Advances in Atomic and Molecular Physics, Volume 17

Advances in

ATOMIC A N D MOLECULAR PHYSICS

VOLUME 17

CONTRIBUTORS TO THIS VOLUME M. YA. AMUSIA C. H. CHEN D. S. F. CROTHERS P. L. DUFTON SERGE FENEUILLE G. W. FOLTZ H. M. GIBBS G. S. HURST PIERRE JACQUINOT A. E . KINGSTON C. D. LIN M. G. PAYNE D. POLDER PATRICK RICHARD M. F. H. SCHUURMANS Q. H. F. VREHEN

ADVANCES IN

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 OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 17

@

1981

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

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COPYRIGHT @ 1981

BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OK BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATlON STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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Cont en ts

ix

LIST OF CONTRIBUTORS

Collective Effects in Photoionization of Atoms

M . Ya. Amusici I. Introduction 11. RPAE and Many-Body Perturbation Theory 111. Calculation of Characteristics of Photoionization

IV. Collective Effects near Inner-Shell Thresholds V. Collectivization of Vacancies VI. Conclusion References

2 4 13

32 40 51

52

Nonadiabatic Charge Transfer

D. S . F. Crothers I. Introduction 11. Phase Integrals and Comparison Equations 111. Perturbed Stationary States and Electronic Translation IV. Nonmolecular Three-Body Analysis V. Summary References

55 63 83 91 93

93

Atomic Rydberg States Stv-gr Feneirillr and Pierre Jacqirinot I. Introduction 11. Preparation and Detection of Rydberg States 111. Spectroscopy IV. Rydberg Atoms in External Fields V. Radiative Properties of Rydberg States References V

99 101 119 131 157 161

vi

CONTENTS

Superfluorescence M. F. H. Schuurmans, Q. H . F. Vrehen. D. Polder, and H. M. Gihhs I. Introduction 11. Semiclassical Theory

111. Quantum Mechanical Description of SF IV. The Effect of Homogeneous and Inhomogeneous Broadening on SF V. Three-Dimensional and Multimode Effects VI . Experimental Techniques VII. Experimental Results VIII. Conclusions Appendix 1 Appendix I1 References

168 172 176 193 202 206 213 222 223 223 226

Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics M. G . Payne, C. H . Chen, G. S. Hurst, and G . W. Foltz 11. Multiphoton Excitation with Broad Bandwidth Lasers 111. RIS Studies of Inert Gases

229 23 1 239

IV. Experiments Combining RIS and Pulsed Supersonic Nozzle Jet Beams References

262 272

I. Introduction

Inner-Shell Vacancy Production in Ion-Atom Collisions C. D. Lin and Patrick Richard I. Introduction Experimental Measurements of Inelastic Ion- Atom Collisions 111. Theory of Inelastic Ion- Atom Collisions IV. Comparison of Theories and Experiments V. Concluding Remarks References

275

11.

277 303 326 347 348

Atomic Processes in the Sun

P. L. Diiftron and A . E . Kingston I. Introduction 11. Atomic Spectra

355 359

CONTENTS

vii

111. Bound-State Wave Functions IV. Spontaneous Decay of Bound States V. Electron Excitation VI. Proton ‘Excitation VII. Applications of Atomic Data to Solar Plasmas VIII. Conclusions References

36 1 370 381 403 406 414 415

INDEX CONTENTS OF PREVIOUS V O L U M E S

419

429

This Page Intentionally Left Blank

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

M. YA. AMUSIA (l), A. F. Ioffe Physico-Technical Institute of the Academy of Sciences of the USSR, Leningrad, USSR C. H. CHEN (229), Chemical Physics Section, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 D. S. F. CROTHERS ( 5 3 , Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland P. L. DUFTON* (353, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland SERGE FENEUILLE (99), Laboratoire Aime Cotton, Centre National de la Recherche Scientifique, 91405 Orsay Cedex, France G. W. FOLTZ (229), Chemical Physics Section, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 H. M. GIBBS (167), Optical Sciences Center, University of Arizona, Tucson, Arizona 85721 G. S. HURST (229), Chemical Physics Section, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 PIERRE JACQUINOT (99), Laboratoire Aime Cotton, Centre National de la Recherche Scientifique, 91405 Orsay Cedex, France A. E. KINGSTON (359, Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 lNN, Northern Ireland C. D. LIN (275), Department of Physics, Kansas State University, Manhattan, Kansas 66506 M. G. PAYNE (229), Chemical Physics Section, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 * Present address: Department of Pure and Applied Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland. ix

X

LIST OF CONTRIBUTORS

D. POLDER (167), Philips Research Laboratories, 5600 MD Eindhoven, The Netherlands PATRICK RICHARD (275), Department of Physics, Kansas State University, Manhattan, Kansas 66506 M. F. H. SCHUURMANS (167), Philips Research Laboratories, 5600 MD Eindhoven, The Netherlands Q . H. F. VREHEN (167), Philips Research Laboratories, 5600 MD Eindhoven, The Netherlands

Advances in

ATOMIC A N D MOLECULAR PHYSICS

VOLUME 17

This Page Intentionally Left Blank

.

ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS VOL . 17

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS M . Ya. AMUSIA A . F . lofle Physico-Technical Institute of the Academy of Sciences of the USSR Leningrad. USSR

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I1 . RPAE and Many-Body Perturbation Theory . . . . . . . . . . . A . Derivation of RPAE Equations . . . . . . . . . . . . . . . B . The Technique of Diagrams . . . . . . . . . . . . . . . . C. Correspondence Rules of Perturbation Theory and the Choice of Self-Consistent Field . . . . . . . . . . . . . . . . . . . . D. Significant Formula and General Relations . . . . . . . . . . 111. Calculation of Characteristics of Photoionization . . . . . . . . . A . Total Cross-Section Calculations . . . . . . . . . . . . . . . . B . Partial Cross Sections and Few-Electron Shells Collectivization . . C. Single Charged-Ion Formation . . . . . . . . . . . . . . . . . D. Angular Distribution of Photoelectrons . . . . . . . . . . . . . . E . Combination of Collective and Relativistic Effects . . . . . . . . F. Collective Effects in Atoms with Open Shells . . . . . . . . . . . G . Double-Electron Photoionization . . . . . . . . . . . . . . . . H. Collective Oscillations . . . . . . . . . . . . . . . . . . . . . IV. Collective Effects near Inner-Shell Thresholds . . . . . . . . . . . . A . Static Rearrangement . . . . . . . . . . . . . . . . . . . . . B . Inner Vacancy Decay and Postcollision Interaction . . . . . . . . V. Collectivization of Vacancies . . . . . . . . . . . . . . . . . . . . A . The Vacancy Wave Function . . . . . . . . . . . . . . . . . . B . The Vacancy Energy and Width . . . . . . . . . . . . . . . . . C . Collectivization of 4p Shell in Xenon . . . . . . . . . . . . . . . D. “Shadow” Levels . . . . . . . . . . . . . . . . . . . . . . . E . Interaction with the Channel Two Electrons-Two Holes . . . . . . VI . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . You still cannot d o everything: example -the

’ “The Theory of Fundamental Processes.

”

2 4 5 6 9 11 13 13 IS

17 18 20 25 30 31 32 33 36 40 41 43 45 46 48 51 52

many-electron atom R . P. FEYNMAN’

W. A . Benjamin. Inc., New York. 1961 .

1 Copynght C 1981 by Academic Press. lnc All nghts of reproduction in any form reserved . ISBN 0-12-003x17-X

2

M . Yu. Amusia

I. Introduction The interaction of the electromagnetic field with atoms is not only a widespread natural process but also a convenient and effective method for studying their structure. The field, if it is not too intensive, weakly perturbs the atom, and photoionization cross section permits one to obtain information on atomic wave function in the ground and excited states. Naturally, the atomic electron structure and the motion and interaction of electrons determine the wave function and therefore the photoionization cross section. For a long time it was generally accepted that each electron moves in the atom almost independently in some average, the so-called selfconsistent field created by nuclei and all other electrons. That part of the interelectron Coulomb interaction which does not contribute to this field (we will call it “direct”) is considered to be small. The Hartree-Fock (HF) approximation based on this idea presents the atomic wave function as an antisymmetrized product of one-electron wave functions. As to direct interaction, it leads to deviations from this model. The analysis of photoabsorption data in the energy region from tens to hundreds electron volts, which were first obtained about 16 years ago (Lukirskiiet al., 1964; Ederer, 1964), demonstrated that the direct interaction influence is not small, but qualitatively alters the electron shell behavior under the action of the electromagnetic field. It proved to be that the direct interaction is essential for photoelectron energy E less than or of the order of the direct interaction energy of a pair of atomic electrons. Estimating the interaction energy2as r;’, where r, is the mean interelectron distance, given by r, Z -2’3 in the statistical model, we come to the inequality

-

< 22‘3 (1) where Z is the number of electrons. Therefore, corrections to the oneelectron picture may be essential for small photon energies, which means for outer-shell photoionization and for the threshold region for intermediate and inner shells. The direct interaction leads to correlated instead of independent atomic electron motion. It was suggested long ago (Bloch, 1933) that the correlations are so strong that they lead to collective oscillations of electrons in atoms analogous to hydrodynamic or sound vibrations. However, the H F success in describing the ground and excited states of atoms make this idea improbable. Only during the last 10-15 years have qualitative manifestations of E

’ In this contribution the atomic system of units e = h

=

rn

=

I is used, energy in Ry.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

3

strong atomic electron correlations been obtained, mainly by studying photoionization, which demonstrates essential ordering in atomic electron motion. The consideration of such so-called “collective” effects is the purpose of this contribution. As concerns “collective,” it is natural to treat everything which cannot be described in the best one-electron approximation. But which of them is the best? Initially the one-electron hydrogen-like approximation serves. It leads to a “teeth-like” cross-section curve, at which every shell and subshell threshold is marked by a jump. Above each threshold the cross section rapidly decreases. The first experiments in the energy region from tens to hundreds of electron volts (Lukirskii et al., 1964; Ederer, 1964) demonstrates the defects of this picture: the cross section at threshold very often starts to increase, then goes via a minimum vdde (called the “Cooper minimum”) and reaches a second maximum far from any thresholds. It proved to be that some of these peculiarities, at least qualitatively, may be described in a one-electron approximation, if one is careful in choosing the common electron potential. Thus, a simplified version of HF was used (Manson and Cooper, 1968), exchange being included by a localized Slater potential. But for some cases large deviations from experiment (see Fano and Cooper, 1968) still remain, and these might be attributed either to the crudeness of the common potential or to the role of correlations. The use of an accurate HF approximation improves the result, but leads to such difficulties as dipole sum rule violation and nonequivalence of the length ( “ r ” ) and velocity (“V”) forms of electron-electromagnetic field interaction operators. Also it was nontrivial to choose the one-electron field in which the electron leaves the atom, either in the ion field with a vacancy in a strictly determined state or with averaging over all possible vacancy states. This difference is very essential because by a proper choice of the field an essential part of many-electron correlations may be included in a oneelectron approximation. Therefore, the separation of one- and manyelectron effects becomes complex in some cases. This is a consequence of the limited number of atomic electrons and of H F many-body nature, the self-consistent field of which is formed by all atomic electrons. In this article we consciously understate the correlation role, considering as collective or correlational only those effects which cannot be described in one-electron approximation with any choice of self-consistent field. Double-electron photoionization, i.e., the removing of two electrons by one photon, which is impossible in a single-electron approximation, is such an eBect. We shall briefly discuss this process also. Most attention will be paid, however, to the one-electron photoionization because the aim of this article is to demonstrate that the corrections to one-electron picture

4

M. Ya. Amusia

due to direct interaction are very large. As to double-electron photoionization data, they can be discussed mainly in the lowest order of perturbation in direct interaction. It is attractive to study correlations using a hydrodynamic type approximation, which proved to be very successful in describing the electron gas in simple metals. Of course, the nonhomogeneity of electron density, as well as quantum corrections to the hydrodynamic description, are to be taken into account. The corresponding equations (Bohm and Pines, 1953) form the basis for the so-called random phase approximation with exchange, RPAE (Altick and Glassgold, 1964; Amusia et al., 1967, 1970; Wendin, 1971, 1974), which permits one first to describe the photoionization of the noble gas outer and intermediate shells satsifactorily. The essential deviation from the one-electron picture permits one to come to a conclusion about the collective nature of the photoionization process of the outer and intermediate shell, at least for noble gas atoms (Amusia et al., 197 1). Then photoionization was studied using many-body perturbation theory (MBFT) (Kelly, 1971) and R-matrix theory (Burke, 1978). In a number of cases good results were obtained with the configuration mixing method (see Armstrong and Fielder, 1980). However, not all collective effects in photoionization can be described satisfactorily by RPAE and its analogs. Deviations from RPAE were observed near the thresholds of some intermediate and inner shells, as well as in the autoionizing states of the noble gas outer shells. The description of these effects is possible only if one includes the atomic relaxation due to vacancy formation and the structure of the vacancy itself. The investigation of vacancy position and structure, which mainly utilizes electron spectroscopy methods, permits one to observe an essential deviation of vacancy energy from its HF value and spectroscopic factors from unity, which demonstrates the essential collectivization of atomic levels (Gelius, 1974; McCarthy and Weigold, 1976). Some of many-electron effects beyond RPAE frame will be also discussed in this contribution.

11. RPAE and Many-Body Perturbation Theory RPAE is a generalization of the HF approximation to systems in a weak alternating external field (Thouless, 1961). Therefore, it should be as satisfactory to describe excited states and ionization in the same way that the HF approximation does in ground state properties.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

A.

5

DERIVATION OF RPAE EQUATIONS

The H F atomic ground state wave function 4 is an antisymmetrized product of single-electron functions cpi(r), where i denotes all quantum numbers that characterize the electron state. The HF ground state energy is

Here (ilTli) = /rpi(r)(-A/2

-

Z/r)cpi(r)d 3 r

( i j ( U ( i j )= (ij(V1ij) - ( i j l V / j i )

(3) (4)

and V = I r - r‘( is the interelectron Coulomb potential. The summation in Eq. (2) is performed over occupied states (isF). The variation of Eq. (2) under the condition SEo/GpT= 0 leads to HF equations for cpi(r):

The occupied states functions are determined by Eq. ( 5 ) w i t h i s F. Then Eq. ( 5 ) is solved for vacant states m ( m > F), which, together with pi ( i s F ) , forms a complete orthonormal set, which will be used as a basis in perturbation theory. Including the external electromagnetic field A ( r , t ) = A(r)e-’”‘ + A*(r)e’”‘

(6)

we consider the atomic wave function to be at each moment a Slater determinant, composed from perturbed states t,bi ( i s F ) (Amusia and Cherepkov, 1975):

Here Cmi(t)are unknown, and summation m > F implies integration over the continuous spectrum. Coefficients C,, characterize the probability of transition to excited states under external field action, and are considered to be small. It is natural to assume for Cmi(t)the same time dependence as in Eq. (6): Cmi(t)= Xrnie-”‘+ Y*mi ehf

(8)

M . Ya. Amusia

6

Then, calculating the mean value of the atomic Hamiltonian with the Slater determinant function constructed from llri up to terms C k i , and demanding the variational derivatives in X and Y of it to be equal to zero, we come to the system of equations

-

It is convenient to introduce M(w):

(mlM(w)li)= -(ern

-

(ilM(w)lm)= -(ern

-

- w)X,t Ei

(10)

+ w)Ymz

which permits one to present the first equation (9) in the form

-

(jlD(w)ln)( mnl Ulij) w

+ En

- Ej -

is

1,

6++0

(11)

The second equation may be obtained from Eq. (11) by permutation of m and i.

B. THETECHNIQUE OF DIAGRAMS It is convenient to present analytical equations in the form of diagrams, which is possible using the notations in Fig. 1. The direction of time development of the process is from left to right. Equation (1 1) is given by Fig. 2. Each intermediate state in Fig. 2 (with n and J’) is represented by the energy denominator ( w - E,, + E j + for so-called “time forward” (Fig. 2b, c) and ( w + E , - E j - is)-’ for “time backward” (Fig. 2d, e). Furthermore, summation and integration are performed over all internal line states n,j. Equation (1 1) determines the matrix elements of the effective field, which acts upon atomic electrons. The first term describes the direct action, while the second gives corrections to it due to self-consistent H F field variation under the action of external field upon all atomic electrons.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS <m\ im>

-

-

G)-

I(>
7

-

- --<;

- ;< .

Therefore, it describes the induced field provoked by electron atomic shell polarization under external field A action. This term therefore describes correlative or collective corrections in RPAE. It is convenient to introduce the effective interelectron interaction r in RPAE:

F I G .2. The diagrammatical representation of RPAE equation.

M . Ya. Amusia

8

FIG.3. The diagrammatical definition of effective interaction r.

which is given diagrammatically by Fig. 3. The matrix T(w) is determined by

It means that T(w) corrects the Coulomb matrix element because of many-electron correlations. Being independent of external field, I-( w) is determined by the electron shell structure. The difference between r and U ( D and A) increases with an increase in atomic electrons number (the sum over j s F becomes larger) and with the strengthening of interelectron interaction as compared with their interaction with the self-consistent field. Thus, the difference between T and U ( D and A ) characterizes the degree of atomic electrons collectivization. Iterating Eq. (1 1) and presenting it diagrammatically (Fig. 4), we see that RPAE includes the transition of an electron-hole pair from one state (n,j ) to another [(k, 1 ) and then (m,i)] (Fig. 4) and also some more complex terms, which include the simultaneous excitation of several n

_m

m

1.

L

.+ ... +

respective exchange terms.

FIG.4. Iterative representation of RPAE equations.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

9

pairs, but only those which, when “stretched” in time, contain at each moment only one electron-hole pair. The RPAE equations may be generalized to include relativistic effects (Johnson and Cheng, 1978). It is achieved by using in Eq. (2) the relativistic form of the kinetic energy operator c . (cupi) + pc‘ instead of -Ai/2, where cu, p are Dirac matrices and p i = - [ V i , c being the velocity of light. Then the relativistic version of HF is obtained, and, repeating calculations which lead from Eqs. (5) to (13), the relativistic generalization of RPAERRPA is obtained. If it is necessary, Breit corrections to the interelectron Coulomb interaction may be included (Johnson and Cheng, 1978).

C. CORRESPONDENCE RULESOF PERTURBATION THEORY AND CHOICE OF SELF-CONSISTENT FIELD

THE

Naturally, from Fig. 1 elements diagrams that are beyond RPAE frame may be constructed. Using the correspondence between Figs. 2-4 and Eqs. (11)-(13), we present without proof (e.g., March ef al., 1967) the rules according to which every diagram is connected with an analytical expression and vice versa. Besides Fig. 1 elements, each intermediate state (i.e., a state between two interactions nearest in time) is described by an energy denominator being a difference between the sums of intermediate holes and intermediate excited electron energies to which the photon energy is added. The general sign of the diagram is given by (- l)p+q,where p is the number of holes and q is the number of closed electron-hole loops (e.g., for Fig. 4a,p = 3 and q = 2). The summation is performed over all electron and hole intermediate states. Very often it is sufficient to use several first-order terms of many-body perturbation theory (Kelly and Carter, 1980) instead of solving the RPAE equations. Because of a lack of good small parameters, which characterize electron correlations in the atom, it is difficult to decide before calculating whether it will be sufficient to include one or two first terms of a perturbation series or whether it is necessary to solve the RPAE equations. It is worth noting that solving RPAE is not more dii3cult than calculating the perturbative approach terms. It is necessary also to bear in mind the fact that pure perturbation theory is applicable in studying many-electron correlations in atoms only if some important infinite sequences of perturbation theory diagrams are taken into account by an appropriate choice of self-consistent field. In fact, not only diagonal [withn = m a n d j = i, see Eq. (9)] but all quasi-diagonal matrix elements of the type (milUlni) for any n may be taken into account by choosing the self-consistent field. It is achieved by

M . Ya. Amusia

10

L

L

i-

FIG.5 . “Time-forward” RPAE diagrams diagonal in the hole state.

summing the “time-forward” sequence in Figs. 2 and 4 in which the hole state i remains fixed (see Fig. 5 ) . Analytically it is done by the summation of a sequence of expressions:

the wave function

I&)

being determined by

which may be transformed into differential form by multiplying Eq. (15) by ( f l H F - Em) [see Eq. ( 5 ) ] :

where H i ; differs from H H F by neglecting the term withj = i in the sum ( 5 ) . Therefore, @m describes the electron HF state in the field of an ion with a vacancy in thei state. The last term in Eq. (16) orthogonalizes ‘Pm to all cpJ ( j s F). Far from the atom, the self-consistent fieldHEi. decreases as -(r-I), which is quite different from the finite range potential for m > F states given by Eq. ( 5 ) . Therefore, in many-body theory language the outgoing electron wave function calculation in the ion, instead of a neutral atom field, corresponds to an infinite sequence of diagonal hole state i diagrams of RPAE. Equation (16) determines only the excited states m > F wave functions, while cpj f o r j s F are obtained from the ground state calculations (“frozen” core approximation).

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

11

D. SIGNIFICANT FORMULA A N D GENERAL RELATIONS In this contribution mainly closed-shell atoms are considered. Therefore, one-electron wave functions may be presented (due to the spherical symmetry of the atom) as a product of radial, angular, and spin parts. The integration over angles and summation over spin in RPAE can be performed easily (Amusia and Cherepkov, 1975), and a number of independent equations for angular momentum components of D - D1and r - rl are obtained. After separating of angular and spin parts, we can sum a sequence of expressions analogous to Eq. (14), in which, however, n, n’,andi denote only the wave function radial part. Then all “time-forward’’ RPAE diagrams with all hole states of the same energy (‘7” determines only a subshell, instead of a definite state) but with different angular momentum and spin projection are taken into account by a choice of radial oneelectron equations analogous to Eqs. (15) and (16). The self-consistent field may be chosen so that the electron angular momentum and spin together with that of the vacancy form total momentum and spin equal to the respective photon values. This is achieved (Amusia and Cherepkov, 1975) by solving the following radial equation for the excited or continuous spectrum electron:

= cn’1’Pncl,(r) (17) Here H f , is the radial part of the HF Hamiltonian with a given angular momentum I ’ ; Pnl(r)is the radial part of the wave function; coefficients a and P, which result from integration over angular parts, are determined by the following expressions:

2(21,

+

1)(21‘

+

1)

li

I‘ I

= (21 + 1) (0 0 0) Such a choice of the outgoing electron self-consistent field is in accord with an intuitive one-electron picture, according to which it is natural to

M. Yu. Amusiu

12

consider the photoelectron as moving in an ion field, the total angular momentum and spin of vacancy and excited electron being equal to that of photon. In the remaining part of this article the solutions of Eq. (17) as oneelectron functions will be used although they include, in fact, a prominent part of many-electron correlations. If the functions of Eq. (17) are used, it is necessary to avoid a double count of the same diagrams in solving Eq. (11). In the photon energy region considered in this contribution, the dipole approximation is valid. This is correct for o = kc << c/reffwhere reffis the effective radius of the ionized shell. There are two forms of the interaction operator between an electron and electromagnetic field in the dipole approximation: length (‘‘r”) and velocity (“V”). The one-electron formulas for photoionization cross section are (see Fano and Cooper, 1968) u[(o)= 47r2a~~omll( m(zli)1’

(19a)

Here wmi = Em - q is the photon energy, a = I/c = (137)-l, a, being the Bohr radius; z and V, are z projections of r and V. In the dipole approximation the Golden Sum rule is fulfilled: SE n,l=F

Lc

n‘PW

fnt.n’l’

+ (27r‘aaf)-’

where Znl is the nl-shell ionization potential and fnl.n,l, is the oscillator strength of n,l -+ n ’ ,l’ transition. For HF functions qrnand (pi, contrary to the case for precise wave functions, u‘ # uv and S # 2 . This is a consequence of H F Hamiltonian (5) nonlocality, due to which equality is violated: (flvlg) # i(E,-

4

(flrlg)

(21)

for matrix elements between HF statesf, g. In RPAE the cross section is determined by Eq. (19), in which matrix elements of z = r cos 6 and V, are replaced by (rnlD(w)li)from Eq. ( l l ) , which are solved with (rnldli) equal to (rnlzli) or (rnlV,(i) instead of (mlAli). Equations (19) imply that continuum wave functions Im) are normalized by the 6 energy function. For discrete Im)states, Eq. (19), after dividing by 47r2a&, determines the oscillator strength. It has been proved (Amusia and Cherepkov, 1975) that uris equal to uv, and the sum rule (20) is fulfilled in RPAE. The RPAE equations are usually solved numerically, and integration over continuous spectrum in Eq. ( 1 1)

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

13

is essential. The numerical accuracy of calculations is checked by a comparison of crr and uv.

111. Calculation of Characteristics of Photoionization The equation for photoionization amplitude ( 1 1 ) permits one to distinguish intrashell and intershell correlations. Thus, if in the sum over j F in Eq. (11) only those states are included that belong to the same subshell, the solution of Eq. (1 1) describes intrasubshell correlations or collective effects. If both transitions from the nl subshell, nl + 4 1 + 1 ) and nl + € ( I - I ) , are taken into account or all terms related to two (or more) shells in the s u m j d F in Eq. (11) are included, then one is referring to “intershell” correlations. Unfortunately, because of purely technical reasons the solution of Eq. ( 1 l) with the inclusion of more than three terms in the sum o v e r j s F is very difficult. But this is also almost unnecessary because of the generally good energy separation of different shells and the relative weakness of their mutual interaction. A. TOTALCROSS-SECTION CALCULATIONS For many-electron shells it is usually sufficient to include only intrashell correlations. The action of other shells and transitions is almost negligible if the photoionization of many-electron shells is considered in the region where it is large. The first successful inclusion of intrashell correlations in RPAE was performed a decade ago for the 3p6 shell of argon by this author and his colleagues. Good agreement with experiment was achieved, and the role of correlations was proved to be large (see Amusia and Cherepkov, 1975, and references therein). The one-electron results, especially those obtained with a simplified Herman-Skillman version of HF (Manson and Cooper, 1968), prominently differ from experiment for the 4d shell in xenon. (Elderer, 1964; Haensel et al., 1971). The RPAE correlations may be partially included using the wave functions (17). It is sufficient to include correlations only within a single 4d + ~f transition, whose cross section is very large and practically uneffected by other transitions and shells. The use of wave functions ( 17) for a single-transition correlation leaves only “timereverse” diagrams to be taken into account. This is achieved by neglecting all but one term in the sum in Eq. ( l l ) , after integrating over angular variables and summation over spin variables. This term corresponds to

M . Ya. Amusia

14

the 4d shell. Also the interaction matrix element ( m i ( U ( k n must ) be replaced by (mil Ulkn )( 1 -

nislilk *

(22)

6,,,)

where n, is the Fermi step function nl = I for t S F and n, = 0 for t > F. In this substitution the “time-forward” interaction matrix elements equals zero within the same subshell with given n and I . The use of Eq. (17) underestimates correlations but essentially simplifies the numerical solution of Eq. (1 1) because it eliminates the diagonal matrix elements (rilUlin),which diverge as In(€ - E ’ ) due to the long-range nature of Coulomb potential. The intrashell correlations when compared to a simple one-electron approximation lead to very specific qualitative variation of the cross section: it is broadened, and its maximum is shifted to the higher energy o side. However, if solutions of Eq. (17) are used as one-electron functions, the situation becomes different. The direct “time-forward’’ diagrams (see Fig. 5 ) lead at large distances to a Coulornbic behavior of the field in which the outgoing electron leaves the atom. Therefore, the threshold value of the cross section becomes finite, whereas the bubble “time-forward’’ diagrams (see Fig. 5 ) shift the cross-section curve, to the higher energy side, as may be demonstrated. Thus, the inclusion of only “time-forward’’ diagrams stretches the cross section and leads to a prominent difference between the “length” and “velocity” cross sections. Therefore, the inclusion of collective effects within one transition, i.e., “time-reverse” diagrams, leads mainly to a narrowing of the cross-section curve maxi-

M(Ry) FIG.6. Photoionization cross section for the 4d shell of xenon: Haensel e / a / . (1971).

(1971); (---)

(-)

Amusia e / al.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

15

mum and to equivalence of the r and V operators. Figure 6 presents the results of 4d-shell photoionization cross-section calculation including intrashell correlations only, together with experimental data, and demonstrates satisfactory agreement. The one-electron Herman- Skillman calculation leads to a spikelike cross section as high as 140 Mb with a maximum at an energy of 1 Ry lower than experiment. The length and velocity cross sections coincide within 3-5%, which proves that the influence of all other shells and transitions upon the main one (4d + € 0 is weak. A number of calculations performed for many atoms demonstrate that the inclusion of intrashell correlations is sufficient to describe only the manyelectron shells in regions where their photoionization cross section is sufficiently large.

B . PARTIAL CROSSSECTIONS A N D FEW-ELECTRON SHELLS COLLECTIVIZATION In this section we shall consider only those partial cross sections which are significantly affected by intershell interactions. The inclusion of intershell or intertransition correlations in RPAE is achieved by solving Eq. (1 1) with two (or three) terms in the sum overj =s F. For each of these terms the excited states and continuous spectrum is described by solutions of Eq. (17), so that “time-forward’’ intrashell diagrams are included automatically. The intershell interaction should include both “time-forward’’ and “time-reverse” diagrams; as to the intrashell interaction, only the “time-reverse” diagrams are included. Let us study the effect of inner electrons upon the photoionization of the outer shell. Equation (11) may be simplified because the valence shell radius is much larger and the ionization potential much smaller than that of the inner shell. Due to the large difference of ionization potentials, the exchange with inner electrons is negligible. With these remarks, Eq. (1 1) in coordinate representation may be reduced to

In the sum over j the contribution of valence shell is omitted. Using the definition of dipole polarizability ad via the effective dipole moment matrix element (Amusia and Cherepkov, 19751,

M. Ya. Amusia

16 we derive from Eq. (23)

D

=

r

- (r/r3)ad(w)

(25)

where ad(w) is the core dynamic polarizability. For small frequences, w in a d may be neglected, and instead of Eq. (25) we obtain

D

=

r - (r/r3)ad(0)

(26)

derived first by Bersuker (1957). In this article the calculations are performed by solving Eq. (1 1) instead of employing Eq. (25), but Eq. (25) is convenient for qualitative discussions. Because a d is positive for small w, the inner-shell action diminishes the valence electron cross sections or its excitation oscillator strength. If a d is large, the second term in Eq. (23) becomes larger than the first. With growth of w due to decrease of q,(o), both terms in Eq. (24) may become equal but of opposite sign, and the valence electron photoionization cross section reaches zero. However, the situation may be more complex because of a possible sign change of matrix element of z = r cos 0 with energy growth (Manson and Cooper, 1968). As an example we consider photoionization of the 6s electron in cesium (Fig. 7). In the broad energy region from the 6s ionization threshold up to w , which exceeds the threshold of 5p, the 5p electrons action dominates and completely governs the ionization of the 6s electron. Recently, Kelly and Carter (1980) obtained good results in the description of photoionization of 4s electrons in zinc. The intrashell interactions of 4s2 electrons, together with the effect of 3d1° upon 4s2, proved to be very strong. All intrashell “time-forward” diagrams were included, using

F I G .7. Photoionization cross section for 6s electron of cesium: Amusia and Cherepkov (1975); (---) Hudson and Kieffer (1971).

(-,

-.-,

-.

.-)

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

17

the solutions of Eq. (17) as single electron, while "time-reverse" diagrams and intershell correlations were accounted for using perturbation theory up to the third order. The interaction between two 4s electrons diminishes the HF cross section twice, but only the influence of 3dI0shell, especially that of its discrete excitations, leads to satisfactory agreement with experiment (Marr and Austin, 1972). Approaching the inner-shell ionization threshold, its polarization increases, while the direct photoionization amplitude decreases and the role of the second term in Eq. (11) [or Eq. (2511 becomes dominant. In the cases considered above the second term far exceeds the first and leads to an increase of valence electron photoionization cross section. The outer shell, in turn, may drastically alter the inner-shell photoionization. This effect is very strong if the outer shell is a many-electron shell with a large cross section, while the object of its influence is a few-electron shell, with a small H F cross section. At first, such an effect was predicted for 3s electron photoionization of argon (see Amusia and Cherepkov, 1975). Close to the 3s' threshold, virtually excited by photons, all 3p6 electrons act upon s2 electrons and ionize them. With an increase of photon energy, the correlational (describing 3p6 influence) amplitude decreases more rapidly than the direct one. For some energy they both become equal, but of opposite sign. Therefore, has a deep minimum observed experimentally (Houlgate et al., 1974; Samson and Gardner, 1974). The same situation prevails for a number of other few-electron shells. In the photoionization of xenon 5s electrons, consideration of not only the outer 5p6but the inner 4d"' shell is essential, the influence of the latter being rather large, even at the 5 s ionization threshold (Amusia and Cherepkov, 1975). This is an example of three-shell correlation, and calculations together with experimental data are given in Fig. 8. Agreement up to the minimum is satisfactory, but prominent differences exist near the 4d'O threshold. The nontrivial reasons for this behavior will be discussed at the end of Section V. The variation of few-electron shell cross sections under the action of the neighbor is so strong that it is possible to say that it is completely collectivized. CHARGED-ION FORMATION C. SINGLE The experimentally observed increase of single charged-ion formation cross section c+in xenon (Van der Wiel and Wight, 1975)in the vicinity of 4d shell ionization threshold is a prominent collective effect. The growth of c+above the 4d shell threshold may be attributed either to excitation of the 4d electron at a discrete level with subsequent 4d-' vacancy Auger

M . Ya. Amusia

18

i.0 -

---, -.-, F I G .8. Photoionization cross section for the 5s-subshell of xenon: (-, Amusia and Cherepkov (1975). Experiment: (0) Samson and Gardner (1974); ( X ) West et a / . (1976); (0) Adam P I a / . (1978).

-.

.-)

decay or to the effect of the dynamic polarization of the 4d electrons upon the photoionization of the 5p2and 5s2 subshells. The first mechanism leads to a narrow maximum, its width being equal to the discrete level ionization energy. Because oscillator strengths of 4d + nf and 4d + n'p excitations are small, the contribution to cr+ by this mechanism may be neglected. On the contrary, the dynamic polarizability of 4d'" is large and, if included using Eqs. (1 1) or (17), qualitatively alters the photoionization cross section not only of 5 s 2 but also 5p6electrons. This is an example of strong action of one many-electron shell upon the other, however far from the latter threshold, i.e., where its photoionization cross section is small. Under 4d shell action the u+cross section acquires a broad and powerful maximum of collective nature (see Fig. 9).

D. A N G U L A RDISTRIBUTION OF PHOTOELECTRONS The photoelectron angular distribution in LS coupling for closedshell atoms absorbing nonpolarized photons (Cooper and Zare, 1969) is given by

where crnlis the partial cross section of the nl subshell, and Pz(cos 0) is the Legendre polynomial. The anisotropy parameter Pnl(01 is expressed via the photoionization amplitudes Dl+l given by Eq. (11) and phases of electron elastic scattering in the ion field:

19

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS I

/

1;_ \

‘.

FIG.9. Single-electron photoionization cross section d of xenon: (-, and Cherepkov (1975); (0) West ef ul. (1976). P d W )

= ((21 + 1)[lDl-1l2+

-6[1(1

+

1~1+1121}-’{(1

- 1)1Dl-1I2 +

(1 +

---)

Amusia

2)1~l+Il2

1)]1’2~Re[Dl+l * DT-lei(61+l -61-1’]}

(28)

The parameter pis more sensitive to the behavior of the amplitude than is the total or even the partial cross sections because of the interference term, containing the product of I l l + ] and Dl-, . Intershell correlations are especially important in the energy region where the denominator of Eq. (28) is small. The dependence of /3 upon correlations is illustrated by /3 for 5p electrons of xenon (see Fig. 10). The intrashell 5p correlations are not

0

2

4

6

8

10

I2

14

F I G .10. The angular anisotropy parameter p for the 5p subshell of xenon: (-, Amusia and lvanov (1976); (0) Dehmer ef ul. (1975); (0) Torop er a / . (1976).

---)

M . Ya. Amusia

20

I

ReD,

1

4

,

wLthout actton

6

I

I

04 4d‘O

40

8

12

I4

FIG.11. The amplitude of 5p -+ Ed transition in xenon.

too strong and only slightly shift the position of the minimum to lower energies. On the contrary, the influence of 4d’O electrons on pSPproved to be very strong (Amusia and Ivanov, 1976). They affect the 5p + cd transition amplitude, which acquires two additional zeros and a prominent imaginary part (Fig. 11). This is reflected in pSp(w), which acquires an additional broad and large maximum. This strong variation is a manifestation of collective effects in angular distributions. The variations in the 5p + cd amplitude may be understood qualitatively if instead of the continuum 4d rf one considers a single pseudolevel with frequency R . Using Eq. ( l l ) , the 5p + cd amplitude may be presented in the following form:

-

(SplDled)

=

(5pldlcd) + (4dlDIO) X

[2O/(w’

-

R2)](4d,RIU15p, E d )

(294

Because the second term in Eq. (29a) is larger than the first for w - R, (5plDIc) acquires two extra zeros, changing its sign at w = R and then at such value of w at which the direct and correlative terms are equal but opposite in sign. The amplitude (Spldlcd) changes its sign itself (“Cooper minimum”) at w , much smaller than R. Therefore, (SplDle) goes three times via zero. OF COLLECTIVE A N D RELATIVISTIC EFFECTS E. COMBINATION

I. s- Electron Angular Distribution In the nonrelativistic approximation p for s electrons according to Eq. (28) (1 = 0) is energy independent and equal to two.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

21

However, due to the spin-orbit interaction the total angular momentum of the outgoing electron may b e j = 3/2 o r j ' = 1/2. The respective wave functions differ in radial and energy dependence. Therefore,j = 3/2 and j ' = 1/2 waves may interfere, leading to energy dependence of the anisotropy parameter (Dill, 1973):

Here D 3 / 2 ( 1 / 2 )and 83/2(1/2) denote the transition amplitude and elastic scattering phase of photoelectrons withj = 3/2 (1/2). Let us consider the 5s' subshell in xenon, which is strongly affected by the outer 5p" and inner 4d'" shells. The amplitude (5slDl~p)changes its sign, which is reflected by the minimum in ( T (Fig. ~ ~ 8). The zeros of (5slDlqP,/,) and ( S s l D l q ~ , ,are ) at different energies. Therefore, a strong variation in ps, appears under the influence of neighboring shells but is manifested because of spin-orbit interaction. In Fig. 12, experimental (White et LII., 1979) and theoretical RPAE (Cherepkov, 1978) and RPPA (Johnson and Cheng, 1978) data are presented; the agreement is satisfactory. In the RPAE calculation the simplifying assumption was used that the 5p6and 4d"' influence upon 5s' is the same as in the nonrelativistic case. The anisotropy parameter p for valence photoelectrons of alkalies or alkali-earth atoms because of relativistic effects is also energy dependent. For valence electrons of alkalis, p has a deep minimum, even in oneelectron approximation (Ong and Manson, 1978b, 1979). However, the inner shell essentially alters the valence cross section, as is demonstrated

F I G .12. The angular anisotropy parameter p for the 5s subshell of xenon: (-) CherWhite ci epkov (1978);(-.-)Johnson and Cheng (1978);(---) Ong and Manson (1978a); (0) ul. (1979); (0) Dehmer and Dill (1976).

M . Yu. Amusiu

22

by the cesium example. Therefore, it is undoubtedly the case that p will be also strongly changed under the influence of inner-shell electrons. 2. Branching Ratios An interesting manifestation of an intershell interaction is the cross section ratio 7 for subshells with the same I, but different total angular momentum, j = I + 1/2 a n d j ' = I - l / 2 . Without spin-orbit splitting 77 is equal to I + 1 / / . If relativistic wave functions are used, the ratio becomes energy dependent and sensitive to the details of electron wave functions. In Fig. 13 7 is given for 5p6subshell of xenon. It is seen that near the 4d'O shell the HF calculation (Ong and Manson, 1979) contradicts experiment. This is not surprising because there were a number of examples presented above of strong 4d'O action upon outer shells. Taking this action into account by calculating the RRPA photoionization amplitudes of the 5p6 shell (Johnson et a f . , 1980), reasonable agreement with experiment is achieved (Wuilleumier et uI., 1977). 3. The Polarization o j Photoelectrons

Not all photoionization amplitude variations are prominantly reflected in the total cross section or angular distribution. To study the details of such a complex function, as is the amplitude (e.g., Fig. 1l), additional information must be used. This can be obtained from photoelectron polarization.

F I G . 13. The

2P112 branching ratio for the 5p' subshell of xenon (from Johnson ef

a / . , 1980): (0) Wuilleumier P f ul. (1977).

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

23

It is characterized by the preferential spin direction of the photoelectron, and is determined by the ratio P

=

(NI

-

NI)/(Nt + N J

(30)

where N t and N Lare the number of photoelectrons with spin directed up and down, respectively. Evidently the polarization is equal to zero if the spin-orbit interaction is neglected. It was demonstrated (see Cherepkov, 1979, and references therein) that photoelectrons leaving the atom at any angle are spin polarized. The photon energy dependence of P is given by a complex expression, which is simplified for closed-shell atoms. For / shell ionization P may be expressed via matrix elements Dl,l and elastic scattering phases of the photoelectron in the ion field &,. To obtain additional information on the photoionization process, it is convenient to consider polarization in such conditions for which P is determined by D and in different combinations than in the partial cross section or in p. One such is the degree of photoelectron polarization in a closed-shell atom ionized by an unpolarized photon (Cherepkov, 1979):

Here s, q, x are unit vectors directed along the electron spin and momentum of the photon and electron, respectively. In the derivation of Eq. (31) it was assumed for simplicity that matrix elements and phases may be calculated in LS coupling, the spin-orbit interaction leading only to the splitting of the 1 shell into two subshells with j = / + 1 / 2 and j ’ = I - 1 /2. While p [see Eq. (28)] is determined by a sum of several terms, P j contains the interference term directly and in another combination than in p. For a given w, P Jreaches its maximum at the angle where the denominator in Eq. (31) is a minimum. The maximal values of P Jas a function of photoelectron energy are given in Fig. 14 for the 5p1,z subshell of xenon. It is seen that P J very closely follows the variations of the 5p + Ed amplitude. The recent experimental data (Heinzmann et a / . , 1979) agree with calculations, well confirming the strong action of 4d electrons upon the 5p6subshell. It is interesting to note that totally relativistic RRPA results (Huang el a / . , 1979) are very close to those obtained in RPAE. The investigation of photoelectron polarization contributes (and in some cases exhausts) the program of the so-called complete experiment: a complex of measurements including total and partial cross sections and

M. Ya. Amusia

24

(-)

F I G .14. The degree of polarization PJ of photoelectrons from the xenon Sp,,, subshell. see Cherepkov (1979);(---) Re D from Fig. 1 1 .

angular distribution measurements, which permits one to obtain the amplitudes directly from experiment, checking the quality of the theoretical models used in calculations as carefully as possible. 4. Auger Electron Anisotropy

Intershell correlations manifest themselves in the anisotropy of Auger electrons if it appears in the decay of a vacancy with j > I / 2 ( I > 0). The Auger electron angular distribution is given by (Fluge et al., 1972) W

- [ 1+ A:”(w)P,(cos

e)]

(32)

where 0 is the angle between photon and Auger electron directions, j being the total momentum of the vacancy. The anisotropy is a consequence of the fact that the probability of photoionization which leaves the atom with the given angular momentum projection depends on its projection on photon direction. The parameter AijYw) may be expressed via D M (Berezhko rt al., 1978): A:‘Yw)

= (-

I)’+j+”’(21

+ 1){+[3(2j + I)]}”’

1; ;rt

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

0 1

2

4

6

25

8

F I G . 14. Angular anisotropy parameter A2 of Auger electrons in the ionization of the barium Sp,,, subshell: (-, ---) Berezhko er a / . (1978).

Here (2 ,“ ;} is the 6 j Wigner symbol. Both A : ” ( w ) and Pj differ from p and are therefore a source of additional information on D l r l . The numerical coefficient of IDl-$ in Eq. (33) is larger than that of 1Dl+,)2 (contrary to the case of p). Therefore, Aij) is more sensitive to the 1 + 1 - 1 amplitude than the total and partial cross section. In order to ascertain the influence of collective effects in A V), its calculation was performed for the Sp,,, subshell of barium, the vacancy in which decays by a transition and removal of both 6s electrons. Just as in xenon, in cesium and barium the 4d shell essentially acts upon 5p6 electrons. The 5p intrashell correlations do not affect D,,, and A2 too strongly, but the 4d’” alters both the 5p + Ed and 5p -+ ES transitions, and therefore A g ) . The results of the calculations (Berezhko et al., 1978) are presented in Fig. 15. It is seen that, due to intershell correlations, an additional oscillation appears in A 2).

F. COLLECTIVE EFFECTS I N ATOMSW I T H OPENSHELLS The collective effects in open-shell atoms may be even stronger than those in noble gases. This is explained by some increase of mean radius as compared to the nearest noble gas atoms, and by a decrease in their ionization potentials. As a result, the influence of the self-consistent field upon outer electrons is weakened, and the relative role of direct interaction increases. The difference between the ionization potentials of the

M . Y a . Amusia

26

outer and the next shells decreases. All this leads to a conclusion that intershell and intertransition correlations in these atoms must be strong. The investigation of open-shell atoms by many-body theory methods is complicated by the degeneracy of their ground states. The first-order correction in interelectron interaction to the ground state wave function is given by

where $, and Go, En and Eo are the ground and excited state wave functions and energies, respectively. If a state +A exists (+A Z tho) which has an energy equal to E,, the correction (34) diverges if for some reason the matrix element (O'IWIO) is not equal to zero. The advantage of RPAE in generalizing it to open-shell atoms is due to the fact that the terms divergent in the sense of Eq. (34) may be included by a proper choice of self-consistent field. All other divergent terms are out of the RPAE framework. In fact, most dangerous from a divergency point of view are the processes of angular momentum exchange between electron or hole in the RPAE electron-hole pair with the rest of electrons that form the residual ion. Denoting the angular momentum exchange with the ion by a cross, we present in Fig. 16 diagrams that describe divergent RPAE processes. It should be borne in mind that irrespective of this exchange the total angular momentum and spin of the electron-hole pair is conserved and determines the photoionization channel. Considering open-shell atoms in RPAE it is necessary to keep in mind the fact that the residual ion together with the outgoing electron are characterized by total angular momentum L and spin S and their projections. These quantities are unaltered by the interelectron Coulomb potential. Therefore, RPAE correlations are taken within each channel separately. The excited state wave functions after summation of Fig. 16 diagrams are determined by channels L and S as well as by the angular momentum and spin of the ion. But the angular momentum exchange taken into account by the double line in Fig. 16 leads, strictly speaking, to the destruction of angular momentum as a good quantum number of every excited electron or hole. The diagrams in Fig. 16 are "time forward" and may be taken into

LS, -

--

7 +

nt

+-

t

"e

net'

=

= /=-

L,5

"e'

F1c8. 16. T h e momentum exchange interaction of an electron-hole pair in open-shell atoms.

27

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

J (RY)

F I G .17. Photoionization cross section for the 3p" shell of chlorine: and Chernysheva (1977); (---) Starace and Armstrong (1976).

(-)

Cherepkov

account by a proper choice of self-consistent field (Cherepkov and Chernysheva, 1977)just as is done in deriving Eqs. (151417). The excited state functions are projected upon definite angular momentum states and are calculated in the "frozen" core field. The corresponding RPAE equations were derived and applied to the 3pJ shell photoionization of chlorine (Cherepkov and Chernysheva, 1977), which is illustrated by Fig. 17. The correlations proved to be even stronger than in the 3p6 shell of argon, especially the interaction between 3p + 3d type transitions, corresponding to different angular momenta and spins of the residual ion. The same Fig. 17 presents results obtained by another version of RPAE generalization to unclosed shells (Armstrong, 1974; Dalgaard, 1975; Starace and Armstrong, 1976). These approaches differ from that developed by Cherepkov and Chernysheva (19771, mainly by the neglect of Fig. 16 processes. The significance of these processes explains why, according to Starace and Armstrong (1976), the role of RPAE correlations in chlorine is small. Much stronger than that in argon is the influence of 3 p upon 3s2 (Kelly and Carter, 1980) in chlorine. Thus, the 5s2 electrons in iodine will be collectivized due to 5pi and 4d"' action, which may be even stronger than that in xenon. In connection with the collectivization of few electron shells of open-shell atoms it would be interesting to study the angular distribution of photoelectrons from the ns' subshells. Due to angular momentum exchange with the residual ion, the photoelectron wave function acquires additional components with other angular momentum, instead of being a pure p wave. Being different for different scattering angles, this admixture

M . Ya. Amusia

28

depends on photoelectron energy. Consequently, p for s electrons becomes energy dependent even in the LS approximation (Starace er d., 1977). It seems that the details of complex energy dependence of 5 s + E P amplitude, which is a result of its collectivization under the action of 5p5 and 4d’O electrons, will be manifested in the 5 s photoelectron angular distribution. The half-filled shell atoms, about 20 elements, form a special group. According to the Hund empirical rule, the spin of these atoms in the ground state is maximal, i.e., the spin projection of all half-filled shell electrons is the same. Neglecting spin-orbit and spin-spin interaction, it is possible to consider electrons of different spin projections as different particles, which may be called “u” (“up”) and “d” (“down”) electrons. From such a point of view a group of electrons with the same n, 1 and spin projection s forms a closed subshell. Therefore, each atomic shell except the half-filled one is separated into two “u” and “d” subshells. The “u” (“d”) electrons may exchange with each other, while “u” and “d” cannot. Because the Coulombic interaction does not change the spin projection, it cannot mix “u” and “d” shells. Therefore, the RPAE may be applied to half-filled shell atoms if it is generalized to treat two kinds of particles, the shells for them being closed. It is easily achieved in the matrix form. The photoionization amplitude is a sum of “u” and “d” amplitudes, 0,and DC, without interference. Then Eq. (1 1) is transformed into

Here U describes the interaction between particles of any kind, xu and and d energy denominators of Eq. (11). The interaction Uuu(Udd) also contains exchange: (iklUlh)uu = ( i k ~ V ~ / m ) u-u ( i k ~ v ~ m / ) , , , ,

x(Ibeing the u

((Id)

(dd)

((Id)

whereas Udu( U u d ) includes only the direct term, (ik/V[m/)uci (dU)

It is necessary to bear in mind the fact that the RPAE electron-hole pair is either “u” or “d,” because in the low-energy region spin flip by the photon is improbable. Analogous to Eq. (13) an equation for f is derived:

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

29

The HF ground state structure is also altered, because after separation into “u” and “d” subshells the ionization potentials become spinprojection dependent. This approach was used in the calculations of the manganese atom (Amusia et u/., 1981), where a powerful maximum was experimentally observed in the vicinity of 3p subshell and interpreted (Connerade ef a/., 1976; Bruhn et ul., 1978) as a manifestation of discrete resonance transition 3p + 3d (to the empty level in the half-filled 3dJ shell), which decays into the continuous spectrum excitations 3d + Ef. Considering the 3d.’ electrons as an “up” subshell, we obtain a strong energy separation of ‘‘u” and “d” electrons. Therefore, Z3,,?= 5.455 Ry, while &,, = 4.46 Ry, which is much larger than the corresponding spinorbit splitting. It is noticeable that even for 4s electrons Z6, = 0.547 and I,& = 0.452 Ry. The cross section is dominated by 3pl + 3dl transition, the unperturbed energy of which w, is 3.87 Ry. The RPAE 3pf correlations lower the 3pl + 3di energy by 0.04 Ry, while interaction with 3d7 lowers w, by a far greater amount, 0.129 Ry. As a result we have obtained w, = 3.70 Ry, which agrees well with experiment. The calculations were performed using Eqs. (11) and (39, where transitions 3pL+ 3, n , Edl and 3d, + E f T were considered. All other transitions proved to be small. Figure 18 presents experimental and calculational results. Intertransition correlations are very strong and in fact determine the total photoionization cross section below the 3p ionization threshold, leading to a broad maximum instead of a discrete line. It is interesting to inquire whether the HF u and d subshell energy splitting describes the real atom structure. It is quite possible that this I

I

I

I

FIci. 18. Photoionization cross section near the 3p threshold of manganese: (-, Amusia r / 111. (1980a): (---) Bruhn r / t i / . (1978).

-.-)

M . Ya. Amusiu

30

splitting is decreased or even eliminated by correlational corrections which are out of the RPAE frame. Because only several calculations exist, it is far from clear whether RPAE (or several first orders of perturbation theory) describes satisfactorily the open-shell atom photoionization data. The polarizability of these atoms is larger than that in noble gases and therefore may alter the electron motion significantly. It will lead to deviations of electron and hole states from being purely HF and of the interaction between them from being purely Coulombic.

G . DOUBLE-ELECTRON PHOTOIONIZATION The probability of double-electron photoionization (DEP) is a measure of “direct” interaction. This process has been reviewed recently (Amusia, 1981). To understand the strength of direct interaction it is necessary to bear in mind the fact that the ratio of double- to single-electron photoionization probability varies from 4% in He up to about 80% in xenon. For atoms with not very high 2 (helium, neon, and even argon) the main features of this phenomenon may be described in the first nonvanishing approximation in the direct interaction. As to heavier elements, especially starting from the krypton region, the role of intershell interaction rapidly increases. This is manifested by an increase in DEP near the 4d’” threshold in xenon (Samson and Haddad, 1974). The DEP proceeds due to the direct interaction between 5p6 and (or) 5s’ electrons and under the influence of 4d“’, which may knock out a pair of outer-shell electrons even if their mutual interaction is neglected. A careful extrapolation (Van der Wiel and Chang, 1978) demonstrates that the 5p6and 5 s 2 electron interactions alone cannot explain the increase of DEP probability in the 4d“’ threshold vicinity (see Fig. 19). Just as strongly affecting single-electron ionization of 5p6 and 5s’ electrons (Fig. 91, the 4d shell polarized due to absorption of a photon may lead to a prominent increase of DEP. The role of 4d tends to increase the effective dipole moment (25), which increases approaching the 4d ionization threshold because the polarizability of 4d is large for w Z4. Therefore, the 4d shell behaves like a resonator which amplifies the absorbed photon in some frequency region. This “amplified” photon is then absorbed by a pair of correlated 5p6or (and) 5s2 electrons. The DEP probability is also very large for barium (Brehm and Hofler, 1975; Holland et al., 1981), where the essential role of 5p electrons is evident but the concrete mechanism is not clear (Connerade et ul., 1979). It would be interesting to study the intershell interaction effects not only upon total DEP cross section, but also on energy and angular photoelectron distributions. Unfortunately only few calculations exist in the field, and the number of experimental data are inadequate.

-

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

/ I

I '

/

31

I

I

FIG. 19. Double-electron photoionization of xenon: (0) Samson and Haddad (1974):

(0, x 1 Van der Wiel and Chang (1978).

H. COLLECTIVE OSCILLATIONS

The collective effects discussed above were connected with one, two, but not more than three nearest shells. It was suggested by Bloch (1933), however, that collective oscillations of an atom may exist in which all (or at least the majority of) atomic electrons participate. Whereas in classical hydrodynamics the oscillations are stable, in an atom they may decay, with the emission of one or several electrons. If the probability of such decay is large, the collective level will become unobservable. It is also possible that, due to the inhomogeneity of the atomic electron density and the large energy separation of the shells, the atom has no characteristic frequencies at which all atomic electrons oscillate. The equation describing collective Bloch-type vibrations may be obtained from Eq. ( 1 1 ) . The expression [(n, - n,)/(w + Eq - ~,)1(41d(p) determines the density variation in a single-electron approximation. Multiplying Eq. (11) by (n, - n,)(w + E , - E , ) - ] and denoting [(n, - n,)/ ( w + E , - ~ , ) ] ( 4 1 0 ( w ) l pas ) pqp(w ) an equation is obtained:

which determines the electron density variation under the outer-shell action. The eigenoscillations of an atom are determined by Eq. (37) without the external field term (qldlp). Such an equation gives a set of eigenfrequencies w t of collective atomic oscillations. For an infinite homogene-

32

M. Ya. Amusia

ous electron gas, neglecting exchange in matrix elements (qrlUlps), it leads to the equation for plasma oscillations, which in the limit of longwave vibration has a plasma frequency solution wo = (47rpd pobeing the equilibrium electron gas density. By neglecting the exchange part in U and in a statistical approximation, several solutions of Eq. (37) for eigenoscillations were obtained. Accordingly dipole oscillations were predicted (Kirzhnits et al., 1975) with frequencies w1 = 2 Ry and w2 z 2.652 Ry, their widths being as small as r, = 2.2 x 10-42Ry and r2= 7 x 10-s2 Ry, respectively. The oscillator strength of the first level proved to be very large: fi = 0.1Z. However, we have studied the photoionization cross section of xenon in the first-level w1 vicinity, w1 = 54 Ry, which is -5 Ry above the 3d ionization threshold. The solution of Eq. ( l l ) , even not in a statistical approximation, but using H F wave functions, leads to prominent deviation from experiment in this energy region. Due to the large energy separation of different shells and the comparatively weak interaction between them, the RPAE influence of all other shells upon 3d1"is negligible. The explanation of the difference between theory and experiment will be discussed in Section IV,A. However, no maximum except one directly connected with the 3d-shell threshold exists either in calculation or in experiment. Walecka (1976) has calculated in a statistical approximation the RPAE density vibration frequencies of different multipolarities, not only for singlets (which may absorb a photon) but also for triplets. The equation for triplet oscillations is obtained from Eq. (37) only if the exchange term is retained in U . However, it was demonstrated (Amusia and Cherepkov, 1975) that in RPAE for closed-shell atoms, apart from dipole, monopole correlational effects are also essential. However, the strength of both of them is insufficient for total collectivization of all or even the majority of atomic electrons.

IV. Collective Effects near Inner-Shell Thresholds In a number of cases RPAE describes experimental data unsatisfactorily, while the deviation from the one-electron picture is still strong. One evident source of this difference is a direct consequence of the deviation of RPAE (HF) ionization threshold from experimental values. However, not only the threshold position but the form of the cross section near intermediate- and inner-shell thresholds is reproduced incorrectly-the RPAE usually gives values larger than in experiment. Both defects have the same origin: the neglect of electron shell rearrangement under the influence of inner-shell vacancy. The RPAE does not describe the fine structure of the cross section either, which is a manifestation of more complex

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

33

excitations than the simplest one-electron-hole RPAE excitations. All these defects may be eliminated by a generalization of RPAE which takes the following into account: (1) The outgoing electron and vacancy wave functions differ from their HF values because the electron and hole polarize the atom. (2) The electron-hole interaction is not purely Coulombic (+ exchange term) but includes corrections determined by virtual excitation of more complex configurations, with two and maybe even more electronhole pairs. (3) The electromagnetic field interaction operator differs due to interaction with complex configurations from its unperturbed value z or V,. (4) The simultaneous removal of two and even more electrons contributes to the photoionization amplitude.

The correct accounting of all these factors is very difficult, and therefore such a program of RPAE generalization requires that corresponding approximate approaches be developed, which is the aim of this article. Photoionization cross-section peculiarities will be considered mainly near inner-shell thresholds. In this region alone rather serious deviations in HF and RPAE results from experiment exist. This is not surprising because HF and RPAE neglect the electron shell rearrangement due to vacancy creation in photoionization. This rearrangement leads to a prominent collective effect-to variation of all electron states, which alters the self-consistent field acting upon the photoelectron. The inner-shell vacancy may decay radiatively or via the Auger effect. If the photoelectron is so slow that the decay time is insufficient for it to leave the atom, it moves not in the field of the initial vacancy i but in the rearranged ion field. The time scale T~ of rearrangement due to vacancy i creation is determined by the difference of ionization potential l i and HF Ei energy: Ti lei + lil-’.The time of rearrangement due to vacancy decay is given by Ti y r ’ . yi being the vacancy i width. For the electron with velocity V% the time of leaving the atom (or crossing the ionized shell) is t = R/&, where R is the atomic radius (or the radius of ionized shell Ri Z;”?. The rearrangement is important if t is larger than Tiand T ~ For . ve, rearrangement has sufficient time to go locities for which t >> Ti and T ~ the to completion; if t Ti or (and) Ti it is performed only partly.

-

-

-

-

A. STATIC REARRANGEMENT The situation is rather simple if t >> Ti,but Ti is sufficiently large. It is natural in this case to calculate self-consistently the ion with vacancy i instead of the initial ion and to consider the outgoing electron in this

M. Yu. Amusia

34

rearranged field. Atomic electrons attracted by the vacancy will increase its screening. It will lead to a decrease of the cross section at threshold. The one-electron wave functions are determined by the following equation instead of by Eq. ( 5 ) :

where the existence of a vacancy in state i is taken into account in the sum o v e r j : j # i, and “k” denotes occupied ( k s F ) and vacant (k > F ) states. If one uses the wave functions (38) instead of ( 5 ) in Eq. ( l l ) , a generalization of RPAE which includes the existence of vacancy i is derived, GRPAE (Amusia, 1977). The photoionization cross section of the barium, cesium, and lanthanum 4d shell was satisfactorily described by GRPAE (Amusia et al., 1980b),especially near the 4d threshold, where the RPAE cross section is much larger than that obtained experimentally. Figure 20 displays the results of RPAE, GRPAE, and experimental data (Rabe et al., 1974) for barium. It is seen that rearrangement significantly effects the cross section. In GRPAE calculations the experimental value of the ionization potential was used. Figure 21 presents the photoionization cross section above the xenon 3d’” threshold, where GRPAE permits one to achieve satisfactory agreement with experiment. A strong disagreement of both experimental and theoretical data with predictions of the collective oscillation model (Kirzhnitz er al., 1975) is demonstrated.

7

8

10

F I G .20. Photoionization cross section for the 4d shell of barium: Amusiaer rrl. (1980b); ( - - - - ) Rabe et 01. (1974).

(--,

-.-.

---)

COLLECTIVE EFFECTS IN PHOTOlONlZATlON OF ATOMS

35

-. .-) F I G .21. Photoionization cross section near the 3d shell threshold of xenon: (-, Amusia et u l . (1980b): (---) Deslattes (1968): (-.-) collective statistical model prediction, spread by “apparatus” width = 0.2 Ry.

A number of diagrams which are out of the RPAE framework are included in GRPAE. These are the diagrams which describe the hole energy shift from its H F value. A simple example of such diagrams is presented in Fig. 22a. The GRPAE takes into account the screening of Coulombic electron-hole interaction due to virtual or real excitation of other atomic electrons. An example is given in Fig. 22b, where the electron and the hole interact not directly but via excitation of an additional electron-hole (“eh”) pair. The GRPAE includes also some corrections to the interaction of eh pairs between each other: together with the Coulombic, compex indirect interaction via virtually excited other eh pairs, for example, those given in Fig. 22c, are included. Evidently, GRPAE does not take into account all high-order diagrams. Thus, the hole energy correction even in

FIG.22.

Examples of GRPAE diagrams.

M. Ya. Amusia

36

second order is determined not only by the Fig. 22a term, but by the sum over all intermediate electron and hole states p > F, q s F and all i’ s F. The residual ion polarization caused by the outgoing electron, which alters the wave function of the latter, is neglected in RPAE. This is justified for those atoms whose positive ion polarizabilities are small. GRPAE is better than RPAE in describing those inner vacancy creations, whose ionization potential shifts from HF value are determined by atomic shell polarization without hole state variation (an example is given in Fig. 22a). If the hole energy shift is strongly affected by configuration mixing with a variation of hole state (e.g., by mixing between i and k, p , q states-see Fig. 22a), then GRPAE may be incorrect.

B. INNER

VACANCY

DECAY A N D POSTCOLLISION INTERACTION

In GRPAE rearrangement due to the decay of vacancy i with the possible creation of two or even more vacancies is neglected. If the decay proceeds fast enough, the slow photoelectron may find itself not in a single, but in a double (or even higher) charged-ion field, which significantly changes the electron wave functions. Because the slow electron spends a significant time near the decaying atom, it may influence the decay process. These two effects are called postcollision interaction (PCI). The peculiarities of the cross section near the decaying state excitation threshold were first observed in heavy atomic particle collisions (Barker and Berry, 1966) and then in the excitation of autoionizing resonances by electrons (Hicks et al., 1974). Later effects of the same type were observed in inner-shell photoionization near threshold (Van der Wiel et al., 1976). The wave function variations may be so strong that the continuous spectrum electron in the deep vacancy field may find itself in a discrete level in the new field created due to vacancy decay. However, not only the outgoing electron wave function but also the vacancy decay energy changes significantly. The diagrams which describe such near-threshold effects are given in Fig. 23. It is seen that they are out of the RPAE and GRPAE framework. The interaction between the “slow” ( E ) and the Auger decay “fast” electrons is neglected in Fig. 23. This is justified if the decay energy is much larger than both E and the Bohr energy unit, 1 Ry. If this is not so, it is necessary to solve the three-body problem, to describe the motion of two interacting electrons in the self-consistent HF field. The analytical expression of the sequence in Fig. 23 is given (Amusia et al., 1979) by

COLLECTIVE EFFECTS IN PHOTOlONlZATlON OF ATOMS

FIG.

37

23. Postcollision interaction sequence of diagrams.

CT - ~ I & i ) p , k ~ ( g ) \ % ( ~ -

< - l k p - eP) dED dE

(39b)

The summation in Eq. (39a) also implies integration over continuum excited states. Here ( ~ ’ l )i is the overlap integral of the electron wave function in the atomic field with vacancies i and kq, respectively. The RPAE correlations may be included in PCI, substituting (ildlE’)with (ilD(E’) I d ) from Eq. (1 1) and ( ipl Ulkq) with the effective interaction r from Eq. (13). If the variation of the field in which the slow electron moves is small, i )&(E’ - E ) and D(i)(E)reduces to the ionization amplitude then ( ~ ’ l = with the creation of vacancy i, which then decays into the k . 4 , p state. Neglecting the dependence of the denominator on E ‘ and using the completeness of functions, the following is obtained for D(i)(i):

- I( i(dli)

lD(i)G)p

12

(40)

This expression takes into account the fact that the vacancy i decays so fast that the escaping electron feels at once the field of k , q vacancies instead of i. The cross section at threshold increases due to an increase of the field. The field variation also alters the slow electron angular distribution (Amusia et ul., 1979). The field increase leads to an energy redistribution between “fast” and “slow” electrons, the “slow” electrons becoming slower and the “fast” faster. Therefore, the total process of inner vacancy creation and decay cannot be separated into two stages, first the vacancy creation and then its decay. On the contrary, the energy transfer connects both stages in a united process. This is manifested very clearly in the “fast” electron spectrum, which becomes asymmetric (sharper from the lower eP energy side) instead of being Breit-Wigner, and oscillative, its maximum being shifted to higher energy. The last effect is a consequence of the overlap integral ( E ‘ I Z ) oscillation and of interference of two ampli-

M. Ya. Amusia

38

(-,

FIG.24. Single-electron photoionization cross section near the 2p threshold of argon: -.-, -. .-) Amusia et al. (1977); (---) derived from Van der Wiel et a / . (1976).

tudes, the resonance, proceeding via intermediate state i, and the nonresonance with the same final state but without initial creation of vacancy i (Morgenstern et u l . , 1977; Amusia et c i l . , 1980a,f). The “fast” electron spectrum maximum shift may be estimated (Barker and Berry, 1%6) taking into account the fact that during the decay time Ti y;’ the “slow” electron travels far from the ion, R e1j2/yi> y:”. At such a distance the field variation W ( R )due to vacancyi Auger decay is W ( R ) R - ’ , and the energy shift AE is

-

-

-

AE

- yiZ-“2 < yf“

(41) The consideration of this section is valid ifyi << 1 because only in this case does the sequence in Fig. 23 give the main contribution. The calculation using Eqs. (39) for) ; 1 being a discrete level was performed for photoionization in the vicinity of the argon 2p shell (Amusia et al., 1977). It appeared that single-electron photoionization acquires a maximum above the 2p threshold (see Fig. 24), in agreement with experiment (Van der Wiel et al., 1976). The width of the maximum is much larger than ym. It is interesting that RPAE leads here to a dip instead of a maximum! The deep vacancy decay is so fast that the escaping electron will move at once in the field created by the Auger effect. This is illustrated by the argon 1s electrons (Fig. 25). The RPAE cross section at the threshold is too large, whereas GRPAE strongly underestimates it. This means that the static rearrangement is not completed during the 1s vacancy decay. Using Eq. (40), where I<) is calculated in the field of rearranged two 2p and (or) 2s vacancies created in the Auger decay of the Is vacancy, satisfactory

39

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

- ) bM ( 6

006

~

I GRPAE

FIG.25. Photoionization cross section of the Is’ electrons of argon: (---) (1%3).

Schnopper

agreement with experiment (Schnopper, 1963) is achieved. Although the role of RPAE in 1s argon photoionization proved t o be small, the process is of a collective nature because it is determined by rather complex rearrangement and vacancy decay in which many atomic electrons take part. The energy shifts according to Eq. (41), just as other X I effects increase with growth of yi. However, for y i ? 1, many other diagrams besides those in Fig. 23 are important. As a result the entire process becomes indistinguishable from double-electron photoionization. In Eq. (39a) it was assumed that y i is independent of electron energy E . However, if E is small and the photoelectron is near the atom for a sufficiently long time, it may change the vacancy b e h a ~ i o rThis . ~ is described by the sequence of diagrams in Fig. 26. Analytically they are represented by the following expression for photoionization amplitude: b(i)p.kp(z)=

(ild[Eif w-

H O P

%(Ei+

w-

@]-‘I;)(

iplulkq)

(42)

where % ( e i + w - b) is the self-energy part of the vacancy Green’s function, denoted by insertion in the hole line of Fig. 26, with account being taken of the electron wave function E ’ variation; kZo and H are H F Hamiltonians with vacancies i and k q , respectively; H - b~= W ( r ) .For 2 we have

These results were obtained with V. K . Ivanov and V. A. Kupchenko. The following results of this section were obtained with M. Yu. Kuchiev.

M . Ya. Amusia

40

FIG.26. Strong postcollision interaction with variation of the vacancy decay.

+ w - @I<) (43) If the dependence of 2 on (ci + w - k)is neglected, ( E’ 12il<)is diagonal = (E”Xi(Ei

and independent of E ’ . Then Eq. (43) reduces to Eq. (39), substituting ei with ei + Re X i , y i being equal to 2 Im The 2 dependence on Halters the “fast” electron energy distribution and the cross section near the threshold. Expanding 2(ei + w - k)into Taylor series, one obtains

zi.

which leads to the following expression instead of Eq. (42): &i)p.kq(W)

(ild(€i

w -

A, - Re

- ityi

+ w(r)f)-’lZ)

x (iPl U l M

(44)

It can be demonstrated thatf is limited -0 < f G 1. Even for smallf Eq. (44) essentially differs from Eq. (39). Therefore, i f f = 0.1, the PCI maximum shift in the fast electron spectrum becomes two times smaller than that forf = 0.

V. Collectivization of Vacancies In this section we shall consider the structure of vacancies created in inner-shell ionization process. In the preceding section the collective effects near thresholds, where the photoelectron energy is small, was discussed. Here we shall assume that the photoelectrons are so fast that their direct interaction with the residual ion is negligible. By investigating the

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

41

photoelectron energy spectra, information on atomic shell structure can be obtained which includes the vacancies energies, their widths, and the existence and nature of complex many-electron excitations. Experimentally, the study of vacancy spectra and their structure is possible using electron spectroscopy methods, i.e., investigating the photoionization cross section not only as a function of photon energy w, but at fixed was a function of electron energies E . Whether the HF description of atomic structure is absolutely accurate, the electron spectrum for fixed w would be a Set of &like spikes at energies equal to w - l k , Ik being the ionization potentials of different atomic shells. However, the direct interelectron interaction leads to level shifts and, because of nonstationarity of excited states, to finite-width maxima instead of 6 spikes. Also, complex states appear which include at least two vacancies and one excited electron. All those deviations from the HF picture can be considered collective effects. A. THEVACANCY WAVEFUNCTION The corrections to vacancy wave function are presented by a sequence of diagrams (Fig. 27), in which the block I: describes virtual and (or) real formation of more complex excitations (e.g., Fig. 28). Summing up the diagrams of Fig. 27 we come to the equation for vacancy wave function t,bi:

Physically Z describes polarization of atomic shells by a vacancy i. Contrary to an ordinary potential, Z is energy dependent. If in Eq. (45) the main contribution comes from the diagonal matrix element (ilX.(E)li) = X i i ( E ) , then Eq. (45) can be solved easily:

FIG.27. The diagrammatical equation for the vacancy state I$,.

M . Ya. Amusia

42

i

9

n

j

F I G . 28. The sequence of diagrams describing the self-energy part of the vacancy Green's function.

The new state energy is determined by zeros in the denominator in Eq. (46): ~i - Z ; i i ( E i ) -

Ei

=

0

(47)

If Z ( E ) contains poles, Eq. (47) does not have one, but several solutions. The nondiagonal matrix elements in Eq. (45) describe the admixing of other HF states to the considered one. From Eq. (45) it is seen that the admixture of k state toi is determined by the expression & ( E i ) / ( E i - E ~ ) "k" belonging to a discrete or continuous spectrum. The vacancy energy correction to ei is derived from Eq. (47):

To clarify the physical meaning of fi, it is convenient to consider only the second-order term in C:

It is seen that

However, ( i p I V l k q ) ( ~+~cq - e p - ei)-' is the admixture of complex state X , 4 , p to i. and thereforeJ; is the probability of spreading i overk, 4, p x e s . In fact, the multiplier ( 1 + J1) demonstrates according to Eq. (46) that in rcli the old state pi is represented with probability ( 1 +.fi)-' less than one. The remainder part of qbi is contributed to by more complex states included in Z . It can be said that (1 + .h-'= F i , which is called the spectroscopic factor, determines that part of the time during which rcli is a pure single-electron state. The remaining part of Jli is spread over more complex atomic excitations.

,

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

43

The function X has [see Eq. (49)] an imaginary part which is determined by real intermediate states (see Fig. 28), i.e., by vacancy i decay into a more complex state, e.g., k , 4,p . As a result, Ei also acquires an imaginary part. Thus, I;&) determines the energy shift, the vacancy width y i = 2Fi Im Zii, and spectroscopic factor Fi. The vacancy i behaves as a well-defined atomic excitation only if Fi is close to 1, and yi is not too large, and definitely much less than q. For large Z i it is better to call a quasi-particle wave function, emphasizing that complex atomic excitations participate in its formation. The quasiparticle state may be quite stable. But ifFi is far from 1 or yi is large, the statei is collectivized; it either decays or strongly mixes with other neighboring states, and thus loses its individuality. Both possibilities may be united by the term “melting” (proposed by U. Fano, private communication). If the state is “melting,” its description by Eq. (45) becomes meaningless, and the structure of states, the connection with which leads to large yi or small Fi , evidently must be taken into account.

B. THEV A C A N C Y ENERGY A N D WIDTH The difference A I , between experimental ionization potential and HF energy E , does not exceed 10-15% of E , . Therefore, to obtain AIi it is sufficient to calculate only second-order diagrams in Zii and also those of higher orders, which are included by choosing the self-consistent field [see Eq. (1611. For inner shells it is normally sufficient to include only diagonal terms in hole state corrections (Fig. 28 with only X = i). Thus, for the argon 2p shell A l , is equal to 0.76 Ry, while the second-order diagrams give 0.80 Ry for Zii. The main contribution to this figure is presented by monopole excitations of atomic electrons. On the contrary, for outer shells not only diagonal in vacancy state but also many other secondorder diagrams (and maybe higher orders also) contribute to X (see Fig. 28), including “time-reverse’’ diagrams, with monopole, dipole, and quadrupole excitation of atomic electrons. Thus, the shift of the 3p level (f;;” = 1.16 Ry) from its H F value (leg,,/ = 1.18 Ry) is small not because each diagram of X is small but because of the almost total compensation of “time-forward’’ and “time-reverse’’ cont ri bu ti ons . The energy shift of the P I S ’ outer shells of noble gases is comparatively large. So A13, is equal to 0.4 Ry, lebl = 2.55 Ry in argon. Three-quarters of this Ala, comes from the interaction of 3s-’ with 3 ~ - ~ 3 d ( n Ed) d , states, mainly with the 3p-’3d state. The remaining 0.1 Ry is a result of the interaction 3s ’ with 3s-’3p-’np(ep). For the inner 4p electrons of xenon and neighboring atoms the large shift of about I Ry is determined not by

44

M. Ya. Amusia

diagonal in the hole state corrections (see Fig. 28, with k = i ) , but by the strong interaction of 4p-' with 4 d - 2 n f ( ~ 0(Wendin and Ohno, 1976), which leads to large dipole excitation of the many-electron 4d shell, the correlations in which are very strong. The total vacancy width is determined by its radiative and Auger decay probability. Let us consider the first one. Of equal importance as the simple mechanism of photon emission in the transition of an electron from an occupied j to a vacant i level, a more complex mechanism exists in which the transition energy is absorbed by other electrons, the deexcitation of which leads, in turn, to photon emission. The radiative decay probability Wit is given by W,,

= 4w6ci1)(ilDb)12

(50)

where D is determined in RPAE by Eq. (11). In HF one must substitute ( i ID1 ) with Y i ldb ) . The difference between W:F and W$YAE in some cases proves to be very large (Amusia and Cherepkov, 1975). Thus, for the krypton 4s vacancy the interaction with 4p shell excitations (mainly with 4p + Ed) decreases WQl,,edfrom 1.034 x lo-' eV to 0.16 x lo-" eV; the outer-shell electron excitations essentially supress the radiation. The same situation applies to the Auger effect. The decay energy is transferred not directly to the emitted electron, but via some other atomic electrons, the deexcitation of which leads to the emission of an Auger electron. Such a process significantly alters the vacancy width. The Auger decay in a one-electron approximation is proportional to the square of the Coulombic interaction matrix element:

where E , is determined by equation e p = li - l q k , Iqk being the ionization potential of the state with q and k vacancies. The RPAE corrections may be taken into account by substituting the Coulomb matrix element in Eq. (51) with the effective interaction (13). We have considered (Amusia and Cherepkov, 1975) the role of correlations in some transitions in xenon, which proved to be large. Thus, the partial width of the 4s-1 + 4 ~ - ~ 5 s - ' ~ p transition in the one-electron approximation is 0.08 Ry, while with inclusion of 4d1"shell virtual excitations it becomes about ten times smaller, only 0.007 Ry. Normally, Auger and radiative width in atoms are sufficiently small and Fi is sufficiently close to one so that the conception of a strictly defined vacancy and shell is quite meaningful. However, there are also examples of rather large Auger widths-even exceeding 1 Ry (McGuire, 1974; Wendin and Ohno, 1976).

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

45

C. COLLECTIVIZATION OF 4p S H E L L I N XENON Significant spread and fast decay may lead to total collectivization of a vacancy. A striking example of such a situation is given by the 4p shell in xenon, especially by its 4p”l level. In electron spectroscopy experiments no traces of the 4p112subshell were observed (Gelius, 1974). This “disappearance” was attributed to a strong collectivization of the 4p shell due tc its interaction with 4dI0 electron excitations, which lead to extremely rapid decay and very strong spread (Wendin and Ohno, 1976). The HF energy of 4p is 12.016 Ry. The relativistic correction to the 4pl12 level and the correction due to monopole polarization of atomic shells are important for this level. They both shift the energy to 12.48 Ry. However, the role of dipole polarization of the atom by 4p-’, which interacts with 4d-%f (McGuire, 1974), proves to be very significant. Just this interaction contributes mainly to Z, which is determined by Eq. (49). If the threshold of the 4d“’ shell double-electron photoionization is above the 4p shell ionization threshold, the denominator is positive and therefore Z given by Eq. (49) is positive. This leads to a decrease in the 4p1/2 ionization potential. If is larger than Ida-z,the denominator in Eq. (49) changes its sign, and therefore the expression (49) may be of either sign. It appears (Wendin and Ohno, 1976)that Z is negative, which increases the 4p ionization potential. Then the 4p1/,level finds itself deep in the 4d-’Efcontinuous excitations, which leads to a huge spread of about 2-3 Ry. Over this very broad region of mainly 4d-%f excitations the level 4p,/, is smeared out. The 4pl12spectroscopic factor is close to zero. Both facts mean complete collectivization of 4p1/2 vacancy. For the calculation of photoionization cross section in this region not the 4pII2vacancy but continuous spectrum excitations 4d-2d(rp)(Wendin and Ohno, 1976) must be taken into account. It proves to be necessary to substitute the Coulombic interaction between 4p-’ and 4d-%f with the effective [see Eq. (13) ]. The 4p3/, level is also significantly collectivized. It interacts and mixes with discrete 4d-”f and continuum 4d-%f excitations. Therefore, Eq. (47) has a solution connected with 4d-24f but strongly shifted in energy and intensity from its unperturbed values. It has also a second solution in the continuous spectrum region. The calculated cross section, together with experimental data, are given for the 4p-I-4s-l region in Fig. 29, which is taken from Wendin and Ohno (1976). The sharp maximum at 10.62 R y is, in fact, the level 4d-”f strongly shifted from its HF position. Let us note that calculational and experimental cross section data are normalized at the 4s level maximum. Although a general agreement exists, there also exists a difference-the calculations lead, instead of a 4pU2level, to a broad but symmetric maximum, while in experiment it is evidently asymmetric. This

M. Yu. Amusia

46 h

v,

,

I5

,

1

14

13

12

I

Binding energy E (Ry) Fib. 29. Photoelectron spectrum in 4s-’4p-’ region of xenon (from Wendin and Ohno, 1976): (---) Wendin and Ohno (1976); (-) Gelius (1974).

difference may be attributed to neglect of the direct excitation of complex states, such as 4d-‘d, without the initial creation of “4p.” The 4s level is also collectivized due to interaction with 4p-’4d-’dor, to be precise, with 4d-%fdf excitations. It loses a prominent part of intensity (F4s --- 0.8), its position being shifted by 0.7 Ry to lower binding energies. D. “SHADOW” LEVELS

Because of essential mixing of one-vacancy and complex states, the latter acquire some properties characteristic of the former. Let us explain this by the example of two levels with JI1 and t,b2 wave functions, the first of which-the “simple” one-may be photoionized, while the second, of more complex structure, does not interact directly with the electromagnetic field. The mixing leads to formation of new levels $, and $2. Both of these may be photoionized, but the cross section is determined by admixture of $, in & and $,. The argon 3s vacancy serves as an example. It was mentioned in Section V,B that the 3s vacancy shift is mainly due to interaction with 3 ~ - ~ 3 d state. This matrix element is large and leads to a prominent deviation of F,, from unity- Fxs = 0.6. The remaining part of the level strength (1 F3,) is transferred mainly to 3p-“d, nd states. Experimentally, in an ( e , 2e) reaction a prominent level was observed (McCarthy and Weigold, 1976) with energy 2.9 Ry, which is interpreted as 3p-*3d. The (e, 2e) cross

COLLECTIVE EFFECTS IN PHOTOIONlZATlON OF ATOMS

47

0.1

FIG.30.

Photoionization cross section of 3 s "shadow" level of argon.

section dependence on the momentum transferred to the atom proves that the angular momentum of the level is zero. Accepting a significant part of 3s level strength, it acts as its shadow, and therefore we call it "shadow" level, 3s. Assuming that its direct excitation in photoionization process is improbable, we? have calculated the cross section of photoionization which leaves the atom in 3p-?3d, i.e., in the 3s state. Neglecting the influence of the 3p subshell dynamic polarization upon the photoionization of 3s or 3s levels, the relation

is obtained. The 3s and 3s thresholds are naturally different. The influence of 3p" electrons prominently alters the relation ( 5 2 ) , just as ma,. the "shadow"-level photoionization cross section, whimsically depends on photon energy, which is illustrated by Fig. 30. The collectivization of 5s subshell in xenon is even stronger. Here FSsis about 0.34 (McCarthy and Weigold, 19761, and the vacancy 5s proves to be mixed not only with nearest states like 5p-'5d but also with comparatively far removed excitations, up to t h e vicinity of the 4d shell, and maybe even further (Giardini-Guidoni pt t i / . , 1979). If collectivization is a result of interaction with continuum excitations (e.g., 5p-?cd in xenon), it is possible that the properties of a "shadow" level will belong to a region of continuous spectrum. In this case a relation between double-electron photoionization cross section and uSs will exist, Obtained together with M . Yu. Kuchiev and S. A. Sheinerman

M . Ya. Amusia

48

analogous to Eq. (52), or analogous to that with the inclusion of outershell influence. The deviation of F from unity affects single-electron photoionization. The inclusion of 2 in the hole line not only leads to a change of ionization potential and thus to a shifting of the curve as a whole, but also multiplies the cross section by F : (T + Frr. The remaining part -- ( 1 - F)rr will describe photoionization cross section of the “shadow,” the existence of which leads to deviation of F from unity. The fact that F for the ns2 subshells of noble gas atoms is much less than unity was not taken into account in the RPAE calculations presented above. Therefore, previously obtained agreement with experiment (see Fig. 8) will be violated. It can be reestablished by correcting, together with hole states or other RPAE elements, the vertex and electron-hole interactions. They are modified due to the linking of at least two channels: “one electron-one hole-(one eh)” and “two electrons-two holes-two eh.” This modification will be considered in the next section. E. INTERACTION

WITH T H E

CHANNEL. TWO ELECTRONS-TWO HOLES

The strong postcollision interaction and variation of the hole structure under the influence of the direct interaction are in fact examples of an interaction between one electron-hole pair (one-eh) and two electronstwo holes (two-eh) channels. This is evident from Figs. 23,26, and 28. The precise accounting of this interaction is very difficult because even the motion of two interacting electrons in the self-consistent field is a threebody problem. It becomes much simpler if only several two-eh configuration contributions dominate. It was mentioned above that the ionization potential of the outer us2 subshells of noble gases is determined mainly by the interaction of / I S - ’ and np-5id states. In this section we shall assume that just this contribution is most important in the variation of both the electron-hole interaction and photovertex (Amusia and Kheifetz, 1981). As an example, the autoionization of 3s-’4p ’ P argon level is considered, the characteristics of which are altered very strongly due to interaction with two-eh configuration. Diagrammatically, the corrections to Coulombic electron-hole interaction between 3s-’4p ‘P discrete level and continuous spectrum excitations 3p + Ed, E S are presented in Fig. 3 1. Figure 31a-d describes the corrections to the interaction between all 3p nd, ~ d ( n sE, S ) states and 3s-’4p ‘P, while Fig. 31e and f corrects the interaction of 3s + 4p transition with 3p .+ 3d only, which is distinguished because of strong interaction of 3s-’ and 3p-’3d states. Analogous diagrams-combination of vertices on the left-hand side of Fig. 31 with the remainder part of Fig. 31 diagrams-are responsible also for the variation of the photon interaction with the level 3s-’4p ‘P.

-

I

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

I

(4

49

(f)

F I L .31. The corrections to electron-hole interaction and to the vertex due to 3p '3d configuration excitation in argon.

The photoionization cross section near an autoionizing level is described by the Fano formula (Fano and Cooper, 1968): (53) where 5 = ( w - G ~ ~ ) / w, t y is , the resonance excitation energy, and y is the resonance width; 4 and p2 are parameters which determine the resonance line, and the form and intensity of the resonance. In Eq. (53) cr is the cross section far from resonance. The parameters in Eq. (53) are determined by the following relations (Amusia and Cherepkov, 1975):

In Eq. (54) ".s" denotes the resonance state 3s-'4p IP and t marks the continuous spectrum 3 p - l ~and ~ 3p-'cd excitations. The amplitudes D,? and D,and matrix elements of the resonance with continuum rYt and the resonance with resonance via continuum Tss effective interaction are determined, respectively, by Eqs. ( 1 1) and (13) in the sums of which t h e term with IZ = 4p and j = 3s is omitted. The summation in Eq. (54) is performed over all those continuum states the transition to which from 3s-'4p IP is allowed by energy, momentum, and spin conservation. The inclusion of Fig. 31 corrections and those of the same kind to the vertex is achieved by substituting in Eqs. ( 1 1 ) and (13), instead o f t / and U , the values 2 and U which differ from d and CJ by the contribution of the

M . Yu. Amusia

50

TABLE I PARAMETERS OF AUTOIONIZATION LEVEL3s-'4p 'P I N ARGON 4

y. lO-'Ry

P'

1.76 2.05 6.23 5.86 0.37

0.78 0.94 0.89 0.86 0.04

1. RPAE:

HF 3s energy Experimental 3s energy 2. RPAE + 2 eh 3 . Experiment (Madden et ul., 1969)

2.13 1.15 -0.27 -0.22 -t 0.05

L

?

interaction with two-eh states (Fig. 31). The intrashell interaction is purely Coulombic in our calculations. The self-energy part 2 is also included in the s hole line and in Z (see Fig. 28a); the contribution of only one configuration 3p-'3d was taken into account. For the 3s level shift from egswe obtain 0.3 Ry instead of the experimental value 0.4 Ry. For the spectroscopic factor we obtain FgS= 0.7 instead of 0.56 in experiment (McCarthy and Weigold, 1976). Because in RPAE and H F the vacancy energies coincide, the RPAE calculation also was performed with 3s energy experimental value. The results are presented in Table I. It is seen that two-eh excitations change y and q very substantially. The interaction with the two-eh configuration is also rather strong in the continuous spectrum region, e.g., in the photoionization of outer ns'subshells of noble gases. It was mentioned in Section V,D that on correcting the hole line "s" by 2 , the photoionization cross section is multiplied by F and therefore becomes smaller. This destroys the coincidence with experiment near n s threshold that was achieved within RPAE. Our preliminary consideration demonstrates that the agreement is restored if the interaction with two-eh configurations is included. If the correction in Fig. 31 to the RPAE Coulombic electron-hole interactions are taken into account, the intershell interaction becomes stronger, which compensates the F factor in the 17s' cross section at the threshold. However, the role of these corrections decreases very fast with increase of w , while the influence of F is energy independent. Therefore, far above the ns' threshold the cross section will be Fnsai:AEinstead of a::"". Apparently, this explains why a::"" in the region of the 4d shell is about two times larger than the experimental value because F 5 , = 0.34 (McCarthy and Weigold, 1976). The purpose of the examples presented is to demonstrate how important it is to complete RPAE by interaction with two-eh configurations. This problem is now at the very beginning of consideration.

COLLECTIVE EFFECTS IN PHOTOIONIZATION OF ATOMS

51

VI. Conclusion In this article a number of collective effects in photoionization were considered, which demonstrate essential deviations from the one-electron HF picture. Without exaggeration it may be said that the role of collective effects in the photoionization and in the decay and rearrangement of vacancies created in photoionization proves to be very important, and sometimes decisive. The photoionization experimental data promote a deeper understanding of the electron structure of complex atoms and help to create comparatively simple methods for their theoretical description, such as RPAE. Now it is necessary to go out of this approximation frame. New experimental data have helped and are helping us now to discover and clarify the role of rearrangement, the influence of the so-called postcollision interaction, and the consequences of the decay and spread of some vacancies, up to their complete collectivization. Taken together, all this must lead to the development of models capable of describing the behavior of manyelectron atoms much better than RPAE does. The inclusion of twoelectron-two-hole states is only the first step in this direction. We have studied the collective effects that are connected with small deviations of electron density from its equilibrium value. Just in this approximation [the coefficients Cmiin Eq. (7) considered to be small!] Eq. ( 1 1 ) was derived, which was used in obtaining almost all results in this article. In this same approximation the nonexistence of plasma-like collective oscillations was demonstrated. However, it does not exclude the possibility of the existence of such atomic excitations, the electron density of which significantly differs from the ground state density. Such excitations might be considered as large-amplitude density oscillations. Their decay may be hampered because of their monopole character or (and) smallness of excitation energy. All this may be correct not only for normal density but to a larger extent for spin density. The large-amplitude spin density oscillations may appear to be much easier because the increase of spin density is not prevented directly by Coulombic interelectron repulsion. Indications of the possible existence of such collective long-lived excitations, which it is proper to call isomer states, are obtained in numerical (Band and Fomichev, 1980) and model (Amusia, 1980) HF calculations. These “isomer” states may be excited in heavy atomic particle collisons and detected using photoionization cross-section peculiarities, in particular, its very low thresholds and rapid decrease with w . The development of methods which take into account collective effects in isolated atoms permits one to begin the consideration of photoionization of more complex objects such as molecules and solid bodies, which, at

M. Ya. Amusia

52

least for intermediate and inner shells, is determined mainly by the constituent atoms.Thus, in molecules it is meaningful to expect in analogous conditions strong intershell correlations, few-electron shell collectivization, and “melting” of some shells. Due to the additional compression of constituent atoms in solid bodies, it is also reasonable to expect there that intershell correlations and collective effects play an even stronger role than in isolated atoms, perhaps with the participation of many shells, even up to the existence, in some cases, of collective oscillations. It is very interesting but difficult to investigate many-electron atom behavior, including ionization, in a strong electrical or magnetic external field. The direct attack by huge computers on such a problem is hardly effective, and some new physical insights are necessry to achieve success in this field.

ACKNOWLEDGMENTS The author thanks his colleagues N. A. Cherepkov, L. V. Chernysheva, V. K. Ivanov, M. Yu. Kuchiev, S. I. Sheftel, and S. A. Sheinerman, together with whom many results presented in this article were obtained.

REFERENCES Adam, M. Y., Wuilleumier, F., Sandner, N., Krummacher, S., Schmidt, V., and Mehlhorn W. (1978). Jpn. J. Appl. Phys. 17, 170. Altick, P. L., and Glassgold, A. E. (1964). Phys. Rev, 133 A632. Amusia, M. Ya. (1974). I n “Vacuum Ultraviolet Radiation Physics” (E.-E. Koch, R. Haensel, and C. Kunz, eds.), p. 205. Pergamon Vieweg. Amusia, M. Ya. (1977). At. Phys. 5, 537. Amusia, M. Ya. (1980). Applied Optics 19, 23, 4042. Amusia, M. Ya. (1981). Comments At. Mol. Phys. X, 4, 155. Amusia, M. Ya., and Cherepkov, N. A. (1975). Cusp Srud. A f . Phys. 5, 47. Amusia, M.Ya., and Ivanov, V. K. (1976). Phys. Lett. 59A, 194. Amusia, M. Ya., and Kheifetz, A. S. (1981). Phys. Lett. 82A, 407. Amusia, M. Ya., Cherepkov, N. A., and Sheftel, S. I. (1967). Phys. Leu. 24A, 541. Amusia, M. Ya., Cherepkov, N. A., and Chernysheva, L. V. (1970). Phys. Lett. 31A, 553. Amusia, M. Ya., Cherepkov, N. A., and Chernysheva, L. V. (1971). Zh. Eksp. Teor. Fiz. 60, 160.

Amusia, M. Ya., Kuchiev. M. Ya., Sheinerman, S.A., and Sheftel, S. 1. (1977). J . Phys. E 10, L535.

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Amusia, M. Ya., Ivanov, V. K., Sheftel, S. I., and Sheinerman, S. A. (1980a). Zh. Eksp. Teor. Fiz. 78, 910. Amusia, M. Ya., Kuchiev, M. Yu., and Sheinerman, S. A. (1980b). I n “Coherence and Correlation in Atomic Collisions” (H. Kleinpoppen and J. F. Williams, eds.), p. 297. Plenum, New York. Amusia, M. Ya., Ivanov, V. K., and Chernysheva, L. V. (1981). J. Phys. E 14, L19. Armstrong L., Jr. (1974). J. Phys. B 7 , 2320. Armstrong L., Jr., and Fielder, W. R., Jr. (1980). P h y s . Scr. 2 I , 3/4, 457. Band, I. M., and Fomichev, V. I. (1980). Phys. Lett. 7SA, 178. Barker, R. B., and Berry, H. W. (1966). Phys. Rev. 151, 14. Berezhko, E. G., Ivanov, V. K., and Kabachnik, N. M . (1978). Phys. Lett. MA, 474. Bersuker, I. B. (1957). Dokl. Akad. Nauk SSSR 113, 5 , 1017. Bloch, F. (1933). Z. Phys. 81, 363. Bohm, D., and Pines, D. (1953). Phys. Rev. 92, 609. Brehm, B., and Hofler, K. (1975). Inr. J . Mass Spectrom. Ion Phys. 17, 371. Bruhn, R., Sonntag, B., and WOE, M. W. (1978). Phys. Lett. A 69A, 9. Burke, P. G. (1978). J . Phys. (Orsny, Fr.) 39, Colloque C4, 27. Chang, T. N. (1977). Phys. Rev. A 15, 2392. Cherepkov, N. A. (1978). Phys. Lett. A 66A. 204. Cherepkov, N. A. (1979). J . Phys. E 12, 8, 1279. Cherepkov, N . A., and Chernysheva, L. V. (1977). Phys. Lett. A HA, 103. Connerade, J. P., Mansfield, M. W. D., and Martin, M. A. P. (1976). Proc. R . Soc. London. Ser. A 350, 405. Connerade, J. P., Rose, S. J., and Grant, I. P. (1979). J . Phys. E 12, L53. Cooper, J. W., and Zare, R. N. (1969). Lect. Theor. Phys. IIC, 317. Dalgaard, E. (1975). J. Phys. B 8, 695. Dalgarno, A., and Victor, G. A. (1966). Proc. R . Soc. London, Ser. A 291, 291. Dehrner, J. L., and Dill, D. (1976). Phys. Rev. Lett. 37, 1049. Dehmer, J. L., Chupka, W. A., Berkowitz, J., and Jivery, W. T. (1975). Phys. Rev. A 12, 1966.

Deslattes, R. D. (1968). Phys. Rev. Lett. 20, 483. Dill, D. (1973). Phys. Rev. A 7 , 1974. Ederer, D. L. (1964). Phys. Rev. Lett. 13, 760. Fano, U., and Cooper, J. W. (1968). Rev. Mod. Phys. 40, 3. Fliigge, S.,Mehlhorn, W., and Schmidt, V. (1972). Phys. Rev. L e f t . 29, 7. Gelius, U. (1974). J. Electron Spectrosc. 5 , 985. Giardini-Guidoni, A,, Fantoni. R., Markonero, R., Camilloni, R., and Stefani, G. Int. Con$ Phys. Electron. A t . Collisions. Ilth, 1979, Book of Abstracts, p. 212. Haensel, R., Keitel, G., Kosuch, N., Nielsen, U., and Schreiber, P. (1971).5. Phys. (Ors(iy. Fr. ) 32, Colloque C4, 236. Heinzmann, U., Schonhense, G., and Kessler, J. (1979). Phys. Rev. Lett. 42, 1603. Hicks, P. J., Cvejanovic. S., Comer, J., Read, F. H., and Sharp, J. M. (1974). Vacuum 24, 573. Holland, D. M. P., Codling, K., and Chamberlain, R. N. (1981). J . Phys. E . 11. 5, 839. Houlgate, R. G., West, J. B., Codling, K., and Marr, G. V. (1974). J . Phys. E 7, 17, L470. Huang, K.-N., Johnson, W. R., and Cheng, K. T. (1979). Phys. Rev. Lett. 43, 1658. Hudson, R. D., and Kieffer, L. J. (1971). A t . Data 2, 205. Johnson, W. R., and Cheng, K. T. (1978). Phys. Rev. Lett. 40, 1167. Johnson, W. R., Lin, C. D., Cheng, K. T., and Lee, C. M. (1980). Phys. Scr. 21, 409. Kelly, H. P. (1971). A t . Phys. 2, 227. Kelly, H. P., and Carter, S. L. (1980). Phys. Scr. 21, 3/4, 448.

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Kirzhnits, D. A., Losovik, Yu. E., and Shpatakovskaya, G . V. (1975). Usp. Fiz. Nuuk 117, 3. Lukirskii, A. P., Brytov, 1. A., and Zimkina, T. M. (1964). O p / . Spektrosk. 17, 438. McCarthy, I. E., and Weigold, E. (1976). A d v . Phys. 25, 5 , 489. McGuire, E . J. (1974). Phys. Rev. A 9, 1840. Madden, R. P., Ederer, D. L., and Codling, K. (1969). Phys. Rev. 177, 136. Manson, S. T., and Cooper, J. W. (1968). Phys. Rev. 165, 126. March, N . H., Young, W. H., and Sampanthar, S.(1967). “The Many-Body Problem in Quantum Mechanics.” Cambridge Univ. Press, London and New York. Marr, G. V., and Austin, J. M. (1972). J. Phys. B 2, 168. Morgenstern, R., Niehaus, A., and Thielmann, U. (1977). J. Phys. B 10, 1039. Ong, W., and Manson, S. T. (1978a). J . Phys. B I I , L65a. Ong, W., and Manson, S. T. (1978b). J. Phys. B I I , L163. Ong, W., and Manson, S. T. (1979). Phys. Rev. A 20, 2364. Rabe, P., Radler, K., and Wolf, H.-W. (1974). I n “Vacuum Ultraviolet Radiation Physics” (E.-E. Koch, R. Haensel, and C. Kunz, eds.). p. 247. Pergamon, Vieweg. Samson, J. A. R., and Gardner, J. L. (1974). Phys. Rev. Lett. 33, 671. Samson, J. A. R., and Haddad, G. N. (1974). Phys. Rev. Lett. 33, 875. Schnopper, K. (1963). Ph,vs. Rev. 131, 2558. Starace, A. F., and Armstrong, L., Jr. (1976). Phys. Rev. A 13, 1850. Starace, A. F., Rast, R. H., and Manson, S. T. (1977). Phys. Rev. Let/. 38, 26, 1522. Thouless, D. J. (1961). “The Quantum Mechanics of Many-Body Systems.” Academic Press, New York. Torop, L., Morton, J., and West, J. B. (1976). J. Phys. B 9, 2035. Van der Wiel, M. J., and Chang, T. N. (1978). J. Phys. B 11, L125. Van der Wiel, M. J., and Wight, G. R. (1975). Phys. Leu. 54A, 83. Van der Wiel, M. J., Wight, G. R., and Tol, R. R. (1976). J . Phys. B 9, L5. Walecka, J. D. (1976). Phys. Lett. A 58A, 83. Wendin, G. (1971). J . Phys. B 4, 1080. Wendin, G . (1974). In “Vacuum Ultraviolet Radiation Physics” (E.-E. Koch, R. Haensel, and C. Kunz, eds.), p. 225. Pergamon Vieweg. Wendin, G., and Ohno. M. (1976). Phys. Scr. 14, 148. West, J. B., Woodruff, P. R., Codling, K., and Houlgate, R. G. (1976). J . Phys. B 9, 407. White, M. G., Southworth, S. H., Kobrin, E. D., Poliakoff, E . D., Rosenberg, R. A., and Shirley, D. A. (1979). Phys. Rev. Lett. 43, 22, 1661. Wuilleumier, F., Adam, M. Y., Dhez, P., Sandner. N., Schmidt. V., and Mehlhorn, W. (1977). Phys. Rev. A 16, 646.

I

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 17

NONADIABATIC CHARGE TRANSFER D. S.F. CROTHERS Depurtment of Applied Mathematics and Theoretical Physics The Queen’s University of Belfast Belfast. Northern Ireland

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Background . . . . . . . . . . . . . . . . . . . . . . B. Specific Background . . . . . . . . . . . . . . . . . . . . . . XI. Phase Integrals and Comparison Equations . . . . . . . . . . . . . A. Stueckelberg’s Matrix . . . . . . . . . . . . . . . . . . . . . B. Exponential Model and Applications . . . . . . . . . . . . . . . 111. Perturbed Stationary States and Electronic Translation . . . . . . . . A. Homonuclear Collisions . . . . . . . . . . . . . . . . . . . . B. Heteronuclear Collisions . . . . . . . . . . . . . . . . . . . . IV. Nonmolecular Three-Body Analysis . . . . . . . . . . . . . . . . . V.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 63 63 78 83 83 88 91 93 93

I. Introduction A. GENERAL BACKGROUND Charge transfer is a fundamental and fascinating three-body process, with many and varied applications, whether in, for instance, man-made tokamaks (cf. Gilbody, 1979) or in the naturally occurring phenomena throughout the universe, including the upper atmosphere (Bates, 1978a), interstellar clouds (Black and Dalgarno, 1977), and astrophysical plasmas (cf. Butler rt al., 1979). It concerns the transfer of one or more electrons between two colliding heavy particles. In this article we shall review some of the principal theoretical models and shall assume that each heavy particle is an ion o r an atom, that the impact energy is nonrelativisitic, and that the electron is confined to outer shells. 55 Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN C-IZM)3817-X

56

D. S. F. Crothers

As a fundamental process, charge transfer is clearly important whether the target is being neutralized or being stripped. Of considerable and recent interest is the question of scaling of cross sections with respect to projectile charge (cf. Olson, 1979); we shall return to this in Section IV. It is also a fascinating process, in that it is particularly amenable to analytical methods, provided that the variation of the Massey parameter (Massey, 1949; Bates, 1978b; Shushin, 1978) is such as to ensure well-localized, nonadiabatic transition regions. We shall review these methods in Section I1 with particular reference to phase integrals and comparison equations. Previous reviews relevant to this topic are Child (1974, 1978, 1979), Janev (1976), and Nikitin (1968a, 1970, 1974) on nonadiabatic transitions and Berry and Mount (1972) and Olver (1978), not to mention the texts of Froman and Froman (1965) and Heading (1962), on phase integral and semiclassical methods, which have been applied by, for instance, Delos (1974), Burns and Crothers (1976), and Barany (1979). At low velocities, whether or not the transitions are nonadiabatic the appropriate method is that of perturbed stationary states with all its difficulties. We shall review these in Section 111, including both homonuclear and heteronuclear collisions. Recent reviews include those of Briggs (1976), with special reference to inner-shell processes, and McCarroll (19761, with special reference to thermal energies. At higher velocities, a variety of methods have been employed, including Oppenheimer-Brinkman-Kramer (OBK), first, second, and higher Born, distortion, distorted wave, Glauber, eikonal, impulse, continuum distorted wave, continuum intermediate states, Fadeev, second-order potential, classical, and refined-orthogonal close coupling of a linear combination of atomic orbitals (LCAO), together with a number of hybrids including pseudostates. In particular, it is well known that at high velocities, the minimum requirement is a second-order theory and the reader’s attention is drawn to two recent and excellent review articles (Belkic et af., 1979; Shakeshaft and Spruch, 1979). We shall review in Section IV the continuum distorted wave method. On a more general level, the reviews of Moiseiwitsch (1977) and Bates and McCarroll (1962) are influential, the latter being devoted entirely to charge transfer. Of course, in general it is not possible to isolate charge transfer and simultaneously ignore elastic scattering, excitation, or ionization.

B. SPECIFIC BACKGROUND Some preliminary mathematics is required. If we are to treat differential and total cross sections with equal facility, then a quantal wave treatment

NONADIABATIC CHARGE TRANSFER

57

of the heavy-particle relative motion is necessary. On the other hand, the calculation of transition amplitudes is greatly facilitated by the assumption of a classical path, in that first-order equations then replace second-order equations. Our opinion is that this is best achieved via the semiclassical JWKB treatment of Bates and Crothers (1970) with its fundamental strategy of forcing a common classical turning point, a minimum requirement for a classical path treatment. A similar treatment has been given by Gaussorgues et ul. (1979, as reviewed by McCarroll (1976), albeit within an adiabatic representation which is not Galilean invariant (cf. Section 111). For simplicity, let us assume a diabatic representation, by which we mean a description of the internal motion of the electron in terms of separated-atom wave functions. This is not unreasonable even at low velocities, provided that the nonadiabatic transitions occur at sufficiently large internuclear separations. More general definitions of “diabatic” are, of course, possible (Smith, 1969a; O’Malley, 1971; Nikulin and Guschina, 1978; Cimiraglia and Persico, 1979; Heil and Dalgarno, 1979; Delos and Thorson, 1979). Again for simplicity and for the time being let us restrict attention to two s states and let us postpone the question of electronic translational effects and nonorthogonality (Bates and McCarroll, 1962) and non-Hermitean behavior (Crothers, 1979b) to Sections I1 and 111. Then following a partial-wave analysis the two-state coupled wave equations for the radial-wave amplitudes uJ2are given, as for excitation, by (Mott and Massey, 1965) (1)

where K J ( R )= k j ( m ) - Ujj(R) - ( I

= kj(R) - ( I

+

t)2/R2

+ ))‘/R2

(3) ( 4)

Here we have already incorporated the Langer modification in anticipation and the differential cross sections may be expressed in terms of al and P I , where U l m

= UZl(0) =

0

(5)

D . S. F. Crothers

58

It is tempting to plunge straight into a semiclassical JWKB solution, treating Ulz = U2, as a perturbation in a variation of constants treatment (Bates and Crothers, 1968; Berson, 1968). However, this involves the neglect of transitions between the inner and outer classical turning points ( R I l and R 2 , , where K ~ ( R , = [ ) 0), where one function is exponentially decreasing and the other is highly oscillatory (Eu, 1971). Clearly if RIland R z l differ at all substantially, this is a dangerous procedure, and indeed Bates and Crothers (1968) obtained errors of 100% in the total cross section, albeit within a distorted-wave context. By forcing a common classical turning point on the semiclassical diabatic JWKB functions given by Rot the outermost zero of R2k,(R)kz(R)- (1 +

4)'

(8)

Bates and Crothers (1970) were able to obtain generalized impact parameter equations for the state amplitudes c, given by

where =

l,"

-

I-'"

( I + t)' R2k1(R)kz(R)

dR

according as we are on the ingoing or outgoing half of the trajectory and where a tilde distinguishes a dummy variable of integration. By analogy with purely classical motion, the impact parameter p is given by P 2 k * ( W ) k 2 ( 4 = (1

+ 4)'

(12)

that is (Bates and Sprevak, 1970), P2 =

PlP2

(13)

If the energy is sufficiently high, we may set

k,(R) = kz(R) = Mu

(14)

where M is the reduced mass and u the impact velocity, except where their difference occurs. We may then deduce the familiar straight-line impact parameter diabatic equations

59

NONADIABATIC CHARGE TRANSFER

where c 2 ( - = ) is zero and where

R'

= p? + U't'

(17)

and

H

=

(2A4-W

+ diag(E,(m), E z ( m ) )

(18)

The method of Bates and Crothers then gives the differential cross sections for a scattering angle t), namely, vlm= [4k:(m)]-'

I

x(21+ l)(Ylm - tilm)P1(cost))l'

(rn

=

I , 2) (19)

l:o

where

Ylm = exp{i(q;1 + q$,)}Sfm qj' = -

s:,

=

Jam

(20)

{ H j , - E,( +=)} di

(21) (22)

cm(+m)/cl(-=)

Naturally the usual rotation-matrix transformation (Stueckelberg, 1932) may be performed to obtain the standard straight-line impact parameter adiabatic equations

where T is the Stueckelberg variable given by

T

= (Hzz

- H11)/(2H12)

However, we now have (cf. Crothers, 1978b)

where the Em are the adiabatic energies, given by 2Em =

H22

+ H1l + (-

I)."[4H:, +

(HP2-

H11)']'r2

(27)

Thus the differential cross sections are invariant (Delos and Thorson, 1972c) in that now, with a z ( - m ) = 0, 9 1 ,

qy =

s:l,

+ q",ls:'m

(28)

{Ej - E , ( 4 ) d ?

(29)

= ew{i(q; -

= u,(+=)/a,(-m)

(30)

D . S. F. Crothers

60

Unfortunately this formal invariance does not carry over to the generalized impact parameter equations, namely, the diabatic equations (9) and (10) and the adiabatic equations:

where 2&(R)

=

k:(R) + &(R)

* {(ki(R)- / c ~ ( R +) ) 4U’ ~ }’” 12

(33)

These equations may be deduced from Eqs. (8) and (9) of Crothers (1976). A riiinimirm requirement for formal invariance would be that the arithmetic and geometric means of k,(R) and k2(R)are equal at all R (cf. some remarks of Delos and Thorson, 1972b). This is unlikely to be the case at low energies, particularly if the diabatic diagonal potentials differ substantially in Coulombic charge (Bates and Crothers, 1970). The question is: Which is the more reliable? The answer would appear to be the adiabatic formulation since it treats all three potentials ( U l l , U Z 2 ,UI2)on an equal footing (Stueckelberg, 1932; Crothers, 1971; McLafferty and George, 1975) and since it more readily generalizes to adiabatic quantities (Crothers, 1971, cf. 3.2), which in principle are exact but in any case are amenable to variational calculation. In fact, Bates and Sprevak (1970) had already extended the forced common turning point method to the question of rotational coupling between perturbed stationary states in homonuclear collisions and in an adiabatic representation which is Galilean invariant following Bates and McCarroll (1962) and Bates and Williams (1964). Now although Delos and Thorson (1972b) agree that the Bates-Crothers classical trajectory is useful, Delos et (11. (1972) have expressed the critical opinion that the corrections so induced, relative to any other reasonable trajectory, are either unimportant (and therefore unnecessary) or else invalid. However, it is worth pointing out that the relative error made by Bates and Crothers in approximating Eqs. (1) and (2) is of the order of h / ( M u 2 ) ) no , error having been made in - K : ~and where A€ is the maximum classically accessible potential difference. It is worth stressing that because of the nature of expression (8), the forced common turning point ROlis always greater than any zero of kl(R)and k,(R), so that the effective velocity u in Ae/(Mu2)cannot approach zero. This means that, at the low velocities at which these classical trajectories are applied, the error is considerably less than I ( K ~ -~ K ~ I ) / ( K+~ K~ I J 1 (Delos and Thor-

61

NONADIABATIC CHARGE TRANSFER

son, 1972b): even though the latter is of order he/(Mv2),the effective velocity can become very small, particularly if the tw-ning points are close. Moreover, Delos and Thorson (1972b) regarded the evidence of Bates and Crothers as tenuous, since their calculation involved a crossing problem. This is surprising in view of the latter’s detailed analysis for U I 2= 2M exp(-2R). In any case, Olson has frequently applied the method successfully to noncrossing problems (cf. Olson, 1972; Olson and Smith, 1973). Moreover, due to the failure of the rudimentary formulas of Landau (1932b) and of Zener (1932) for curve crossings and of Demkov (1964) for noncrossings, at and beyond certain critical impact parameters, there has developed what might be termed a predeliction with the so-called close curve crossing problem of BNO (Bykhovskii rt al., 1965), and DT (Delos and Thorson, 1972a,c) interesting though these models are. As we shall see in Section 11, curve fitting at the close curve crossing is not generally a good approximation: it is much more important to let the double Stokes line joining the complex transition points bend into the classically accessible region on the real internuclear axis and to maintain analytical accuracy along that Stokes line. A small error in the location of the classical turning point, as in the above classical trajectory model, is therefore not quite the catastrophe (Thorn, 1975; Berry et ul., 1979) it might otherwise appear in relation to the BNO-DT models. However, this is hardly to say that the (radial) turning point(s) could be removed (Child, 1974, p. 178) for practical calculations. It is also sometimes thought (cf. Gaussorgues et ul., 1975) that Eqs. (9)-(12) do not generalize to three or more states. Although no calculations have been performed to my knowledge, a possible generalization to three s states is

+

+

1

Z

c3 exp[ - i 2k]pL(R)kA’2(R) U13(R)

(k,

j (k,

-

k3) d Z }

(35)

-

k3) d Z }

(36)

z

U23(R)

2 k:“ ( R )kAiL( R )

c3 exp[ - i

D. S. F. Crothrrs

62 where

This formulation has the correct limits, whether elastic ( k , = k2 = k3) or two state (k3 += +to), and is subject to a relative error similar to that for Eqs. (9)-(12). For a generalization to non-s states a more sophisticated partial-wave analysis is required (Gaussorgues er al., 1975), and indeed for a generalization to radial and rotational couplings between perturbed stationary states in an adiabatic representation, whether for homonuclear (Crothers and Hughes, 1978, 1979a,b,c) or heteronuclear (Crothers and Todd, 198la,b) collisions, a consideration of translation factors is also required (Thorson and Delos, 1978a,b) if Galilean invariance and accuracy are to be preserved. Nevertheless, the concept of a classical trajectory with forced common turning point carries through in all these cases. To finalize the preliminaries, we note that Eqs. (15) and (16) and Eqs. (23) and (24) may be written respectively as

(

. dc, - - c2exp - 2 ;

I

ds

L8

T ( s ' )ds'

]

(39)

where, as before, T is the Stueckelberg variable given by Eq. (25) and where the new independent variables (Delos and Thorson, 1972c),with or without the prime, is given by s =

J,: HI2(r')dt'

(43)

It should be noted that this reduction in the number of dependent variables is quite general and not in any way dependent upon any particular or special model, such as the BNO or DT models of close curve crossing mentioned above. The only obvious constraint is that the t + s mapping should be isomorphic, the usual and sufficient assumption being that H 1 2 does not change its sign. Moreover. it is possible to parameterize the S

NONADIABATIC CHARGE TRANSFER

63

matrix of Eq. (20) in terms of three real numbers (Bateset ul., 1964; Delos and Thorson, 1972c; Barany, 1979), for instance,

s = G-G-

(44)

where

subject to

IGii(0, --M)/' + JGz,(O, -m)1'

= 1

(46)

and where G,,(O, -m) and G2,(0, -m) are the complex transition amplitudes c,(O) and c2(0) obtained by integrating inwards either numerically or analytically to the turning point from the initial values c , ( - m ) = 1, cz(-") = 0. In the context, of course, - m refers strictly to the time variable since s(--x) is negative but finite and since 7Y-m) may be k x .

11. Phase Integrals and Comparison Equations A . STUECKELBERG'S MATRIX

Independently, Landau (1932b) using an adiabatic impact parameter treatment, Zener (1932) using a diabatic impact parameter treatment, and Stueckelberg ( 1932) using a JWKB semiclassical phase integral treatment all examined strong coupling in heavy-particle collisions. The first two authors concentrated on the effect of pseudocrossings of potential-energy curves on total cross sections. Stueckelberg, on the other hand, followed Landau (19324 and attacked the case of accidental adiabatic curve crossings as well as diabatic noncrossings. The power of Stueckelberg's formulation is that it readily provides accurate differential cross sections, including quantal interference effects associated with competing classical trajectories. Such effects have been rightly termed Stueckelberg oscillations (Smith, 1969b). Crothers (1971) critically reviewed the ZwaanStueckelberg method and applied it to pseudocrossings, noncrossings, and rotational coupling. The method itself involves the analytic continuation and tracing of phase integrals in the complex internuclear separation ( R ) plane. However, unlike Zwaan (1929), who apparently initiated the method, Stueckelberg appealed only to the Stokes phenomenon, analyticity, unitarity, and symmetry. By the Stokes phenomenon, we mean that

64

D. S. F. Crothvrs

upon crossing a Stokes line (on which only exponentially dominant and exponentially subdominant phase integrals are defined) the coefficient of the subdominant term suffers a discontinuity of a magnitude proportional to the coefficient of the dominant term. A corollary is that a subdominant term is only continuous in crossing a Stokes line when the dominant term is absent. Although branch cuts are inevitably involved, it is worth pointing out that the Stokes phenomenon is not specifically associated with branch cuts as such but rather with the nature and meaning of asymptotic expansions (Heading, 1962; Froman and Froman, 1965; Dingle, 1973; Olver, 1974, 1975b). In particular, Stueckelberg’s choice of branch cut, intersecting the real axis and repeated by Dubrovskiy and FischerHjalmars (1974, 1979), is not convenient for the comparison equation or variation of constants method, which is essential for a parameterization of the phases of the Stokes constants. In summary, Crothers (1971) found the two-state scatterings matrix for an isolated pseudocrossing to be given, within the Stueckelberg phase integral approximation, by a full wave treatment generalization of Eq. (28), namely,

Here the adiabatic wave numbers are given by

+ &)‘/R2

(52)

+ 2M(~5(03)- Ej(R))

(53)

u ~ , ( R=) $ ( R ) - ( I v ~ ( R=) vj(w)

where M is the reduced mass, 1 the azimuthal quantum number associated with the relative motion, ( l / Z M ) v ? ( x ) the impact energy, and AE = c2(m) - el(m) = ( I /2W( v:(m) - v:(m)) the excitation energy. The Langer correction is included and the adiabatic eigenenergies are assumed to have square root branch point degeneracy at Rc and R,$. The latter complex transition points are assumed to lie in the first and fourth quadrants of the complex R plane, respectively, and to lie adjacent to the real avoidedcrossing point. The classical turning or transition points associated with the

NONADIABATIC CHARGE TRANSFER

65

vjI are labeled R5,. The four branch cuts (strictly, we should count those from Rc and R$ as one passing through x ) emanating from the four transition points are assumed to recede to 30 away from the classically accessible portion of the real R axis, and arg vII and arg ( v l l - v z I ) are all taken to be zero at +m. The complex contour integrals in Eq. (51) are assumed to avoid the branch cuts so that with the above conventions the phase lag u and the Stueckelberg-Nikitin nonadiabatic parameter 6 are positive. In general the real transition point is not given by the avoided-crossing point but more precisely by R x , where u is given by Eq. (51) and where

In effect Rx lies on the intersection of the double Stokes line (ioining Rc and R F ) with the real axis, for, of course, Rc and RF each lie on one of the other’s three Stokes lines. In general, it should be noted that u includes an off-the-axis contribution, and particularly as / increases, the double Stokes line is likely to bend convexly toward + m so that Rx moves farther and farther beyond the avoided-crossing point, and u remains reasonably large and does not tend to zero, as is usually assumed (Child, 1978, 1979). Thus, even when min{Rll, R z I }3 Re R c , the so-called close curve-crossing regime (Bykhovskii et a/., 1965; Delos and Thorson, 1972a,c; Child, 1974), the phase integral method gives unequivocally a well-defined transition point and a nonzero transition probability. Moreover, the idea (Miller and George, 1972) that the transition actually occurs at Rc (or R $) is somewhat restrictive and misleading. It is worth observing also that the concept of tunneling (cf. Olson ef al., 1971 ; Nikitin and Reznikov, 1972; Nikitin and Ovchinnikova, 1972; Gaussorgues, 1973; Ovchinnikova, 1973; Shushin, 1976; Bandrauk and Miller, 1979) in relation to the curve-crossing problem is not a useful one. Obviously such a concept is essential in relation to one or more real transition-classical turning points. However, when the transition points are complex and the Stokes line bends, the concept of tunneling is unnecessary and indeed unduly restrictive in that it leads to Airy functions fitted at the crossing and decaying exponentially and therefore rapidly (Nikitin, 1961; Bykhovskii et a/., 1965; Kotova, 1969; Ovchinnikova, 1971; Kotova and Ovchinnikova, 1971; Nikitin and Ovchinnikova, 1972; Nikitin and Reznikov, 1972; Elisenkov and Nikulin, 1973; Saxon and Olson, 1975; Child, 1979). A further advantage of the phase integral method is that it firms up the rather loose concept of the Massey parameter since it may be closely identified with the nonadiabatic parameter 6 of Eq. (51). Thus, if 6 is too small, the collision is no longer nonadiabatic, but simply diabatic in the

D. S . F. Crothers

66

true sense of the word. If 6 is too large, the collision simply evolves adiabatically in the absence of other nonadiabatic regions. Of course, the smaller the effective adiabatic term difference, the smaller must be the impact velocity for an effective 6 and an effective collision, which accounts for the success of the Nikitin school in applying the method to fine-structure transitions at thermal and electron volt energies (cf. Nikitin, 1965, 1974; Dashevskaya et al., 1969; Zembekov and Nikitin, 1971; Gordeev et ul., 1977; Nikitin and Reznikov, 1980). At the other end of the scale, when velocities are quite large, the collision may still be nonadiabatic, provided that the term difference is correspondingly large and u is not unduly small. Returning to our basic Eq. (47)-(53) it is worth remarking that the phase integral method, whether starting from an adiabatic or from a LCAO diabatic formulation (Crothers, 197l), consistently and necessarily parameterizes the S matrix in terms of adiabatic quantities. Of course, it is possible to factorize the S matrix, using Stueckelberg’s “nonphysical” branch cut, which crosses the real axis. The difficulty is that such a factorization necessarily involves not diabatic phase shifts but pseudoadiabatic phase shifts, corresponding to curves which follow one adiabatic curve up to the real transition point and then jump discontinuously to follow the other adiabatic curve. It is sometimes thought that this is splitting hairs. Indeed, it is sometimes stated (Child, 1978; Barany, 1979) that the phase lag u may be determined in a mixed diabatic-adiabatic formulation thus, for instance:

where the superscript d clearly indicates “diabatic.” This is simply not true in general, as the following counterexample proves: consider the exponential model (Nikitin, 1962a,b; Bayfield et ul., 1973; Crothers, 1978b) and in particular consider the LCAO straight-line impact parameter approximation at zero impact parameter. Then 6 is given by 35, where 5 [cf. Eq. (26)] is given by expressions (56) and (57) of Crothers (1978b), thus: Cr= &YCOS 19- & A + - f h cos I3 -

;A In[(s

In[(s

+

)S

+y

-

A cos O ) / { A ( I

A - y cos O ) / ( 2 A)]

-

cos d ) } l (56)

where s = { y2 + A’ - 2 h y cos d } and where the parameters are defined in Eqs. (103),(104), (110), and (127). On the other hand the exact u is given by

NONADIABATIC CHARGE TRANSFER (7

=

s -

+

A In{(s

+

y cos H

+

A cos H In{(.s

y

-

--

67

h ) / ( ysin 0))

A cos O)/( A sin 0))

(57)

A ln(y/h)

(58)

Of course, equality does hold, that is, cr = ir = y

-

A

-

in the Landau-Zener linear model limit, namely, in the limit as y and A + x and H -+ 0 in such a way that A( 1 - cos 0) remains finite. Clearly Eqs. (56) and (57) otherwise and quite generally disagree by terms of order 1/21 = M / v l ( m ) , and if such a discrepancy exists for the exponential model, discrepancies can hardly be expected to be absent for other realistic potentials. Of course, more generally an impact parameter interpretation of our Eqs. (471453) will hardly itself be reliable at low velocities, particularly for the elastic phase shifts [Eq. (48)]. The principal defect of the phase integral method, however, is its inability to specify the phase (o in Eqs. (49) and (50). In short, it is known to be an indeterminate phase of a Stokes constant (Heading, 1962; Crothers, 1971; Thorson et u / . , 1971). Within the parabolic model (Crothers, 1971, 1972, 1975, 1976) it is determined by ( ~ ( 6=) 7 ~ / 4- 6/n-

+

( S / x ) ln(S/n)

-

arg r ( l + iS/.rr)

(59)

This is outside the realm of the strict phase integral method; rather it lies in the realm of the comparison equation method, whereby the general two-transition point problem is compared in its essence with the special two-transition point problem covered by the parabolic cylinder functions. This aspect has been well reviewed by Crothers (1975) and Barany (1978, 1980). The principal difference between the parabolic model of Crothers (1971, 1972, 1975, 1976) and that of other authors (Zener, 1932: Dubrovskiy, 1964a; Kotova, 1969; Bandrauk and Child, 1970; Child, 1971; Dubrovskiy and Fischer-Hjalmars, 1974) lies in the nature of the asymptotic expansions of the parabolic cylinder functions. Thus, the latter use weak-coupling asymptotic expansions for the D J z ) for which it is assumed that

I4 >> max( 1, ]PI)

(60)

which in terms of the BNO-DT model parameters /3 and E [see Eq. (62)] corresponds in general to 161- >> 1 and /3 k 1 and hence distant crossings, that is, small impact parameters. The former, on the other hand, uses strong-coupling asymptotic expansions for the D,(z) (Crothers, 1972, 1978a, 1979a) for which it is assumed that IZP

- IPI >> 1

(61)

D. S.F. Crothers

68

-

which corresponds in general to ( € 1 I and 1/31 >> 1 and hence to close crossings, that is, impact parameters close to the avoided crossing distance. This would appear to have been misunderstood (Dubrovskiy and Fischer-Hjalmars, 19791, particularly since cp given by Eq. (59) was found by Crothers and Hughes (1977) to give excellent agreement with exact numerical results (Delos and Thorson, 1972c) and noting that cp accounts for most of the discrepancy between u and fi in Fig. 4 of Barany (1979). A recent review of Crothers (1971) by Bandrauk and Miller (1979) unfortunately is restricted to classically accessible diabatic crossings since, as pointed out by Barany and Crothers (19811, the Landau-Zener interpretation of 6 in Eq. (59) for q is inadequate for E < 0. The further advantage of the phase cp [given by Eqs. (51) and (5911 is that it uniformly interpolates between the weak-coupling and strong-coupling limits. Thus, the strongcoupling derivation extends the validity of cp over a wider range of / values, even though the independent fundamental solutions (Heading, 1962)change their algebraic form as the condition (60) gives way to condition (61). In particular, strong coupling expansions are inherently more difficult because they generally have, unlike weak coupling expansions, large and small terms even on an anti-Stokes line (Crothers, 1979a). Returning to the exact analytical evaluation of a and 6 for the BNO-DT models, in which Eqs. (39) and (40) hold with the Stueckelberg variable given by T(s) =

-

E

+ 4s2//32

(62)

we note that several expressions have been given. Crothers and Hughes (1977) obtained a

+ i 6 = /3 e x p ( ~ i / 4 )I ( - i ~ )I ( + i ~ ) " ?

(- 1/2,

x

1/2; 2;

(I -iE) -

Barany (1978, 1980) obtained a + i6=

$p(1 + €')"'"[{(I

+ i{(l

-

-

kZ)K(k) + (2k2

K')K(/;.) + (2k2 - l)E(K)}I

where k

=

-

[+ +

+

E2)-I/Z]l/2

I)E(k)}

69

NONADIABATIC CHARGE TRANSFER /$

+

= [& -

(67)

E2)-191!*

and where K and E are complete elliptic integrals of the first and second kind; and finally Bandrauk and Miller (1979) obtained

314, -114; 312; 9 €- + I

The fact is that Eqs. (63)-(65) and (69) are all correct, being analytical continuations of each other, and involving always two independent hypergeometric evaluations for each of u and 6. For the evaluation of both, Eqs. (63), (64) and (69) require two whereas Eq. (65) requires four. However, Eq. (65) is particularly neat, depending as it does (A. BArany, private communication) on the following representation: U(E) =

6 ( - ~ )= ( P / 2 f l )

,/

(1+<2)'/2

-t

ds(I

+ 'E

+

- S ~ ) ' ~ ~ / ( SE )"'

(70)

The trick for reduction to Eq. (65) is the rationalization of the numerator and removal of the quadratic term by forming an exact differential of the cubic under the square root sign in the denominator. The deduction of Eq. (68) from (63) by Bandrauk and Miller appears to depend upon the application of two quadratic transformations, namely, Eq. (15.3.28), followed by Eq. (15.3.261, both of Ambramowitz and Stegun (1970). Thus, it is not surprising that Eq. (68) is incorrect. The reason is that the intermediate representation v + is

=

$[(1

+

€')(I

+

i ) P ~ r ] ~ F ~ ( 15/4; / 4 . 2; 1

+ e2)

(71)

does not converge and so is not defined. More especially the conditions under which quadratic transformations apply are thus not fulfilled (cf. Bateman Manuscript Project, 1954, p. 65). Thus, neither quantum nor semiclassical mechanics can possibly yield Eq. (68). Before leaving the BNO-DT model, a final word of warning: if fl and E are both small, especially P, it is unlikely that the collision will be nonadiabatic. It is then likely that transitions ensue at all internuclear separations (Bates, 1960) so that in a real problem curve fitting of the matrix elements at specific turning or crossing points is unrealistic.

D . S. F. Crothers

70

Let us now anticipate to some extent the exponential model of the following section and consider the closely related model He+-Ne problem of Olson and Smith (1971), albeit adopting the model parameter values of Cho and Eu (1974): H2,

=

(al/R

Hll

=

(al/R) exp(-R/a,)

-

ug)exp(-R/u,)

HI, = a4 exp(-R/ad

+

AE

(72) (73) (74)

where in atomic units a, = 21.1, a2 = 0.678, a3 = 12.1, a5 = 0.667 and AE = 0.61742, while,?, I , and M take the values 2.606 (70.9 eV), 310, and 6089, respectively. The fact that a, # a5 causes a small variation on the exponential model. The results of Cho and Eu (1975) and our results for transition probabilities p12(= ~S12[2) using Eq. (52) are compared in Table I as a function of a4. It is worth mentioning that the computations took less than 1 sec and that the path integrals in Eq. (51) were evaluated along straight lines connecting Rjl and Rc. The table clearly demonstrates that for this close curve-crossing collision, the full Stueckelberg formula gives the maximum at a4 = 1.8 considerably more accurately than the Magnus approximation of Cho and Eu (19751, with or without the stationary phase approximation. The other turning values at a4 = 0.6 and 1.4 are admittedly not quite as good as the full Magnus approximation but are marginally better than the stationary-phase Magnus approximation. We conclude that the full Stueckelberg model of Crothers (1971) competes very favorably in this sensitive test and that the naive Landau-ZenerStueckelberg approach is misleading in that it does not permit the Stokes line to bend. This is an important conclusion in that having severely criticized the Stueckelberg approach (Eu, 1971), a considerable number of articles on the uniform WKB and Magnus approximations has been published(cf. Eu, 1972, 1973a, b; Eu and Tsien, 1972, 1973, 1974; Cho and Eu, 1974, 1975, 1976; Ouerin et a/., 1974; Eu and Guerin, 1974; Eu and Zaritsky, 1978, 1979). This is not intended to denigrate the value of this work, merely to do justice- to the Stueckelberg approach: thus, the criticism of Eu (1971) is misdirected in that, although the zero-order Stueckelberg semiclassical JWKB functions are not singular at the complex transition points, their first-order perturbation corrections are (Crothers, 197 1 , cf. 3.2)! This is analogous with the case of the single real transition point and does necessitate the tracing of the semiclassical JWKB functions in the complex plane to avoid close proximity to the complex transition points (Thorson et af., 1971). Moreover, the conclusion of Child (1978), that Eu and Tsien (1972) had clearly demonstrated the inadequacy of the Stueckelberg-Landau-Zener model at general interaction strengths in the

71

NONADIABATIC CHARGE TRANSFER TABLE I COMPARISON OF T R A N S I T I O PN R O B A B I L I TpI1E2S= ISI2l2B Y V A R I O UMETHODS S

u4 (a.u.)

Exact"

0.01 0.1

0.0008 0.0732 0.2704 0.5320 0.7813 0.9494 0.9958 0.9177 0.7458 0.5301 0.3219

0.2 0.3 0.4 0.5

0.6

0.7 0.8 0.9 1 .O

1.2 1.4 1.6 1.8 2.0 2.2 2.4 "

"

0.0551

O.OOO6 0.0249 0.0292 0.0127 0.0015 0.0001

UWKB exact" 0.0

0.0861 0.2881 0.5439 0.7859 0.9501 0 9948 0.9163 0.7446 0.5292 0.3218 0.0548 0.0007 0.02% 0.0288 0.0121 0.0013 0.0002

U WKB modified Magnus"

0.0003 0.0832 0.2852 0.5426 0.7881 0.9531 0.9985 0.9207 0.7519 0.5415 0.3375 0.0651 0.0005 0.0391 0.0642 0.0439 0.0121 0.0

UWKB modified Magnusd stationary phase

0.0008 0.0761 0.2783 0.5407 0.7843 0.9435 0.9828 0.9029 0.7342 0.5234 0.3185 0.0517 0.0026 0.0044 0.0587 0.0349 0.0192 0.0052

LZS'

LZS '

0.0002 0.0211 0.0977 0.2429 0.4337 0.6099 0.7069 0.6948 0.5909 0.4414 0.2921 0.0926 0.0192 0.0025 0.0002

0.0009 0.0871 0.3134 0.5943 0.8380 0.9782 0.9884 0.8803 0.6928 0.4767 0.2786 0.0405 0.0022 0.0282 0.0298 0.0120 0.0012 0.0002

0.0 0.0 0.0

Numerical solution of second-order coupled equations (Cho and Eu, 1975). Numerical solution of uniform WKB first-order coupled equations of Eu (1971,1972,

1 973a,b) .

' The modified Magnus approximation of Cho and Eu (1974,1975). As for c but with stationary phase approximation (Cho and Eu, 1975). ' The naive Landau-Zener-Stueckelberg theory as applied by Cho and Eu (1975). 'The full Stueckelberg theory of Crothers (1971).

"

threshold region, should also be interpreted in the light of Table I. It would therefore be interesting to apply the full Stueckelberg treatment to the three-state He+-Ne model of Cho and Eu ( 1976) and of Laing and George (1977) since their semiclassical treatments predict transition probabilities which behave erratically in the threshold region [see Figs. 3 and 4 of Laing and George (197711. There arises the natural question: What is the source of error in the full Stueckelberg model of Crothers (1971)? First, by inspection of the second and seventh columns of Table I we are only considering an error of 0.06 at most and even then only for weak coupling, that is, small u4. Second, the error is not due to any correction to the parabolic model, arising from the

D . S. F . Crothers

72

exponential model, in that for this model problem of Olson and Smith the mixing angle 0 is very small, being given approximately by tan-’(2u4/u,). Third and most importantly it might reasonably be assumed that, given u and 6 are both small, the remedy lies in a uniform generalization of the parabolic model in the sense of Barany (1978, 1980), Connor (1981), and Connor and Farrelly (1981). However, the intelligent application of Barany’s uniformisation to allow for variations from the BNO-DT models and for cp # n/4 (which is certainly the case, except for u4 = 0.01) shows little change for u4 = 0.3. Thus, replacing sinyu + p) by

produces a (slightly worse) value of 0.605. Incidentally replacing 6 by has certainly no affect on either p or p12since 6 is small and u is positive. This conclusion appears to remain true even if one uniformly generalizes in terms of

rather than in terms of u appears to be

=

T1 - Tz. Actually the best uniform formula

p12= 4ne-’*(I - e-2*)[(3a/2)”tiAAi(-(3a/2)2’3)COS(+T - p)

+ ( 3 ~ / 2 ) - “ ~ A-i(’3( ~ / 2 ) ” ~ sin(+n ) - cp)]’

(78)

(cf. Connor, 1973, 1981; Child, 1974), which gives 0.56 and 0.75 for u4 = 0.3 and 0.8, respectively, and which represents a considerable improvement, particularly at u4 = 0.8. One intuitive reasoning for preferring Ai’ to Bi is simply that the latter is exponentially dominant in the classically inaccessible region (in the context, negative u, albeit a hypothetical regime given bending of the Stokes line). Nor is this t o say that expression (75) will have no application, say, even nearer to threshold. However, there still remains an error of 0.03 at a4 = 0.3 unaccounted for and in any case Eqs. (75)-(78) must be regarded as empirical. A close consideration of the work of Eu (1971, et seq.) shows that his Wronskian n W(Al, A,) in the classically accessible region may be approximated by

73

NONADIABATIC CHARGE TRANSFER

Because the energy is low, the precosine term is small but not negligible. It may well be that this coupling between in- and outgoing waves accounts for the above small discrepancies, in that the parabolic model of Crothers (1976) neglects such coupling. Indeed, one of the principal advantages of Eu’s formulation is the use of real basis functions. Also since his work on coupfed second-order differential equations generalizes the work of, for instance, Olver (1954, 1956, 1958, 1975a, 1978) on single second-order equations, it would seem that his work merits closer attention. In particular, it would be interesting to attempt an adiabatic version of the exact diabatic formulation of Bates and Crothers (1970) in terms of Green’s functions. One final comment before leaving the question of uniform approximations: in a very real sense the exact parabolic model formula of Crothers ( 1972, 1975) represents a uniform approximation, namely, PlZ

=

where CY

= ~ - ~ - exp(-3ri/4)1~~1fi) ~ ~ ( 2

P

=

(81)

exp(3ri/4)I~~Ifi)

(82)

~ i y ( 2

Here the parabolic cylinder functions have not been approximated or expanded asymptotically. One possible phase integral interpretation involves y and ITo[,given by y

ITol(I + ITo12)”’ + In{(l

+ lTolz)’r2+

=’

s/r

ITol} = .rr(+/26

(83)

(84)

For y and ITol reasonably small, CY and p may be evaluated in terms of J,’s, using Kummer’s relation and Maclaurin expansions. For y very small (a4very small), a may be expressed in terms of the well-known Fresnel integrals C andS. Quite generally, the denominator of Eq. (80) is a simple exponential being based on a simple Wronskian relation. In Table I1 we present the results for p12given by Eq. (80). Clearly they are not as good as the full Stueckelberg formula of Table I at high a,; nevertheless, for a4 s 0.7 the maximum error is less than 0.03, so that Eq. (80) competes favorably with Eq. (78) in this region. Before leaving the curve-crossing model, let us mention further the work of Dubrovskiy. Following early consideration (Dubrovskiy, 196413) of the one-dimensional, three, real transition point problem using the equation of the parabolic cylinder function as comparison equation, Dubrovskiy and Fischer-Hjalmars (1974, 1979) have generalized to resonant scattering in the two-state, curve-crossing context. They distinguish be-

D . S. F . Crothers

74

TABLE I1 UNIFORM PARABOLIC C Y L I N D EFUNCTION R FORMULA a4 (a.u.)

pIz

0.01 0.1 0.2 0.3 0.4 0.5

0.0007 0.0683 0.2541 0.5061 0.7539 0.9314

I

I

a4(a.u.)

plz

0.6 0.7 0.8 0.9 1.0 1.2

0.9961 0.9396 0.7868 0.5827 0.3764 0.0890

I

a4(a.u.)

pI2

1.4 1.6 I .8 2.0 2.2 2.4

0.0024 0.0064 0.0089 0.0016 0.0007 0.0050

I_

tween an inner crossing, resulting from diabatic curves of similar slope, and an outer crossing (Child, 1969), resulting from diabatic curves of dissimilar slope (but note remarks of Eu and Zaritsky, 1978). In both cases the upper diabatic and therefore the upper adiabatic curve becomes classically inaccessible at infinite separations: hence resonant scattering. They also distinguish overbarrier and underbarrier cases, corresponding to classically accessible and inaccessible curve-crossing points, respectively. They apply the Zwaan-Stueckelberg and parabolic comparison equation methods to derive a wide range of formulas covering all four cases. In the overbarrier cases, the upper and lower classical turning points lie inside the crossing distance; however, the energy minimum of the (upper) potential well lies above or below the crossing energy according to whether the crossing is outer or inner, respectively. Their work on nonresonant inner crossings has been critically reviewed by Barany (1980). Using the impact parameter (or parametric) method, Dubrovskiy (1964a, 1965, 1970) has also applied the Zwaan-Stueckelberg and comparison equation methods from another angle, which is not generally satisfactory, in that the attempt to allow for the pole (which occurs in the exponential model, for instance) in the s plane [Eq. (43)] at infinity does not permit a uniform treatment of the three transition points. In particular, Crothers (1971) has pointed out that the treatment does not apply to certain types of potential. The case of noncrossing diabatic curves is also amenable to the Zwaan-Stueckelberg and parabolic comparison equation methods. The S matrix is still given by Eq. (47) but now (Crothers, 1971) with theS matrix given by S,, = S% =

exp{-26 - 2ia) + exp{2ip} 1 + exp{-26}

NONADIABATIC CHARGE TRANSFER

75

where F and 6 are still given by Eq. (51) in terms of the adiabatic curves and cp is given (Crothers, 1976) within the parabolic model by

(87)

p = o

Virtually everything carries over conceptually and analytically from the crossing case, except for the phase cp and the single transition probability, which is given by

P12= (1

+ exp[26]}-'

(88)

as against

P12= exp{ -26)

(89)

for the crossing case. The original impact parameter model of Demkov (1964) involves exponential coupling between parallel curves at zero impact parameter and predicted Stueckelberg oscillations, and in the limit as the resonance defect (excitation energy) vanishes, symmetric resonance obtains. For more general couplings and nonzero impact parameters, this limit is still obtained, so that it is reasonable to use the term perturbed symmetric resonance, the perturbation being the distance between the parallel curves. Crothers (1973) showed that Eqs. (85) and (86) are also exact for the Rosen-Zener and Callaway-Bartling models and that they reconcile the noncrossing theories of Stueckelberg (1932), Nikitin (1968b), and Demokov (1964) in that each may be deduced from Eqs. (85) and (86) as a limiting case. Moreover, it is worth mentioning in passing that here the strong-coupling asymptotic expansions of the DJz) are essential (Crothers, 1972) in that weak-coupling expansions yield nugatory expressions. Equation (88) for a single transition probability is to be preferred to Eq. (107) of Child (1979), namely,

P12= 1 sech{is}

(90)

which is generally incorrect except at 6 = 0. Indeed, Eq. (88) was deduced by Meyerhof (1973) from Demkov's ISlz12and applied at zeroimpact parameter to account for inner-shell vacancy sharing probabilities, while J. S . Briggs (private communication, 1974) and Stolterfoht (1980) used Eqs. (51) and (88) successfully at nonzero impact parameter. Once again, however, the full phase integral interpretation is often discarded, resulting in unduly restrictive concepts and expressions. It is often postulated (cf. Demkov, 1964) that the transition point is given by

I&(&)

- HlI(RX)l

= =lzw'Y)

(91)

This is unfortunate because no contributions from impact parameters greater than Rx can arise. Barany and Crothers (1981) have shown, by

D . S . F . Crothers

76

considering the model problem of Olson (1972), that the bending of the Stokes line is equally important in noncrossing collisions as in crossing ones if total cross sections are to be obtained without errors of typically 100%. Moreover, they noted that the amplitudes of the oscillations in the transition probability were obtained more accurately in the full Stueckelberg treatment than by Dinterman and Delos (1977), probably because the latter curve-fitted their nonadiabatic parameter without reference to the singularity in the radial velocity. The concept of tunneling (Olson, 1972) is once again inappropriate and leads to inaccuracy and unnecessary empiricism (Olson and Smith, 1973; Matic et al., 1980). Our final application of the Zwaan-Stueckelberg and parabolic comparison equation method is to rotational coupling. First there is rotational coupling arising out of perturbed united-atom degeneracy, such as 2p u2p rr coupling in proton-hydrogen collisions (Bates and Williams, 1964; Knudson and Thorson, 1970; Bates and Sprevak, 1970). At a typical energy of 500 eV, there is a large angle peak in the curve of the H(2p) transition probability Px plotted against impact parameter. It occurs at small impact parameters and arises because the molecular cloud can not keep up with the rapid rotation of the internuclear axis (Bates and Sprevak, 1970). It requires the use of an effective Coulomb trajectory for an accurate description. There is also a small angle peak associated with an impact parameter of approximately 1 a.u. Crothers and Hughes (1979d) have shown that both peaks are well described by the Stueckelberg parabolic noncrossing formula, derived by Crothers (1971), namely, px

=

41&12

+

(92)

where S12is given by Eqs. (86) and (87) and where the represents the initial probability of being in the 2puUstate. In fact, Crothers and Hughes found the full wave interpretation of (T and 6 unnecessary: they used an impact parameter treatment with Coulomb path, thus: u

+ i6

=

lozc[(AE/v)* + (4p2/R")G2]"2dZ

(93)

where AE is the adiabatic term difference, G is the angular momentum matrix element, Z is the path length, and Z , the most effective complex transition point. It should be noted that Crothers and Hughes found it necessary to curve-fit AE and G as follows: AE

G-1

0.1R2 - hR'-

cR'

C U R z - p R 4 + yR6

(94) (95)

in order to preserve accuracy. However, unlike Demkov et al. (1978), they did not consider the elastic phase shift. Here we would suggest that it is given by arg S,,where

77

NONADIABATIC CHARGE TRANSFER

[l

eli

Sll=-

\/z

+ exp{-2ia [1

26}]

-

+ exp{-26)1

and where <=

jOm [4(d0/dZ)‘ + ( A E / v ) ~ ] ’ dZ / ‘ - jOm ( A c / v ) dZ

(97)

+ (AE/v)’]”’ dZ

(98)

a = /oz” [4(dB/dZ)‘

The extra phase 5 [cf. Eq. (26)l is necessary to take out the pseudoadiabatic phase already included in Eq. (85) (Crothers, 1971, 1978b). In interpreting d e / d Z as p G / P , the curve fit of Eq. (95) would be inappropriate and indeed unnecessary once Z , is determined by interpolation from Eqs. (93) and (98). The full Galilean-invariant treatment of G by Crothers and Hughes is clearly essential in evaluating Eq. (97). Interestingly enough the peculiar discontinuity found by Demkov et al. (1978) arises, namely, in the limit 6 << 1: arg S,,(mod 274

:I

(0

5

=

(T

< 7r/2) (99)

(a = T / 2 )

+r

(T/2 < u s

7T)

Of course in the sudden limit (Demkov et nl., 1978) we have (T

= 7T -

e(+q

(100)

where e(+t.) is the center of mass deflection angle. Demkov et nl. actually find 6 to be - x / 6 in the sudden limit and a curious infinity in (T at e ( + m ) = n/?.The limit of 6 = (T = rr is the straight line trajectory limit of Bates and Williams (1964). It should be stressed that although Demkov et ril. generalized to general charges the principal quantum numbers, they took 6 . c . a , B, and y of our Eqs. (94) and (95) to be zero. Thus, it is difficult at this stage to relate their Massey parameter approach to our Stueckelberg approach. Secondly, there is rotational coupling associated with a curve crossing at finite internuclear separations (Russek, 1971). Here we consider the model Ne+-Ne problem of Fritsch and Wille (1978), later considered by Namiki ef ul. (1980). in which

H2z

-

H I 2= 0.71 pv/R2

(101)

H , 1 = 2.71(R-’ - 1.5-I)

(102)

We have intentionally adopted ‘‘diabatic” notation because our earlier full Stueckelberg curve-crossing model suffices to describe this collision rather accurately. Indeed, even though a wave treatment is unnecessary,

D.S. F. Crothers

78

we used the same program to produce Tables I and 111, merely setting M = 18,370.There are two possible choices of complex transition point in this model, but that corresponding to the positive square root, in the limit as pv + 0, is the more effective and is preferred. Once again the bending of the Stokes line is absolutely crucial for this close curve-crossing collision. The results presented in Table I11 were produced in 4 sec and may be compared very favorably with the exact results of Fritsch and Wille and Namiki et al. for u = 0.2 and 0.5, respectively. At u = 0.5, the detailed characteristics of the transition probability are much better produced by the Stueckelberg curve-crossing formula than by any of the proposed Airy approximations, notably the shape of the major peak.

MODELA N D APPLICATIONS B. EXPONENTIAL The advantage of the exponential model over the parabolic models of the previous section is that it contains an extra parameter. Its disadvantage to date is that a purely phase integral derivation, albeit in terms of indeterminate Stokes constants, is not yet at hand. We must therefore rely on comparison equation derivations supplemented by ZwaanStueckelberg phase integral interpretations and abstractions. Fortunately, the parabolic models are all special limits so that generalized abstractions are not too difficult. The original model introduced by Nikitin (1962a,b) is essentially posed by H Z 2- H l l = A€

-

A cos H exp(- a R )

H 1 2= +A sin 0 exp(- a R )

(103) (104)

The three independent parameters are A , 0, and A€, with a merely a scaling quantity. In summary the weak-coupling derivation of Bayfield et al. (1973) and the strong-coupling derivation of Crothers (1978b) may be uniformly interpreted (Crothers and Todd, 1978; Nikitin and Reznikov, 1978) to yield S l l = Sgz = ( I

-

P12)exp{2icpl} + Plzexp{2i(cp2- d}

S l 2 =S z l = -2iP121’2( I -

sin(o+ cpl p} /sinh

-

pz)

(105) (106)

TABLE 111 FULLS T U E C K E L B E R C T R A N S I T I O N a4 (a.u.)

PIZ( C = 0.2 a.u.)

plr ( u

=

P R O B A B l L l T l E S FOR

FRITSCH-WILLE MODEL"

0.5 a.u.)

~~~~

0.05

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.OO 1.05 1.10 1.15

1.20 I'

0.0010 0.0036 0.0012 0.0131 0.0167 0.0016 0.0316 0.0376 0.0416 0.0311 0.1227 0.0119 0.0742 0.2099 0.1111

O.oo00 0.1234 0.3292 0.3648 0.2089 0.0376 0.0128 0.1630 0.4081

0.0021 0.0102 0.0105 0.0240 0.0061 0.0834 0.0942 0.0216 0.00% 0.1129 0.2552 0.3362 0.3139 0.2135 0.0947 0.0145 0.0063 0.0754 0.2065 0.3732 0.5480 0.7077 0.8365 0.9262

1.25 1.30 1.35 1.40 1.45 1.50 1.55

I .60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40

0.9749 0.9852 0.%24 0.9132 0.8443 0.7622 0.6729 0.5816 0.4927 0.40% 0.3348 0.2694 0.2139 0.1678 0.1304 0.1004 0.0768 0.0584 0.0442 0.0333 0.0250 0.0187 0.0140 0.0104

0.6428 0.7970 0.8514 0.8218 0.7355 0.6142 0.4741 0.3342 0.2153 0.1285 0.0723 0.0398 0.0205 0.0106 0.0054 0.0027 0.0014 0.0007 0.0003 0.0002 0.0001 0.0000

O.oo00 O.oo00

Nakamura, H., and Namiki, M.,(19801. J . P l r y . ~S. O C . Jtrptrri 49, 843, have obtained similar results for

L' = 0.5

a.u.

D . S.F . Crothers

80

As for the parabolic model, the function cp is given by Eq. (59), while u and 6 are given by Eq. (51) with the usual adiabatic interpretation of the wave numbers. The extra phase integral given by Eq. (1 11) is a function of R & the next most important transition point after Rc, the primary transition point. Both points are of course branch points of v l l - vzl. The single-passage transition probability is clearly given by Plz in Eq. (107). Specifically, for the exponential model at zero impact parameter the nonadiabatic parameters are given by 6

=

( r A / 2 ) ( I - cos 6)

(112)

+ cos 6)

(1 13)

p = (rA/2)(1

Accordingly, Eqs. (105) and (106) reduce correctly to the parabolic model for perturbed symmetric resonance in the limit as 6 + r/2. They also reduce correctly to the parabolic model for curve crossing in the limit as 6 + 0 and A + + ~4 in such a manner that A( 1 - cos 6) remains finite, and for the Rosen-Zener, Callaway-Bartling, and symmetric resonance models (Crothers, 1978b). Perhaps of more importance is that the above interpretation allows for bending of the double Stokes line, which, as we have seen, is essential if we wish to calculate accurately the outer oscillations in the impact parameter transition probability and, ips0 facto, differential and total cross sections. The latter phenomenon is well illustrated by the impact parameter calculations of Crothers and Todd (1978) on the curve-crossing collision Be2++ H(ls) -+ Be+(2s)+ H +

(1 14)

for which the diabatic crossing is given by

Rd = 5.81 and the primary complex transition point by Rc

=

6.503

(1 15)

+ 0.951i

As recorded in their Table 1, R,, the real transition point increases as a function of impact parameter pfrom 6.63 at 5.0 to 6.93 at 6.6 to 8.22 at 8.2, thus always keeping ahead of p. They also found that for such a curvecrossing transition, p is well approximated by p

I r X - 61

(1 17)

Here as in Eq. (110) A is essentially the residue of a simple pole of the Stueckelberg variable [Eq. (25)] in the plane of the Delos-Thorson vari-

NONADIABATIC CHARGE TRANSFER

81

able [Eq. (43)l (Crothers and Todd, 1980a). Indeed, this is one of the attractions of the exponential model, namely, that the Whittaker functions involved give a canonical representation of a three transition point problem: two conjugate branch points plus one simple pole in the adiabatic term difference. Other applications include H + + He(1s"S) Mg2+ + H(ls)

+ He+(ls)

+

H(ls)

+

Mg'(3s)

+ H+

(118) (1 19)

both of which are noncrossing and therefore of perturbed symmetric resonance character, and Zn2+ + H(ls) Znf(4s) + H + ( 120)

-

Cd'+ + H(1s) + Cd+(Ss) + H + B"

+ H(ls)

C6++ H(1s)

-+

-+

(121)

B+((2s)9 + H+

(122)

Cs+(430) + H'

(123)

all of which are essentially curve-crossing transitions. The helium transition has been discussed by Bums and Crothers (1976) and Hughes and Crothers (19771, while the others have been discussed by Crothers and Todd (1980a). The helium and carbon transitions are instructive in that a is obtained using accurate adiabatic eigenenergies rather than LCAO approximants. The helium and boron transitions also cover an energy range where it is essential to include electronic momentum translation factors, a circumstance well facilitated by the phase integral method, provided that problems concerning the lack of symmetry are resolved (Crothers, 1979b). The applications to beryllium and carbon are also particularly interesting in that curve-fitting matrix elements on rhe real axis to the exponential model was found to be successful, the more obvious procedure being to analytically continue off the axis matrix elements derived from simple wave functions. Perhaps the least satisfactorily understood and most puzzling of the above transitions is the magnesium one, in that the phase integral interpretation based on the middle root is somewhat blurred by the supposedly ineffective outer diabatic root. The exponential model is also attractive for K vacancy sharing applications since P I 2of Eq. (107) generalizes Eq. (88) in a satisfactory manner (Boving, 1977; Meyerhofet al.. 1978). It would also seem to be relevant to the strong violation of adiabaticity arising from the promotion of the 3 d a orbital in N+-Ar collisions (Afrosimov er al., 1975). It is also nice to relate Stueckelberg, the exponential model, and Eu's uniform WKB work, in that Shushin (1978) has obtained good agreement with the exact exponential model single-transition probability (for A 0.5) using a Magnus approximation, the maximum error being 0.05.

82

D.S. F. Crothers

This limitation of the Magnus approximation to weak coupling appears to be quite general (see Table I; also Korsch and Kriiger, 1980). A different type of limitation applies to the well-known Landau-Teller harmonic oscillator problem:

where typically m = 3, a = 0.3, and E = 10 in dimensionless units, and where r is the vibrational coordinate of the H2 molecule and R is the distance from the center of mass of the H 2 molecule to the He atom projectile (Secrest and Johnson, 1966). Apart from the exact numerical solution of the latter, there is the rather impressive semiclassical action angle analysis of Miller, Marcus, Connor, and others, which has been well reviewed by Child (1974) and more generally by Nikitin (1974). More recently Child and Hunt (1977) have developed a rather elegant uniform semiclassical Laguerre theory which gives excellent results. Superficially one would think that a two-state calculation in a coordinate representation based on asymptotic oscillator states would facilitate a Stueckelberg phase integral exponential model formulation. For instance, we obtain a value of 0.235 for pol(compared to the exact 0.218). Certainly rn = 5.242 and 6 = 1.179 are typically nonadiabatic, and indeed Galilean invariant with bending of the Stokes line. The difficulty, of course, is that the multiquantum transitions are virtually zero since 0 = 7~ [cf. Eqs. (107) and (113)]. The truth is that even Demkov’s simple model (1964) has not been generalized to three coupled states and that coupled equations have to be solved even with optimized basis functions in order to solve Eq. (124) in a coordinate representation (Cross, 1979; Baer et al., 1980; Harvey and Truhlar, 1980). Returning to ion-ion collisions, the remarks of Bates (1970) on H+-Hmutual neutralization at low energies are encouraging in that analogy with the beryllium transition is made, essentially in connection with the bending of the Stokes line. A further review by Bates (1974) merely served to underline the idiosyncrasies of the process, while nevertheless drawing attention to the asymptotic Landau-Herring method since reviewed by Janev (1976) and exploited by the Smirnov school, the latest production of which includes the articles by Chibisov (1979), Duman and Men’shikov (1979), and Kereselidze and Chibisov (1979). Indeed, the last mentioned have reinvestigated the H +-H- process using a new exchange interaction for the n = 2 channel with some success. Some of our own preliminary calculations applying the exponential model to both n = 2 and n = 3 channels are encouraging. Finally, it is worth noting that just as Eq. (80) is a uniform approximation to pI2 = ISI2l2,so also (Crothers, 1978b) is

83

NONADIABATIC CHARGE TRANSFER

sin’ O pI2 = 4(1 -

d [&y211Fl(l + (ih/2)(I

-

cos 0); 3

+ ih;

iy)li]’

(125)

+

4 exp{xh cos O} sinh{(nh/2)(1 + cos O)} sinh{(nh/2)(1 - cos 0)) sinh’{ zrh} x [ 2 Re F f ( q ,b)FL(r).

(126)

where L

=

ih/2,

6

=

y/2

=

A/2vcu,

7) =

( A cos 0)/2

(127)

The function FL(r), 6) is a Coulombic wave function with real r ) and fi (Abramowitz and Stegun, 1970) but complex (pure imaginary) angular momentum L . It follows that the Coulombic phase shift is complex and that the hyperbolic functions in Eq. (126) partly arise from

The Coulomb functions FL essentially give rise purely to Stueckelberg phase interference, except of course if y is too small, for instance, y << 1 gives plz

=

( y 2sin2 O)/( 1

+

A‘)’

(129)

111. Perturbed Stationary States and Electronic Translation A. HOMONUCLEAR COLLISIONS Thus far, we have strictly been considering nonadiabatic transitions without attending too much to specific problems concerning charge transfer. The very term “nonadiabatic” implies adiabaticity to zero order. So indeed when the impact velocity is sufficiently small compared to the orbital velocity of the active electron, the intuitively most acceptable basis set comprises the stationary states of the Hamiltonian with the internuclear distance fixed. The perturbation caused by the relative motion of the nuclei then gives rise to so-called perturbed stationary states (Mott, 1931; Massey and Smith, 1933). Let us restrict attention initially to the impact parameter treatment and homonuclear collisions. Bates et al. (1953) showed that Mott’s formulation failed to take account of the electronic translational momentum and that consequently his cross sections depended on the choice of origin of coordinates, that is. Galilean invariance was violated. Bates and McCarroll

D . S . F . Crothers

84

(1958) overcome this difficulty by introducing traveling molecular orbitals, expanding the total wave function as a function of time t and electronic coordinate r thus:

where the gerude and irngernde wave functions are given by (Df =

+{(xi + x;) exp(-&v

* r )2

(x:

-

x;)

exp(+iv o r ) }

(131)

The xi are the stationary states with eigenvalues egf(R) and the exp(k$iv r) are the electronic momentum translation factors: the Qu2 is just a consequential translational kinetic energy. Equations (130) and (131) reduce correctly in the limit of large impact parameter (Bates, 1958). However, at small impact parameters and small internuclear separations, Eq. (13 1) does not behave satisfactorily, the molecular functions being contaminated by the translation factors (Schneiderman and Russek, 1969). In principle this defect can be remedied by replacing exp(k.biv * r ) in Eq. (131) by e x p ( k f a r , R)v ‘ r ) with suitable choices of.f;, frequently with opposite sign (switching functions), so thatf; + 0 as R + 0, and Eq. (131) obtains as R -+ m(Levy and Thorson, 1969; Schneiderman and Russek, 1969; Lebeda ef ul., 1971; Taulbjerg et ul., 1975; Fritsch and Wille, 1977; Vaaben and Briggs, 1977). Mostly,f$ were chosen empirically and independently o f s . Firmer criteria were adopted by, for instance, Rankin and Thorson (19781, who determined each .f.: to minimize the coupling of states to the continuum, given that ionization should have low probability at the energies of interest. However, this entails much labor. An alternative criterion is provided by the Euler-Lagrange variational principle of Sil (1960). Riley and Green (1971) adopted and abandoned such an approach, since it led them into the realm of coupled nonlinear equations for the amplitudes c $ and the switching functions f ; . Ponce (1979) also adopted a similar approach but with neglect of couplings between states. This appears to have led to some strange effects. In particular, having chosen

f=

g(z’. f )

+ W(f)X‘Z’

(132)

where z’ and .Y’ are rotating-frame electronic coordinates, he appears to at small R which are inconsishave found values of g(2p0,) and iv(2p7-rnl,) tent with the rotational coupling sudden limit theories of Bates and Sprevak (1970) and Demkov et nl. (1978) and therefore with experiment. The neglect of nonadiabatic couplings by Ponce means that his switching

85

NONADIABATIC CHARGE TRANSFER

function is erroneously and undesirably trying to compensate. The variational criterion had, however, already been successfully applied by Crothers and Hughes (1978), who used as trial functions

CP:

=

~CCX;

+ xi)exp(-ifv

- r)

2

(x,'

-

xi)exp(ifv 1-13

(133)

withfrestricted to be a function of R only and not of either r ors and with as impressed kinetic energy. Of course, such averaging would be unnecessary in an exact and complete set for which the uncertainty principle would be automatically satisfied. Such an approach is impractical, and the intuitive remarks of Crothers and Hughes justifying the averaging off appear to have been misunderstood (Ponce, 1979). Allowing for coupling between the two resonant states to second order in o Crothers and Hughes determinedf to be given by 3u2.f

where the dynamic quadrapole 9;; and hybrid rotational-radial matrix elements Yo" are given by

9;:

=

9,

-

[XiZ'X;

dT

(135)

ZXz (d/dZ,) X i d T

(136)

+*-I

where r has the usual significance (Bates and McCarroll, 1962). In Table IV we recall Crothers and Hughes's successful description of the location of the turning points in the curve of the capture transition probability Pc against impact energy for scattering at a fixed angle 3" in proton-hydrogen atom collisions. Crothers and Hughes ( 1979a,b,c) subsequently extended their expansion up to 10 molecular states, calculating differential and total cross sections for H(2s) and H(2p) formation, and indeed fractional charge exchange and the polarization of Lyman a radiation. In particular, Crothers and Hughes (1979~) demonstrated conclusively the affect of varTABLE IV v41 [ I F 5 OF T H E

ENERGY E,/keV

A1 IHE

T U R N I NPGO I N T 5

OF

Pc(E)

Type

max

min

max

min

max

min

max

Experiment" Theoryb

0.78 0.81

1.11 1.11

1.57 1.59

2.39 2.40

3.92 3.92

7.69 7.40

20.1 19.8

"

Experiment of Lockwood and Everhart (1962).

' Variational theory of Crothers and Hughes (1978).

86

D.S. F. Crothers

iational switching functions on radial and rotational coupling: not only the enforcement of Galilean invariance and the avoidance of spurious coupling at infinity, but also the affect at finite distances. However, it may be noted that controversy still surrounds the cross section curve for H(2s) production (Hill et d., 1979: Morgan et ul.. 1980). The agreement on 2s fractional charge exchange (Crothers and Hughes, 1979b)with experiment (Bayfield, 1970) is nevertheless satisfactory as are elastic and inelastic differential cross sections (Crothers and Hughes, 1979a; Houver rt d., 1974; Helbig and Everhart. 1965). Let us now pause to consider five types of process which are certainly not homonuclear: He+(ls)+ H(ls1-t He(ls2s +

-

-, He" + H( Is)

-+

ls2p '.3P)+ H +

I%,

He+(ls) + H(2s, 2p) He'(2s)

+ H(Z)

He+(2s)+ H

+

He+(Z)+ H +

(137) (138) (139) (140) (141)

Gilbody's group has considered all of these processes experimentally except possibly process (137). Process (139) appears to be a genuine twoelectron problem, while process (140) and (141) are genuine one-electron heteronuclear problems of a type discussed in Section III,B below. However, provided that we regard the He +(1 s) electron as a spectator (Fayeton et d., 1976), a good approximation at both large and small separations, and provided that we scale the mass and energy appropriately we can interpret processes (137) and (138), respectively, as

The total cross sections for these two latter processes are well nigh equal since gerade coupling is unimportant. The exchange cross sections should of course be apportioned in the ratio 3 : 1 as between triplets and singlets and an extra factor of two arises since the Hz(lsvg)in processes (142) and (143) has no real analog in processes (137) and (138). We plot the five molecular state results of Crothers and Hughes (1979~)suitably scaled in Fig. 1. Agreement with experiment is satisfactory and markedly better than the other theories which are not intended for such an energy range. The assumption of a spectator role for He+(ls) in process (138), of course, automatically excludes double transitions (Bell and Kingston, 1978), which is not unreasonable in this low energv range.

NONADIABATIC CHARGE TRANSFER

87

T

i

O lo

5

d 10

I

I

20

25

3(

Energy ( k e V )

FIG.I . Cross sections (cm') (rzl,(H)and t r ~ , ( Hfor ) formation of H(2p) and H(2s) atoms in He+-H collisions plotted against impact energy (keV). (rZl1(H): ( 0 )McKee et a / . (1977); ((3) Young et ul. (1968): (0) Flannery (1969) theory: (-) present theory. az,(H): (M) McKee et ti/. (1977); (0) Flannery (1969) theory; (-) present theory.

D. S. F . Crothers

88

B . HETERON uc L E A R COLL ISIONS We now consider genuine heteronuclear collisions with special reference to the typical asymmetric charge transfer processes (140) and (141) and derive variational switching functions thus generalizing the work of Crothers and Hughes (Section II1,A) on homonuclear collisions. In particular, we continue to average over r the electronic coordinate (Crothers and Todd, 1981a,b), and to measure it from the midpoint of the internuclear axis. However, on the grounds that any two states only couple at specific nonadiabatic separations, unlike the symmetric resonance case in which the electron resonates to and fro continually between the two principal states, we do let each state have its own switching function. In principle, the total wave function is given by

W, t ) =

2 cj(i)Qf(r, t )

(144)

x A ~ R)lj(R, , r)Ej(r)

(145)

j

Qj(r, t ) =

where xr is the exact adiabatic wave function, Ef is an energy phase factor, and tj is an electronic translation factor given by tj= exp{if,(R)v

r}

( 146)

wherefi is the switching function. Using Sil’s variational principle we obtain (

(

fj = -(xjlz

a/azlxj> +

~k(XjIZIXk)(Xkla/azlXj)

(147)

where z is i r and Z is the impact parameter path length. Equation (147) allows for coupling, in principle, to all other states to first order, whereas the homonuclear expression (134) allows for coupling between the two resonant states to full order. It is satisfactory in that, neglecting the sum over k , it may be written

fj =

(X,lZIxj)

(148)

from which it is clear that vfj represents the velocity of the mean position of the electron. It is also satisfactory in that

depending to which nucleus simple, being given by

dissociates. The energy phase is now also

NONADIABATIC CHARGE TRANSFER

89

where ej is the usual eigenvalue and

The switching function may be conveniently written as

where the radial and transverse functions BJand .Tj are determined by rotating-frame matrix elements. All 91jand TJare found to have the centre of charge limit - Q in the united-atom limit. Green’ (1978a,b) has proposed a vectorized form of fr, but this has the same difficulties inherent in Ponce’s homonuclear choice, and in practice T. A. Green (private communication, 1979) averages his vector components u posteriori. He has confirmed the center of charge limit in the Cfi+Hproblem. In applying Sil’s variational principle to obtain and solve the coupled equations for the amplitudes cj, we have synthesized the strategies of Crothers and Hughes (1978) and of Crothers (1979b), that is, we have symmetrized the matrix elements to second order in u . In Fig. 2 we present some of our results for processes (140) and (141), using five states (2pa, 2p7r, 3da, 3dn, 2 s ~ in) the coupled equations (but considerably more in determining the fj). The 2po-3do noncrossing and the 2 p a - 2 ~rotational ~ coupling are, of course, predominant mechanisms. Within the energy range 3-20 keV ( ”He2+ impact) the total capture cross section is in excellent agreement with experiment. The He+(2s) production cross section is not so good, and like other theories (Winter and Hatton, 1980) lies above experiment. In Table V we compare with both Winter and Hatton and Vaaben and Taulbjerg (19791, the former using Bates-McCarroll plane waves (fr = ki),the latter using an empirical switching function. Considering that Winter and Hatton retained all powers of u while Vaaben and Taulbjerg and Crothers and Todd only retained first- and second-order terms, respectively, the agreement is very good. However, we hasten to add that, as the energy and the number of included states increase, the cross sections show greater changes accordingly. Nevertheless, all the evidence is that the inclusion of translation factors, with avoidance of spurious long-range couplings and enforcement of Galilean invariance, greatly increases the rate of con-

’ See also Green, T. A.

(1981a). Phys. Rev. A 23, 519; Green, T. A. (1981b). Phvs.

Rev. A 23, 532: and Green, T. A., Shipsey, E. J., and Browne, J . C. (1981).Phys. Rev. A

23. 546.

D. S.F. Crothers

90

He' ( 2 s ) production

5.0

2.0 4

10

20

50

He2' impact energy ( k e V)

FIG.2. Total capture and Hef(2s) production cross sections ( c d ) plotted against impact energy (keV) in He2+-H collisions. The two full curves are the variational calculations of Crothers and Todd (1981b). Experimental results: (0)Bayfield and Khayrallah (1975); (0) Shah and Gilbody (1978); (+) Nutt et a / . (1978).

vergence with respect to the number of included states (cf. Piacentini and Salin, 1974, 1977; Winter and Lane, 1978; Hattonet af., 1979).The specific choice of translation factor, variational or not, is, of course, much more crucial in detail for the calculation of large-angle capture probabilities (cf. Table IV), but we must postpone further consideration of this to another article (Crothers and Todd, 1981b).

91

NONADIABATIC CHARGE TRANSFER TABLE V

( x lo-"' cm') RESULTS FOR V A R I O UIMPACT S ENERGIES (keV)",b TOTALCROSS-SECTION

He '(2s) production Energy 'He" impact (keV)

Total capture

Four state, WH (1980)

Five state, this work

Three state, WH (1980)

Three state, this work

Three state, VT (1979)

0.432 1.49 2.91

0.395 1.46 2.90

I .45 5.93 10.1

1.24 5.10 10.09

1.25 5.40 10.36

3 8 10

The initials W H stand for Winter and Hatton and VT stand for Vaaben and Taulbjerg. The asterisk denotes that the straight line approach is not used, but a 2pu trajectory is used [see Vaaben and Taulbjerg (1979) and Winter and Lane (1978)J. " Three state 2pv. Zpr, 3da: four state = Z p , 2pr. 3dc. 2sa: five state = 2pu. 3d 7. I'

-

Other approaches are of course possible (Mittleman, 1969, 1974; Mittleman and Tai, 1973; Schmid, 1977; Thorson and Delos, 1978a,b), but practical calculations in the keV energy range have not yet been reported. Nor have applications been made to two or more electron systems; here we would suggest that averaging over r facilitates the avoidance of practical problems concerning molecular orbitals and the Pauli exclusion principle, a minimal prerequisite for vectorizing static correlation diagrams and the associated promotion models (cf. Barat and Lichten, 1972; Eichler et l i l . , 1976; Nikulin and Guschina, 1978).

IV. Nonmolecular Three-Body Analysis In Section 111 we reviewed the state of the art in employing traveling molecular orbitals. Despite the complications, the PSS method is attractive in that it explicitly treats the three Coulombic interactions simultaneously and on an equal footing. At higher velocities, close-coupling calculations based on traveling atomic orbitals are often more appropriate (Bates, 1958; Bransden et d., 1980). However, particularly at high velocities, it is known that second-order effects involving the (atomic) continuum are essential for an accurate description of charge transfer. Suitable theories are second Born, continuum intermediate states and continuum distorted waves (CDW). Recently we have used a number of theories, including the latter two, for an investigation of the scaling of charge transfer cross sections with respect to projectile charge (Crothers and Todd. 1980b: Crothers. 1981). A simple classical continuum theory of

D . S. F. Crothers

92

Bohr and Lindhard (1954) was adapted to give total cross sections proportional to 4 3 , thus:

(154)

QBohr/na$ = 8q3/v7

in reasonable accord at intermediate energies with both experiment and encouragingly CDW. Of course, both CDW and Bohr theories understandably fail at lower energies. The point is that to the best of my knowledge the only two theories other than PSS which provide a uniform three-body analysis are CDW and the classical trajectory Monte Carlo (CTMC) theory of Abrines and Percival (1966a,b), frequently applied by Olson (for a recent application, see Salop and Olson, 1979). Thus, both CDW and CTMC intrinsically treat the three-body Coulombic interactions simultaneously and equally. The CTMC theory is clearly variational, involving the solution of 12 Hamilton’s equations, but is difficult to apply without statistical error and is limited to reasonably high velocities and nonresonant events. The CDW theory up to now has not been variational (cf. Belkic et al., 1979) and has therefore been the “odd man out.” In fact, it is possible to formulate CDW simply and variationally as we now briefly outline. Consider the reaction Bq’

+

A(1s)

-+

B‘q-L’+(nlm) + A+

(155)

In the two-state impact parameter CDW theory (Cheshire, 1964), we set Ip+) = pi(rA)exp{-+iv*r + ( i q / u ) ln(vR - v’t) - Qiv‘t - i c f t ) x exp{nq/2v}r(l

-

i q / u ) lFl(iq/u; 1 ; ivr,

I p - ’ = pr(re) exp{iiv *r - ( i q / u ) In(uR x exp{n/2v}r(l

+ i/v)

+ u’t)

-

+ iv

rd

(156)

Qiv’t - ic)’t}

lFl(-i/u; 1; -ivrA - iv-r,)

(157)

We adopt generalized nonorthogonal coordinates regarding rA, rB, and t as independent, so that the full Hamiltonian may be written as

H

=

+ VA(rA)+ v87rg) + W ( R ) - i ( a / d t ) - +iv*VrA+ iiv*VrB -

VrA*VrU - fV;B

(1 58)

It follows that

(H

-

i d/dt,)*Irj+’

=

{-VrApi(rA)Vr,, ,F,(iq/v; 1; iwB+iv rB)} x exp{-+iv *r + ( i q / v ) In(vR - v’t) - Qiv’t - i&t

x

r(l - i q / v )

+ srq/2u} ( 159)

NONADIABATIC CHARGE TRANSFER

93

and that all matrix elements necessary to apply Sil’s variational principle may be written down to produce two-state coupled equations. Unlike, say, the two atomic state theory of Bates (19581, qi+)and qj-)are not normalized except at t = fm, where both functions obey the correct, exact boundary conditions. The full wave treatment may be similarly formulated, and both are promising in that allowance for back-coupling (and indeed distortion of the distorted waves) should cause the cross section to peak and then reduce, as the energy decreases, unlike the usual one-state CDW theory.

V. Summary As the title “nonadiabatic charge transfer” suggests, this review was concerned with two principal aspects of charge transfer: first, the choice of optimized basis set, depending on energy, for example, perturbed stationary states or continuum distorted waves, and second, the identification and description of nonadiabatic couplings between these basic functions, whether crossing, noncrossing, rotational, or whatever. Suffice it to say there are still many interesting and unsolved problems, a few of which may have caught the reader’s attention. Nevertheless, we have demonstrated the power of the Stueckelberg phase integral method, suitably reinforced by the parabolic or exponential model, and indeed the power of the traveling molecular (or atomic) orbital expansion.

ACKNOWLEDGMENTS Thanks are due to J. G. Hughes and N. R. Todd for their assistance with Figs. 1 and 2 and to A. Barany, J.N.L. Connor, T. A. Green, G . J. Hatton, N. F. Lane, and T. G. Winter for prepublication copies of their papers. The author is especially grateful to Professor Sir David Bates, F.R.S., and Professor P. G. Burke, F.R.S., for computer time on the Daresbury Cray-1 machine under a Science Research Council contract GWB26206 and to A. Bkany and N. R. Todd for a critical reading of the original manuscript.

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D . S. F. Crothers

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Dubrovskiy, G. V., and Fischer-Hjalmars, I. (1974).J. Phys. B 7, 892. Dubrovskiy, G. V., and Fischer-Hjalmars, I. (1979). Phys. Scr. 19, 318. Duman, E. L., and Men’shikov, L. I. (1979). Sov. Phys.-JETP ( E n d . Trans/.) 50, 433. Eichler, J., Wille, U., Fastrup, B., and Taulbjerg, K. (1976). Phys. Rev. A 14, 707. Elisenkov, F. A., and Nikulin, V. K. (1973). SOL*.P h y s . J E T P (Engl. Transl.) 37, 65. Eu, B. C. (1971). J. Chem. Phys. 55, 5600. Eu, B. C. (1972). J. Chem. Phvs. 56, 2507, 5202. Eu, B. C. (1973a). J. Chem. Phys. 58, 472. Eu, B. C. (1973b). J. Chem. Phys. 59, 4705. Eu, B. C., and Guerin, H. G. (1974). Can. J . Phys. 52, 1805. Eu, B. C., and Tsien, T. P. (1972). Chem. Phys. Lefr. 17, 256. Eu, B. C., and Tsien, T. P. (1973). Phys. Rev. A 7, 648. Eu, B. C., and Tsien. T. P. (1974). Phys. Rev. A 9, 995. Eu, B. C., and Zaritsky, N. (1978). J. Chem. Phys. 69, 1553. Eu, B. C., and Zaritsky, N. (1979). J. Chem. Phys. 70, 4986. Fayeton, J., Houver, J. C., Barat, M., and Masnou-Seeuws, F. (1976). J . Phys. B 9, 461. Flannery, M. R. (1969). J. Phys. B 2, 1004. Fritsch, W., and Wille, U. (1977). J . Phys. B 10, L165. Fritsch, W., and Wille, U. (1978). J. Phys. B I I , L43. Froman, N., and Froman, P. 0. (1%5). “JWKB Approximation.” North-Holland Publ., Amsterdam. Gaussorgues, C. (1973). J. Phys. B 6, 675. Gaussorgues, C., Le Sech, C., Masnou-Seeuws, F., McCarroll, R., and Riera, A. (1975). J . Phys. B 8, 239. Gilbody, H. B. (1979). A d v . Ar. M o l . Phys. 15, 293. Gordeev, E. A., Nikitin, E. E., and Shushin, A. I. (1977). Mol. Phys. 33, 1611. Green, T. A. (1978a). Sandiu Lab. [Tech. R e p . ] SAND780235. New Mexico. Green, T. A. (1978b). Sandia Lab. [Tech. R e p . ] SAND78-0158. New Mexico. Guerin, H. G., Tsien, T. P., Eu, B. C., and Olson, R. E. (1974). Phys. Rev. A 9, 995. Harvey, N. M., and Truhlar, D. G. (1980). Chem. Phys. L e f t . 74, 252. Hatton, G. J., Lane, N. R., and Winter, T. G. (1979). J. Phys. B 12, L571. Heading, J. (1962). “Phase Integral Methods.” Methuen, London. Heil, T. G., and Dalgarno, A. (1979). J . Phys. B 12, L557. Helbig, H. F., and Everhart, E. (1%5). Phys. Rev. 140, 1715. Hill, J., Geddes, J., and Gilbody, H. B. (1979). J. Phys. B 12, L341. Houver, J. C., Fayeton, J., and Barat, M. (1974). J. Phys. B 7, 1358. Hughes, J. G., and Crothers, D. S. F. (1977). J. Phys. B 10, L605. Janev, R. K. (1976). A d v . Ar. M o l . Phys. 12, 1. Kereselidze, T. M., and Chibisov, M. I. (1979). Opt. Specrrosc. (Engl. Transl.) 46, 356. Knudson, S. K., and Thorson, W. R. (1970). Con. 1. Phys. 48, 313. Korsch, H. J., and Kriiger, H. (1980). Mol. Phys. 39, 51. Kotova, L. P. (1969). Sov. Phvs.-JETP (Engl. Transl.) 28, 719. Kotova, L. P., and Ovchinnikova, M. Ya. (1971). Sov. Phys.-JETP 60, 2026. Laing, I. R., and George, T. F. (1977). Phys. Rev. A 16, 1082. Landau, L. P. (1932a). Phys. Z. Sowjefunion 1, 89. Landau, L. P. (1932b). Phys. Z. Soyietunion 2, 46. Lebeda. C. F., Thorson, W. R., and Levy, H. (1971). Phys. R e v . A 4, 900. Levy, H.,and Thorson, W. R. (1969). Phys. Rev. 181, 252. Lockwood, G. J., and Everhart, E. (1%2). Phys. Rev. 125, 567. McCarroll, R. (1976). In “Atomic Processes and Applications” (P. G. Burke and B. L. Moiseiwitsch, eds.), p. 467. North-Holland Publ., Amsterdam.

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McKee, J. D. A., Sheridan, J. R., Geddes, J., and Gilbody, H. B. (1977).J. Phvs. B 10, 1679. McLaBerty, F. J., and George, T. F. (1975). J. Chem. Phys. 63, 2609. Massey, H. S. W. (1949). Rep. Prog. Phvs. 12, 248. Massey, H. S. W., and Smith, R. A. (1933). Proc. R . Soc. London, Ser. A 142, 142. Matic, M., Sidis. V., Vujovic, M., and Cobic, B. (1980). J. Phys. B 13, 3665. Meyerhof, W. E. (1973). Phys. Rev. Lett. 31, 1341. Meyerhof, W. E., Anholt, R., Eichler, J., and Salop, A. (1978). Phys. Rev. A 17, 108. Miller, W. H., and George, T. F. (1972). J. Chem. Phys. 56, 5637. Mittleman, M. H. (1%9). Phys. Rev. 188, 221. Mittleman, M. H. (1974). Phys. Rev. A 9, 704. Mittleman, M. H., and Tai, H. (1973). Phys. Rev. A 8, 1880. Moiseiwitsch, B . L. (1977). Rep. Prog. Phys. 40, 843. Morgan, T. J . , Stone, J., and Mayo, R. E. (1980). Phvs. Rev. A 22, 1460. Mott, N. F. (1931). Proc. Cambridge Philos. Soc. 27, 553. Mott, N. F.. and Massey, H. S. W. (1965). “Theory of Atomic Collisions.” Oxford Univ. Press (Clarendon), London and New York. Namiki, M., Yagisawa, H., and Nakamura, H. (1980). J. Phys. B 13, 743. Nikitin, E. E. (1961). Opt. Spectrosc. (Engl. Trunsl.) 11, 246. Nikitin, E. E. (1%2a). Discuss. Faraduy Soc. 33, 14. Nikitin, E. E. (1%2b). O p t . Specrrosc. (Engl. Transl.) 13, 431. Nikitin, E. E. (1%5). J . Chem. Phys. 43, 744. Nikitin, E. E. (1968a). In ”Chemische Elementarprozesse” (H. Hartmann, ed.), pp. 43-77. Springer-Verlag, Berlin and New York. Nikitin, E. E. (1%8b). Chem. Phys. Lett. 2, 402. Nikitin, E. E. (1970). A d v . Quantum Chem. 5, 135. Nikitin, E. E. (1974). “Theory of Elementary Atomic and Molecular Processes in Gases.” Oxford Univ. Press (Clarendon), London and New York. Nikitin, E. E., and Ovchinnikova, M. Ya. (1972). Sov. Phys. -Usp. (Engl. Transl.) 14, 394. Nikitin, E. E., and Reznikov, A. I. (1972). Phys. Rev. A 6, 522. Nikitin, E. E.. and Reznikov, A. I. (1978). J . Phys. B 11, L659. Nikitin, E. E., and Reznikov, A. I. (1980). J . Phys. B 13, L57. Nikulin, V. K., and Guschina, N. A. (1978). J . Phys. B 11, 3553. Nutt, W. L., McCullough, R. W., Brady, K., Shah, M. B., and Gilbody, H. B. (1978). J . Phys. B I I , 1457. Olson, R. E. (1972). Phys. Rev. A 6, 1822. Olson, R. E. (1979). In “Electronic and Atomic Collisions’’ (N. Oda and K. Takayanagi, eds.), p. 391. North-Holland Publ., Amsterdam. Olson, R. E., and Smith, F. T. (1971). Phys. Rev. A 3, 1607. Olson, R. E., and Smith, F. T. (1973). Phys. Rev. A 7, 1529. Olson, R. E., Smith, F. T., and Bauer, E. (1971). Appl. Opt. 10, 1848. Olver, F. W. J. (1954). Philos. Trans. R . SOC. London, Ser. A 247, 307. Olver, F. W. J. (1956). Philos. Trans. R . SOC. London, Ser. A 249, 65. Olver, F. W. J. (1958). Philos. Trans. R . Soc. London, Ser. A 250, 479. Olver, F. W. J. (1974). “Asymptotics and Special Functions.” Academic Press, New York. Olver, F. W. J. (1975a). Philos. Trans. R . Soc. London, Ser. A 278, 137. Olver, F. W. J. (1975b). In “Theory and Application of Special Functions” (R. A. Askey, ed.), pp. 99-142. Academic Press, New York. Olver, F. W. J. (1978). Philos. Trans. R . Soc. London. Ser. A 289, 501. O’Maliey, T. F. (1971). A d v . A t . Mu/. Phys. 7, 223. Ovchinnikova, M. Ya. (1971). Sov. Phys.-JETP (Engl. Trunsl.) 32, 974. Ovchinnikova, M. Ya. (1973). Sov. Phys. -JETP (Engl. Trunsl.) 37, 68.

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Piacentini, R. D.. and Salin. A. (1974). J. Phvs. B 7. 1666. Piacentini, R. D., and Salin, A. (1977). J. Phys. B 10, 1515. Ponce, V. H. (1979). J . Phys. B 12, 3731. Rankin, J., and Thorson, W. R. (1978). Phys. Rev. A 18, 1990. Riley, M. E., and Green, T. A. (1971). Phys. Rev. A 4, 619. Russek, A. (1971). Phvs. Rev. A 4. 1918. Salop. A.. and Olson. R. E. (1979). Phvs. Lett. A 71A. 407. Saxon, R. P., and Olson, R. E. (1975). Phys. Rev. A 12, 830. Schmid, G. B. (1977). Phys. Rev. A 15, 1459. Schneiderman, S. B., and Russek, A. (1%9). Phys. Rev. 181, 311. Secrest, D., and Johnson, B. R. (1966). J . Chem. Phys. 45, 4556. Shah, M. B., and Gilbody, H. B. (1978). J . Phys. B 11, 121. Shakeshaft, R., and Spruch, L. (19791. Rev. Mod. Phvs. 51. 369. Shushin, A. I. (1976). Chem. Phys. Lett. 43, 110. Shushin, A. I. (1978). Chem. Phys. Lett. 55, 535. Sil, N. C. (1960). Proc. Phys. Soc., London 75, 194. Smith, F. T. (1969a). Phys. Rev. 179, 111. Smith, F. T. (1969b). Lert. Theor. Phvs. 1 IC. 95. Stolterfoht, N. (19801. J. Phys. B 13, L559. Stueckelberg, E. C. G. (1932). Helv. Phys. Acru 5, 369. Taulbjerg, K., Vaaben, J., and Fastrup, B. (1975). Phys. Rev. A 12, 2325. Thom, R. (1975). “Structural Stability and Morphogenesis.” Benjamin, Reading, Massachusetts. Thorson, W. R., and Delos, J. B. (1978a). Phvs. Rev. A 18, 117. Thorson, W. R., and Delos, J. B. (1978b). Phvs. Rev. A IS, 135. Thorson, W. R., Delos, J. B., and Boorstein, S. A. (1971). Phys. Rev. A 4, 1052. Vaaben, J., and Briggs, J. (1977). J . Phys. B 10, L521. Vaaben, J., and Taulbjerg, K. (1979). Electron. At. Collisions, Inr. Congr. Phys. Electron. At. Col1ision.s. Contrib. Pup.. 11th. 1979 p. 566. Winter, T. G., and Hatton, G. J. (1980). Phvs. Rev. A 21, 793. Winter, T. G., and Lane, N. F. (1978). Phvs. Rev. A 17, 66. Young, R. A., Stebbings, R. F., and McGowan, J. W. (1968). Phys. Rev. 171, 85. Zembekov, A. A., and Nikitin, E. E. (1971). Chem. Phys. Lett. 9, 213. Zener, C. (1932). Proc. R. Soc. London, Ser. A 137, 6%. Zwaan, A. (1929). Thesis, Utrecht.

ADV4NCES IN AlOMIC A N D MOLECULAR PHYSICS. VOI.

17

ATOMIC RYDBERG STATES SERGE FENEUILLE

tirid

PIERRE JACQUINOT

Ltihor(iioire Aitne C o t t o t i ' : Crritri, Ntiiioritrl do In Recherche SiYmt(jiqric Or.$tiy, Friitwe

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Preparation and Detection of Rydberg States

. . . . . . . . . . . . A. Preparation of Atoms in Rydberg States . . . . . . . . . . . . . B. Detection of Rydberg States . . . . . . . . . . . . . . . . . .

99 101

103 I12 . 1 I9 . . . . . . . . . 120 . . . . . . . . I26 . . . . . . . . . I30 . . . . . . . . . 131

H I . Spectroscopy . . . . . . . . . . . . A. Alkali Metals . . . . . . . . . . . . . B. Two-Electron Spectra . . . . . . . . . . . . . C. Complex Atoms . . . . . . . . . . . . . . . IV. Rydberg Atoms in External Fields . . . . . . . . A . Rydberg Atoms in Magnetic Fields . . . . . . . . . B. Rydberg Atoms in Electric Fields. Field Ionization . . . . C. Rydberg Atoms in Crossed Fields . . . . . . . . . . . . V. Radiative Properties of Rydberg States . . . . . . . . . . . A. Lifetime Measurements . . . . . . . . . . . . . . . . . B. Superradiance and Maser Oscillation on Transitions between Rydberg States . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

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

131

. . . .

136 1 S7 157

. . .

IS8

. . .

I s9 Ihl

I. Introduction Since so much work has been done recently on atomic Rydberg states, it is appropriate at this time to present a review of several aspects of this field. The study of atomic states with high principal quantum numbers is far from new, since the first observations of long series in the alkalis appeared in an article by Liveing and Dewar more than one hundred vears ago (1879). Eleven years later Rydberg (1890) proposed the famous formula u = C - R(n - 6)V2,where the quantum defect 6 is approximately constant in a given series. The effective quantum number n x = n 6 ~

Laboratoire aswcie a I' Universite Paris-Sud, France. 99 Copbright r' 19x1 h\ 4i.idemiL Pie\< In< All righra ot reprinluiiion In dn\ torm r a e i r e d I \ R C 0 I ? lY)?X17 %

100

Serge Feneuille and Pierre Jacquinol

therefore increases by integers along the series. Of course, all the states obeying the Rydberg formula may be called Rydberg states, but the practice of calling only atomic states with high values of I I Rydberg states seems to have appeared rather recently, in the mid-1970s. From the very beginning of quantum mechanics it was possible to predict the approximate variations of the main properties of Rydberg states with / I (atomic radius as n', radiative lifetime as ri3, scalar electric polarizability as n', diamagnetism as n', collisional cross section as n', and so on) and to understand the origin of quantum defects. During the following 40 years, however/ while the atomic spectroscopy (and, more generally, atomic physics) of valence states (small values o f n ) was developing as an active and fruitful field, nothing new was revealed about Rydberg states. Of course, very long Rydberg series were measured in many elements, which can be found in Charlotte Moore's tables, but such measurements were only a part of the then current spectroscopic work, the principal thrust of which was the determination of ionization potentials. However, during this period, Amaldi and Segre (1934a, b) observed interesting broadening and shifts in high members of series in alkalis by foreign gases, and Jenkins and Segre were responsible for some beautiful experiments on the effect of strong magnetic fields. During that time experimental data were obtained from absorption spectra as in the days of Rydberg, and as a result it was difficult, or impossible, to progress further. However, using quite different methods, radio astronomers discovered, by the observation of a line at 5.4 GHz, a transition between the states I I = 110 and I I = 109 of hydrogen in the Orion nebula (Kardashev, 1960: Hoglund and Mezger, 1965). The situation changed radically with the advent of tunable lasers since it then became possible to prepare Rydberg atoms in definite states and sufficient quantities, even in such low-density media as atomic beams, permitting very pure conditions for study without collisions or Doppler broadening. At the same time rapid progress was being made in the detection of the states, permitting almost every Rydberg atom to be detected. It thus became possible not only to determine the positions of the states (spectroscopy of Rydberg states) with high precision, but, moreover, to study the properties of Rydberg atoms, i.e., the effects of external actions such as magnetic or electric static fields, radiation, and collisions (physics of Rydberg states), in great detail. Although, as already stated, the study of Rydberg states does not begin with the tunable lasers, this review deals essentially with what has happened since 1975. Section I1 is devoted to the production and detection of Rydberg states. This section is developed rather extensively since every experiment starts and ends with these processes. The main results of spectroscopic studies are analyzed in Section 111. A rather strong em-

ATOMIC RYDBERG STATES

101

phasis is placed on the effects of external fields in Section IV, as this is probably the domain where the most interesting and sometimes surprising results have been obtained. The radiative properties are studied in Section V, and in addition to lifetime measurements, recent experiments on superradiance and maser effects in Rydberg states are analyzed. A sixth section, could (or should?) have been written that was devoted to the physics of collisions of Rydberg atoms. This is a vast and important field in itself, however, and since a review has already been published (Edelstein and Gallagher, 1978) that deals mainly with this question, we prefer to treat the other sections more extensively, and not to develop the latter. Even so, such a review cannot pretend to be complete, and there are some points that are not covered-not because they are less important but because we have not been able to include them in the plan we have adopted. This is particularly the case for some prelaser works, or studies on ionization with a very large number of microwave photons and interpretation of the results in terms of semiclassical theory. Furthermore, the list of reference is not intended to be exhaustive but only representative. In conclusion, during the last few years, considerable and often unexpected advances have been made in our understanding of the properties of Rydberg atoms. In the same way as liquid crystals in the physics of condensed matter, the study of Rydberg atoms is a beautiful exac7ple of a very old subject which had been considered for many years to be fully completed, but, nevertheless, has been completely revived in a very short time. Various applications concerning, for example, laser isotope separation, microwave detectors, or infrared lasers have already been suggested. From a more fundamental point of view, many new phenomena have been observed and understood. However, some problems are still open to interpretation, and one can easily predict that the study of Rydberg states will remain a very active field of atomic physics for some time to come.

11. Preparation and Detection of Rydberg States Any experiment on Rydberg states can be divided into three phases: (1) The atoms are first prepared in more or less isolated states, with selective or nonselective preparation. ( 2 ) A modification of this situation then arises, either spontaneously (radiation) or under the influence of external perturbations such as collisions, external fields, or coupling with external radiation.

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Serge Feneuille and Pierre Jacquinot

(3) Finally atoms in the different Rydberg states are then more or less selectively detected.

If nothing intervenes during Phase ( 2 ) , one has simply made an experiment of pure spectroscopy, provided that at least one of the processes (1) and (3) is highly selective. Phases (1) and (3) do not even have to be distinct from each other. This was the case in the first experiments, where an absorption spectrum from a continuous source was studied with a spectrograph. Although this method has given many excellent results in early experiments (see, e.g., Jenkins and Segre, 1939) and even quite recently (see, e.g., Garton and Tomkins, 1969),we shall not treat it further since it is well known. The following properties are of importance for the preparation and detection of atoms in Rydberg states: (a) T k i r ( t ~ t w r g y )positions. Aid1 tire ~ I o s c t o thr litnit o f ionixiIn the case of optical excitation, UV or far-UV transitions, multistep, or multiphoton processes are necessary if one starts from the ground state. (b) The distance het\zveen neighboring stntes. For large t i this distance, in the absence of external fields, is 2R(n*)-" between states of principal number differing by one unit, and 2R(n)-:'(ti,- ti,,) between states of the same n and different values of/ ( 6 being the quantum defect). Of course, for states differing only by m , it is zero in the absence of external fields. This means that a high selectivity in energy may be required for the preparation or detection of states with a well-defined set of quantum numbers, or that use must be made of selection rules or of external fields. (c) Their sttrhility. In the absence of any external perturbation the radiative lifetime for a given I scales as tz?:;'and is, for instance, as long as about 1 psec for a lop state and 60 p e c for a 40p state in rubidium; in addition, it increases greatly with I . This is an important property for many experiments, especially when pulsed excitation is used. One must be very careful, however, as this lifetime can be greatly altered (lengthened or shortened) by external causes such as collisions or fields. In particular, if atoms have been prepared in an ( i d ) state, they can be rapidly transferred to ( i d ' ) states ( I ' > I ) by collisions with rare gas atoms (see Section V). Furthermore, due to the large dipole moments between neighboring states, Rydberg atoms are strongly coupled to electromagnetic fields, and they can easily make transitions to t i ' / ' states, which are induced by the blackbody radiation emitted by the walls of the apparatus at room temperature (Cooke and Gallagher, 1980). One thus can believe that t i / states have been prepared, whereas the population is actually composed in large part of states with different values ofif andl. which lie a fioii.

ATOMIC RY DBERG STATES

103

little higher (or lower). The lifetime of the states is of importance for evaluating, in the case of optical excitation, the intensity necessary to excite a given proportion of atoms in a given state in pulsed regime’ since the energy received by the atom during the pulse must be about the same as the energy it should receive during its lifetime in continuous regime. This is the essential reason why higher states are less efficiently populated than lower ones. (d) Their iiidtli. The natural (or radiative) width 6u = I / r scales as 2 3 - 3 , and may be extremely small for high-energy members, for instance, 6 u < 1 MHz forti > 10. It is thus usually smaller than the Doppler width, even in highly collimated atomic beams (a collimation ratio l000il corresponds to a residual Doppler width of the order of 1 MHz). Thus, except for very rare experiments involving two-photon, Doppler-free excitation with extremely narrow continuous lasers (Lee rt (11.. 1979), the resonance width is determined not by the Rydberg state itself, but by experimental conditions such as (residual) Doppler width or laser width. This is important when evaluating the intensity required to excite a given proportion of atoms. (e) Their trcrnsitioti pvobiihilitics to lonw states. Here also the transition probability to (or from) the ground state scales as I?-:’ and is extremely low for high states. But this does not necessarily imply that it will be more difficult to optically excite higher states, since the probability of returning (by spontaneous as well as by induced emission) is also smaller. This transition probability A L R between a Rydberg state IR) and a low state IL) is small because in the relation ALR 0: & l ( f ? l r l L ) 1 2 the dipole moment is small, the wavefunctions of H and L having very little overlapping. As concerns the transition probability between two high Rydberg ~ ( R ~ r ~ Rit ’is)still ~ 2 much , smaller in spite of the fact that states A R ’ R i)iR, H and H have a large overlap. This is due to the factor Y : ’ , the effect of which is more important than the effect of the dipole moment. The . s p o t i tirtrrorrs transitions between Rydberg states thus have very little probability; this is, of course, no longer true for induced transitions.

Even now, it is not always possible to selectively prepare atoms in isolated ( t i . 1. m,)Rydberg states. and. of course. this was the general case before the advent of lasers. This selective preparation is not always necessary, however, and many experiments can be made with a mixture of Rydberg states if the rest of the experiment is selective.

’ In fact, most commonly the atoms experience a pulsed regime even when a continuuuh wave (CW) laser is used since. due to their motion in a vapor or in a beam, they see the laser beam during a tinite time. trsually shorter than their lifetime.

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Serge Feneuille und Pierre Jucquinot

Let us now note, however, that it is possible to isolate, for instance, in a broad mixture of n states, the effect of a narrower band of n values. An example was given by Bayfield and Koch (1974) in the study of multiphoton ionization of highly excited hydrogen atoms. They switched a static electric field F between two different field strengths and measured the difference between the signals observed in the two cases. The first field Fn, quenches, by field ionization (see Section V,B), all states with ti > n’ while the second one, F , . quenches all states withn > 1 1 ” . The effect of the band n ’ n” is then obtained by difference; in the quoted casen‘ = 63,n” = 69. A n improved variant of this method (in the case of CW excitation) consists of modulating the quenching field around the value F,, at which the states ti are ionized. The modulated signal is then measured by synchronous detection and gives the effect of states n only. This method has been used by Riviere and co-workers [see II’In (1973), where useful references are given]. Of course, it is much better, and in fact most often necessary, to prepare well-defined states, and this is now possible by optical excitation with highly monochromatic lasers. It may not be sufficient, however, to excite atoms with a well-defined energy to obtain a well-defined nlml state since states with different values of1 (I >> 1) and m lhave the same energy. An interesting method for exciting states with well-defined, high values of 1 was given by Freeman and Kleppner (1976) and used several times by Gallagher and his co-workers (see, e.g., Gallagher and Cooke, 1979; Cooke et ril., 1978). This “Stark-switching’’ method consists of exciting the atoms by a light pulse in the presence of a static electric field such that the Stark levels are well separated and 1 is no longer a good quantum number. Definite ti, n , n,, Im( Stark states (see Section IV,B) can then be populated (there is no longer any 1 selection rule) (see Fig. 6, Section IV,B). If, after the pulse, the Stark field is switched to zero, the atoms can be found in a definite 1 state. Since m always remains a good quantum number, the tn values obtained are still subject to the selection rule Am = 0, + 1 , according to polarization. ~

I . Preptiration by Electron Trunsfer

When ions are sent with an energy of several kiloelectron volts (e.g., 10-100 keV) through a gaseous target of neutral atoms or molecules (hydrogen, noble gas, cesium, etc.), a certain fraction of them (e.g., 0.05-0.3) are neutralized by electron capture and left in the ground state or in an excited state. This phenomenon was theoretically studied long ago by Oppenheimer (1928) and then by Jackson and Schiff (1953) in the case of protons colliding with atomic hydrogen. They showed that the number of

105

ATOMIC RYDBERG STATES TABLE I" T Y P I C PARAMETERS ~L FOR HYDROGEN RYDBERG STATES EXCITATION ELECTRON TRANSFER Parameter H + beam intensity Ion source Gas target

NeutraVion conversion efficiency H (n = 10) fraction in neutral beam H (n = 10) beam flux I'

Real experiment 0.6 pA Rf discharge Xenon (thickness 5 x atoms/cm') 0.07 2 x 10-4 5 x 10' atoms/sec

BY

Improved source >10 PA Duoplasmatron Cesium

0.3 4 x lo-:'

7 x

atoms/sec

From Bayfield (1976).

atoms in excited states scales as n-" and that essentially s and p states are produced. The experiments made since then have confirmed these conclusions up to n = 80 (Bayfield and Koch, 1974). This mode of preparation is thus absolutely nonselective. It was used by Riviere and Sweetman (19641, II'In (1973), and Bayfield and Koch (1974) for up ton = 70 Rydberg states of hydrogen. Since a larger part of the ion beam is not neutralized, the ions remaining after the traversal of the gas target must be deflected by an electric field. Typical values of experimental parameters, results effectively realized, and what could be made in an improved experiment have been given by Bayfield (1976) for a 10 keV proton beam, and are reproduced in Table I. The number of atoms produced in states withn > 10 can be found by using the n P scaling rule. This use of a fast H Cbeam and charge exchange can be combined with laser excitation, the optical excitation starting from a low-lying Rydberg state, for instance, n = 10 (Bayfield, 1976), as we shall see later on. 2 . Preparation by Electron Bombardment

This method, which is not more selective than the preceding one, has been used for many years by various authors. It seems that the first observation that electron bombardment could leave atoms in high Rydberg states was made more or less by chance by Cermak and Herman (1964). They were studying collision reactions of rare gas atoms in metastable states excited by electron bombardment (0-100 eV) in an atomic beam. As the collision chamber was located some centimeters away from the excitation region, some observations could be explained only by assuming that long-lived states of much higher energy than the metastable

106

Serge Feneuille and Pierre Jacquinol

state were produced, which they were able to identify as Rydberg states. Later on, Hotop and Niehaus (19681, Kuprianov (19691, and Shibataet ril. (1974) made the same type of collision studies with the Rydberg states of hydrogen, carbon, nitrogen, etc. The most extensive use of this mode of preparation was made by McAdam and Wing (1975, 1977) in their study of the fine structure of highly excited states of helium. The experimental setup was quite simple; the helium pressure was about lo-” Torr, the electron current 100-500 PA, and the electron energy 30-150 eV adjusted to the most efficient value for exciting the states under study. The excitation process itself was studied by Schiavoneet al. (19771, who measured the cross sections of excitation by low-energy electron impact on helium atoms and the effect of I transfer to higher I states. Helium atoms were excited by 0-500 eV electrons, in an excitation chamber filled Torr, the electron current being about 30 with helium at a pressure of FA. Rydberg atoms with ti = 15-80 were detected by field ionization in another chamber 20 cm away so that the mean flight time was about 100 psec. The Rydberg states were initially created by the electron impact with an I value differing by 0 or +-1 from the ground state. In these experiments, however, an important role was played by the transfer to higher I states due mainly to collisions of Rydberg atoms with the electrons and to a lesser extent with atoms in their ground state. Excitation curves Z ( E , E ) have been measured as a function of the electron energy E for various bands ofn values aroundE = 46,36,27, . . . (mean value). All of them show a step threshold at E = 25 eV and then decrease more or less slowly. In those experiments the variation of the cross section u as a function of n was not really measured since too many parameters were involved in the observed signal. It was only assumed that u,,,(n, E ) = u,,,(l, E ) n - “ , in accordance with the usual assumptions of Rydberg orbital overlap with the ground state. The effect of n was taken into account by putting an “analyzing” field F between the collision chamber and the detection chamber so that all the states withn 2 n, and F = (2nc)-‘ (in atomic units, see Section IV,B,4,b) were ionized and by measuring the signal due to the remaining Rydberg states Z(F). If the scaling of a,,, had been very different from n +, the observed results would have been self-contradictory. Finally an absolute value of the cross section was given: u,,,(n. E

=

100 eV)

=

(9 2 5)10-’c3 cm2

One should also remember the importance of the transfer to high I values by collisions with electrons. The absolute cross section that was measured was & - = 5 x lo-il.;?il.:3l In( 1OOEn ‘)n?! - I cm I/

ATOMIC RYDBERG STATES

107

( E in eV). Here a : is not precisely defined. It includes the transfer to all states with higher 1 states (i.e., with longer lifetime) but with the same / I (the processes with A H # 0 are much less probable). This huge cross section is of importance in all cases where Rydberg states and electrons are both present (e.g., plasmas, interstellar space). 3. Opticd Excitcitioti

In contradistinction to the preceding modes of preparation, optical excitation may be highly selective since it is a resonant process. In fact, it has been used since the beginning of the history of Rydberg states but has become most extensively used since the advent of lasers. The nature of the excited states is dictated by the selection rules AI = 1 and Am = 0, ? 1, according to the polarization used. However, it is easy to overcome the A1 rule by applying a static electric field, which can be small since Rydberg states are very sensitive to this perturbation. This field can be chosen small enough so as not to displace or split the levels but still to produce a mixing of states such that the s or d states can be excited from an s state. It can also be chosen large enough to produce a wellresolved Stark splitting so that “Stark switching” can be used to populate states with higher values of 1.

*

Diferetit schemes .for loser excitatio/i. If the optical excitation is made with a laser, a very high resolution can be achieved. To take full advantage of the narrow bandwidth of single mode lasers (currently 1 MHz, sometimes even 2 kHz!) atomic beams or Doppler-free, two-photon excitation are frequently used. With lasers now available, a great variety of excitation schemes can be used. They differ in (1) the type of laser used-CW or pulsed; ( 2 ) the number of excitation steps: (3) the one- or two-photon character of the transition; or (4) the fact that the laser excitation is combined with another type of excitation. It is impossible to describe all the processes which have been used; only a rough classification and a few significant results are given hereafter. One-step excitation from the ground state is possible only for those atoms possessing low ionization potentials, and thus is well adapted to alkalis but excludes hydrogen, and noble gases, for example. Even for alkalis short-wavelength pulses are required (e.g., A < 300 nm for rubidium), usually obtained by frequency doubling. This process was used mainly for rubidium by Freeman and Bjorklund ( 1978)(external frequency doubling), Ferguson and Dunn ( 1977) (intracavity frequency doubling), and Pinard and Liberman (1977) (external doubling of a single-mode, pulsed laser).

N.

I08

Serg e Fen e id ille u nd Pierre Jacqu in ot

Multistep excitation from the ground state is the most frequently used scheme since states as high as a few electron volts can be reached by means of wavelengths easily produced by currently available lasers. The first step usually populates the resonance level (it has been sometimes made with a conventional source emitting the resonance line; see, e.g., Svanberg et d., 1973). Some examples are given in Fig. 1. Two points are of importance in these multistep processes. ( 1 ) Polrrrixtiotis of’ tlw different becrms. By a proper choice of the polarization of the laser beams it is possible to populate states with given inl or lmll and hence with given1 since I 2 lmll. For instance, if only (ml(= 2 states are excited, this excludes the s and p states. Circular polarizations are sometimes used. Due to the Am, = 51 rule for circular polarization (J?, the I states that can be populated depend upon whether the different beams have the same sense of circular polarization or not. For example, in excitation from an s ground state in N steps with all the same sense of

1

p

5pns 5pnd

6s75

3a FIG.1. Different examples ofpure/>?o p tir d multistep excitation of Rydberg states. (a) Two-step excitation a sodium used by almost all the authors (e.g., Haroche et a / . . 1974; Ducas er u / . , 1975; Ambartsumyan ct d . . 1975; Leuchs and Walther, 1977; Vialle and Duong, 1979; Jeys et ul., 1980). Quite similar schemes have also been used for alkaline earths; see, for example, the excitation of 6sns:’S1and 6sttd3D in ytterbium by Camus t’r d . (1980). (b) Three-step, three-laser excitation used by Littman el a / . (1978) for lithium. A similar scheme was used by Eshenck er a / . (1977) for alkaline earths [(rns)2’So-,msmp3P, -+ tns(m + I)s3Sl+ rnsng’P]. See also Bekov et N / . (1979) for ytterbium and gadolinium, and Solarz et (I/. (1976) for uranium. (c) Three-step, two-laser excitation, including an intermediate step by spontaneous decay, used by Zimmerman et a / . (1979) for cesium. A similar scheme was used by Fredriksson er d . (1980) with the sequence 6s’ -,6s7p -+ 6sSd -+ 6snf. (d) Three-step, three-laser excitation of autoionizing Rydberg states. In the last step a second electron is excited, while the Rydberg electron acts as a spectator. (e) Another way of exciting autoionizing Rydberg states of strontium. Here the first step is a two-photon transition, and two electrons are excited in the second step (Freeman and Bjorklund, 1978).

ATOMIC RYDBERG STATES

109

polarization, in the final state (ml(= N , and it can only be a l = N state. If one of the steps has a different sense of polarization, several I states can be populated. These properties have been exploited, for example, by Vialle and Duong (1979) and by Ducas et al. (1975). (2) Delay between pulses. In pulsed operation the pulses corresponding to the different steps can be delayed with respect to each other by a time shorter than the lifetime of the intermediate states. In effect, it may be advantageous to have a short delay between the pulses instead of shooting them exactly at the same time. Different spurious effects, such as the broadening-or splitting-of the intermediate level under the radiation of the first pulse or multiphoton transitions, are thus avoided. But if the delay is too long, while remaining shorter than the lifetime, the preceding results ( 1 ) concerning the effect of polarizations may become false if the atom has a nuclear spin. In effect, the selection rule for the first transition is AmF = ? 1 , but the orientation of I is not immediately affected by the excitation so that Am, = 0 and Am, = AmF = ? 1 (rn, and rn, are no longer good quantum numbers). If the second pulse arrives immediately thereafter, there is no effect the nuclear spin, and the conclusions of ( 1 ) are still correct. But if the second pulse is delayed, the atom has time to make its own evolution at the hyperfine period 1/Avhfs, which can be of the order of a few nanoseconds so that several rn, states are present when the second pulse amves. In this case, several I Rydberg states can be produced, even with steps having the same sense of circular polarization. This has been verified by Ducas e f al. (1975) in the two-step excitation of Na: 3 ‘Sl12+ 3 2P3,2+ Rydberg state. When the second pulses were delayed by 6t < 3 nsec, only ?D states were produced, whereas and ‘D states were produced for 6t > 8 nsec. Two-photon excitation is adapted to the same type of atoms as the one-step, one-photon excitation with frequency-doubled laser pulses. The need for a well-located relay level is always more or less satisfied in the case of high Rydberg levels, so that the excitation is relatively easy to obtain. According to selection rules, only states of the same parity as the initial state can be prepared in this way. This method has been used either at low resolution without attempting to suppress the Doppler broadening [e.g., Popescu et al. (1974) for cesium and Esherick e f al. (1976) for calcium and strontium] or at high resolution with cancellation of the Doppler broadening by reflecting the laser beam back on itself so as to have counterpropagating photons. A good example has been given by Harvey and Stoicheff (1977) and Stoicheff and Weinberger (1979a, b), who excited the ?Sand ‘D states of rubidium up ton = 116 and 124, respectively, with a cw dye laser ( A -- 595 nm) of 50 mW and 10-15 MHz bandwidth (see also

110

Serge Feneuille and Pierre Jacquinot

Harper and Levenson, 1976, for krypton up ton = 19). The most striking example, however, is that of Lee et ul. (19791, who excited rubidium n 'D levels (n = 30) in an atomic beam with a CW dye laser ( - 3 5 mW) of extraordinarily narrow bandwidth (-2 KHz) and observed resonances as narrow as 17 KHz in a Ramsey fringe arrangement (two excitation regions along the beam, separated by 4.2 mm). A two-photon transition can also be used as one of the steps of a multistep excitation (see Fig. 1 ) . b. Combinution with other types q f excitution. In the case of elements with a high ionization potential (noble gases, alkaline earths), it is possible to populate a metastable state in a discharge or in an atomic beam by electron bombardment, and from there to use a one-step or a multistep or a two-photon transition to reach Rydberg states. A discharge was used, for instance, by Camus et uI. (1979) to excite, levels of the type 5dtd in barium by a two-step process from a metastable state. The excitation of ions in a metastable state in an atomic beam by a low voltage dc discharge taking place immediately at the exit of the oven was used by Barbier and Champeau (1980) to excite 6snd:'D1 levels of ytterbium from the metastable 6s6p:'P0 state, by means of a UV, pulsed, single-mode dye laser. The excitation of metastable ions in an atomic beam by electron bombardment was used by Stebbings et ul. (1975) in the case of xenon, for instance. A small fraction of atoms was excited to the metastable :3Po(5p6s)state by an electron gun placed around a xenon beam at the exit of the beam source, and states 2P312np and 'P312nfwere excited from there by a pulsed dye laser. In the experiments of Popescu ef d.(1973) on cesium, the atoms in a cell were excited by electron impact in a low-voltage diode to 6p states, and from there to nd states by optical excitation. The states from which the optical excitation starts can also be populated by electron transfer to ions (see Section II,A,l). This has been done in fast hydrogen beams with a fixed-frequency laser beam merged collinearly with the atom beam over a length of 1 m (to ensure a sufficient interaction time). The tuning to the wanted transition was obtained by adjusting the voltage accelerating the initial protons so as to obtain the correct Doppler shift. Two types of excitation were used by the same group. In the first, the U V Ar' laser line at 364 nm made the direct transition from the metastable 2s to 40 < t f < 55 (Koch et d., 1976). In the second (Bayfield, 1976), a CW COz laser (2.5 W) was used to excite transitions from n = 10 states (percentage in the neutral beam, 2 x lo-') to n = 44-50 states with an efficiency of 1%; according to the author, with an improved device, 3 x 10'" atoms per second at n = 44 could be obtained. These experiments

ATOMIC RY DBERG STATES

111

seem to be the only ones in which Rydberg states have been selectively prepared in hydrogen.

c. Light intensities necessury to popiilute Rydberg stutes. It is usually said that higher intensities are necessary to populate higher states since the transition probability, scaling as n F L . is smaller. This agrees with experiment and seems quite natural. However, the reason for this behavior is not so obvious if the phenomenon is analyzed carefully. The essential point is that, for high states, the natural width y = T - I (e.g., 10 kHz for a n = 30 p state) is very small compared to the width of usual lasers used in this type of work. This, of course, is no longer true for lower states or in very rare cases (see, e.g., Lee et a l . , 1979; rubidium n = 32 states excited with a laser of only 2 kHz bandwidth). The most frequent case is then a case of “broad-band excitation” (6v,>> y ) , and the phenomenon can be described by rate equations. If it were possible to work in a truly stationary regime, one would find that the intensity needed to excite a Rydberg state does not depend on n since the deexcitation probability decreases as the excitation probability. In fact, however, this condition of stationary regime is practically never fulfilled, even with a CW laser, since the transit time 8 of the atoms across the laser beam is short compared to Rydberg states lifetimes T. In this case ( H / T << l ) , it is easy to find that the intensity I,, (in number of photons per second and unit area) necessary to excite a fraction a of the atom is This formula is valid only if there is no Doppler broadening, which is the case for experiments made with atomic beams. 4, is, of course, inversely proportional to the interaction time 8 (i.e., the shorter of the pulse duration and the transit time), but it is also proportional to T , which scales as n3. We thus find the known result. The preceding formula also shows that what counts is I J S v , , i.e., the spectral density, which was to be expected in broad-band excitation. d. Lasers t o be used.

Dye lasers have been used in practically all the experiments. Pulsed dye lasers follow, to a great extent, the arrangement first proposed by Hansch (1972), where the pumping is made by the light pulses delivered by a nitrogen laser. Typical conditions for the most simple arrangements are: pulse durations 3-10 nsec; peak power, a few kilowatts; bandwidth A v l , 5-10 GHz. These conditions usually permit, This remains approximately true if one takes into account the probabilities of spontaneous transitions to all intermediate states.

112

Serge Feneuille and Pierre Jacquinot

according to the formula given in the preceding paragraph, a comfortable excitation of high-lying Rydberg states. External amplifying cells, excited by the same nitrogen laser, are sometimes used. Dye lasers oscillating in CW regime are now commercially available. It is easy to make them work on a single mode and to scan and stabilize them. They are then well adapted to high-resolution spectroscopy. Their power is largely sufficient for any kind of work on Rydberg states, even for two-photon excitation, provided, in that case, that the beam is concentrated in a waist of the order of 10 pm. In the case of one-step, one-photon excitation, frequency doubling is necessary. This is made by means of an optical nonlinear crystal (ADA, ADP, KDP, etc.) placed inside the laser cavity. For generating single-frequency, tunable, pulsed U V light, an interesting and original device has been set up by Pinard and Liberman (1977), and it has been shown to be quite useful in the extensive high-resolution studies of Rydberg states. It consists of a single-mode, adjustable dye laser, which is submitted to both CW and pulsed excitation. Under the proper conditions, it has proved to remain single-mode operating when so adjusted in CW regime. Such an arrangement can be thought of as an injection-locking laser in which the CW single-mode oscillation plays the role of injected light, while pulsed excitation provides high-power laser action. An advantage of the system compared with injection locking lies in the fact that it uses the same laser cavity and thus avoids the delicate problem of coupling two independent cavities on the same mode frequency. The single-mode pulses thus delivered by the laser can be amplified through one or two external dye cells and then frequency doubled through a nonlinear crystal. Pulsed excitation of the laser as well as of the amplifiers is provided by a frequency-doubled Nd-Yag laser. Single-mode peak power higher than 200 W has been obtained in the visible range, with a pulse duration of 120 nsec, at repetition rates of up to 50 Hz. After frequency doubling, one obtains U V single-frequency pulses of a few tens of watts peak power and a duration of about 6&70 nsec. The U V linewidth thus obtained is less than 50 MHz.

B. DETECTION OF RYDBERG STATES The detection may be used not only to get a measure of the populations in Rydberg states; it may also be useful to determine in which kind of nlm states are the atoms as a result of the preparation or of any change intervened between preparation and detection: this is the selective detection. This selectivity may be relative to the energy or to the nature (essentially

ATOMIC RYDBERG STATES

113

1 and m ) of the states. We shall see methods permitting both types of selectivity, as well as nonselective methods of detection. Since the population of Rydberg states is usually rather poor (low atom density to avoid collisions, plus a small fraction of atoms excited in high states), an essential quality of the detection is its sensitivity. There are methods permitting one to detect and to count almost every atom in a Rydberg state. The various methods that we shall see now rest directly upon the destruction of the Rydberg states. This is due to the fact that they make use only of the change of internal atomic variables produced by the excitation to Rydberg states. I . Detection bv Fluorescence

Except for the old method of absorption, which is not very sensitive, the most direct way of detecting Rydberg states is to measure the fluorescent light emitted by them either directly toward the ground state or by some cascade. In effect, every excited atom will emit, sooner or later, one photon (or several different photons in the case of cascades) by radiative deexcitation unless it is quenched by some completely nonradiative process (increasingly probable with increasing n ) . The energy stored by a Rydberg population may thus be completely refunded in the form of photons, regardless of the value of n (in the absence of a nonradiative process). But the sensitivity of the method depends on the rate of fluorescence, which is proportional toNy, N being the number of Rydberg atoms and y scaling asn-3. In a true stationary regime (CW excitation, atoms not escaping from the observation region), the sensitivity would thus scale as n -3. Of course, the effect of this factor can be compensated for by counting the photons during a time scaling as n 3, the duration of the experiment being thus proportional to the lifetimes of the Rydberg atoms. In addition, if the Rydberg atoms spend a part of their lifetime outside the observation region (which happens in the case of a fast atomic beam or a thermal beam for very long-lived states), a part of the emitted photons is lost. In the limiting case of an observation region of a given length 1 (along the beam) small compared to the lifetime length L , the fraction of observed photons l/L would also scale as n -3 since L scales as n 3 . In this case the sensitivity would scale as n - 6 ! We thus see that this type of detection will be limited to the lower Rydberg states. The same considerations hold in the case of pulsed operation. For a given value of the number N of Rydberg atoms produced by each excitation pulse, the number 9 of counted photons will be independent of y (and thus of n ) if the counting time 0 is long compared to lly. If 0 is matched to a small fraction k << I of lly, k being kept

114

Serge Feneuille and Pierre Jacquinot

constant as n varies the sensitivity scales as n-". This is particularly the case when a time resolution is wanted, for instance, for lifetime measurements or quantum beat experiments. Instead of observing the direct fluorescence from the Rydberg state to the ground state, one sometimes observes the last step of a cascade. This does not change the preceding considerations. Of course, the absolute value of the sensitivity depends on the quantum efficiency of the detector and on the ratio W4n, 0 being the solid angle in which the photons are gathered; the product of these two factors can In difficult cases the photon-counting techbarely be better than niques are very useful. It is also necessary to get rid of the laser light directly scattered by the walls of the apparatus. Whenever this is possible, it is good to take advantage of a cascade to separate the observed wavelength from the excitation wavelength by means of a filter, a monochromator, or the response of the detector. Examples of detection by fluorescence are given in Section 111. Although detection by fluorescence could, in principle, be energy selective, this is very rarely used. The main reason is that usually the radiation that is observed is the last step of a cascade, the wavelength of which does not depend on the Rydberg state. Much more interesting is the state selectivity since the transition probability for the first (or unique) step may strongly depend on quantum numbers other than n . This is the principle of the experiments discussed in Section 111,A,3. 2 . Detection by Field Ionization

Rydberg atoms are easily ionized by a dc electric field, and the ions or electrons thus produced can be counted almost without loss. The minimum field necessary to ionize a state of effective quantum number n* is upproximately F, = 3.21 x 10' x n*-'(in V/cm) (this phenomenon will be studied in detail in Section IV,B,4,b). Field ionization detection (FID) is thus well adapted to very high states and appears as complementary of the detection by fluorescence. As every Rydberg atom is very rapidly ionized when F is applied and every ion (or electron) can be counted, the sensitivity of this method is the highest possible. Moreover, with some precautions it may be used for selective detection. The first use of FID seems to have been made by Riviere and Sweetman (1964); see also Riviere, 1968) on Rydberg hydrogen atoms. Since then, it has been used in almost all the experiments on high-lying states made with atomic beams. Only a few examples are quoted here: sodium, Ducas et al. (1975); xenon, Stebbings et al. (1975); rubidium, Duong et al. (1976); sodium, Leuchs (1979).

ATOMIC RYDBERG STATES

115

The experimental arrangement may be quite simple. A dc electric field F is applied to the atoms ccfrer their excitation to Rydberg states. Of course, a delay (shorter than the lifetime) is necessary (otherwise the atoms would be excited in Stark states, not in zero field states). In the case of an atomic beam, this delay can be obtained by placing the ionization region downstream of the excitation region (Riviere and Sweetman, 1964’)). In this case both excitation and ionization can be made in CW regime. However, most often, the two regions coincide, and both excitation and ionizing field are pulsed, with a delay T of the order of 1 psec. Usually the field F is applied between two plates so that it is perpendicular to the atomic beam. The ions or electrons are counted with an electron multiplier gated during a time T ’ (e.g., 1 psec). In this way one detects all states for which the ionization rate is larger than 1 / ~ (e.g., ’ lo6 sec-’) at field F . 4This method is not really selective since it does not distinguish between states whose ionization threshold is smaller than F . Whenever true selectivity is needed, FID may be improved in different ways. One way, as proposed by Riviere and Sweetman (19641, consists of using a modulated field F(r) = Fa + N cos w t (cr << F a ) . The ionization signal is then modulated (at frequency w ) only if Fa coincides with the ionization threshold of a populated state. The modulated part of the signal is measured by means of a synchronous detection. By varying Fa one can thus detect successively the different states. Of course, this method requires a continuous excitation, for instance, in the case where the excitation region is spatially separated from the ionization region, as can be obtained with an atomic beam. Another way used by Gallagher et 01. (1977b) and by Fabre et ul. (1977) consists of making a time resolution of the ionization signal during the rise time of the ionizing pulse (here it is preferable to detect the electrons rather than the ions since their transit time is shorter and better defined). Figure 2 shows signals corresponding to two states a and b with threshold values Faand Fh. Instead of recording the ionization signal as a function o f t , one can also gate the detection at the time ti corresponding to a state of threshold Fi so as to detect only a state i , for instance, to study a transition falling on (or starting from) i. This selective mode of FID is more or less equivalent to a “Rydberg ionization spectrometer. The practical resolution of this spectrometer permits one to separate states the n“ of which differ by more than 0.5, around n = 30. ”

In these experiments, the field was parallel to the beam and was applied between two bored electrodes. In the special case of spectroscopy of “stable” Stark states in an electric field Fo, it might be useful to distinguish these “stable” states from states spontaneously ionizing in the presence of Fo. In this case the detecting field pulse F must be delayed by a time T after the laser pulse so that the states with ri > 7 - ’ have disappeared when the detecting pulse arrives.

’

Serge Feneuille and Pierre Jacquinot

116

______--_--_----

Electric field pulw F(t)

/

Ionization signal

t ,

I

1

1

1~

t

Boxcar gate

Rc,. 2. Selective detection by time resolution of the ionization signal. Fck,and Fcbare the critical fields for states a and b, respectively; they are not necessarily in the same order as lEztl and [&I. Only state a is detected if a gate is open at f a .

Once the states i corresponding to given values Fi have been unambiguously identified, this method is valuable and has been used with success for the detection of radiofrequency resonances between Rydberg states. Field ionization detection has also been used with success by Leuchs (1979) to detect quantum beats between close Rydberg states. Although ionization by a dc field seems to be the most convenient, ionization by a microwave field has been used by the Bayfield group not only as a subject of study but also as a means of detection. Let us remark that it would be more logical to consider this as a detection by photoionization (microwave photons). Since this is a rather peculiar method, the reader should refer to the different publications of this group (e.g., 1977) for more details. Bayfield et d., 3. Detection by Collisiontrl Ionization in a Space Charge Limited Diode

This mode of detection, applicable only to gases or vapors in a cell, rests upon collisions between Rydberg atoms and atoms in their ground states. Different ionization processes are possible, for instance, in the case of cesium

cs**+ Cs

j

Cs:

+ e-,

CS"" T Cs

+

Cs;

+

Cs+ + Cs-

ATOMIC RYDBERG STATES

117

Of course, this method of detection is by no means selective since it detects all the states high enough to be ionized by these collisional processes. A strong amplification of the ion signal can be obtained if the ions are created in the region surrounding the cathode of a diode working in the space charge regime. This is due to the fact that, when an ion is created in this region, it is imprisoned by the space charge during a time which may be of the order of milliseconds. During this time the space charge is lowered so that more electrons can escape toward the anode, giving rise to a current pulse, the total charge integrated over this pulse being of the order of 10*'-106electrons. The amplification phenomenon was known long ago (Mohlerer al., 1926), but was used for the detection of Rydberg states for the first time by Popescu et al. (1966). Since then it has been used by several authors. The device is quite simple and very sensitive. The cathode of the diode is usually a tungsten wire heated by an electrical current (Popescu et al., 1969, 1973; Harvey and Stotcheff, 1977; Camus et al., 1978). When the vapor of the element under study is produced in an oven (or a heat pipe) at high temperature, the cathode does not need to be se;arately heated and can be either a filament (Aymar et al., 1978) or a plane electrode (Armstrong et al., 1977). The anode is placed very close (-0.5 cm) to the cathode and can be either a cylinder (Aymaret al., 1978) or a plane electrode (Popescu et al., 1973; Armstrong et al., 1977). The best operating conditions (filament temperature, pressure, anode voltage) must be found experimentally in each case; usually the best anode voltage is of the order of 0.5 V and the pressure of the order of a Torr. If the excitation is produced by a pulsed laser, the current pulses are transformed into voltage pulses in a resistor and transmitted to an amplifier and integrator by means of a condenser. Different variants have been proposed. For instance, to secure an interaction region between the light and the atoms free of electric field, Harvey and Stoicheff (1977) divide the anode into two compartment (each 2 cm long and 7 x 7 mm2cross section) separated by a fine nickel mesh, the hot filament being centered along the axis of one compartment (detection compartment). The laser beam propagates along the axis of the other (equipotential) compartment, and the ions diffuse toward the detection compartment. Another variant allows a two-step excitation and uses two similar diodes very close to each other so that the ions formed in one of them can diffuse in the space charge of the other. In the first diode the atoms are excited to the first resonant state by electron impact and from there to high states by laser excitation; the ions are detected by the other diode (Popescu et al., 1973). The extreme sensitivity of this method of detection is due to the fact that the ionization efficiency of Rydberg atoms is close to unity and that every ion is counted with very high gain.

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Serge Feneuille and Pierre Jacquinot

4 . Optogulviinic Detection

The optogalvanic effect (Meissner and Miller, 1953; Green et al., 1976) consists of producing a change in the voltage-intensity characteristics of a discharge, produced by an illumination resonant with a transition possible in the discharge. It can be used as a means of detecting transitions between states populated by the discharge and high Rydberg states. In that case the discharge serves both as a means of producing the first step of a multistep process and as a means of detecting the last step. The optogalvanic effect is due to the fact that states which are easily ionized in the discharge are either depleted or overpopulated by the optical transition, thus changing the voltage (at constant current) across the tube. Depending on the states involved in the transition, the voltage can be increased or decreased (“positive” or “negative” signal). The interesting fact is that the change is surprisingly high, several volts or tens of volts, and thus very easy to detect. The first use of the optogalvanic effect for a Rydberg study was reported by Bridges (1978) on cesium in CW (chopped) excitation. Quite recently Camus et ul. (1979) used it for the study of Rydberg states of barium, with J values ranging from 0 to 5. A two-step pulsed laser excitation was used, starting from the 5 d 6 ~ : ’ D , ,metastable ~,~ levels populated by the discharge. In their experiment the barium pressure in the tube was -2.5 Torr, and a stable discharge was obtained with a source of 1.45 kV and a ballast resistor of 50 kR. The tube operates at 20 mA and 400 V, and the changes of voltage across the tube are transmitted to an electronic device through a capacitor. This device measures the amplitude of the transient following each laser pulse. This mode of detection has very attractive features: simplicity and a good signal-to-noise ratio. Of course, it is not selective. The fact that the excitation takes place in a discharge has some advantages (essentially the possibility of exciting from several intermediate states) but also some shortcomings for high-resolution work (the conditions are not as pure as with an atomic beam or a low pressure cell without discharge). 5 . Detection bv Photoionizu lion The cross section of the photoionization process is usually rather small, and by no means larger for high-lying states than for lower ones. This mode of detection would then not be sensitive, unless very intense laser beams be used. Fortunately, the frequency required to ionize from Rydberg states is low, and very powerful C 0 2 lasers exist at 10.6 pm. This was used for the detection of Rydberg states in uranium by Solarz et cil. (1976). It may also happen (Stebbings ef al., 1975) that photons from the

J

ATOMIC RYDBERG STATES

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exciting beam produce ions-with low efficiency-from the Rydberg states. Ions can also be produced by photoionization by one of the exciting beams in a two-step process if the intermediate level lies higher than midway in the ionization limit, thus giving rise to a spurious ionization background. Such a background can also be caused by multiphoton ionization from the ground state.

111. Spectroscopy As noticed before, the famous Rydberg formula describing the wavenumbers of a spectral series in an alkali spectrum was proposed as far back as 1890. The origin of the quantum defect was understood, at least qualitatively, from the very beginning of quantum mechanics (Waller, 1926; Hartree, 1927), and the deviations observed in many electron spectra with respect to the Rydberg formula were rapidly recognized as resulting from configuration interaction with doubly excited valence states (see, for example, Edlen, 1964). This means that the basic spectral properties of atomic Rydberg states were actually known for a very long time. Furthermore, conventional absorption techniques allowed one, during several decades, to record hundreds of spectral series and thus to analyze thousands of Rydberg levels. Under those conditions, it would seem that laser spectroscopy could not bring new significant information about the spectroscopy of Rydberg states, which thus appeared to be a fully completed subject. However, even for alkali metals, some problems actually remained open. For example, quantum defects were measured with rather poor precision, especially for high members (large values of n ) of spectral series and for states with high angular momentum, which prevents one from extracting useful information about the interaction between the Rydberg electron and the atomic core. Moreover, it has been known for a long time that low members of some spectral series exhibit anomalous fine structure patterns, in the positions as well as in the relative intensities of the lines, and the question arose as to whether these anomalies still appeared for highly excited Rydberg states despite their quasi-hydrogenic character. Obviously, the corresponding data could be obtained only by using ultrahigh-resolution techniques, which, in a way or another, require the use of tunable lasers. Furthermore, it gradually appeared that localized configuration interactions between Rydberg states and perturbing valence states were insufficient to really describe highly excited many-electron spectra, and there-

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fore many perturbers were not identified unambiguously. The solution was given theoretically by Seaton and his colleagues in 1966 (see Seaton, 1966) through so-called multichannel quantum defect theory (MQDT), which was reformulated in a more efficient way by Fano and his coworkers (Lu and Fano, 1970; Fano, 1975). Unfortunately, such a parametric approach requires a very large number of data, and preliminary investigations showed that, at the beginning of the 1970s, the number of spectral series sufficiently completed to permit a MQDT analysis was extremely small. Therefore, there was an actual need for new spectroscopic data, especially in two-electron spectra. High resolution was not really useful in this case, but many excited levels can be reached only through multiphoton transitions or multistep excitations, which again require the use of laser techniques. A . ALKALI METALS As mentioned above, only high-resolution studies can now provide exciting data in optical spectra of alkali metals. In laser spectroscopy, high resolution can be reached essentially through three types of different techniques. In the first, high resolution is achieved directly on optical transitions. This requires the simultaneous use of monomode lasers and Doppler-free techniques. In the two other ones, on the contrary, the Doppler effect introduces no limitation since high resolution concerns only radiofrequency or microwave transitions. The observation can be done either in the time domain (time-resolved spectroscopy) or in the frequency domain (double-resonance spectroscopy). In this case, the laser is only used to strongly populate some excited levels, either in a coherent way (time-resolved spectroscopy) or in a selective way (double-resonance spectroscopy). These three types of techniques have been used in the spectroscopic study of highly excited states of alkali atoms, and the corresponding results will be discussed separately. I . High-Resolution Optical Spectroscopy

The only studies of this type concern essentially the rubidium atom. This is not surprising if one remarks that half of the frequency required to ionize the rubidium atom just falls in the spectral range of rhodamine 6-G tunable dye lasers, which are the most convenient to work with in single-mode operation. Thus, with such lasers, highly excited n 2S and n 2D states of rubidium can be populated by two-photon transitions, and this technique is particularly interesting since it allows one to eliminate the

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Doppler effect. It was first used by Stoicheff and his co-workers (Harvey and Stoicheff, 1977) to measure the n 2Dfine structure from n = 1 1 ton = 85. Up to n = 32, optical resonances were detected by cascade fluorescence from the 8p level. More recently, Stoi'cheff and Weinberger (1979a, b) extended that study by investigating n 'S and n *D terms up ton = 116 and n = 124, respectively. Above n = 30, fluorescence detection was replaced by ionic detection. In all cases, the resolution was essentially limited by the commercial laser bandwidth to -2 MHz. It must be noticed that a similar, but not so refined technique, had been previously used by Curry el al. (1976) to measure the n2D fine structure of the cesium atom, but, as they used a multimode pulsed dye laser, the resolution was rather poor. Therefore, they were able to investigate only rather low members of the spectral series, namely, up to n = 19. Similar results were obtained in potassium by Shen and Curry (1977). Dopplerfree, two-photon spectroscopy using fluorescence detection has also been applied to the study of various Rydberg levels in potassium (Harper and Levenson, 1976, 1977; Harper el al., 1977), and in sodium (Salour, 1976). Another way to populate highly excited Rydberg states of rubidium with a rhodamine 6-G tunable dye laser is, of course, to double the laser frequency. Before the advent of ring lasers, the power delivered by CW lasers was too weak to allow such a doubling, and it is not easy to obtain the single-mode operation of a pulsed dye laser with sufficient power. As noticed before, the solution to this problem was first given by Pinard and Liberman ( 1977). When using one-photon transitions, the most convenient way to avoid the Doppler effect is to utilize a transversely illuminated atomic beam. With this technique Jacquinot et al. (1977) and Liberman and Pinard (1979) were able to resolve the fine structure of n 2P states of rubidium up to n = 60. The optical resonances were detected by field ionization. The resolution was limited to -40 MHz by residual Doppler broadening in the moderately collimated atomic beam. The most striking result obtained in the latter experiment, is the appearance in fine structure patterns of anomalous intensities which are nearly independent of n . 2 . Time-Resolved Spectroscopy

In quantum beat spectroscopy, the role of the laser is only to create a coherent superposition of excited states, and therefore the use of short laser pulses is required. Tunable dye lasers pumped by a nitrogen laser are well adapted to this purpose. After coherent excitation, the atomic system oscillates at the frequencies characteristic of the spacings between the various excited states. This oscillatory behavior appears, in particular, on the time evolution of the fluorescence emitted from the excited levels.

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Haroche and his co-workers first noticed that Rydberg levels are well suited for such a spectroscopic technique, and many fine structures were measured by these authors in the Rydberg n 2D states of sodium (Haroche et al., 1974; Fabre and Haroche, 1975). The precision is mainly determined by the time resolution of the detection device, and in the latter case was approximately 1 MHz. A very similar technique was applied respectively by Svanberg and his co-workers to an extensive study of the fine and hyperfine structure of alkali s, p, and d states (Fredriksson and Svanberg, 1976). However, the fluorescence intensity decreases as n -",and therefore, fluorescence detection cannot be used to study the high members of the spectral series. For example, in sodium, the corresponding studies of Haroche and his co-workers were limited to n = 16. This difficulty was recently solved by Leuchs (19791, who showed that quantum beats can also be observed on field ionization signals if one records in a given direction the number of produced electrons as function of the delay of the ionizing field pulse with respect to the laser excitation pulse. In such experiments, which have been performed on atomic beams, highly excited levels can be investigated and indeed, Leuchs measured fine structure intervals of n2D levels of sodium for values of n as large as 31. Let us remark that all quantum beat measurements also provide the lifetime of the considered excited level. The lifetimes of atomic Rydberg states are discussed in Section V. 3. Double Resonance Microwave Spectroscopy

The common origin of all the experiments of this type is to be found in the pioneering work of Wing and Lamb (1972) on the moderately excited states of the helium atom. In this work, which was recently extended by McAdam and Wing (1975, 1976, 19771, no laser is used since the first excitation is achieved by electron bombardment. As noticed in Section 11, this type of excitation is not selective, but the s, p, and d states are preferentially populated. Microwave resonances on the second step are detected by observing the fluorescence emitted from either one of the levels connected by the microwave transition. The strongest resonances observed in helium involve only one photon and are of the type n 'L -+ n 'v3(L + l ) , but An = 0 two- and three-photon transitions also appear, leading to a large number of precise data (uncertainties less than I MHz) on quantum defects and fine-structure intervals for values of n varying between 6 and 18, and values of 1 as high as 4. Similar techniques have been used by Gallagher and his co-workers (1976a,b, 1977a,b; Cooke et al., 1977a,b) to study A n = 0 transitions between Rydberg levels of sodium and lithium, the only difference being the use of multistep pulse laser excitation instead of electron bombardment.

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Here again, one-, two-, and three-photon resonances have been observed, leading to extensive and precise data on d , f, g, and h levels but, once again, fluorescence detection prevented the study of values of rz larger than 20. For more highly excited levels, selective field ionization detection must be preferred. This has been actually utilized by Haroche and his coworkers (Fabre et al., 1977, 1978) to systematically investigate An = I , 2 one- and two-photon microwave transitions, again between Rydberg levels of sodium, but now for values ofn between 23 and 41. The precision achieved to date is better than lo-" for microwave frequencies varying around 75 GHz, and recently allowed Haroche and his co-workers to precisely determine the energy dependence of the quantum defect of the n S series. Their final result is (Goy c't d.,1980) 6 J n ) = 1.3479692(4) + 0.06136(10)/(n - 1.348)'

A similar technique was also used by Gallagher and Cooke (1978) for measuring fine-structure intervals of highly excited 'D states of potassium. Let us finally notice that microwave spectroscopy on transitions between Rydberg states with high angular momentum could be a competitive technique for measuring the Rydberg constant. This was first suggested by Kleppner (1975), but to date, some particular difficdties still remain because of the extreme sensitivity of highly excited Rydberg states to external fields. 4 . Qrrantum Dcfect Theory for Alkali Atoms

As already noticed, the origin of quantum defect has been understood for a long time. Because of the extension of the atomic core, penetration and polarization effects appear, and thus, the valence electron, even highly excited, does not move in a purely Coulombic potential. In a central field model, calculation to first-order penetration and polarization energies involves, respectively, electron density at the nucleus and mean (quadrupolar polarizability) values of Y -' (dipolar polarizability) and for the outer electron. Taken between hydrogenic states, all these quantities behave as n -'J. Relativistic corrections have the same dependence, and this peculiar dependence actually justifies the validity of the Rydberg formula since obviously ( n can be approximated by n-' + 261n-''. Quantum defect theory has been extensively used for many years in the calculation of various atomic parameters, but, rather paradoxically, the first ub initio determinations of quantum defects appeared only a few years ago, following the nonrelativistic polarization model suggested by Deutsch ( 1970, 1976) for the particular case of helium. In this very simple

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model, penetration and interaction between singlets and triplets are neglected, and, actually, only dipolar polarizability is introduced. These approximations, which lead to a 1-5 dependence for the quantum defect, are particularly valid for Rydberg states of alkali metals with high angular momentum. It was shown, nevertheless, by Freeman and Kleppner ( 1976),that penetration effects and quadrupolar polarizability must be also introduced to reproduce the precise data of Gallagher et al. (1976a, b) on sodium in a satisfying way. The approach of Freeman and Kleppner is essentially parametric, and, because of the dynamic effects introduced by the motion of the Rydberg electron, the parameters they obtained are only phenomenological quantities. Such dynamic effects have been studied for the 4f states of sodium by Eissa and Optik (1967), but their n dependence is not yet known, and further investigations are certainly needed before significant comparison can be made between experimental data and ab initiu calculations. Such ab initio calculations, including exchange and relativistic effects, have been performed by Chang and Poe in 1974 for the particular case of helium, leading to excellent agreement with available experimental data, but much remains to be done with many-electron atoms. 5. Relativistic Effects

Anomalies in fine-structure patterns of some spectral series have been known for a long time. Concerning intensities, for example, the first observations were done in 1930 on the resonance lines of cesium for which the ratio R, of the intensities of the two components differs strongly from 2, that is, from the ratio of the statistical weights of the n 2P3,2and n 2P1,2 excited levels. Fermi (1930) gave the first qualitative explanation of this anomaly by introducing spin-dependent configuration interactions. The observation of anomalously small or even negative fine structure splittings is also very old, and these anomalies were known to be particularly dramatic on the lowest members of the n d series of sodium and the n f series of cesium. In the latter case again, one can understand the deviations observed with respect to hydrogenic predictions within the framework of spin-dependent configuration interactions. The most striking result in the considerable body of work experimentally done during the last few years by laser spectroscopy on the fine structure of alkali Rydberg states is that fine-structure anomalies appear even for very highly excited states despite their quasi-hydrogenic character. For example, all the experimental data (up to n = 16) on the finestructure interval of the sodium n 2Dlevels obey the empirical formula (Fabre et al., 1975)

ATOMIC RYDBERG STATES ANa

(MHz)

=

-

965,60O/(n

+

125

+ 498,50O/(n +

which is to be compared with the hydrogenic formula

AH (MHz) = 29,200/n3 Another example is provided by the ratio R,, previously defined, which, in rubidium remains approximately constant (R, = 5.9 & 1.4) for large values of n ( n > 30) (Liberman and Pinard, 1979), and which, in cesium, increases continuously withn up to the largest observed value (n = 30) for which R, = 1170 & 200! (Raimond et af., 1978). Therefore, although these phenomena already had been investigated theoretically in great detail, they were recently reinvestigated by many theoreticians. Two types of approach were used. In the first, following the pioneering work of Fermi, the Hamiltonian is restricted to the low Z Pauli operators, but the wavefunctions are correlated in one way or another. Concerning fine-structure intervals, it seems that the most efficient method is the one introduced by Sternheimer and his colleagues (1976; Lee et af., 1976; Foley and Sternheimer, 1975) using an exchange core-polarization model, spin-orbit interaction being taken into account for determining first-order core wave functions. It must be also noticed that multiconfigurational wave functions (Beck and Odabasi, 1971) and many-body theory formalism (Holmgren ef af., 1976) have also been used with success. Concerning intensity ratios, the most reliable results derived in this type of approach seem to be the ones obtained for cesium by Norcross (1973), who utilizes a semiempirical model potential introducing spin-orbit and core polarization effect. The second approach, which was introduced by Luc-Koenig (1976; Koenig, 1972; Luc-Koenig and Bachelier, 1978), is apparently quite different since it starts from an independent-particle (central field) relativistic model. Thus, the wave functions are not correlated, and despite this rough approximation, the agreement with experimental data is excellent, as well fine-structure intervals of sodium n 2Dlevels (3 d n S 16) and intensity ratios of principal series of cesium and rubidium. In the latter case, the calculations were carried out up to n = 80, and they show that R, increases rapidly with n toward a limiting value, in agreement with the data of Liberman and Pinard (1979). This result is also in agreement with the old prediction of Fermi (see zu Putlitz, 1969), but it must be noticed that other theoretical investigations lead to different behaviors for R,; some authors predict a maximum (Weisheit, 1972), while others obtain a continuous increasing with n (Norcross, 1973). Further theoretical and experimental work is certainly needed. In any case, it is rather surprising at first sight that two such different

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approaches can lead to such similar results. Actually, it has been shown formally that these two methods are rigorously equivalent, at least to order a* (Feneuille and Luc-Koenig, 1977), and this shows that the distinction between relativistic and spin-dependent correlation effects is somewhat arbitrary, a point which has often been missed in the literature. Let us finally remark that precise data on hyperfine structure intervals of highly excited levels of alkalis would be of great theoretical interest with respect to the previous matters. Of course, these quantities which also behave as n -3 are very small (of the order of a few kilohertz for n = 50 in rubidium), and therefore they are very difficult to measure. Some data are available for excited states of cesium and rubidium, but they concern only the lowest members of the spectral series. B. TWO-ELECTRON SPECTRA

For many years, absorption from the ground state has been the main technique used to study experimentally highly excited spectra of alkaline earths. Therefore, with the exception o f J = 1 odd levels, these spectra were sparsely known until recently. Moreover, alkaline earths have much more complex, highly excited spectra than the alkali metals. This is due, first, to the variety of possible couplings between the Rydberg electron and the unpaired s electron of the ionic core and, second, to the interaction between some doubly excited valence states and either the quasicontinuum of Rydberg levels or the ionization continuum, which leads in the latter case to autoionization. Multichannel quantum defect theory takes into account these various effects, but its practical application requires very numerous data. Therefore, one understands that, only a few years ago, the analysis of the highly excited spectra (bound or autoionized) of alkaline earths was quite incomplete and, in many cases, ambiguous. The situation was strongly modified during the last few years by the use of two experimental techniques: multiphoton laser excitation and ionic detection of resonances. Since these two methods have been described in Section 11, we focus here on the results. I . Obser\ved SerieJ

Most of the results recently obtained come from two-photon transitions starting from the ground state. Therefore, they concern even-parity levels, with J = 0 and J = 2 mainly belonging ton sn's and n sn 'd configurations. The first observation was made by Esherick and his co-workers

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on calcium in 1976. Seventy-two new states were observed and classified without requiring the introduction of new perturbers. Similar results were then obtained in strontium by Ewart and Purdie (1976) and by Esherick (19771, but in this case they showed the necessity of treating both configuration interactions and recoupling between the ID, and “D, series simultaneously. Since it exhibits many perturbers, the barium two-photon spectrum, observed first by Camus and Morillon in 1977 and completed in 1978 by Aymar et d . , is more complex. In the latter case, broad signals were observed in the spectral range of fluorescein (5350-5650 A) and interpreted as single-photon transitions, starting from the resonant 6s6p !PI level populated by the fluorescent light of the dye. The levels observed in this way are of the same type as those studied by photographic absorption spectroscopy from odd levels selectively populated with a dye laser (Bradley et d . , 1977; Rubbmark et ( I / . , 1977). Three-step processes have also been used in calcium, strontium, and barium (Armstrongat al.. 1979)for studying odd-parity/ = 1 levels and, in particular, the nsn’p”P:‘levels, which have been observed up to 1 1 ’ = 14 in calcium, n’ = 21 in strontium, and n’ = 43 in barium. In barium, in states, :jF! and ‘ G j states have also been measured and addition to the :%Po tabulated by the same technique. The observation of highly excited levels of barium with higher values of J (up to J = 5 ) has been recently made by combining two-step laser excitation with an optogalvanic detection (Camus et ( I / . , 1979). In the latter case, the excitation process starts from metastable levels, which are populated in a heat-pipe disthe Sd6s :1D:1,2,3 charge. The Sdnl bound configurations have been observed, as have the 6sn’l‘ level that they perturb, via the Sd6p intermediate levels. Despite the fact that ytterbium is classified as a rare earth, let us include in this paragraph mention of a very complete study recently performed by two-photon spectroscopy on even-parityJ = 0 a n d / = 2 levels of this element (Camus et ul., 1978, 1980). Indeed, a large part of the ytterbium spectrum has the characteristics of alkaline-earth spectra, since the only differences come from perturbing excited levels resulting from the excitation of the 4f closed shell. This point will be discussed later in Section III,C. Finally, some members of the doubly excited autoionizing series 6pnl were recently observed on an atomic barium beam (Gallagher et d . , 1979) by using three-step laser excitation 6s2 + 6s6p + 6snl + 6pnl ( n = 10- IS), and by detecting the ejected electrons. Similar observations had been previously made on strontium (Gallagher et ( I / . , 1978), and in both cases, interesting information was obtained on the I?*’ and I dependence of the autoionization rate.

Serge Feneuille and Pierre Jacquinot

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2 . MQDT Analysis The basic ideas of the MQDT introduced by Seaton (1966) and reformulated by Fano and his co-workers (Lu and Fano, 1970; Fano, 1975) are the following. Beyond a certain distance ro from the nucleus, the electron moves in a Coulombic potential and hence has an analytically known wave function; for smaller distances, it interacts with the ionic core, and the wave function of a discrete level is represented as the superposition of the wave function of “collision channels” i , which are characterized by a definite state of the ion, the state of the outer electron and their coupling. The mixing coefficients depend on two types of parameters, pa and uiu, which vary slowly with the energy and represent, respectively, the quantum defects associated with the “close-coupling channels” a of the system and the coefficients of the unitary transformation between the collision channels i and the close-coupling channels. If one denotes (Armstrong er al., 1977) the number of interacting channels by M and the number of different series limits involved (that is, the number of distinct states of the ionic core) by N , the N effective principal quantum numbers ut of a given level obey the following relations:

I,

-

R/v:

=

.. .

-

I,

-

R/v: =

* *

= IN - R/V$

( I i being the ith ionization limit), which determine a line 2 in the N-dimensional space of the u i s . Moreover, the asymptotic behavior of the wavefunction at r = = requires

detlurusin r ( u i +

CL,)~=

0

which describes a surface Y in the same space. Therefore, the positions of bound levels (i.e., the u / s ) are given by the intersection of the line 2’with the surface Y.Graphical presentations of the projection of Y and 2’on a given plane ( u i , u j ) are called Lu-Fano plots (Lu and Fano, 1970). The considered spectrum is analyzed by a fit of the adjustable parameters CL, and uru,which minimizes the energy differences between calculated and observed values. When these parameters are known, the wave functions are also determined unambiguously. It is beyond the scope of this review to describe in detail the numerous MQDT analysis which have been successfully performed on two-electron spectra during the last few years. Indeed, most of the references given in the previous paragraph contain a very complete MQDT analysis of the observed series. As an example, let us just briefly discuss the most complicated case studied to date. It concerns t h e J = 2 even-parity spectrum of the barium atom (Aymar and Robaux, 1979). Nine channels and four

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129

limits had to be introduced to accurately fit the energy values of allJ = 2 even-parity levels known between 30,237 cm-' and the first ionization limit, 42,035 cm-' (1 12 levels, rms deviation 2.44 cm-I). The accuracy of the interpretation of the high-lying levels was further maximized by excluding levels lower than 40,987 cm-' (6s13d3Dz). Using the corresponding parameters, the rms deviation between calculated and measured energy values is reduced to 0.2 cm-', 89 levels being considered. The mutual interaction between the 6snd'D2 and 6snd3D2, as well as the perturbations induced by the 5d7s, 5d8s, 5d6d, 5d7d, and 6p2configurations, have been clearly understood, leading to new assignments of some ID2 levels. Moreover, many energy values especially for the high-lying 3D2 levels have been predicted. Let us also notice that, starting from a MQDT analysis, various atomic quantities such as Lande factors (Armstrong et al., 1977) or oscillator strengths (Armstrong et al., 1979) have been calculated and successfully compared with experimental data. 3. High-Resolution Studies

In all the experiments described in Section 111,B,2, the resolution is Doppler limited. High-resolution investigations on highly excited twoelectron spectra have been much less numerous up to now, since they actually concern only two series belonging, respectively, to the spectra of barium and ytterbium. In both cases, the technique is the same: excited levels are populated by single-photon UV transitions starting from metastable levels in a transversely illuminated atomic beam. The technique utilized to populate the metastable levels is similar to the one described by Brinkman et al. (1969), and the single-mode, pulsed laser is the same as the one described in Section I1 with regard to the single-photon excitation of rubidium (Pinard and Liberman, 1977). The resonances are detected by field ionization. In barium, the only observed resonances were 6snd3D, + 6snf3F2from n = 30 and n = 70, and no information was obtained on the fine structure of the high-lying 6snf3F terms, which, in any case, do not seem to be significantly perturbed (Camus et af., 1977). The results obtained for ytterbium (Camus et al., 1977; Barbier and Champeau, 1980) are much more exciting since the observed transitions, which are of the type 4f146s6p 3P0-+ 4P%snd, exhibit a structure that strongly varies with n . For high values of n ( n > 50), the structure is mainly determined by the hyperfine structure of the ionic core 4P%s, and then one obtains a single line (4P46s6p3P0 4P46snd 3D,) for each even isotope and two lines for each odd isotope. For the odd isotopes (171, 173), it has been verified that the

-

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Serge Feneuille and Pierre Jacquino t

observed splitting is in agreement, within experimental uncertainty, with the hyperfine structure of the ground level 4P46s of Yb+ (Chaiko, 1966). This result illustrates the possibility offered by Rydberg states of studying the structure of the ground level of the corresponding ion in neutral atoms. For smaller values ofri (Barbier and Champeau, 1980), the residual interactions (spin-orbit and Coulombic energy) are sufficiently large to lead to a supplementary splitting of the resonances observed for the odd isotopes. The corresponding fine-hyperfine-structure patterns have been recorded and interpreted in a parametric way from n = 24 to 17 = 53, showing a perturbation in the vicinity of n = 26. This perturbation is induced only by hyperfine interactions and does not appear for even isotopes. This provides a new insight into configuration interaction in highly excited two-electron spectra, and a lot of work certainly remains to be done with respect to this matter. ATOMS C. COMPLEX With the exception of a study of xenon (Stebbings el a/., 1975), the only spectroscopic investigations of highly excited complex spectra by means of laser techniques concern lanthanides and actinides. A considerable body of work, partially unpublished, has been done in the Lawrence Livermore Laboratory in connection with laser isotope separation (Solarz et a/., 1976; Worden et al., 1977a, b). The experimental technique utilizes stepwise, pulsed laser excitation of an atomic beam, optical resonances being detected either by IR photoionization or field ionization. Because of their long lifetimes, unperturbed Rydberg levels can be distinguished from valence levels by delaying the ionizing pulse. Thousands of lines have been recorded by this way, but to our knowledge, their analysis still remains rather preliminary. However, this method has permitted a determination of ionization limits 10-100 times more precise than values available previously. Furthermore, MQDT analysis should be performed, but this is particularly difficult in heavy atoms with complex spectra because of the large number of channels and limits which must be introduced. Moreover, perturbing configurations with several open shells appear. They spread over a large range of energy, and they contain a great number of levels which do not correspond usually to a definite coupling. Therefore, it seems that the interpretation of highly excited spectra of complex atoms requires the use not only of MQDT but also of the Slater-Condon parametric method for analyzing the perturbing configurations. Such a procedure has been successfully applied in a recent interpretation (Aymar et a / . , 1980) of the J = 0, 2 even-parity and J = 1 odd-parity levels of ytterbium observed by two-photon and two-step spectroscopy (Camus el

ATOMIC RYDBERG STATES

131

al., 1980), but, of course, ytterbium is a relatively simple case, and it is not obvious that application of such a technique to more complex spectra is actually possible.

IV. Rydberg Atoms in External Fields Beyond the spectroscopic investigations described in the preceding section, much attention has been recently paid to highly excited atoms in the presence of “strong” magnetic and electric fields. Atoms excited in the vicinity of their field-free ionization limit allow one to observe, with rather small field strengths, some phenomena which would appear in unexcited atoms, but, of course, with huge field strengths; this corresponds to a situation such that the field effects become comparable to those of the Coulombic interaction. Such intense field regions are of astronomical interest, especially in the magnetic case, since the magnetic fields in a neutron star are estimated to 10”T, and they are also of a great theoretical interest since they cannot be fully understood within the framework of standard perturbation methods. Excitons in semiconductors and electron layers outside the surface of liquid helium have been and remain good candidates for investigating such “intense” field regions since the corresponding Coulombic interaction is very weak, but the use of highly excited atoms appears to be much more versatile, especially since the advent of laser spectroscopy. A. RYDBERG ATOMS I N MAGNETICF I E L D S

The respective behaviors of Rydberg atoms and of atoms in their ground state are not basically different when they interact with an uniform magnetic field. However, as mentioned before, in highly excited atoms, some phenomena appear for reasonable values of the field, let us say a few tesla, while, for atoms in their ground state, they appear only for huge field intensities. This was recognized very early by Van Vleck (1932) and by Jenkins and Segre (19391, who first observed diamagnetic shifts and state mixing on n 2P sodium levels (10 9 n 3s) with a field intensity smaller than 3 T. More surprising is the observation by Garton and Tomkins (1969) of equally spaced structures in the u absorption spectrum of barium in the presence of a rather weak magnetic field, varying between 0 and 2.4 T. These structures extend across the zero-field series limit into the continuum, with a spacing of about 1 .Shw,, o, being the cyclotron frequency (see Fig. 3). This unexpected behavior restored the interest of theoreticians in the old but still unresolved problem of one electron in-

--

2--

$4

m

i ? i

-7 I

1

a--+ W

!

l

J

" I

ATOMIC RYDBERG STATES

133

teracting simultaneously with a Coulombic electric field and an uniform magnetic field (see, e.g., Garstang, 1977).

I. Theoretical Investigations The main difficulty in solving the problem comes from the fact that the corresponding Hamiltonian is not fully separable. Then, the only constants of motion are the magnetic quantum numbers ml,m, and the parity. The mixing of atomic states with different n and 1 values comes only from the diamagnetic interaction, which is proportional to B z. To first order, diamagnetic corrections vary as n 4, while paramagnetic corrections are independent of n , and one understands that for highly excited states, diamagnetic shifts become larger than paramagnetic shifts, even for very weak fields. The I mixing depends strongly on the quantum defect of the considered Rydberg states, but for high angular momentum states, it appears for vanishingly small fields; n mixing must be taken into account only for values of the field intensity larger than a limit, B,, varying as n -7’2. For n = 30, B, can be estimated to approximately 1 T. For values of B not exceeding a few times B,, the energy levels and the wave functions of the perturbed system in an energy range corresponding to the unperturbed levels n - 1, n ,n + 1 can be theoretically derived by diagonalizing the Zeeman Hamiltonian on an atomic basis, including all the states with a given parity, a fixed magnetic quantum number, and a principal quantum number varying between n - 3 and n + 3. Such a diagonalization procedure has been successfully used by Zimmermanet al. (1978b) for interpreting their data on diamagnetic structure of odd-parity m l= 1 sodium Rydberg states in the vicinity of n = 28 for field intensities varying between 0 and 6 T (see Fig. 4). It must be noticed that the corresponding numerical calculations suggest that for quasi-hydrogenic states (high angular momenta), the levels from different terms may actually cross. This would imply that an approximate dynamic symmetry exists, but to date it has not yet been found. A similar but more extensive calculation has been also performed by Lu et al. (1978a) on odd-panty ml= 1 Rydberg states of lithium. For values o f B much larger than B,, such an approach cannot be used since the Coulombic field and the magnetic field become equally important in determining the motion of the electron. This is particularly true in the vicinity of the field-free ionization limit, E,. For a given value of n , the “diabatic” states cross this limit when the magnetic field is close to another critical value Bn, (>&), defined by - ( Z * e 2 / 2 n 2 a o )+ ( e 2 a ~ / 8 r n Z 2 ) n 4 B =~0,

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Serge Feneuille and Pierre Jacquinot

MAGNETIC FIELD

(TESLA)

FIG.4. Experimental and theoretical diamagnetic structure of odd-parity rnl = 1 sodium states in the vicinity of n = 28 (from Zimmerman er NI., 1978b).

For n = 50, B,, is approximately equal to 4 T, while for n = 1, it would be as high as 5 x 10’ T. In such a strongly perturbed region, one faces, in principle, a two-dimensional ( 2 , p ) nonseparable problem. However, it has been recognized for many years (Schiff and Snyder, 1939) that a cylindrification procedure can overcome the lack of separability through a factorization of a two-dimensional transverse harmonic oscillator with a longitudinal Coulombic interaction averaged over the previous one. By using virial considerations, Angelie and Deutsch ( 1978) recently showed that this cylindrification procedure is well based in the whole B range and that actually the spectrum may be worked out in WKB approximation. With such an approximation, the quantization condition is 2(2p)”‘

1””[E

-

V ( p ) ] ” ’ d p = (n,,+ B)h

(I

where p is the electron mass and V ( p )= - e’/p + bpwzp’, po being given by E - V(po)= 0. The previous equation was integrated numerically by Edmonds (1971) and Starace (1973), and the corresponding results show that, around E = E, = 0, the levels are roughly equally spaced, the interval being close to ( 3 / 2 ) h w ,, and that, in the same energy range and for

ATOMIC RYDBERG STATES

135

field strengths of a few teslas, n,, is large, namely a few tens. This result was most directly understood by Rau (1979), who, considering that n,, is large for E = 0, writes

After elementary integration, one actually gets dE/dn,, = ( 3 / 2 ) h w , . It must be noticed that this value had already been derived through semiclassical arguments (O'Connell, 1974; Rau, 1977). Of course, for large positive values o f E , the spectrum becomes again periodic, but the spacing is now hw, , that is, the well-known spacing predicted by Landau for a free electron interacting with an uniform magnetic field. All the previous calculations give the position of the resonances but they do not provide any information about their structure, their intensity, and their width, and, for example, nothing is really known about the ionization limit of an atom in the presence of a magnetic field. Therefore, there is still need for a full quantum treatment of the problem. Such a calculation was recently undertaken by Fano ( 1977), who utilized spheroid functions to reduce the two-dimensional ( r , 8) eigenvalue problem to a system of coupled ordinary equations in the radial functions. To our knowledge, this system has not been solved, but this approach allowed Fano ( 1977) to qualitatively understand the influence of light polarization on the intensity of the observed resonances in absorption spectroscopy. 2 . Recent Experimental Stirdies

Rather surprisingly, the experimental study of Rydberg atoms in the presence of an uniform magnetic field has reaped a rather small profit from laser spectroscopy, since most of the data actually have been obtained by the Argonne group (Lu et al., 1978a, b) using basically the same standard absorption techniques as those in the work of Garton and Tomkins (1969). Such techniques were, in fact, sufficient to observe n and 1 mixing, and quasi-Landau resonances in the spectra of strontium, barium and alkalis, and lithium through cesium. In all these spectra, the quasi-Landau phenomenon occurs in CT polarization, but not in 7~ polarization. This result is in agreement with the predictions of Fano (1977). Moreover, in most of the latter spectra, the quasi-Landau spacing is indeed ( 3 / 2 ) h w , , as expected, but in lithium, the observed spacing is only one-half of the cyclotron frequency (Lu et al., 1978a). Such a value is predicted (Rau, 1979) from simple classical arguments when the atom interacts simultaneously with crossed electric and magnetic fields, but, in the lithium experiment, the motional electric field (approximately 100 V cm-' for B = 5 T) is too

136

Serge Feneuille and Pierre Jacquinot

small to make this explanation relevant. However, Crosswhite et al. (1979) have recently shown that, when combined with the 1 , n mixing terms, symmetry breaking resulting from the motional electric field leads also to an )tiw spacing. Such a value does not appear in heavier atoms, since the motional field has a smaller value. Quasi-Landau resonances also were observed in the even-parity channels of barium and strontium (Fonck et al., 1978, 1980) by using twophoton spectroscopy and thermoionic detection. In the latter case, the resonances appeared only for rnl = 0 states, a result again in agreement with the predictions of Fano (1977). Moreover, in both cases, a (3/2)hw, spacing was observed, but resonance profiles were found to be species dependent. A very similar experiment was recently carried out by Camus and Morillon (1981), on barium again, but using the Doppler-free version of two-photon spectroscopy. The optical resolution (-150 MHz) was sufficient to permit the observation of diamagnetic shifts of then 'Dz levels (n 50) forB strengths as small as 0.04 T. Similar techniques, but using laser photodissociation of molecules followed by single-photon excitation of resulting metastable states (hybrid resonances), have been also utilized in cesium (Gay et al. 1980); Landau condensation of the IMI = 3 odd-parity spectra were investigated for fields up to 8 T and for energies ranging *120 cm-' around the threshold. The spacing of the resonances near the threshold is shown again to be (3/2)hw,. To date, the only experiment performed on an atomic beam concerned the diamagnetic structure of sodium Rydberg states in the vicinity of n = 28 (Zimmerman et al., 1978b). Two-step laser excitation was used, and resonances were detected by field ionization. Such clean techniques are rather difficult to achieve, but no doubt they are those best adapted for studying in detail the structure of Rydberg atoms in the presence of external fields.

-

B. RYDBERCATOMSI N ELECTRIC FIELDS. FIELD IONIZATION The problem of one electron interacting simultaneously with a central field and an uniform electric field is at the same time less and more complicated than in the magnetic case. It is less complicated since, at least for the hydrogen atom, the corresponding nonrelativistic Hamiltonian is fully separable in parabolic coordinates (7 = r - z , 5 = r + z), and thus the problem can be exactly solved in principle for this particular case. However, the situation is more complicated than that in the presence of a magnetic field since it is well known that any atomic state ionizes spontaneously in the presence of an uniform electric field. Actually, as already

ATOMIC RYDBERG STATES

137

FIG. 5 . Potential energy surface of an electron interacting simultaneously with a Coulombic field and a uniform electric field (only one-half of the surface is represented).

noticed in Section 11, this ionization process exhibits a threshold behavior. This is related to the existence of a saddle point in the potential energy surface of an electron interacting simultaneously with an attractive central field and an uniform electric field. This is illustrated in Fig. 5 . In the case of a Coulombic central field, the energy of the saddle point, ESP,and the uniform field strength, F (both, expressed in atomic units), obey the following equation: ESP= El - 2 P 2 , where E, is the ionization potential of the unperturbed atom (from now on, we shall write El = 0). We shall see later that for an energy lower than ESP,the excited electron remains essentially bound, and the corresponding Stark states are quasi-stable with respect to ionization, while, for an energy higher than ESP,a nonhydrogenic atom ionizes. In other words, for a given value of the energy, the atom ionizes if and only if the field strength is higher than a critical value given approximately by F, = $,tp= ( 2 n * ) - 4 There are essentially two ways to experimentally investigate the structure of the atomic spectrum in the presence of an uniform electric field. In

138

Serge Feneuille and Pierre Jacquinot

the first (Section IV,B,4), a given atomic energy level (or sometimes a given atomic state) is selectively excited and the electric field is applied only a certain time after the excitation pulse, the number of created ions (or electrons) being counted as a function of the strength of the electric field. In the second, on the contrary, the field is kept at a fixed value during the excitation, the number of ions (or electrons) being counted as the function of the excitation frequency, which is continuously tuned. The first experiments performed on field ionization of atomic Rydberg states were of the first type, and it is usual to begin with them when introducing the subject. However, the corresponding results do not depend only on the atomic structure in the presence of the electric field, since their interpretation (see Section IV,B,4) requires the introduction of dynamic (or transient) effects, depending, in particular, on the manner of applying the field. Therefore, we have decided to adopt another presentation, first describing our current knowledge (theoretical and experimental) of atomic structure in the presence of an uniform electric field, and then giving some indications on dynamic effects. 1. Atomic Structure in the Presence of a Uniform Electric Field

a . Theoreticul and experimental investigations for the hydrogen atom. As already noted, the Hamiltonian is fully separable into parabolic coordinates for hydrogen, at least in the nonrelativistic approximation. Therefore, the corresponding states can be labeled by lmll and one of the two separation constants Z , (for the .$ motion) or Z2(for the q motion) ( Z , + Z2 = Z). However, as the 6 motion is always bound, one prefers to utilize, instead of the separation constants, an integer number, n , , which is nothing other than the number of nodes in the .$ bounded motion. In any case, the previous quantum numbers are not sufficient to fully characterize quasi-stable states, and thus one usually introduces two other quantum numbers: n , and n2 = n - Jmll- n1 - 1, which are good quantum numbers in the zero-field limit, but, in the general case, are only approximate. Their use is, in fact, justified only as long as the corresponding state remains quasi-stable. Unstable states are fully characterized by lmll, n , , and the energy (see, e.g., Luc-Koenig and Bachelier, 1979). Although exact solutions were only obtained a few years ago (Alexander, 1969; Hirschfelder and Curtiss, 1971; Damburg and Kolosov, 1976, 1977, 1978; Luc-Koenig and Bachelier, 1979, 1980) because of computational difficulties, most of the important features of the Stark spectrum were pointed out in rather old studies using either perturbation theory or WKB approximation, which are described in detail in the famous book by Bethe and Salpeter

ATOMIC RYDBERG STATES

139

(1957). In particular, the most striking result is the presence, far above the saddle-point limit of quasi-stable Stark levels. This is not surprising if one notices that, in parabolic coordinates, the condition for ihe existence of a potential barrier for the q motion depends not only on the excitation energy and on the field strength but also on lmll and on the separation constant Z z . For example, for lmll = 1 , this condition is 4FZz/E' < 1. Thus, the ''classical'' ionization limit is, in fact, given by E, = - 2 ( F Z z ) ' i 2 . Z, depends on F in a complicated way, but one knows that, for highly excited states, Zzcan take very small values (of the order of l/n ), and thus some quasi-stable Stark states exist far above the saddle-point limit (4F/EP,, = 1). Of course, because of tunneling effects, a significant ionization probability can appear even if there is a potential barrier for the 77 motion, but this is not related to the saddle-point limit defined in cylindrical coordinates. However, this saddle-point limit retains some meaning, even for the hydrogen atom, since the separation constant Z, is always very close to 1 for one (the lowest one in energy) of the various Stark levels of the manifold corresponding to the same unperturbed n level. For lmll 1, the condition for the existence of a potential barrier in the q motion cannot be expressed in a compact way (see Herrick, 19761, but qualitatively, the previous conclusions remain valid. Therefore, in hydrogen, the atomic structure can be schematized in the following way (see Fig. 6): below a limit, which depends on Iml(but which is, in any case, very close to the saddle-point limit, all the atomic states are quasi-stable with respect to ionization (region A), while, above this limit (region B), quasi-stable levels

+

FIG.6. Stark manifolds in hydrogen.

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Serge Feneuille and Pierre Jacquinol

are superimposed on ionization continua and ionization resonances with various widths. Of course, this structure is quite complicated, and it may be asked whether some regularities exist as in the presence of a magnetic field, especially in the vicinity of the field-free ionization limit. The answer was first given by Freeman et al. (1978), who, from a study of the classical motion of an electron in the potential V = - l/r + Fz, showed that nearly closed orbits exist, even for energies greater than E , , and that, around E,, they are equally spaced in energy, at least in a one-dimensional model. Then, Rau (1979) showed that the origin of the equally spaced structures can be understood using parabolic factorization within the framework of WKB approximation exactly in the same way as in the magnetic case. A periodic structure is obtained for any ml,and the spacing is found to be proportional to F3’*across the field-free ionization limit, whereas it would be proportional, respectively, to F2I3and to Ffar above and far below this limit. Of course, as in the magnetic case, such semiclassical calculations do not provide any information about the width of the resonances. More recently, exact calculations of photoionization cross sections of hydrogen in the presence of a uniform electric field were reported by Luc-Koenig and Bachelier (19791, who showed that the possibility of observing such structures is related to the lAmll value for the studied transition rather than the Imll value of the upper states. This result is quite important since it points out that the photoionization cross section (or the excitation spectrum) is not directly related to the spectral density of the atomic states, because of possible cancellation effects in the oscillator strength. Exact calculations have also been recently reported for states lower in energy. Most of them concern the determination of the ionization width of various Stark levels across and above the saddle-point limit. For fields weaker than “the classical ionization limit” derived in parabolic coordinates, no significant deviations were obtained with respect to calculations based on the WKB approximations (Lanczos, 1930a, b, 1931; Rice and Good, 1962; Bailey et al., 1965; Yamabe et al., 1977). . Of course, Rayleigh-Schrodinger perturbation theory has been also extensively used to theoretically investigate the Stark structure of the hydrogen atom, especially, in the weak field limit far below the saddle-point limit. Actually, the first-order term showing that the separation between the two extreme Stark levels of a given n manifold varies as n has been known from the very first calculations of Schrodinger. Later on, the perturbation expansion was carried out up to higher order, and recently, a 160th-order calculation was even reported (Alliluev et al., 1979)! It is well known that the perturbation series does not converge in the usual sense and exhibits an asymptotic character (see, e.g., Kato, 1966), which leads us to define a maximum useful field strength (see, e.g., Silverstone, 1978).

ATOMIC RYDBERG STATES

141

However, since the discovery of a dispersion relation between the ground state energy shift and its ionization rate by Herbst and Simon (1978), and its generalization to excited state by Silverstone et af. (1979), perturbation theory appears now to be a usable method in also calculating ionization rates, but the corresponding calculations actually are not simpler than numerical integrations of the separated equations. Completely different methods using, in particular, complex-coordinate methods (for references, see Benassi and Grecchi, 1980) and variational principles (Herrick, 1976; Macias and Riera, 1978, 1979) have also been proposed and extensively studied from a purely theoretical and even mathematical point of view. Their main practical interest comes from the fact that they can be used in principle for nonhydrogenic atoms since they do not necessarily require the separability of the Hamiltonian. However, to date, most of the corresponding actual calculations concern weakly excited states and rather weak fields. Therefore, it is quite difficult to claim that these rather sophisticated methods can be practically used in theoretical investigations of highly excited states in the presence of an external electric field. It must be pointed out that until now, no relativistic effects have been introduced in the various calculations; of course, it is well known that relativistic effects become very weak for highly excited atoms, but, in any case, they break the supersymmetry related to separability in parabolic coordinates. Thus, they may have important effects, especially on the ionization rate of the states, which, in the nonrelativistic approximation, would remain quasi-stable far above the saddle-point limit, since a coupling can be relativistically induced between these quasi-stable states and the underlying ionization continua. It seems that the only experiments on hydrogen have been made by Koch (1978). Rydberg states of n = 10 were produced as explained in Section II,A, 1, and a C 0 2 laser beam was merged with the H beam. The frequency of the laser, as well as the velocity of the beam, was kept fixed. The resonance was produced by varying the transverse electric field to which the H beam was submitted. By using this method Koch measured the difference between the energies of the Stark levels (25, 21, 2, 1) and (10, 8, 0, 1) [in the notation (n,n,, n 2 , Irnll)] in one experiment, and (30, 0, 29, 0) and (10, 0, 9, 0) in another experiment, for electric fields around 2.5 kV/cm and 0.69 kV/cm, respectively. This experiment was specially intended to check the validity of perturbation calculation S"'as a function of the order N of perturbation up to N = 25. An excellent agreement was found in the second case, but in the first case, S'" oscillates about the experimental result and finally diverges from it when N becomes larger than 1 1 . The minimum discrepancy, obtained for S"j' is more than ten

142

Serge Fen e uil le and Pierre Ja cquin o t

times the estimated error. It is to be noted that the (25,21,2, 1) state in the field was “stable” (r < lo5 sec-I) although it lay 197 cm-’ above the SP line, in conformity to what was explained earlier. b. Spectroscopy of stable Stark states in alkulb. In these experiments the Rydberg states are excited by one of the schemes in Fig. 1 in a constant electric field F, and detected by field ionization. An electric field pulse of about 10 kV/cm, is applied with a delay T of a few microseconds after the laser pulse. In this way only “stable” states, i.e., states with r < T - I (-105-6 sec-I) are detected. For each value o f F the signal is recorded as a function of the excitation frequency. Essentially two types of measurements have been made by two different groups. They are more or less complementary. The first group has explored limited regions at high resolution (6a --- 0.002 cm-’) for high values of n (around n = 50) for one element, rubidium, whereas the other group has made an extensive study of two elements, lithium and cesium, at lower resolution (6a= 0.2 cm-’) and at lowern values, namely, n = 15 (the structures are then less intricate) so as to obtain a complete mapping of the Stark structures for fields varying from zero to the ionizing field. In both cases the experimental results have been compared with complete calculations. In nonhydrogenic atoms, perturbative treatments using a power series expansion in the field are fundamentally inadequate because of anticrossings between Stark levels characterized by a given lmll value. However, as in the magnetic case, for relatively low electric fields ( F 5 F,), the Stark structure of Rydberg states can be calculated by diagonalizing the energy matrix built on a truncated basis set of unperturbed states. The more intense the electric field, the larger is the dimension of the required basis. This technique was utilized by Luc-Koenig er ul. (1979) in the theoretical investigation of the Stark structure of rubidium around n = 46 for fields varying up to 20 V/cm (FIF, d 0.36). In this particular case, 156 unperturbed states (obtained from a numerical central field calculation and expressed, of course, in the spherical representation) had to be introduced for reproducing experimental data (positions and oscillator strengths from the ground state) on the m l= 0 Stark levels located between the unperturbed 48p and 49p levels. Similar but more extensive calculations, including fine-structure effects, were recently reported by Zimmerman et ul. (1979) for all the alkalis in the vicinity of n = 15 for fields varying up to 6 kV/cm (FIF, 5 0.95). Here again, excellent agreement was obtained with experimental data. However, it must be noticed that, for a given value of FIF, , and for a given precision, the dimension of the required basis increases as n 2 , and therefore that the method is very cumbersome for highly excited states.

ATOMIC RYDBERG STATES

143

In the experiments at high resolution on rubidium around n = 50 (LucKoenig et a/., 1979), it was possible to follow the phenomenon from low fields ( 5 Vicm), in which each s, p, or d state shows its own Stark effect, to “high” fields (20 V/cm!), where one sees almost only a periodical structure. The positions of the components are almost hydrogenic in contradistinction to the intensities. For the hydrogenic approximation one would obtain only a very small number of components instead of the periodical structure, which was observed with almost equal intensity for all the components in (T polarization (/mil= 1). An extensive study on different alkalis at lower resolution has been made by Zimmermaner ul. ( 1979). In these experiments the region around n = 15 was chosen as being best adapted to the type of results wanted. The average density of Stark states (scaling as n ’for ml = 0 ) is one state in 3 cm-’, and the basis set required for the computation (scaling as n’) extends over seven values ofn, whilen is high enough for the “strong field” to be reached with a few kilovolts only. By suitably choosing the polarizations, the lmll = 1 or m i = 0 states were excited in lithium (the fine structure being negligible) and the mj = 1/2 states in cesium. Figure 7 shows the experimental results for lithium, and those for cesium are shown in Fig. 8. Many interesting features are observable in these maps, which are in excellent agreement with the calculated ones. Particularly interesting is a detailed study of a level anticrossing made in lithium with a narrower grid of field values and a narrower excitation (a pulse-amplified CW laser was used to excite the final step with a linewidth of about 0.01 cm-’). Figure 9 shows the result; one should notice that the line strength for one of the states vanishes, due to the symmetry properties of the wave functions at the exact position of the avoided crossing. The computations have been made as explained previously for lithium and sodium ( [mi( = 1, 01, potassium ( (mi( = 2, 1, 0 ) , rubidium and cesium (ImjI = 5/2, 3/2, 1/2), the fine structure being taken into account for the two last cases. Complete maps are given, encompassing the energy and electric field ranges 440-540 cm-’ and 0-6 kV/cm, respectively, the energies being calculated at 100 V/cm intervals. These maps exhibit a rich variety of interesting features, the principals of which are summarized hereafter. ( 1 ) As expected, in weak fields, states with nonnegligible quantum defects 6 exhibit a quadratic Stark effect, whereas states with negligible 6 form manifolds corresponding to the different values of n ; these manifold exhibit a linear Stark effect. At higher field the different manifolds start to overlap, and one observes repulsions between the different levels. These mutual repulsions exhibit the most dramatic differences with respect to hydrogenic behavior. (2) For a given element the effect of /mil is very important. This is

144

Serge Feneuille and Pierre Jacquinot

FIG.7. Lithium experimental Stark structure at n Zimmerman er a / . , 1979).

=

15: (a) (rnl = 1; (b) m

=

0 (from

145

ATOMIC RYDBERG STATES

0

1000

2000

3000

FIELD

4000

5000

6000

(V/cm)

FIG.8. Cesium experimental Stark structure Imjl

=

112 (from Zimmerman et a/., 1979).

illustrated in Fig. 7, giving the maps for lmll = 1 and ml = 0 in lithium. For the higher value of lmll (where no s state is involved) the map is almost hydrogenic. For ml = 0, the presence of the s level with its large quantum defect 6, = 0.35 profoundly modifies the whole map, and large anticrossings appear everywhere. This is, of course, true for all alkalis, as shown,

FIELD FIG.9. Detail of an anticrossing recorded in lithium between states (18. 16,0, 1 ) and (19, 1 , 16, 1) at 321.5 cm-', 943 V/cm (from Zimmerman er a / . , 1979).

146

Serge Feneuille and Pierre Jacquinot

for instance, in the case of potassium, for which maps are given for Imll = 2 , lmll = 1, and ml = 0. Due to these anticrossings, each Stark level above a certain electric field starts to slalom between the others almost horizontally. This behavior is of prime importance in the understanding of ionization phenomena by an electric field varying more or less rapidly from zero to higher than the critical value. When the fine structure is not small, as is the case for cesium, the map for a given ml shows, at high fields, two separate manifolds corresponding to the two different values of lmll, Imjl &1/2. For mj = 1/2, the manifold with lrnjl + 1/2 is nearly hydrogenic, while the other shows the usual repulsions due to the quantum defects. (3) The anticrossings are very sensitive to the quantum defects and are well accounted for by the calculations. In the special case where only a single 6 differs from zero, the calculated values obey a simple law for the variation of the minimum distance A between the two anticrossing levels. If 6‘ is the difference between 6 and its nearest integer, for small values of 6’ one finds A = 1.9n-%‘.This relation has been found for n = 15, 16 and also checked to be valid for n = 25, 26; it is then probably universal. For heavier alkalis the maps may be very different, and peculiar situations can occur, for instance, if 6, - 6, or 6, - 6, happens to be close to an integer plus or,e-half. In this case, levels coming for the s or the p state can traverse a long part of the whole pattern without anticrossing, giving a general aspect quite different from the other cases. Because of these features it is impossible to predict even qualitatively the Stark structure without exact calculations. Although some features do not seem to obey any simple scaling law, some properties can be easily extended to other values of n , essentially those depending only on the quantum defects. Some scaling laws are recalled in the following: Anticrossing gap

A

a

nP

Density of Stark states First crossing between two manifolds at Number of crossings (avoided or not) before a given Stark state reaches the ionization threshold

F1 a n-” a n 2 for for

[a n

ml = O lmll = n - I

From these laws it is evident that for higher values of n the first crossings appear for smaller values of FIF,. Therefore, the Stark states appear strongly mixed for smaller values of F / F , , as shown in the experiments of Luc-Koenig et al. (1979) at n = 50.

147

ATOMIC RYDBERG STATES

2 . The Efect of’ Electric Fields upon Stable und Airtoionizing States in Alkdine Eurths

The structure of Stark spectra of barium around the 6 ~ 1 2 manifold 1 has been studied by Kleppner (1977), with the same technique as that for alkalis, up to 10 kVicm. This study is interesting because the doubly excited valence state 5d7d lies in the same region; it was thus possible to see how this configuration mixing radically alters the Stark structure. In particular, there appears at high field (-6 kVicm) a strong repulsion between the 6s 121 manifold and the valence state, which otherwise would exhibit negligible Stark shift. A study on strontium has also been made by Freeman and Bjorklund (1978). It showed the effect of anticrossings between Stark states on their widths in the case of autoionization of doubly excited states 4d(2D,,2)nl above the first limit of ionization 5s. The excitation of these states was made according to the scheme e in Fig. I , and the ions produced by autoionization were counted without the help of an ionizing electric field pulse.” The transitions close to 5s7s + 4d12p and 5s7s 4d12f were recorded for several values of the electric field from 0 to 10 kV/cm; for increasing fields the 121 manifold close to 12f and the 91 manifold close to 12p appear with strong intensity. The interesting feature appears for the 4d12p state at 6 kV/cm, where there is an avoided crossing with the nearest Stark state of the 91 manifold. The 12p component becomes sharper than that for lower or higher fields (1.2 cm-’ instead of 2.7 cm-’ at zero field). This effect is interpreted as follows. The state 4d12p, owing to its large quantum defect (2.93, is more strongly coupled to the Sr’ 5 s continuum than the states with I > 2, which have negligible quantum defects, and are thus more broadened; at the anticrossing the states share their properties, and this results in a narrowing of the 4d12p state. This experiment shows how the coupling of a discrete state with a continuum can be controlled by means of an electric field.

-

3. Photoionization Spectra in the Presence qf an Electric Field (PIPEF Spectra ) These spectra are obtained by scanning the excitation energy beyond the limit E = -2F”2 in the presence of a dc field, and collecting the ions or the electrons produced through a time gate of duration T (of the order of 2 psec) after the laser pulse, without applying any supplementary field, in contrast with the spectroscopy of “stable” Stark states. In a diagram This case is quite different from the next one since here the ionization is due to the coupling with a continuum which exists in the absence of the electric field.

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(E, F) the observed resonances correspond to the intersection of a F = cst line with levels of the Stark manifold the ionization rate r (or width) of which is greater than T - I . This intersection may happen at places where the hydrogenic Stark levels are already broader than 7-l (Type I).' In this case an ionization resonance should be observed in the general case (Coulombic field as well as non-Coulombic field). The intersection can also happen at places where the hydrogenic level is not yet broadened more than T-' (Type II).' In this case ionization should not be observed in the Coulombic case. However, in the case of a non-Coulombic field, the coupling of such a level with the continuum due to other states-coupling which does not exist in the pure Coulombic case because of the supersymmetry-gives it a supplementary width (or r ) so that ionization can be observed. Each Stark state contributes to this continilurn in a region defined by F 3 E 2 / 4 Z z(cf. Section IV,B, 1,a). As the lowest level of each manifold corresponds toZ, = 1 (at least for lmll = I), the region in which such contributions exist is defined by F 2 - E2/4,i.e., above the SP line (region B, Fig. 6). Ionization should thus normally happen in this B region for alkalis, in contradistinction to the case of hydrogen. This explanation was first proposed by Littman et al. (1978), who thus made a decisive step in the understanding of ionization phenomena in alkalis. In the case of Type I1 resonances, since the ionization is due to the coupling with a continuum, it should bear the signature of this coupling, i.e., present a Fano profile. The results of this discussion is summarized in Fig. 10 (Fig. 10a is purely hypothetical since no such experiments have been made on hydrogen). The first experiment showing the existence of sharp resonances in a PIPEF spectrum was made by Feneuille et al. (1977) by exciting rubidium atoms directly from the ground state in an atomic beam with the laser described in section II,A,3,d. The frequency was swept - 2F''*Jwith F = 80 Vkm, which correabove the limit v = h-'IEground sponds to states around n = 50. Figure 11 shows recordings made in the u and x excitations, respectively. The sharp resonances observed in (+ excitation are about 50 MHz wide, which corresponds to the residual Doppler width of the atomic beam. Their true width is then smaller than 50 MHz, which was quite unusual for photoionization resonances. In contrast, the resonances in x polarization are broader and less intense; in addition, one can perceive a slight indication of an undulating background and of Fano profiles (this will be shown more clearly later on). The difference between the u and the 7~ spectra is easily understandable since the u spectra correspond to (mil = I for the upper level and the x spectra to m l= 0. The (mil = I levels are more hydrogenic (because of the absence of s contribution), the coupling is smaller, and the levels narrower. The same fi The notation I is now used instead of (Y (broad resonances) or p (narrowerresonances) in a preceding article (Jacquinot, 1979) and I1 instead of y.

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149

I (a) H

I[

FIG. 10. PIPEF spectrum (al and b,) and spectrum of stable states (a, and b,) for hydrogen (a) and alkalis (b). The Stark spectra are recorded by applying a detecting pulse after a delay r (only states with r < r-I are recorded); the PIPEF spectra are recorded without a detecting pulse, the ions being collected after the laser pulse through a time gate of width r(only states with r > r-' are recorded). The spectrum b shows both resonances I and 11, while I and I1 are separated between the spectra a , and a,, respectively.

-/

excitation U-

excitation n

--

-excitation a

FIG. 11. Photoionization spectra in the presence of an electric field F = 80 V/cm for a separated rubidium isotope. One-step excitation from the ground state. (a) Excitation with polarization u (parallel to F). (b) The central part of the recording is made with rr excitation. The width of the sharp resonances observed in u excitation is about 50 MHz and is limited by the residual Doppler width of the atomic beam (from Feneuille et a / . , 1977).

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Serge Feneuille and Pierre Jacquinot

V

= ,,

F-62 V/cm

(Tpolarization

I I

I

I I I I

Fib. 12. Stark spectrum of stable states (a) and PIPEF spectrum (b) for rubidium, in u polarization in the presence of an electric field F = 62 V/cm (Laboratory Aime Cotton, unpublished data).

phenomenon of sharp PlPEF resonance was observed more recently by Littman et al. (1978) in an extensive study of lithium at lower resolution. The results are quite similar and show that these resonances appear on a very large spectral range but disappear as soon as [El < 2F-’I2. This last fact is also illustrated by Fig. 12, which is to be compared with Fig. lob; the transition between the two regions is not quite abrupt, but the width of the transition is only 5 GHz, which corresponds in the formula E = -2F’” X K to a variation of K of only 0.3%. Freeman et ul. (1978) have also reported PIPEF resonances in rubidium around n = 20. with a limit of resolution of 0.5 cm-’ in an article essentially devoted to broad undulations above the field free ionization limit. The signature of the coupling of “stable” Stark levels with the continuum in which they are embedded was shown in a later experiment by Feneuille et al. (1979) on a TT PIPEF resonance in Rb (Fig. 13). The 7~ polarization had been chosen in order to have a sufficient coupling. If the excitation frequency is increased, at fixed F, the sharp resonance disappear as one goes further from the SP line. At the same time, however, one sees the appearance of much broader resonances in the form of a wavy structure extending beyond the field free ionization limit. This phenomenon was discovered by Freeman et ul. (1978) in rubidium (see also Freeman and Economou, 1979), who showed that the undulations

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151

F I G .13. A typical Fano profile for a PIPEF resonance in rubidium (TT polarization) (from Feneuille ef a / ., 1979).

appear only in i~ polarization (the initial state being the ground state ‘SllZ) and have a period scaling as F”’4.The authors gave a classical theory agreeing with the experiment. In the Stark levels scheme (Fig. 6) these resonances correspond to the intersection of a vertical line with very broad Stark levels (type I intersections) and should also be observable in hydrogen. A complete quantum theory has been given by Luc-Koenig and Bachelier (1979). The same types of experiments have been made more recently by Liberman et al. (1980) at much higher resolutions (0.002 cm-’ instead of 0.5 cm-’) so that it is easier to distinguish between sharp and broad resonances in the vicinity of the SP line. 4 . Electric Field Ionization

Two types of studies can be made concerning the ionization by a static electric field.

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( 1 ) Individual Stark states are prepared in the presence of a given electric field F, and one measures how many ions, or electrons, are produced as a function of time after the preparation of the states by a short laser pulse. This gives the ionization rate. In fact, these experiments are not fundamentally different from the previous ones since one measures here a lifetime instead of a width, but both are measured in the presence of the field. (2) Individual states are prepared, usually without field, and, after that, a more or less rapidly varying field F(t) is applied. One then measures the ions or electrons produced as a function of the values (instantaneous or final) of F. This gives a determination of ionization thresholds (or critical fields) F, . This is fundamentally different from (1) since here the dynamic effects play an important part. To our knowledge there are no experimental determinations of ionization rates of hydrogen Rydberg states by a static electric field, although there are many calculations. The only experiments seem to have been made on sodium by Littman et al. (1976), who measured the ionization rate of a few individual Stark sublevels for n = 12, 13, 14; Irnl = 2. Decay from the excited state was monitored by measuring the distribution of arrival time of the first ion after excitation by a 5 nsec laser pulse with a time-to-digital converter. It was possible to measure rates r in the range from lofito lo8 sec-' with this method. The lower limit was essentially due to radiative and collision decay and the upper one to timing resolution. The main results are as follows: a. Studies on ionization rates.

( I ) For the state (14, 0, 1 1 , 2), which is the lowest of its manifold, the curve r = f(Z9is exponential and is in agreement with a curve extrapolated from the calculations of Baileyet al. (1965) for hydrogen. This was to be expected since, because of Irnl = 2 and n, = 0, this state has a hydrogenic behavior. (2) For the state ( 1 2 , 6 , 3 , 2 ) a sharp increase, by more than two orders of magnitude above its values 106-107 in a field around 15.7 kV/cm, is observed when it crosses the state (14, 0, 1 1 , 2), which is rapidly ionizing (more than 10'" sec-') in this field. This is a striking example of mixing properties between two states. b. Studies on ionization thresholds in alkalis. The simplest technique consists of applying a field step, with a maximum value Fo at a time T after the excitation of the atoms by a laser pulse, and counting the ions (or electrons) produced N(Fo)through a time gate of duration T' according to Fig. (aN(F)/dF)dF.IfN(Fo) 14a. In this way, what is measured is N(Fo) =

fi

ATOMIC RYDBERG STATES

IN j

153

,

I

I

F++ AF

FIG.14. Timing sequences used to measure critical ionization fields. (a) The delay T after the laser pulse is adjusted to take account of the transit time of the ions or electrons; it should not be longer, in order to avoid the redistribution of states under the influence of blackbody radiation (Cooke and Gallagher, 1980). The gate is open during a time T' (- 1 psec) so that one determines the field Fo required to produce an ionization rate of about T ' - I = lofi sec-I. (b) Differential method. f' is chosen so that all atoms which ionize at rate 7 I - I at fieldF are completely ionized when the field step AF arrives. The step AF ( A F / F < 0.02) is maintained constant, while F is varied. rand T' are the same as in sequence (a). In both cases A and B refer to single and multiple thresholds, respectively.

is plotted versus F o , shapes A or B are obtained, according to whether there is only one very sharply defined threshold or several thresholds. This is the way most of the experiments on critical fields have been made. For instance, Stebbings er al. (1975) in xenon, and Ducas et al. (1975), Van Raan et al. (1976), and Duong et al. (1976) in different alkalis have found that a law F , = K ( 2 t ~ * ) = - ~ -E2,/4 was obeyed with 1 < K < 1.1 for states ns, np, and nd, n ranging from -15 to -60. In addition Gallagheret al. (1977b) have shown on sodium ( n = 15-20) the existence of several thresholds related to the different (mil states excited in the experiment. They found that the preceding law was true with the following values of K(lmll): K(0) = 1.00,

K(1) = 1.03,

K(2) =

1.19

More refined techniques (differential methods) can be used to obtain the different thresholds directly by measuring aN(F)/ a F . (1) For instance, Vialle and Duong (1979) used the time sequence shown in Fig. 14b. In this way only the ions produced by fields between F and F + A F are counted, and one obtains dN(F + iAF)/dF. The different thresholds thus appear well separated on a zero background. In this

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Serge Feneuille and Pierre Jacquinot

way multiple thresholds related to different values of lmll were studied in detail. Vialle and Duong also showed the appearance of some multiple thresholds due to a partially adiabatic passage from low field to the high ionizing field. (2) Another method consists of making a time resolution of the ion (or electron) production during the rise time of the electric field F(t) (in this case, it is preferable to count electrons rather than ions because the transit time from the interaction region to the collector is shorter and better defined). This is the method that was used by Gallagher ef a / . (1977b) (together with the integral method) and more recently by Jeyset al. (1980). In this manner Jeys and his co-workers have shown that quite different thresholds are obtained according to whether the passage from the zero field state to the very high field state is diabatic or adiabatic. The most naive theory, based on the saddle-point model, would lead to F, = E $ / 4 , Eobeing the field free energy of the Rydberg state considered; but, of course, this is not acceptable. It is more realistic to write F, = E g / 4 , E, being the energy of the Stark state coming from the initial Rydberg state, at the high field where it ionizes. This is acceptable since we know that, except for hydrogen, a state is normally ionized when it lies in the B region above the SP line. Here we must remember that, in fact, there are several SP lines, rather close to each other, corresponding to the different possible values of Iml(, so that we should not be surprised to observe a lmll structure in the thresholds. The law F, = Eg/4 would be identical to the first only if Es could be taken as equal to E,, but, of course, at this point of the analysis, there is no reason for that. We now understand that E, , and then F,, depend on the path followed, during the rise time of the field, by the point P representative of the state in a diagram (E, F ) from the zero field state to the state close to the ionizing field. Because, for high values of n, many Stark states coming from different n manifolds experience many more of less close anticrossings, the path followed by the point P depends on how fast the region of anticrossings is traversed. Figure 15 illustrates what happens in an anticrossing between two states 11 ) and 12), of energies E l andE,, asymptotic to states I+) and I-), of energies E , and E - , respectively. The probability of a diabatic passage from 1 I ) to (2) is given by a formula of the Landau-Zener type,

where AE is the gap (E+ - E J m i nand dF/dr the slew rate of the field F. This means that the passage will be completely adiabatic (P remains on the line I+) or I- )) or diabatic (Pjumps from the line I+ ) to the line I T ) ) ,

ATOMIC RYDBERG STATES

I

I

155

_-

F F I G .15. Diabatic or adiabatic passage through a level crossing. See text.

according to whether At << h / A E or At >> h/AE, At being the time during which the field varies by the quantity AF shown on the figure. For the intermediate values the passage is only partially adiabatic (the atom is left in a superposition of states I I ) and 12)). Because of the exponential formula when At varies around the value h / S , one changes rather abruptly from the adiabatic to the diabatic passage. Since any Stark state undergoes many anticrossings from its zero field starting point to the point where it meets the ImlJSP line, a large variety of situations can occur, depending on the slew rate and on the closeness of the different anticrossings: (1) all the anticrossings may be gotten through adiabatically (very slow rate) or diabatically (very fast rate); (2) some may be crossed adiabatically, some diabatically, and others partially. The measured thresholds may thus be multiple and very different, the greatest difference being between the all-adiabatic and the all-diabatic situation. If one remembers that the closeness of the anticrossings increases with Iml((more hydrogenic states) and withn(AE a K ~ ) ,one sees that diabatic passages will be encountered more easily for higher lmll states and for higher n values. In the case of all-adiabatic passage, which occurs most often, the path followed by the different states may be schematized as in Fig. 16 for sodium, as shown by Gallagher et id. (1977b). It is then apparent that the critical fields forns, (n - l)p, and (n - 1)d are almost the same and equal to E : / 4 , where E, is very close to &I - #)-2 = Bt?T-2 since, in the case of

156

Serge Feneuille and Pierre Jacquinot

I

I

F

0

FIG.16. Schematic representation of the paths followed by the Rydberg s, p. or d states of sodium in the case of an all-adiabatic passage. This explains why the simple low F, = (2n*)--‘is approximately true. (The position of the SP line is slightly different for the different values of lrnll.)

sodium, the quantum defect Ss is 1.34. Surprisingly, the simple law F , = E % / 4 = ( 2 n a)-4 derived from the too naive saddle-point model is now justified, in conformity with the coarse experiments cited above. Summarizing the above discussions we see that: (1) There can be multiple rather close thresholds due to different values of Jrnll. (2) There can be a great variety of more or less distant multiple thresholds due to the “baticity” (diabatic or adiabatic character) of the passage from low to high fields. By judiciously choosing the states and the slew rate of the field, it is possible to follow the change between the two extreme cases, as demonstrated by Jeys et ul. (1980), who have shown the first example of an all-diabatic passage.

It is, of course, possible to imagine and create complicated or bizarre situations by varying the rate and the law of the electric field rise. Now, however, the phenomenon seems to be well understood-which was not the case at the time of the first experiments-and all the experiments made thus far are well accounted for by the above analysis.

ATOMIC RYDBERG STATES

I57

c . RYDBERG ATOMSI N CROSSED FIELDS At the end of this section, a few words must be said about recent predictions (Rau, 1979; Crosswhite et al., 1979; Gay et al., 1979) concerning the behavior of Rydberg atoms in crossed fields. We have already noticed that in such a situation, quasi-Landau structures are expected, with a spacing equal to V z h w , . It must be also noticed that, in the direction of the electric field, the potential energy surface exhibits, in addition to the inner Coulombic valley, an outer well whose position, depth, and width depend essentially on the respective strengths of the two fields. The various consequences of this property have been qualitatively examined by Rau (19791, and the most amazing one concerns the possibility of creating atoms with an electron temporarily trapped in an eccentric orbit far from the nucleus. The observation of such asymmetric atoms has not yet been achieved, but several research groups seem to be interested in it.

V. Radiative Properties of Rydberg States Radiative properties of hydrogenic Rydberg states are well known (see, e.g., Bethe and Salpeter, 1957); in particular, the radiative lifetime of an ( n l ) level increases for a given 1 as n 3 when n increases. Moreover, the average lifetime of all the nl states corresponding to a given n increases as n ‘ , 5 . However, because of interactions between the outer electron and the ionic core, on the one hand, and of dramatic relativistic effects on some transition probabilities on the other hand (see Section 111), some deviations from the hydrogenic values could be expected for Rydberg states of nonhydrogenic atoms. Moreover, the behavior of lifetimes versus the vapor pressure provides significant information about collisional properties of Rydberg states in the presence of a foreign gas or of the same species in its ground state. Therefore, during the last few years, a lot of attention has been paid, both experimentally and theoretically, to the lifetimes of Rydberg states, especially in alkali atoms. Furthermore, Rydberg states appeared recently as good candidates for studying superradiance, both in the far infrared (Gounander al., 1979) and in the microwave regions (Gross et d., 1976, 1979). In the latter case, they allow also an easy observation of the maser oscillation with a very small number of active atoms and with extremely low radiated energy per pulse (Gross et a/., 1979). These very recent experiments open fascinating perspectives in the coupling between radiation and highly excited atoms and thus deserve a relatively detailed description, especially since some results are still unpublished.

Serge Feneuille and Pierre Jacquinot

158

A.

LIFETIME

MEASUREMENTS

The technique utilized for measuring Rydberg state lifetimes is essentially the same one in all the experiments. The Rydberg state(s) under study is (are) stepwise excited by pulsed tunable lasers (or sometimes by a spectral lamp and a pulsed laser or by pulsed lasers and rf), and then one observes the evolution in time of the fluorescence emitted, either from the initially excited level or from some levels, lower in energy, populated by cascades. As mentioned before on several occasions, such a technique can only be actually applied to Rydberg states with relatively low n Values. However, for highern values, fluorescence detection can be replaced by field ionization. The number of high Rydberg atoms in a given state is now measured as a function of the time delay between the laser pulse and the application of the ionizing pulse (Stebbings et al., 1975). In all cases, care must be taken to make sure that only one state is being observed, since, at room temperature blackbody radiation is able, through absorption and stimulated emission processes, to redistribute the Rydberg population among nearby levels (Cooke and Gallagher, 1980). If one takes as exceptions some early measurements on the np[ 1 /21 and n q 3 / 2 ] states of xenon (Stebbings et d., 1975), the most extensive data obtained with the previous techniques concern alkali Rydberg states: the 'S and 'D states of sodium (Gallagher et al., 1975a), 'S and 'D states of rubidium and cesium (Lundberg and Svanberg, 1973, 'P states of rubidium (Gounand et al., 1976), ?3 and 'D states of cesium (Deech et al., 1977), and 'F states of sodium (Gallagher et al., 1978) and rubidium (Hugon et al., 1978). In some cases, extrapolation to zero alkali pressure and correction for thermal escape effects (Curtis and Erman, 1977) had to be performed before obtaining actual radiative lifetimes. Most of the data obtained in this way obey the (n*)3scaling law within the experimental , which is uncertainties, but different scaling laws, for example, (n*)2.52*0.06 verified for the n 'S states of cesium (8 < n < 14) (Deech ct al., 1977), have also been found. Moreover, although, in sodium, experimental lifetimes for the 2S and 2Dstates are in good agreement (Gallagher et al., 1975a) with Coulombic approximation calculations (see, e.g., Gounand, 1979), some significant deviations appear (Gounand er al., 1976; Lundberg and Svanberg, 1975) in many cases. This is not very surprising, especially for the 2S and 'P states, since for such states, relativistic and core polarization effects (see section 111) should certainly be introduced in the calculation for permitting a significant comparison with the observed results. Spin-orbit and polarization effects have recently been introduced by Theodosiou (1980) in a calculation of emission oscillator strengths of Rydberg n 'D states to lower 'P states of potassium. Sharp minima analogous

ATOMIC RYDBERG STATES

159

to the well-known “Cooper minima” in photoionization cross sections have been found. They are absent in the 2S and 2P states and in sodium and rubidium, but they appear in the 2F states of cesium. We have already noticed that measuring radiative lifetimes of Rydberg states often requires an extrapolation to zero pressure. The study of the variations of the observed lifetimes with pressure provides cross sections for the collisional depopulation of the considered Rydberg level induced by thermal collisions with ground state atoms or molecules (Marek and Niemax, 1976; Gounand et al., 1977; Deech et al., 1977; Gallagher et al., 1978). Usually, the inverse of the lifetime increases linearly with pressure, but in some cases, the apparent lifetime of a Rydberg state can be dramatically lengthened by introducing a collision partner. This phenomenon was shown for the first time by Gallagher et a/. (1975b) on ‘D states of sodium in the presence of rare gas. It has been interpreted by a collisional mixing between the initially populated 2D state and nearly degenerate higher angular momentum states of the same n. Therefore, the quantity actually measured in an average lifetime can be much longer than the lifetime of the state initially populated. Such depopulation and /-mixing effects have been extensively studied both experimentally and theoretically, but nothing more will be said here about these subjects since the collision studies of Rydberg atoms cover a very broad domain, which is out the scope of this review. Many references can be found in a recent review published by Edelstein and Gallagher (1978). B. S U P E R R A D I A ANNCDEMASEROSCILLATION O N TRANSITIONS BETWEEN RYDBERG STATES Dicke superradiance is presently a very active field of research. Of course, we cannot enter the details of the corresponding theory here (see, e.g., Bonifacio and Lugiato, 1975; McGillivray and Feld, 1976, and references therein), but let us recall that superradiance is a collective process such that the fluorescence emitted by a collection of excited atoms no longer decreases in time according to the standard exponential law but is concentrated in a pulse delayed with respect to the excitation pulse. The peak intensity of the superradiant pulse is proportional to thesquare of the number, N. of active atoms while the delay is proportional to W ’ .Because of quantum fluctuations, these properties are valid only on an average. Actually, it has been shown (Bonifacio and Lugiato, 1975; McGillivray and Feld, 1976), within the frame of a nondegenerate two-level model, that superradiance may occur in a pencil-shaped sample of lengthl, only if the dephasing time T $ of the atomic dipoles is much longer than a charac, is given by teristic time T ~ which

Serge Feneuille and Pierre Jacquinot

160

where N is the atomic density in the upper state, and A andA are, respectively, the wavelength and the Einstein coefficient of the superradiant transition. In an atomic beam, TZ is proportional to A-', and therefore the ratio T $ / T , is proportional to A'. Therefore, one understands that transitions between Rydberg states are especially suitable for observing superradiance since their wavelength varies according to the excited level from the infrared to the microwave domain. This was first noticed by Gross et 01. in 1976. In a cell, the situation is apparently less favorable since TZ is essentially determined by the Doppler effect and therefore is proportional to A, while A is proportional to A?. Therefore, T $ / T Ris apparently independent of A. The previous conclusion remains valid, however, because of unusually large dipoles moments of Rydberg atoms. If one excepts an early experiment performed in HF vapor on transitions at A 80 pm (Skribanowitz et al., 1973), most of the recent experimental studies have been realized in the near infrared (Gross et al., 1976; Gibbs et al., 1977; Crubellier et a/., 1978) or in the visible (Brechignac and Cahuzac, 1979). Such wavelength ranges are very convenient for the direct observation of the superradiant pulse and testing of its collective properties, but, on the other hand, the range of time available for varying the delay between the excitation and the superradiance pulses is very narrow (a few nanoseconds), and therefore, quantitatiL e comparisons with theoretical predictions are difficult to achieve. In the far infrared, such a difficulty disappears, but the problem now is to observe the light pulse. This new difficulty can be overcome by measuring, in some way or another, the evolution in time of the population of the lower level of the transition, or even of lower energy levels populated by cascades (cascading superradiance). Such a technique was utilized first by Gross et al. (1976) to obtain indirect evidence of the superradiant character of the 4 'P + 3 'D sodium line ( A 9.1 pm) after stepwise excitation of the 4*P levels and superradiant emission on the 5 ' S + 4*P line ( A 3.4 pm). A related experiment was recently performed on transitions between more excited Rydberg levels (n - 10) of rubidium (Gounand et al., 1979). The corresponding results clearly demonstrate that superradiance emissions drastically modify the radiative behavior of Rydberg levels and that the lifetime measurements described in the previous section indeed require great care. More exciting again are very recent atomic beam experiments performed by Haroche and his co-workers on superradiant microwave transitions between the high Rydberg levels (n 25) of cesium (Haroche et al., 1979; Gross et al., 1979). An n 'S state is stepwise excited by pulsed dye

-

-

-

-

161

ATOMIC RYDBERG STATES

lasers, and the evolution in time of the resulting populations of then 2S and (n - 1)'P levels is observed by using time-resolved field ionization techniques (see Section 11). For n = 24, for example, superradiance appears above the threshold N 5 x lo5, which is very low with respect to the thresholds observed in the infrared. The corresponding delay, which is in fair agreement with theory, is of the order of 1 psec. A slightly modified setup allowed the same authors to observe the maser oscillation on microwave transitions between the highly excited Rydberg levels ( n 25) of sodium. Here again, time-resolved field ionization technique is used, but, of course, the illuminated part (length 5 mm, diameter -1 mm) of the atomic beam is inside a microwave Fabry Perot cavity exactly tuned by auxiliary double-resonance spectroscopic experiment (see Section 111). With the finesse used (-200), the threshold N value is as low as lo3,and it could certainly be reduced again by increasing the finesse. Let us also notice that in this case, the radiated energy per pulse is extremely low J). It must be also pointed out that in both cases, the processes are not initiated by spontaneous emission, which is extremely weak in such a spectral range, but by blackbody radiation. This is not the least surprising aspect of the physics of Rydberg states.

-

-

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M . F . H . SCHUURMANS. Q . H . F. VREHEN. and D . POLDER Philips Research Laboratories Eindhoven. The Netherlands

and H . M . GIBBS Optical Sciences Center University of Arizona Tucson. Arizona

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Semiclassical Theory . . . . . . . . . . . . . . . . . . . . . . 111. Quantum Mechanical Description of SF . . . . . . . . . . . . . . A . Equations of Motion in the Initiation Time Regime . . . . . . . B. Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . C. Correlation Functions . . . . . . . . . . . . . . . . . . . . D. The Stochastic Variables Description . . . . . . . . . . . . . . E . The Initiation of SF . . . . . . . . . . . . . . . . . . . . . F. Classical Behavior and the Interpretation of Single-Shot Outputs . G . Average Behavior; Effective Initial Tipping Angle . . . . . . . . H . Fluctuation Behavior . . . . . . . . . . . . . . . . . . . . . IV. The Effect of Homogeneous and Inhomogeneous Broadening on SF . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B . Equations of Motion and Their Solution . . . . . . . . . . . . C. The SF to ASE Transition . . . . . . . . . . . . . . . . . . D. The Effect of Inhomogeneous Broadening on the Delay Time . . . E . Homogeneous Broadening . . . . . . . . . . . . . . . . . . V. Three-Dimensional and Multimode Effects . . . . . . . . . . . . . VI . Experimental Techniques . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B. Conditions for SF Experiments . . . . . . . . . . . . . . . . C . The Cesium Experiment . . . . . . . . . . . . . . . . . . . D . The Effects of Inhomogeneous Broadening . . . . . . . . . . . E . The Inversion Profile . . . . . . . . . . . . . . . . . . . . . VII . Experimental Results . . . . . . . . . . . . . . . . . . . . . . A. Pulse Parameters . . . . . . . . . . . . . . . . . . . . . . . B . Spatial and Temporal Coherence . . . . . . . . . . . . . . . . . C. Initiation and Fluctuations . . . . . . . . . . . . . . . . . .

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167 Copyright 0 1981 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 0-12-W3817-X

M . F . H . Schuurmans et al. VIII. Conclusions Appendix I . Appendix I1 References .

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I. Introduction Spontaneous emission by excited atoms has been commonly viewed as a random process in which the stored energy is released in the natural lifetime T , of the excited state. In 1954 Dicke predicted that under certain conditions the energy is released cooperatively and in a much shorter time T~ = T J N . whereN is the number of excited atoms. Correspondingly, the emission intensity is proportional to N Z ,instead of to N , as expected for random individual emission. The phenomenon predicted by Dicke is now called superfluorescence (SF). Such anNZintensity behavior is well known from the coherent emission of driven electric or magnetic dipoles. For instance, when the dipoles are all in a volume small compared to the cube of the wavelength of the driving field, these dipoles all oscillate in phase, and the emission is thus proportional to the square of the number of dipoles. Observations of such cooperative effects of atoms go back as far as Hahn’s spin echo experiment of 1950. The peculiarity of SF lies in the fact that the atoms which are initially all in the excited state le) then have no dipole moment, i.e., (el&) = 0. Classically the atoms would not even radiate. Also, the spontaneous emission of an excited atom is a random process in which the phase is undefinable. At first sight one does not expect cooperative emission from such a system. If, nevertheless, SF exists, one expects it to occur when the excited atoms are all put together in a “point” sample, i.e., a sample with dimensions smaller than the wavelength of the emitted field. This was the first possibility considered by Dicke in his 1954 article. However, it was argued much later that a point sample will not give rise to S F except with very special geometric arrangements of the atoms. This is because of the near field interaction of the atoms, i.e., the dipole-dipole interaction,

* In this article the following terminology is adhered to. Radiation by coherent dipoles with N’ intensity dependence is called superradiance (SR) when the coherence has initially been induced by some macroscopic means, e.g., by an optical field. The name superfluorescence (SF)is reserved for the special case in which the coherence evolves from the completely inverted state through quantum fluctuations. Unfortunately, either of these terms has occasionally been used in the literature to indicate amplified spontaneous emission.

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which gives rise to strong mutual detuning of the atoms so that a cooperative dipole cannot build up in the sample. This is demonstrated in Appendix I. In order to observe SF, one must therefore consider “extended” samples, i.e., samples in which the mean interatomic distance is much larger than the wavelength of the emission. Such samples were also discussed by Dicke. It must still be demanded that dephasing processes such as atomic collisions and atomic motions are so slow that the build up of a collective dipole is not hindered. In 1973 SF was first observed by the MIT group of Feld (Skribanowitz et al., 1973). Their system was optically pumped HF gas. The SF occurred on a transition between two adjacent rotational levels in the first excited vibrational state corresponding to infrared radiation. Since then, S F has been observed on a number of far-infrared, near-infrared, and visible transitions. The spatial and temporal characteristics of SF are summarized in Fig. 1. In general a pencil-shaped sample is used. The emission is then highly anisotropic: almost all of the radiation is emitted into a small solid angle around the pencil axis in the forward and backward directions. This emission pattern is to be contrasted with the isotropic spontaneous emission pattern of an unpolarized atom. The temporal behavior shows incubation of the emission characterized by a delay time T ~ ) ,defined as the time at which the emission intensity reaches its maximum. The maximum intensity is proportional to N2.The buildup of the cooperative emission is

SPATIAL

TEMPORAL

0

XD

t-

FIG. 1. Sketch of the temporal and geometric emission characteristics of a superfluorescing pencil.

170

M . F . H . Schuurrnans et al.

-

characterized by a “collective decay” time TR ( ~ , / N ) ( 4 n / A a where ), b0/4~r is the fraction of solid angle into which the SF emission goes. T R is also the time in which the first photon is emitted into Aa. The delay time T,)is typically 10-100 times larger than T ~ depending , o n N . Note that SF is not a fast form of spontaneous emission. Superfluorescence shows incubation behavior because time is needed for a collective dipole to build up in the medium. The SF output fluctuates from shot to shot, i.e., when the sample is repeatedly prepared with the atoms in the excited state. These fluctuations, both in delay time and shape, are of quantum-mechanical origin; they correspond to the initial quantum uncertainties in the state of the field and the atomic system. Superfluorescence thus offers the unique possibility of studying macroscopic quantum fluctuations in the time domain (Vrehen er al., 1980). The energy of the atomic system is in general not emitted in a single pulse but in the form of a series of pulses of diminishing size. This phenomenon is called ringing. The amount of ringing strongly depends upon transverse effects such as diffraction. Single-shot pulses have also been observed (Gibbs et ul., 1977a, b). The SF output pulses are temporally coherent. This is demonstrated by an experiment in which the SF outputs of two different samples with slightly different S F transition frequencies produce beats (Vrehen, 1979). When the delay time of the emission is varied so as to come close to a dephasing time T,, instead of being much shorter as required for SF, the emission gradually changes its character as was first observed by Okadaet ul. (1978a, b). The emission becomes stationary in nature with a correlation time T 2 . We know now that the SF has then changed into amplified spontaneous emission (ASE) (Schuurmans and Polder, 197913;Ikedaer al., 1980; Schuurmans, 1980). The emission no longer occurs via a large macroscopic dipole as in the case of SF but directly via the excited state population of the atoms. The above shows clearly that theory has a lot to explain. In view of the initial remarks of this introduction it should explain first and above all how it is possible that a macroscopic dipole and thereby SF arises in the system of initially excited atoms. The theory of SF has a long history. Many of the first attempts to understand SF considered point samples, ignoring the dipole-dipole coupling. Later on, detailed discussions of “SF” in point samples in the presence of dipole-dipole coupling appeared in the literature (Rehler and Eberly, 1971; Friedberg et ul., 1972; Friedberg and Hartmann, 1974a,b). Some early important contributions to the theory of SF in extended systems are due to Ernst and Stehle (1968), Agarwal (1969, 1970), Arecchi and Courtens (1970), Rehler and Eberly

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171

(1971), and Degiorgio (1971). Later work on SF in extended systems can be divided into two categories: (1) quantum-mechanical but “mean field” theories, and (2) semiclassical theories, which describe the field envelope variation along the pencil axis. Contributions to the theories of the first kind are mainly due to Bonifacio and collaborators (Bonifacio et af., 1971 ; Banfi and Bonifacio, 1974, 1975; Bonifacio and Lugiato, 1975a, b). These theories greatly contributed to the understanding of the initiation of SF. However, they could not describe the details of the emission since the field envelope variation along the sample was disregarded. The semiclassical theories did account for this variation. Skribanowitz et af. (1973) and MacGillivray and Feld (1976) first launched a semiclassical theory in an attempt to understand their experimental results for the SF infrared emission in HF gas. The initiation of SF in this theory is simulated by an ad hoc-introduced polarization noise source. The theory revealed many important aspects of the phenomenon of SF. Several attempts have been made to bridge the gap between the quantum-mechanical and the semiclassical theories (Haroche, 1978; Ressayre and Tallet, 1978). Recently, fully quantum-mechanical theories of SF, which also describe the field envelope variations along the pencil axis, have appeared in the literature. This article describes recent systematic experimental and theoretical investigations of the phenomenon of SF. The experimental work that is mainly referred to is due to two of the authors ( Q . V. and H. G.) and concerns atomic cesium in an atomic vapor or an atomic beam. The conditions for “pure” SF are easily met in the cesium system, and much of the above-mentioned behavior was first observed in this system. The theoretical work is due to Glauber and the Essen group of Haake (Glauber and Haake, 1978; Haake, 1979; Haake et al., 1979a, b), and, independently, to three of the authors (D. P., M. S., Q. V.) (Schuurmans et al., 1978; Polder et al., 1979; Schuurmans and Polder, 1979a,b; Schuurmans, 1980). It is interesting to note that some aspects of this theory are already in a paper on a related topic: the small signal pulse growth in a swept-gain homogeneously broadened amplifier (Hopf et al., 1976). Other recent experimental and theoretical work will be indicated whenever appropriate. A more complete list of references to theoretical work on SF can be found in the articles by Haake et al. (1979a,b) and Polder et al. (1979). An extensive review of the present experimental status of SF has been given by two of the authors (Vrehen and Gibbs, 1981). It should be stressed already at this stage that, although the present-day theory of SF provides, in particular, a good understanding of the initiation of SF, it is still essentially a one-dimensional theory, which is not capable of explaining all the experimental results. Ringing is a prominent example. Experimentally one observes much less ringing than predicted by the

M . F . H . Schuurmans

172

el al.

numerical evaluations of the theory in the time regime of nonlinear evolution (MacGillivray and Feld, 1976; Gibbs et ul.. 1977b; Saunders et ul., 1976; Bullough et al., 1978). We shall briefly discuss the first attempts toward a three-dimensional and multimode theory, which eventually should also remedy the shortcomings of the one-dimensional theory.

11. Semiclassical Theory In order to get a feeling for some of the physical aspects involved in the phenomenon of S F we start with a semiclassical theory in which the radiation field is treated classically and the atomic system quantum mechanically. The theory is based upon a number of suppositions whose origin will only become clear later on. Let the SF pencil of length L , cross-sectional area S , and volume V contain N two-level atoms or molecules. The levels are coupled by a transition dipole operator p = pz, perpendicular to the pencil (x) axis. The transition frequency is wo. Now suppose that the SF emission takes place in the form of two plane-field waves propagating along the pencil axis to the right (toward positive x ) and to the left. For simplicity also suppose that left- and right-traveling waves are uncoupled. We consider the emission to the right. The response of the atomic system to the classical plane wave field is well known to be described by the Maxwell-Bloch equations. Let the polarization P(x, t ) vary according to P(x, r )

=

iB(x, t ) exp[-iwo(t - .u/c)]

+ C.C.

(1)

and let the electric field vary as E(x, t ) = i 8 ( x . r ) exp[-iwo(t - .u/c)] +

C.C.

(2)

The amplitudes 8 and B are complex valued. Let the inversion of the atomic system be described by n ( x . t ) ; when all atoms are excited, n = 1 . Suppose further that 8, n , and 9 are slowly varying envelope amplitudes (SVEA), i.e., they vary slowly inx on the length scale hoand slowly inr on the time scale l/wo. The Maxwell equation then reads

where ko = wo/c and p = N / V is the atomic number density. The Schrodinger equation leads to a9/at dn/at

= -yzB =

-yl[(n

+

i(plzgm/h I)/?]

+ ( 4 / h ) Im 8*9

(4)

(5)

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173

The decay rates y 1 and y z describe the decay of the upper-level population ( n + 1)/2 and the atomic polarization or dipole moment, respectively. Note that n and 9 pertain to quantum mechanically averaged quantities. The decay may include decay due to collisions and, in a rough way, atomic motions. The latter will be dealt with more precisely in Section IV. We finally need the initial and boundary conditions for Eqs. (3)-(5). Suppose that the inverted state of the two-level transition is obtained by irradiation with a right-traveling 6 function light pulse resonant on a transition different from but coupled to the S F transition (see also Section VI). Then n ( x , t ) = 1 at t = x/c. Since the expectation value of p in the excited state vanishes, P(x, t ) = 0 at t = x/c. The field at the left end facex = 0 does not change due to emission to the right, and so % ( O , t ) = 0. The solution of the equations of motion is now a trivial matter. One finds n ( x , t ) = 1,

9 ( x , 2)’

0,

8(x, t ) = 0

(6)

The system does not radiate because there is no field incident on it, nor is there an initial polarization. This is the result we had anticipated since it is well known that spontaneous emission cannot be described in terms of a semiclassical theory (Sargent et d.,1974). It is instructive to compare the situation with that of a needle put upright in the gravitational field. The needle is in a metastable state, and in vacuum it will not move. If the needle is put in a gaseous atmosphere, atoms collide with it and provide a stochastic force, which turns the needle away from its upright position. Tben gravity takes over, and the needle is pulled further away from its metastable initial position. By analogy one may conclude that what is needed in the semiclassical theory is noise that makes the atom-field system go. To this end MacGillivray and Feld ( 1976) introduced an ud hoc polarization noise source into the equations of motion in order to simulate the initiation of the emission. In the next section we shall see how this noise of quantum-mechanical origin is properly introduced. Anticipating that later discussion, it is instructive to consider the behavior of the atom-field system when its “motion” is triggered by an incident field %(x = 0, t ) . In the next section we shall see that % ( O , t ) has a definite meaning in terms of quantum field noise. For the moment we are only interested in the gain of energy of the electric field in the sample out of the inverted atomic system. We shall limit the discussion of that subject to the case where the decrease in upper level population can be disregarded, i.e., we put n = 1 in Eq. (4). The solution of the resulting linear equations of motion is facilitated by the introduction of the retarded time coordinate T = t - x i c . With the initial condition P(x, T = 0) = 0 and boundary condition %(.r= 0, T ) = % d ~ one ) , then obtains the field equation

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M . F . H . Schuurrnans et al.

Since the natural lifetime is

T,,

TR =

= h/(4/3 Jpl'k$one easily verifies that

(877/3)(Tn/Ph%)

(9)

In terms of the number of atoms N in a pencil of length L and crosssectional area S , one can also write

i.e., T R is the time for the collective decay into the diffraction solid angle Ag/S out of 8x/3 available for linear dipole emission. The solution of Eq. (7) with g(0, 7) = go(^) can be obtained by Fourier transformation (cf. Crisp, 1970). One finds

where -

g o ( v )=

lom go(T)e-iuT dT

(12)

The Fourier transform can be done explicitly using exp[&z(t+ I / t ) ] C~=,-mf'Z~(z)(cf. Abramowitch and Stegun, 1972). We find

=

where I. is the modified Bessel function of zeroth order. Let us first consider the case where S F behavior is expected, i.e., y2 0 formally. For an incident field of constant amplitude goin time one then finds from Eq. (131

-

Since lo(z) = [exp(z)]l(2~z)"'for z + m , the electric field in the medium grows exponentially both along the time axis and along the pencil axis. Next we consider the case of a finite y 2 . Careful analysis of Eqs. (11) or (13) then shows that for a prolonged incident field pulse of constant amplitude go,the field in the medium attains a stationary value. Putting g 0 ( v )=

SUPERFLUORESCENCE

175

27r6(v)8,,,one finds from Eq. (11) that the stationary value is described by 8(x,

T)

=

goexp ad;

(15)

where the linear gain aoL

= T~Z/TH

and T2 = 1/y2. The above analysis shows that the incident pulse is amplified dynamically for short times when the damping of the polarization is not yet effective. This is described by Eq. (14). Then after some time a stationary state is reached in which the field grows exponentially along the pencil axis. This is described by Eq. (16). Clearly the gain is only appreciable as long as aJ.

>> 1

(17)

and this will be assumed henceforth. Decreasing inversion limits the above behavior. The dynamic amplification behavior will turn out to be typical for SF; then the stationary amplification behavior is never reached because before that happens the energy will have escaped out of the system. Only when T2is sufficiently short can one observe the stationary amplification before the inversion decreases appreciably. This is the regime of ASE. The above discussion pertains to the amplification of plane-field waves propagating along or almost along the pencil axis. More generally one can discuss the amplification of plane field waves along other directions. Equations (14) and (15) clearly show that the amount of amplification depends on the distance travelled by the waves through the active medium. As a result the waves in the geometric solid angle S/L' around the pencil axis gain most from the active medium. This explains why SF emission goes mainly into these solid angles and is thus highly anisotropic. If, in addition, we require that the geometric solid angle S / L z be about equal to the diffraction solid angle G/S, i.e., that the Fresnel number F = S/(AoL)will be near unity, the broad features of SF can be described in terms of only two counterpropagating SVEA plane-wave, end-fire field modes, thereby justifying the above one-dimensional analysis. When F >> 1 , the pencil also supports off-axial modes, and a purely one-dimensional theory is impossible. For F << 1, the end-fire modes will be strongly damped due to diffraction. The condition F = 1 represents a compromise. For later use in the quantum-mechanical theory we also note here the relation between the macroscopic field 8 and the atomic variables ?? and n on the one hand and the corresponding microscopic variables on the other hand. Consistent with the plane-wave, end-fire mode assumption the mac-

176

M. F . H . Schuurmans et al.

roscopic variables are expressed in terms of the microscopic variables by averaging over the positions of the atoms in a thin slice perpendicular to the pencil axis. The slice is thin compared to ho and contains many atoms N , ; note that pSho >> 1 since rH<< T, (see also Appendix 11). The necessity of this equiphase plane type of averaging is obvious since the mean interatomic distance is much larger than ho/2r (cf. Kramers, 1938). To begin with we shall restrict ourselves to the initiation time regime where decreasing atomic inversion can be disregarded and to systems where atomic motion and collisions can be disregarded. These demands will be relaxed later on. In the initiation time regime left- and righttraveling waves are decoupled since the pertaining equations are then linear. Henceforth, we shall discuss only the right-traveling waves. Introducing the dimensionless coordinates

and the dimensionless variables

P

=

Pip*

(20)

then, in the absence of damping, the linear Maxwell Bloch equations take the simple form dP/dT

=

E

(21)

dE/dX

=

P

(22)

These equations form the basis for the quantum-mechanical discussion of SF in the initiation time regime.

111. Quantum Mechanical Description of SF A. EQUATIONS OF MOTION IN

THE

INITIATION TIMEREGIME

The derivation of the equations of motion and the commutation relations for the macroscopic field and polarization variables from the Hamiltonian for the microscopic atomic and field Schrodinger operators is far from trivial. The transition from microscopic to macroscopic variables is made by slice averaging, as indicated in the last section. However, the transition also involves difficulties associated with the operator character

177

SUPERFLUORESCENCE

of the variables. Appendix I1 contains a brief sketch. Here we shall content ourselves with some general arguments and we borrow from the appendix when necessary to present the final equations. The correspondence principle tells us that if we adopt E , E " , P , and P* as used in the last section as Heisenberg operators E , E ' , P , and P i in order to step over from the semiclassical theory to the quantum field theory, the Heisenberg operators still satisfy the same linear equations of motion, i.e., dP/aT

=

E

BE/dX

=

P

and the Hermitean conjugated variables satisfy the corresponding Hermitean conjugated equations. The Heisenberg operators are equal to the Schrodinger operators at t = 0, not T = O! The quantum mechanics is hidden in the commutation relations for the Schrodinger operators E ( X ) , E ' W , PLY), and P ' ( X ) . The expression for the polarization in terms of a slice averaging over the microscopic atomic variables as given in Appendix I1 shows that [ P ' ( X ) , P ( X ' ) ]= 6 ( X

-

X')/N

(25)

and [ P i ,P'l = [ P ,PI = 0. The appearance of the number of atomsN finds its origin in the discrete nature of P. The 6 function signifies the obvious uncorrelated polarization in different slices. The commutation relation is of the Bose type. This is at first sight unexpected since the two-level atoms are to be described by spin-3operators. However, since the number of atoms in a slice is large, the operator P corresponds to a spin operator of large magnitude (>>1). We thus use the same approximation as in spin wave theory (Walker, 1963) and may consider the Schrodinger operators P and P i as Bose operators satisfying Eq. (25) (see also Appendix 11). The Schrodinger electric field envelope operators E and E t satisfy [ E ( X ) ,E'(X')l

= (CT,/L)[&X

-

X')/N]

(26)

and [ E , El = [ E ' , E t ] = 0. It is instructive to rewrite this expression in terms of the original field envelope amplitude via E = -i(pZ?/h)TR.The N dependence in Eq. (26) is then seen to be merely due to the particular dimensionless field operators involved in Eq. (26). One finds [ 8 ( X ) ,%'(A'')]

=

(hj/4rS)2h~k$[6(X - X')/L]

(27)

which depends, as it should, only on field quantities. Note that the righthand side of Eq. (27) can also be written as (27rhw0/V)G(X- X ' ) , i.e., an energy density. The present form (27) clearly shows that E and % corre-

178

M.F. H . Schuurmans et al.

spond to the end-fire modes of the Schrodinger field with propagation vectors k E h i / S . Only these modes couple to the slice averaged collective polarization P in the Heisenberg equations of motion. The Schrodinger operators E and P are of different physical origin, and so

[ E , PI

=

[ E ,P ' ]

=

0

(28)

The operators act upon the initial state of the atom-field system I$), describing that all atoms are inverted and the radiation field is empty (no photons). From Appendix I1 it will be clear that the Schrodinger operator P i is a raising spin operator, and so (29)

P'lql) = 0

The Schrodinger operator E corresponds to photon annihilation and so El$)

=

(30)

0

Clearly E and P', and not E and P , are comparable in their Bose behavior. This has to do with the fact that the polarization Bose operator refers to the excited state of the atomic system and not to its ground state. It will have important consequences later on. Note also that the equations of motion (23) and (24) can be derived as Heisenberg equations from the Hamiltonian

x=(~ifi/T~)

+(E~E - E+E,)

+ L-'(EIP - P+E)

(31)

using the properties (26)-(28) of the Schrodinger operators E, E', P , and P' . This shows immediately that the above commutation relations for the Schrodinger operators also hold as equal time (t not T!) commutation relations for the Heisenberg operators. In order to describe the evolution of the atom-field system from an initial state, produced as described in the last section, by a right-traveling light pulse resonant on a transition coupled to the S F one, we effectively let the interaction between E and P start at t = x/c, i.e., at T = 0. Henceforth, we will not explicitly indicate this by modifying the equations of motion (23) and (24) and the Hamilton operator (31). Instead, we specify the properties of the Heisenberg operator for the initial envelope polarization P(X, T

=

0)

=

P,(X)

(32) and the Heisenberg operator for the envelope of the vacuum field waves incident on the left end face of the pencil

E(X

=

0, T ) = E,(T)

(33)

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179

and then solve Eqs. (23) and (24) with these initial and boundary conditions. The initial polarization Po can be obtained from aP/BT = 0 in the interaction free regime. One finds

[ P i ( X ) ,P , ( X ' ) ] = 6 ( X P;l+)

=

-

X')/N

(34) (35)

0

and [PO, Pol = [ P i , Pi] = 0. Equation (35) shows that the initial Heisenberg operator Pois randomly phased, ( P o )= 0. Note also that (P,,Pi) = 0, signifying that all atoms are in the excited state; PP' measures the decrease in upper level population (cf. Section 111,C). The average ( P : ( X ) P o ( X t)) = 6 ( X - X ' ) / Nis due to the commutation relation (34). Of course this relation describes the quantum-mechanical uncertainty in the polarization when the atomic system is prepared in an exact eigenstate of the inversion. The corresponding Po in the semiclassical theory took the value zero. The Heisenberg operator Po being different from zero as a consequence of the quantum mechanics presents one element of the collective initiation of SF. The other element involved in the collective initiation is the Heisenberg operator of the envelope of the incident vacuum field Eo(T ) being different from zero. Using a E / d X = 0 in the interaction free regime and for X s 0, one finds

[ E , ( T ) ,E ; ( T ' ) ] = 6 ( T - T ' ) / N EON)

=

0

(36) (37)

LEO,EO]= 0 and [ E i , E:] = 0. Clearly this shows that ( E o ) = 0, i.e., Eo is randomly phased and (EbE,) = 0, i.e., the vacuum field contains no energy [see also after Eq. (46)l. However, ( E , ( T ) E i ( T ' ) )= 6(T - T ' ) / N , corresponding to the existence of virtual excitations in the photon vacuum. Of course, the commutation relation (37) stems from the commutation relation of the electric field and the vector potential, the conjugate variables in the Maxwell theory. The commutation relation thus signifies the quantum-mechanical uncertainty related with a measurement on the field system. Finally we mention that since the Schrodinger operators are unrelated, as exhibited by Eq. (28), the Heisenberg operators Eo and Po are also unrelated, i.e., [ E o , Pol

=

[Eo, GI

=

0

(38)

Equations (231, (24), and (33)-(38) fully describe the quantummechanical evolution of the SF system in the initiation time regime. We can calculate arbitrarily ordered correlation functions of both the field and

M . F. H . Schuurmans et al.

180

the atomic variables. The above description pictures the initiation of SF, not as being due to the spontaneous emission of a photon by a single atom, which is then passed on to the other atoms, but as being due to the end-fire modes of the vacuum field and the initial polarization of the system. Loosely speaking, one can also say that SF starts off by photons spontaneously and collectively emitted by the atoms along the pencil axis.

B. SOLUTIONS The equations of motion (23) and (24) for the Heisenberg operators, being linear and hyperbolic, have solutions which can be most easily expressed in terms of the Green’s function G ( X , TI. It is the solution of d 2 G / ( 8 T 8 X )= G

+ 6 ( T )6 ( X )

(39)

which vanishes for all negative X and/or T. For X > 0 and T > 0, we have G ( X , 7 )= 1 0 [ 7 ( X T ) 1 1 2 ]

(40)

Since Zo(0) = 1, the Green’s function is discontinuous at X = 0 and T = 0. The first derivative of G with respect to X or T thus contains a 6 function contribution. The solution of Eqs. (23), (24), (321, and (33) reads

E(X, T ) =

lo1 Irn G(X

+

-

X ’ , T ) P o ( X ’ )dX’ ( X , T - T ’ ) E o ( T ’dT’ )

0

(42) Since G ( X , T ) exp [WT)”’’] for X T 2 1 , the electric field and the polarization in the medium grow cumulatively and dynamically at the expense of the inversion in the medium. At the right end of the sample, at x = L, the growth starts at T = T ~ We . come back to this behavior of the solution later on in greater detail. A straightforward calculation using Eqs. (41) and (42) shows that Q

[ E ( X . TI. P + ( X l ,TJ1 + [ P ( X , T I , E + ( X , , TI)] = 0

(43) In combination with d P / a T = E this means that the commutator [ P ’ ( X . T ) , P(Xl, T1)]is conserved under equal time translations T’ - T and T i Tl along the characteristic lines (Courant and Hilbert, 1962) X = constant

181

SUPERFLUORESCENCE

and XI

=

constant, respectively. We thus also have [ P ' ( X , T I , P ( X ' , T ) ]= 6 ( X - X ' ) / N

(44)

Similarly, [ E M , T I , E ' ( X , , T I ) ]is conserved for equal position translations along the characteristic lines T = constant and TI = constant. We thus have

[ E ( X , T ) , E t ( X , T')]= 6 ( T - T ' ) / N

(45)

The conservation rules (43)-(45) extend the equal time ( t ) commutation rules of the previous subsection. Note that (43)-(45) are due to the fact that initial and boundary conditions are given on characteristic lines of the hyperbolic equations. Note also that [ P ' ( X l , TII, P ( X , TI1 f 6 ( X - X , ) / N . The commutation relations show that in the initiation time regime we deal with field harmonic oscillators coupled to atomic harmonic oscillators. However, the commutation relations describing them involve different ordering of the operators. The ordering would have been the same, as noted in Section III,A, if the atomic oscillators had referred to the ground state of the atomic system. Now that they refer to the inverted state, the oscillators are inverted oscillators; we shall see shortly that unlike the field oscillators their energy is bounded from above. This different ordering and correspondingly different oscillator character will play an important part in the next subsections.

c.

CORRELATION FUNCTIONS

What kind of correlation functions do we need to describe the emission behavior of the SF system? It is easily verified that the energy emitted per second is described by the operator J ( f )= (NhWO/TR)(E+E)(X = L,

1)

(46)

The normal ordering (crosses appear to the left) corresponds to the fact that a photon counter first annihilates a photon in the detection (Glauber, 1970). From the equations of motion and Eqs. (44) or (45) one finds the energy conservation equation in dimensionless form to be given by

This confirms that the operator 614 = PP' describes the decrease in upperlevel population in the atomic system. The energy of the atomic system itself is described by I - PP'. The ordering in this expression is of the antinormal type (crosses appear to the right). The different ordering in

182

M . F. H. Schuurmans et al.

J and 6u relates to the different harmonic oscillator character of field and polarization. The field oscillators are of the common type; the energy E'E is bounded from below. The polarization oscillators are of the inverted type; the energy -PPt is bounded from above. This different harmonic oscillator behavior has important consequences. In view of P&) = 0 all antinormally ordered (all crosses appear to the right) correlation functions can be fully expressed in terms of correlation functions of E0 and EL, only. Similarly, in view of Eel$) = 0, all normally ordered correlation functions can be fully expressed in terms of P o and P i only. For example, the average decrease in upper level population is given according to Eq. (42) by

x (Eo(TlE;(Tz))

(48)

Using Eqs. (36) and (37) one thus finds ( 6 u ) = (I/N) =

I,' I i [ Z ( X T ' ) ' p 2dT' ]

(T/N){z9[2(XT)'py- z ' : [ Z ( X T ) ' q }

(49)

The average emission intensity ( J ) can be calculated similarly or by directly using the energy conservation relation (47). Using the commutation relation (44) all equal-time ( T ) correlation functions of the polarization can be expressed in terms of antinormally ordered correlation functions of the polarization and thus in terms of E o ( T ) only. A similar statement holds for the equal position (XI correlation functions of the field. Of course, for the evaluation of more complicated correlation functions like mixed field polarization and different-time, different-position correlation functions we need both E o ( T )and P o ( X , . In the following we restrict ourselves to the calculation of equal-time and equal-position antinormally ordered correlation functions. They determine directly the response of the atomic system and after proper use of the field commutation rule (45) also the radiation field behavior. Of course, one can also take the opposite starting point and calculate normally ordered correlation functions. This has been done by Haake (1979) and Haakeet af. (1979a, b). Dealing with antinormally ordered correlation functions means that we can formally put Po = 0. In view of the linearity of the equations of motion in the initiation time regime the ultimate consequence is that the statistical behavior of our system can be conveniently represented in terms of classical (c number) stochastic variables.

SUPERFLUORESCENCE

183

D. THESTOCHASTIC VARIABLES DESCRIPTION The stochastic variables description suited for the calculation of antinormally ordered correlation functions is obtained by putting Po = 0, since Podoes not contribute to such functions, and by treating E , E ' , P , and P ' a s complex-valuedc numberse, e * , p , andp*, by considering (Eo, EL) as a classical fluctuating field source ( e o ,e ; ) ( T ) and by identification of the quantum-mechanical average with the ensemble average over stochastic variables. The stochastic variables description pictures the atom-field system as being driven by an incident bivariate field noise source ( e o , e $ ) ( T ) . Since the original operator Eo satisfies E o ( $ ) = 0 and [ E , ( T ) , Eb(T')] = 6(T - T ' ) / N we find the (1) all correlation functions containing an odd number of functions e, and eg vanish, (2) the second-order correlation function

( e , ( T ) e $ ( T ' ) )= 6 ( T

-

T')/N

(50)

and (3) all even higher order correlation functions factorize according to

( ~ o ( T I.) .

. e0(Tak$(C). . . d ( T h ) )

2n a

=

P

(eo(Th&TbJ))

(51)

J=l

where the sum is over all possible permutations of time. The property (51) makes ( e o ,e : ) ( T ) a bivariate Gaussian noise source, and this is due to the Bose character of the operator E o ( T ) . The equations of motion and the initial and boundary conditions are very similar to the Maxwell-Bloch equations of Section I1 in the initiation time regime. There are two differences: (1) the variables are now stochastic variables and not quantum-mechanical averages over the atomic part of the system, and (2) the incident field e o ( T )is now a stochastic variable different from zero. This noise source e o ( T )already anticipated in Section I1 initiates in this description the SF and accounts for the quantum uncertainty in the electric field. A s we have seen before, classically or semiclassically the system would not radiate. In a different terminology the noise source e o ( T )may be said to consist of the zero-point fluctuation waves of the electromagnetic field in vacuum as they propagate to the right in the solid angle h$lS around the pencil axis. In terms of the original field variables this is shown by the relation ( %(T)%*(T'))

=

( h i / 4 r r S ) 2 h k % 6 (-~ T')

(52)

which is easily derived from Eq. (36). Note that the strength of the zeropoint fluctuation waves is twice the usual one (cf. Landau and Lifshitz, 1960) since, adopting our viewpoint of antinormally ordered correlation

184

M . F . H . Schuurmans et al.

functions, this strength must assure the spontaneous emission initiation all by itself. If symmetric ordering is used, Po and Eo contribute equally to the initiation, and the corresponding p o and eo have the usual strength. In a moment we shall see that Eq. (52) indeed gives the correct singleatom spontaneous emission behavior for 0 s T << T ~ . E. THE INITIATION OF S F Physically the initiation of SF can now be described in classical terms as follows. Zero-point fluctuation waves pass through the pencil in all directions. These waves induce electric dipoles in the medium, which then, in turn, start to radiate. As long as the gain in the system can be disregarded, , emission from the system is both temporally and i.e., for 0 S T << T ~ the spatially incoherent. It is just single-atom spontaneous emission. For T > T ~ the , field modes in the end-fire solid angles A%/S around the pencil axis start to dominate since they acquire the largest gain. The R-field modes, as described bye , couple to a collective polarization described by p . The emission of the system then becomes highly anisotropic and, as we will see, temporally coherent. For each position along the pencil axis the state of the atomic system can at each time T be described by the Bloch vector p(X, T ) which has as its components the complex-valued collective polarization p ( X , T ) and the atomic inversion n(X, T ) . The Bloch vector can also be described by a tipping angle 8 and a phase angle 4. The collective polarizationp = 4 sin 8 exp ic$ and the inversionn = cos 8. For 8 K 1, as is the case in the initiation time regime, we have p = +O exp ic$ and the decrease in upper level population 6u = a#. According to Eq. (42) the behavior o f p is described by p ( X , T ) = I T d T ’ Z o ( 2 [ X ( T- T’)]’p2)eo(T’) 0

(53)

The corresponding average decrease in upper-level population is described by Eq. (48). At T = 0 all atoms are inverted and correspondingly all Bloch vectors are upright, i.e., 8 = 0. Due to the zero-point fluctuation waves as described by e o ( T )the Bloch vectors start to jitter in a kind of two-dimensional Brownian motion. Indeed, we find from Eq. (53) that ( p ) = 0, and the mean-squared tipping angle grows proportionally to time T (diffusion-like) for T << 1, i.e., (0’) = 4 T / N

Accordingly, for T << T given by

~ the ,

(54)

average decrease in upper level population is

( 6 u ) = (~/~,)(3Ai/8nS)

(55)

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185

This is precisely the result expected for single utom spontuneous emission in the solid angle A%/S out of girl3 uvailuhle fiir linear dipole emission. On the time scale 0 T << T~ the atoms still radiate as single entities, and they do so isotropically in all directions, not only in the one selected here in view of the high gain in that direction and F = I . After a time T~ amplification is predominant, and an exponentially growing tipping angle directs the Bloch vector downwards on the Bloch sphere. On account of Eq. (53) and the asymptotic behavior of Z,, for large arguments, we find (6u ) =

f (H’)

=

e~p[4(XT)”~1/8irNX

(56)

Energy conservation then shows that on the average the energy emitted per second through the end face is given by ( J ( t ) )= ( h w 0 / 8 n t )exp[4(t/~,)”~]

(57)

and is released fully via the large macroscopic dipole formed in the medium. Later on, when we consider the case of additional homogeneous or inhomogeneous broadening, the energy will only partly be released via the macroscopic polarization. The Bloch vector length is then no longer conserved. So far we have not discussed the meaning of the Bloch vector trajectories corresponding to different eo(T ) source functions. Also it is not clear how we can extend the linear theory in order to describe decreasing inversion. Both subjects will be treated in the following section after the introduction of the notion of classical behavior.

F. CLASSICAL BEHAVIOR AND

THE

INTERPRETATION

O F SINGLE-SHOT OUTPUTS

Equation (53) shows that for T >> T~ the zero-point fluctuations become less and less important for the further time development of the polarization. The Bloch vector then follows the trajectory it has chosen once and for all on the Bloch sphere, and the time behavior of the system becomes deterministic or classical in nature. The latter conclusion follows more generally from the operator solutions (41) and (42) of the complete and quantum-mechanical equations of motion for the system development in the initiation time regime. For increasing field and polarization amplitudes the operator character of both the atomic and the field variables become less and less important; the commutators like Eqs. (43)-(45) which govern the quantum mechanics of the system remain constant in value while the polarization and the field themselves grow exponentially for X T >> 1. The field and polarization operators are thus going to behave like c numbers in the course of time,

186

M . F . H . Schuurmans et a l .

i.e., the state vector describing the system remains eventually at the same position in the Hilbert space. The tendency toward classical behavior can be exemplified, though not proved, by the following relations that hold for XT + x : (E'E)

-

( ( E - Eo)'(E

Eo))

(58)

Po)')

(59)

( ( E - E o ) ' ( P- P o ) ) + ( ( P - Po)(E - E o ) ' )

(60)

(PP')

+

( ( P- PONP

-

-

These results can be simply derived from Eqs. (41) and (42) using the asymptotic behavior of lo for large arguments. The tendency toward a well-defined deterministic relationship between E and P is illustrated by the relation

for X T + x . A detailed analysis of the way the classical regime is entered has so far not been made. Estimates show that the quantum noise is outweighted by a factor of 100 for X T = 4, i.e., T is equal to a few T~~ almost everywhere in the sample. Near the left end face of the pencil the emission to the right must always be dealt with quantum mechanically. The meaning of the different Bloch vector tip trajectories on the Bloch sphere, corresponding to different eo(T 1 functions, is now clear. Each trcdectory corresponds t o the behavior of the system afrer a single-shot preprirlition of the excited state of the utoms in the sample. Correspondingly, ensemble averaged quantities like ( J ) , for example, correspond to experimental quantities averaged over many such single shots of preparation of the inversion. Quantum mechanically this result is also rather obvious. If, after a few T ~ { i.e., , in the classical regime, the emission intensity is measured for the first time, the vector describing the state of the system in the Hilbert space is projected onto the eigenspace of the measuring operator. In the classical regime, whose very existence is due to the high gain in the system, the projected state vector stays at the same position (in the same subspace) of the Hilbert space. and so subsequent measurements give strongly correlated results. One measures a classically behaving, deterministic SF pulse. However. one cannot possibly predict the shape of the pirlse even though the pulse dynamics is clcrssical.for the times relevant to its observation. The particular shape depends on the particular projection involved in that first measurement in the course of time. The various possible shapes are determined by the quantum-mechanical uncertainties

SUPERFLUORESCENCE

187

in the field and the atomic system. Averaged over many single shots one thus observes the quantum-mechanical average. In our stochastic variables description different pulses thus correspond to different trajectories on the Bloch sphere, each trajectory being associated with a particular element of the ensemble of noise source functions e d T ) which effectively represents the quantum uncertainty in the field. One may wonder here whether a different stochastic variables description, for example, one based upon normal ordering and with po(X)as a noise source, would provide a different set of single-shot outputs. However, as already indicated by Eqs. (58)-(60), and in particular by Eq. (61), the ensemble of classically behaving single-shot outputs is generated equally well by whatever stochastic variables description is used. Outside the classical regime, for example, for 0 s T < T ~ the , stochastic descriptions have only a formal but no physical meaning. Let us once again return to the Bloch vector description of the initiation of SF. After a few T~ a Bloch vector follows its chosen trajectory, i.e., the phase becomes fixed and 0 behaves deterministically in the course of time. The single-shot emission output thus also behaves deterministically. The SF emission is temporally coherent. This is beautifully illustrated by the beat experiments described in Section VII. Note that 4 is a random variable which can take values in between 0 and 2ir. Therefore,.formally, on the civercige the system produces no jeld and no polarization! Decreasing inversion limits the validity of the linear theory. However, if the system behaves classically before we enter the regime of nonlinear evolution, then the solutions in the linear and classical regime can be continued into the nonlinear regime by the nonlinear Maxwell-Bloch equations an

711T =

dt

-4 Re[P,,Ef + P,ERl

The + and - signs correspond to the R and L waves again. The nonlinear evolution starts at times of the order of the delay time 7,) of the emission pulse. Since, as we will see in a moment, T,) (rR/4)[lnN I 2 , the classical regime is approached before the nonlinear regime when In N is sufficiently large. Haake (1979) and Haakeet cil. (1979a,b) have numerically integrated the nonlinear Maxwell-Bloch equations with a polarization noise source

-

188

M . F. H . Schuurmans et al.

p o ( X ) ;they work with a stochastic variables description based upon normal ordering. These calculations provide full single-shot J ( t ) trajectories. We should stress here once again that this is only possible when the system behaves classically before the nonlinear regime is entered, i.e., for sufficiently large N. When this is not the case, one runs into difficulties in particular because the Bose type commutation relation (44) is no longer valid in the nonlinear regime. Under these conditions the linear theory, although still valid, becomes less useful too. A Holstein-Primakoff transformation (Walker, 1963) may then guide the way to enter the nonlinear regime, as is also indicated in Appendix 11.

G. AVERAGE BEHAVIOR; EFFECTIVE I N I T I ATIPPING L ANGLE Superfluorescence is particularly interesting because quantum fluctuations manifest themselves macroscopically in the time domain. From excitation shot to excitation shot the SF emission output fluctuates in shape. The strength of the initiating quantum noise as described by the field and the polarization commutation rules determines the average delay time. The Bose character (or Gaussian character in a stochastic variables description) of the quantum noise determines the fluctuations in the delay time, the amplitude, and in fact the shape of the emission pulse. We shall now first describe the emission behavior to the extent that this is possible using the linear theory of the preceding subsections. The aver) be estimated from the average decrease in upper age delay time ( T ~ may level population ( S u ) as described by Eq. (56). We take (7,))to be the time at which ( S u ) at the end face is equal to 4, i.e., the mean-squared tipping angle ( P ) , according to ( S r r ) = b(f3”), becomes equal to 1 at x = L. We find from Eq. (56) This result is only slightly modified by a different choice for ( S u ) (x = L , T = ( T ~ ) since ) only the argument of the logarithm is changed and In N is assumed to be very large. Equation (65) may then also be expected to describe the experimental ( T ” ) , which is defined as the time at which the emission intensity reaches its first maximum. The argument of the logarithm does not depend on the end-fire solid angle AgS, as was predicted by some of the earlier theories (Rehler and Eberly, 1971; Banfi and Bonifacio, 1974, 1975). The reason for these erroneous predictions lies in the implicit assumption that the spontaneous emission fully goes into the end-fire solid angle A%/S.The dependence of ( T , ) ) on the logarithm is quadratically unlike the linear dependence predicted by the mean field theories (Bonifacioer ul., 1971; Banfi and Bonifacio, 1974, 1975; Bonifacio

SUPERFLUORESCENCE

189

and Lugiato, 1975a, b). The reason for this difference is that in the meanfield theories the amplification of the emission at the end face goes as exp(tl.r,), whereas propagation modifies this into exp(rlTR),liB as shown by Eq. (56). One can simulate the avercrge behavior of the S F emission in the semiclassical Maxwell-Bloch theory by giving all the Bloch vectors an initial tilt over the angle O0. Equivalently one may send a 6 function light pulse of area 8,, through the atomic system. A simple calculation similar to the one made in Section I1 then shows that for T >> T~ the angle 8 increases at the right end face according to 6 = [ 8 0 / ( 4 ~ ) ’ r 2 ] T - ’e’xp( 4 2 VT)

(66)

The T-”4 dependence makes the increase with time a bit slower than exponential. This is due to the fact that the effective initial tipping angle can be considered to simulate a noise source acting for a finite time only. We can now calculate 13,from Eqs. (65) and (66) by taking it to be the value at which we get equal delays from these equations. We find 1 3 ~= ( 2 / f i ) ( g In 2 n ~ ) l ’ ~

(67)

The value of O0 can also be estimated in the following simple way. According to Eq. (55) the average decrease in upper level population due to emission alongx in hg/S is ( 6 u ) = ( T / T ~ ) UN). ( So at T = T ~ one , photon is emitted collectively in the end-fire solid angle Ag/S. At that time amplification starts to set in, and the quantum noise becomes less and less important. The effective initial tipping angle O0 can be estimated from the value of (0’)”’’ at T = T ~ One . then finds B0 = 2/v%. Thus, B0 can also be interpreted as the average angle over which the Bloch vectors are tilted when one photon is emitted collectively into the end-fire solid angle G / S . Corresponding to the different predictions for ( 7 , ) ) there have been different predictions for O0. Recent experiments by Vrehen show that, at least at N = lOW, there is agreement with the prediction of Eq. (67). These experiments are described in Section VII. Numerical calculations of the average emission intensity as a function of time based effectively upon the nonlinear semiclassical Maxwell-Bloch equations with an effective initial tipping angle B0 for the Bloct- vectors have been performed by MacGillivray and Feld (1976), Gibbs et [ I / . (1977b), Saunders rt u /. (1976), and Bullough et [ I / . (1978). These calculations were strongly influenced by the pioneering work of Burnham and Chiao ( 1969), who studied the propagation of small-area pulses through absorbers. The numerical results exhibit strong ringing due to local exchange of energy between the atomic system and the field in the time regime of nonlinear evolution. However, this ringing is much stronger than observed. We return to this point in Section V.

M . F . H . Schuurmans et al.

190

H. FLUCTUATION BEHAVIOR

In this section we first give the statistics of the SF emission in terms of the statistics of the Bloch vector behavior in the linear regime. Then we discuss the statistics of the delay times. Finally we come to numerical calculations which yield both the delay time statistics and the pulse amplitude statistics. We are interested in the probability density n(a,a*) for the stochastic variablep to take the value a and forp* to take the value a*. All ensemble averages can be expressed as moments of n, i.e., (p”p””l)

=

I ana*”’n(a,

a*) dZa

(68)

Using the Gaussian properties of the noise source e , ( T ) ,and Eq. (531,one finds (Polder et al., 1979) by standard linked averaging techniques (Lax, 1968) that 11 is the two-dimensional Gaussian probability density described by

where (pp*)

=

f ( 8 ’ ) = (I/N)

I’ 0

12,[2(XT‘)’”]dT’

(70)

Equation (69) shows that, in accordance with our expectations in view of the Gaussian character of the initiating noise, the stochastic variation of the collective Bloch vector can be described by a Fokker-Planck process in two dimensions on the Bloch sphere. In fact, one easily proves that

Introducing the amplitude q and the phase x of a in accordance with a = $77 exp ix, one finds that the probability fi dq2 d x for IY to be in between q2 and q 2 + dq‘ and 4 to be in between x and x + dx is given by

This again illustrates that, starting from its upright position, there is no preferential direction of motion for the collective Bloch vector. Moreover, it shows that 7’ is exponentially distributed with mean value ( P ) . The fluctuation behavior of the SF pulse can now be analyzed. From Eq. (72) one finds (04)= 2(g2)’, i.e., a spread in O2 values of 100% in accordance with a two-dimensional Gaussian distribution. Since- 8 in-

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191

creases exponentially with time, the corresponding spread in delay times is much smaller. The average delay time ( T , ) ) was estimated in the linear theory from the time at which the mean-squared tipping angle at the right end face reaches the value 1. A similar definition can be used for the single-shot delay time T , itself. It is the time at which the tipping angle at the end face reaches the value 1, i.e., TI)= T D / T H satisfies O(TD, 1) = 21p(T,,. 1 ) )

= 1

(73)

Using Eq. (53) we may also write

here This equation can hardly be solved analytically to give T1,({eo(T}); {eo(T)}for a given eo(T)function denotes the collection of its values taken for 0 T T , . Even if Eq. (74) can be solved, then it will still be very hard to do the functional integral over the probability density n ( { e 0 ( T ) } ) , which expresses that e o ( T )for each T is Gaussian and independently distributed. Of course, one can do the analysis numerically. Such an approach was adopted by Haake et (11. (1979a, b). We shall discuss it in a moment. It does turn out to be possible to find the delay time spread in the following way. In an excellent approximation we have

Here we use the finding that early eo(T)contributions yield the dominant contributions to the polarization at the right end face. We rewritep(1, 7‘) as follows: p ( I . T ) = $ g ( T )exp(2v‘T)

(76)

where g ( T )is defined by comparison with Eq. (75). From Eq. (73) and (76) one finds that the dimensionless delay time TD satisfies TI) = &[Inlg(T,,)1212

(77)

At first sight this relation seems to be no more useful than Eq. (74). However, the point is that the distribution of g ( T ) , as defined here, is independent oJ’T for T >> &. By means of linked averaging techniques one finds that Ig(T)IZis exponentially distributed according to

A simple calculation shows that

M . F . H . Schuurmans et al.

192

where n is a nonnegative integer and T, is given by Eq. (77). One finds ( T ~ = ) (T:))

( ~ ~ / 1 6 ) (+ y ‘7r2/6)

+ ( ~ ~ / 1 6 ) * [ + 7 r+9 ’8 5 ( 3 ) y + (11/90)7r4]

= (T,))~

(81) (82)

where y = y E + ln(27rN). Euler’s constant yE = 0.57721. and the Riemann function ((3) = 1.20205. . . . Note that although (T,) as obtained in Eqs. (65) and (81) is differently defined, the resulting values for ( T,) differ only slightly. The relative standard deviation AT, is defined as

AT”

= [((T;)

-

(TD)’)/(TD)*]”~

(83)

From Eqs. (81) and (82) one finds that AT” vanes roughly (error
(84)

Similar logarithmic behavior was derived by Degiorgio (1971) but the proportionality constant was 1.3 instead of 2.3. The difference is due to the fact [see also the discussion below Eq. (631 that Degiorgio used a mean field type expression for T ~ ) . Note that exactly the same results are obtained by approximating p ( l , T ) very roughly by p ( I , T ) = poet*

(85)

and letting the variable Ipol’ have an exponential probability density

where (Ipol*) = 1/(8 T N ) . Clearly Eq. (85) constitutes a family of “congruent” emission pulses, whereas Eq. (74) yields a larger amount and a greater variety of different pulses. Nevertheless, for the calculation of ( T D ) and A T D the description (85) is useful although it certainly does not adequately describe the whole family of emission pulses that can be generated. Note also that AT^, can be estimated directly from Eq. ( 8 5 ) , Eq. (65) for ( T ” ) , and ( lp14) = 2 (lp12)2.An approach of that type is used in the next section when homogeneous and inhomogeneous broadening are taken into account. Finally note that Ar, is obtained from the linear

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193

theory. Nonlinear effects may affect to some extent the results obtained here. As noted in one of the previous sections, Haake et al. (1979a, b) have numerically integrated the nonlinear Maxwell-Bloch equations with a polarization noise source p o ( X ) .These calculations provide single-shot J ( t ) trajectories, which can be used to study single-shot emission behavior in great detail. Here, of course, one has to realize that the theory only applies to idealized conditions since it is a one-dimensional theory in the absence of inhomogeneous and homogeneous broadening. Using the fact that p o ( X )for each X is Gaussian and independently distributed, Haake and co-workers have calculated both AT,, and the relative standard deviation AA in the maximum emission amplitude for N = 4 X lo9, which applies to typical experiments done by Vrehen (1979). They find A T , , = 12% and A A --- 18%. The earlier discussed linear theory yields AT,, = 12.5% at N = 10’. In Section VII theory and experiment will be compared on this point (see also Section V, however).

IV. The Effect of Homogeneous and Inhomogeneous Broadening on SF A. INTRODUCTION

When the atoms in the active medium have slightly different transition frequencies, e.g., in the presence of inhomogeneous broadening, the atoms in a thin slice, as introduced in the previous sections, get slightly out of phase, even though they feel the same field in the slice. This dephasing hinders the buildup of a collective macroscopic polarization in the medium. Correspondingly, the SF delay time will increase. Eventually, for a large amount of inhomogeneous broadening, only a small part of the emission goes via the macroscopic polarization, the main part being governed by rate processes involving the upper-level populations of the atoms themselves. The rate of growth of the polarization with respect to time then becomes zero, and the emission can be characterized as amplified spontaneous emission (ASE). This type of emission is usually observed in systems in which the inversion is kept constant by supplying energy at a constant rate, for instance, in a gas discharge. In our case, the duration of the ASE process is limited by decreasing inversion. In this section we present the equations of motion which describe the initiation of the emission in the presence of inhomogeneous broadening.

194

M . F . H . Schuurmans et al.

These equations are then solved and used to describe ( 1 ) the transition between SF and ASE and (2) the effect of inhomogeneous broadening on ( T , , ) and AT^. Finally, the effect of homogeneous broadening is very briefly discussed (Schuurmans and Polder, 1979a, b; Schuurmans, 1980).

B . EQUATIONS OF MOTIONA N D THEIR

SOLUTION

Let the transition frequencies of the atoms be spread around the center frequency wo with the probability density u(Aw), where Aw is the detuning from line center with respect to wo. The probability density has a width 1 /Ti and is normalized according to .f ~(40) d A w = 1. The width I/Tg is the Doppler width in case the transition frequencies are Doppler shifted due to atomic motion. We assume that T R << Tg << T,,, implying that the , the pencil axis is much larger small-signal linear gain a0L = T ~ / Talong than 1 . This, combined with F = 1, allows a one-dimensional description again. Introducing the envelope P, of the collective polarization of atoms with dimensionless detuning A = A o T R , the correspondence principle then tells us that in the initiation time regime the Heisenberg operator envelopes E and P, of right-traveling waves satisfy aPJaT

=

aE/aX

=

iAP,

+E

P

(87) (88)

where P is the total polarization as given by

P

=

1V(A)P,dA

(89)

and V ( A ) = u(hw = h / T R ) T R - ’ . We further proceed by andogy with Section 111. The initial and boundary conditions are P,(X, T E(X

=

=

0)

=

P,,o(X)

0, T ) = Eo(T)

(90) (91)

where the vacuum field envelope Eo satisfies Eqs. (36) and (37) and commutes with and The initial polarization envelope satisfies

and Pi,ol+) = 0. The total polarization Po as defined by Eqs. (89) and (90) satisfies Eqs. (34) and (35). The above constitutes a complete quantum-mechanical description of the atom-field system in the initiation time regime. The solution to Eqs. (87) and (88), together with Eqs. (90) and (91), is obtained by

195

SUPERFLUORESCENCE

means of a double Fourier transformation with respect to the time T and the position X. The solution can be expressed in terms of a function A ( u ) defined in the lower half-plane of the complex u plane by

This function is analytic in that plane. The response to the boundary field E o ( T )is most easily expressed in terms of the Green’s function

where the Heaviside function H ( X ) = 1 for X > 0 and H ( X ) = 0 for X < 0. The function F, satisfies the Green problem

and vanishes for X < 0 and/or T < 0. The response to E o ( T ) is thus = 0) described by (formally P,(X, T ) =

Iom

F,(X, T

-

T’)Eo(T’) dT’

(96)

and E = a P , / a T - iAP,. The response to P,,,(X) can also be expressed in F,(X, T ) . One finds

E(X. T ) =

Im dX’ Im dA’V(A‘)F,,(X

and P , is obtained from P,

=

P,,oe‘AT +

--m

-

X ’ , T)P,<,o(X’)

I,’E ( X . T ’ )exp[iA(T

-

T ’ ) ]d ~ ‘

(97)

(98)

Explicit calculations with these solutions show that [ E ( X , T ) , E + ( X ,T ’ ) ] = 6(T - T ’ ) / N

(99)

i.e., the initial and boundary commutation relations are conserved when penetrating the first quadrant of the X-T plane along the characteristic lines T = constant and X = constant, respectively. This is reassuring in the sense that we do expect that the quantum aspects of the problem, as expressed by the Bose character of the field E and the polarization P locully, should not change upon the admission of a spread in atomic resonance frequencies. A direct consequence of Eqs. (99) and (100) is the operator energy conservation relation

196

M . F. H . Schuurmans et al.

a

- 6~ =

dT

a

-E+E

ax

where the operator

describes the decrease in upper-level population. The emitted intensity is given by Eq. (46). In the following, the transition from SF to ASE will be analyzed from the response of the atomic system in terms of antinormally ordered correlation functions of the polarization. As in the previous section, this response can be fully expressed in terms of E,(T) only, i.e., we only need Eq. (96). Also one can formally introduce a stochastic variables description and thereby describe the response of the atomic system in a Bloch vector picture. As before, stochastic variables will be referred to by means of small letters.

C. THESF TO ASE TRANSITION In the stochastic variables description the dynamical behavior of the atomic system can, for each position X along the pencil axis and for each detuning A, be described by the motion of a Bloch vector p,(X, T ) of length 1 with tipping angle 8,(X, T ) and azimuthal angle +,bY, T ) . We have m, = n , / V ( A ) = cos 8, and 2p, = sin 8, exp i+, , where m , is the inversion of atoms with detuning A. The detuning integrated inversion is n = n, d A. Under the influence of the zero-point fluctuation waves as described bye&T)a polarizationp,(X, T ) is introduced in the system and this polarization in turn generates an electric field. The behavior of the Bloch vector p,(X, T ) is described by the Maxwell-Bloch equations am,/dT dp,/aT

=

-4 Rep*,e

(103) (104)

= m,e

&/ax = J p , V ( b )

db

(105)

with the initial condition PAW, 0)

=

0

and the boundary condition 40, T ) = edT)

(107)

Note that strictly speaking these equations are only valid in the initiation time regime. Their extension toward the nonlinear regime is based upon

SUPERFLUORESCENCE

197

the eventual deterministic behavior of the system; the noise should be of no influence on the further time development of the system. In the ASE limit, however, the noise remains important. It is doubtful whether Eqs. (103)-(107) can then be used to describe the ASE in the time regime of nonlinear evolution. Initially, all the Bloch vectors are upright, i.e., along the inversion axis. As is well known from the behavior of the solutions to the Maxwell-Bloch equations, the Bloch vectors start to describe circles on the Bloch spheres. Initially this is described by Eq. ( 9 6 ) , and we then have m , = 1 - 2 p g ; . The decrease in total upper level population is described by Eq. (102). The total polarizationp as defined by Eq. ( 8 9 ) can also be written as p(X, T ) =

1; g ( X , T

-

T ’ ) p o ( T ’dT’ )

(108)

where g(X, T) =

V(.l)F,(X. T ) d A ~

7.

In the absence of inhomogeneous broadening V(A) = 6(9), A ( v ) = l / ( i v )and thus g ( X , T ) = lO(2m). Comparison with the previous section shows that the expression for p ( X , T ) describing pure SF is indeed regained. Also 614 = p p ” , and the atomic system loses its energy via the collective polarization p. In the presence of inhomogeneous broadening a simple application of Schwarz’s inequality to the eqs. (108) and ( 9 6 ) yields ( p p ” ) < (611), i.e., not all the energy is radiated collectively via the fotal polarization p. Correspondingly one verifies that g ( X , TI + 0 as T + x , i.e., the Bloch vector p shrinks in length due to the dephasing corresponding to the circular motion commenced by the Bloch vectors p l . For small enough T$ we have ( p p ” ) << ( 6 u ) , and the energy of the atomic system is released via single atom spontaneous emission amplified in the inverted medium, i.e., by ASE. A detailed discussion of the emission characteristics as a function of time is difficult for general V(A). In the following we therefore specify V ( A ) in the form of a Lorentzian profile: V(9)

=

YIT ~

.I2 + Y 2

(110)

where y = T ~ / T = $ l / ( a 0 L ) ThenA(v) . = I / ( i v + y ) and contour integration in the UHP of the complex v plane yields

198

M . F. H . Schuurmans et al.

g ( X , T ) = Zo[2(XT)’”l exp(- y T )

Contour integration in the UHP of the complex average decrease in upper level population

u

(112)

plane shows that the

where the “collective” part is (pp*)

=

N-’ I T g 2 ( X ,T ’ ) d T ’

(114)

We consider the system’s behavior for XT >> 1. Then the exponential behavior o f l i ( 2 m ) for large arguments leads to a “sharp” maximum of the integrand g U , T ) at To = X / y 2 . For T < To, the main contribution to the integral in Eq. (1 14) comes from T’ = T. We then find (PP*) =

For T > To, expansion of integral, and therefore

exp[4(XT)’” - 2yT] 8irNX g2

around T

=

(1 15)

To results in a Gaussian

From energy conservation and Eq. (113) one can easily calculate the emission intensity ( J ( t ) )f o r t >> T R and t >> L / c . For t << T f 2 / r R= (a&)Tf = ( ( Y , & ) ~ T ~one , finds ( J ( t ) ) = ( h ~ o / 8 i r te)x p [ 4 ( t / ~ , ) ’-~ ~2t/T41

(117)

Comparison with Eq. (57) for the case of pure SF shows that this expression describes SF emission damped by inhomogeneous broadening. Since ( 6 u ) - ( p p * ) is much smaller than ( p p * ) , most of the energy is radiated collectively. The high gain overcomes the dephasing during the early stages of the evolution. In the later stages, f o r t >> T;’/TR, the intensity ( 4 2 ) ) reaches a stationary value ( J , ) = ( h w , , / 2 ~ r T ,exp(2aol) ~~~) where TEff = Tf(2a0L/7r)”!‘

(1 18)

199

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describes the bandwidth narrowing in the linear high gain system. This result may also be stated more generally, i.e., for arbitrary V(A). Deriving E viaE = d P , / a T - iAP from Eq. (96) and using the commutation relation (99) one finds that the autocorrelation function j ( T l , T2)= ( N z l w , / ~ , ) ( E ' ( X . T , ) E ( X . Tz))exp[iw,T,,(T', - T2)]for TI x and T2 + x. depends only on T , - T2,i.e., the emission becomes stationary. The intensity emitted into a frequency band [w, w + d w ] can then be described by J,dw and J , = JrmJ,d dw. The noise spectrum J , is the Fourier transform of the autocorrelation function j ( t - t l ) and one easily verifies that

-

dJ,/dx

=

2a(w)[J,+ (hw/Zn)]

(120)

where the absorption coefficient a ( o ) obeys a ( w ) f . = Re A [ . =

(W

-

Wo)T,]

=

nv[A

= (w

- OO)TH]

(121)

Equation (120) is exactly the equation derived by Einstein (1917) for the growth of the intensity of spontaneous emission in a medium with a frequency-dependent "absorption" coefficient a ( w ) . This is what we call ASE. The incident noise spectrum J z = h w / 2 n of the amplifier corresponds to twice the contribution of the intensity per unit area and per unit c~] and Lifshitz, 1960) of the zerosolid angle 1; = I h c 0 ~ / [ ( 2 n ) ~(Landau point field fluctuations in vacuum; we haveJz = 2Z;(hi/S)S. As we know by now, such a number of zero-point fluctuations is necessary in a classical theory to simulate the spontaneous emission. Only a tiny amount of collective polarizationp builds up in the system; ( 6 u ) >> ( p p * ) ,and a shrinking Bloch vector starts to move down to the south pole while making small stochastic excursions. From an analysis of the autocorrelation function j ( t ) of the emission one finds that the correlation time of the emission is roughly TE". The noise source e J T ) remains important at all times. The emission behavior is not deterministic us in the case of pure SF. The number of photons emitted in a correlation time TEffis estimated to be of the order of ( 1 / 2 T ) exp ( 2aoL). So once a0L approaches 1 , the output becomes spontaneous emission in nature. The pure ASE process is limited by decreasing inversion. As soon as the tip of the Bloch vector p approaches the center of the Bloch sphere, there is no more gain in the system and the emission becomes spontaneous (absorbed) emission in nuture. The emission always starts as supertluorescence and eventually develR ops into ASE. The question is: Does this change around t = T z 2 / ~occur before or after the inversion in the system decreases appreciably? To put it differently: When is the emission SF and when is it dominated by ASE? To answer this question we note that on the average thepure SF emission reaches its maximum intensity at the delay time ( 7 " ) . The emission is thus

200

M . F. H . Schuurmans et al.

“damped” SF for T ~ ‘ / T R >> ( r I , )or Tg >> ( T (~f n ) ) l 1 ‘ (see also MacGillivray and Feld, 1976) and the emission is dominated by ASE for TB << ( T R ( T D ) ) ” ~ ; the emission still starts as SF. In view of Eq. (65) for ( T )~the emission is SF for 2aoL >> 4 In 2nN and is dominated by ASE for 1 << 2aoL << 4 In 2 r N . Note that the latter condition can in general only be satisfied for weak atomic transitions and a very large number N of active atoms. Similar results have been obtained by the group of Matsuoka using Maxwell-Bloch equations with a polarization noise source chosen ad hoc (Ikeda el ul., 1980). Finally we mention that the restriction to the linear theory is more severe in the ASE limit than in the SF limit. Straightforward calculation using Eqs. ( 1 17) and ( 1 18) shows that by the time the atoms at the end face have lost one-fourth of their energy, The atomic system as a whole has lost a fraction of its energy equal to& = ~ ) ( T R / ( T D ) ) I ’ *in the SF limit and fASE= Q ( T R / T ; ) in the ASE limit. Since in the SF limit TZ >> ( T R ( T ~ ) ) ) ~and ~ ’ thus fsF >>fASE, the far greater part of the ASE occurs in the nonlinear regime, where the present theory is not valid. Extension to the nonlinear regime is difficult because the behavior of the system is not really deterministic in the nonlinear regime. It is therefore not obvious that the stochastic variables description (103)-(107) can even formally be used there. The nonBose character of the polarization in the nonlinear regime may deserve a more complicated description. BROADENING D. THEEFFECTOF INHOMOGENEOUS ON T H E DELAY TIME We consider the effect of TZ being finite on the average delay time and the spread in delay times for Tb >> ( T R ( T I ) ) ) ~ ’ ~ , i.e., in the damped SF regime. Since the energy is then still mainly radiated via the collective polarization one has ( 6 u ) = ( p p * ) with ( p p * ) given by Eq. (115). The defined so as to match the pure SF average delay average delay time (in), time ( T ~ at ) TZ = 30, is given by

Note Z . also that Eq. (122) is neglecting higher order terms in ( T R ( T ~ ) ) ) “ ~ / T only valid for TZ >> ( T R ( rD) ) I r 2 or aOLTZ >> ( T )~. Estimates for (7”) valid ) be obtained from Eq. ( 1 1 3 ) . for aoLTZ 2 ( T ~ can The spread in delay times can in principle be calculated as in Section 111,H,5, Eqs. (85) and (86). Equation (85) is replaced here by p

=

po e x d 7 V T - T / ( ~ , L ) ]

(123)

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20 1

and Ipol' is distributed according to Eq. (86). The dimensionless delay time fll= thus satisfies 4

fl,

-

2 f 1 , / a o L= -In 4/p012

(124)

The calculation of the integrals over fI(lp,,12)of the explicit expression for PI) are quite involved now. Therefore, we will calculate Atl) by a less accurate method but one which at least indicates the order of magnitude of the effect. This is done by considering the relative increment 6fl) of TI) when Ipo12decreases an amount 61pol' = Ipol' (Ipol' is exponentially distrib) in the resulting equauted with ( Ipol') = 7 ( Ip0l2))".Replacing fllby (fL) tion one finds for Tz >> (( T ~))T,<)"' A?,)

=

AT,){^

+ $(T~(T,)))"'/T~}

(125)

where AT^) is the relative standard deviation for the case of pure SF. If we compare Eqs. (122) and (125), then we find that the Tz-induced relative change in the average delay time is twice the relative change in the spread in the delay times. and A?,) has also been studied analytically The effect of TZ on and numerically by Haake rt (11. (1980) for N = 1, 5 x 10' active atoms and for several distributions W A ) . They find that the effect of TT on and A?l) is much smaller for a Gaussian distribution, i.e., pure inhomogeneous broadening, than for a Lorentzian distribution as used above in obtaining Eqs. (122) and (125); the reason for this difference seems to be the relative importance of high frequencies in the case of a 1981). Lorentzian as compared to a Gaussian (see also Haake rt d., BROADENING E. HOMOGENEOUS

In a linear theory as valid here in the initiation time regime the effect of homogeneous broadening should be the same as the effect of inhomogeneous Lorentzian broadening. Thus, all the results previously derived for the latter case can be taken over directly to the case of homogeneous broadening. Explicitly this can be shown phenomenologically by introducing a damping term - y P into Eq. (23) for pure SF and by adding to that equation an external Langevin operator force A in order to preserve the commutation relations (44) and (45). We thus obtain instead of Eq. (23) dP = E - y P + A dT

where y

= T&$.

202

M . F . H . Schuurmans et al.

It turns out ( I ) that A and A% must commute with P o and E o , (2) that = 0 , and (3) that the commutator

ATlJI)

[At(X, T I , A ( X ’ , T ’ ) ] = ( 2 y / N )6 ( T - T ’ ) 6 ( X

-

X’)

(127)

Property (2) ensures that A does not contribute to antinormally ordered correlation functions. If we do the analysis in terms of the latter functions, we may as well remove A from Eq. (126). The resulting equations of motion are then simply solved with the above anticipated result. For details, see Schuurmans and Polder (1979b).

V. Three-Dimensional and Multimode Effects The theoretical description of SF in this article has so far been a onedimensional one. We have assumed that the emission goes into two plane-wave, SVEA, end-fire modes along the pencil axis. The onedimensional, one-mode theory was based upon (1) the presence of the largest gain for field waves in the solid angles S / L 2 around the pencil axis; (2) the description of these waves in terms of one forward- and one backward-traveling diffraction-limited mode in view of S / L 2 -- h % / S ,i.e., F = 1; and (3) the replacement of these modes by two SVEA plane-wave, end-fire modes, discarding diffraction altogether. Correspondingly, the initiating zero-point fluctuation waves were taken to consist of diffraction-limited modes coherent over the cross section of the pencil. For Fresnel numbers larger than one, the situation is completely different. The initiation then occurs by zero-point fluctuation waves traveling through the medium in all directions. The zero-point fluctuation waves can no longer be considered as coherent over the cross section. The question then is: What mode pattern is formed in a single-shot emission pulse from the gain in the active medium of the pencil? This question is by no means a trivial one, not even in the linear initiation time regime. The emission will have a multimode character, which may vary from shot to shot. One approach to solving the problem may be to expand the field and the polarization in terms of the Gaussian modes commonly used to describe multimode situations (cf. Sargent er al., 1974). So far this has not been done, and it probably requires numerical analysis because of the large number of modes needed. Less involved is the problem of describing the behavior of the emission in terms of the solution of the propagation of an incident small-area, uniform-plane-wave coherent pulse through an inverted medium of finite

SUPERFLUORESCENCE

203

lateral dimensions. The pulse area is the effective initial tipping angle B o . Such a three-dimensional, one-mode study would be particularly interesting in view of the problem of the ringing: one-dimensional numerical simulations (Section 111) exhibit ringing to the extent that each ring can be 50% as intense as the preceding ring, but ringing that pronounced is neither observed in the cesium experiments nor in the H F experiments. Mattar et ul. (1981) have started a study of the effect. They allow one extra transverse degree of freedom in simulations of the SF output. One must then add to the right-hand side of the SVEA Maxwell equation (64)

where i. = r / r p and r, is the radius of the initial inversion density at half maximum. E is, of course, complex so that phase variations introduced by diffraction can be included consistently. The results are illustrated in Figs. 2 and 3. Ringing from a uniformly inverted cylinder (Fig. 2a) is largely removed by diffraction if F is less than 0.4. Now F = 03 describes the one-dimensional, one-mode situation. Radial averaging by a Gaussian inversion (Fig. 2b) of very large F (i.e., insignificant diffraction) eliminates most of the ringing, resulting in an asymmetric pulse with a long tail. Reducing F of the Gaussian profile reduces the asymmetry since the outer portions of the cylinder are stimulated to emit earlier by diffraction from the inner portion. Figure 3 illustrates the appearance of appreciable ringing if one observes the output with a detector much smaller than the output diameter. The extended cylinder of unity Fresnel number does not emit its energy in one cooperative burst after all. Simulations reveal that the ringing becomes washed out into smaller and smaller radii for decreasing Fresnel number. The discussion of three-dimensional effects is only now beginning. The abo