic expansion in A. Because A is large, the integrand is rapidly oscillating, and so the numerical evaluation of F is difficult. However, this can make an approximate analytic treatment relatively easy. The integral is dominated by contributions from the neighborhood of the real singular points, xk(c),k = 1, . . . , t i , where a f / d x , = 0,

i = 1,.

. . ,N

(28)

184

A . S. Dickinson and D . Richards

Then the usual stationary phase analysis gives

F = Fsp= (27r/A)"* N

where H is the determinant of the Hessian matrix

where A: are the eigenvalues of the Hessian matrix. This approximation is valid if the singular points are well separated. If there are no real singular points, then, in general, little is known about the evaluation o f F . For one-dimensional integrals ( N = I ) , the method of steepest descent may be used (see, e.g., Dingle, 1973). Here the principal contribution comes from some of the saddle points in the comp1ex.r plane; the global properties of f ( x ) determine which saddle points are relevant, and this can be a difficult problem. For many-dimensional integrals (N 2 2), much less is known about such integrals and there are no standard techniques available (see, e.g., Ursell 1980). However, in the context of scattering theory we usually know on physical grounds that F decreases exponentially with A when there are no real singular points. The singular points x k of Eq. (27) are clearly functions of the control parameters c; as these vary over their domain, two or more singular points may coalesce; here H = 0 and Eq. (29) is invalid. If one requires some mean ofF(A; c) over a domain of c, including some ( M - 1)-dimensional region where two stationary phase points coalesce, then usually F sp is integrable, and often the replacement of F by F sp in the integrand, ignoring all complex singular points, gives a reasonable approximation. On the other hand, if one requires just F , and if two or more singular points are close, then usually a better approximation than Fspis necessary. In some circumstances this can be obtained by a uniform approximation. For clarity we first discuss these for one-dimensional integrals. There are two stages to finding a uniform approximation to Eq. (27). First we find a new variable 11' such that w ( x )and its inverse x ( w ) are single valued. This variable is defined implicitly by a relation of the form f(W) = f ( x )

(32)

INELASTIC HEAVY-PARTICLE COLLISIONS

185

so that

The second stage is to choosef(w) and approximations tog, 6, 2) so that

z, andg (say,

ci.

can be evaluated in terms of standard functions. For isolated singular points this yields the one-dimensional form (29); for two close singular points, the most common case in practice, we obtain the conventional Airy-function uniform approximation (see, e.g., Chester et a/., 1957), and for three close singular points we obtain the Pearcey uniform approximation (see, e.g., Ursell, 1972; Connor and Farrelly, 1981a). While the comparison functionT(w) of Eq. (32) must mimic f ( x ) , it is not unique. Elementary catastrophe theory (see, e.g., Poston and Stewart, 1978; Stewart, 1981) shows thatfexists and that for a given number of singular points we may choose a polynomial of a particular form, known as the universal unfolding. The fold,? = 1 t 1 3 + a w , and cusp,? = + w 4 + aw2 + pu., unfoldings produce the Airy and Pearcey comparison integrals, respectively. The fold catastrophe is commonplace in potential scattering (Berry and Mount, 1972), while the cusp catastrophe has been used to study potential scattering from potentials having both a maximum and a minimum (Connor and Farrelly, 1981b). In principle the same method may be applied to many-dimensional integrals, but there are three difficulties, one theoretical and two practical. The first problem is that the classification of the comparison functions is not complete unless only a few singular points coalesce. For example, with four close singular points the classification is complete, and with N = 2 the relevant universal unfoldings are (see Stewart, 1981; Connor, 1976) the elliptic umbilic (three saddle points and one maximum or one minimum) and the hyperbolic umbilic (two saddle points and one maximum and one minimum). In practice, four coalescing singular points is probably the maximum that is required. The two practical difficulties are as follows: first, in deciding which unfolding is appropriate [this is particularly difficult if f(x) is known only numerically], and second, the evaluation of the comparison integral itself is nontrivial, although presumably standard “packages” will eventually become available.

186

A . S . Dickinson and D . Richards

The power of elementary catastrophy theory is that it reduces many types of integral to a similar form, whose behavior can be understood once and for all. The Airy function is well understood; the Pearcey function can now be relatively easily calculated (Connor and Farrelly, 1981a) and the behavior of the elliptic umbilic has been described in some detail (Berry er a/., 1979). In many cases g(x) is periodic in each xi and

U(xl, . . . , xj + 2rr, . . . , x N ) = 2nmj + U(x) m j = 0,

kl,.

..

(35 )

Then elementary catastrophe theory, being a local theory, does not provide a classification scheme. Sometimes this does not matter; the periodicity may be ignored and the above theory used, for example, the usual uniform Airy approximation to the ordinary Bessel functions. Otherwise, different comparison functions are needed; for some integrals of this type, approximations based upon Bessel functions of integer order (Stine and Marcus, 1973) and noninteger order (Connor and Mayne, 1979a,b) have been used. Leubner (1981) has investigated a uniform approximation to the generalized Bessel function. The idea of a uniform approximation is very general and finds a variety of applications. Approximate solutions of differential equations, in particular, Schrodinger’s equation, may be found (see, e.g., Child, 1974; Dingle, 1973). Kriiger (1979, 1980, 1981) and Connor (1981) use the idea to evaluate Franck-Condon factors; Child (1975) uses this idea to evaluate one-dimensional matrix elements, and Child and Hunt (1977) use the known forced harmonic oscillator vibrational transition probabilities in the same spirit.

V. Semiclassical Theories A. INTRODUCTION Many semiclassical methods exist, and here, apart from Section V,E, we discuss methods that use a classical description of both target and relative motion. First, we describe the methods based upon exact classical trajectories; then we consider the use of approximate trajectories. In order to use exact classical trajectories, both relative and internal motion must be described classically. Depending on the nature of the

INELASTIC HEAVY-PARTICLE COLLISIONS

187

target, this can produce significant errors if the target quantum numbers and their changes are small. This is the main disadvantage of describing the collision classically; we discuss this problem again in Section V,F. For heavy-particle collisions, classical dynamics provides a sound basis for calculating observable quantities, although often some quantal flesh must be sewn on to these classical bones. For many systems the most important quantal effects are interference and tunneling. As the origins of these are quite distinct, different techniques are required for their description. In general tunneling is more important for soft potentials, for example, proton-polar molecule collisions, than for hard potentials, for example, atom-molecule collisions. Interference effects are due to two or more classical paths producing the same effect; a dynamical symmetry together with interference often leads to selection rules not present in a purely classical treatment. B. PRIMITIVE SEMICLASSICAL S-MATRIX THEORY

Primitive semiclassical S-matrix (PSC) theory enables interference effects to be obtained from the same classical trajectories giving the classical probability of Eq. (6b). This is done by associating a phase with each of the classical paths leading to the required transition, the roots of Eq. (8); the transition amplitude for each path is just this phase multiplied by the square root of the classical probabilities [Eq. (7)]. The probability for a transition is the square of the sum of these amplitudes, rather than the sum of squares, Eq. (6b). To be specific, and to simplify the notation, we now specialize to a one-dimensional system with particular reference to the collinear excitation of a diatom. Then, if (Q, P) are the position and momentum of the projectile, the primitive semiclassical S-matrix element is (Miller, 1970a)

:1

Q, = -

dr(QP

+ eZ),

=

(T~

sgn

(e)

(37)

the sum being over all real roots of Eq. (8). The action 0 is the generating function in momentum representation for the canonical transformation from the precollision to the postcollision coordinates (see, e.g., Goldstein, 1980). A more general formalism for multidimensional targets is given by Miller (1970b, 1975); see also Child (1976).

A . S . Dickinson and D . Richards

188

Comparing the PSC and classical [Eq. (9)] transition probabilities m

Ppsc =

2 Pf' + 2

xz

(f'f1f't1)1'2 cOS[@l -

@j

+ (7~/4)((Tl- a))] (38)

1>I

1-1

which differs from the purely classical expression by the addition of the interference terms. To illustrate this method, we show the results due to Miller (1970a) for the Jackson-Mott model for collinear vibrational excitation (Jackson and Mott, 1932) for which the Hamiltonian in angle-action representation is H

=

I

+ ( 2 m ) - P 2 + exp(-aQ +

cos 0)

(39)

In Fig. 4 the action &(c#JJis shown. It is clear that Eq. (8) has two real roots for 0 s nf s 4 but none for nf 2 5 , the classically inaccessible region. The probabilities for excitation out of the ni = 1 and tii = 2 states are given in Table I. Comparing the classical and PSC results we see that for some transi2, 2 + 1, and 2 3, interference effects are very tions, for example, I important. Also for nf near to the classically inaccessible region, for ex0, 2 + 0, 2 + 5 , the PSC approximation is poor, ample, 1 + 4, 1 generally overestimating the probability. Note that quantum probabilities for transitions into the classically inaccessible region, for example, 1 += 5 , 2 6, can be larger than those for transitions into the accessible region, for 3. example, 1 += 2, 2

-

-

-

-

-

4 0.2 0.4 0.6 0.8 1.0 +,/2r FIG.4. Graph showing the final action 12(4Jfias a function of the initial phase 4, (see Section II,A) for the Jackson-Mott model, [Eq. (39)] in the case m = 2/3, a = 0.3, total energy E = 10, and with initial action I , = 3h/2 corresponding to the first vibrationally excited state. The dashed line at I2 = 5h/2 intersects this curve at the two real classical paths contributing to the sum (36). (From Miller, 1970a.)

INELASTIC HEAVY-PARTICLE COLLISIONS

189

TABLE I TRANSITION PROBABILITIES I N T H E JACKSON-MOTT MODEL, Eq. (39), F O R THE PARAMETER rn = #, a = 0.3, E = 10"

Transition

Classical

PSC

Airy uniform

1+0

1-2 1-3 1-4 1-5

0.356 0.130 0.128 0.159 0

0.423 0.009 0.168 0.285 0

0.011 0.174 0.240 0.062

2+0 2- 1 2+3 2-4 2+5 2+6

0.212 0.131 0.105 0.114 0.169 0

0.416 0.009 0.020 0.165 0.262 0

0.381 0.011 0.017 0. I70 0. I94 0.045

0.21 1

Quanta1 0.218 0.009 0.170 0.240 0.077 0.366

0.009 0.018 0.169 0.194 0.037

From Miller (1970a).

These marked interference effects are not always observable. Generally, probabilities are averaged over one or more parameters, for example, the impact parameter and molecular orientations, and this can reduce these effects (Miller, 1971). However, for a target possessing a symmetry interference, effects may not average out. Particularly interesting effects arise when a near symmetry occurs leading to propensity rules; for atom-rigidrotor collisions, this effect has been systematically studied by McCurdy and Miller (1977); see also the discussion by Bhattacharyya and Dickinson (1982). The PSC method has been applied to atom-rotor excitation (Miller, 1970b; McCurdy and Miller, 1977) and to three-dimensional vibrational excitation of H2 by He (Doll and Miller, 1972) and by Li+ (Raczkowski and Miller, 1974). In general, for classically accessible processes the PSC method is quite accurate, but it fails near the inaccessible boundary. In the inaccessible region the theory outlined above gives zero probabilities, but by the introduction of the complex roots of Eq. (8) nonzero probabilities can be obtained (Miller, 1972). We discuss this further below. The main problem with the implementation of PSC theory is the purely classical problem of obtaining the solutions to Eq. (8); this is equivalent to solving a boundary value problem for a set of 2N coupled, nonlinear, differentialequations. For one-dimensional systems this presents no difficulty, but it becomes rapidly more difficult with increasing degrees of

190

A . S . Dickinson and D . Richards

freedom. McCurdy and Miller (1980) have suggested that the application of relatively sophisticated linear optimization techniques makes this root-finding problem tractable.

C. INTEGRAL REPRESENTATIONS OF T H E S MATRIX There are a variety of independent integral forms for semiclassical S-matrix elements, all of which give the same result if evaluated using the stationary-phase approximation, Eq. (29). The simplest and easiest to use is the "initial value representation" (IVR) integral form due to Miller (1970a). For one-dimensional systems, this is

-

S(ni nr) = (2nI-l /02Td4ila42/t@ilexp i A / h A

= &(I2

- Zi) - @

+h

~ / 2

(40) (41)

where all quantities are defined in Section V,B. Because of the emphasis on the initial state, this form of the transition amplitude does not satisfy detailed balance, although in practice this produces serious error only for the smallest probabilities (see Table I1 below and also Section V,D). By changing the independent variable to 42[Eq. ( S ) ] we obtain the final value representation (FVR) of Marcus (1972), who also gives an alternative expression which does satisfy detailed balance, but is not so easy to use. Double-integral forms of S(ni n f ) have also been given. Miller (1972) gives S as an integral over both initial variables (ai,I,); Pechukas and , in Child (1976) give S as an integral over the initial and final pair ( c # I ~42) the particular case of a forced linear oscillator. In the latter case it is instructive to derive Eq. (40) by doing one of the integrals by the stationary phase approximation. We then see that the integrand of Eq. (40) is invalid in the neighborhood of the point where a4,/84, = 0, where two singular points coalesce. However, this region of integration is not usually important. The integral (40) is of the form (27) so that the methods described in Section IV may be used for its evaluation. Since @ is the action in momentum representation, a@/a12= 42 and consequently A is stationary at the roots of

-

(W2/Wi"2(I,,

4d

-

If1 = 0

(42)

Usually the first factor of this equation is nonzero, and so we regain Eq. (8). After some manipulation (Miller, 1975), we find that the stationary-

INELASTIC HEAVY-PARTICLE COLLISIONS

191

phase approximation to Eq. (40) gives the PSC S matrix. If the roots of Eq. (8) are too close for a stationary-phase approximation to be valid, the more general uniform approximations of Section IV may be used. In these approximations, because the initial and final states are treated symmetrically, the integral (40) and the symmetrized versions of Marcus (1972) give the same result and both satisfy the detailed balance requirement. In the classically inaccessible region the roots of Equation (8) are complex. For one-dimensional problems the method of steepest descent may be used; this is how the complex paths mentioned in Section V,B originate. Clearly, use of the integral form (40) necessitates a qualitative understanding of the behavior of &(4J.For weak collisions classical perturbation theory shows that 42 - 4, is a small periodic function of so that 84,/841~# 0. As the collision strength increases, geometric arguments, similar to those described by Pechukas and Child (1976; See Fig. l), show that c $ ~= (sin &)/a,a being a measure of the collision strength. Thus, for strong collisions the numerical evaluation of Eq. (40) will be difficult as usually the position of the square-root singularity is not known. This is also a problem in the many-dimensional case and is discussed by Kreek and Marcus (1974) in their calculations on rigid-rotor excitation. If the roots of 842/84iand Z2(+J = Zfdo not coincide, then a uniform approximation to Eq. (40) is preferable. Otherwise, Eq. (40) is invalid, and a uniform approximation to one of the many-dimensional integral representations mentioned above is necessary; however, the occurrence of this coincidence is rare and in most cases of little physical significance. For classically inaccessible transitions the roots of Eq. (8) are complex, and so a uniform approximation to the integral necessitates integration of Hamilton’s equations into the complex time plane. Since the analytic structure of the solutions to Hamilton’s equations is usually unknown, in order to avoid unknown singularities it is necessary to stay fairly close to the real time axis, meaning that the transition must not be too inaccessible. A further constraint is that for a piecewise continuous potential the radius of convergence of its analytic fit may be small. Nevertheless, it now appears feasible to integrate stably into the complex plane (McCurdy and Miller, 1980, and references therein). In Table I1 we compare the Airy-function uniform approximation to the integral (40) with a direct numerical evaluation and with exact quanta1 results for the Hamiltonian (39) (Secrest and Johnson, 1966). At the low energy the direct numerical integration is most accurate, while at higher energies the uniform approximation is, paradoxically, better. Connor and Mayne ( 1979b) have made an extensive comparison of different uniform approximations to this problem.

A . S . Dickinson and D . Richards

192

TABLE I1 TRANSITION PROBABILITIES FOR THE JACKSON-MOTTMODEL,Eq. (39). FOR T H E PARAMETERS m = f , (Y = 0.3" Transition E

E

3: I-tO 1 4 2

Uniform

Integral

Quanta1

0.012b

0.25b 0.001'

0.022b 0.0009b

0.1 176 0.04Sb 2.1(~5)~

0.108b

=

0.013b

= 4:

1 4 0

0.108"

1 4 2 1 4 3

0.044"

''

1.5(-5Y

0.042b I.5(-W

All transitions are classically inaccessible. From Miller (197Oa). Mean of the I -+ 2 and 2 + 1 transition probability. Stine and Marcus (1973).

The advantage of the uniform methods of approximating the integral (40) is simply that the value of the integral is given by the few trajectories satisfying the boundary values: in one-dimensional problems usually only two trajectories are required. Of course, extra trajectories are calculated in solving this boundary value problem. The numerical evaluation of the integral (40) requires a number of trajectories proportional to the number of oscillations in the integrand, and this is roughly proportional to the number of classically accessible states, which can be quite large. For many-dimensional systems both of these problems are more serious. The computational effort required to find the relevant trajectories increases as N N being the number of degrees of freedom, and Q 4, while the effort required to evaluate the integral numerically increases exponentially with N . Thus, the uniform methods of evaluating these integrals are important, but, as discussed in Section IV, uniform approximations for N 2 3 do not yet exist, and for N = 2 the generic integrals are not readily calculated. This difficulty is seen in the recent calculations of McCurdy and Miller ( 1980) on the three-dimensional problem of the simultaneous vibrational and rotational excitation of a diatomic molecule. There the PSC results agree well with quanta1 calculations, except near the classically inaccessible region, where they are too large; this is the region.where a uniform approximation is necessary. For the two-dimensional problem of the excitation of a rigid rotor suitable comparison functions for the uniform integrals do exist and applications are in progress (Uzer and Child, 1982; Uzer et d., 1982).

193

INELASTIC HEAVY-PARTICLE COLLISIONS

D. THESTRONG-COUPLING CORRESPONDENCE PRINCIPLE While classical S-matrix theory has usually provided satisfactory results, there remain serious problems in its implementation for systems of many degrees of freedom. The direct evaluation of the integral is difficult because of the large number of trajectories required, and the appropriate uniform approximations are difficult to apply or nonexistent. These problems may be avoided by using classical perturbation theory (CPT) to approximate the motion. This is described in Section III,F,2for rotational excitation; the general method is similar. We suppose that the interaction between the target and projectile can be written in the form + V , , where t depends only upon the separation of the target and projectile, and V, depends upon these coordinates and the internal target coordinates. The first approximation is to suppose that the relative motion is determined by V; then V , depends upon the time and internal coordinates only. Using this approximation and Heisenberg's form of the correspondence principle, we obtain the strong-coupling correspondence principle (SCCP) [Percival and Richards, 1970; see also Clark et al., 1977, for an alternative derivation, and Miller and Smith, 1978, for the connection with Eq. (40)]

v

I

S SccP(nl+ nr) = (2n)+' d + exp i [ +

(nf - n3

-

Al/hl

(43)

where A, is the first-order change to the target action: Al

dtV,(ot

= -

+ +, I, f),

I = (Ii + I3/2

(44)

--I

where V , is the time-dependent potential expressed in terms of the angleaction variables of the target, where = dii, [Eq. (411. Note that the SCCP is an approximation to the time-dependent Schrodinger equation (Section V,E), while the methods discussed in Sections V,B and V,C approximate the solution of the time-independent Schrodinger equation. Both types of approximation are generally valid only if all quantum numbers are large, but the SCCP has the additional restriction, due to the use of CPT,

+

IniJ>> Inf

-

nil

(45)

Like the integral form, Eq. (40), Eq. (43) does not satisfy detailed balance; also the additional dynamical approximations destroy energy conservation. However, the condition ( 4 9 , if satisfied, means that these laws are not too seriously violated. A priori adjustments to enforce these conditions can often improve the accuracy of this method (see, e.g., Percival and Richards, 1971 ; Dickinson and Richards, 1974).

194

A . S . Dickinson and D . Richards

The advantage of this approximation over the classical S-matrix theory of the previous sections is that the action is much easier to calculate; often it can be obtained in terms of elementary functions, or a few onedimensional integrals. When an orientation-averaged cross section is required, these integrals are usually orientation independent; this consequence of CPT leads to considerable computational savings. Thus, although Eq. (43) is similar to Eq. (40), a direct numerical evaluation is usually feasible, thus avoiding the problem of uniform approximations and complex trajectories. The SCCP has been successfully applied to a variety of collision processes. The excitation of hydrogen atoms by protons has been considered by Lodge et ul. (1976, and references therein) and by electrons (Percival and Richards, 1975, and references therein). Clark and Dickinson (1971) have applied it to the forced harmonic oscillator, comparing it to exact quantal results and to various quantal approximations. The excitation of rigid rotors by atoms is considered by Dickinson and Richards (1974), where results for He-N, are compared with exact quantal calculations. The relation of the SCCP to other approximations and comparisons with other quantal results are given by Dickinson and Richards (1976). Simultaneous rotational-vibrational excitation of H, by He has been considered by Clark (1977). Application to rotor-rotor collisions is in progress (Munoz, 198 1; McFarlane and Richards, 1982). Comparisons with collision-induced fluorescence :xperiments on Liz have been made by Bhattacharyya et a/. (1980). Usually the S-matrix depends upon an impact parameter, but on using Heisenberg’s correspondence principle this may be converted directly to momentum transfer representation for inelastic differential cross sections. This had been applied to small-angle vibrational excitation of CO and COz by protons (Richards, 1982b). The first application of the SCCP to nonseparable targets, atom-triatom collinear collisions, has been reported by Farantos and Murrell (1981). Good agreement with quantal calculation for the He-CO, system was obtained but results for Kr-CO, were less satisfactory.

E. TIME-DEPENDENT METHODS One of the oldest semiclassical approximations in atomic and nuclear physics uses a classical path for the relative motion, often, but not necessarily, a straight line, yielding a time-dependent perturbation acting on the target. The resulting time-dependent Schrodinger equation for the target is solved numerically, thus retaining exact target quantization. (For the special simplifications when the sudden approximation is valid see Section

INELASTIC HEAVY-PARTICLE COLLISIONS

195

V,F.) The relationship between S-matrix elements and classical path amplitudes is examined in detail by Broglia et al. (1974) and Gaussorgues et 01. (1975). The basic conditions for the validity of the classical-path method are that the change in energy AE and angular momentum AL of the relative motion be small compared to their initial values. These changes need not be small compared to the corresponding initial values of the target. Except perhaps for close collisions, both conditions are usually readily satisfied in heavy-particle collisions. Partly because the numerical work remains substantial this approximation has not been widely used in inelastic atom-molecule collisions, and no systematic comparisons with full quanta1 solutions exist. Its most widespread application has been to processes dominated by distant collisions for which straight-line paths may be employed-see Jamieson et al. (1975) for H+-CN rotational excitation, Bottcher (1979) for Na+-HCN collisions, Bouloy and Omont (1977, 1979), Elitzur (1977a,b), Sakimoto (1980, 1981), and Takayanagi (1978, 1980) for charged-particle dipole collisions, and Strekalov for rare-gas CsF rotational excitation. The calculation of proton-induced fine-structure transitions using hyperbolic paths is considered by Doyle et al. (1980), who refer to earlier work on fine-structure transitions. Curved trajectories for Ar-HC1 collisions have been employed by Neilsen and Gordon (1975) and Band (1979), for He-NH, by Davis and Boggs (1978), and, with subsidiary approximations, by Saha et af. (1974), for He-TlF by Saha and Guha (1975), for Ar-TlF by Bhattacharyya et al. (1978), and for He-HD+ by Ray et al. (1978). Schinke (1980) has used a body-fixed approximation in a model rainbow investigation. Vibrational excitation with a classical-path approximation has been investigated by Schinke (1977), McKenzie (1977), and Iwamatsu et 01. (1981). Related, more complex, calculations making some allowance for back-coupling between the target and the trajectory have been performed by McCann and Flannery (1978, and references therein). Billing (1974) extended the classical path method, treating rotational coordinates classically, while vibration was treated quantally. Application has been made to a variety of collisions not easily treated by other means-see, for example, Billing (1980a, He + HD; 1980b, CO + Nz; 1980c, N2 + Nz); (1981a, Li+ + COz, N,O); (1981b, Ne + COz). F. THESUDDEN APPROXIMATION Here we briefly discuss the sudden approximation applied to rotational transitions (Section 111), emphasizing its potential importance in quantizing classical trajectories. We ignore vibrational transitions partly for lack

I96

A . S . Dickinson and D . Richards

of space and partly because physical conditions leading to sudden vibrational transitions are less common (see, e.g., DePristo, 1981, and references therein; Dickinson, 1981b). Heavy-particle collisions usually involve large values of orbital angular momentum, and so it is natural to use a classical description for the relative motion. Ideally we should like to treat the target quantally and the relative motion classically; unfortunately, such a theory cannot be made consistent without making additional approximations. For example, a perturbative treatment of the relative motion leads to the time-dependent Schrodinger equation (see Section V,E). However, for sudden collisions the coordinates of the internal and the relative motions may be treated separately. In a purely classical treatment this means that changes in the target may be inferred from the relative motion. For rigid linear molecules, angular momentum conservation may be used (Cross and Hershbach, 1965), but for other targets a more sophisticated method is needed. In a quanta1 treatment the analogous approximation is that of Drozdov (1955) and Chase (1956). This relates the elastic amplitude of the projectile moving in the anisotropic field of the “frozen target”.?(O, @: R), R being the target orientation, to the inelastic-scattering amplitude:

f(n

+

n’:8,a) = (n’f((8,

a: R)Jn)

(46)

Generally the values of the orbital angular momentum contributing tof are large so that it may be approximated semiclassically, avoiding any approximation to the target quantization even for low quantum numbers. We are not aware of work using exact classical orbits to evaluatef, although work in this direction is in progress. However, several approximations tof have been studied. Any 10s approximation (see Section 111,F,5)for atom-linear rotor scattering using WKB phase shifts (the routine form) implicitly uses Eq. (46). For differential cross sections the usual semiclassical theory for spherical potentials (see, e.g., Berry and Mount, 1972) may be used to find7 and hencef using Eq. (46). This is equivalent to the approach of Korsch and Schinke (1980), who started from the 10s inelastic theory; these authors also used a uniform approximation for the matrix element (46)-in this case a one-dimensional integral-to obtain a very simple result. Unfortunately, only for atomic scattering from linear targets does this approach employ the well-known semiclassical results for isotropic scattering; otherwise, more complex problems remain after using the 10s approximation. When the isotropic part of the interaction potential dominates the relative motion, classical perturbation theory may be used. This gives another

INELASTIC HEAVY-PARTICLE COLLISIONS

197

approximation tof (Richards, 1981a; Dickinson, 1981a; Stolte and Reuss, 1979) similar to that arising in isotropic potential scattering. This should be useful in the intermediate angle region, and is not restricted to linear targets. The Kramer-Bernstein (Kramer and Bernstein, 1964)form of the sudden approximation also uses CPT in an impact-parameter representation (see Nyeland and Billing, 1981, and references therein). In the small-angle region, straight-line trajectories may be used to approximate the relative motion. Thenf may be obtained from the Glauber ( 1955) approximation. This has been applied extensively to electron-atom scattering (see, e.g., Byron and Joachain, 1977) and electron-molecule scattering (Ashihara et a/., 1975; Gianturco et al., 1978; Gianturco and Rahman, 1978), but has found little use in atom-molecule collisions, although it is an ideal theory for studying small-angle inelastic and total cross section, provided the sudden approximation is valid (see Dickinson and Richards, 1980; Reuss, 1980; and references therein). An alternative approach via individual quantal S-matrix elements, with successful application to electron-dipole collisions, has been described by Allan and Dickinson (1 98 1).

G . OTHERMETHODSOF QUANTIZING CLASSICAL TRAJECTORIES If the quantization of the target is the only quantal effect expected to be important, then a purely classical calculation suffices. Exact or approximate dynamics may be used, but almost always additional approximations are needed to calculate the classical transition probabilities. Consider the collision described in Section II,B with a transition probability given by Eqs. (6a) and (6b). An additional approximation is usually made in evaluating the integral (6a); this may be avoided by using Eq. (6b), which involves solving a boundary-value problem. As discussed above, this is usually less practical than approximating Eq. (6a). The usual approach is to replace each 6 function by a high, narrow rectangle of unit area and to perform the subsequent integral using a Monte Carlo technique. This method, described by Pattengill (l979a), has the advantage of being very easy to implement; it is often called the “boxing” or “histogram” or “quasi-classical” method. Clearly, a 6 function may be approximated in infinitely many ways; in practice there is little to choose between those approximations which are practical to use. When results depend on the choice made, probably this quantization technique is inappropriate (see Clark er a/., 1977, for a discussion). Truhlar et al. (1981) have considered triangular and parabolic, as well as rectangular approximations, and noted no significant difference.

198

A . S . Dickinson and D . Richards

A different approach to the approximation of Eq. (6a) is to express Z, in terms of a complete set of functions, the coefficients of which are given by 1977, p. 76). This method has been a simpler integration (see Clark et d., applied to a variety of problems by Truhlar and Blais (1977), who conclude that for a given number of trajectories it is generally no more accurate than the histogram method. Nesbet and Clary (1979) approximate Eq. (6a) by putting the system in a box, thus converting the collision to a bound-state problem. Then they use the standard EBK theory and techniques to find the invariant tori of these bound states (Percival, 1977). Further work has been reported (Bruinsma and Nesbet, 1981a,b). This method has been successfully applied to the collinear vibrational excitation of H2 by He, but clearly further work is needed, in particular, the application to many-dimensional systems. Other methods for obtaining transition probabilities from classical trajectories, avoiding Eq. (6a), have been proposed, but all have a very weak theoretical foundation. Ramaswamy and DePristo (1981) use a “continuous quantization procedure”; this cannot be recommended as it is wrong in the weak-coupling limit. Truhlar and Duff (1975) obtain the transition probabilities for collinear vibrational excitation using the equation 11, +I11

(AEk)=

~

n ‘=nl

I

AE,k,,P(n-*n’),

k

=

1,.

..

,m

(47)

calculating the energy transfer moments ( A E k ) , k = 1, . . . , m , classically. In their application this o n m z appears better than the direct application of Eq. (6a). Variations upon this theme are discussed by Truhlar et a/. (1981). A completely different approach has been tried by Lee and Scully (1980), who combine classical trajectories with Wigner distribution theory to obtain the transition probabilities of the collinear vibrational excitation of a harmonic oscillator. They obtain surprisingly accurate results, even for some classically forbidden processes. The reasons for this are not clear, but the work of Heller (1976), Heller and Brown (1981), and Sebastian (1981) suggests that it may be partly because the problem is close to a linearly forced harmonic oscillator. Clearly this method needs further study before it can be used reliably.

VI. Conclusions Classical and semiclassical methods are widely used in inelastic atomic and molecular collisions partly because, for many applications, they are

INELASTIC HEAVY-PARTICLE COLLISIONS

199

cheaper than any approximate quantal calculation of comparable accuracy and partly because they provide greater insight into the collision dynamics. Purely classical methods are relatively simple to use and are particularly useful when highly averaged quantities are required. Often simple quantization of classical trajectories is necessary; although there is a variety of methods available (see Section V,G),we recommend that the computationally most convenient should be used, there being little theoretical justification for any other choice. Other, more sophisticated, rnethods are not so easy to apply routinely (see Section V), and a compromise between approximate dynamics, approximate quantization, and computational convenience is usually necessary. It is worth noting that existing applications are to targets having separable Hamiltonians and for which WKB quantization is very successful, even for small quantum numbers. Not all separable targets are so easily quantized; for the asymmetric top or hindered rotations there are problems with quantization in the neighborhood of the separatrix dividing librational from rotational motion. Most semiclassical scattering approximations break down in this region, and little is known about the accuracy of trajectory methods under such circumstances. However, only very detailed observations would be able to probe this sensitive region. Another caveat concerning existing experience with semiclassical methods is that much attention has necessarily been paid to excitation of harmonic oscillators. This target is atypical because of the correspondence identities, and so semiclassical methods applied to it are usually particularly accurate. Accordingly, success for this problem should never be taken as sufficient grounds for the adoption of a semiclassical procedure. The methods discussed here are also valid, in principle, for nonseparable but integrable targets and in the regular region of nonintegrable systems where EBK quantization may be used (Percival, 1977). However, very little work has been carried out on such systems. In the irregular region there are difficulties with any description of the collision process. Areas where work remains to be done are: (1) quantization using the sudden approximation; (2) quantal effects on rainbow singularities where some averaging occurs; and (3) a classical study of reorientation cross sections, particularly with the 10s approximation. In view of the success of this dynamical approximation, it is striking that few analogous classical calculations appear to have been performed. When fully quantal calculations are warranted, we recommend a preliminary classical investigation of any proposed dynarnical approximation. Validity of the dynamical approximation in a classical calculation would confirm its applicability in the quantal calculation, where direct verification is often impossible.

200

A . S . Dickinson and D . Richards ACKNOWLEDGMENTS

We thank Dr. F. J. Wright for very valuable comments on a first draft of Section IV. A. S. D. thanks Professor R. McCarroll and Universite de Bordeaux I for hospitality while writing this article, and the Royal Society for making possible this visit.

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A . S . Dickinson and D . Richards Doll, J. D., and Miller, W. H. (1972). J . CIietri. Phys. 57, 5019. Dove, J. E., Raynor, S., and Teitelbaum, H. (1980). Clietn. Phys. 50, 175. Doyle, J. G., Kingston, A. E., and Reid, R. H. G. (1980). Astron. Astrophys. 90, 97. Drolshagen, G., Mayne, H. R., and Toennies, J. P. (1981). J. Clierii. Phys. 75, 196. Drozdov, S. I. (1955). So\*. Pliys. JETP 1, 591. Eastes, W., Ross, U., and Toennies, J. P. (1979). Chern. Pliys. 39, 407. Elitzur, M. (1977a). Astron. Astrophys. 57, 179. Elitzur, M. (1977b). Astrofi. Astrophys. 59, 173. Farantos, S. C., and Murrell, J . N. (1981). / f i t , J . Q . Cliem. 19, 95. Flower, D. R.. and Kirkpatrick, D. J. (1982). J . Pliys. B 15, L11. Flower, D. R., Launay, J. M., Kochanski, E., and Prissette, J . (1979). C/iern. P h y s . 37,355. Gaussorgues, C., Le Sech, C., Masnou-Seeuws, F., McCarroll, R., and Riera, A. (1975) P h y s . E 8, 239. Gelb, A., and Alper, J. S. (1975). C/iem. Phys. Lett. 31. 245. Gelb, A., and Alper, J. S. (1979). Cliern. Phys. 39, 141. Gentry, W. R. (1974). J . Cliem. Phys. 60,2547. Gentry, W. R. (1979). / ! I “Atom-Molecule Collision Theory” (R. B. Bernstein, ed.), p. 391. Plenum, New York. Gentry, W. R., and Giese, C. F. (1975). J . Clrem. Phys. 63, 3144. Gianturco, F. A., and Rahman, N. K. (1978). J. Phys. B 11, 722. Gianturco. F. A., Lamanna, U. T., and Rahman, N . K. (1978). J. Chuii. Phy.s. 68, 5538. Giese, C. F., and Gentry, W. R. (1974). P k y s . Rei.. A 10, 2156. Gislason. E. A . (1976). Chc,m. P h y s . Lett. 42, 315. Gislason, E. A., and Sachs, J. G. (1977). Chcvii. Phys. 25, 155. Glauber, R. J. (1955). P h y s . Rev. 100, 242. Goldstein, H . (1980). “Classical Mechanics,” 2nd Ed. Addison-Wesley, Reading, Massachusett s. Green, S. (1977). Chvni. Phys. Lett. 47, 119. Green, S., Ramaswamy, R., and Rabitz, H. (1978). Astrophys. J . Sirppl. 36, 483. Heidrich, F. H., Wilson, K. R., and Rapp, D. (1971). J. C l i o i i . P h y s . 54, 3885. . 375. Heil, T. G., and Kouri, D. J. (1976). Cliern. Pliys. L ~ t t 40, Heller, E. J. (1976). J. Chcvrr. Plrys. 65, 1289. Heller, E. J., and Brown, R. C. (1981). .I. Cheni. Pliys. 75, 1048. Itikawa. Y. (1975). J. Phys. S o r . J p . 39, 1059. Iwamatsu, M.. Onodera, Y . . Itoh, Y.,Kobayashi, N., and Kaneko, Y. (1981). Clicwi. P/ry.c. Lett. 77, 585. Jackson, J. M., and Mott, N. F. (1932). Proc. R . Soc. Loridori Ser. A 137, 703. Jarnieson, M. J., Kalaghan, F. M.,and Dalgarno, A. (1975). J. Pliys. B 8, 2140. Khare, V., Fitz, D. E., and Kouri, D. J . (1980). J. Chetii. Phys. 73, 2802. Kouri, D. J. (1979). I r i “Atom-Molecule Collision Theory” (R. B. Bernstein, ed.), p. 301. Plenum, New York. Korsch, H. J., and Richards, D. (1981). J. P h y s . B 14. 1973. Korsch, H. J., and Schinke, R. (1980). J. C/icui. Phy.s. 73, 1222. Korsch, H. J., and Schinke, R. (1981). J. Cliem. P l i ~ ~75, s . 3850. Kramer, K. H., and Bernstein, R. B. (1964). J . Chcni. Pliys. 40, 200. Kreek, H., and Marcus, R. A . (1974). J . Clierri. Phys. 61, 3308. Kriiger, H. (1979). Theor. Chini. Artcr 51, 31 I . Kriiger, H. (1980). Tlicwr. Ch;iii. A ~ t t r57, 145. Kriiger, H. (1981). T / r t w . Chifir. A m 59, 97.

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LaBudde, R. A., and Bernstein, R. B. (1971). J . Chem. Phys. 55, 5499. Lee, H.-W., and Scully, M. 0. (1980). J . Chem. Phys. 73, 2238. Leubner, C. (1981). Phys. Rev. A 23, 2877. Lodge, J . G., Percival, I. C., and Richards, D. (1976). J . Phys. B 9, 239. Loesch, H. J. (1976). Chem. Phys. 18, 431. Loesch, H. J . (1980). Adif. Chem. Phys. 42, 421. McCann, K. J . , and Flannery, M. R. (1978). J. Chem. Phys. 69, 5275. McCurdy, C. W., and Miller, W. H. (1977). J . Chem. Phys. 67, 463. McCurdy, C. W., and Miller, W. H. (1980). J. Chem. Phys. 73, 3191. McFarlane, S. C., and Richards, D. (1981a). J. Phys. E 14, 3423. McFarlane, S . C., and Richards, D. (1981b). J . Phys. B 14, 3643. McFarlane, S. C., and Richards, D. (1982). J. Phys. B. In preparation. McGuire, P., and Kouri, D. J. (1974). J . Chem. Phys. 60, 2488. McKenzie, R. L. (1977). J . Chem. Phys. 66, 1457. Maitland, G. C., Vesovic, V., and Wakeham, W. A. (1981). Mol. Phys. 42, 803. Marcus, R. A. (1972). J . Chem. Phys. 56, 311. Miller, W. H. (1970a). J. Chem. Phys. 53, 3578. Miller, W. H. (1970b). J . Chem. Phys. 53, 1949. Miller, W. H. (1971). J. Chem. Phys. 54, 5386. Miller, W. H. (1972). J . Chem. Phys. 56, 5668. Miller, W. H. (1975). Adv. Chem. Phys. 30, 77. Miller, W. H . , and Smith, F. T. (1978). Phys. Rev. A 17, 939. Monchick, L., and Schaefer, J. (1980). J . Chem. Phgs. 73, 6153. Muckenfuss, C., and Curtiss, C. F. (1958). J . Chem. Phys. 29, 1257. Muckerman, J . T. (1981). In "Theoretical Chemistry" (D. Henderson, ed.), Vol. 6A, p. 1 . Academic Press, New York. Mulloney, T., and Schatz, G. C. (1980). Chcm. Pliys. 45, 213. Munoz, J. M. (1981). PhD thesis, Stirling University. Neilsen, W. B., and Gordon, R. G. (1975). J . Chem. Phys. 58, 4131. Nesbet, R. K., and Clary, D. C. (1979). J . Chem. Phys. 71, 1372. Nyeland, C., and Billing, G. D. (1978). Chem. Phys. 30, 401. Nyeland, C., and Billing, G. D. (1981). Chem. Phys. 60,359. Olson, R. E. (1980). Proc. I n ! . Cur$ Phys. Elecrrun. Atom. Cull. I l t h , p. 391. Parker, G. A , , and Pack, R. T. (1978). J. Chem. Phys. 68, 1585. Pattengill, M. D. (1979a). IN "Atom-Molecular Collision Theory" (R.B. Bernstein, ed.), p. 359. Plenum, New York. Pattengill, M. D. (1979b). J. Phps. Chem. 83, 974. Pechukas, P., and Child, M. S . (1976). Mol. Phys. 31, 973. Percival, 1. C. (1977). A h . Chem. Phys. 36, I . Percival, I . C., and Richards, D. (1970). J. Phys. B 3, 1035. Percival, I. C . , and Richards, D. (1971). J . Phys. B 4, 932. Percival, I . C., and Richards, D. (1975). A h . Arum. M o l . Ph.vs. 11, 1 . Poppe, D. (1978). Chem. Phys. 35, 151. Poppe, D., and Bottner, R. (1978). Chem. Phj9.s. 30, 375. Poston, T., and Stewart, 1. N. (1978). "Catastrophe Theory and its Applications." Pitman, London. Preston, R. K., and Pack, R. T. (1977). J. Chem. Pliys. 66, 2480. Raczkowski, A. W., and Miller, W. H. (1974). J . Chem. Phys. 61, 5413. Raczkowski, A. W.. Lester, W. A., and Miller, W. H. (1978). J. Chem. Phys. 69, 2692.

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Rahman, N. K., Gianturco, F. A., and Lamanna, U. T. (1978). fhys. R e ) * .A 18, 74. Ramaswamy, R., and DeF’risto, A. E. (1981). Chem. f h y s . Lett. 77, 190. Ramaswamy, R., Rabitz, H., and Green, S. (1978). Chem. fhys. 28, 319. Ray, S., Saha, S., and Barua, A. K. (1976). J . fhys. B 9, 2341. Ray, S., Saha, S., and Barua, A. K. (1978). J . f h y s . B 1 1 , 1953. Reuss, J. (1980). Chem. fhys. 54, 139. Richards, D. (1981a). J . f h y s . B 14, 1465. Richards, D. (1981b). J . Phys. B 14, 4799. Richards, D. (1982a). J. Ph.vs. B 15. In press. Richards, D. (1982b). J . P h y . B 15. In press. Saha, S., and Guha, E. (1975). J. f h y s . B 8, 2293. Saha, S . , Guha, E., and Barua, A. K. (1974). J . Phys. B 7, 2264. Sakimoto, K . (1980). J. f h y s . Soc. Jap. 48, 1683. Sakimoto, K. (1981). J . Phps. Soc. J o p . 50, 1668. Sathyamurthy, N., and RaE, L. M. (1980). J . Chem. Phys. 72, 3163. Schaefer, J., and Lester, W. A. (1975). J . Chem. fhys. 62, 1913. Schatz, G. C. (1980). J . Chem. f h y s . 72, 3929. Schinke, R. (1977). Chem. fhys. 24, 379. Schinke, R. (1980). Chern. f h y s . 47, 287. Schinke, R., and Bowman, J. M. (1982). In “Molecular Collision Dynamics” (J. M. Bowman, ed.). Springer-Verlag, Berlin and New York. Schinke, R., and McGuire, P. (1978). Chem. f h y s . 31, 391. Schinke, R., Muller, W., Meyer, W., and McGuire, P. (1981). J . Chrm. f h v s . 74, 3916. Sebastian, K. L. (1981). Chem. fhys. Lett. 80, 531. Secrest, D. (1975). J. Chem. Phys. 62, 710. Secrest, D., and Johnson, B. R. (1966). J . Chcm. f h v s . 45, 4556. Stace, A. J., and Murrell, J. N. (1978). J . Chem. f h y s . 68, 3028. Stewart, I. N. (1981). fhpsicu D 2, 245. Stine, J. R., and Marcus, R. A. (1973). J . Chem. f h y s . 59, 5145. Stoke, S., and Reuss, J. (1979). In “Atom-Molecule Collision Theory” (R. B. Berstein, ed.), p. 201. Plenum, New York. Strekalov, M. L. (1981). Opt. Spec. 49, 18. Takayanagi, K. (1978). J. f h y s . Soc. Jpn. 45, 976. Takayanagi, K. (1980). Comments Atom. Mol. f h y s . 9, 143. Tarr, S. M., Rabitz, H., Fitz, D. E., and Marcus, R. A. (1977). J . Chem. f h y s . 66, 2854. Thomas, L. D. (1977). J . Chern. fhys. 67, 5224. Thomas, L. D. (1981). I n “Potential Energy Surfaces and Dynamics Calculations” (D. G. Truhlar, ed.). Plenum, New York. Thomas, L. D., Kraemer, W. P., and Diercksen, G. H. F. (1978). Chem. f h y s . 30, 33. Thomas, L. D., Kraemer, W. P. and Diercksen, G. H. F. (1980). Chem. f h y s . Lett. 74,445. Thompson, D. L. (1981a).J. Chem. f h y s . 75, 1829. Thompson, D. L. (1981b). ChPm. P h y . Left. 84, 397. Truhlar, D. G., and Blais, N. C. (1977). J. Chem. f l i y s . 67, 1532. Truhlar, D. G., and Duff,J. W. (1975). Chem. fhys. Lett. 36, 551. Truhlar, D. G., Reid, B. P., Zurawski, D. E., and Gray, J. C. (1981).J. fhys. Chem. 85,786. Tully, J. C. (1980). Ann. Rev. flrys. Chem. 31, 319. Turfa, A. F., and Knaap, H. F. P. (1981). Chem. fhys. 62, 57. Turfa, A. F., and Marcus, R. A. (1979). J. Chem. fhys. 70, 3035. Turfa, A . F., Liu, Wing-ki, and Marcus, R. A. (1977). J . Chem. Phys. 67, 4400. Udseth, H., Giese, C. F., and Gentry, W. R. (1974). J . Chem. f h y s . 60, 3051.

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I1

ADVANCES 1N ATOMlC AND MOLECULAR PHYSICS.VOL . 18

RECENT COMPUTATIONAL DEVELOPMENTS IN THE USE OF COMPLEX SCALING IN RESONANCE PHENOMENA B . R . JUNKER Deprirttnetit of tlir

Nmy Office of Nmval Resrcrrch Arlittgfoti. Virginio

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Gamow-Siegert States . . . . . . . . . . . . . . . . . . . . . . A . Biorthogonal Sets . . . . . . . . . . . . . . . . . . . . . . B . Incoming-Outgoing Waves . . . . . . . . . . . . . . . . . . I11 . Complex-Coordinate Theorems and Properties of the Wave Functions A . Complex-Coordinate Theorems . . . . . . . . . . . . . . . . B Wave-Function Properties . . . . . . . . . . . . . . . . . . C . TimeReversal . . . . . . . . . . . . . . . . . . . . . . . . D . Hypervirial Theorems . . . . . . . . . . . . . . . . . . . . E . Analytical Models . . . . . . . . . . . . . . . . . . . . . . IV. Variational Principle . . . . . . . . . . . . . . . . . . . . . . . A . Variational Functional . . . . . . . . . . . . . . . . . . . . B . Polar Representation . . . . . . . . . . . . . . . . . . . . . V. Variational Calculations . . . . . . . . . . . . . . . . . . . . . A . Configuration Interaction Expansions . . . . . . . . . . . . . . B . Self-Consistent Field Calculations . . . . . . . . . . . . . . . VI . Many-Body Theories . . . . . . . . . . . . . . . . . . . . . . VII . Nondilation Analytic Potentials . . . . . . . . . . . . . . . . . . A . Stark and Zeeman Effects . . . . . . . . . . . . . . . . . . . B . Multiphoton Ionization . . . . . . . . . . . . . . . . . . . . C . Cubic Anharmonic Oscillator Model . . . . . . . . . . . . . . VIII . Complex Stabilization Method . . . . . . . . . . . . . . . . . . A . Complex Stabilization Procedure . . . . . . . . . . . . . . . B . Configuration Interaction Calculations . . . . . . . . . . . . . C . Self-consistent Field Calculations . . . . . . . . . . . . . . . D . Many-Body Techniques . . . . . . . . . . . . . . . . . . . . IX . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

208 210 210 211 214 214 217 220 221 222 227 227 228 229 229 240 243 244 244 245 246 247 249 251 255 256 256 260

207 Copyright 0 1982 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 0-12-003818-8

208

B . R . Junker

I. Introduction Many interesting phenomena can be described in terms of a resonance structure. Figure 1 illustrates some atomic physics phenomena of this nature. The observed structure is associated with a position EHand width r, which are assumed to be related to the complex energy E , of a pole of the scattering matrix (see, for example, Taylor, 1972), that is,

E r = E R + iE,= ER - i(r/2) = lEPlexp(-2iP)

(1)

with E, < 0, r > 0, and p > 0. The terms ER and may, in addition, depend on other parameters such as the internuclear separation in the case of electron-molecule resonances in the Born-Oppenheimer approximation or a field such as in the case of electric field ionization. The poles of the S matrix can rigorously be shown to correspond to the poles of the resolvent for potential scattering if certain assumptions are made about the potential. For the N-particle problem a number of advances have been made in this direction. Hagedorn (1978, 1979) has

-

A(qJ FIELD A1q

+ 1J +

e

A(q) + w(n)-A(q-lJ +e' +w(n-l) (9) Examples of resonance phenomena: (a) e- atom resonances: (b) e- molecule resonances: (c) Penning ionization; (d) ion-atom collisions: (e) predissociation; (0 field ionization: (9) photoionization. Hc. I .

RESONANCE CALCULATIONS USING COMPLEX SCALING

209

proved the connection between the poles of the S matrix and the resolvent for two-body channels for systems with four of less particles, while Sigal (1978a,b) has proved this result for the general N-body system when only one channel is present. In both of these cases the potentials are assumed to fall off exponentially. Balslev (1980a,b) has also shown that in general the poles of the S matrix should also be poles of the resolvent below the lowest three-body threshold, even for certain potentials which fall off more slowly than exponentials. We shall assume that this result is also valid for the Coulomb potentials encountered in atomic and molecular physics. Numerous techniques have been developed to compute these poles. These include close coupling, R matrix, Feshbach, stabilization, and others. In some methods a complete scattering calculation is performed, follqwed by a fitting of the result to a Breit-Wigner formula to extract the resonance parameters, while the pole of the resolvent is computed directly in others. In all of these methods the asymptotic form of the wave function plays a critical role in performing the calculation and/or extracting the scattering information. On the other hand, the complex coordinate theorems (Aguilar and Combes, 1971; Balslev and Combes, 1971; Simon, 1972, 1973) which apply to dilation analytic potentials have defined the conditions under which the transformation to complex coordinates yields a square-integrable resonance wave function. The square integrability of the exact complex-coordinate resonance wave function naturally invited the application of normal bound-state techniques to the determination of approximate resonance energies and widths. In Section I1 we discuss biorthogonal basis sets and the characteristics of Gamow-Siegert states. These two concepts form a convenient basis on which to formulate the rest of the material. Section I11 contains a discussion of the complex coordinate theorems; the properties of the bound, scattering, and resonance states; and analytical examples to illustrate these properties. A variational principle is derived in Section IV, which is followed in Section V by a discussion of the various types of configuration interaction and self-consistent calculational approaches which have used this variational principle. The many-body formalisms which have been suggested are considered in Section VI, while the successful application of these computational techniques to nondilation analytic potentials is considered in Section VII. The generalization of these stabilization techniques for configuration integration, self-consistent field, and many-body calculations to what we have called the complex stabilization method for the real Hamiltonian is discussed in Section VIII. The concluding remarks in Section IX reiterate those points in the article which are mathematically rigorous and those which appear to be plausible extensions, as well as

B. R . Junker

210

noting a number of significant pertinent topics which were not considered here. [Reinhardt (1982) prepared a review on the use of complex coordinates in atomic and molecular structure and dynamics.] Atomic units are assumed throughout except where explicitly noted otherwise.

11. Gamow-Siegert States In potential scattering a pole of the S matrix implies that the corresponding wave function will be a purely outgoing wave. The allowed values of k can be shown to lie on the positive imaginary axis, ik,, and the lower half, k, - ik, (kI > 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 , proved superpositions of Yukawa potentials. Then, for 0 < 8 < ~ / 4 they 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 than 57/2 to a value infinitismally less than ~ / 2 is , to discontinuously 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
(89)

Application of Green's theorem to the last term on the right-hand side of Eq. (89) then gives 6F( on(8)) =

an(8) dT

{ 6@F( 8)[ %?(8) - En]

+ I[x*(e) - ~;)@:(e)]*@,,(e)

d7

228

B. R. Junker

ForF(X) to be a variational functional through first order, all three terms in Eq. (90) must be zero. The first two terms vanish if a,,(@ and @A(@ form a biorthogonal set (see Section 111,B).In addition, if 8, > 8 > p, where 8, is some angle at which the cut associated with a higher threshold may cross E n , Eqs. (37) and (50) imply that the surface term will vanish since W z / d n tends to zero as well as @;(8). Thus, under these conditions 6F is zero and Eq. (88) provides a variational expression for the calculation of approximate complex poles of the resolvent. Equation (88) reduces to the well-known variational principle for bound states when X ( r ) is used. However, unlike the case for bound states, Eq. (88) only provides a stationary principle without upper or lower bounding properties, and, even more important, rigorous convergence criteria do not exist. Note that in spherical coordinates @A* just corresponds to complex conjugating the angular part of Qn, but not the radial part (Herzenberg and Mandl, 1963).

B. POLARREPRESENTATION Certain (1980) has proposed a polar representation for the rotated Schrodinger equation

X(8)V

=

[ER- i(r/2)]V

By writing =

+ exp(iS)

one can replace Eq. (91) with the pair of equations

RESONANCE CALCULATIONS USING COMPLEX SCALING

229

Holmer et ol. (1981) give the following variational functionals for approximate 6 and 3 :

p{-+z[v: - (vS)21+ W R } W

F($)

=

m

(964

4

Unfortunately Eq. (96a) is a variational functional for only for the exact S, while Eq. (96b) is a variational functional for only for the exact Jl. On the other hand, since the operator in Eq. (%a) is real self-adjoint, the usual bounding properties are obtained.

V. Variational Calculations The variational principle derived in Section IV has been used as the basis for both configuration interaction (CI) expansions and self-consistent field (SCF) techniques. In addition, numerous techniques have been suggested for performing these calculations. The following discussion is not an attempt to review all variational calculations that have been reported, but instead to discuss the basic techniques and lay the groundwork for the complex stabilization method in Section VIII. A. CONFIGURATION INTERACTIONEXPANSIONS Certainly the largest number of calculations employing complex scaling have used a straightforward CI expansion in some basis set. Initially, the functional dependence of the exact resonance wave function on r and 0 only through the quantity r)

= r exp(i0)

(97)

(for the present we shall surpress the real scaling factor, a) and the fact that this was the reason that the resonance poles of the resolvent were independent of 8 was not appreciated. Although the argument that the use of basis functions depending on r ) would yield only real eigenvalues is true for real basis functions, the resonance wave functions, as noted above, are innately complex functions. This initial use of real basis functions only for calculations involving X ( 6 ) led to two apparently different types of approaches-Siegert and complex-coordinate calculations.

B . R . Junker

230

In the Siegert approach the trial wave functions were required to contain a term satisfying a Siegert boundary condition explicitly. In this case, the variational wave function for a resonance of X ( r ) was expanded as M

w = I] c f x f ( r l ., . . , rN+J + c,+~ d h ( r l , . . . , rN) I= 1

x r - 1 ( 1 - exp(-ar)) exp[i(k, - ik,)r.v+lI

(98)

The x i ' s were represented in terms of a square-integrable basis as was the . The calculations were iterated until the input value of target function k,< - ik, satisfied Eq. (16) and the complex coefficients ciwere determined from Eq. (88). The complex scaling (28) was used to define the divergent integrals involving the Siegert functions as originally suggested by Dykhne and Chaplik (1961). On the other hand, in the alternate approach the wave functions for the resonances for X ( 0 ) were expanded as

+,.

M

tp = I] ci(e)Xf(r2.. . . , rN+J

(99)

i=l

where the x i ' s were again represented in terms of a square-integrable basis. The ~ ~ ( 0 )were ' s also determined by Eq. (88). Since the x i ' s did not depend on q , the approximate resonance eigenvalue, Er(O ) , did.

I,

Siegort

Cdciil(itions

Bardsley and Junker (1972) used a variational wave function of the form of Eq. (98) to compute the resonance parameters for the ' S H- Feshbach resonance. Later, in a study of the two approaches discussed above, Bain c't (11. (1974) applied the Siegert method to a model potential problem and the (ls2s2)*SHe- Feshbach resonance. In the latter case, the resonance parameters appeared to converge in a well-behaved manner, while the resonance parameters for the model potential did not appear to converge to more than a couple of significant figures for the Slater-type orbital (STO) basis used. Subsequently, Isaacson et 01. (1978) repeated the model potential Siegert calculation using generalized Lagurre polynomials as basis functions and obtained completely stable converged results. [Recent calculations of Junker (1981, 1982), again using a STO basis but a complex version of the QZ algorithm (Moler and Stewart, 1973) for solving the complex eigenvalue problem as opposed to the projection technique of Bain et al. (1974), also converged to the correct result. Diagonalization of the overlap matrix indicates that the earlier difficulty arose from the nearlinear dependence of the basis set with respect to the finite arithmetic of digital computers. The solution of the general complex eigenvalue prob-

RESONANCE CALCULATIONS USING COMPLEX SCALING

23 1

lem can be an extremely numerically unstable problem. Loss of accuracy due to such problems as just discussed generally enters very subtly and can easily go undetected unless these problems are explicitly investigated.] Isaacson er d . (1978) also studied the convergence with respect to basis-set size of the resonance parameters for the (2s)*'S H-,( 2 ~ )He, ~~s and (2s2p)'P He resonances and reported well-converged results. Isaacson and Miller (1979) have applied the Siegert method to the Penning ionization of H by 21s3S He in the BO approximation. Due to the numerical problems noted in Section III,A, the integrals cannot be evaluated by use of Eq. (28). Instead they evaluated the integrals for Im k > 0 and analytically continued the results to the region Im k C 0. The results appear to agree well with calculations using other techniques. An alternate suggestion for avoiding the diverging integrals associated with the use of Siegert functions in atomic and molecular calculations has been given by Yaris et ul. (1979). If the square-integrable basis is chosen such that all of the nonlinear parameters are greater than -1m k , all off-diagonal matrix elements with the Siegert function will converge. While they note that this is not much of a restriction since one is generally interested in resonances close to the real axis where Im k is small, this is, in fact, no restriction since one can always define (both rigorously and practically) a complete basis satisfying this criterion, although the criteria for convergence of this type of variational calculation has not been established. The diagonal matrix element involving two Siegert functions is evaluated noting that k must satisfy Eq. (16). Requiring the Siegert function to satisfy a differential equation also incorporating any long-range potentials, they show that the diagonal matrix element, (Ol(X - E)lO), between two Siegert functions reduces to (OlAlO), where A is a shortrange operator, depending on the form of the potential. The latter method has not yet been employed in multiparticle calculations to determine whether any computational difficulties arise. Finally, Nicolaides and Beck (19784 and Nicolaides et a/. (1981) have discussed strategies for performing Siegert calculations on N-particle systems with emphasis on incorporating correlation effects. Again, however, no calculations of sufficient size to determine whether computational difficulties might arise have been performed. Again, it should be noted that the imposition of a Siegert boundary condition for theN-particle case is an crd hoc condition extrapolated from the two-particle problem. 2 . Dooleti-Type Basis As noted above, the use of a variational wave function of the form given in Eq. (99) leads to 0-dependent approximate resonance energies, Er(0).

B. R. Junker

232

Doolen rt (11. (1974) used several different variational wave functions constructed from a Hylleraas basis to determine the resonance parameters for ~s They simply continued to increase the number the ( 2 ~ )H-~ resonance. of configurations in the wave functions until variational wave functions with different nonlinear parameters and for different values of 8 converged to approximately the same complex eigenvalue. Doolen (1975) proposed a technique which has subsequently been used in many calculations on one- and two-particle systems. He proposed first calculating a 8 trajectory, that is, a set of approximate resonance eigenvalues for a given wave function as a function of 8. He then argued that since the exact resonance eigenvalue is independent of 8, the best approximate eigenvalue is that eigenvalue, along the 8 trajectory, which varies least with respect to variations in 8 . Figure 3 (Doolen, 1975) illustrates such a calculation for the ( 2 ~ ) resonance ~ ~ s in H- using a 95-term Hylleraas wave function. Curves A, B, and C correspond to different nonlinear parameters. All three 8 trajectories have a region of stability near the calculation of Bardsley and Junker (1972). Doolen notes that the calculation is simplified since for Coulomb potentials the Hamiltonian matrix elements scale as

(31X ( a exp(i8)r)lll) =

(Y-~

exp(-2ie)(&l~l$)+

a-1

exp(-iO)

($Iv~$)

(100)

Thus, the matrix elements need to be computed only once. Also he notes that the other eigenvalues appear to lie on rays extending from the various thresholds and rotated down by approximately 28 (see Fig. 3 of Doolen, 1975). This permits a reasonably simple separation of approximate resonance eigenvalues from the approximate nonresonance or scattering-state eigenvalues . The behavior of the 8 trajectory can be understood in the following manner (Junker, 1980b, 1981). The basis functions span a &independent vector space. On the other hand, the exact resonance function depends on 8 as discussed above. Consequently, as 8 in X(8)is varied, the overlap between the exact resonance wave function and the vector space spanned by the &independent basis changes. Stabilization of Er(8) can then be expected to occur when this overlap is largest. Theta is thus simply playing the role of a nonlinear variational parameter. This interpretation of the behavior of the theta trajectory has recently been reinforced for a model potential calculation by Atabek rt ul. (1981). They directly numerically integrated the Schrodinger equation as well as performed variational calculation for the model potential V(r) = (3r' - J) exp(-Arz)

+J

(101)

RESONANCE CALCULATIONS USING COMPLEX SCALING

-0.001

-

-0.002

-

-0.003

I

233

B

aw m

9

E Iii

I

I

I

- 0.2980

- 0.2975

- 0.2970

-

- 0.2985

E R (RYDBERGS)

F I G .3 . 0 Trajectories for a 95-configuration wave function for the (2s)*'S H- Feshbach resonance. Curves A , B , and C correspond to wave functions with different values for the nonlinear parameter (Doolen, 1975).

They then computed the functional distance (Lowdin and Shull, 1958) defined as =

J:ml$(r,

8) - +(r,

+

dr

(102)

where 4 is the variational function and is the exact wave function derived from the numerical integration. For the resonances they studied, B,(@)stabilized in the region of the minimum of I(0). This suggestion by Doolen (1975) was a very important result which motivated much of the subsequent research in this area, including the ideas leading to the complex stabilization method in Section VIII. However, several points should be made. The slow convergence arises primarily because the basis set does not properly incorporate the &induced kinematic oscillations which occur in the exact wave function due to its dependence on r exp(i0). This forces this nonlinear 8 dependence into the linear coefficients. In addition, the exact resonance functions are innately complex functions. Since only real basis functions were used in these calculations, this innate nonlinear complex nature of the resonance functions is also forced into the linear coefficients. On the other hand, if one uses basis functions which depend on r exp(i8) and includes at least some

B . R . Junker

234

complex functions, even the crudest wave function will produce &independent eigenvalues. Then one is faced with the problem of developing some means of determining which eigenvalues correspond to resonances. We shall return to this point in Section VIII. Finally, thus far, no theorem exists which proves that as the basis size is increased, the variational wave function approaches the exact function and

Er@)+ Er

( 1 03)

Brandas and Froelich (1977) have suggested that the complex virial theorem be used to determine the best value of 8. As noted in Section III,D the expectation values of T ( 0 ) and V ( 8 ) are independent of 6 for the trial functions with the proper dependence on 0. In addition for the exact wave function

x(e)+(e) = E+(o)

(104)

X(8*)1&(8*) = E+(8")

(105)

since E is independent of 0 as opposed to

x(e*)+(e*) =~*+(e*)

(106)

as Brandas and Froelich (1977) suggest. Thus, for a wave function with the proper dependence on I ) , Eq. (61) does not provide a means of determining an optimum value of 8. The difficulty in their analysis arises in their assertion that for 8 = 0, X is real, implying is real. This led them to state that

+

$*@) = +(@*)

(107)

On the contrary, as noted in Section III,B, the essential difference between the resonance wave functions and the bound-state wave functions is that the former are innately complex functions, even for 6 = 0. Thus, while Eq. (107) is valid for bound states where

Ei = E,

(108)

it is not valid for resonances as is obvious from the complex structure of Eqs. (82) and (83), for example. to be a nonlinear variational parameter, one However, considering has that their results are correct and do not rely on Eq. (107). Instead for

ad = (+Y?-l)l =

(++

x~~)l+(?-l))/(++(~-l)l+(?-l)) (10% I x(?)+)/(++( +)

Brandas and Froelich (1977) and Yaris and Winkler (1978) have shown that for the first-order variation of E ( v ) to vanish, 77 must be chosen to

RESONANCE CALCULATIONS USING COMPLEX SCALING

235

satisfy the virial expression 277(JltlT(4lljl)

= - ($+IV(mJ)

(1 10)

for Coulomb potentials. The same result follows in a straightforward application of the Hellman-Feynman (Feynman, 1939) theorem to Eq. (109), as is done in bound-state calculations (Lowdin, 1959). Note that if $ is the exact wave function, q must be one clearly indicating that the phase of q should not be confused with 6 in Eqs. (28) and (32). The large phases previously obtained for q are a result of the use of only real basis functions to represent a wave function that is necessarily complex. Thus, while the previous interpretation of Eq. (110) needs to be altered, the use of it to define a global complex scaling of all nonlinear parameters may still be useful. This scale will, however, depend critically on the basis functions chosen. The following is the essence of an iterative procedure suggested by Brandas and Froelich (1977). First, one chooses some complex scale q o , and some set of configurations for representing $, , and solves the secular equation to determine $,o. One then employs the relation

- ($;o(q61)l W771)l +q0(vi1)) M o ( V i 1 ) l$ v 0 ( v i 1 ) )

(111)

to obtain a polynomial in q1 * q1 is then chosen such that &!?,o(qoql)/ 871, is zero, that is, the virial expression (110) is satisfied for q 1 with respect to $,o(qo).Finally qo is replaced by qoql and the procedure repeated until qf 4 1. Carrying this procedure to convergence ensures that

as well as that the virial theorem is satisfied. Here q is qoq1q2. . * . Stopping at the first iteration only ensures that the virial theorem is satisfied for $," and here

'The latter restricted scaling is often used in bound-state problems. Doolen's technique has been used in numerous resonance calculations. Bain et d.(1974) compared this technique to the Siegert approach of

B . R . Junker Section V,A,I and a combination of the two, that is, the addition of a Siegert function to a Doolen type variational wave function. They studied a model potential, the lowest 'S state of H-, and the lowest 2S resonance of He-. In all cases, except for the model potential (but see Section V,A, l), the Siegert approach was superior. In particular, the Doolen-type function never yielded any results for the three-particle ,S He- resonance. Nonetheless, Doolen's technique has been successfully applied to a wide variety of one- and two-particle systems of which the following are representative, though not exclusive, examples. Hickman et cil. ( 1976) have applied the technique to the determination of resonance parameters for the IS and 1,3P resonances in H- and He. Winkler (1977) computed the resonances in H-. Ho resonance parameters for a number of IS and 1*3P (1977, 1979a, 1980, 1981) computed the positions and widths of a number of IS, IP0,and 3P0 resonances below the n = 2. 3, and 4 hydrogenic thresholds in the helium isoelectronic series up to Z = 10. Moiseyev et cil. (1978b) required that the complex scale factor a exp(i0) be chosen such that the complex virial theorem is satisfied in their calculation of the resonance parameters for the lowest IS resonance of helium. As noted in Section III,A, the BO Hamiltonian is not dilation analytic. Nevertheless, several calculations have been performed on molecular systems and molecular-like model potentials. McCurdy and Rescigno ( 1978) suggested one could use the unrotated Hamiltonian while scaling all nonlinear parameters by exp( - 2 i B ) for Gaussian-type basis functions I which would imply exp(-iB) for STOs since for spherically symmetric systems this gives a scaling of the radial coordinate in the basis functions by exp(-iO)]. This, in turn, is equivalent to scaling the Hamiltonian radial coordinates by exp(i0) and using real basis functions. Since this equivalence is also approximately true for their diffuse Gaussian basis functions, which determine the asymptotic form of the wave function, they argue their calculation should converge properly. They successfully applied this to the bound states of H: and the resonance states of a nonspherical model potential. Subsequently, Moiseyev and Corcoran ( 1979) simply transformed the electronic coordinates of the BO Hamiltonians for H, and H;, keeping the nuclear coordinates real. McCurdy (1980) has shown that this calculation can be interpreted as a realization of Simon's (1979) external complex scaling. A general justification of these calculations has been given by Morgan and Simon (1981) and Junker (1980a) (see Section VIII). Resonances in systems involving positrons have also been studied with this method. Doolen rt cil. (1978) verified the existence of a e+-H S-wave resonance at -0.257374 a.u. with a width of 0.000134 a.u. using a variety of variational wave functions with 200-600 configurations. Ho (1978) in-

RESONANCE CALCULATIONS USING COMPLEX SCALING

237

vestigated the ground state and a resonance state in positronium hydride. While he found some indication that a resonance did exist, the results for this three-particle system were too unstable to provide conclusive resonance parameters. Finally, Ho ( 1979b) investigated the lowest S-wave resonances in the series of systems from positronium negative ion to Hby varying the mass of the positively charged particle from that of an electron to infinity. Chu (1980) and BaEic and Simons (1980) have both applied this technique to rotational resonances in atom-diatom scattering. Due to the approximations used in the actual calculation and systems studied, these calculations are just single-particle potential scattering calculations. Moiseyev ( 198la,b) has performed similar calculations for predissociation in diatomic systems with real internuclear potentials as well as complex potentials to describe autoionization. Holmer ef cil. (1981) applied the polar representation to the calculation He’ sresoof resonance parameters for a model potential and the ( 2 ~ ) ~ nance. While they obtained reasonable convergence for the model potential, the results for the He resonance became worse when they improved the approximate wave function. This cast doubt on their claim that the polar representation may provide a good initial estimate of resonance parameters. If we consider as an example a singlet state of a two-electron system, we can gain some insight into the structure of a polar representation. For such systems the independent-particle picture represents a good starting point. In addition, as noted in Section 111, the rotated wave function is, in fact, a function of I’ exp(i0) while also being necessarily complex. Taking this into account, we have, for a given independent-particle configuration, the following polar representation:

x

=

(1 + PI2)&(1) exp(-iSI(l))4d2) exp(-iS2(2))

=$(I,

2) exp(-iS(1, 2))

(114)

where

IL( 1 .

2)

=

[MI)d$(2) + 442)#4(1)

+ 2 ~ 1 ( 1 ) ~ 2 ( 1 ) ~ 1 ( 2 ) ~cos(SI(1) 2(2) + SAI) + Sl(2) + S2(2))]”2

(115)

S(1, 2) = tan-’

and PI2permutes particles 1 and 2. $ and S are very complex functions in this approximation, although more experience might yield better choices

238

B . R . Junker

for the basis functions. On the other hand, the problem of slow convergence which motivated their study of the polar representation has been successfully addressed, as discussed below in Sections V,A,3 and VIII. Moiseyev and Weinhold (1980) reported a study of a number of criteria for assessing the quality of trial wave functions, but applied them only to Doolen type bases. They considered the following four criteria:

( 1 17c)

( 1 17d)

where q was chosen such that the virial theorem was satisfied, Eq. ( 1 17b) is the biorthogonal product of ( X - E)IU/),and E,is the exact resonance energy. For the IS helium resonance they studied, none of the above criteria correlated significantly better than any other. They suggested that a better criterion might be the stabilization length, that is, the degree of stability of €, with respect to variations in a about q o = a. exp(i8,). Unfortunately, long shallow stabilization lengths did not necessarily correlate well either (see Moiseyev and Weinhold, 1980, Figs. 2 and 3) besides the fact that long stabilization lengths imply a larger basis than may be necessary. The difficulties in developing criteria are probably at least in part due to the poor choice of basis functions. Moiseyev et t i / . (1981) have discussed the shape of the 8 trajectories in the vicinity of the stationary point in terms of the expansion of €,(TI in a Puiseux series in (q - T ~ )where ~ , qo is the stationary point a. exp(i8,) and p is a positive rational number. They find that if p is an odd integer, the 8 trajectory will be a smooth curve going through the stationary point q o ;if p is an even integer, the 8 trajectory will have a zero angle cusp about q o ; if p is a rational number, n / m > 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, , . . . , R'(H, 41

9

.

*

9

,.

.

.)+, =

A*]+,

( 124a)

hj4t

( 124b)

1 $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 this is not valid for resonances. The necessarily the limit H -,O, 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 , would have the correct properties for operators, O ( r ) and O + ( r ) which 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-

B. R. Junker

246

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) where

w

( 134)

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. A. COMPLEX STABILIZATION PROCEDURE 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<

I (9p) 14.547784 7810 I (7P) I( 5 ~ ) 7741 7960 11 (9P ) 11 (7P ) 8000 7005 11 (5P )

-El x 10' 0.9357 0.9373 0.9704 0.9382 0.9510 1.0905

0 = 40"

-EH

- E , x 10'

14.547710 7715 7713 7774 7786 7464

0.9438 0.9444 0.9559 0.9359 0.9370 0.9555

e=w

0 = 50"

-EH 14.547422 7425 7424 7720 7714 7692

-El

X

10'

0.9815 0.9831 0.9936 0.9342 0.9341 0.9312

-El(

-El

X

10'

14.545577

1.431 I

6194

1.3739

" See the text for the distinction between type I and type I1 calculations. The quantity in parenthesis indicates the number of p functions used. Units are a.u. These results can be compared with E , = - 14.54733 - 0.009371' a.u. (McCurdy e / NI., 1980).

for three of the wave functions of type I and three of the wave functions of type 11. McCurdy and Mowrey (1982) have applied these techniques to compute the position and width as a function of the internuclear separation of the shape resonance in H; at the SCF level. Their ab initio results are in excellent agreement with the empirical results of Wadehra and Bardsley (1978, 1979). McCurdy et al. (1980) note that one must use multiconfigurational SCF theory to compute Feshbach-resonance parameters.

'c:

D. MANY-BODY TECHNIQUES The only comment we wish to make concerning many-body techniques is that since SCF orbitals are generally used as a starting point for manybody calculations, the discussion above clearly indicates how to develop many-body calculations for X ( r ) as opposed to X ( 0 ) without explicitly using Siegert functions. That is, one could either incorporate complex basis functions in the SCF calculation or complex scale some of the nonlinear parameters.

IX. Discussion In the preceding sections we have presented a number of concepts, properties of functions, properties of operators, assumptions, etc. At this

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257

point we shall recap these in order to clearly distinguish between those for which there is a rigorous mathematical basis and those which are assumed to follow either due to analog-to-potential scattering, similarity to boundstate properties, or results of limited computations. The association of poles of the S-matrix (resonances) with poles of the resolvent and the relationship of these to Gamow-Siegert functions for potential or two-body scattering in Section I1 is rigorous. Neither, however, is rigorously proven for the general N-particle problem (Simon, 1973). The spectral properties of dilation analytic potentials discussed in Section I11 is, of course, rigorously mathematically founded. This is also true of the functional dependence of the exact bound and resonance states on 1’, a , and 0 only as LYT exp(i0), that is, these functions are just analytical continuations of certain functions associated with X ( r ) . In addition, the nondependence of the functions for the cuts on (YT exp(i0) for 0 > 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 1971 ; Taylor and above the threshold for complete breakup (Walton et d., 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

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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

x+x

( 146a)

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

x+x p

+

(147a) 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.

REFERENCES Aguilar, J., and Combes, J. M. (1971). Commun. M a t h . Phys. 22, 269. Atabek, O., Lefebvre, R., and Requena, A. (1981). Chem. Phys. Lett. 78, 13. Avron, J., and Herbst, I. (1977). Commun. M o t h . Phys. 52, 239. Avron, J . E . , Herbst, I. W., and Simon, B. (1978). Duke Math J. 45, 847. Avron, J. E., Herbst, I. W., and Simon, B. (1981). Commun. M o t h . Phys. 79, 529. BBciC, Z., and Simons, J. (1980). Int. J. Quantum Chem. 14, 467. Bain, R. A., Bardsley, J. N., Junker, B. R., and Sukumar, C. V. (1974).J . Phys. B . 7, 2189. Balslev, E. (1980a). A n n . Inst. Henri PoincarP 32, 125. Balslev, E. (1980b). Commun. M a t h . Pfivs. 77, 173. Balslev, E., and Combes, J. M. (1971). Commun. M a t h Phys. 22, 280. Bardsley, J. N. (1980). Sanibel Quantum C h r m . Symp., 14th Invited talk. Bardsley, J. N., and Junker, B. R. (1972). J. Phys. B 5, L178. Bardsley, J. N., and Wadehra, J. M. (1979). Phys. Rev. A 20, 1398. Berggren, T. (1968). Nucl. Phys. A 109, 265. (Note: there are several errors in this paper.) Brandas, E., and Froelich, P. (1977). Phys. Rev. 16, 2207. Brandas, E., Froelich, P., and Hehenberger, M. (1978). I n t . 1. Quotitrim Chem. 14, 419. Caliceti, E., Graffi,S. , and Maioli, M. (1980). Commun. M a t h . Phys. 75, 51. Cerjan, C., Reinhardt, W. P., and Avron, J. E. (1978a). J. Phys. B 11, L201. Cerjan, C., Hedges, R., Holt, C., Reinhardt, W. P., Scheibner, K., and Wendoloski, J. J. (1978b). I n t . J . Quantnm Chem. 14, 393. Certain, P. R. (1980). Chem. Phys. Lett. 65, 71. Chu, S. I . (1978a). C h r m . Phys. Lett. 58, 462. Chu, S. I. (1978b). Chem. Phys. Lett. 54, 369. Chu, S . I. (1979). Chem. Phys. Lett. 64, 178. Chu, S. I. (1980). J . Chem. Phys. 72, 4772. Chu, S. I. (1981).J. Chem. Phys. 75, 2215. Chu, S. I., and Reinhardt, W. P. (1977). Phys. Rev. Lett. 39, 1195. Damburg, R. J., and Kolosov, V. V. (1976). J Phys. B 9, 3149. Deift, P., Hunziker, W., Simon, B., and Volk, E. (1978). Commun. Muth Phys. 64, I . Donnelly, R. A., and Simons, J. (1980). J . Chem. f h v s . 73, 2858. Doolen, G. D. (1975). J . Phys. B 8, 525. Doolen, G. D. (1978). I n t . J . Quctntum Chem. 14, 523. Doolen, G. D., Nuttall, J.. and Stagat, R. W. (1974). Phys. Rev. A 10, 1612. Jholen,G. D., Nuttall, J., and Wherry, C. J. (1978). Phys. Rev. Lett. 40, 313. Dykhne, A. M . , and Chaplik, A. V. (1961). J E T P 13, 1002. Feynman, R. P. (1939). Phys. Rev. 56, 340. Froelich, P. (1981). Quantum Theory Project Preprint T F 546. Froelich, P. Hehenberger, M., and Brandas, E. (1977). Int. J. Quctntrtm Chent. Qlrclntrrm Chem. Symp. 11, 295. Gamow, A. (I93 I). “Constitution of Atomic Nuclei and Radioactivity.” Oxford Univ. Press, London and New York.

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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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267

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

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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).

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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 orientation tensor characterizing the excited level; h") and are geometrical 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 analogously P4 = ~ / I ' ~ ) ( A+; 'A) ~g 1 ) / I X Ps , = 0 , and Ps = 0, whereI" = 4 + /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)

= a2. Nienhuis (1980) Again, reflexion symmetry requires a, = - a 1 , 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 (<800 eV) where mechanism (i) dominates, and proposed a qualitatively correct interpretation. These experiments were followed by further investigations, also with K beams, by Diiren et al. (1974) and, especially, by Kempter and collaborators (1974a,b; Mecklenbrauck, 1976) who also measured polarizations (Alberet al., 1979, later followed by studies of the Li, Na, Rb-rare-gas systems (Mecklenbrauck et al., 1977; Speller et al., 1979; Staudenmayer and Kempter, 1980). Experiments with alkaline earth ions (Mg+, Ca+, Sr+, Ba+) were taken up by Shpenik and collaborators (Ovchinnikov ef a / ., 1974, 1977; Zapesochnyi et al., 1975). For the heavier ions, capture is no longer negligible, especially for Kr and Xe targets. At higher energies, where mechanism (ii) dominates, Andersen et al. (1975, 1976) studied total cross sections and polarizations of the simple Be+,

288

N . Andersen and S. E. Nielsen

Mg+-rare-gas systems. Later the Li, Na-He, Ne systems (Olsen e f d . , 1977; Nielsenet d . , 1978) and the K , Zn+-He, Ne, Ar systems (Andersen e f al., 1979d,e) were investigated. Closely coordinated with these studies was the theoretical program of Nielsen and Dahler (1976, 1977, 1980; Wahlstrand rf d . , 1977; Manique e f NI., 1977). Kadotaet al. (1978) applied the knowledge obtained about total cross sections for the development of a new method to probe spatial electron density profiles of plasmas by means of neutral Li beams. Differential cross sections were determined for Be+-He (Olsen and Andersen, 1975; Fayeton, 1976; Olsen e f d . , 1979b) and Mg+-rare gas (Fayeton et d . , 1976), later followed by studies of the Li, Be+, Na, Mg+, K-rare-gas systems by (polarized) photonscattered ion coincidence techniques (Pedersen e f a/., 1978; Zehnle et ol., 1978a,b, 1979; Blattmannet al., 1980; Menneret NI., 1981; Andersenef d., 1979a,b,c, 1980, 1982; Andersen and Pedersen, 1981). We restrict the discussion mainly to systems which have been studied both theoretically and experimentally, since this offers the best basis for firm conclusions. Emphasis will accordingly be given to results for the lighter projectiles Li, Be+, Na, and Mg+ colliding with He and Ne, and primarily in the energy region where the simple mechanism (ii) dominates. Detailed results for mechanism (i) will be given for the only case, Na-Ne, where elaborate cih inifio calculations have been performed (CourbinGaussorgues rt d . , 1979). A. OPTICALSPECTRA

Figure 11 shows optical spectra in the 3000-8200 8, region for 10 keV Li-He, Ne collisions. This spectral region contains most of the important spectral lines of Li, He, and Ne, except the rare-gas resonance lines which are in the vacuum ultraviolet (VUV) region. The intensities are given as percentages of the Li(1) 22S-22P resonance line intensity (peak hights have been corrected for the wavelength dependence of the detection efficiency, but not for polarization effects or variations in beam current). The resonance line dominates the spectra completely, followed by other lines from the Li(1) Rydberg series. Lines corresponding to core excitation of Li, He, or Ne are very weak indeed, supporting the qualitative concept of a quasi-one-electron system with essentially only one active electron, the valence electron. Studies of the VUV light at lower energies by Staudenmayer and Kempter (1980) lend further support to this picture. The optical spectra of the isoelectronic Be+-He, Ne systems display very similar features. Figure 12 shows analogous spectra for the Na-He, Ne systems. The

289

DIRECT EXCITATION IN ATOMIC COLLISIONS

-1

(a’

LF X

(b)

L

25

-

s -

5

FII 32p-325

C

3’s -2’P

32D-22P

2.

52 r

7 62D 5’0 1

1

1

L’D-L’P

I

I

---

-~

L‘S-22P b

0-

MI

I

Wavelength

FIG. I I . Optical spectra from the (a) Li-He and (b) Li-Ne collisions. The ordinate indicates intensities in percent of the Li(1) 22S-22P resonance line emission.

Na-He spectrum is similar to the Li-He, Ne spectra, but the Na-Ne spectrum is very different (notice the change in ordinate scale). Although excitation of the Na(1) lines is still very prominent, a multitude of Ne(I), Ne(II), and Na(I1) lines corresponding to core excitation show up, which, when added, has a nonnegligible magnitude compared to Na I excitation. VUV measurements of Mecklenbrauck et al. (1977) show a similar difference. This can be ascribed to the quasi-symmetry of the two L-shell cores (Fayeton et d.,1976) in the Na-Ne system and is also evident in the optical spectra of the isoelectronic Mg+-rare-gas systems (Ovchinnikov et a / . , 1974). B. TOTAL CROSS

SECTIONS

In Figs. 13 and 14 the experimental cross sections for the Be(I1) and Na(1) resonance line emission in collisions with He and Ne are shown as functions of impact energy. From the spectral information (e.g., Fig. 12), it is known that the observed resonance emission has a cascade contribution of 10-30%, depending on energy. The cross sections all display the

N . Andersen and S . E . Nielsen

290 1

(a)

1

‘T L’P -325

3’D-3’P

I

I

5 x

=ul P

6‘0

5-D-32P

I

0 3000

LOO0

-

5000

6000

7000 8200

so00

6000

7000 820C

8

I

s6

-

x

ul

S L C

I

2 0

3000

LOO0

Wavelength IAI

F I G . 12. Optical spectra from the (a) Na-He and (b) Na-Ne collisions. The ordinate indicates intensities in percent of the Na(1) 3’S-3’P resonance line emission. Notice the change in ordinate scale for the Na-Ne system.

same gross energy dependence, a smooth distinct maximum with a halfwidth of about one decade of energy positioned at an energy Ell which, according to the Massey criterion [Eq. (2)1, scales as E\I = h1,,u2\, W I , , ( A E Q )From ~. the projectile mass m,,,the energy defect A E , and a simple estimate ofn from the orbital radii of the projectile and the target, one may predict the relative positions of E\I for the collision systems studied (Andersen c>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 the one-electron model potentials VIIb(-), V:,k (---), and the many-electron model (-.-I, 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 and the cross-section mechanism (ii) (cf. Section II,A,2,c) predict 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

'G 15

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 1976; Nielsen and Dahler, noted for Mg+-He collisions (Andersen pt d., 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-

-

0 '5-

A

2, 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 1976). This simultaneous target excitation is important (Fayeton et d., illustrates how the light coherence is destroyed when several excitation channels contribute to the emitted resonance line radiation. The asymmetNe

Ar

-

ZI

.-

3n 0.5 d

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.

DIRECT EXCITATION IN ATOMIC COLLISIONS

299

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

x

1.0 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

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303

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

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

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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 sequence (Ganas, 1980b) transition 2p43P+ 2 ~ ~ ( ~ S ) nbut s ~for S ,the 2p43P -+ 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.

where

x

1.)

+ ( Z - 411

21+1

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

[I - j / ( 2 /

(8)

+ 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 the sum of the energy deviations of appropriate states IEF"" - E;""I a 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 force the radial functions {Pi} of { &} to have zeros at these same radial 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

is a core orbital and ( I , is a constant, the substitution of Eq. (22) where 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. so that xl should have no radial nodes Their aim is to choose the {cI,} (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 of potassium in terms of Gaussian Goddard (1974), in representing x4% 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)

+

h(r) =

-gv2 +

(44)

{
i=l

where

I: r;'

x m

Ut(r)ll)(11

(45)

1=0

with U , = U , for I > 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;

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

0.225

-

0.197

0.207

0.192

0.189

A € (a.u.)

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 . and Schwarz radial core functions and therefore potentials V N ( r ) Hafner (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 } do not dependbn j. The set of in Eqs. (64) so that the radial functions {Pi 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 for example, the relativistic contraction of the s and p solution of Eqs. (64): 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. Ganas, P. S. (1979~). J . Opt. Sot. A m . 69, 1140. 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. Szasz, L., and McGinn, G. (1967). .I Cliem. . Phys. 47, 3495. 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 experimental data (u,, ) are from thermal-energy swarm measurements. (Reprinted from 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 molecule tends to infinity ( k ' k ) , yielding

-

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 at the origin. It would pertain even for an extrct behaves incorrectly as r 2 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 )

=

A

n'

Fn,nr(r)+nW

(10)

where r is the radial coordinate of the scattered electron, X refers to the totality of molecular coordinates, and A implies antisymmetrization. The asymptotic form of Fn,nt(r) is related to the scattering amplitude .fn,n, for 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): r t l , R) =

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

=0

(18)

The approximate vibrational and rotational eigenvalues are

EL,=

w,(u

+ $1,

c

=

0, I , 2, . . .

(21)

Ej = B , j ( j + 11, j = ILI. (A,/ + 1 , . . . (22) and the total energy of the molecule in state n is En = E J R ) + E,. + E j . The set of radial functions i i b F ( r )satisfy the infinite set of coupled equations {(d2/dr2)+

ki

- [/(I

+

l)/r2]}u$.(r)

where Vt;Fm(r) and W;;,,(r)are direct- and exchange-matrix elements (the latter being an integral operator), and the second index p ' is appended to

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

353

designate a particular linearly independent solution of Eq. (23). The channel energies are related by

X.2, -

=

2[E,,(R,,) + E,

+ Ej - Eu,(Ra,)

-

Ev, -

Ey]

(24)

where the electronic energies are now evaluated at the equilibrium separations. 2 . Tlir Body-Fi.v-Pd Coordincite Frcrme

An equivalent representation is possible in a coordinate system fixed to the body of the molecule (BF), most conveniently with its Z axis along the molecular symmetry axis. In this frame Eq. (16) becomes

with

a,"'((;, rll, R) =

Y I ~ ( ~ ) R M % & ;( R ~ )X I

(26)

The subscripts inl and i n in Eqs. (17) and (26) are distinguished by reason of the fact that I? in Eq. (26) now refers to the BF frame. The channel functions remain eigenfunctions of J 2 and J , , but now also of Jab with eigenvalue A = m + A,'. The radial equations analogous to Eq. (23) are { ( d 2 / d r 2+ ) ki - [/(/ =

+

l)/r*]}~,B$(r,R)

2C, [V,B$(r, R ) + W,B$(r. R ) + (91H,&'') q,,

where 4

=

+ (91ffmlq''>I~X,(~, R) (dm; J). The channel energies are related by ki, - k i

=

2[E,(R)

-

E,,(R)]

(27)

(28)

with the vibrational and rotational components of the total energy represented by the last two terms in square brackets in Eq. (27). The first of these is diagonal in A , but the second is not. The eigenfunctions Rk,.JR) and RA,,( R ) in Eqs. (17) and (26) are a normalized form of the rotational D matrix element (Rose, 1957), e.g.,

~ i , , . , ~ =f i [(2j ) + I ) / ~ T I * / ~ D ~ 0, ., ,o)* ,,(~,

(29)

3 . The Frritne Transformation

At this stage, the representations in the two frames are equivalent, but solutions in the BF frame are most conveniently transformed into the LF

354

D. W . Norcross and L . A . Collins

frame in order to derive explicit expressions for cross sections. In this frame, for an electron incident along the -2 axis, the scattering amplitude is as follows (Arthurs and Dalgarno, 1960):

x C( j1.I; rnjO)C(j’l’J;r n j m l r ) ‘~FT;,Y~,mc,(&,) (30)

The elements of the LF-frame transition matrix ‘,FT are related to the asymptotic form of the solutions of Eq. ( 2 3 ) . These are usually taken to satisfy the boundary condition

-

ubL,(r)

-x.

kA’2rjl(k,r) 8;.

+ /i;’2r-vl(kpr)‘ X ; .

(31)

where j l and yf are spherical Bessel functions, ‘dFK;,is an element of the and LFK are related by reactance matrix ‘.FK, and L1 S= 1-T

=

(1

+ iK)(I

-

iK)-I.

(32)

The elements of l,FK could also be obtained from the solutions of Eq. ( 2 7 ) and the transformation

where the elements of the orthogonal transformation matrix A are A,, = [(2j + 11/(25 + 1)]”*C(jlJ;A U m )

(34)

and I, A are preserved, and the index u is that In this transformation a , /,. associated with p . For every linearly independent solution ukj$(r) required, a different linearly independent solution of Eq. (27) must be used in Eq. ( 3 3 ) . The solutions uiF(r, R ) must also be obtained over a sufficiently broad range of R that the integrals in Eq. (33) can be performed. With asymptotic forms analogous to Eq. (31) in the BF frame, and resulting reactance matrix elements RFK$(R),the relationship between l,FK and RFK that follows from Eqs. (33) and (34) is

‘dFK = A(ulBFK(R)Iu’)AT

(35)

Since A is an orthogonal matrix and the vibrational functions are orthonormal, it follows that Eq. (35) also holds for T. Another, but also equivalent, BF-frame representation can be defined (Chang and Fano, 1972) by expanding r ~ $ ~ R ( r), over a complete set of vibrational wave functions in Eq. ( 2 7 ) . Then the channel index 4 carries the additional index u , the dependence on R in Eqs. (27) and (28) vanishes, the difference 2[Ev - E,,] is added to Eq. ( 2 8 ) , and the radial integrals in Eqs. (33) and (35) become superfluous.

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

355

B. FIXED-NUCLEI APPROXIMATIONS It has long been recognized that one representation is more appropriate than the others to a particular region of coordinate space. Convergence of numerical calculations in the L F frame may require the coupling of a very large number of rotational and vibrational terms in Eq. (16) at small distances, in spite of the fact that Hvi0 andor Hrot make small contributions relative to the electrostatic interaction. Conversely, at large distances the inclusion of the relatively larger and physically meaningful effects of Hvib and H,,, in BF-frame calculations may require a very large number of terms in the expansion over m in Eq. (25). In practice, the complete solution of either Eqs. (23) or (27) is currently numerically intractable. It is at this point that the powerful and elegant frame-transformation theory of Chang and Fano (1972) comes to the rescue. They suggested that Eq. (33) be applied not at the asymptotic point, but at some intermediate value of 1’, say r, , and that the resulting functions ub;,(r,)be used to initialize the propagation of Eq. (23) over the remaining distance. The immediate consequence is to permit maximum advantage to be taken of the most appropriate representation for the expansion of Eq. (10). The real power of the frame-transformation theory rests, however, on the use of additional approximations based on the same physical arguments that suggested it. This leads to the working hypothesis that there is a range of collision parameters for which Hviband/or H,,, can be completely neglected (the FN approximation for these interactions) in solving Eq. (27), and that for the balance, the exchange term Wb$(r) can be neglected in solving Eq. (23). It is at this point that a theory leads to a tractable method, and we will explore several in the following sections. By taking full account of the rovibrational dynamics for large collision parameters, these methods all avoid the essential breakdown of the FN approximation for polar molecules. By neglecting, or approximating, these interactions for some range of collision parameters, however, they may ultimately fail at threshold. We will pay particular attention to efforts to minimize the consequences of this failure, and thus to maximize the energy range over which FN approximations are useful. Before proceeding, we would like to emphasize the importance of a particular perspective. The choice of coordinate representations is not an automatic c o t ~ s ~ q i ~of~the t~w decision to neglect Hviband/or H,,, (more precisely their contributions to channel energies) in a certain regime of collision parameters, as the results would be formally identical had we remained in the LF frame throughout (Choi and Poe, 1977b). The choice of coordinate frame is merely a computational device for taking maximum advantage of the approximations made. The development of

356

D. W.Norcross an.d L . A . Collins

computational tools has been an important part of the theoretical progress that concerns us here, but it should not be allowed to obscure the physical picture. I . The Raditrl Frrrme- Trcrtisfimntrtioti Mrrliod

This method uses the scattering radius I' as the collision parameter of choice, and thus follows directly from the frame-transformation theory with the approximations discussed above. Neglect of H,,, for r s r1 eliminates coupling in A , and neglect of H,iheliminates an awkward integral operator. Neglect of W::,,(r) in Eq. (23) forr 2 r 1 eliminates the requirement for a set of i / b ; , ( r ) at d/r d rt from Eq. (33). If H,,,,is to be retained for some range of r d rt ,the BF-frame equations can be recast in the alternative form mentioned above. It has been suggested that an approach with at least the sophistication of the radial frame-transformation (RFT) method of Chang and Fano (1972) is required for a comprehensive treatment of electron scattering by a polar molecule (e.g., Chandra and Gianturco, 1974; Chandra, 1975a). The proper choice of the transformation radius r, is not well defined. Had no approximations been made to the interaction potential, the results would be totally insensitive to the choice of r t . To the extent that the FN approximation is valid and the RFT method useful, it should therefore be possible to design a transformation analogous to Eq. (33), the results of which are insensitive to rt over a reasonable range. Le Dourneuf ot al. (1979) pointed out that sensitivity to r , can be dramatically reduced by the proper choice of boundary conditions for the radial functions used in Eq. (33), since these are presumed to transform according to Eq. (35). The form Eq. (31), for example, may lead to awkward variation for small r1. A transformation using the R matrix instead (Wigner and Eisenbud, 1947) will break down for large r t . They argue that the choice most consistent with the FN approximation for r s rt is that which corresponds to the weakest energy dependence near I' = 0, and they suggest the use of functions with boundary conditions 8s;

r/;;,(r)-k;'r&r) ,-I

+ kyI'y&J)

LFM$

(36)

where in matrix notation M

=

k-(i+l/2)Kk-(/+l/2)

(37)

What might be gained can be appreciated by considering the limit r,+ x . The choice of boundary conditions Eq. (31) leads to the transformation Eq. (35) for ',FK, whereas the choice of Eq. (36) leads to the transformation

LFK= XA(uIHFK(I?)Iu')ATXT

(38)

ELECTRON SCATTERING BY HIGHLY POLAR MOLECULES

357

X = kEl/2Ak-l-l/2AT Q

(39)

where Generalized diagonal matrices of channel momenta are to be understood in Eq. (39). With the FN approximation for Hviband/or H,,l, the channel momenta k , are ill defined (e.g., for L Y U = L Y ' U ' , are we to associate k-, = k,. with k, or k,. , where these refer to different rotor states?); but when short-range interactions dominate the scattering process, as treated in the BF frame, RFMwill be slowly varying near the threshold and thus LFK as defined by Eq. (38) is relatively insensitive to the channel momenta k, and has the correct threshold energy dependence. In the RFT method this translates into weak dependence on rl. The assumed dominance of short-range interactions suggests difficulties, however, for highly polar molecules. For I ' = I f 1, limk,=k,,+OK;t constant in the FBA (Chandra, 1975a), which we expect to be correct near threshold. In this case some of the elements of LFKobtained using Eq. (38) will be strictly divergent in the limit k, = k-,, +. 0 and r1+ x . Thus even if sensitivity to rl is reduced in the RFT calculations, it may come at the expense of sensitivity to the choice of the channel momenta k,. While generalization to include the effects of long-range interactions 1979),it has been suggested that a more may be possible (see Green rt d., direct and extensive modification of the basic RFT method is required for polar molecules (Vo K y Lan PZ a / . , 1979). It was noted that a choice of rI sufficientlysmall to satisfy the essential criterion (Ik - k'lr, << 1) for validity of the RFT method may severely complicate solution of Eq. (23) beyond r1 near threshold, symptomatic of the difficulty noted above. They suggested use of the /,-conserving approach of Choi and Poe (1977b); i.e., retaining in the BF-frame equations only the diagonal part (in A) of H,,, , thereby accounting for the average change of the relative kinetic energy of the colliding electron as it follows adiabatically the rotation of the molecule. This achieves the breaking of the rotational energy degeneracy of the BF-frame equations for I' s r I ,thereby restricting the strong dipole coupling to nondegenerate channels. This accomplished, the advantages of the M matrix based transformation approach to the RFT method are restored for vibrationally elastic scattering; indeed, Le Dourneuf rr a/. (1979) show that it becomes less necessary. For vibrational excitation, however, the I, -conserving modification fails to solve the problem for polar molecules. Again, appealing to the FBA for I ' = I f 1 , limk,,+,, BFK$ k'6;e1'z/k',+1'z (Chandra, 1977),but on the scale of the vibrational spacings k , k , , thus yielding again divergent '.FK, unless HVlhis explicitly taken into account in the BF-frame calculations.

-

-

D. W . Norcross 2 . Tlir Angirltit.

and L . A . Collins

Frtrt?ir-T,.trti.~~~rtn~itioti MtJthod

The RFT method, while conceptually simple, is not trivial to apply. At this writing only one calculation fully employing it has been published (Chandra, 1977), although others have been reported (e.g., Vo Ky Lan et d., 1979). An alternative has been suggested that involves minimal additional approximation but which results in considerable computational simplification (Collins and Norcross, 1978). It relies for its claim to validity on the same physical arguments but proceeds from an angular rather than radial partitioning of space. The working hypothesis in this angular frame-transformation (AFT) method is that there is a range of angular momenta I s I, for which the FN approximation is reasonable for t i l l r , but that for 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 x reduce to what is referred to as the AFT method in the limit I, 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 With Eq. (42), Eq. (13) becomes by the result of Eq. (35) applied to HFT. 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 work to date, HVib and H,,, have been completely neglected. The methods 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 As has been shown for the strongly polar system LiF potential (-r4). (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 - y A / r A + ’ , where y h is the Ath multipole moment have the form u A ( r ) 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 a lOI,,,,,/k. for matching to the asymptotic boundary conditions, i.e., rmax 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 AFT method (Siege1 of d., 1980), or I' > I-, in the RFT method (Clark, 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 PARTIAL A N D TOTAL( S U M M EPARTIAL) D INTEGRATED CROSSSECTIONS" FOR LiF FROM 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.

*

D D

2.59 a.u. 2.47 a.u. Interpolated. =

=

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 both D and the short-range interaction. The ERT predicts that C 1 IOD = 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 modDCO ( 0 . 3 ) : ---,DCO (0.50); els and from the CCHS model ( 0 )of Rudge (1978): --, , 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

30

I

I 60

1

1

I

90

387

I

I

1

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 1982), elaborate calculations to date on scattering by HCI (Padial ef d., 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.

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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

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

. . . . .

. .

. . .

. .

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)

G. W.F. Drake

402

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. His result therefore indicates that Mohr's values for G S E ( Z ~are ) probably 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 Lambshift measurement and is the Lamb shift calculated as in Table IV, 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) 3339(35) 3343.1(1.6) 3405(75) 3339.5( 1.6) 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 measuremicrowave resonance: (A) anisotropy measurement: (+) quench-rate ment according to: (0) 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 term and the 2pll, - Is,,, - 2p3/, cross term. The from the 12p,/, - ls1/2~2 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 SPIN ANISOTROPY OF IONS WITH ZERO NUCLEAR

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, quench-rate measurements had been extended to C5+,07+, and Fn+using 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

\

-‘\$Or

LASER , BEAM

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 beam of 07+ ions from the Kansas State University tandim van de Graaff 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 measurement in the exotic system K - - ~ H ~by~ Bertin + et d.(1975). Their experi-

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,<= (ezZ4/12at)( r 2) = 0.80979 X 10-6Z4(p/trz,)3(r2) (meV/fm2) (58) and the muon self-energy (including anomalous magnetic terms, but excluding the vacuum-polarization terms corresponding to A l l and AX: which are included in the numerical integration results in Table I),

where [a(Za)4m,czl/6.rr= 0.1 1599 ’meV, AESE= SSE(2p,) - SSE(2sllZ), and is defined as in (19). The expression for AEsE assumes a point nucleus. It can be corrected for finite nuclear size by replacing the wave function at the origin I$(0)Izby the expectation value of the nuclear charge density (ps(r)), as done for example by Rinker (1976) and Borie and Rinker ( 1978). The corresponding correction for electronic systems leads to the S u term discussed in Section II?B, Eq. (40). Also, the Bethe logarithms with finite nuclear size included have been calculated by Klarsfeld (1977) for heavy nuclei. The largest effect not included above is the Kallen-Sabry correction of order a 2 ( Z a ) 4to the vacuum-polarization part. The leading term gives the BIZ contribution to the electronic Lamb shift, but for muonic systems a numerical integration must be done similar to the one for the Uehling term (Barbieri rt NI., 1973; DiGiacomo, 1969). There is also a relativistic recoil term analogous to the second term of Eq. (42), but with me replaced by m, , together with a finite radius recoil correction to the 2s1,, state given by

426

G. W. F. Drake

(Rinker, 1976; Friar and Negele, 1973)

for a uniform nucleus of radius R o . These terms plus other small corrections are discussed by Borie and Rinker (1978, 1982). Their results for P - - ~ H ~ , +are shown in Table VIII. These calculations are particularly significant for electronic Lamb-shift measurements in 4He+ because they provide an independent check on the value (r2)l12= 1.674 2 0.012 fm for the nuclear radius obtained from electron-scattering data (Sick rt 01.. 1976; Rinker, 1976). The 3p-3d transition frequencies (Borie and Rinker, 1980) and hyperfine structure in 3He muonic states (Borie, 1976) have also been calculated. Since Lamb-shift measurements in other light muonic atoms may become possible, estimates obtained from Eqs. (57)-(59), together with the nuclear radii in Table IV, are given in Table IX. The higher order terms discussed above could also be included, but the uncertainties are dominated by the nuclear radius correction. The important conclusion is that C\E(2p,I,-2s,,,) for 6Li(X = 6900 A), and AE(2p,,,-2~,~,)for eBe(h = 6000 A) lie in the visible region of the spectrum accessible to tunable dye lasers. The calculation of transition energies in heavy muonic systems is reviewed by Brodsky and Mohr (1978), and detailed tabulations have been given by Engfer et t i / . (1974) and Rinker and Steffen (1977). Self-energy corrections for Is,,, levels have been calculated to all orders in the external field by Cheng c’t cd. (1978). The effects of electron screening (von Edidy et a/.. 1978) and the Breit interaction (Rashid and Fricke, 1980) on the fine structure of excited muonic states have also been discussed.

IV. Two-Electron Systems High-precision calculations for two-electron atoms and ions are complicated by the necessity of simultaneously taking into account relativistic, QED, and electron-correlation effects. There is no unique way of specifying an exact relativistic two-electron Hamiltonian analogous to the Dirac Hamiltonian without at the same time including QED effects to all orders. The rigorous starting point is generally accepted to be the fully covanant Bethe-Salpeter equation derived from the Feynman form of QED by Salpeter and Bethe (1957), and from quantum field theory by Gell-Mann and Low (1951). Exact solutions to this equation have not been obtained. All practical calculations are based (explicitly or implicitly) on expansions

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS TABLE VIII TO T H E 2s-2p SPLITTINGS I N THE @-*He)+ SYSTEM' CONTRIBUTIONS

Electronic VP: Uehling aZa first iteration

1666.57 - 0.76rz = 1664.44 1.70 higher iterations 11.55 Kallen-Sabry a2Za a ( Z a ) " .n t 3 -0.02 a 2 ( Z a )* 0.02 Muon VP a Z a 0.33 .0.02 p-e VP a 2 a 0.15 Hadron VP E1tol: 1680.32 - 0.76r* = 1678.19

Vertex corrections:

+

-10.90 0.23r = -10.52 -0. I 6 a ( Z a ) " ,n P 2 -0.03 dZCY Totul: - 1 I .09 + 0.23r = -10.71 Recoil, Breit +0.17r = 0.28

CXZa

Two-photon

-0.44 Total: -0.44 0.17r = -0.16

+

Point Coulomb (fine structure) 145.70 Totul QED: 1814.49 + 0.40r - 0.76r2 = 1813.02 3.1 ? 0.6 Nuclear polarization - 105.46r' + 1.40r3 Finite size Totul:

Experiment

= -288.9 ? 4.1 1817.6 + 0.4r - 106.2r2 + 1.4r3 = 1527.2 2 4.2 1527.5 f 0.3

1666.30 - 0.76rz = 1664.17 I .70 11.55 -0.02 0.02 0.33

0.02 0.15 1680.05 - 0.76r' = 1677.92

- I I .23

+ 0.23r

= -10.85

-0.16 -0.03 - 11.42 + 0.23r = -11.04 0.17r = 0.28 -0.44 -0.44 + 0.17r = -0.16 0.00 1668.19 + 0.40r - 0.76r' = 1666.72 3.1 2 0.6 -105.46r' + 1.40r3 = -288.9 f 4.1 1671.3 + 0.4r - 106.2r' + 1.4rs = 1380.9 2 4.2 1381.3 0.5

*

From Borie and Rinker (1978, 1982); r, root-mean-square charge radius of 'He in fm: all formulas evaluated using the value r2= 2.802 fm*.

427

G. W . F. Drake TABLE 1X ESTIMATES OF ENERGY DIFFERENCES IN LIGHTMUONlC ATOMS"

'H 4He ELi sBe IlB 1 2 c

0.2050 1.666 4.665 9.256 15.376 22.999

0.0017 0.01 1 0.032 0.064 0.11 0.16

-0.0038 -0.289 -3.593 -11.21 -24.99 -54.14

-0.006 -0.01 1 -0.051 -0.148 -0.336 -0.653

0 0 0 0 0 0

0.2023(I ) 1.377(4) I .053(70) -2.037(89) -9.8( 1 .O) - 3 I .63(22)

E(2~312) E(2~i/z) -

IH

fiLi sBe I'B

l2C

0.2050 1.666 4.665 9.256 15.376 22.999

0.0017 0.01 I 0.032 0.064 0.1 1 0.16

-0.006 -0.01 1 -0.049 -0.143 -0.323 -0.626

-0.0038 -0.289 -3.593 -11.21 -24.99 -54.14

0.0085 0.146 0.795 2.368 5.797 12.04

0.2108( I ) I .523(4) 1.800(70)

0.336(89) -4.0( I .O) - 19.53(22)

" In meV.

* The Kallen-Sabrey

term. The values for "Li to 12C are estimates.

of the Bethe-Salpeter equation in powers of a , a Z , andZ-l. Early work of this type for two-electron atoms was done by Sucher (1958) and Araki (1957). Unlike the Lamb shift in one-electron systems, there are no QED effects which manifest themselves directly in lowest order. It is first necessary to subtract the nonrelativistic two-electron energies and relativistic corrections before specifically QED effects such as the Lamb shift are revealed. Distinctly different types of approximations are useful in the low- and high-Z regions. For low Z , the best strategy is to obtain highly accurate solutions to the nonrelativistic Schrodinger equation and then include relativistic effects by perturbation theory. For high Z , better results are obtained by starting with exact solutions to the one-electron Dirac equation and then including the electron-electron interaction and higher order radiative corrections by perturbation theory. Because of developments in beam-foil spectroscopy, and other highly stripped ion sources such as tokamaks and fusion plasmas, high-precision energy-level measurements in heavy two-electron ions are becoming

429

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

available for comparison with theory. The two types of approximation in the low- and high-Z ranges are discussed separately in the following sections, together with comparisons with experiment. A. C A L C U L A T I O AN ND S RESULTSFOR He

AND

Li+

I. Fine-Strirctrrw Splittings Until recently, most calculations for low Z have been based on an approximately covariant generalization of the one-electron Dirac equation proposed by Breit (1929). The nonrelativistic energy ENR is of order azmc2 and the fine-structure splittings are of order a4mc2.The terms to this order are given correctly by the eigenvalues of the 16-component spinor equation HB$=

(61)

EB$

where HB = Hi>(I)+ HD(2)

+

V12

+B

H,(i) = c a i - p i + pmc' - Z e 2 / r , VI2 = e2/rI2 a and p are 4 x 4 Dirac matrices (Bjorken and Drell, 1964), and B is the Breit operator representing the retarded Coulomb and magnetic interactions between the two electrons. For diagonal matrix elements in lowest order, B reduces to (e.g., Akhiezer and Berestetskii, 1965)

B

=

-(~2//2r12)(a,*a2 + al-ilZcr2*il2)

(65)

where rI2 = Ir, - r21. Mittleman (1971, 1972) has derived a state-dependent operator which is valid for off-diagonal matrix elements and which also includes higher order Coulomb retardation corrections. Defining two-particle matrix elements of an operator p1,2by Pnr.n,r,

=

11

dr, ~

~

2

~

~

~

~

1

~

~

~

(66) ~ ~

then Mittleman's operator for the matrix elements (V12 + B)nl,n,l,can be written in the form (Drake, 1979) Vlz + B

=

(e')/rl2)[F(rd- cul*a2C(r12)1

(67)

2

~

~

G. W.F . Drake

430

with F(1.12) =

1 + (E1,[*/2En,n,)(l- cos

+ (En,n,/2E,,lt)(l - cos C(rl2) = ~ C O S + cos fl[,p)

fln,n,)

fir.^)

(68)

(69)

where Q n , n o = En,n,rlz/hcand En,,,, = E , - E n . . For diagonal matrix elements, Eq. (66) reduces to the more familiar form V12 + B = (e2/r12)(l- a I * a Z cos )

(70)

used in Dirac-Hartree-Fock calculations (Bethe and Salpeter, 1957; Mann and Johnson, 197 1). If the cos part of Eq. (70) is expanded into replaced by the double coma power series and the term (eEn,n!/hc)2r12 mutator -(e/hc)' [HI)(I), [H,,(2), rlz]], then Eq. (70) reduces to the stateindependent form of Eq. (65). Further reducing Eq. (62) to the twocomponent Pauli form results in

Hp

=

HO

+ Bp

(71)

where Ho is the nonrelativistic Schrodinger Hamiltonian H o = -(h2/2m)(Vf

+ Vf)

-

(Ze2/rl) - ( Z e 2 / r 2 )+ ( e 2 / r , , )

(72)

and B,, is a sum of five operators of O(a4mc2)representing the well-known spin-orbit, spin-spin, etc., terms [e.g., Bethe and Salpeter, 1957, Eq. (39.14)]. High-precision calculations of the eigenvalues of H o and the matrix elements of B , have been done by Accad er N / . (1971) and Schiff rt ti/. (1973) for the lsns 'S, lsnssS, lsnp 'P, and lsnp3P states with n up to 5 and 2 up to 10. The most accurate calculations for the ls3d 'D and 3D states are those of Blanchard and Drake (1973), Drake (19811, and Sims et t i / . (1982). Results for higher D and F states are given by Chang and Poe (1976) and Brown and Cortez (1971). These corrections, together with mass-polarization effects, must be subtracted from experimental transition-frequency measurements before higher order QED effects can be observed. The experimental measurements in helium most sensitive to higher order effects are the fine-structure splittings J = 0 + 1 and J = 1 + 2 in the ls2p 3PJlevel (Lewis er a/., 1970; Kpanou et a/., 1971; and see Fig. 9). The experimental precision of about 1 ppm is sufficient to determine the fine-structure constant to the same accuracy if the theoretical splitting can be calculated. Since corrections of O(a6mcz)to B , can be expected to contribute about 100 ppm, they must be included in the calculation. Schwartz (1964) initiated a program to carry the theoretical calculations to the 1 ppm level of accuracy.

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

43 1

I I

I I

I

I

I

1S2P 3P

/

\

T\\

\

\

\

J =1 J =2

229f.196 f 0.005 MHz

t

F I G .9. Energy-level diagram showing the fine-structure splitting of the ls2p 3Pstate of neutral helium; J = L + S is the total angular momentum.

The necessary corrections of O ( a h c 2 )cannot be obtained directly from Eq. (61) because the equation becomes ill behaved if relativistic effects are treated as other than a first-order perturbation. As pointed out by Brown and Ravenhall (1951), Salpeter (1952), and recently discussed by Sucher (1981), the Breit equation is inconsistent with Dirac hole theory in that it allows transitions into negative-energy states, which in hole theory are filled. To avoid these difficulties, Douglas (1971) and Douglas and Kroll ( 1974) carried through a systematic reduction of the Bethe-Salpeter equation to obtain all the spin-dependent terms which contribute to the 2 3P, fine structure up to order a6mc2.The results can be expressed symbolically in the form

EJ = Eo

+ a 2 ( B p ) , + a4(Bp(E0 - H0)-'Bp)J + a4(H6),

(73)

where Eo is the nonrelativistic energy, the second term is the usual Breit interaction, the third term represents the Breit interaction taken to second order, and the fourth term containing is the sum of higher order QED corrections to B p obtained by Douglas and Kroll. The terms included are represented by the Feynman diagrams in Fig. 10. (Their result is not completely general in that only the spin-dependent terms are retained.) In addition, there are anomalous magnetic-moment corrections of leading order a5mc2(Araki et a / . , 1959; Stone, 1961, 1963; Schwartz, 1964; Daley et a / . . 1972) and small relativistic recoil corrections of order (rn/M)a4mc2

G. W . F . Drake

432

F I G .10. Feynman diagrams contributing to the a%ic2 fine-structure splitting of twoelectron systems. The dashed line represents a Coulomb interaction between the two bound electrons and the wavy line a transverse photon interaction.

(Stone, 1961, 1963; Daley rt written in the form

41

id.,

1972). If the spin-dependent part of Br, is

= Hz,,,,

+ He,,, + He,,,

(74)

where If,,,,, is the spin-orbit interaction with the nucleus, If,,,,,is the spin-orbit interaction between the electrons, and H,,,, is the spin-spin interaction between the electrons, then the anomalous magnetic-moment terms are

H5

=

(a/2n)(2Hz,w

+

We,,,,

+ 2He.J

(75)

and the spin-dependent relativistic recoil terms are (Stone, 1961; Daley et

d.,1972) H,

= -(m/MHe)[Hz,w

-t

3He.w

+ 3He,,,

-

( Z a 2 / r ? b i * ( r i Pz)] (76)

The next higher anomalous magnetic-moment terms are included in H 6 , which contains a total of 16 terms that are written out in full by Daley ct id. ( 1972).

The terms in Eq. (73) were calculated to high accuracy with correlated variational wave functions (containing up to 455 terms) in Hylleraas coordinates. The second-order perturbation term was calculated variationally by solving an inhomogeneous perturbation equation according to the method of Dalgarno and Lewis (1956). The results of these extremely lengthy calculations are summarized in Table X (Lewis and Serafino, 1978). If the theoretical value for vol is taken as correct, then the derived value for a-l is 137.03608(13), in accord with other measurements as discussed by Lewis and Serafino (1978). The fine-structure splittings of the helium ls3p 3P states are discussed

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

433

TABLE X THEORETICAL CONTRIBUTIONS TO T H E F I N ESTRUCTURE OF 2 3 H ~ ELIUM~ Termb

Vlt

Vo1

29564.577(6) 54.708 - 10.707 1 1.657(42) -3.331(4) 29616.904(43) 296 16.864(36)' 0.040 = 1.35 ppm

23 17.203(2) -22.548 I .952 - 6.866(8 I ) 1.542(7) 2291.283(81) 2291 .1%(5)d 0.087 = 37 ppm

" In MHz. From Lewis and Serafino (1978). The values of a - l , c, R,, and mlM used are 137.035987(29), 2.99792458(12) x 1O1O cm/sec, 109737.3143 cm-I, and 1.370934 x W4,respectively. Kponou 1'1 t i / . (1971). Lewis PI ol. (1970).

by Tam (1979). He recommends the values v m = 8772.55 2 0.04 MHz (Kramer and Pipkin, 1978) and v12= 658.67 5 0.09 MHz. The best theoretical values available for comparison are the calculations of Schiff et a/. (1973), which include a correction for mixing between ls3p 3P, and ls3p 'P, but not for mixing with other intermediate states introduced by the third term of Eq. (73) or for the H6 terms. The theoretical values are 8772.5 and 658.04 MHz, respectively. The poor agreement for v12relative to vo2 indicates that the J = 1 state may be substantially shifted by further intermediate mixing terms. In Li+, the 2 3P,, fine-structure splittings have recently been measured to high precision by Bayer et a/. (1979) and Holt e? al. (1980). Both experiments involve crossing a laser beam with an excited 'Li+ 2 %, ion beam to measure the 2 3S,-2 3PJtransition frequencies. In the former experiment, the laser is tuned, while in the latter, Doppler-shift tuning is used. The fine-structure splittings of the 2 3PJmanifold are obtained by first extracting the effects of hyperfine structure (Jette et a/., 1974; Aashamar and Hambro, 1977) and then taking differences of the transition frequencies to the common 2 3 S , state. The results for a hypothetical spin-zero nucleus are compared with each other and with theory in Table XI. The experimental results for vo2 do not lie within the quoted error limits of each other, although the vol and v2, intervals agree reasonably well.

G . W.F. Drake

434

TABLE XI COMPARISON OF THEORY A N D EXPERIMENT FOR T H E FINE-STRUCTURE SPLITTING OF T H E ls2p 3P0,1,pSTATE" IN Li+

Theory AERI Total Experiment

A yo*

A yo1

3.1046 0.0131 3.1177 3.1051(12) 3.1028(2)

0.0104 5.2048 5.1948(12) 5.1934(8)

5.1944

Reference

AW*l

2.0898 -0.0027 2.0871 2.0897(12) 2.0906(8)

Schiff ef a / . (1973)

Holt et a / . (1980) Bayer er a / . (1979)

In cm-'. Relativistic corrections of order a 4 Z Band a'Z5.

The calculation of the a6mc2terms in Eq. (75) has not yet been extended to Li+. The theoretical results of Schiff et al. (1973) shown in Table XI include the a 2 ( B p ) J term (of order a4mc2), together with a 2 'P,-2 3P1 mixing correction and the lowest order anomalous magnetic moment terms. These values are in reasonably good agreement with experiment. However, addition of the term AE,, containing the leading corrections of order a6Z8mc2and a6Z5mc2in Eq. (73) puts the totals into strong disagreement with experiment. The source of the discrepancy appears to come from higher order terms in Z-l. For example, AEreIfor the 0 + 2 transition is approximately AE,.,,,(O + 2)

=

-0.01953dZ6~c2( 1 - 7.4192-')

The large contribution from the Z-l term will presumably be reduced by higher order terms in the series. The accuracy for low Z could be improved by introducing a screening approximation. 2. Lamb Shifts A number of transitions in helium have been measured to sufficient accuracy to be sensitive to the Lamb shift (Accadet nl., 1971). The leading terms in the Lamb shift of two-electron systems have been discussed by Araki (1957), Ermolaev (1973, 1975), and Aashamar et al. (1976), among others. The results can be expressed in the form

435

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

where E;,,2is the proper Lamb and vacuum-polarization correction given by El,,,(nLS) = &8a5Zmc2)(63(r,) + a3(r2))

and ELs2(nLS) is an electron-electron interaction correction given by EY.,2(nLS) = ~ ( a s m c 2 ) [ ( 8 3 ( r 1 2 ) ) x ( ~ l n a + - 178 - - s l *40 s2)

15

-yo]

(79)

3

where Q is the principal part of ( r ; , 3 ) , and E ; I j ( n L S J ) is the anomalous magnetic-moment correction given by the expectation values of the operator H, defined by Eq. (75). Specifically,

with pe = eh/2mc. If (a3(r1))in Eq. (78) is replaced by the hydrogenic valueZ3/m3(in atomic units), then Eq. (78) reduces to the corresponding expression for the one-electron Lamb shift. The (a3(r12))terms vanish for triplet states. The greatest uncertainty in evaluating Eq. (77) arises from the value of the two-electron Bethe logarithm ln(ews/Rm). It has been accurately calculated by Schwartz (1961) for the ground state of helium with the result ln(cloo/Rm)= 4.370 5 0.004. The resulting ground-state Lamb shift of about 1.35 cm-I, combined with the accurate calculations of Pekeris (1959) brings the theoretical ionization potential for helium, Jtheo= 1983 10.685 k 0.005 cm-I, into agreement with the experimental value J,,, = 198310.82 2 0.15 cm-I (Herzberg, 1958). More recent calculations by Aashamar and Austvik (1976) have confirmed Schwartz's value and extended the calculations to higher values of 2. Their values are given in Table XII, along with the hydrogenic approximation values. Agreement with the hydrogenic approximation is remarkably good for low 2 but tends to get worse for Z > 6. The only other results of comparable accuracy are from a calculation by Suh and Zaidi (1965) for the ls2s 'S and ls2s 3S states of He. They used

G. W. F. Drake

436

TABLE XI1 VALUESOF In(e,,/R,) FOR T H E G RO UND STATEOF ISOELECTRONICSEQUENCE"

2 3 4 5 6

4.37 c 0.01 5.21 ? 0.01 5.777 ? 0.003 6.214 c 0.002 6.565 2 0.002

4.370 5.181 5.757 6.203 6.568

7 8 9 10

THE

HELIUM

6.864 t 0.002 7.115 2 0.002 7.334 c 0.002 7.525 ? 0.002

6.876 7.143 7.379 7.589

Aashamar and Austvik (1976).

an elaboration of an oscillator-strength sum method proposed by Pekeris (1959) to obtain In(c,,/R,)

=

4.345

4

0.020 (2 lS)

ln(rzlo/R,)

=

4.380

k

0.020 (2 3S)

For Li+, Dalgarno (quoted by Accad et d.,1971) estimated that the Lamb-shift corrections to the ionization potentials of the 2 lS and 2 3S states are I , = -0.69 and - 1.14 cm-', respectively. (These values differ in sign from the corresponding energy-level shifts.) Berry and Bacis (1973) used instead a Z 4 scaling of the Suh and Zaidi (1965) calculation for the 2 3S, state of He to obtain ZI. = -0.99 cm-I for Li+. They then derived from the experimental 2 3S1-2 3P1transition frequency, the value 0.28 k 0.05 cm-I for the residual Lamb shift of the 2 3Pstate. Although this value is unexpectedly large, Ermolaev (1975) showed that it can be explained by an anomalously low electron density at the nucleus in the 2 3P state. The values calculated by Ermolaev for Li+ are 1,-(23S1) = - 1.025 5 0.055 cm-l and Z1(23P,)= 0.291 2 0.041 cm-'. The predicted Lamb-shift contribution to the 2 3S1-2 3P, transition frequency of 1.316 k 0.069 cm-' agrees with the value 1.2543 k 0.0016 cm-I derived from the measurements of Holt et d.(1980). Experimental and theoretical values for higher Z ions are discussed in the following section. B. HIGH-ZE X T R A P O L A T I O N S I . TIieot?

For higher values ofZ, it becomes advantageous to perform a doubleperturbation expansion in powers ofZ-' as well asaZ. Early calculations of

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

437

this type were done by Dalgarno and Stewart (1960), Layzer and Bahcall (1962), and Doyle ( 1969). Extensive numerical results have been obtained by Labsovskii (1970), Klimchitskaya e f nl. (1971), Ermolaev and Jones ( 19741, Ivanov e f rrl. (1973, and Vainshtein and Safranova (1978). A useful review of this work and the experimental data for helium-like ions in the range Z = 11-18 has been published by Martin (1981). The discussion in this section is based on a formulation by Drake (1979). The origin of the Z-' expansion can be seen as follows. If the units of distance and energy are rescaled so that E = E / Z ' , p = Z r , then the nonrelativistic Schrodinger equation becomes (e.g., Dalgarno and Stewart, 1957) Hod,

=

(81)

€04

with HO =

+ Z-WP

(82)

H,O = -(h2/2rn)(V& +

VA) -

(e*/pl) - (e2/pz)

(83) (84)

HY = e ' / p , ,

Thus, Z has been scaled out of the zero-order Hamiltonian H8. The term Z-WP can now be treated as a perturbation with 2-1playing the role of a perturbation parameter; Eo and C$ then have the perturbation expansions

+ Z-*E$ + . . . +] 4 = 4o + Z-l+] + z-24, + . . .

Eo = Z2[E8 + Z-'EY

(85)

(86)

such that (H! - E8)+, = 0

(87)

(H8 - E,O)41 + HPC$o = EP40

(88)

etc. Accurate variational solutions to the above perturbation equations have been used to calculate a wide range of atomic properties in the form of a Z-I expansion series (e.g., Sanders and Scherr, 1969; Dalgarno and Drake, 1969; Doyle e f (11.. 1972; Aashamaref cil. 1970). Using the 4 expansion to evaluate the matrix element 03,) in Eq. (73) results in an expansion of the form aZ(B,) = aZZ4[Ei

+ Z-IET + Z-*E$ +

. . .]

(89)

and the a4 terms in (73) can be expanded a 4 ( B I , ( H0 E0)-'BI,) + a 4 ( H 6 = ) a4Z6[E,4+ z-1~;

+ z-"q +

.

.

(90)

G . W . F . Drake

438

and similarly for higher order terms. Thus, the total energy can be written in the form E

=

ZZEg + a2Z4Eg + a4Z6E$ + . ' .

+ Z'EP + a2Z3Ef + a4Z5Ej + + Z0E$ + a2Z2EZ + . . . + Z-'E! + a2Z'Ei + . . .

... (91)

excluding Lamb-shift and nuclear-motion-type corrections. These can be added in separately at the end. For convenience, the anomalous magnetic-moment correction Eq. (80) is included in B P . The first-column sum is the exact nonrelativistic energy EO, the second-column sum is a z ( B , , ) ,the first-row sum is the sum of the exact single-particle Dirac energies for the two electrons, and the second-row sum corresponds to the matrix element ((e2/r12)+ B ) of the operator [Eq. (67)], using products of hydrogenic Dirac spinors as wave functions. Both the rows and the columns are summed to infinity, and off-diagonal mixing effects are included, by diagonalizing the matrix

H

=

(Ho +

Bp)L.y

+ R(HI1 + V12 +

B)jjR-'

-

A

(92)

in the basis set of states which are degenerate in zero order for a given total angular momentum J = L + S (or J = j, + j, in j j coupling) and LS recoupling transformation and A is the doublepanty: R is the j j counting correction for the four terms in the upper left-hand corner of Eq. (91) which are otherwise counted twice. For consistency, we define

-

H I ) = H,)(l) + HD(2)- 2mc'

(93)

Consider as a particular example the zero-order degenerate states ls2p 3P, and ls2p 'P,, which have the same panty and total angular momentumJ = 1. Then each of the terms in Eq. (92) is a 2 x 2 matrix in the basis set of states 1 ls2p 3P1)and I ls2p 'P,). The subscript LS on the first term means that the matrix elements are to be evaluated with the best available LS-coupled variational wave functions. For Z 5 10, the results tabulated by Accad et t i / . (1971) can be used. The subscript j j on the second term means that the matrix elements are to be calculated exactly, using hydrogenic products of jj-coupled Dirac spinors for wave functions; i.e., ~ l s l ~ z 2 p , 11 2) ,and Ils1122p312,1). For this example, t h e j i -+ LS recoupling transformation is

OED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

with

v5

R=-(1

f

For the states 3 3D,and 3

'4,it

i

l

-1 f

)

i

439

(95)

is

with

For states which are nondegenerate in zero order, such as ls2 lS,,, ls2p 3P0, ls2p 3P2, . . . , Eq. (92) contains only 1 x 1 matrices, and R = 1. The matrix A in Eq. (92) subtracts those contributions which are counted twice in the first two terms. Specifically, t h e s e b e the contributions to the matrix elements of orders Z2,Z , a2Z4and a2Z3displayed in Table XIII, calculated in LS coupling. The entries up to ls2p 3P2are equivalent to those evaluated by Doyle (1969) i n j coupling. As examples, A(ls2 'So) = -2'

+ 0.6252 - a c ~ ~+ Z0.4801396a2Z3 ~

(98)

and A for the ls2p 3P, and 'P, states is a 2 x 2 matrix with elements

5

A1.1

= - jj Z 2 -t 0.2257277852 -

A2.2

= --Z2

A1*2

=

5 8

A2,'

=

59 384a2Z4+ 0. 1304287a2Z3

(99)

55 + 0.2598689222 - 384a2Z4 + 0 . 0 5 5 4 0 3 0 ~ ~ ~(100) 2~

d2 -96 02Z4+ 0 . 0 2 8 8 5 0 8 ~ ~ ~ 2 ~

(101)

In the latter example, the states ls2p 3P,and 'P, are progressively mixed by the spin-dependent interactions in Has Zincreases until ultimately the &coupling limit is reached at high Z. The eigenvectors of H (labeled by the eigenvalues E , , A = 1, 2 ) can be written

G . W .F . Drake

440

TABLE XI11

LEADING TERMS I N THE Z-'

AND

( a Z ) 2EXPANSIONS OF T H E H A M I L T O N IMATRIX AN

IN

IS, ls2s 'So 1 s2s 3s, Is2p 3P, Is2p 3P, I s2p ,PI 23~,-21~, I s2p 3P2 ls3s IS, ls3s ls3p I s3p 3P, 1 s3p 'PI 33P1-31P, I s3p 3P2 ls3d 3D, ls3d 3D2 ls3d ID2 33D,-31D, ls3d SD3 33S,-33D, Is2

a

-I - 518 - 518

- 518 - 518

- 518 0 - 518 - 519 - 519 - 519 - 519 - 519 0 - 519 - 519 - 519 - 519 0 - 519 0

LS COUPLING^

- 114 -211128 - 2 11128 -211128 - 591384 - 551384 - fiI96 - 171128 - 5/36 - 5/36 - 5/36 - 11/81 - 431324 - \/5/324 - 7/54 - 7/54 - 521405 -231180 - fi/ 1620 -411324 0

0.62500oooO 0.23 18244 I7 0.187928669 0.225727785 0.225727785 0.259868922 0 0.225727785 0.105255127 0.093719482 0.104293823 0.104293823 0.1 13357543 0 0.104293823 0.110775757 0. I 10775757 0.111270142 0 0.110775757 0

0.4801396 0.1694782 0.0769352 0.2 197682 0.1304287 0.0554030 0.0288508 0.0406387 0.0581620 0.0331906 0.0715549 0.0460209 0.0269123 0.0088266 0.0189765 0.023862 I 0.0 1405I 6 0.0108856 0.0030447 0.0044280 -0.00090986

In atomic units.

with

T=(

cos 8

sin 8 (103)

-sin 8 cos 8

and 8 is the singlet-triplet mixing angle. In the limit of low Z , T + 1 ( L S coupling) since the second and third terms of Eq. (92) nearly cancel, and HxR>> Bp in the first term. In the limit of largeZ, T R-' ( jcoupling) since the first and third terms of Eq. (92) nearly cancel, and Ho >>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 measureand F7+. ments for 06+ 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 For ions are in reasonably good agreement, except for N5+,F7+,and S4+. 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 OF HIGH-Z L I T H I U M - L IIONS KE FREQUENCIES

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,
(1 25)

Edlen chose to calculate screening parameters separately for each term in Eq. (124) from the equation s = -6E:/4SE$

( 126)

where SE: (n = 0 , l ) is the change in E : when the outer electron is added to the Is2 'So core. Using the data in Table XIX, this yields (in atomic units) Ar('S) = -(5a2/I28)(Z - 1.2808)4 + O(a4Z6)

A,('P,)

=

-(7a2/384)(2 - 2.2410)4 + O(a4Z6)

A,(2P) = (a2/32)(Z - 1 .7449)4 + O(a4ZS)

(1 27)

(128) (129)

for the leading terms in powers of a'. Edlen also included the hydrogenic terms of order a4(Z - sl6 and a 6 ( Z - s P with the same screening parameters as in Eqs. (127- 129). The screening parameter in Eq. (129) is slightly different from his empirically determined value, s = 1.7415 + 0.633(Z 0.80)-,, but this does not substantially alter the discussion to follow. The dominant QED correction comes from AI,('S). Edlen used the hy-

453

QED EFFECTS I N FEW-ELECTRON A T O M I C SYSTEMS

drogenic expression of Eq. (15) with an adjustable nuclear charge Z - s,. to obtain sla= 1.60 for the 'S state and sL= 2 for the 'P state. The resulting total Lamb shifts shown in Table XVII are quite different from the unscreened values used by Berry et a / . (1980), even though the final transition frequencies are nearly the same. The additional empirical corrections introduced by Edlen into AENRand A#P) are too small at high Z to account for the discrepancy. The source of the discrepancy lies in the values of the coefficients E%, E ; , . . . , in Eq. ( 115) implied by the screening approximations. Expanding Eqs. (127- 129) yields A E & ~ ,=G -Ar('S) ~

+ Ar('P,)

-

3 A,('P)

+

= a2(OZ4 0.10869Z3 - 0.545392'

+ 0.935072 - 0.54777)

(130)

Similarly expanding Eq. (123) yields the terms of order a', AE:,,.,,,.

- 0.50858)4 -

-&$(Z

=

+ $a2(Z - 0.47366)4

(5a2/128)(Z - 1.7944)4

-

(5a2/128)(Z - 1.3224)4

+

a2(OZ4 0.10869Z3 - 0.396252'

=

+ 0.566722 - 0.28967)

(131)

TABLE XX

RELATIVISTIC A N D L A M B - S H I F T CONTRIBUTION\ I S22p 2Pl/2- l S 2 2 S 's,,,TRANSITION FREQUENCY"

C O h l P 4 R l b O N OF LOWEST ORDER

0.870 1.604 2.667 4.120 6.023 8.439 1 1.426 15.047 19.363 24.434

10 12

14 16 18 20 22 24 26 28

0.736 1.402 2.383 3.740 5.534 7.825 10.675 14.145 18.296 23.189

_ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ __ _ _ _ _ _ _ ~

~

-0.077 -0.129 -0.197 -0.285 -0.396 -0.534 -0.706 -0.922 -1.192 - I .535

0.134 0.202 0.284 0.380 0.489 0.614 0.75 I 0.902 1.067 1.245 ______

~ _ _ _ _ _ _

TO T H E

.

~

Calculated with the screening approximations used by Berry r t n l . (1980) and Edlen (1979); 1000 cm-I. J = 1.60 in the calculation of Edlen (1979). "

454

G . W.F . Drake

Thus, the two screening approximations agree up to order a2Z3as expected, but the higher order coefficients are different. The difference is AEgrrrs- 9E2,,,,bn= (0.149442' - 0.368672

+ 0.25809)

x 11.687 cm-'.

( 132)

The above values for AE2 and their differences are tabulated in Table XX. For high Z , where the Lamb shift is large enough to be noticeable, the differences largely compensate for the different Lamb shifts used in the two calculations. Each has allowed the error in AE2 to be absorbed by the Lamb shift, resulting in very different values for sL. It is therefore not possible to draw firm conclusions about the true behavior of the Lamb shift from experimental data without knowing the value of AE; to an accuracy of a few percent.

VI. Concluding Remarks and Suggestions for Future Work Both theory and experiment have reached a high state of refinement for one-electron systems. In terms of theoretical significance, the highprecision measurements of Lundeen and Pipkin (1981) for neutral hydrogen ( 2 9 ppm) have now been nearly matched by a laser-resonance measurement for C P + (Wood trt d.,1982). While the C P + measurement is in good agreement with Mohr's calculated value, the interpretation of the hydrogen measurement is clouded by uncertainties in the proton-radius correction and the additional Borie (1981) correction (see Fig. 3). Additional theoretical work on other relativistic terms which might cancel the Borie correction, together with a high-precision measurement for deuterium would help to clarify the situation. In addition, the calculation of the G"(Za) term should be repeated in order to resolve the discrepancy between the results of Erickson and Mohr. For two-electron systems, the good agreement between the refined calculations of Lewis and Serafino (1978) and high-precision measurements of the ls2p 3Pfine structure has confirmed the O(a6mc2)spin-dependent corrections to the Breit interaction derived by Douglas and Kroll (1974). However, no attempt has been made to extend these very lengthy calculations to higher values of Z , even though high-precision measurements are now available for Li+ and Ne8+. Also, the spin-independent part of the correction has not been derived. It may very well be profitable to extend the screening-approximation techniques discussed in Section V,B to the

QED EFFECTS IN FEW-ELECTRON ATOMIC SYSTEMS

455

O(a6mc2)terms. The theory of two-electron Lamb shifts is hampered by a lack of accurate Bethe logarithms for excited states. There is now an extensive body of experimental data on ls2s 3S,-ls2p 3PJ transition frequencies for helium-like ions up to FeZ4+,which is sensitive to the Lamb shift. The results tend to support the hypothesis that two-electron Lamb shifts can be constructed from sums of one-electron Lamb shifts, provided that the electron density at the nucleus is properly taken into account. The theory of relativistic and QED effects in systems containing three or more electrons is still in a very rudimentary state. Although accurate variational wave functions have been obtained for the lower members of the lithium-like sequence, they have not been used to calculate matrix elements of the Breit interaction. The screening approximations discussed in Section V,B are not sufficiently accurate, even at high Z, to draw meaningful conclusions about the behavior of the Lamb shift from the experimental data. Dirac-Fock calculations with correlation corrections contain errors of about the same magnitude. An accurate calculation of the Eg and E; coefficients in Eq. ( 1 15) for three-electron systems would be a major step forward. Finally, the recent high-precision atomic beam measurements of the F-G, F-H, and F-I intervals for then = 7-1 1 levels of neutral helium by Cok and Lundeen (1981) and Farley ef cil. (1979) deserve special mention. While some of the intervals have been measured to an accuracy of 10 ppm, there are no corresponding high-precision calculations. The experimental values deviate systematically by 1% or more from the polarization model of Deutsch (1976) and the perturbation calculations of.Cheng and Poe (1976). Considerably more theoretical work is required to improve the agreement. Of particular importance is a QED retardation correction to the usual long-range -&a,,/r4 polarization potential experienced by the high-nl electron in the field of the He+ 1s core (Kelsey and Spruch, 1978). The effect can be regarded as arising from the correlation in the displacements of the two electrons by the fluctuating zero-point fields of the electromagnetic vacuum. For \vry high n and I states, Kelsey and Spruch have obtained the energy correction

where a,l = &a3 is the core polarizability. For some of the intervals, AE,, can account for as much as half of the 1% discrepancy between theory and experiment. However, it is not clear what corrections should be added to Eq. ( 133) for moderately low values of n and I, or how it should be incorporated into a full two-electron calculation not based on a polarization potential model. Also, the nonrelativistic contributions to the intervals must be

G . W.F. Drake understood to better accuracy before further progress can be made in testing (6.1) against the experimental data.

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INDEX nonresonant multiphoton ionization of, 97-160 “virtual” states of, 98 Atom-atom scattering, in external field, 5 Atomic collision experiments, collision geometry in, 278-284 Atomic collision physics, planar symmetry of, 282 Atomic collisions, see also Atomic collision physics; Collision(s); Collision physics; Quasi-one-electron systems coherence analysis in, 283-284 direct excitation in, 265-304 experimental techniques in, 279-287 first- and second-generation experiments in, 280-282 light polarization studies and, 281-282 on molecular orbital basis, 278-279 optical spectra in, 288-289 polarizations in, 293-2% qualitative considerations in, 268-271 reflexion symmetry in, 283 Stokes parameters in, 285 theoretical models in, 271-279 third-generation experiments in, 282-287 total cross sections in, 289-293 1 + 2 transition in, 270-271 D + P transitions in, 286-287 P 4 S transitions in, 285-286 tunable laser in, 269-270 types of, 268-271 violent vs. soft, 268-269 Atomic energy-level calculations, beamfoil spectroscopy in, 309 Atomic hydrogen, see also Hydrogen Lamb shift in, 399 multiphoton ionization rates for, 109-1 11 two-photon ionization cross sections for, 112

A

Above-threshold ionization angular distribution measurements of, 156 electron spectroscopy in, 153 multiphoton, 152 potential puzzles in, 159 Adiabatic nuclear vibration amplitude, 359 Adiabatic nuclei approximation in cross section specification, 372-374 in electron scattering by polar molecules, 359-360 Airy-function uniform approximation, 191 Alkali atom, ns -+ np resonance transition of, 268 Alkali atom-alkali ion systems, 273 Alkali atom-closed-shell systems, 267 Alkali-like-rare-gas systems, 273-278 Analytical models in Gamow-Siegert states, 222-226 with resonances, 224-226 Angle-action variables classical collisions in, 169 of free rotor, 171 in inelastic heavy-particle collisions, 167-169 isolated bound systems in, 167-168 Angular frame-transformation method, 358-359 Antimatter-matter interactions, in positron-atom scattering experiments, 53-57 Asymptotic states Coulomb waves in, 30 dressed-target states and, 17-23 ATI, see Above-threshold ionization Atom(s) eigenstates of, 98 461

462

INDEX

Atomic radiation theory, atom-field interaction in, 19 Atomic states, Stark shift in, 118 Atomic structure, model potentials in, 309-338 Atom-Molecule Collision Theory (Bernstein), 166 Atom-molecule inelastic collisions, 166, see also Inelastic heavy-particle collisions Atom-plus-field system, Floquet Hamiltonian of, 119 Atom-vibrating rotor collisions, in rotational excitation. 181-182

B BII approximation, 371-372, see also Born approximation Be+-Ne collisions, Be+(2s + 2p) experimental excitation probability in, 297 Be+(2s 2p) excitation probability, 297 Beam-foil spectroscopy, 309 Bebb-Gold method, in nonresonant multiphoton ionization, 104-105 Beryllium sequence 2s2 ‘S-2s2p’p transition in, 314 transition probabilities in, 333 Bessel functions in electron-atom scattering, 21 vanishing, 32 Bethe logarithm calculated values for, 407 estimates of, 441 Bethe theory, inelastic cross sections from, 78 BF frame, see Body-fixed coordinate frame Bielefeld system, in total scattering crosssection experiments, 60-64, 73-76 Binary atomic collisions, inelastic processes in, 266, see also Atomic collisions Biorthogonal sets in Gamow-Siegert states, 210-21 1 Hilbert space and, 213 Bloch-Nordsieck analysis, of spontaneous infrared radiation problem, 37 Bloch-Nordsieck sum rule, 29 -f

Bloch-Nordsieck theory coherent states and, 10-13 of nonrelativistic potential scattering, 6 Body-fixed approximation, in rotational excitation, 179- I80 Body-fixed coordinate frame, 353 Body-fixed sudden approximation, in rotational excitation, 180-181 Born amplitude, in scattering-theory formulation, 25, 81 Born approximation first, see First Born approximation in polar molecule calculations, 371-372 scattering amplitude and, 12 unitarized, 372 Born elastic scattering amplitudes, 25, 81 Born-Oppenheimer molecular states, 272 “Boxing” method, Monte Carlo technique and, 197 Bremsstrahlung, 6-13 Bloch-Nordsieck analysis of, 6 electron-atom scattering and, 1 infrared divergences in, 6 plasma heating and, 29 quantum analysis in, 7 single-photon, 13, 41, 44-45 spontaneous, 6 Bremsstrahlung photon distribution, Poisson law in, 9, 12 C

Calcium, two-electron potential in, 310 Campeanu-Humbertson measurements, of total scattering cross sections, 69-72 Catastrophe theory, in inelastic heavyparticle collisions, 185- 186 CEM, see Channeltron electron multiplier Cesium four-photon ionization of, 112, 132 three-photon ionization of, 1I2 two-photon ionization of, 112, 125, 143148 Channeltron electron multiplier, 60-61 Classical collisions, angle-action variables in, 169 Classical description, of inelastic heavyparticle collisions, 167 Classical methods, for polar molecule calculations, 370-371

463

INDEX Classical perturbation theory for polar molecules, 370-371 S-matrix theory and, 193 Classical trajectories, quantizing of, 197I98 ClebschGordan coefficients, 155 Close-coupling approaches, in polar molecule calculations, 361-364, 368-369 Close-coupling class, 347 Close-coupling hard-sphere method, 379382 Coherence analysis, in excitation process, 298-302 Coherence effects, in NRMIA, 115-1 16 Coherent states Bloch-Nordsieck theory and, 10-13 scattering amplitude and, 12 Collisional approximations, in polar molecule calculations, 368-372 Collisional excitation, in quasi-one-electron systems, 271-279, see also Atomic collisions Collision geometry cylindrical symmetry and, 281 planar symmetry and, 282 Collision-induced radiation, optical spectrometric studies of, 266 Collisions atomic, see Atomic collisions atom-molecule inelastic, 166 description of, 170-172 heavy-particle, see Inelastic heavyparticle collisions Collision spectroscopy, 266 Collision systems, 267-268 Complex-coordinate theorems, 209 “rotated” Hamiltonian and, 214-216 wave-function properties and, 214-226 Complex scaling configuration interaction calculations and, 251-255 developments in, 258-259 in resonance phenomena, 207-259 Complex-scaling transformations, 259 Complex stabilization method configuration interaction calculations for, 251-255 Doolen method and, 250 many-body technique in, 256 procedure in, 249-250 in resonance calculations, 247-256

self-consistent field calculations in, 255256 Configurational interaction calculations, complex stabilization method in, 251255 Configurational interaction expansions, 229-240 Doolen-type basis in, 231-238 modified Doolen-type basis in, 238-240 in resonance calculations, 227 Siegert approach in, 229-231 e trajectory in, 232 Conversion efficiency, defined, 57 Coordinate frame, laboratory- and bodyfixed, 352-353 Copper resonance lines, oscillator strengths for, 334 Core polarization, model potentials and, 327-332 Core projection operator, 324 Coulomb approximation potential, defined, 311 Coulombic potentials, in Gamow-Siegert states, 223-224 Coulomb N-particle problem, 226 Coulomb potential, effective-change, 273 “Coulomb problems,” in impact ionization in presence of external field, 49, 226 Coulomb tail, in potential scattering, 29-30 Coulomb waves, in potential scattering. 30 Cross-section specification AN approximation in, 372-314 closure application in, 374-376 MEAN approximation in, 376 Cross-section sum rule, 30-32, see also Sum rules Cubic anharmonic oscillator model, 246 Cylindrical focusing, vs. spherical, 127

D DECEN method, for forced harmonic oscillation, 181 MWHM (energy width), in slow e + energy distribution, 57-60 Density-of-states correspondence principle, 169 Destructive interference effects, in nonresonant multiphoton ionization of atoms, 140-151

464

INDEX

Detroit system, in total scattering crosssection experiments, 60-64, 73-76 Differential e+-Ar scattering results, for different positron mean energies, 86 Differential scattering cross sections, in positron-gas scattering measurements, 84-86 Dipole cutoff potential, in polar molecule calculations, 367-368 F .pole interaction, perturbation theory and, 345-346 Dipole moments, as molecular constants, 345 Dipole potential, number of, 347 Dirac-Coulomb-Green's function, 407, see ulso Green's function Dirac equation, for electron scattering, 14 Dirac-Fock calculations, for model potentials, 310, 335 Dirac-Fock equations, model potentials or pseudopotentials for, 336 Dirac hole theory, 43 1 Direct excitation excitations probabilities and, 296-298 experimental techniques in, 279-287 Direct excitation mechanisms, 287-302, see also Excitation process Direct method, in nonsaturated multiphoton ionization experiments, 126-128 Distorted-wave method, for polar molecules, 370 Doolen technique, in resonance calculations, 235-237 Doolen-type configuration interaction expansions, 231-238 modifications in, 238-240 Double multiphoton ionization of atoms, 156-157, see also Multiphoton ionization experiment Dressed-target states, in electron-atom scattering, 17-23

E e--Ar collisions, QT measurements for, 65-66 e+-Ar scattering, positron formation cross sections for, 88

e+-atom (molecule) scattering, in e--atom (molecule) scattering, 56 e+ beam attenuation of in gas scattering region, 60 energy analysis of, 60 low intensities of, 60 e+-beam energy distribution UFWHM in, 57-59 in small-angle energy distribution, 63 e+-beam energy spectrum, narrow lowenergy peak in, 59 Effective coupling strength, in multichannel scattering, 36 Effective range theory for LiF molecule, 384 for polar molecules, 371 e-H scattering, resonant transitions and, 35 e+-He collisions Qel measurements for, 70 QT measurements for, 64-65 e--He collisions, Qcl measurements for, 70 e+-He differential elastic cross sections, e+-He measurement errors and, 78 e+-He scattering, positronium formation cross sections for, 88 e+-He scattering, positronium formation cross sections for, 88 e+-He total cross section, intermediateenergy, 76 e--Kr collisions, QT measurements in, 6567 Electric dipole approximation, in targetfield states, 25 Electron-atom core, effective-change Coulomb potential for, 273 Electron-atom scattering, see olso Electron scattering; Scattering asymptotic states in, 14-23 bremsstrahlung and, 1, 6-13 electron recoil effects in, 42 free-free vs. bound-free transitions in, 27 in gas breakdown, I gauge transformation in, 5 generalized low-frequency approximations in, 37-49 intermediate coupling in, 43-46 in laser-driven fusion, I multichannel, 33-37

INDEX multiphoton process in, 5 in plasma heating, I potential scattering in, 28-3 I in radiation field, 1-50 scattering theory in, 23-37 strong coupling and, 46-49 Electron-atom scattering resonances closed-channel type, 23 cross-section sum rule and, 32 Electron-atom system, field intensity experienced by, 3 Electron-atom total cross section measurements, first, 53 Electron-field interaction, in intermediate states, 13, 43-44 Electron multiplier, in nonsaturated ionization experiment, 126 Electron-polar molecule collisions, rotational excitation in, 388 Electron recoil effect, in electron-atom scattering, 42 Electron scattering, see also Electronatom scattering; Polar molecules; Scattering adiabatic nuclei approximation in, 359360 angular frame-transformation method and, 358-359 application of theoretical tools in. 377389 fixed-nuclei approximations in, 348, 355360 radical frame transformation method and, 356-357 Electron scattering equations, solution to, 361-372, see also Scattering equations Electron scattering theory, highly polar molecules and, 341-392 Electron self-energy, Bethe's estimate of, 399 Electron translational factors in collisional excitation, 271 correlated valence orbital, 277 expansion of, 273 in perturbed-stationary-state theory, 278 Electrostatic analyzer, 90°, 60 Electrostatic-quenching-anisotropy method, in Lamb-shift measurement, 417 Elementary catastrophe theory, 185- 186

465

EMA approximation, see Energy-modified adiabatic approximation e+ moderators, high conversion coeficiency type, 59 e--Ne collisions, QT measurement for, 66 Energy-loss analysis excitation probability vs. impact parameter in, 2% of scattered particles, 266 Energy-modified adiabatic approximation, 359-360 Energy-transfer processes, inelastic collisions in, 166 Energy width, in slow e+ energy distribution, 57-60 e+-Ne scattering, positronium formation cross section for, 88 e N e system, resonant states of, 33 c (conversion efficiency), 57 ERT, see Effective range theory e+ scattering, moderators in, 57 ESE approximation, see Exact staticexchange approximation ETF, see Electron-translational factors e- total cross-section measurements, Phillips cathode in, 61-62 Exact static exchange defined, 361-362 in lithium fluoride study, 378 Exact static-exchange approximation, for polar molecules, 361-364 Exchange potential, in polar molecule calculations, 365-366 Excitation and ionization cross sections, in inelastic scattering investigations, 8991 Excitation probability, vs. impact parameters, 2%-298 Excitation process, see also Atomic collisions coherence analysis in, 298-302 for molecular mechanisms, 303 opposite time symmetry of matrix elements and, 295 optical spectra in, 288-289 polarizations in, 293-2% probabilities in, 2%-298 total cross sections in, 289-293 e--Xe collisions, QT measurements of, 65-67

466

INDEX

Exponential well potential, in GamowSiegert states, 222-223 Exterior complex scaling method, 216 External field, scattering in, see Scattering in external field External plane wave field, electron interaction with. 14

F FBA, see First Born approximation Feshbach-Yennie low-frequency approximation, 41, 43-46 Few-electron systems, quantum electrodynamic effects in, 446-454 Feynman diagram, for electron self-energy and vacuum polarization, 402 Field-free scattering cross section, 8, 10 Field-free scattering operator, in multichannel scattering, 36 Field-free scattering region, 60 Final value representation, in S-matrix integral representations, 190 Fine-structure intervals, of ls2p3po,,., states, 445 Fine-structure splittings energy-level diagram for, 43 I Feynman diagrams and, 432 in He and Li’, 429-434 Finite-dimensional photon Fock space, 32 First Born approximation, 345-347, 378, see also Born approximation defined, 346 fixed-nuclei approximation and, 348 MEAN approximation and, 376 Five-photon ionization, of sodium, I12 Five-photon ionization cross sections, calculated vs. measured values in, 137

Fixed-nuclei approximation, 348-350, 355360, 391

in cross-section specification, 373 Floquet Hamiltonian, of atom-plus-field system, 119, see also Hamiltonian Floquet theorem, 245-246 FN approximation, see Fixed-nuclei approximation Fock space, finite-dimensional photon, 32

Forced harmonic oscillator, quanta1 solution for, 181 Fourier coefficient function, in scattering theory, 23 Four-photon ionization, of cesium, 112, 132 Four-photon ionization cross sections, calculated vs. measured, 137 Frame transformation, 353-354 Free-electron-gas potential, 365-366 Free-free transitions vs. bound-free, in electron-atom scattering, 27 multiple absorption of photons by, 31 “Full width at half-maximum,” in e + energy distribution, 57-59

G Gamow-Siegert functions, S-matrix poles and, 257 GarnowSiegert potential, Coulombic potentials and, 223-224 GarnowSiegert states, 209-214 analytical models of, 222-226 biorthogonal sets in, 210-211 exponential well potential in, 222-223 hypervirial theorems in, 221-222 incoming+utgoing waves in, 21 1-214 time reversal in, 220-221 wave-function properties in, 217-220 Gas breakdown, in electron-atom scattering, 1 Gauge transformation, in electron-atom scattering problem, 5 Generalized low-frequency approximations, in electron-atom scattering, 3749

Glauber approximation, for electron-polar molecule collisions, 371, 379 Gold resonance lines, oscillator strengths for, 334 Green’s function, 3 energy denominator in, 12 in nonresonant multiphoton ionization, 108-109

in three-photon ionization cross-section calculation, 137- 138 in resonance calculations, 227

467

INDEX

H Hamiltonian alkali atom, 273 Dirac, 310 “Hermitian,” 247 model, 312 of model potentials, 326 multivalent, 326 nonrelativistic Schrodinger, 430 N-particle atomic and molecular, 249 one-electron, 277 reorganization of, 3 10 “rotated,” 214 total molecular, 216 two-electron valence eigenfunction in, 321 unrotated, 249 valence, 325 for valence electron in field of two closed-shell cores, 274 zero-order. 329, 437 Hamiltonian matrix, in LS couping, 440 Hartree-Fock approximation, 310, 317 relativistic, 448-450 Hartree-Fock computer codes, 266 Hartree-Fock formalism, potentials based on, 317-327 Hartree-Fock ground state, of N-electron rare-gas atom, 274 Hartree-Fock operator, 243 Hartree-Fock orbitals, 335 Hartree-Fock ratio, 316 Hartree potential, 329 defined, 317 Heavy atoms, core electrons in, 337 Helium fine-structure splittings in, 429-434 three-photon ionization of, 114, 129 Helium Is3p’P states, fine-structure splittings of, 432-433 Helium isoelectronic sequence, ground state of, 436 Hellmann potential, 325 “Hermitian” Hamiltonian, 247 Hessian matrix, 184 HFEGE, see Free-electron-gas potential High-Z extrapolations, in two-electron systems, 436-446

High-Z ions laser resonance experiments in, 421-422 quenching anisotropy measurements in, 422-424 High-Z measurements, Lamb shift and, 419-424 Hilbert space, biorthogonal sets and, 213 Hillman-Feynman theorem, in resonance calculations, 235 Hydrogen atomic, see Atomic hydrogen light polarization in multiphoton ionization of, 109, 114-115 Penning ionization of, 231 two-photon ionization of, 108 Hydrogen atom stark problem, 244 Hydrogenic ions, Lamb shifts and, 401, 413-424 Hydrogen-rare-gas systems, as quai-oneelectron systems, 272 Hydrogen states, resonant transitions in, 35 Hypervirial theorems, in GamowSiegert states, 221-222

I Impact excitation, one-electron model potentials and, 275-276 Impact ionization, “Coulomb problems” in, 49 Impact parameter, vs. excitation probability, 2%-298 Implicit summation techniques, in nonresonant multiphoton ionization, 106-109 Incoming-outgoing waves, in GamowSiegert states, 211-214 INDECENT method, for forced harmonic oscillator, 181 Induced resonance, in scattering in external field, 4-5 Inelastic heavy-particle collisions angle-action variables in, 167-169 classical and semiclassical methods in, 165-199 description of, 170-172 primitive semiclassical S-matrix theory in, 187-190

468

INDEX

rotational excitation in, 170-183 semiclassical theories of, 186-198 uniform approximations in, 183-186 Inelastic processes, in binary atomic collisions, 266-267 Inelastic scattering investigations, see also Electron-atom scattering; Scattering excitation and ionization cross sections in, 89-91 positronium formation cross sections in, 86-89 Inert gases at intermediate energies, 76-79 positron and electron comparisons for, 79-80 QT measurements of, 64-65 small-angle scattering in, 79-80 Initial value representation integral form, 190 Integrated rotational cross sections, for lithium fluoride molecule, 380-381 Interaction potential, approximations to, 364-372 Interference minima, in MPJ process, 141 Intermediate coupling, in electron-atom scattering, 43-46 Intermediate-energy 'e+-Ar total crosssection results, 77 Intermediate-energy e +-He cross-section results, 76-77 Intermolecular collisions, energy-transfer processes and, 166, see also Inelastic heavy-particle collisions Ionization multiple, see Multiple ionization saturation, 125-126 two-photon, 108, see also Nonresonant multiphoton ionization 10s approximation for rotational degrees of freedom, 182 WKB phase shifts in, 1% Isolated bound systems, angle-action variables and, 167-168 Isolated quanta1 system, representation of, I67 ITFITS method, for forced harmonic oscillator, 181

1. K Jackson-Mott model, transition probabilities in, 189, 192 KallenSabry correction, 425

L Laboratory-fixed coordinate frame, 352 Lamb shift(s) in atomic hydrogen, 399 calculated vs. experimental, 410-413 electrostatic quenching anisotropy in, 417 for FeZ4+,442 finite nuclear size effects in, 409 helium transitions and, 434-436 high-Z measurements and, 419-424 in hydrogenic ions, 412-424 in tight muonic systems, 424-426 low-Z measurements and, 413-419 Mohr's values for, 412 of muonic systems, 403 precision measurement of, 415 quantum electrodynamic effects and, 400 quenching anisotropy method in, 4164 18,422-424 in Rydberg frequency units, 402 uncertainties in calculated values of, 412 unscreened hydrogenic, 452 Lamb-shift calculations, 405-412 to I S ~ ~ ~ ~ P ~ transition ~ ~ I S fre~ ~ S ~ S , , ~ quency, 453 Lamb-shift measurements, sequence of, 399-400 Lamb-shift theory, in one-electron systems, 401-405 Laser beam section, isodensitometric mapping of, 122 Laser-driven fusion, electron-atom scattering in, I Laser energy, measurement of, 122 Laser field, eleventh-order moment of, 124 Laser intensity, multiphoton ionization probability and, 99 Laser intensity distribution measurement, improvements in, 132-133

469

INDEX Laser intensity range, in MPI process, 131-132 Laser light polarization, in multiphoton ionization process, 138-140, see also Light polarization Laser radiation, coherence of, 123-125 Laser resonance moments, in high-Z ions, 42I -422 Laser temporal coherence, in multiphoton ionization, 123-125 Light muonic atoms, energy differences in, 428 Light muonic systems, Lamb shifts in, 424-426 Light polarization atomic collisions and, 281 -282 laser, 138-140 in NRMPI, 114-115 in two-photon ionization of cesium, 143148 Linearly polarized light, 143-146 absolute cross sections in, 133-138 Lippman-Schwinger integral equations, in scattering theory, 24-25, 34 Lithium “difference orbitals from different spins” scheme for, 322 energy levels of relative to ionization limit, 313 fine structure splittings in, 429-434 Lithium Is22sZS-IsZnpZ, oscillator strengths for, 314 Lithium (2s+ 2p) total cross section, as function of impact energy, 277 Lithium fluoride molecule comparative study for, 378-388 differential cross section for, 385-388 as electron scattering model, 377 integrated cross sections for, 380-381 momentum-transfer cross sections for, 382-383 total differential cross section for, 386387 vibrational excitation in, 388-389 Lithium-like ions, relativistic energy expansion coefficients and Snyder screening parameters for, 45 I Lorentz invariance, in electron-atom scattering in radiation field, 37

Low-energy e+
M Magnesium, oscillator strengths in, 326 Magnetohydrodynamics, as power source, 342 Many-body techniques, SCF orbitals and, 256 Many-body theories, in resonance calculations, 243-244 Many-electron models, in alkali-like raregas systems, 276-278 Massey parameter or criterion, 271,290 characteristic time and, 295,297 in excitation process, 290, 293 scaling by, 303 vs. udu,,294 MEAN approximation, see Multipoleextracted adiabatic nuclei approximation MHD, see Magnetohydrodynamics Mode-locked lasers, in NRMIA, 116 Model potentials advantages of, 337-338 in atomic structure, 308-338 core polarization and, 327-332

470

INDEX

from Dirac or Dirac-Fock potentials, 336 frozen-core assumption by, 328 Hamiltonian of, 326 in Hartree-Fock formalism, 317-327 vs. pseudopotentials, 310 relativistic, 332-336 Schrodinger and Dirac Hamiltonian potentials as, 310 simple semiempirical, 31 1-3 16 Model potential theory in three-photon ionization cross-section calculation, 137-138. 150 in two-photon ionization of cesium, 145146 Moderators conversion efficiencies of, 59 low-energy positrons from, 57-58 OFHC copper tube, 61 single-crystal, 59-60 “venetian blind,” 59 Modified Doolen-type basis, in resonance calculations, 238-240 Modified perturbation theory, in generalized low-frequency approximations, 40-43 Modified plane waves and scattering in external field, 2-4 and scattering in laser field, 16-17 Molecular gases, low-energy QT measurements for, 82-84 Molecular mechanism, primary excitation for, 303 Molecular orbital model of inelastic processes in “molecular” region, 266 primary excitation in, 303 Momentum-transfer cross section, 342 dipole moment and, 342 for LiF molecule, 382-383 MPI, see Multiphoton ionization MSM, see Multiple-scattering method Multichannel scattering, 33-37, see ulso Scattering effective coupling strength in, 36 Multiphoton ionization, see ulso Multiphoton ionization cross section(s); Multiphoton ionization process; Nonresonant multiphoton ionization defined, 98

double, 156-157 history of, 98 laser temporal coherence in, 123-125 light polarization and, 114-1 15 in nondilation analytic potentials, 245246 nonresonant, see Nonresonant multiphoton ionization of atoms qualitative studies of, 120 Multiphoton ionization cross section(s) absolute measurements of, 119-133 for circularly to linearly polarized light, 139 defined, 122 destructive interference in measurement of, 142 experimental methods in determination of, 130 experimental values of, with linearly polarized light, 134 Multiphoton ionization experiment ion yield determination in, 121-123 Nth-order interaction volume in, 123 principle of, 121-125 resonances in, 125 setup for, 119-120 Multiphoton ionization generalized cross sections, for alkalis, 113 Multiphoton ionization probability calculation of, 106-109 laser intensity and, 99 Multiphoton ionization process, see also Multiphoton ionization background effects in, 131-132 different calculational methods for, 138140 intensity range in, 99 interference minima in, 141 laser intensity range for, 131-132 laser light polarization in, 138-140 limiting cases in, 159 schematic diagram of, 99 Multiphoton ionization studies, extensions to small molecules or collective effects, 159-160 Multiphoton radiation, laser radiation properties in, 123 Multiple complex-scale factors, *P Becalculations in, 254-255 abovethreshold, 151-156, 159

47 1

INDEX Multiple ionization electrons, separation from parasitic electrons, 151-152 Multiple-scattering method, for polar molecules, 369-370 Multipole-extracted adiabatic nuclei approximation, 376, 391 (F'H~)*system, 2s + 2p splittings in, 427 Muonic systems, Lamb shift in, 403-404

N Na(3s -+ 3p) excitation, in Na-Ne collisions, 292 Nd-Yag laser, in six-photon ionization of Xe, 153-154 Nondilation analytic potentials cubic anharmonic oscillator model and, 246 multiphoton ionization in, 245-246 in resonance calculations, 244-246 Stark effect and, 244 Zeeman effect in, 244-245 Nonrelativistic potential scattering, BlochNordsieck analysis of, 6, see also Scattering Nonresonant multiphoton ionization of atoms, 97-160, see also Multiphoton ionization experiment above-threshold ionization in, 151-155 and absolute cross sections in linearly polarized light, 133-138 angular distributions in, 155-156 Bebb and Gold method in, 104-105 and breakdown of lowest order perturbation theory, 117-1 19 coherence (photon statistics) effects in, 115-1 I6 destructive interference effects of, 140151

experimentally measured cross sections in, 158 experimental results vs. theory in, 133140

general formalism in, 101-104 Green's function method in, 108-109 of hydrogen, 109-1 11 implicit summation technique in, 106109

light polarization and, 114-1 I5 lowest order perturbation theory and, 101-103

Nth-moment measurement in, 116 numerical results in, 109-1 14 QDM or model potential in, I 1 I saturation methods in, 128-131 theory of, 101-1 I9 truncated-summation method in, 105106

Nonsaturated ionization regime, direct method in, 126-128 Non-TOF experiments, in total scattering cross-section measurements, 63 N-particle problem, S matrix and, 208 N-particle systems, Siegert calculations in, 23 1 NRMPI, see Nonresonant multiphoton ionization of atoms ns --t np resonance transition, of alkali atoms, 268 Nth-moment measurement of field, in NRMIA, 116 Nth-order interaction volume, 121-123, 126 saturation method in, 129 0 One-electron Dirac theory, 2s,1~ and 2p,, states in, 399 One-electron model potentials, 274-275 One-electron systems, 267, 401-424 as charge-transfer systems, 272-273 Lamb-shift theory for, 401-405 Opposite time symmetry, in excitation process, 295 Optical model formalism, elastic cross sections from, 78 Optical spectra, for atomic collisions, 288289 Orbital angular momentum quantum number, 267 Orbital energies, as functions of internuclear distance for Na-Ne, 279 Orthogonalized-static-modelexchange, 366, 378-379 Orrho-positroniumformation cross section, energy dependence of, 87

412

INDEX

Oscillating field, Poisson distribution for, 9, 12

OSME,see Orthogonalized-static-model exchange

P Parasitic electrons, MPI electron separation for, 151-152 Pauli principle, in atomic collisions, 275 Penning ionization, Sigert method in, 23 1 Periodic potentials, quasi-energy method and, 18 Perturbation theory breakdown of, 99-100, 117-119, I59 dipole interaction and, 345 potential scattering and, 29 in rotational excitation, 177 Perturbed-stationary-state theory, ETFs and, 278 Photoionization, schematic diagram of, 208 Photomultiplier tube, in total cross-section experiments, 61 Photon distribution, Nth moment of, 116 Photons, spontaneous emission of, 3 Photon statistics, in NRMIA, 115-1 16 Plane wave field, electron in, 14-17 Plasma heating, inverse bremsstrahlung process in, 39 PMT, see Photomultiplier tube Point dipole potential, in polar molecule calculations, 368 Polarization potential, in low-energy elastic scattering, 364-365 Polarized photon-scattered-particle coincidence experiment, 283 Polar molecule calculations, applications of, 377-389 Polar molecules approaches and approximations in calculations for, 360-377 Born approximations for, 371-372 classical and semiclassical methods for, 370 classical perturbation theory for, 37037 1 close-coupling approximation for, 368369 collisional approximations for, 368-372

computational treatment of, 345 distributed-wave method for, 370 electron scattering by, 341-392 exact static-exchange calculations for, 363 general formulas for, 350-360 Glauber approximation for, 371, 379 interaction potentials and, 364-372 interest in, 343 multiple-scattering method for, 369-370 scattering equations and, 351-354 Polar representation, in variational principle, 228-229 Positron-atom scattering experiments, 5357 Positron-atom total cross section measurements, first, 53 Positron-beam production, in total crosssection measurements, 57-60, see also e + beam; e- beam Positron-electron comparisons, for inert gases, 79-80 Positron-gas scattering experiments, 5393, see also Total scattering crosssection experiments differential scattering cross sections in, 84-86 future directions for, 92-93 inelastic scattering investigations in, 8691 and inert gases at intermediate energies, 76-78 molecular gases and, 82-84 research groups and references in, 55 resonance searches in, 91-92 summary of, 54 sum rule in, 80-82 Positronium formation cross sections, 8689 Born values for, 89 Positrons, low-energy, 57 Potassium core polarization effect in 4s-np oscillator strengths in, 330 three-photon ionization of, 112, 138, 149- I51 Potassium-rare-gas collisions, 287 Potential scattering, 28-3 I , see also Scattering Coulomb tail in, 29-30

473

INDEX Primary excitation, from molecular mechanisms, 303 Primitive semiclassical S-matrix theory, in inelastic heavy-particle collisions, 187-190 Pseudopotential model potential schemes, core treatment in, 327-328 Pseudopotentials, from Dirac or DiracFock equations, 336 pss theory, see Perturbed stationary-state theory

Q

QDM, see Quantum defect method QT experiments, see Total scattering crosssection measurements Quantum analysis, heuristic treatment of, 7 Quantum defect method in nonresonant multiphoton ionization calculations, 104, 108, 111-113 theoretical predictions based on, 137 in three-photon ionization of potassium, 150-151

in two-photon ionization of cesium, 145146

Quantum electrodynamic effects corrections to, 444 in few-electron atomic systems, 399-456 in light muonic systems, 424-426 in one-electron systems, 401-424 in two-electron systems, 426-446 Quantum electrodynamics, beginnings of, 399 Quantum mechanical effects, RamsauerTownsend minima in, 65 Quasi-energy method, periodic potentials and, 18 Quasi-one-electron systems defined, 267 in direct excitation mechanism, 303 hydrogen-rare gas systems as, 272-273 studies of, 265-304 Quenching anisotropy measurements, in Lamb-shift measurements in high-2 ions, 422-424

R Radial frame-transformation method, 356357 Radiation field coherent state of, 9 target states and, 35 Rainbows, rotational, see Rotational rainbows Ramsauer-Townsend minima, quantum mechanical effects and, 65, 69, 71 Rare gases, multiphoton ionization of, I14 Real scaling factor, in variational calculations, 229 Reflexion symmetry, in atomic collision experiments, 283 Relativistic Hartree-Fock calculations, in few-electron systems, 448-450 Relativistic model potentials, 332-336 Resonance, external, see External resonance; see also Resonance process Resonance calculations complex-coordinate theorems and properties of wave functions in, 214-226 complex stabilization method in, 247256 GamowSiegert states and, 210-214 many-body theories in, 243-244 nondilational analytic potentials in, 244245 self-consistent field techniques in, 240242 variational principle in, 227-229 Resonance process analytical models with, 224-226 complex scaling in, 207-259 cross-section sum rule and, 32 in multiphoton ionization experiment, 125 “two-state,” 33 in positron-gas scattering experiments, 9 1-92 of e-Ne system, 33 RFT method, see Radial frametransformation method Riemann zeta function, 406 Rigid-shell scattering, in rotational excitation, 174-175 Rotated Hamiltonian, complex-coordinate theorems and, 214

414

INDEX

Rotational excitation approximate methods in, 176-181 atom-vibrating rotor collisions and, 181182 body-fixed approximation in, 179-180 body-fixed sudden approximation in, 180-181 classification in. 176-177 in electron-polar molecule collisions, 388 in inelastic heavy-particle collisions, 170- 183 numerical calculations in, 175-176 perturbation theory in, 177 rigid-shell scattering in, 174-175 rotational rainbows and, 172-174 rotor-rotor collisions in, 182-183 sudden approximation in, 178-180 Rotational rainbows, 172-175 averaged, 173 unaveraged, 172-173 Rotational transitions, sudden approximation in, 195-196 Rubidium, three-photon ionization cross section for, 137-138 Rydberg frequency units, Lamb shift and, 402-403

S Saturated ionization regime, 128-131 Saturation methods, in nonresonant multiphoton ionization, 128-131 Scattered particles, energy-loss analysis of, 266 Scattering, see also Electron-atom scattering e+ vs. e-, 56 in external field, 2 by highly polar molecules, 341-392 inelastic, see Inelastic scattering in low-frequency domain, 49 in low-frequency external field, 38 multichannel, 33-37 positron-atom, see Positron-atom scattering experiments potential, 28-3 1

in presence of long-range Coulomb interaction, 3 small-angle elastic, 63-64 time-dependent, 2 Scattering equations polar molecules and, 351-354 solution to, 361-372 Scattering in external field vs. free-field scattering, 2 modified plane-wave states in, 4 wave functions as modified plane waves in, 16 Scattering region, gas number density in, 62 Scattering resonances, in scattering theory, 31-33 Scattering theory, 23-37 approximation techniques in, 28-37 formulation of, 23-28 multichannel scattering in, 33-37 potential scattering in, 28-31 scattering resonance in, 31-33 SCCP, see Strong-coupling correspondence principle SCF, see Self-consistent field Schrodinger equation analytical continuations for GamowSiegert state, 217 commutation relation and, 1 I defined, 3 1 I for large atoms, 310 nonrelativistic, 14, 38, 437 reduction from large atom to twoelectron problem, 312 replacement with one-electron wave equation, 312 in scattering theory formulation, 23 single-particle, 219 solution of, 14 time-dependent and time-independent, 211, 275 S + D excitation, coherence study of, 30 1-303 Self-consistent field calculations, 227, 240242 complex stabilization method and, 255256 for 2P Be- resonance, 256 Self-consistent field orbitals, 256

475

INDEX Self-consistent field techniques, 229 in resonance calculations, 227, 240-242 Semiclassical approximations sudden approximations and, 195-197 time-dependent methods in, 194-195 Semiclassical methods in inelastic heavy-particle collisions, 165-199 for polar molecule calculations, 370-371 Semiclassical sudden matrix, 379 LiF differential cross section and, 385 Semiclassical target-field wave, 34 Semiclassical theories, of inelastic heavyparticle collisions, 186-198 Semiempirical model potentials, 31 1-316 Shot-to-shot laser energy, 122 Shot-to-shot variations, in four-photon ionization of cesium, 132 Siegert approach, in configurational interaction expansions, 229-231, see also Gamow-Siegert states Siegert boundary condition, 258 uM, see Momentum-transfer cross section uN, see Multiphoton ionization cross section Silicon(l), oscillator strengths of 3s23p2 3P + 3s23p4s3postate of, 326 Silver resonance lines, oscillator strengths for, 334 Single-crystal moderators, in positron total cross-section experiments, 59-60 Single-photon absorption, in weak-field limit, 31 Single-photon bremsstrahlung, 41, see also Bremsstrahlung low-frequency approximation for, 44-45 Single-photon emission, in weak-coupling limit of low-frequency approximation, 44 Six-photon ionization, of xenon, 153 Slater functions, in atomic calculations, 324 Slater orbitals, sums of, 325 Slow many-electron atomic collisions, in molecular orbital basis, 278-279 Small-angle elastic scattering, 63-64 S-matrix Gamow-Siegert functions and, 257 integral representations of, 190-192 poles of, 208-209, 257

S-matrix theory classical perturbation theory and, 193 primitive semiclassical, 187-190 strong-coupling correspondence principle and, 193-194 Snyder screening parameters, for lithiumlike ions, 451 Sodium five-photon ionization of, I12 3s2Sn2Ptransitions of, 319 Sommerfeld formula, 450 22S-22P Be(I1) emission cross sections, 291 S + P excitation, coherence studies of, 299-300 Spherical focusing, vs. cylindrical, 127 Spin-orbit interaction, in two-photon ionization of cesium, 148 3S-3’P Na(1) emission cross sections, 291 Spontaneous infrared radiation problem, Bloch-Nordsieck analysis of, 37 SSSM,see Semiclassical sudden S matrix Stark effect, nondilation analytic potentials and, 244-245, 257 Stark shift, of resonance position, 33 Static potential, in polar molecule calculations, 366-367 Stokes parameters, in atomic collisions, 215 Strong coupling, in electron-atom scattering, 46-49 Strong coupling correspondence principle, 193-194 Sudden approximation body-fixed 180-181 Kramer-Bernstein form of, 197 in rotational excitation, 178-179 in rotational transitions, 195-196 Sum rules cross-section, 30-32 in intermediate coupling, 45-46 in strong coupling, 46-47 tests of, 80-82 SUPERSTRUCTURE code, 319

T Texas time-of-flight spectrometer, for differential scattering cross-section measurements, 85

476

INDEX

tl Trajectories

Doolen technique in calculation of, 232 points of stability on, 257 shape of, 238 Three-photon ionization of cesium, 112 of helium, 114, 129 of potassium, 112, 149-151 Three-photon ionization cross sections, calculated vs. measured values of, 136 Time-dependent methods, in semiclassical approximations, 194-195 Time-dependent scattering theory, 2-3 Time-of-flight spectrometer measurements, of differential scattering cross sections, 85 Time-of-flight system, 60-63 Time reversal, in Gamow-Siegert states, 220-22 1 T-matrix element, see Transition matrix element TOF system, see Time-of-flight system Total elastic cross section measurements, for e’-He and e--He collisions, 70 Total momentum-transfer cross sections, as function of dipole moment, 342 Total scattering cross-section experiments, 60-64 Detroit setup for measurement in, 62 general setup for, 280-281 positron-beam characteristics for, 58 Total scattering cross-section measurements, 57-64 Bielefeld setup for, 60-61 percentage of errors in, 74-75 positron-beam production in, 57-60 Total scattering cross section results, 6484 inert gases at low energies in, 64-76 Transformed electron-field states, 26 Transmission matrix element, in scatteringtheory formulation, 24 Truncated-summation method, in nonresonant multiphoton ionization process, 105-106

Two-electron systems high-precision calculations for, 426-446 high-Z extrapolations in, 436-446 singlet state of, 237

Two-photon ionization of atomic hydrogen, I12 calculated vs. measured values in, 135 h e a r polarization methods in, 143-146 quantum defect method in, 145-146 Two-photon ionization cross section of cesium, 125, 145-146 interference minima in, 141 Two-state resonance process, 33 U

Uniform approximations, in inelastic heavy-particle collisions, 183- 186 Uranium, 5f36d7s2state of. 335

V Vacuum-polarization correction, 403 Variational calculations configuration interaction expansions in, 229-240 resonance calculations and, 229-242 self-consistent field calculations and, 240-242 Variational functional, in resonance calculations, 227-229 Variational principle, in resonance calculations, 227-229 Variational wave function, expansion of, 230 “Venetian-blind” backscattering moderator, 59 Vibrational excitation, in HF and HCI molecules, 388 Vibrationally elastic and inelastic scattering, differential cross sections for, 377-378 Vibrational transitions, quanta1 calculations for, 182

W Wave functions complex-coordinate theorems and, 2 14226 evaluation of, 317 in Gamow-Siegert states, 217-220 as modified plane waves in scattering theory, 16-17

477

INDEX Weak-field limit, single-photon absorption in, 31 Weierstrass transformation, 248 Wigner distribution theory, 198

eleven-photon ionization of, 124 six-photon ionization of, 153 Z

X

Xenon atom above-threshold ionization of, 153-154

Zeeman effect, in nondilational analytic potentials, 244-245 2-' expansion calculations, 450-454

Contents of Previous Volumes

Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A . T . Amos Electron Affinities of Atoms and Molecules, B . L . Moiseiwitsch Atomic Reamangement Collisions, B . H . Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies. H . Pauly and J . P . Toennies High-Intensity and High-Energy Molecular Beams, J. B . Anderson, R . P. Andres, and J . B . Fenn

AUTHORINDEX-SUBJECT INDEX

Volume 2 The Calculation of van der Waals Interactions, A . Dalgarno and W . D. Davison Thermal Diffusion in Gases, E . A . Mason, R . J . Mum, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A . R. Samson The Theory of Electron-Atom Collisions, R . Peterkop and V . Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . deHeer 478

Mass Spectrometry of Free Radicals, S . N . Foner

AUTHORINDEX-SUBJECT INDEX Volume 3 The Quanta1 Calculation of Photoionization Cross Sections, A . L . Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H . G . Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B . Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H . C . Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas CrystalSurface van der Waals Scattering, F. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood

AUTHORINDEX-SUBJECT INDEX Volume 4 H. S. W. Massey-A Sixtieth Birthday Tribute, E . H . s. Burhop Electronic Eigenenergies o f the Hydrogen Molecular Ion, D. R . Bates and R . H . G . Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E . Gal Positrons and Positronium in Gases, P. A . Fraser

CONTENTS OF PREVIOUS VOLUMES

Classical Theory of Atomic Scattering, A. Burgess and 1. C . Percival Born Expansions, A . R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C . B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R . G . W . Keesing Some New Experimental Methods in Collision Physics, R . F . Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J. Seaton Collisions in the Ionosphere, A . Dalgarno The Direct Study of Ionization in Space, R. L. F . Boyd

AUTHORINDEX-SUBJECT

INDEX

Volume 5 Flowing Afterglow Measurements of Ion-Neutral Reactions, E . E. Ferguson, F . C . Fehsenfeld, and A . L. Schmeltekopf Experiments with Merging Beams, Roy H . Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A . Ben-Reuven The Calculation of Atomic Transition Probabilities, R . J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s's'mpq, C . 5 . H . Chisholm, A . Dalgarno, and F . R. Innes

479

Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

AUTHORINDEX-SUBJECT

INDEX

Volume 6 Dissociative Recombination, J . N. Bardsley and M. A . Biondi Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A . S . Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazyu Itikawa The Diffusion of Atoms and Molecules, E. A . Mason and R . T . Marrero Theory and Application of Sturmain Functions, Manuel Rotenberg Use o f Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A . E. Kingston

AUTHORI ~ D E X - S U B J E C T INDEX Volume 7 Physics of the Hydrogen Master, C . Audion, J . P. Schermann, and P . Grivet Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C . Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Paunez, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules-QuasiStationary Electronic States, Thomas F. O'Malley Selection Rules within Atomic Shells. B. R . Judd

480

CONTENTS OF PREVIOUS VOLUMES

Green’s Function Technique in Atom- Volume 10 ic and Molecular Physics, G y . Csanuk, H. S . Taylor. and Robert Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr Yaris and Serge Feneuille A Review of Pseudo-Potentials with Emphasis on Their Application to The First Born Approximation, K. L. Bell and A . E. Kingston Liquid Metals, A . J . Greenfield AUTHORINDEX-SUBJECT INDEX Photoelectron Spectroscopy, W . C . Price Dye Lasers in Atomic Spectroscopy, Volume 8 W. L a n g e , J . L u t h e r , and A . Steudel Interstellar Molecules: Their FormaRecent Progress in the Classification tion and Destruction, D. McNal1.v of the Spectra of Highly Ionized Monte Carlo Trajectory Calculations Atoms, B. C . Fawcett of Atomic and Molecular Excitation in Thermal Systems, James C . A Review of Jovian Ionospheric Chemistry, Wesley T . Huntress, Jr. Keck INDEX Nonrelativistic Off-Shell Two-Body SUBJECT Coulomb Amplitudes, Joseph C. Y. Chen and Augustine C . Chen Photoionization with Molecular Beams, Volume 11 R. B. Cairns, Halstead Harrison, The Theory of Collisions between and R. 1. Schoen Charged Particles and Highly ExThe Auger Effect, E. H . S. Burhop cited Atoms, I. C. Percival and D . and W . N . Asaad Richards AUTHORINDEX-SUBJECT INDEX Electron Impact Excitation of Positive Ions, M. J. Seaton Volume 9 The R-Matrix Theory of Atomic Process, P. G . Burke and W . D . Correlation in Excited States of Atoms, Robb A . W . Weiss Role of Energy in Reactive Molecular The Calculation of Electron- Atom Scattering: An Information- TheoExcitation Cross Sections, M. R. retic Approach, R. B . Bernstein H. Rudge and R . D. Levine Collision-Induced Transitions be- Inner Shell Ionization by Incident Nutween Rotational Levels, Takeshi clei, Johannes M . Hansteen Oka Stark Broadening, Huns R. Greim The Differential Cross Section of Low-Energy Electron- Atom Col- Chemiluminescence in Gases, M . F. Folde and B. A . Thrush lisions, D. Andrick AUTHOR INDEX-SUBJECT INDEX Molecular Beam Electronic Resonance Spectroscopy, Jens C . Zorn and Thomas C . English Atomic and Molecular Processes in Volume 12 the Martian Atmosphere, Michael Nonadiabatic Transitions between B . McElroy Ionic and Covalent States, R . K. Janev AUTHORINDEX-SUBJECT INDEX

CONTENTS OF PREVIOUS VOLUMES

48 1

Recent Progress in the Theory of Atomic Isotope Shift, J . Bauche and R. - J . Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M . Brover, G . Gouedard, J . C. Lehman, and J . Vigue Highly Ionized Ions, Ivan A . Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C . Reid

(e, 2e) Collisions, Erich Weigold and Ian E . McCarthy Forbidden Transition in One- and Two-Electron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in HeavyParticle Collisions, M . S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobashev

Volume 13

UV and X-Ray Spectroscopy in Astrophysics, A . K . Dupree

Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M . Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I . V . Hertel and W . Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K . Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W. B . Sommerville AUTHORINDEX-SUBJECT INDEX

Volume 14 Resonances in Electron, Atom, and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C . Webster, Michael J . Jamieson, and Ronald F. Stewart

AUTHORINDEX-SUBJECT INDEX

Volume 15 Negative Ions, H . S. W. Massey Atomic Physics from Atmospheric and Astrophysical Studies, A . Dalgarno Collisions of Highly Excited Atoms, R. F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J . W. Humberston Experimental Aspects of Positron Collisions in Gases, T. C. Grifith Reactive Scattering: Recent Advances i n T he o r y a n d E x p e r i m e n t , Richard B . Bernstein Ion- Atom Charge Transfer Collisions at Low Energies, J . B. Hasted Aspects of Recombination, D . R . Bates ‘The Theory of Fast Heavy-Particle Collisions, Bransden Atomic Collision Processes in Controlled Fusion Research, H.B . GilbodY Inner-Shell Ionization, E. H . S. Burhop

482

CONTENTS OF PREVIOUS VOLUMES

Excitation of Atoms by Electron Impact, D. W. 0. Heddle Coherence and Correlation in Atomic Collisions, H . Kleinpoppen Theory of Low-Energy ElectronMolecule Collisions, P. G . Burke

AUTHORINDEX-SUBJECTINDEX

Volume 16 Atomic Hartree- Fock Theory, M . Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren Sources of Polarized Electrons, R. J. Celotta and D. T . Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M . H . Key and R . J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch

Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N. Fortson and L. Wilets

INDEX Volume 17 Collective Effects in Photoionization of Atoms, M . Ya. Amusia Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Supeffluorescence, M. F . H . Schuurmans, Q. H . F . Vrehen, D. Polder, and H. M . Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G. Payne, C. H . Chen, G . S . H u m , and G . W . Foltz Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D . Lin and Patrick Richard Atomic Processes in the Sun, P . L . Dufton and A . E. Kingston INDEX