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)

SUPERFLUORESCENCE

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 above preliminary exploration already shows large effects on the average SF emission shape. Clearly, the fluctuations in the emission observed with a large area detector depend strongly on the number of emission modes, which is roughly proportional to F 2 . In particular, one expects AT,, to decrease strongly with increasing number of modes. A comparison between experimental results and the predictions of the above onedimensioncrl theory as regards fluctuation properties can thus have hardly any meaning unless it is confirmed experimentally that the emission goes mainly into one diffraction-limited mode. The present one-dimensional theory has only limited capabilities of describing experimental results yucrntitath~ely.

M. F. H.Schuurmans et al.

204

Lz

Y B

c

3

a

t 3

0

n W

Y _J

a E

Lz 0

z 1

0

I

I

I

50

I

100

I

I

I50

T/rR

(a)

FIG.2. Normalized SFoutput power vs. T h n , O l = 2 X lo-'. 'Il = TI = 7; = X J ; L/crn = 3.9. (a)F = x . For the uniform profiles the excited state density no(r) = n: and Oo(r) = 0:; for Gaussian profiles n d r ) = ns exp[-In 2(r/rp)2]and Odr) = 0: expi0.5 In 2 ( r / r p ) 7 .[(i) no, Oo uniform; (ii) no Gaussian, Oo uniform; (iii) no, Oo Gaussian; (iv) no uniform, Oo Gaussian.] (b) Same as (a) but with diffraction (F = IrrE/XoL)included and uniform n o @ )and Oo(r):F = XJ (-), F = 1.0 (---), F = 0.4 (---), F = 0.06 (--).

SUPERFLUORESCENCE

205

b) F I G . 3. Influence of diffraction on SF pulse shapes. Parameters are the same as in Fig. 2a, with no Gaussian and B0 uniform. (a) Emitted power: F = x , 11 (-); F = 1.0 (---); F = 0.4 (----): F = 0.1 (--). (b) Isometric graph of intensity vs. r/r,, and r / r Rfor F = 1 case of Fig. 3a.

M. F. H . Schuurmans et al.

206

VI. Experimental Techniques A. INTRODUCTION

The first observation of superfluorescence was reported by Skribanowitz et al. in 1973, nearly 20 years after the first discussion of superradiance by Dicke (1954). It should not be concluded that SF is an exotic phenomenon, only rarely to be observed. On the contrary, the many reports of superfluorescence in atomic systems (Gross et al., 1976; Flusberg et al., 1976; Crubellieret al., 1978; Okadaet al., 1978a, b; Marek, 1979; Brechignac and Cahuzac, 1979; Cahuzac et al., 19791, published since 1976, have made it abundantly clear that SF can easily occur, even inadvertently. Therefore, some familiarity with the basic features of SF seems important for all those who use strong selective excitation of atomic or molecular systems in their experiments. A review of SF experiments through 1979 has recently been prepared by two of the present authors (Vrehen and Gibbs, 1981). Such a survey will not be given here. Instead attention will be focused on the experiments with atomic cesium performed at the Philips Research Laboratories from 1976 until 1980. A description of those experiments may illustrate to what extent the various theoretical concepts discussed in the first part of this article have been brought to the test, and what the accomplishments and shortcomings of both theory and experiment are. It is hoped that by this approach the article may help to define areas of future research interest. For a meaningful comparison with theory, SF experiments must meet a series of rather stringent conditions. Those conditions are the subject of Section VI,B. Various details of the cesium system are described in Section VI,C. Some of the most nearly ideal cesium experiments have been done in an atomic beam in order to reduce inhomogeneous (Doppler) broadening to a negligible amount. However, several other experiments have been made in cells, in which the dephasing time T 8 equals about 5 nsec, shorter than the SF delay time, which is usually chosen to be about 10 nsec. An experiment that explores the effect of inhomogeneous broadening on SF peak intensity is presented in Section VI,D. Superfluorescence theory so far has usually been based on the approximation of a uniform plane wave propagating through the sample. Several experiments indicate that transverse variations of polarization and electromagnetic field may play a very significant role. As discussed in Section V, efforts are now being made to include such effects in the theory. In Section VI,E, the excitation mechanism and its consequences for the

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207

transverse inversion profile are discussed. Section VII deals with the major experimental results. The most easily accessible experimental quantities are the pulse shapes, peak intensities, and delay times, and these are considered in Section VI1,A. Spatial and temporal coherence of the SF emission are the subject of the following Section VI1,B. Finally in Section VII,C, two experiments are described that were performed to study the average strength of the quantum fluctuations as well as the fluctuations in the delay times. FOR SF EXPERIMENTS B. CONDITIONS

In their articles on the theory of SF Bonifacio and Lugiato (1975a, b) formulated some experimental conditions for the observation of “pure superfluorescence.” (The distinction between “pure” and “oscillatory” superfluorescence occurs in the mean-field treatment of SF, but it plays no role in the Maxwell-Bloch theories.) Those conditions may be somewhat too restrictive, but they give a useful guidance, nevertheless. (1) The two levels involved in the SF radiation should be nondegenerate. Degeneracy of the levels does not suppress the SF, but beats may be formed which highly complicate the emission process, and its analysis (Vrehen rt al., 1977). (2) The excited sample should have a Fresne; number F = S/ AL = 1, where S is the cross-sectional area of the sample, L its length, and A the SF wavelength. Behind this condition is the assumption that with F = 1 only the axial mode can grow without appreciable loss, while all off-axis modes are damped sufficiently for them to be ignored (see also the theory sections). In Section VI,E, it will be argued that it is not always easy to determine what the F value of an excited volume is. (3) Various relaxation times should be longer than the delay time. In SF a coherent polarization grows a s e x p [ 2 ( t / ~ , ) ~ ’ ~where ], T~ is the characteristic radiation time. On the other hand, the coherent polarization decays through energy relaxation ( T I ) ,through transverse relaxation (Ti) and through inhomogeneous dephasing ( T z ) . Bonifacio and Lugiato require all these relaxation times to be longer than the delay time 7 ”. MacGillivray and Feld (1976) have argued that T2 is effectively lengthened by ) sufficient; see the amplitude gain aoL in the sample, so that a0L7; > T ~ is also Section IV. Since a& can be large, the condition on TF is considerably relaxed. In Section VI,D, an experiment is discussed which gives some support to the analysis of MacGillivray and Feld. However, for a detailed study of the pulse shape after the first intensity maximum, it is ) fulfilled. certainly desirable that T $ > T ~ be

208

M . F. H . Schuurmans et al.

(4) The escape time T~ = L / c , where c is the velocity of light, should be shorter than the characteristic time T ~ This . condition, which is equivalent to the requirement that the sample length shall be small compared to the cooperation length (Arecchi and Courtens, 1970), guarantees that electromagnetic disturbances can propagate over the full length of the sample before appreciable amplification can occur. If all atoms are excited simultaneously (transverse excitation), this condition is indeed necessary. In practice the pumping is realized with a short optical pulse propagating along the axis of the sample (swept excitation). Such excitation is usually also assumed in the theory (see Sections 111and IV). The forward S F pulse (i.e., the one propagating in the same direction as the pumping pulse) is then described in retarded time, and in a first approximation it is not affected by the length of the sample. The backward pulse, however, will be affected, and, through its nonlinear coupling with the backward pulse, the forward pulse will eventually be affected as well. ( 5 ) Finally, it must be required that the excitation shall occur in a time T ~ much , shorter than the delay time T ~ It. would be even safer to require T~ < T H , but simulations suggest that such a stringent condition is unnecessary. If T~ >> T E , as has always been the case in the cesium experiment, swept excitation is equivalent to transverse excitation.

C. THECESIUM EXPERIMENT Prior to the MIT experiments (Skribanowitz er al., 1973) one of the major obstacles to the observation of SF had been the difficulty of preparing a completely inverted system. Any macroscopic dipole moment produced at the S F transition by the pumping process might easily dominate over the quantum fluctuations in the initiation of the cooperative radiation. Skribanowitz et al. (1973) introduced the three-level technique that has been used in all subsequent experiments. The technique may be illustrated with the excitation used in the cesium experiments (Gibbs er al., 1977a; Vrehen et al., 1977), as shown in Fig. 4. A short laser pulse raises a or the number of atoms from the 6 2S1,2ground state to either the 7 TlI2 7 *P3/2excited state. This leads to complete inversion on the system 7 'P3/2 (or 7TlI2)to 72S1/2,and thus superfluorescence may develop on that transition. In order that the 7P-7s system can be considered as an isolated two-level system, it is necessary that the coupling of either of these levels to all other levels is sufficiently weak. More precisely, the characteristic time for cooperative radiation on the 7P-7s transition must be (much) shorter than those on the 7P-6S, the 7P-6D, and the 7s-6P transitions,

209

SUPERFLUORESCENCE PULSE FROM DYE AMPLIFIER

7s

f,# l 1.36\ 1.L7

1-

'k.36prn -5D

-6P

cs BEAM OR CELL

4 FILTER

InAs DETECTOR AND AMPLIFIER

cs

FIG.4. Simplified energy-level diagram of atomic cesium and sketch of the apparatus used to prepare a sample and to observe the SF emission (Gibbs ct d.,1977a).

when all the times are calculated with the same initial excited state density. The condition is well satisfied (Gibbs, 1977). The excitation proceeds with a short nitrogen-laser-pumped dye laser pulse at 455 nm. The pulse duration is 2 nsec, the bandwidth between 500 and 1200 MHz. The long focus of the beam defines the thin pencil of excited atoms in either an atomic beam or a cell. The pump power density is of the order of 10 kW/cm'. In order t o prepare a truly nondegenerate two-level system, a transverse magnetic field of about 0.28 T is applied transversely to the atomic sample. It is then possible to excite only a single Zeeman level of the initial state, and if suitably chosen that level will only decay to one single Zeeman level of the final state. In most experiments excitation has been from 6'Ss/2 ( mJ = - 1/2, m, = -5,/2) to 72P312(mJ = -3/2, m, = - 5 / 2 ) with subsequent superfluorescence from that level t o 72S1,2(rnJ = - I / 2 , m, = - 5 / 2 ) . In some experiments the nuclear quantum number was chosen to be m, = -3/2 by applying the slightly different magnetic field of 0.33 T. In both cases the transitions had comparable strength, and no significant differences were found for the two sets of levels. The S F pulses, which have a wavelength A = 2.931 pm, are focused onto an InAs photovoltaic detector, which has a diameter of the sensitive area of 150 p m and a quantum efficiency of 0.2. For a sample length of 20 mm, a Fresnel number 1 and a delay time of 10 nsec, the number of excited atoms amounts to about loH.Roughly 3 x lo7photons can then be expected in the first peak of the SF pulse emitted in either direction. With a FWHM pulse width of 5 nsec, the peak detector current is estimated to

2 10

M . F. H . Schuurmans et a/.

be 190 PA, in reasonable agreement with experiment. These large signals allow single-shot observations. The detector current is amplified in a 50 R wideband amplifier and recorded with a Tektronix Transient Digitizer. The pulses are stored on a hard disk. In the cesium experiment the inhomogeneous dephasing time was increased to T$ = 32 nsec by the application of an atomic beam. From atomic data (Gibbs, 1977) it follows that T , = 70 nsec and T i = 80 nsec. By a variation of the density the delay time was varied from 5 to 40 nsec. It can be checked that the Bonifacio-Lugiato conditions are indeed fulfilled. A pure nondegenerate two-level system is completely inverted initially. With a sample length L = 20 mm the escape time equals T E = 0.067 nsec. For T~ = 0.3 nsec, a delay time of about 10 nsec is obtained, clearly shorter than all relaxation times mentioned above. With T~ = 2 nsec, the pump pulse is sufficiently shorter than T ~ . D. THEEFFECTS OF

INHOMOGENEOUS

BROADENING

An atomic beam has been used in a number of crucial experiments to study the detailed pulse shapes, in particular, for times longer than the delay time T,, , which is experimentally defined as the time at which the SF pulses reaches its first maximum. In other experiments, such as those discussed in Sections VI1,B and VII,C, cells were used for greater convenience. In the cesium vapor of about 70°C the inhomogeneous dephasing time at the SF transition amounts to T5 = 5 nsec if the full Doppler profile is inverted. This rather short dephasing time limits the maximum observable delay time to about 20 nsec, as compared with >40 nsec in an atomic beam. For delay times much larger than T $ the peak intensity of the SF pulse is seriously affected by the dephasing (Vrehen et al., 1978). This can be seen in Fig. 5 , where the peak intensity I,, , multiplied by T ; , is plotted as a function of T ~ < If. relaxation played no role at all, one would expect Z,,,T~ to be constant, since I, is proportional to the density squared and T~~ is inversely proportional to the density. The plot shows I , to decrease faster than I / T ; for T,< > 0.3 nsec in a cell. In contrast, Z,T; is nearly constant up to T~ = 1.3 nsec in an atomic beam. MacGillivray and Feld (1976) (see also Section IV) have argued that the condition T,, < TZ for the observation of S F is too restrictive, and that actually T ~ <)< aoLT$ is sufficient, where a0L is the small-signal amplitude gain of the totally inverted sample. Since aoL = T$/ rRand T " / T , ~= 30 for the cesium experiment, the condition becomes ff)<< 30TB' or T,) < 5.5T$.In the cell experiment with T$ = 5 nsec, the peak amplitudeI, begins to fall off steeply for T~~ > 0.5 nsec, or T ~ > ) 15 nsec = 3T$, in reasonable agreement with the

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211

FIG. 5 . The effect of inhomogeneous broadening on the peak intensity of the SF pulse.

In a cell where T$ = 5 nsec, the signal falls off dramatically for rll > 0.5 nsec. No such signal reduction is observed in an atomic beam where Tg = I 8 nsec. The sample length is 30 mm for the cell and 36 rnm for the beam and the Fresnel number equals I in both cases.

qualitative argument given above. It is concluded that, as far as the first pulse is concerned, the results are not too much affected by inhomogeneous broadening as long as TI) s 2 T z . See also the discussion of delay times in Section VI1,A. PROFILE E. THEINVERSION

One of the conditions discussed in Section VI,E is that the pumped volume shall have a Fresnel number 1. Although the available ab initio theories do not take into account the sample diameter explicitly, the implicit assumption is that there exists initially a uniform excited state density within a radius po defined by n p f = A L , while beyond that radius no excitation is present. In this section to what extent that assumption is realized in practice will be discussed. The pump laser beam has an approximately Gaussian transverse intensity profile. Furthermore, it propagates through the sample as a very nearly plane-parallel wave with negligible attenuation in the propagation direction. As to the resulting excitation, several different cases can be distinguished, depending on the intensity of the pump, its temporal coherence, and the relaxation times T ; , T%for the atomic dipole moment on the pump transition in comparison with the pulse length.

212

M . F. H . Schuurmans et al.

(1) The beam intensity very small: The excitation is now proportional to the intensity everywhere. The inversion has a Gaussian profile with the same FWHM as the pump beam. (2) Beam intensity very high, pulse duration long compared to atomic relaxation times: The atomic transition is saturated up to some distance away from the axis. The radial inversion profile is very flat in the center, whereas it falls off exponentially at large radii. The FWHM of the inversion will be larger than that of the pump beam. For example, with the saturation intensity denoted by I,,, , for Z/Zsa, = 10 on the axis, the inversion FWHM is about 1.9 times the pump FWHM. If the latter has been chosen to define a nominal Fresnel number F = 1, the actual Fresnel number is about 3.5 times larger. (3) Beam intensity high, pulse fully coherent, no atomic relaxation: The excitation consists of a rotation of the atomic Bloch vectors (on the pump transition) through an angle 19,which depends on the radial ~ coordinate p , 8 = 8 ( p ) . On the axis one has 2 k r < 8(0) < 2(k + 1 ) for some value of k 2 1 since high intensity has been assumed. With increasing p the inversion, which equals 4(1 - cos O ( p ) ) , will oscillate between the values 1 and 0, before it eventually decays to 0. (4) Beam intensity high, pulse incoherent, no atomic relaxation: For a given radius p the Bloch vector (for the pump transition) can be characterized by 8 ( p ) and 4 ( p ) . The quantities 6 and 4 vary rapidly and randomly with p . In the central region the average inversion corresponds to saturation of the pump transition. In the actual experiment, conditions are somewhere in between Cases (2)-(4) mentioned above. The pulse duration T ~ = , 1.5-2.0 nsec is always much shorter than the homogeneous relaxation time T ; : it is comparable with or shorter than the inhomogeneous relaxation time T ; at the pump transition (T; = 5 nsec in an atomic beam, T ; = 1 nsec in a cell). The pulse is neither coherent nor fully incoherent; in various experiments the bandwidth has varied between 500 MHz and 1200 MHz. The pulse intensity is high; if the pulse were fully coherent, the on-axis intensity would yield a pulse area of about 5n. From shot to shot the pulse delay time fluctuates far more than can be understood from the quantum fluctuations; it may be concluded that the axial inversion varies from shot to shot. It is most probable that the average delay time (i.e., averaged over many shots) corresponds to full saturation of the central region of the sample. The diameter of the inverted sample is not defined very sharply. It may be larger than the FWHM diameter of the pump beam, and therefore the actual Fresnel numbers F may be larger than the nominal ones quoted in this article by possibly a factor of four.

213

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ATOMIC BEAM

L = 2 0 cm n 1.9 x 10” ~ r n - ~

76

x

10’0cm-3

1 H

-\

c

H

0

10 20 t (nsecl-

30

FIG. 6. Normalized single-shot pulse shapes for several cesium densities n : Fresnel number F = I . Uncertainties in the values of n are estimated to be (+60, -40)% (Gibbs et r d . . 1977a).

VII. Experimental Results A. PULSEPARAMETERS

The most characteristic properties of SF are the emission of a delayed pulse, with delay time T,, inversely proportional to the atomic density, and peak intensity proportional to the density squared, radiated within a narrow cone around the axial directions. Typical SF pulses are shown in Fig. 6 . The apex angle of emission has been measured by placing a variable diaphragm just in front of the condensing lens, which focuses the energy on the detector, and by measuring the signal intensity as a function of the

M . F. H . Sckrrirrmans et al.

2 14

loor---

I 0.1

1

1

-

10

T K ~n s e c - * )

F I G .7. Plot of the peak intensity I , of the SF pulse as a function of T H ~ demonstrating , that I,,, depends quadratically on the density. Each experimental point represents an average of many shots (Vrehen er d.,1978).

diameter of the aperture. Let d be the diameter of the pump beam at FWHM intensity, L the sample length, and 8 the full emission apex angle at FWHM intensity. For a series of Fresnel numbers, 8 has been found to be -2dlL. roughly a factor of two larger than expected. This rather large angle might be a consequence of an inversion diameter exceeding the pump diameter, as discussed in Section VI,E. Only for F = 15 was it possible to measure the diameter of the S F radiation source; it turned out to be somewhat smaller, not larger than the pump diameter. That result does not necessarily hold for the smaller Fresnel numbers. The quadratic dependence of maximum signal intensity on the atomic density is demonstrated in Fig. 7, where I, is plotted as a function of 1 / t i for an atomic beam of 36 mm length. For delay times far exceeding the inhomogeneous relaxation time T%the quadratic relationship no longer holds, as has been discussed in Section VI,D, and as is illustrated in Fig. 5 . According to Eq. (651, the average delay time ( T ~ is ) approximately proportional to the characteristic radiation time tR. More precisely, for lo7 < N < lo9, the range in which the cesium experiments have been done, Eq. (65) predicts ( tD) to be proportional to 7gs. In Fig. 8 the delay times measured in a cell of length L = 3 cm have been plotted versus tR. The temperature of the coldest spot in the cell, and thus the atomic density in the cell, was varied, while all other experimental parameters were

SUPERFLUORESCENCE

215

FIG. 8. Average delay time T,) as a function of the characteristic time rRin a 3 cm cell; 7" has been varied by changing the coldest spot temperature of the cell, and therefore the vapor density. The relative densities are rather accurate and thus demonstrate the linear relation between r,) and rH. The absolute densities are estimated to have an error margin of approximately 30%.

held constant. The radiation time T R was calculated on the assumption of complete saturation of the pump transition, with the help of vapor pressure data. It is believed that in this experiment the relative densities are rather accurate, but the absolute densities may contain a sizable error, resulting from uncertainties of the vapor pressure data and the absolute temperature measurements. The slope of the straight line in the logarithmic plot of Fig. 8 amounts to 0.85 and is rather well established. The position of the straight line, however, is somewhat uncertain. The ) 10 best current estimate is ( T , , ) / T , < = (32 k 10) for F = 1 and for ( T ~ = nsec. It is worth noting that the ratio ( T , ) ) / T ~has been found to depend rather strongly on the Fresnel number. Measurements show ( T~ ) / T R to be 1.5-2.0 times smaller for F = 15 than for F = 1 (Vrehen et af., 1978). In the comparison of theoretical and experimental delay times as given above, the possible influence of inhomogeneous broadening has not been considered. In the range of interest, 0.1 nsec < T R < 1 nsec, Eqs. (65) and (122) predict (ill ) to be nearly proportional to TA." for TZ = 5 nsec, as is the case in a cell. The exponent of this power law is clearly larger than observed. The experiments do not yield evidence for an effect of TZ on the

2 16

M . F. H. Schuurmans et al.

observed delay times. It should be realized that Eq. (122) has been derived in the limit of vanishing values of (TR ( T~ ))'"/TZ . Moreover, the definition of the delay time in the linear theory differs from that in the experiments. See also the discussion following Eq. (125). Single pulses have always been observed for T~ >> T E = L / c (Gibbset al., 1977a). These single pulses are often nearly symmetric, but a somewhat asymmetric shape, with the rising slope steeper than the falling one, is ~ , pulses more typical. For shorter delay times, i.e., for T~ S 3 ~ multiple have been observed. Detailed experiments (Vrehen et al., 1978) have shown that the emission of multiple pulses is not related to the so-called Burnham-Chiao (1969) ringing but rather to radiation in several independent transverse modes. So far, this phenomenon has not been explained in terms of the fully quantized Maxwell-Bloch theory (see Section V). Bonifacio et a/. (1979) have interpreted the multimode emission on the basis of the mean-field theory.

B. SPATIAL A N D TEMPORAL COHERENCE In the one-dimensional Maxwell-Bloch description the electromagnetic field in the sample is assumed to be a uniform plane wave. In all points of the sample end face the field has the same amplitude and phase. The amplitude grows from very small values to a maximum many orders of magnitude larger, and then decays again. The phase fluctuates strongly in the very early stages of the evolution, but rather quickly it approaches a stationary value, and it stays close to that value during the remainder of the pulse. Of course, the stationary value is completely arbitrary, and it fluctuates from shot to shot in the interval from 0 to 2x. Thus, the SF pulse is described as a fully coherent pulse of undetermined phase. The question is to what extent the actual SF emission conforms to the very simple picture depicted above. It may be suspected from the outset that the SF radiation process is more complex and shows only partial coherence. Unfortunately, the theory of optical coherence has been considered almost exclusively for the case of stationary ergodic fields (Born and Wolf, 1973, when ensemble averages may be replaced by time averages. Concepts developed for stationary fields, e.g., the degree of coherence, cannot be applied to characterize the coherence of SF pulses without some major modifications. Such a reformulation of the theory of optical coherence for nonstationary fields will not be discussed here. The following analysis may serve as an introduction to the experiment described below. Let us consider a plane normal to the axis, just outside the sample near its end face. For classical radiation the emission can be de-

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217

/

FIG. 9. Apparatus used t o study the coherence properties of SF radiation (Vrehen, 1979).

scribed by the distribution of amplitude and phase in that plane. Now, if in each point of the plane the phase is constant during a given pulse, it may be said that the SF is temporally coherent. If, moreover, the relative amplitudes and phases of different points of the plane are the same from shot to shot, it will be said that S F is spatially coherent as well. The coherence properties have been investigated experimentally by bringing two S F beams to interference (Vrehen, 1979). The experimental apparatus is shown schematically in Fig. 9. Two identical 5 cm cesium cells are simultaneously pumped, and their infrared outputs are brought to interference at the detector. The transverse magnetic fields are adjusted so that in one cell the atoms are excited to 7P3/2 ( m J= - 3 / 2 , ml= -512) and in the other cell to 7P312(mJ = - 3 / 2 . m, = - 3 / 2 ) . Depending on the precise setting of the magnetic fields, the two SF transitions have frequencies differing by 300-600 MHz. The interference of the two beams thus leads to intensity beats at the detector. Optimum interference requires the complete overlap of the two beams. That condition can be realized by means of the rotatable mirrors and beam splitters and with the help of the rotatable plate P. A typical beat signal is presented in Fig. 10, together with its Fourier transform. Beats at about 450 MHz are clearly visible, indicating a certain degree of temporal coherence. Quantitative information can be gathered most easily from the Fourier spectrum after it has been corrected for the finite response time of the detector. The Fourier spectrum shows two bands, one around frequency zero, and one around the beat frequency. Analysis reveals that the width of the v = 0 band is directly related to the shape of the pulse that would result from an addition of the two beam

218

M . F. H . Schuurmans et al.

0

10

20

30 LO 50 Time (nsecl +

la)

'0

u OL

0 0.8 1.2 1.6 Frequency (GHzl +

Ibl FIG. 10. Interference signal showing beats at the difference frequency of the two SF transitions (a) and the Fourier transform of the signal (b) (Vrehen, 1979).

intensities, or, stated differently, the pulse that is obtained by smoothing out the beats. On the other hand, the width of the band at the beat frequency is related not only to the overall pulse shape, but also to the relative phase variations between the two beams. In particular, random phase variations, occurring independently in the two beams, will broaden the sideband in comparison to the central band. Hardly any such broadening was found in the experiments, indicating that random phase variations play only a minor role. The occasional occurrence of large phase fluctuations as considered by Hopf (1979) cannot be excluded, however. The Fourier spectrum of Fig. 10 represents the Fourier amplitude as a function of frequency. If, instead, the square of the amplitude is plotted versus frequency, a measure can be obtained (Vrehen and Andersen, unpublished) of the spatial coherence of the S F emission as defined earlier in this section. That measure consists in the ratio of the area under the sideband of the (squared) Fourier spectrum and that under the central band. For complete spatial coherence the ratio equals 0.25. In the experiments the ratio is

219

SUPERFLUORESCENCE infrared blue

attenuator blue h V

attenuator inf r a r e d II

II

cell 1 cesium

n U

I n 1

II

cell 2 cesium

r\n

vu

+ -

filter

FIG.11. Apparatus for the measurement of the effective initial tipping angle Bo (Vrehen and Schuurmans, 1979).

found to be up to 10 times smaller, which means that emission does not occur in one single mode but rather in as many as 10 modes, even though the sample is nominally prepared with a Fresnel number 1. Since the number of modes varies as F y , it would imply that actually F = 3, or a sample diameter about 1.7 times as large as the nominal value, consistent with the large apex angle of emission, as discussed in Section VI1,A.

c. INITIATION A N D FLUCTUATIONS As explained in the theoretical section of this article, SF can be envisaged as being initiated by fluctuating waves of the electromagnetic field, which rotate the collective Bloch vector away from its vertical position. The strength of the fluctuating fields determines how rapidly the radiation is being started off, and consequently the average delay time depends on that strength. In numerical simulations the fluctuating fields are often replaced by a short coherent pulse at the SF wavelength, which traverses the sample at t = 0 and rotates the Bloch vector over an angle Oo. When O0 is chosen such that the resulting delay time equals the average delay time caused by the fluctuating field, it is called the effective initial tipping angle (see also Section 111,G). The value to be given to Oo had been under debate, until a direct measurement of the quantity (Vrehen and Schuurmans, 1979) resolved much of the controversy. The experimental apparatus is shown in Fig. 11. Two cesium cells are pumped successively by the same pump pulse. In the first cell a rather high cesium pressure is maintained, and as a consequence an S F pulse is emitted with a very small , 1.5 nsec). This pulse is temporally coherent and has a delay time ( T ~ =

M . F. H.Schuurmans et a / .

220

15 -

T

----A

I

OO

I

LO

1

1

I

80

I

120

I

I

160

I

I

200

I1n(o/2nII2 FIG.12. Average delay time as a function of [ln(B/2~)]’.Bo is derived from the interception of the dashed lines. The dashed line with the positive slope is shifted with respect to the data points to correct for the time delay between the pump pulse and the injection pulse (Vrehen and Schuurmans, 1979).

measured area of about TT. This pulse, which will be called the injection pulse, enters the second cell immediately after the excitation of the atoms by the pump pulse. The area 8 of the injection pulse at the entrance of the second cell is much smaller than 7r because of the divergence of the beam and because of a variable infrared attenuator between the two cells. Now, if 8 < O0, the initiation of SF in the second cell will be dominated by the field fluctuations, and therefore the delay time will not be affected by the injection. However, if 8 > Bo, the initiation will be dominated by the injection and the delay time will be reduced. It follows that Bo can be determined by measuring (7,))as a function of 8; 8 itself is varied by varying the infrared attenuator. An experimental result is shown in Fig. 12. From curves like this one B0 has been determined as 1 .O x < O0 < 2.5 x for a sample with F = 1, L = 5 cm and a total number of atoms N = 2 x 1oH. From the quantum theory [see Eq. (67)l one expects O0 = ( ? / m ) [ ( l n 2 7 r N ) / 8 ] ’ ” , or Bo = 2.3 x for the present case. Recently Carlson et d. (1980) have reported an experiment in which the injection forces the energy to be radiated in a slightly off-axial direction, rather than causing a reduction of the delay time. Again B0 is determined by measuring the power level at which the injection begins to dominate for roughly similar the fluctuations. These authors find B0 = 1 x conditions. The measurements thus largely confirm the predictions of the

22 1

SUPERFLUORESCENCE

CELL 1

n

\\

\\

u

CELL 2

N

DELAY

FILTER

Transient digitizer

FIG. 13. Apparatus to measure quantum fluctuations in the delay time. Fluctuations resulting from variations in the pump pulse are compensated for by pumping two identical cells simultaneously (Vrehen, 1979).

quantum-mechanical theory, although the experimental value for Oo tends to be somewhat larger than the theoretical one. The measurement of Oo yields information on the average strengths of the quantum fluctuations. The initiating noise, however, also leads to fluctuation in the delay time, i.e., if the sample is excited repeatedly under completely reproducible conditions, the delay time will nevertheless vary from shot to shot. In principle these fluctuations can be measured very easily by determining the delay times in a large number of shots and by studying their statistics. In practice a complication arises from the fact that the excitation process is not sufficiently reproducible because of variation in the pump pulse. The problem can be overcome in the manner shown in Fig. 13. Two identical cells, with the same vapor pressures and the same transverse magnetic fields, are simultaneously pumped (Vrehen, 1979). For that purpose the pump beam is split into two equivalent beams with the help of a beam splitter. If, as we shall assume, the two samples prepared in the same pulse are equivalent in all respects, any difference in their respective delay times can be ascribed to the quantum fluctuations. The results of a preliminary experiment are shown in Fig. 14, which presents a distribution of the delay time differences. From that distribution the relative standard deviation in the delay time of a single cell, AT^) = ( ( T ? ) ) - ( T ~ ) ) ~ ) ” ~ / (can T ~ )be derived. The result of the preliminary experiment is AT,)= (13 3)%. A meaningful comparison of this result with theory is difficult at present. The theory assumes that a single transverse mode participates in the SF. Experimentally the number of participating modes is not well known (see Sections VI,E and VI1,B). Moreover, a more recent experiment (Vrehen and der Weduwe, 1981) reveals that the

*

222

M . F. H . Schrrrrrmcrns et

-5

cil.

0

5

r,-r2(nsecl

10

FIG.14. Distribution ofdelay time differences as measured with the apparatus in Fig. 13 (Vrehen, 1979).

delay time fluctuations depend strongly on the Fresnel number, i.e., on the number of modes. From a theoretical point of view (Section V) this may also be understandable. These provisions being made, it may be mentioned that Eq. (84) yields AT^) = 12% for the case of cesium. A numerical study of the fluctuations by Haake et al. (1979a, b) arrives at the same result.

VIII. Conclusions The experiments confirm certain qualitative properties of SF emission predicted by theory. In particular it has been shown that a temporally coherent pulse is emitted, with a peak intensity proportional to the density squared and a delay time inversely proportional to the density. In some respects semiquantitative agreement also exists. The ratio ( r D /) T ~ the , magnitude of the delay time fluctuations, and the average strength of the initiating quantum fluctuations either agree with the predicted values within experimental error, or do not differ from those values by more than a factor 2. However, several qualitative discrepancies remain. The lack of spatial coherence and the unexpectedly large emission angle prove that the radiation field does not conform to the uniform plane wave (UPW) assumed in the calculations. The pulse shape differs rather sharply from the one foreseen in the UPW approximation as well. Probably these discrepancies are testimony to the fact that the three-dimensionality of the

223

SUPERFLUORESCENCE

radiation process cannot be ignored. As discussed in Section V, two phenomena have to be considered. An initially plane wave may evolve into a focusing or defocusing wave during the radiation buildup. Moreover, the quantum fluctuations initiate waves in all directions. A selection occurs through the different gain for the different modes. Inclusion of these two aspects into the theory poses a formidable challenge. Performing experiments with very well-defined and measured geometries may be equally difficult. Improvement of the quantitative description of experimental results will probably require an attack on both problems.

Appendix I The inability of a "point" sample to superfluoresce is easily demonstrated. The dipole-dipole dephasing occurs at a rate Ipl'h-'R-3, where p is the dipole moment matrix element of the transition involved and R is the mean interatomic distance. For N atoms in a volume ( h o / 2 r I 3the rate is N I p 1 2 f i - ' ( h o / 2 r ) - 3The . natural linewidth is y,, - Ip.('h-'(Ao/2r)-"and so the rate l / ~ at, ~which the collective polarization builds up is NIp('h-' ( h 0 / 2 ~ ) - 3Clearly . the dipole-dipole dephasing is faster than the SF buildup rate for "point" samples smaller than a wavelength cubed!

Appendix I1 We present schematically ( 1 ) the derivation of the operator MaxwellBloch equations in the initiation time regime and ( 2 ) the commutation relations for the Schrodinger operators involved. For details see Polder et UI. (1979). The two-level atoms labeled by the subscriptj,j = 1, . . . , N , are described by the spin 1/2 angular momentum operators R, . Their components, the raising operator R,+,the lowering operator Ry and the inversion operator Rj3' satisfy [ R ] , R;]

=

2aj1Ri3),

[Ri3', R?]

=

t6jlRT

( 129)

The dipole operator p' = p(RJ + R y ) ; the matrix element p being along the z axis and, for simplicity, real valued. The radiation field is described by a mode decomposition in a quantization volume V,,. The electric field operator E = E(+'+ E'-), where E(-) = (E"')?, and the positive frequency part is

224

M . F. H . Schuirrmuns et al.

The modes are labeled by A = (k, a),where k is the wave vector and (T denotes the state of polarization. The polarization vector is E, . The Bose operators a, and a: satisfy [a,, a:.] = 6 A A , , and annihilate and create a photon in the state A , respectively. The equations of motion for the Heisenberg atomic and field operators are dRi3’/dt = (i/h)(RJf - R y ) p *E(rl, t ) (131) dRy/dt

= -iw&y

-

( 2 i / h ) R j 3 ’ pE(rl, * t)

(132)

and the microscopic Maxwell equation is

where a and p refer to the x, y , and z components and is the transverse 6 function (cf. Power and Thirunamachandran, 1980, and references therein). At t = 0, the Heisenberg operators are equal to Schrodinger operators satisfying Eqs. (129) and (130). The electric field as obtained from the microscopic Maxwell equation (133) at an “aufpunkr” r # rJconsists of two parts: a driven part, finding its origin in the electric dipole sources in the medium, and the vacuum field, i.e., the homogeneous solution of Eq. (133). The latter is taken to be zero in treatments starting from the classical Maxwell equation. However, the operator vacuum field does not vanish. The field E(rJ,t ) acting upon atom j as it appears in Eqs. (131) and (132) is different from the microscopic Maxwell field at an “aufpunkt” in that it contains a radiation reaction part, the dipole field due to the other atoms and the vacuum field. We simplify the equations of motion using the fact that the gain of waves is largest for field waves with k E S / L 2 and the requirement S / L * -A$/S. As elucidated in Section 11, a treatment in terms of two counterpropagating SVEA field modes must then be sufficient to describe the broad features of the SF emission. These modes only couple to the collective atomic operators where N8 is the mean number of atoms in a slice of thickness d oriented perpendicular to the pencil axis H , R = ( R + , R - , R‘3’)and ci), denotes the collection of atoms in the slice with center position x. The thickness d of the slice is chosen such that (1) N , >> 1 and (2) d << Ao/(27r); this is possible since pSA0/(27r)>> 1 in view of T~ >> T R . The electric field E(x, t ) , slice averaged by integrating E(r, t ) over r in the slice at x and

225

SUPERFLUORESCENCE

dividing by Sd, represents the end-fire modes. In the spirit of the planewave, end-fire mode assumption, we then require E(x, t ) and R,(r) to be independent of the particular, randomly assumed, positions of the atoms in a slice. Accordingly we find from Eq. (133) the macroscopic Maxwell equation (E//p now):

where R ( x , t ) is formally adopted as depending continuously on x since N , >> 1 . Diffraction has been disregarded altogether. Before rewriting Eqs. (131) and (132) in a slice averaged form we introduce the second important simplification: the restriction to the initiation time regime. The Schrodinger collective variable N,R represents a spin vector S with magnitude S = bN,. Initially all spins point upwards and the system is in an eigenstate of S ( 3 )with eigenvalue S = ANs. As long as the system develops through states in which the number of deviated spins is small compared to 2 s = N , , the Heisenberg operator Y3'can be replaced by 4NJ and d 3by I I , where I is the identity operator. The validity of this approximation defines the initiation time regime. The atomic system is thus described by Bose operators Yi+ and 3-.Deviations from this behavior can be studied using the Holstein-Primakoff transformation (Walker, 1963). Note that the Bose behavior rests entirely on the introduction of collective variables R (x, t ) for many atoms N , in a slice at .x . A single atom does not exhibit such behavior. Analysis of Eqs. (131) and (132) now shows that, consistent with 9"= AI, we must drop the radiation reaction field in the evaluation of the slice averaging. Since ~ ( h ~ / 2 7<
(136)

where E ( x . t ) is the Maxwell field obeying Eq. (135). The commutation relations for the slice averaged Schrodinger operators are now readily derived. One finds [ Y ~ + ( x ) ,%-(.x')]

=

( L / N ) 6(x

[ E ' + ' ( x )E"(x')] ,

=

( h c k , L / V ) 6(x - x ' )

-

x')

(137) (138)

The final approximations consist of invoking the SVEA approximation and the rotating wave approximation. The first is performed in a manner similar to that described by the Eqs. (1) and (2) (section II), i.e., %?(x, t ) = %;(.x.

t ) exp[T(ikG - iwot)]

( 139)

226

M . F. H . Schuurmans et al.

E ( x , t ) = E k W , t ) exp[t(ik,,x - ioot)l

(140)

we consider only right-traveling waves. The use of the RWA is justified since dipole-dipole interaction is disregarded anyway because of P(A~/~< T<) ~1. Introducing dimensionless variables according to E = -i(v.Ek+’/fi)~, and P = 9?i,one finds the equations of motion (23) and (24) and the Schrodinger commutation relations (25) and (26).

REFERENCES Abramowitz, M., and Stegun, 1. A. (1972). “Handbook of Mathematical Functions,” AMS 55. Natl. Bur. Stand., Washington, D.C. Agarwal, G. S. (1969). Phys. R e v . 178, 2025. Agarwal, G. S. (1970). Phys. Rer,. A2, 2038. Arecchi, F. T., and Courtens, E. (1970). Phvs. R e v . A2, 1730. Banfi, G., and Bonifacio, R. (1974). Phys. R e v . Lett. 33, 1259. Banfi, G., and Bonifacio, R. (1975). Phys. R e v . A 12, 2068. Bonifacio, R., and Lugiato, L. A. (1975a). Phys. Rev. A I I , 1507. Bonifacio, R., and Lugiato, L. A. (1975b). Phys. Rev. A 12, 587. Bonifacio, R., Schwendimann, P., and Haake, F. (1971). Phys. Rev. A 4, 302, 854. Bonifacio, R., Farnia, J. D., and Narducci, L. M. (1979). Opt. Commun. 31, 377. Born, M. A., and Wolf, D. (1975). “Principles of Optics.” Pergamon, Oxford. Brechignac, C., and Cahuzac, P. (1979). J . Phys. Lett. 40, L-123. Bullough, R. K., Saunders, R., and Feuillade, C. (1978). Coherence Quantum Opt. 4, 263. Burnham, D. C., and Chiao, R. Y. (1969). Phys. Rev. 188, 667. Cahuzac, P., Sontag, H., and Toschek, P. E. (1979). Opt. Commun. 31, 37. Carlson, N. W.,Jackson, D. J., Schawlow, A. L., Gross, M., and Haroche, S. (1980). Opr. Commun. 32, 350. Courant, R., and Hilbert, D. (1%2). “Methods of Mathematical Physics,” Vol. 11. Wiley (Interscience), New York. Crisp, M. D. (1970). Phys. Rev. A I , 1604. Crubellier, A., Liberman, S., and Pillet, P. (1978). Phvs. R e v . Lett. 41, 1237. Degiorgio, V. (1971). Opt. Commun. 2, 362. Dicke, R. H. (1954). Phys. Rev. 93, 99. Einstein, A. (1917). Phys. Z. 18, 121. Ernst, V., and Stehle, P. (1968). Phys. R e v . 176, 1456. Flusberg, A., Mossberg, T., and Hartmann, S. R. (1976). Phys. Lett. 58A, 373. Friedberg, R., Hartmann, S. R., and Manassah, J. T. (1972). Phys. Lett. 40A, 365. Friedberg, R., and Hartmann, S. R. (1974a). Opt. Commun. 10, 298. Friedberg, R., and Hartmann, S. R. (1974b). Phys. R e v . A 10, 1728. Gibbs, H. M. (1977). I n “Cooperative Effects in Matter and Radiation” (C. M. Bowden, D. W. Howgate, and H. R. Robl, eds.), p. 61. Plenum, New York. Gibbs, H. M., Vrehen, Q. H. F., and Hikspoors, H. M. J. (1977a). Phys. R e v . Lett. 39,547. Gibbs, H. M., Vrehen. Q . H. F., and Hikspoors, H. M. J. (1977b). Springer Ser. Opr. S r i . 7, 213. Glauber, R. J. (1970). In “Quantum Optics” (S. M. Kay and A. Maitland), p. 53. Academic Press, New York.

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Glauber, R., and Haake, F. (1978). Phys. Lett. A 68.4, 29. Gross, M., Fabre, C., Pillet, P., and Haroche, S. (1976). Phys. Rev. Lett. 36, 1035. Haake, F. (1979). I n “Laser Spectroscopy” (H. Walther and K.W. Rothe, eds.), Vol. 4, p. 451. Springer-Verlag, Berlin and New York. Haake, F., King, H., Schroder, G., Haus, J., Glauber, J., and Hopf, F. (1979a). Phys. Rev. Lett. 42, 1740. Haake, F., King, H., Schroder, G., Haus, J., and Glauber, R. (1979b). Phys. Rev. A 20, 2047. Haake, F., Haus, J., King, H., Schroder, G., and Glauber, R. (1980). Phys. Rev. Lett. 45, 558. Haake, F., Haus, J. W., King, H. and Schroder, G. (1981). Phys. Rev. A23, 1322. Hahn, E. L. (1950). Phys. Rev. 80, 580. Haroche, S. (1978). Coherence Quuntum O p t . 4, 539. Hopf, F. (1979). Phys. Rev. A 2Q, 2064. Hopf, F., Meystre, P. and McLaughlin, D. W. (1976). Phys. Rev. A13, 777. Ikeda, K., Okada, J., and Matsuoka, M. (1980). J. Phys. Soc. Jpn. 48, 1636, 1646. Kramers, H. A. (1938). “Die Grundlagen der Quantentheorie,” p. 492. Akad. Verlagsges., Leipzig. Landau, L. D.. and Lifshitz, E. M. (1960). “Electrodynamics of Continuous Media.” Pergamon, Oxford. Lax, M. (1968). I n “Brandeis University Summer Institute in Theoretical Physics, 1966, Statistical Physics” (M. Chretien, E. P. Gross, and S. Deser, eds.), Vol. 2, p. 270. Gordon 8c Breach, New York. MacGillivray, J. C., and Feld, M. S . (1976). Phps. Rev. A 14, 1169. Marek, J. (1979).J . Phys. B 12, L229. Mattar, F. P., Gibbs, H. M., McCall, S. L., and Feld, M. S. (1981). Phys. Rev. Lett. 46, 1123. Okada, J . , Ikeda, K., and Matsuoka, M. (1978a). Opt. Cornmun. 26, 189. Okada, J., Ikeda, K., and Matsuoka, M. (1978b). Opt. Cornmun. 27, 321. Polder, D., Schuurmans. M. F. H., and Vrehen, Q. H. F. (1979). Phys. Rev. A 19, 1192. Power, E. A., and Thirunamachandran, T. (1980). Phys. Rev. A 22, 2894. Rehler, N. E., and Eberly, J. H. (1971). Phys. Rev. A 3, 1735. Ressayre, E., and Tallet, A. (1978). Coherence Quantum Opt. 4, 799. Sargent, M., 111, Scully, M. O., and Lamb, W. E., Jr. (1974). “Laser Physics.” AddisonWesley, Reading, Massachusetts. Saunders, R., Hassan, S. S., and Bullough, R. K. (1976). J. Phys. A 9, 1725. Schuurmans, M. F. H. (1980). Opt. Cornmun. 34, 185. Schuurmans, M. F. H., and Polder, D. (1979a).In “Laser Spectroscopy” (H. Walther and K. W. Rothe, eds.), Vol. 4, p. 459. Springer-Verlag, Berlin and New York. Schuurmans, M. F. H., and Polder, D. (1979b).Phys. Lett. A 72A,306. Schuurmans, M. F. H., Polder, D., and Vrehen, Q. H. F. (1978).J. Opt. Soc. Am. 68,699. Skribanowitz, N., Herman, I. P., MacGillivray, J. C., and Feld, M. S. (1973). Phys. Rev. Lett. 30, 309. Vrehen, Q. H. F. (1979).I n “Laser Spectroscopy” (H. Walther and K. W. Rothe, eds.), Vol. 4, p. 471. Springer-Verlag. Berlin and New York. Vrehen, Q. H. F., and der Weduwe, J. J. (1981). To be published. Vrehen, Q. H. F., and Gibbs, H. M. (1981). I n “Dissipative Systems in Quantum Optics” (R. Bonifacio, ed.), Topics in Current Physics 27, in preparation. Springer-Verlag, Berlin, New York. Vrehen, Q . H. F., and Schuurmans, M. F. H. (1979). Phys. Rev. Lett. 42, 224. Vrehen, Q. H. F., Hikspoors, H. M. J., and Gibbs, H. M. (1977). Phys. Rev. Lett. 38, 764.

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Vrehen, Q.H. F., Hikspoors, H . M. J . , and Gibbs, H. M. (1978). Coherence Quanrum O p f . 4, 543. Vrehen, Q. H. F., Schuurmans, M. F. H., and Polder, D. (1980). Nafure (London) 285, 70. Walker, L. R. (1%3). In “Magnetism” ( G . T. Rado and H. Suhl, eds.), p. 305. Academic Press, New York.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 17

APPLICATIONS OF RESONANCE IONIZATION SPECTROSCOPY IN ATOMIC AND MOLECULAR PHYSICS M . G. PAYNE, C . H . CHEN, G. S. HURST, and G . W. FOLTZ" Chemical Phy&s Section Oak Ridge National Laboratory Oak Ridge, Tennessee

I. Introduction

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

11. Multiphoton Excitation with Broad Bandwidth Lasers . . . . . . . . 111. RIS Studies of Inert Gases . . . . . . . . . . . . . . . . . . . . .

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Photon Excitation of Ar, Kr, and Xe . . . . . . . . . . . C. Measurements of Photoionization Cross Sections, Collision Rates, and Lifetimes for Excited States of Inert Gases . . . . . . . . . D. Studies of Resonance Radiation Trapping in Inert Gases . . . . . E. Estimates of Three-Photon Excitation Rates in He and Ne . . . . IV. Experiments Combining RIS and Pulsed Supersonic Nozzle Jet Beams A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . B. Isotopically Selective Detection of CO . . . . . . . . . . . . . C. Crossed-Beam Studies of Laser-Induced Collisions . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

229 231 239 239 240

. 249 . 252 .

261

. 262 262 . 265 . 269 272

I. Introduction Resonance ionization spectroscopy (RIS) can be defined as a stateselective detection process in which pulsed tunable lasers are used to promote transitions from the selected state of the atoms or molecules in question to higher states, one of which will be ionized by the further absorption of another photon. At least one resonance step is used in the stepwise ionization process, and it has been shown (Hurst et al., 1979)that the ionization probability of the spectroscopically selected species can nearly always be made close to unity. Since measurements of the number * Postdoctoral Research Appointment through Western Kentucky University Subcontract No. S-7640 with Union Carbide Corporation. 229 Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0- 12-0038 17-X

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of photoelectrons or ions can be made very precisely and even one electron (or under near vacuum conditions, even one ion) can be detected, the technique can be used to make quantitative measurements of very small populations of the state-selected species. The ability to make saturated RIS measurements opens up a wide array of applications to both basic and applied research. In previous reviews of RIS (Hurst er al., 1979, 1980), the subject was treated generally including the underlying photophysics foundations, the ability to use it to count single atoms, and some of the applications of one-atom detection. In a more recent review, Chener al. (1981) treated the resonance ionization of noble gases in some detail, showed how to count with isotopic selectivity the individual atoms of Ar, Kr, and Xe, and again described applications of counting these atoms. In contrast, this review does not deal with single-atom detection or its applications, but rather with photophysics research. Hurst er ul. (1979) described a number of studies involving alkali metal atoms which had been made possible by the high spectroscopic selectivity, sensitivity of detection, and time resolution obtainable by using pulsed tunable lasers in the RIS technique. The atomic and molecular studies emphasized in that paper were: (1) measurements of photoionization cross sections for the excited states of alkali metals; (2) measurements of lifetimes and branching ratios for optical transitions in alkali metals; (3) measurements of photodissociation cross sections for alkali halides; (4) studies of the diffusion of alkali metals in various gases; ( 5 ) measurements of chemical reaction rates of alkali metals with various reaction partners; and (6) high-resolution spectroscopy using resonance ionization techniques. It is our purpose in the present article to discuss further applications of RIS methods to atomic and molecular physics. We examine topics of current interest in which we feel the use of RIS can yield significant contributions. To this end we discuss several previous studies as examples of the methods involved and then suggest a variety of specific experiments, the performance of which, we endeavor to show, should be possible with present-day technology. The areas of consideration are: (1) laser excitation with relatively broad bandwidth lasers; (2) measurements of photoionization cross sections; and (3) studies of collision and transport processes. Particular attention is given to the inert gases and to diatomic molecules. We hope that this discussion will be indicative of the types of investigations in which RIS may be used. We do not claim that this article reviews all laser ionization work. In particular, we are aware of the large body of literature on resonant multiphoton ionization. We view resonance ionization spectroscopy as a

APPLICATIONS OF RIS

23 1

specific type of multiphoton ionization in which the goal is to make quantitative measurements of quantum-selected populations of atomic and molecular systems. This goal is attained by requiring that the selective excitation steps be resonant in nature and involve only one- or two-photon absorption processes, thereby allowing the entire process to be carried to saturation without the loss of spectroscopic selectivity due to level shifting and broadening. We begin in Section I1 with a theoretical description of multiphoton excitation using broad bandwidth lasers. The purpose of this section is twofold. First, the theoretical results are used to estimate excitation probabilities with regard to the feasibility, expected signal magnitude, etc. of the experiments we discuss. Second, the results of these experiments can, in turn, be used to test various aspects of the theory such as atomic structure calculations and the model chosen to describe the laser radiation. In this regard we have considered the effects of broad bandwidth lasers upon the excitation process since such lasers will most likely be used. It should be noted that these tests of the excitation theory are also important to the refinement of the RIS technique. In Section 111 a variety of photophysics and collision physics experiments on inert gases are suggested and described. The results of an experiment on the two-photon excitation of Xe are discussed. In Section IV the possible use of RIS techniques in conjunction with pulsed supersonic nozzle jet beams is illustrated with the proposal of a couple of experiments. One of these, the detection of small concentrations of I4Ci6Ois presently being undertaken by the authors.

11.

Multiphoton Excitation with Broad Bandwidth Lasers

We consider a two-state model of excitation by lasers. The absorption o f N photons satisfies the conservation of energy between a lower state 10) and an excited state 11). as shown in Fig. 1. In Fig. 1 we have defined 6 through h[zr(oLt - o1 + oO]= ha, where hul is the energy of the upper level, fiwo is the energy of the lower level, and hoLl are the individual photon energies at line center for the lasers used for the N-photon excita+ tion. If the N photons are all absorbed from a single laser beam, zj"hwL1 Nho,,, etc. Let I$(t)) be the state vector of the atom in the presence of the laser fields; then Al$(t))= i h a / $ ( t ) ) / a t ,where A = A0 + $'(t). Here A, is the Hamiltonian of the isolated atom and v ( t )represents the interaction between the atom and the laser fields, with the latter being described as = lo), and as classical electromagnetic fields. Initially, we have 1 $ ( - ~ ) )

M. G . Payne et al.

232

nu0 FIG. 1. Energy level diagram for a two-state model of N-photon excitation. The detuning from resonance 6 is defined by N

fi6 =

2

fiu'P)

-

fi(Wl -

wo)

P=l

It is assumed that 8 is much less than the detuning from any other k-photon resonance with k C N and that the detuning from any lower resonance is large compared with the laser bandwidth or the Rabi frequency for any lower order resonance.

the laser pulse arrives at the atom, I+(?)) evolves according to I+(r)) = f('(t)lO), where ?(t) is the time evolution operator. In standard quantum mechanics text books (see, for example, Messiah, 1966) it is shown that if we let f(t)= exp(-iAot/h)3(t)

then

$2)

(1)

satisfies

= exp(ikfot/h)v(t)exp(-ikfot/h). Equation (2) is obtained by where vH(t) using Eq. (1) in the time-dependent Schrodinger equation to give

On integration, Eq. (2) follows. By iterating Eq. (21, we arrive at the following equation, which is equivalent to Eq. (2):

233

APPLICATIONS OF RIS

where

So = 1, and for

k

2

1,

Equations (4)and ( 5 ) hold for any integer p . We would like to find equations of motion for the probability amplitudes for being in states 10) and I I ) , i.e., for Co(t) = exp(iwot)(Ol$(t)) = ( O ( S ( t ) l O ) and Cl(t) = exp(iwlt)( I I$(t))= ( 1 I!$t)lO). In defining Co(t)and Cl(t),the phase factors were introduced for mathematical convenience and do not change the values of lCOl2 and IC1l'. We now derive equations of motion for Co(t)and Cl(t). Equation (3) immediately yields ih(aCo/at)= (olV,(t)S(t)(o)

(6)

We will show how operator methods using Eqs. (2) and (4)can be employed in manipulating the quantity ( Ol~H(t).!?(r)lO)to derive a very general two-state approximation to the laser excitation problem where the required approximations are clearly evident. The situation withN = 1 will be treated separately. In the latter case we note that if &,In) = honln), then the unit operator can be written as i = Snln>( n l , with indicating a sum over discrete states and an integral over the continuum. Thus, for N = 1 we have, using (O(exp(ifi,,t/h) = (Olexp(iw,t) and exp(-ifiot/ h)ln) = exp(-iw,t)ln),

sn

ih(aco/at)= ( o l v H ( t ) l I ) ( lIS(t>lo> +

S (olvH(t)ln>(nlS(t)10) n# 1

=(

o I W 1)

+

exp[i( wo - w l ) t I ~ l ( t )

S (OlV(t)ln) exp[i(wo n# 1

-

wn)t](nl~(t)lo)

When the sum is investigated by applying perturbation theory to ( nlS(t)lO) and there are no other one-photon resonances, it is found that the term is negligible. Starting again with Eq. (3), an equivalent equation for ih(aCl/dt) may be obtained in a similar fashion. Thus, we have

ih(ac,,/at) =

(oJiTt)l~ )c,exp[i(wo - w l ) t ]

i f i ( ~ , / a t )= ( I(P(t)lO)Coexp[-i(wo - wl)t]

Equation (7) is standard and in general simple to derive.

(7a) (7b)

M . G . Payne et al.

234

When N > 1 is considered, the derivation of equations analogous to Eqs. (7) is cumbersome; it is here that the operator method greatly simplifies the derivation. We begin with Eq. (6) and use Eq. (2) to replace !at): ih(aC,/dr) = =

(O l j v H ( r ) [ 1 +

(ifi)-l

j‘ --m

vH(f’)S(t’) dr’]10)

I:,p H ( f r ) $ f ’10)) d f ’

(ih1-I (01 p H ( f )

(8)

The last form follows because (Oli/H(f)lO)= 0 by parity considerations. We let D = 1 - lO)(Ol and insert i between e H ( tand f ) & t ’ ) in Eq. (8): ih(aC,/Br)

=

( o IAV H ( t ) Q H ( t rI)o) Co(rf)dr’ + (ifij-1 ( olvH(r)j‘ v,,(rt)bj(rr) lo) dt’

(ih)-l

L - m

-m

(9)

The first term on the right-hand side has been separated off because it will eventually give rise to the ac Stark shift term. We also expect to eventually get a term involving C , on the right-hand side and a product of N of the p H ( f ) interaction terms because the lowest-order term which conserves energy involves the absorption of N photons. The trick is to now replace S(f‘)by Eq. (4) with p = N - 2. We obtain ifi(dc,/at) = (ifi1-l

I‘ ( o ~ ~ H ( ~ ) ~ ,c,(t!) ,(r~)~o) d:’

--m

+ (ih)-’

k=O

1‘

-a

(olvH(f)vH(f’)bSk(f’)lO) df’

A two-state approximation will hold only if the middle set of terms involving ( O / p H ( t ) v H)(bfS’ k ( f ’ ) p ) can be neglected, and if, after inserting = /1)(11 + S,+lln)(nl between PH(tl) and %tl) in the last In) (nl can term, the non-energy-conserving terms coming from SnZl be neglected. As in the one-photon excitation case, the neglected terms can be estimated. All of the terms, in the absence of any other near resonances of the same or lower order, will be found to be either very small or small and multiplied by a rapidly oscillating complex exponential whose effect averages to near zero unless the laser power density is extremely large. It is advisable to be sure that the neglected terms are

235

APPLICATIONS O F RIS

negligible in any case where the two-state approximation is used. The non-energy-conserving terms arising from inserting Sn+, In)( nl between V&) and $r,) can be estimatcd by developing $r1) as a perturbation series in P d t ) by using Eq. (2). It is important to note that before these terms are thrown out, the relation is still exact. In the two-state approximation, we have then ih(ac,/at) = (ih)-l

1' ( o ~ P H ( r ) ~ H ( r ~ ) ~ dt'o ) C o ( r ' ) --m

+ (ifi)-"+' j t

--m

dtN-1

(ol~H(r)~H(rN-l)b

Equation (1 1) can-be simplified further by noting that for a function F,

1'

--m

F ( t ' ) exp(iat') dt' = [ ~ ( t ) / i aexp(iRt) ]

(12)

provided that exp(iat') oscillates many times, while F(t ') makes only a small change for any t ' . On inserting between the factors and using (nlexp(iEi,t/h) = (nlexp(iw,t), exp(-iAot/h)ln) = exp(-iw,t)ln>, together with the time dependences of the laser fields, it is found that Eq. (12) can be applied to all the time integrations in Eq. (11) in the same situations where the neglected terms are small. The final effect is that COO')and C , ( t , ) can be removed from the integrals and evaluated at t. The same type of scheme with similar approximations also works for the equation for ih(aC,/dt). We get

vH

dC,/dt = iAo(t)Co+ inN([)exp(iSr)C, dC,/dr

= iAl(r)Cl

+ ia*,(t)exp(-i6t)Co

(134 (1%)

where QN(t) = -i(ih)-," exp( - i S r )

and, for i

=

0 and 1 ,

The quantities Cldr)and A,(t) are theN-photon Rabi frequency and the ac Stark shift of li), respectively. In evaluating Eqs. (14) and (13, it must be

M. G. Payne et al.

236

remembered that the same approximations used in removing Co(t)and Cl(t) from the integrals must be employed. Thus, unit operators are inserted between the p,,(ri),and Eq. (12) is applied repeatedly. With this understanding, Eqs. (14) and (15) represent a very compact and general mathematical form in which the lasers can have any state of polarization, any direction of propagation, any bandwidth consistent with our approximations, and N is general. Nearly the same type of operator manipulation can be used to extend the treatment to multiple resonances, ionization, and effects of degeneracy, as well as collisional effects. However, the resulting theories are not two-state approximations. The two-state approximation has been discussed many times and is included in many standard reference books (see, for example, Sargentet al., 1974; Allen and Eberly, 1975). An elementary discussion including the effect of phase changing collisions and the resulting line broadening effects can be found in Hurst et al. (1979). In accord with widely held interest in the use of broad bandwidth commercial lasers, we shall now discuss excitation with such light sources. In general, the use of broad bandwidth lasers will result in a complex value of nN(t) as evaluated by Eq. (14). It is necessary in what follows to resolve Odt) into a real and an imaginary part. Thus, we write n N ( f ) = exp[iq(t)], where aN(t) is chosen to be real and q is determined by the particular model chosen for the lasers. In dealing with stochastic light, we must go to a density matrix formalism (see, for example, Sargent et al., 1974; Allen and Eberly, 1975). We begin by casting the two-state problem in terms of bilinear combinations of C, and Cl, which are measurable quantities. We let

aN(r)

Z,

=

2 Re[C$Cl exp(i6t + i q ( t ) ) l

Z3 = 2 Im[C$;C,exp(i6t

(16)

+ iq(t))]

In this alternative description, Z, measures the population inversion. Equations of motion for Z1, Z,, and Z3are equivalent to Eqs. (13) for the development of the probability amplitudes. Use of Eqs. (13)-( 15) in Eq. (1 6) yields dZl/dt = 2nN(t)z3 dZ,/dt

=

-(6 - Ao(t)

dZ3/dt = (6 - Ao(t)

+

+

Al(t)

Al(t)

+

+

dq/dt)Z3

(17)

dq/df)Z* -211N(t)Z1

Equations (17) apply to an atom which experiences a particular detailed time history of the laser field. However, atoms do not see the same de-

237

APPLICATIONS OF RIS

tailed field from pulse to pulse, and it is forever impossible to describe the laser field in anything but a statistical way. As a result, we would like to determine the average response of atoms to pulses having the statistical properties of the lasers being used. We then need to ensemble average Eqs. (17) over the phase space of the laser, and this averaging procedure leads to the Bloch equations for the system (see, for example, Hurst et al., 1979). It should be noted that (IC,l’) = p l l , ( lCol’) = poo, (C;Cl) = pol, etc., where pii are elements of the density matrix. Equations (17) can be manipulated so that Z , and Z , are eliminated and an integral equation for Z1 is derived. We find

where K(t, r ‘ )

=

-4

ltaN(t’)aN(rf’) cos[Q(t’)

-

I’

Q(t”)]dt“

(19)

with

If the product of the bandwidth of at least one of the lasers and the pulse length is very large compared with unity, it also frequently occurs that the Now, from Eqs. (13) we bandwidths are also large compared with see that l/lfiNlis a measure of the time interval over which lColand (C,I can change appreciably. We then see that in this limit Z1 can only change in a time span which is very long compared with the coherence times of the lasers. The value of Z1 at a time t is, therefore, not correlated with the laser field within a few coherence times oft itself but depends only on an integrated effect of the field over a much longer time span. We note that

InN/.

dZl/dt =

1‘

dt’[aK(t,t’)/arlZl(t’)

(21)

--x

where

t w r , t’)/ar

=

- 4 f i N ( t ’ ) R N cos[Q(r’) (t) - ~ ( t ) ]

(22)

Since ( a K ( t , r ‘ ) / a t ) is a laser field autocorrelation function, we expect that it will only be appreciable when r - t‘ is smaller than a few inverse bandwidths. Nearly all of the contribution to the integral then comes from is close to zl(t).Using the excellent times very close to t where approximation ( Z , ( a K / a t ) ) = y I ( a K / d r ) , we get (2,= ( Z l ) ) :

zl(f’)

M. G. Payne et al.

238

dZ,/dt

=

In this formulation where the pulse length is short compared with the spontaneous decay rate and the ionization rate from 11) is negligible, we have poo + pll = 1. Thus, Eq. (23) is equivalent to dpOO/dt =

dMN(fir

dpll/df =

-IMN(6,

t)(pll t)(pll

POO),

(24)

- POO)

Obviously, R(N)= MN(6, t ) / 2 is the rate at which the population of 10) can be promoted to 11 ). Equation (24) is a rate equation limit. It is well known that relaxation processes such as photoionization, spontaneous emission, or impact broadening of a line enhances the validity of a rate equation description (see, for example, Hurst et a/., 1979). Thus, in a situation where the laser bandwidths are large compared with the pressurebroadened width, the rate of ionization, and the spontaneous decay rate, and where either circular or linearly polarized light is used, we have dPOO/dt = R(N)(Pll

dddr

=

-R("p11

-

Poo),

- Pool

-

TIP11

(25)

for cases where another photon ionizes I 1 ) and MJ changing collisions are ineffective within the pulse length. In the expression above, rl is the ionization rate of 11 ). The fact that the laser bandwidths (at least one) are much larger than laN1does not imply that pll cannot become comparable to poo. On the contrary, it is possible to achieve pll = pOO 1/2 without focusing when broad bandwidth lasers are used for one-photon excitation (see, for example, Hurst et a / . , 1979). For two-photon excitation, the same leveling of the populations can be achieved with modest beam focusing using broad bandwidth (-lO"/sec) commercial lasers (Chen et a/., 1980). For N 2 3, extremely high power densities are required for population leveling, and it may not be achievable due to the fact that amplitude fluctuations operating through Al(t) - Ao(t) (i.e., the ac Stark shifts) are simultaneously leading to a strongly broadened and shifted line as the power density is increased. Resonance ionization spectroscopy (RIS) has excellent application to high-resolution spectroscopy. For instance, with counterpropagating beams, two-photon excitation can be carried out in a Doppler-free manner (Vasilenko et ul., 1970; Grynberger al., 1976). The key mechanism is that at low-power densities, where power broadening and ac Stark shifts are 2:

APPLICATIONS OF RIS

239

small compared with the natural level width, all atoms, regardless of velocity, can resonantly absorb a photon from each beam. Thus, for beam propagation vectors along the z axis, the energy absorbed is ha( 1 + u,/c hw( 1 - u,/c) = 2ho. However, the absorption of two photons propagating in the same direction is off resonance by 2 h o u , / c and is much less effective. Absorption of two parallel propagating photons leads to the “Doppler pedestal,” which can be eliminated entirely by using two counter- wL21 >> Doppler propagating beams centered at and %,, with loL1 width, but - q21/q,l << 1. Such Doppler-free excitation requires near transform-limited bandwidths for the lasers so that the bandwidth can be small compared with the level width. The process must also be carried out at very low concentrations and in the absence of external fields. Resonance ionization spectroscopy can be applied by carrying out excitation as described above and then firing a more powerful laser, which is time delayed relative to the excitation process (to avoid power shifts in the absorption resonance) to selectively ionize the excited atoms. At the low pressures used, the resulting ions can be mass analyzed and detected with a particle multiplier. The detector is gated relative to the firing of the lasers, permitting integration of the signal over many pulses without interference from noise signals. Such a technique permits very high sensitivity and enables one to keep power densities and concentrations extremely low. We shall not concentrate here on spectroscopy, but instead focus our attention on measurements of photoionization cross sections and studies of collision processes, transport processes, and laser excitation with relatively broad bandwidth lasers.

+

111. RIS Studies of Inert Gases A. INTRODUCTION Inert gases have long been of great interest in atomic and molecular physics. Part of this interest is related to the fact that they exist as atoms in the gas phase, while most elements are chemically reactive and exist either as solids or molecular gases except at greatly elevated temperatures. Much of the remaining interest is related to their energy storage properties. The excited states of a rare gas typically lie within 1-5 eV of the ionization continuum, while the ionization potential of the ground state is at least 3 . 5 times larger. In addition, rare gases have long-lived metastable states, and, at concentrations 2 10‘4/cm3,even states having allowed transitions to the ground state have long effective lifetimes due to radia-

240

M . G . Payne et al.

tion trapping (Holstein, 1947, 1951;Payne el al., 1974).Since recombination of rare gas ions also leads to excited states (Bardsley and Biondi, 1970),energy deposited by way of charged particles or a discharge leads with high efficiency to energy storage in long-lived excited states, which, in contrast to ground state inert gas atoms, are very reactive with other species. For these reasons, inert gases have been widely studied in discharges (Phelps and Molnar, 1953;Phelps, 1955, 1959)and with charged particle excitation (Thonnard and Hurst, 1972).They play a key role in the operation of many applied devices such as most gas lasers. In this section we describe some of our past studies of inert gases using RIS and indicate how the techniques which we have developed can be applied to a series of very detailed studies of atomic physics in gases. We cover the following topics: (1)two-photon excitation of inert gases and applications to the production of a line source of excited atoms; (2)use of RIS for detailed studies of radiation trapping in inert gases; (3) accurate measurements of the photoionization cross sections for excited states of inert gases; and (4)lifetimes, branching ratios, and collisional rates for inert gas excited states.

EXCITATION OF k , Kr, B. TWO-PHOTON

AND

xe

We begin by considering the evaluation of M2(6, t ) for two-photon exciare tation. This evaluation cannot proceed until A*(?) ( i = 0, 1) and a2(t) determined, and this, in turn, requires the specification of the radiation field. We allow for the general case in which two different lasers are used and considered the most common situation of plane polarized light with the propagation directions of the laser beam being identical. Thus, we take

E = mio(t)cos(q,lt+ &(t))+ kE20(t) cos(%d+

42(t))

(26)

v(r)

The atom-field interaction term is then = - a * E , in the dipole approximation, where p i s the electronic dipole operator. In our expression for E, q1 is the central frequency of laser 1, q2is the central frequency of laser 2,El,&) is the fluctuating amplitude of laser field 1, E2,(t)is the fluctuating amplitude of laser field 2, +l(t) is the fluctuating phase of laser field 1, ~ $ ~ ( f ) is the fluctuating phase of laser field 2,and the polarization vector is along thez axis. The time scale of fluctuations in amplitude and phase determine the laser bandwidth, which is the inverse of the coherence time. We are considering broad bandwidth lasers so that the coherence time is much shorter than the pulse length. A rather simple model is chosen for the statistical properties of the laser fields: Eio(l) cos(%it

+

+i(t))

(A1 + ATID

(27)

APPLICATIONS OF RIS

24 1

where =

exp(iq.,t)

Bl(w, t ) exp[i(wt +

MU,

t))~ do

(28)

and similarly for Ezo(f)cos[ q 2 f + d ~ ~ ( f )The ] . integral term represents the spread of frequencies about w L f ,and since wLf is the central frequency, BAw, t ) is peaked at w = 0. The time dependence of Bi(w, t ) and &(w, t ) is assumed to be slow-limited largely by the pulse length. The statistical properties of the light enter in this model through our assumption about the phase of the different frequency terms that make up A i . In particular, we assume that the C # J ~ ( ~t), do not repeat from shot to shot and that in averaging any product of laser fields for i, j = 1, 2 (exp(i&(w, t ) ) exp(-i4j(o', t ) ) )= (exp(i4dwl, =

f))

tjij 6(o - w ' )

e x ~ ( i 4 ~t )( )~exp(-i4k(w3, ~, t ) ) exp(-iMw4,

8ik 611 6(wl

- w3)

6(wZ -

+ 6i1 6jk

w4)

6(wl -

w4)

6(w2

1)))

(29)

- w3)

etc., i.e., the phases of all frequency components are assumed to be independent. For i = 1 or 2, the power density is given by Ii(f) =

c(A,Ai)/8~

jmIBi(w, t)I' dw

=(c/~T)

--m

where Eq. (29) has been used. The model used here can be viewed as the continuum limit of a large number of phase-independent modes. One of the consequences of the model is that ( E : " ( t ) )= n ! [ ( E f ( t ) ) ] "The . model thereby yields the observed n ! enhancement of off-resonance multiphoton ionization by a broad bandwidth relative to a coherent laser field (Lecompte et d.,1974; Mainfray, 1978). Several stochastic models of laser light have been studied recently which yield very similar results in the rate equation limit (Lambropoulos, 1976; Zoller, 1979; Zoller and Lambropoulos, 1980), except that they typically lead to Lorentzian line shapes for the laser. Such line shapes are not completely valid for lasers operating far above threshold. In our case the line shape may be specified. However, our assumption of completely independent phases for different frequency components is certainly not totally realistic; we rely to a great extent on the large amount of model insensitivity to this assumption. We are now in a position to evaluate Ai(t)(i = 0, 1) and f12(t).This will serve as an example of how Eqs. (14) and (15) are to be evaluated in more general situations. For At(r) we have

242

where we have divided Ai(t) into a contribution, Au(t) ( j = 1, 2), from each of the laser fields j. Using Eq. (12) and throwing away rapidly oscillating parts, At&)

=

s

(IA#/W

n

( i l p h )( nlpzli>

x [(wn - wi -

qj)-'

+

+ qj)-'] (32)

(wn - wi

The rapidly oscillating terms which have been neglected produce little average effect on C l and C,, while the term that has been retained produces a time-dependent phase change in C1 and C z and is the well known ac Stark shift in the level li) (i = 0, 1) due to laser fieldj. Equation (32) also includes the Bloch-Siegert shift. The ensemble averaged shift is

x =

[(On

-

wi

- qJ-' + (wn - wt

+ %J-']

mr)

(33)

We now consider fl,(t) under the assumption that two-photon resonance cannot be achieved by the absorption of two photons from either individual laser. Instead, resonance is achieved by absorbing one photon from each laser beam and ti6 = i ~ ( q+,q2 ~ - w1 + wo). Thus,

n,(t) = ih-2 =

exp(-iSt)

iti-2 exp(-ist)

(o I Q,(t)i

QH(t1)

dtll I )

S exp(i(wo- wn)t)(OIP(t)ln) n

x

j:mexp(i(o,

-

wl)tl)(nlQ(tl)lI) dtl

= (A1Az/4h2) exp(-i(q,,

x [(wn - wo =

+ qz)t)

S ( oP,In) (nIp,I I ) n

- wL1)-' + (wn - wo - q 2 ) - ' 1

HaAlA2) exp(-i(ql + qdt)

(34)

where we have used w1 = wo + q.,+ q,,in the resonance denominators and only the lowest frequency part has been retained following the use of

243

APPLICATIONS OF RIS

Eq. (12). In all of the above it should be remembered that the time dependences ofB(w, r) and +(w, t ) are assumed to be on the time scale of the pulse envelope, which is very slowly varying compared with the complex exponential factors. Equations (33) and (34) serve to define (Y and p. Equations (32) and (34) are the standard results. We proceed now to evaluate M 2 ( 6 , t ) . We assume here that a single can ionize 11). In the latter case, there is photon with energy fiqlor fiq2 no high-lying discrete state In) such that w, - w1 - q jis small. The existence of such a state would give a greatly enhanced ac Stark shift in the upper state. However, if either laser can give one-photon ionization of ) 1 ) , the ac Stark shift will typically be (l-lO)Zf(r), with the shift being in rad/sec and Zi in W/cm*. For Zi(t) < 10“’W/cm’, X L f < 3000 A, and AALf (= linewidth of laser i) > 0.03 A, the ac Stark shift will typically be smaller than the laser bandwidth. From Eq. (34), we see that to get a2(t), A land A2 must be put in the form A{ = Efo(t)exp i [Rit+ &(t)],where E,,(t) is an amplitude which vanes very rapidly with time and +&) is the fluctuating phase. Then v ( t )is seen to be the total phase of the radiation field, i.e., v(t)= + l ( t ) + 42(t).Efo(t)and +*(t)depend in a complicated way on B , ( o , t ) and all of the + f ( w , t ) in Eq. (28). However, the time scale of Id4,(t)/dtl x/rcf, where rciisthe coherence time of laseri. When l / r c i>> lAf(r)l, the ac Stark shifts can be neglected in Q(t)and hence in the evaluation of M2(6, t ) . We get

-

= a2 Re

1: exp(i6(r’ m

-

t))(A,(r’)A~(t))(A,(t’)Af(r)) dt’

where we have used Eqs. (29) and (30) and the relation Re w’

+ S)T] d r

= x6(w

lom exp[i(w +

+ w’ + 6). A reasonable line shape for laser i is

ab Such a Gaussian The full width at half maximum is A(i) = 2line shape, together with Eq. (29), yields for the field autocorrelation function ( A { ( t’ ) A r ( t)) = [8.rrli(r) / c ]exp{ $[ -(t - t ’)*a:1)

(37)

A Lorentzian line shape, Zi(w, t ) = Zi(r)(rL/r)/(d + I 3 , where 2rL is the full width at half-maximum of the line, yields (A# ’)A&))= (8.rr/c)Zf(t)exp[-rLlr - r ‘ l ] . Most stochastic theories yield an auto-

M . G . Payne et al.

244

correlation function of the latter form and a Lorentzian line shape (Lambropoulos, 1976; Zoller, 1979; Zoller and Lambropoulos, 1980). With the Gaussian line shape, we find

At low power densities the observed line shape for the three-photon ionization via the two-photon resonance has a full width at half-maximum of A, = 2[2 (In 2)(a: + a$)]"' = [A2(1) + A2(2)]1'2 (39) where 4 1 ) and 4 2 ) are full widths at half maximum of the two laser lines. For a given full width at half-maximum for the lasers and S = 0, we find very similar results for Mz(O, t), regardless of whether we assume a Lorentzian or a Gaussian line shape. Also, the two stochastic models lead to nearly equal results. At least in the rate equation limit, there does seem to be model insensitivity for two-photon excitation. We must now evaluate a. In inert gases the ionization potential is quite high, but all the discrete excited states lie in a much narrower band of energy. A very reasonable approximation is achieved for the lower lying excited states for which allowed two-photon transitions from the ground state exist if one replaces w, - wo - wLiby W - wo - wLf,where W is the average frequency corresponding to I n ) . If wL1and wL2are not extremely different, a reasonable estimate of W is to define hG to be the average of the ionization potential I and the energy of the lowest-lying excited state for which (Oli),ln> and (nlf'zll) are nonzero. Then, closure yields = (2h2)-'(OlP~~t)[(W - wo -

wL1)-'

+ (G -

00

- w&']

(40)

If two photons from the same laser are tuned for excitation, Eq. (25) still holds, but M,(S, I) is replaced by (Payne et al., 1980)

with

(Y

replaced by

In this special case, I ( t ) is the power density of the single laser, w, is the /~ w2/2a2]. central frequency, and I ( w , t ) = Z ( r ) ( 2 r ~ 7 - 'exp[In making estimates of a, we have used a single configuration approximation and ( j l ) coupling to estimate p : matrix elements between the np6 ground state and states of the form Inp5(*P3,,)n'p[kIJ), where k is the quantum number obtained by diagonalizing 0' + 1)' with j being the angu-

245

APPLICATIONS OF RIS

lar momentum operator for the core and 1 the orbital angular momentum operator for the excited electron. The quantity in the round brackets is the term designation of the parent core. With plane polarized light and core angular momentum j = 3/2, the possible values ofk are 1/2,3/2,and 5/2, while the total angular momentum quantum numberJ ranges from 0 to 3 in integer steps. Only J = Oand J = 2 states have nonzero matrix elements. The selection rule on M j is A M j = 0 so that only the M j = 0 sublevels can be excited. With circularly polarized light, A M j = 22, and only J = 2 levels would be excited. Pertaining to the case of plane polarized light, the largest matrix element results when 11) = Inp'(2P3/z)n'p[1/2]0). In the latter case,

I(llfil0)l = #Rz(n,1

=

I; n',/' = I)

(43)

where the 2/3 is an angular factor and R2is the radial matrix element of r 2 , which is independent of k and J. Matrix elements between the ground state and other levels of the n'p multiplet will involve a different angular factor times the same radial integral. The angular factor is nearly a factor of two smaller for Inp'(2P3,z)n'p[5/2]z) and almost a factor of six smaller for lnp'('P312)n'p[3/2]2). The radial matrix elements R2 have been estimated for some important transitions in Ar, Kr, and Xe. We let R2 = L2e2u;, where Lz is a dimensionless quantity. All of the inert gases Ar, Kr, and Xe can be excited either by two photons from a single frequency doubled dye laser or by a combination of frequency doubled dye laser output with the strong line from a He-F2 Excitation schemes are as follows (Payne et al., 1980): laser at 1576.3

A.

(a) Ar(3p6) + fiw,,, (A

=

1576.3 A)

Here we have estimated R z (b) Kr(4p6) +

+

=

hw,,

(A

2292.8 A)

=

3262

1.7e'u;.

f i ~ (A, =~ 1576.3 A) + hwl., (A

A) -

=

Ar(3p'(("P,,,)4p[l/210)

Kr(4p'(2P3/z)5p[1/21~)

In Case (b) we find R 2 = 3e'u;. (C)

Xe(5p')

+ 2 f i q (A

=

2496

A)

+

Xe(Sp~(2P312)6p[l/2]e)

The Case (c) value is Rz = 7e2a;. For Xe we could also use (d) Xe(5pfi)+ h u l l (A

=

1576.3 A) + f i ~ , (A , ~= 5995 A) + Xe(5p;(2P3,1)6p[1/2],)

The 1576.3 8, line, obtained by running He-Fz mixtures in excimer lasers, has a linewidth -0.01 A, an energy per pulse of E = 10 mJ, and a pulse length of T = 10 nsec. Using Schemes (a), (b), or (d), we find for the rate of two-photon excitation in Eq. (25) R(2)= Mz(0, f)/2 = K Z i ( f ) Z z ( f )

(44)

246

M. G. Payne et al.

where, for all the inert gases mentioned above, Eqs. (38) and (40) yield K -- 14L$/[Az(l)+ A'(2)]"'

(45)

In Eqs. (44) and ( 4 3 , I , and Z2 in W/cm2, R(2) in sec-', and A ( l ) and A(2) are the full widths at half-maximum of the laser line shapes in rad/sec. In Processes (a), (b), and (d) the dye laser can be tuned to excite other levels, and if circular or unpolarized light is used, a similar quantity K can be obtained for J = 2 levels. If the laser beams are reduced in radius via a telescope arrangement such that beam areas A, where A 0.04 cm', result, a line source of excited atoms can be produced. From Eq. (44) we obtain a probability of excitation at low-power densities

-

P, -- K E , E ~ / A ~ T

(46)

and a number of excited atoms per unit length

N,

=

nAP,

=

~KE,E~/AT

(47)

where n is the inert gas concentration in cmP3, E , and E~ are the pulse energies in joules, T is the pulse length in seconds, andA is the beam area in cm'. With T = = lo-? J, e2 = 10-5 J, A(2) = lO"/sec, sec, L2 = 2, and A = 0.04 cm2, where laser 1 is the He-F2 laser and laser 2 is the dye laser, we get P, = lo-'. At pressures of 1 Torr, we would excite about lo9rare gas atoms/cm, and more than lo8 would be ionized by the 1576.3 A light. Typically, if we want to produce slOx/cm excited atoms and ionize very few, it will be necessary to attenuate the He-F2 laser beam. It is apparent that by using different wavelengths for the dye laser, all of the In pY'Pr)n ' p[k].,) states for which two-photon transitions are allowed can be excited for n ' = n + 1. In the case of Kr and Xe, even higher states can be excited with relative ease. The characteristics of this excitation are that (1) the excited atoms can be produced uniformly along a cylinder having a very small radius, and (2) they can all be produced in a time period of -5 nsec. It is this possibility of producing a rather thin line source of excited atoms in a period of time which is short compared with their natural lifetime that makes a large number of very detailed atomic physics studies of inert gases possible. Very little experimental information exists on the two-photon excitation of inert gases. One relatively recent study (Chen et al., 1980) reports ~)~P[ two-photon excitation from 10) = 15pfi)of Xe to I I ) = I ~ P ~ ( ' P ~ ,I /2],,). In this study the output of a dye laser was frequency doubled and the resulting light frequency summed with the 1.06 pm light from a Nd-Yag laser to obtain A,. = 2496 A radiation with E = 1 mJ and T = 4 nsec. The original dye laser had a linewidth of 0.16 A, and it was inferred that the

APPLICATIONS OF RIS

247

linewidth of the 2496 A beam was between 0.05 and 0.1 A. The radial intensity distribution was reasonably well represented following focusing by ( p = radial coordinate, x = distance from focus along the beam axis) I ( p , x) = (E/T)ACT) exp[-(p -

po(x))'/d2(x)I

(48)

where d ( x ) = [(F+,,,)' + (dfi/F)*I1",po(x) = p o f i / F , p,,,, is the distance from beam center to peak intensity just before the focusing lens, +llz is the half-angle beam divergence of the initial beam before focusing, do is the half-width to c - ' drop in intensity of doughnut-shaped beam before focusing, F is the lens focal length, and

Equation (48) represents the intensity achieved near the focus when a doughnut-shaped beam is passed through a lens and the beam has +l/z >> diffraction limit. The measurements were carried out at a Xe concentration of -3 x 10'"/cm3, and the ions produced within -0.4 cm of the focal point were detected by a channeltron operated at low voltage so as to function in a proportional mode. With E c- 1 mJ and F = 15 cm, the power density is sufficiently large at the focus to strongly saturate both the excitation of the resonance and the ionization of the excited Xe atoms. Since the lifetime of the state being excited is very long compared with I, rate equations-like Eq. (25) with N = 2 were used to calculate an ionization probability P, for an atom at a point p andx. Using R = Mz(O,t ) / 2 and rl = ionization rate where both R and TI depend upon p andx, one finds

where J = (R' + rf/4)I'*, p1 = R + rl/2 - J , pz = R + r1/2+ J. Since R cc Z *( p , x) and TI a I ( p , x), the number of ions observed was N,

=

n

j::/2dx:j 2 n p P 1 ( R ( p x). , Tdp, x)) d p

(50)

where n is the Xe concentration and L is the length of the region observed by the detector. Figure 2 shows the ionization signal observed as a function of E*; the solid curve represents a fit of Eqs. (49) and (50) to the observed data using R = K P , TI = u19= ulZ/hw, where 9is the photon flux, and the parameters K = 2 x 10-9sec-' W-'cm4and CT, = 3 x lo-'# cm'. Theoretical calculations using Rz = 7e 'a f and a laser linewidth of 0.08 8,yielded 1 x 10-9sec-1W-*cm4forthe value ofK. To further check the results, an effective volume of ionization was defined by N , = n AV. One expects that, if the resulting ions are accelerated to high energy, they

M . G . Payne et al.

248

4

3 S

2

0 0

10

20

30

50

40

60

70

80

E*

FIG.2. Ionization signal of Xe versus the square of the energy per pulse in mJ for laser light at h L = 2496 A focused with a lens with F = 14 cm. The beam divergence of the laser beam was ell* = 5 x rad and the linewidth = 0.08 A. The 2496 A radiation is twophoton resonant with the state 15py2P312)6p[ 1/21,,) which has an ionization cross section = 3x cmz.

will be implanted in the channeltron detector and lost from the gas phase. Thus, after N l laser shots, the concentration should decrease to

n = no exp[-Nlp AV/Vl

(5 1)

where p is the efficiency of ion implantation, no is the initial atom concentration, and V is the total volume. Figure 3 shows the ion signal versus time for the laser operating at 10 Hz and V = 50 cm3. The volume of AV obtained from the decay was p AV,,, 2 x cm3, while A V calculated from the values of K and CT, determined from Fig. 2 was found to be 3x cm3. The implantation voltage was such that p should have been near unity. The excitation rate could be more clearly separated from the ionization process by using lower power densities for excitation and time delaying a more powerful ionizing laser of larger beam radius and C T ,~ T >> 1 so as to measure the total number of excited atoms produced by the exciting laser pulse. We regard accurate measurements of K as a function of bandwidth and state of polarization to be a promising area of research. The results will provide stringent tests for atomic structure calculations as well as stochastic theories of the laser light.

-

249

APPLICATIONS OF RIS 2

I

I

I

I

I

I

1

Z

0 l-

a

N 0.5

t

1

;I-

I

W

W

n

i

0.2

0.

0

60

120

480 240 300 RUNNING TIME (rnin)

360

420

FIG.3. Ionization signal versus time for a situation where n = 10'"/cm3 and the ions produced by laser light at A,. = 24% A are accelerated and implanted. The laser characteristics were: e = 0.7 mJ, pulse length -4 nsec, H,,, = 5 x lo-' rad, linewidth -0.08 A, and the light was focused with a 14 cm lens. The volume of the cell was 50 cm3 and the laser repetition rate was 10 Hz.The exponential decay rate implies p A V = 2 x cm3, where A V is the effective volume of ionization and I.L is the implantation probability.

C. MEASUREMENTS OF PHOTOIONIZATION CROSSSECTIONS, COLLISION RATES,A N D LIFETIMES FOR EXCITED STATESOF INERT GASES In Section III,B we indicated how a line source of excited inert gas atoms can be produced at a well-defined time by a commercial laser system. A variety of levels can be excited in Ar, Kr, and Xe, and these in turn decay to other excited states (since transitions back to the ground state are not allawed for dipole transitions). Further, 106-107 excited atoms can be produced per centimeter for n = 10'6/cm3, with very little ionization occurring due to the exciting lasers. Here one would use beam diameters -0.02 cm for the exciting lasers. To measure the photoionization cross section of the excited state, another laser, the pulse of which is delayed by -10 nsec relative to the exciting laser pulses, is used to photoionize the prepared state. The ionizing laser beam is concentric with and has a much larger diameter (-0.1 cm) than the exciting laser beams, as shown in Fig. 4. Using dye lasers pumped by the third harmonic of a

250

M . G . Payne et al.

FIG.4. Experimental arrangement for measurements of photoionization cross sections in inert gases. The U V beam is used for two-photon excitation of atoms along a cylinder of radius -0.02 cm. The larger beam is time delayed relative to the first and is used either to ionize the original excited species by one photon (for photoionization cross sections) or to selectively ionize a population by stepwise two-photon ionization. Present commercial lasers permit ionization probabilities to be near unity for a beam radius -0.1 cm. The geometry lets all excited atoms see the same laser field and thereby simplifies the shape of the ionization saturation curve. Lifetimes and collision studies can be made with the same apparatus.

Nd-YAG laser or by an excimer laser such as XeCl or XeF, about 15-40 mJ can be obtained near the ionization threshold. Thus, the time delayed laser can have a photon fluence 2 4 x 10'' photon/cm'. The number of ions produced per unit length will be given by where alis the photoionization cross section at the photon frequency in question, 4, is the photon fluence at beam axis for the time-delayed laser, N, is the number of excited atoms produced per unit length by the exciting lasers, y is the decay rate of the excited state, and T D is the time delay. Near threshold it will usually be possible to make al+,>> 1 so that N, = N,e-Y7o. Once N, is measured as a function of the output of the exciting lasers, a study of Nl versus should permit measurements of uI to 20% accuracy over a wide range of photon wavelengths. The number of ions produced is sufficiently large that accurate measurements can be made with a parallel plate ionization chamber equipped with low noise amplifiers.

+,,

APPLICATIONS O F RIS

25 1

To illustrate the type of study that can be carried out, suppose that the [ 3 / The 2 ] ~ state ) in question is excited state is 11) = 1 5 ~ ~ ( ~ P ~ / ~ ) 6in~ Xe. believed to have a lifetime of -40 nsec, and for photon energies near the ionization threshold, none of the states populated by spontaneous decay from ( I ) can be ionized by one photon. Thus, as 7D is increased, an exponential decay of N, will be observed, which is a measure of the lifetime of 1 1 ) . Furthermore, after -100 nsec, 11 ) will have decayed to 15p5(('P3,2)6s[3/21,) or 1 5 ~ ~ ( ~ P , , ~ ) 6 ~ [The 3 / 2 J] ~=) .2 state is metastable, and at low pressures its decay rate can be neglected. The J = I state has a lifetime of -5 nsec, but due to radiation trapping (Holstein, 1947, 1951; Payne et a/.. 1974) it effectively lives several microseconds. To determine branching ratios, the ionizing laser can be tuned to resonance between the J = 2 level and one of the 5p5np( n = 7,8) sublevels, which is then excited and also ionized. Thus, atoms in the original J = 2 level are selectively ionized. In this way, one determines the initial population of 1 I ) on an absolute basis, its total decay rate, and finally the absolute population of t h e J = 2 level as a function of time. In Section II1,D we describe how the population observed for the J = 1 level can also be interpreted. Consequently, we see that branching ratios may be determined. At higher pressures the same type of techniques could be used to study processes such as associative ionization, collisional redistribution, and Penning ionization. By selectively ionizing the J = 2 level, information will be obtained concerning the ionization cross section of a higher excited state. Stebbings et a/. (1973), Dunning and Stebbings (1974a,b), and Rundel et al. (1975) at Rice University have combined lasers and metastable inert gas beam techniques to obtain information on the photoionization of some excited states in He, Ar, Kr, and Xe. More recently, Bokor et a/. (1980) have studied the photoionization cross section of an excited state of Kr at a single wavelength. Some recent theoretical predictions of Duzy and Hyman (1980) are given in Figs. 5-7. For other theoretical results, see, for example, Pindzola (1981) and the references contained in Dunning and Stebbings (1974a)and in Duzy and Hyman (1980). In some relatively early studies, Hurst et al. (1975) demonstrated RIS of He(2 IS), and from saturation they inferred a photoionization cross section for He(3 lP), which was close to that of Dunning and Stebbings (1974a) when corrected for MJ changing collisions due to resonance broadening effects (Hurst et a/., 1975). Payne et a/. (1975) carried out a detailed kinetic study of collisional effects on He(2 IS) by studying the ionization as a function of the time delay between excitation and laser detection. In this study, selective detection was obtained by using a long pulse length laser to equilibrate the He(2 IS) and He(3 'P) levels with associative ionization proceeding out of

252

M . G . Payne et al.

FIG.5. Photoionization cross sections versus wavelength for excited s states of inert gases: (---) Ne(2p53s);(---) Ar(3p34s);(-) Kr(4$5s);(- . - . -)Xe(596s). The cross sections are averaged over the rnultiplet and are taken from Duzy and Hyman (1980).

the He(3 'P) and He(3 ID) levels, the latter of which was populated by collisional redistribution. In this way about 80% of the He(2 IS) population can be ionized for n > 1018/cm3with a relatively low-power dye laser, which needs only to saturate a one-photon discrete-discrete transition.

D. STUDIES OF RESONANCE RADIATION TRAPPING I N INERT GASES The trapping of resonance radiation in inert gases was mentioned earlier + l)s[1/2, 3/21,) can as a mechanism by which the states lnp5(2P~,za~2)(n have a system lifetime of several microseconds even though the Einstein coefficient for transitions to the ground state yields a lifetime of a few nanoseconds. The mechanism for the long system lifetime can be understood to some degree from the Einstein relations. Einstein applied the principle of detailed balance (i.e., the condition for thermal equilibrium) to the absorption and emission of radiation associated with a line transition of an atom. Let u J w ) be the absorption cross section at frequency w of the atom as modified by collisional broadening due to other particles, as(w)

253

APPLICATIONS OF RIS !0-'6

E

I

I

480

380

I

I

1

10-49

10-20

580

280

(80

80

h(nrn)

Fic;. 6. Photoionization cross sections versus wavelength for excited p states of inert gases: (---) Ne(2p'3p); (---) Ar(3p54p); (-) Kr(4pVp);(- . - -)Xe(Sp'6p). The cross sections are averaged over the multiplet and are from Duzy and Hyman (1980).

the stimulated emission cross section at frequency w as modified by linebroadening effects, P( w) the frequency distribution function for spontaneous emission, and r,,,the rate of spontaneous emission. The number of absorptions per unit time per unit volume and per unit frequency interval is U ( w ) c N l ~ . J w ) / h . wwhere , U ( w ) is the Planck function and Nl is the concentration of atoms in the lower state. The number of emissions per unit time per unit volume and per unit frequency interval is C/(w)cN,a,(w)/hw + N , , P ( w ) T , , ,where N u is the concentration of atoms in the upper state. Equating these rates and solving for U ( w ) , we obtain

In thermal equilibrium, if w is much closer thankT t o the resonance,N,/N, = ( g l / g , )exp(hw/kT). Using the latter relation and requiring that U ( w )be

M . G . Payne et al.

254 to+ 5 2

k0'7

5

F-

580

480

280

380

I80

80

X(nrn)

FIL. 7. Photoionization cross sections versus wavelength for excited d states of inert gases: (---) Ne(2p3d); (---) Ar(3pj3d); (-) Kr(4P4d); (-.-,-) Xe(5pjSd). The cross sections are averaged over the multiplet and are from Duzy and Hyman (1980).

given by the Planck distribution, we get a , ( 0)= (Ru / R d

4W )

as( W) = (1 ', ,i A 2 / 8 ~ ) PW ()

(53) (54)

Under conditions of radiation trapping at pressures where collisions which mix MJ levels occur many times in the trapping time and nearly all of the atoms are in the ground state with a Maxwellian velocity distribution, it is reasonable that any given atom will absorb and emit with the same line shape on the average as would one of the fantastically rare atoms excited by thermal effects in an optically thick cavity at thermal equilibrium with the same temperature and pressure. This result also pertains to conditions of radiation trapping where the trapping is so strong that the radiation is nearly isotropic. Thus, Eqs. (53) and (54) can be applied under conditions of strong radiation trapping. On integrating Eq. (54) over the resonance,

APPLICATIONS OF RIS

which is assumed to be narrow so that A preciably,

1ad4 d

0

=

=

255

2 n c / w does not change a p

(ru*,a/8n

(55)

If the width of the resonance is rR,then the value of csat resonance If rU,, = 5 x 10X/sec,rH= lO"'/sec, and A. = is u,(wo) ru,lAi/(8nr,J. 10-j cm, we get us(oo) = 2 x lo-'' cm2. Thus, if the absorber concentration is n = lOI7/cm3,the mean free path (for s -+ p transitions) is 1 = 5 x lo-" cm. Consequently, whenever the photon is emitted near line center, it hardly goes anywhere. It is only on the rare occasions when the photon is emitted on the line wing and w - wo is so large that the value of u a ( w ) is thousands of times smaller than the energy actually moves very far without absorption. The key is that nearly all of the atoms are resonant absorbers. In the next several paragraphs we shall discuss the line shape problem in more detail, and it will be seen that the value of rRused above is reasonable for n = 10'7/cmy. We shall first try to give a rather qualitative idea of what determines the linewidth as a function of pressure. The resonance line broadening problem has been treated by a large number of workers. To give an intuitive picture, we shall rely mostly on the work of Holstein et al. (1952). We consider two atoms: one in the ground state Ig) and the other in an excited state ( e ) ,which has dipole-allowed transitions to the ground state, separated by a distance R. If R >> extent of atomic wave functions, the individual Coulomb interactions between the nucleus and electrons of one atom and those of the other can be expanded in a multipole series. Thus, the atom-atom interaction is (for neutral atoms)

-

=

(eZ/P)

2

i I.ir

[xi, xi2 + yi,yi2 - 2zi,zi21

+ higher order multipole terms

(56)

The leading R -:j (i.e., dipole-dipole) term has no nonzero diagonal matrix elements. However, if we consider a collision between the atoms with a large impact parameter b and allow for the possibility that energy might be transferred between them, we write ( + ( t ) )= adr)lg)lle)z+ al(t)le)llg)z; then if atom 2 is originally excited, we have a,(-:) = 1 and al(-m) = 0. The coefficients ad?)and al(t)are determined by H l + ( t ) ) = ifial+(t)>/at with k = h,,+ hzo+ V ( 0 , where hloand hz0 are the Hamiltonians of atoms 1 and 2, respectively, at infinite separation. The term V ( t ) is given by Eq. (56) with R R(t) = ( b 2 + u z f a ) l " , where the relative motion is assumed to be along a straight line with speed u and impact parameter b. The treatment can be made more general by allowing for more states with different M J . The treatment of Holstein derives the q ( t ) at large b by

-

M . G . Payne el al.

256

perturbation theory, and an energy transfer probability is found to be (for J = I + J = 0 transitions)

P(b) = $e.tf2/m2w$b4v2

(57)

where m is the mass of the electron, hw, is the energy difference between the ground and excited states, andf is the oscillator strength of the transition. Equation (57) is used at smaller b until P(b,) = 1, where the approximation is becoming questionable. It is assumed that for b < bo, the energy is transferred back and forth many times so that on the average P(b) = 1for b < b,. With this form for P(b),a rate of energy transfer, r T , can be derived and it is found that

rT= fiAiru,ln/87r2 = 0.022A$I'u,In

(58)

where nearly all atoms are in the ground state and have a concentration n, ru,lis the transition rate from upper to lower state, and A, is the resonant wavelength for transition from upper to lower state. A large fraction of the energy transfers results in a change in M,, so that after a few such collisions any orientation effects associated with the original excitation are lost. Correspondingly, the autocorrelation function of the polarizability of the medium should decay in a time which is close to 1 / r, . The Fourier transform of this autocorrelation function gives the emission line shape; since A w A t 1, one would expect a width of - r T . Berman and Lamb (1969) have treated the line shape problem in great detail for J = 1 + J = 0 transitions. Neglecting Doppler broadening, they find that the frequency distribution function for spontaneous emission, under conditions when the resonance-broadened linewidth Tc >> natural width, is given by

-

where rc= 0.0229 AiT,,p, which is close to the value of rT obtained from the rather qualitative arguments given above. By Eqs. (531, (54), and (59),

According to Eq. (60) the cross section at line center is ua(wo) = I/n A,, corresponding to a mean free path 1 = A, when n A! is sufficiently large. For Eqs. (59) and (60) to be applied to the radiation trapping problem, we must consider the validity of Eq. (60) on the wings of the line. To investigate this question, we note that if, when w - wo is large compared with the inverse time of collision, the impact-broadening theory which led to Eq. (59) is no longer valid. However, when w - w, is this large,

257

APPLICATIONS OF RIS

statistical-broadening theory applies. One can find Born-Oppenheimer type adiabatic states using H = Hl0 + A,, + P, and some of these molecular states are repulsive while others are attractive. The repulsive and attractive states lead to a symmetric statistical broadening about line center. On averaging the attractive potentials, one gets AE(R) = - ( 3 / 3 2 . i r 3 ) ~ r . , ~ ( A 0 / ~with ) ~ , the average repulsive term having equal magnitude. On calculating P ( w ) by statistical theory, one obtains P ( w ) = (A/.ir)T,/(w where A is close to unity. This means that Eqs. (59) and (60) will frequently be found to be valid as much as 20-30 A from line center, i.e., Eqs. (59) and (60) are close to being universal line shapes until lw - woI becomes so large that valence forces come into play. Finally, it is important to consider the effect of Doppler broadening on the line shape and width. The line shape of a Doppler-broadened line is Gaussian, while the pressure-broadening effects discussed above lead to a Lorentzian line shape. Typically the resonance lines of the heavy inert gases exhibit a Doppler width T,, --- ( v ) w o / c -- lO"'/sec at T -- 300 K, where ( u ) is the mean speed of the atoms. Since Tr 10-Hn( n is in cm-" and Tc is in sec-') for n < 10"/cm3 the core of the line has a Gaussian line shape, and the width of the line is dominated by the Doppler width. However, since a Lorentzian line shape drops off much slower than a Gaussian profile away from line center, the wings of the line are Lorentzian in character, and so any absorption and emission at many Doppler widths from line center is still given by Eqs. (59) and (60) under conditions where the trapping is strong. We note that at line center a3(wo) = 3r,,&/ (877rD)= 6 x cm' and even at n 2 IO1,j/cm3,the mean free path is S1OP3 cm. Thus, in order to have a mean free path of the same order of magnitude as the dimensions of a gas cell, the photon must be emitted so far on the wings that w - wo >>To and we can employ a J w ) -3A%G.,lTc/[8~2(w- w,)']. Using Tc = fih:T,,,,n/87?, we see that the mean free path for I >> A. is given by I = Ao(w - wOl2/( fi r:). Thus, for transport of resonance radiation over distances -1 to become important, the photons must be emitted at

-

lW

-

wo~ >

3 1 / 4 r c a>> rD

The onset of radiation transport which is dominated by self broadening should occur when n is large enough so that

Tc > 3TD(xo/l)lr2= 10X/sec where we use 1 = 1 cm and A. = 10-j cm. Since Tc = 10-'n, the pressure-broadened regime should occur for n L lOI6/cm3. We then expect that for n z 1016/cm3the radiation transport which is observed

258

M . G . Payne et al.

should start to be independent of temperature and of concentration. The concentration independence has been observed up to n = 10’s/cms (Payne et al., 1974). To further validate the qualitative discussion given here, the reader will want to compare a qualitative estimate (based on the above physical picture) of the time for transport of photons out to distances I from the point of excitation with the more rigorous results of the next paragraph. We estimate for the rate of passage of resonant photons out of the surface of a sphere of radius 1 centered about the point of excitation: rE=T,,,,F,, where F,= probability of the photon being emitted further from line center than lw - wOln= 31/41‘c(1/AJ1/2. Thus, F, = ( 2 T c / n ) / l w- wOIm = (2/.rr3”4)(A0/l)”2.We have as an estimate r,

-

r,,*,(Ao/o

Under the conditions of strongly trapped resonance radiation where Eqs. (59) and (60) would apply, the emission spectrum of an atom does not depend on where it is located. This led Holstein to write the following integrodifferential equation for the time evolution of the concentration of resonantly excited atoms N ( r , t ) : a N r , t ) / a t = S(r, t ) - r,,.lN(r,t ) +

r,,,,Id3r’G(lr- r’l)N(r‘,t )

(61)

where G ( 3 ) = -( 1 / 4 ~ 9 atla% )

T(&) =

d w ~ ( w exp(-n(+,(w)%) )

(62)

With our assumptions, T ( 3 is just the probability of a photon emitted at % traveling at least a distance 24 before absorption. G ( 3 )is the probability

of a photon emitted at 3 = 0 being absorbed in unit volume a distance 3 from the point of emission. S(r, t ) is a source term representing the rate of production per unit volume of excited atoms at r and t. When 24 is not extremely small, we can evaluate T(%) as (3 >> A”):

T(%) = ( n 3 / 2 / 3”J)(A,/%)”~ (63) The Holstein theory has been studied by observing the time decay of resonance radiation escaping through a window of a gas cell to a detector (Payne et ul., 1974: Holstein et ul., 1952; Mitchell and Zemansky, 1934). However, really detailed studies have not been made. We shall now describe how the transport of radiation can be studied in great detail. We have seen in Section 1II.B how a line source of loHatoms/cm can be + l)p[l/2l0), then after produced. If we choose I I ) = InpY2P112,312)(n 50- 100 nsec nearly all of this population will decay to the resonance states Inp5(’P112,312)(n + l ) ~ [ 1 / 2 . 3 / 2 ]),~ which in LS coupling are referred to as the IP, (for the 2Pllpion core) and the 3P1(for the 2P3,2ion core) resonance

APPLICATIONS OF RIS

259

FIG.8. Experimental arrangement for studies of the trapping of resonance radiation in inert gases. The source laser produces a line of excited atoms by two-photon excitation. Spontaneous emission leads to a tine of atoms in the 'PI or 3P1resonance states in -100 nsec. The detector laser is time delayed relative to the source laser and determines the concentration at a distance po from the line source by selectively ionizing the resonance state. The detection can either be by two-photon stepwise ionization or at pressures above a few Torr by tuning a laser between the resonance state and a high-lying p state, which may lead to associative ionization.

states. After this period of time nearly all of the resonance population will be localized along a line. To study the resonance radiation transport, we use a second small parallel laser beam displaced from the first by a distance p o and time delayed relative to the first by a time t , as shown in Fig. 8. The second laser is tuned to selectively ionize the chosen resonance state. If p o is small compared with the size of the ionization chamber, the presence of the walls will not become important until a rather large time interval has passed. We have used the same technique to study the diffusion of alkali atoms in inert gases (Hurstet ul., 1978). Here the experimental comparison is to be made with a solution to Eq. (61) with the initial conditions corresponding to a line distribution of atoms at = 0. Such a solution yields the density of excited atoms at time t and at a distance p from the axis, following the introduction of N , excited atoms per unit length at t = 0 and p = 0 and was found by Payne and Cook (1970) to be

M. G. Payne et al.

260

Y

H(Y)

Y

H(Y)

Y

H(Y)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

o.oo00 0.0237 0.0430 0.0583 0.0702 0.0791 0.0855 0.08% 0.0920 0.0929

1 .o 1.1 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

0.0925 0.0912 0.0891 0.0832 0.0761 0.0685 0.0610 0.0538 0.0471 0.0410 0.0357

3.0 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 6.6

0.0309 0.0232 0.0174 0.0131 0.0099 0.0076 0.0059 0.0046 0.0036 0.0029

~ ( y =)

jOm

exp(-uy) du

u3.1~(u2)

= (6/y4)[l - (210/y3)

+

*

. .],

y

>>

I

m

= T-’

n=o

sin(3n~r/4)[r(l+ n/4)]’(2y)’’/n!

(65)

A tabulation of H(y) is given in Table I. Clearly, the ionization signal due to the time-delayed laser beam, which is tuned to detect resonantly excited atoms at distance po from the initial line source, will yield a signal proportional to N ( po, t), where t is the time delay. The size of both laser beams must, of course, be small compared with po. It should also be possible to measure N ( p, t ) at lower pressures where the complete redistribution assumption has been questioned (Payne et af., 1974). If Eq. (64)can be completely justified forP k 3 Torr, it will become possible to study collision effects on resonance states. In the presence of a volume collisional destruction rate A, Eq. (64)is modified to N ( p , t ) = exp(-At)No( p, t )

(66)

where A is the collisional destruction rate and No(p, t ) is just the quantity given by Eq. (64) for the excited atom density in the absence of collisional destruction. To measure A it might be preferable to use a concentric laser beam configuration like that shown in Fig. 4. In an alternative experiment, if the state I I ) is chosen to be Inp3(2P3,2) (n + l)p[3/2I2), a large part of the radiative decay will lead to the 3P2metastable state; if the parallel beam is tuned to selectively ionize the metastable state, precise studies can be made of metastable diffusion in complete analogy with previous alkali metal diffusion studies (Hurst et al., 1978).

APPLICATIONS OF FUS

26 1

E. ESTIMATES OF THREE-PHOTON EXCITATION RATESI N He

AND

Ne

There is no commercial laser scheme for achieving unit ionization probabilities for He and Ne. However, if the initial excitation is a three-photon process, we believe that ionization probabilities > 10+ can be achieved over volumes > cm3 with relatively little broadening or shift due to the ac Stark shifts induced by lasers which exhibit amplitude fluctuations. We suggest the following excitation schemes: He(ls'S,,)

+

2fiwp (A = 1576.3A)+ hw, ( A Ne(2p"SJ

- 2260A) +He(ls2p

+ 3hw ( A = 2208 A)

4

'Po)

Ne(2prs IP,)

Rough estimates of rates for three-photon excitation can be based on relatively crude estimates of f13(r). The type of closure arguments used in Section III,B gives

where WL = (wL1 + wL2 + wL3)/3 and K = I if all three photons are absorbed from the same laser beam, K = 3 if two of the photons are absorbed from one laser beam and the other photon is absorbed from a = 6 if the three photons come from different second laser beam, and laser beams. For inert gases, (01821I ) == L3e3a$ where L3 typically satisfies 1 s L3 =s 100. Using ( G - wo - w,)(G - wo - 2aJ == 2 x 10J2/secZ,eaoE,(r)/2h= 10HZir2(r) with Z , ( t ) in W/cmZ, we get 3i3(t)= 5 x 1 0 - ~ K L , [ I , ( ~ ) Z ~ ( ~ ) Z ~ ( ~ ) I ' "

(68)

-

The rate for three-photon excitation R(3) (@)s/( rI2\J, where TI,\Iis the largest laser bandwidth and s is 3! if all three photons are from the same beam, 2! if two are from the same beam, and l ! if all are from different sec): beams. We get for the excitation probability (using 7 == 5 x --P, = R T 2.5 x 10-17TK'~3sz,(r)i2(r)z3(t)/r,,,

-

- lo-'.i@L"~--l(t)z2(r)z3(r)/ rLLI

(69) Since for both excitation schemes another photon from any laser present puts one in the ionization continuum and since high-power densities are anticipated, any atom excited will generally be ionized. Anticipated ac Stark shifts will be on the order of (1- 10) rad/sec, and a similar magnitude is expected for the level width of 11) due to ionization. Illis the power density in W/cm2 for the most powerful laser. If either the mean ac Stark then a rough estimate of P, is shift or the ionization rate is larger than rLh,, obtained by replacing r,, by the larger of the ac Stark shift and the ionization rate. It is relatively easy to get 10 mJ in a 10 nsec pulse and

262

M. G . Payne et al.

- 0 . 5 cm-' linewidth at 1576.3 A. In the 220CL2300 A region, 1-4 mJ can sec and a linewidth -1-2 cm-I. Thus, if the be obtained with T = 4 x beam area is cm2, we can get 10"' W/cm'. The largest of the bandwidths, the ionization rate, and the ac Stark shift would probably be -2 x lO"/sec. Therefore, P , 5 x 10-7g2L$s. Correspondingly, an ionization line width 5 2 cm-' and an ionization probability > over a volume of -lo-" cm3 is reasonable. We have for the number of ions: N , z 10-"n, where n is the gas concentration. In the presence of a similar power density at a wavelength within a few angstrom units of an allowed discrete-discrete resonance between 11 ) and another excited state, ac Stark shifts nearly a hundred times larger could be observed. However, the latter situation does not arise here. Any attempt to focus tightly and achieve Zi(t) = 10l2 W/cm2 would almost certainly lead to a much broadened and shifted line due to the large ionization rate and the fluctuating ac Stark shifts. 2 :

IV. Experiments Combining RIS and Pulsed Supersonic Nozzle Jet Beams A. INTRODUCTION The use of lasers to drive electronic transitions in molecules, as a tool for sensitive and selective detection, is complicated by collisional effects and by the fact that in the gas or vapor phase a high degree of rotational excitation is usually present. Even when only one rovibrational level is occupied, the absorption spectrum of a molecule is far more dense than that of an atom due to the nuclear degrees of freedom. When the population is spead over many rotational levels, the absorption shows relatively few sharp structures until the linewidth of the exciting laser is made so narrow that transitions between pairs of individual rovibrational levels can be resolved. In I2 or NO2 at room temperature and concentrations s101,'3/cm3, the laser linewidth must be 50.01 A to permit selective absorption by a single rovibrational population. With such narrow bandwidths, only a small fraction of the molecules absorb, i.e., those in the selected state, and the absorption is correspondingly weak. Nearly all of the problems described above are absent when one considers the laser spectroscopy of a supersonic nozzle jet beam with the molecules in question being seeded into an inert gas which acts as the carrier. In such a beam, rotational temperatures 5 3 K are fairly typical and the spread in speeds is correspondingly small. In recent years extensive use

263

APPLICATIONS OF RIS

3

6100

'K

-_c

6000 WAVELENGTH

5900

5800

( a1

FIG.9. The absorption spectrum of a molecule is greatly simplified by reducing the rotational temperature. Here the total fluorescence following laser excitation is shown as a function of wavelength for three rotational temperatures. The upper curve was obtained for Torr of NO, in a cell at room temperature. The middle result was obtained when pure NO, was allowed to undergo supersonic expansion, resulting in a rotational temperature -30 K, and the lower spectrum was obtained when I% NO was seeded into He, resulting in a cooled beam with a 3 K rotational temperature. In the last two cases the laser linewidth was -0.5 A, and in the lower curve individual rovibrational states are easily resolved.

has been made of such beams in studies of both electronic (Smalley et af., 1974, 1975), vibrational (Grant ef af., 1977; Coggiola el af., 1977), and rotational transitions (Gentry and Giese, 1977a,b). Pulsed supersonic nozzles have been developed (Gentry and Giese, 1978) which give TR < 3 K while simultaneously yielding low translational temperatures and concentrations -10'5/cm3 in pulses with lengths -10-5 sec and repetition rates -10 Hz. The main advantage of the pulsed jet is that it permits large concentrations to be present when pulsed lasers are fired without requiring enormous gas pumping capabilities to maintain low partial pressures of uncooled gas. As an example (Smalley et af., 1974) of the advantages to be gained in using a supersonic nozzle, Figure 9 shows total fluorescence from NOz as a function of the wavelength of the laser under three con-

264

M . G . Payne et al.

ditions. In the upper scan, NOz was in a static cell at Torr and T 2- 300 K. The middle scan shows fluorescence when pure NOs is fed through a continuous wave (cw) supersonic nozzle resulting in a rotational temperature -30 K, while the bottom scan was obtained when 1% NOz was seeded into He passing through a cw supersonic nozzle, resulting in a beam having a rotational temperature -3 K. With low rotational temperatures, even the relatively broad band laser easily picks out individual rovibrational levels and leads to absorption by a large fraction of the population (Chen et al., 1979). During the last three years commercial laser systems have been developed which provide energies per pulse > 1 mJ, pulse lengths -5-10 nsec, linewidths <0.06 A, and repetition rates -10 Hz at nearly all wavelengths in the range 2200 A s A c 7500 A. Through much of this region, similar outputs can be obtained with a linewidth A. One type of system with these characteristics is a Nd-YAG system in which various harmonics of the 1.06 pm radiation are used to pump one or more dye lasers with part of the fundamental remaining available for special uses such as frequency summing with the doubled output of one of the dye lasers. In addition, recent developments (see, for example, Wallenstein, 1980) on third harmonic generation and four-wave mixing in mixtures of inert gases have A can shown how peak power outputs -10 W with linewidths be obtained over much of the wavelength region between 1050 A s A s 2000 A. It is clear that with modest focusing this source can be used to carry out saturated one-photon excitation in a large number of molecules. The impact of this UV light source on spectroscopy and molecular detection should be great. The > 1 mJ quantities of relatively narrow linewidth UV radiation available from Nd-Yag and XeCl or XeF pumped dye lasers and nonlinear optics are sufficient to saturate many two-photon transitions in molecules (Teets er al., 1976; Bernheim er al., 1977) over volumes -10-5-10-2 cm3, depending on the strength of the transition, the precise energy per pulse, the linewidth, and the pulse length. In some cases more efficient two-photon excitation can be achieved by combining tunable light with the output of a fixed-frequency light source such as the He-F, laser to drive processes such as AB + hwl + hw, + AB*. Nearly any simple molecule with reasonably favorable Franck-Condon factors between its ground vibrational state and a bound excited state could be excited with high selectivity and efficiency in a pulsed nozzle jet beam. As other powerful, narrow bandwidth sources become available, work based on combining pulsed nozzle beams with pulsed lasers should become extensive. The spectroscopic selectivity of excitation in a nozzle beam can be further enhanced by using a tunable IR laser to saturate a vibrational

265

APPLICATIONS OF RIS

transition between the D = 0 and v = I levels in the ground electronic state while simultaneously driving an electronic transition between this u = I , J = I level and a rovibrational level of an electronically excited state. This IR step is particularly helpful in eliminating interference from the uncooled residual gas component and in introducing an enormous degree of isotopic selectivity. A pulsed IR source with tunability in the ranges 4.6 pm s A < 5.4 pm and 9 pm s A s 1 1 pm is achievable with commercially available COz lasers and a doubling crystal. With proper regulation of the power density and a linewidth 50.02 cm-I, anharmonic effects permit the leveling of the u = 0 and u = 1 populations without producing any appreciable population for u > I . Parametric oscillators can also be used to obtain tunable, pulsed, and narrow linewidth light in the IR region. Resonance ionization spectroscopy cannot easily be applied to the detection of some types of molecules. Sometimes it happens that a selective excitation can be carried out, but the Franck-Condon factors present difficulties in promoting further excitation and ionization. However, there are many molecules where a relatively simple scheme can be used. In Section IV,B we see how pulsed lasers, pulsed nozzle jets, mass spectrometers, and time-gated ion detectors can be combined to provide enormous spectroscopic selectivity and sensitivity for simple molecules.

B. ISOTOPICALLY SELECTIVE DETECTION OF CO To illustrate with a particular example how the highly sensitive and selective detection of molecules in a pulse nozzle jet beam can be achieved, we now describe a study which we are currently pursuing. The goal of the study is to see if I4C detection using laser spectroscopic techniques can be competitive with accelerator techniques (see, for example, Purser et ul., 1979). To be competitive, a laser technique should be capable of detecting the presence of - 1 o j I*C160molecules in about a 0.04 gm (i.e., -loz1 molecules) sample of CO. Thus, the technique must discriminate against 10l6times as many l2CI6Oand - l O I 3 times as many I2CIHO molecules while detecting single ions from at least I% of the ‘‘C’60. While it is certainly not obvious that this can be done, we believe that there is a reasonable chance of success. The isotopic effects in CO can be approximately described for small u and J by the following equation (see, for example, Herzberg, 1950):

&,./ = Te +

weo(Po/P)”2(u +

+ B(po/p)JiJ+

1)

4)

- XeoweO(Vo/P)(u

+

&I2 (70)

M . G . Pavne et al.

266

where T, is the term energy of the electronic state in question, we0 is the vibrational constant for l 2 C160, xeow,ois an anharmonic vibrational constant for 1zC160, B, is the rotational constant for I2CI6O,p is the reduced mass of the isotope in question, po is the reduced mass of " C " 0 , J is the rotational quantum number, and u is the vibrational quantum number. Calculations based on spectroscopic constants for 1 2 C 1 6and 0 Eq. (70) suggest the following resonance steps for the RIS of 14C'60starting from X lX+(O, 0) where the numbers in parentheses indicate the values of u and .I. respectively:

+ hw,, (A

=

4.858prn)-X1I:'(l,l)

X L X + ( l , l+) h a t 2 ( A

=

1594.8 A)-A'lI(O,O)

+ nul3 (A

=

4139.8 A)-

XIC'(O,O)

A'n(0,O)

(71)

B'I+(l,I)

B ' I ' ( l , l ) + h w , , ( A s 4500A) - X 2 1 + ( 0 , 0 ) + e + KE

It is relatively easy to get >50 mJ output at A = 4.858 p m by frequency doubling a commercially available, tunable C 0 2laser with 1 J/pulse and bandwidth -0.02 cm-I. Only a few kW/cm2 peak power density is needed at A = 4139.8 A , and the ionizing laser is easily provided. Thus, all the light sources except that at A = 1594.8 A are easily available with characteristics which should allow 10% ionization (by multiple passing of the laser beams) of the "ClhO molecules in a pulsed jet beam having a pulse length of 10 psec and a beam velocity - 1 0 cm/sec. The situation is also encouraging for the X + A transition using 1594.8 A radiation. Figure 10 shows the potential energy curves for the molecular states in question, and it is clear that Franck-Condon factors are very favorable. Since the lifetime of A'lI(0, 0) is --1O-' sec, we believe that a peak power of a few watts with a linewidth -4 x lo-' A of 5 nsec duration would be sufficient to provide a scheme for ionizing -5% of all of the T?60present in a nozzle jet pulse. A laser system is now available which gives -60 mJ of transform-limited bandwidth light at 2496 A (Hawkins et al., 1980). It may be that 2496 A radiation, which is near resonance with a twophoton transition in Xe, can be mixed with tunable red light in Xe to give radiation at 2w (A -- 2496 A) - w ' ( A = 5700 A), thereby giving the 1594.8 A radiation. When the scheme in Eq. (71) is used, the only powerful light is the ionizing laser. By carefully choosing its wavelength to stay away from any near resonances and keeping power densities as low as possible, we estimate that the ionization probability of 1 4 C 1 6can 0 be made >0.5 with that of 12CIf'0 being -lo-". Table I1 gives energies of various molecular states of CO for the various isotopes. A close examination of this table, together with the fact that the power densities of all lasers is to be kept as small as possible and still be consistent with ionizing several percent of the I 4 C"'0. indicates whv the selectivity is excellent.

-

-

APPLICATIONS OF RIS

267

(x 4 0 ~ )

4 .O

I

1

08

1.2

0.9

0.8

0.7

0.6

-I

5

0.5

I

W

0.4

0.3

0.2

0.4

0

0

0.4

4.6

R(B)

FIG. 10. Potential curves involved in the isotopically selective detection of ''C'60. The molecules are rotationally cooled and the v = 1 level of X 'I' is excited by 4.85 pm radiation. While the v = 0 and I' = 1 levels are leveled by the IR radiation, light at -1594 A promotes transitions to the v = 0 vibrational level of A In. A third narrow bandwidth light source drives transitions between A ' I 1 and the I' = 1 level of B 'Z', which is ionized.

A schematic of the apparatus which will be used on CO is shown in Fig. 11. The ionization occurs near the entrance slit of a mass spectrometer which can discriminate against almost all of the most abundant 12C'60+ background ions. Each pulse will contain lo''$CO molecules so that only one pulse in a thousand contains a "C"'0 molecule. Thus, an ionization event for ''CI6O may only occur every 10" laser pulses. Since the ions are all produced within -5 nsec and their time of arrival at the ion detector is highly defined, the detector is only gated on for about 1 psec for each

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M. G . Payne et al.

268

TABLE I1

ENERGY LEVELS IN DIFFERENT ISOTOPICSPECIES O F

co

.En,,(cm-') Isotopic species

n

J

IZCIRO

0 0 0 1 1

0 1 2 0 1

0 0 0 1 1 1

0 1 2 0

lacin0

2 2 2

izciwo

0 0 0

1081.61 1085.47 3224.88 3228.74 1036.65 1040.20

1

3091.65 3095.19

2 0 1 2

5 122.24 5125.78 5132.88

0 1

1

2 0

1

1

1

2 0 1

2 2

x 'I' of co

1055.62 1059.29 3147.87 3151.54

A

In

of CO

65830.08 65833.28 65839.72 67309.49 67312.71

B 'Z+ of CO 87997.8 88001.7 90080.11

65798.82 65801.77 65807.69 67218.12 67221.08 67226.99

87954.04 87957.65 87964.85 89950.80 89954.40 89961.60 91919.63

658 12.OO 658 15.07 6582 1.20 67256.69 67759.75

87972.49 87976.23 90005.36 9oO09.10

90016.57 92009.28 92013.01

pulse. Thus, with present detectors, one spurious count would occur every lo6 shots of the laser. The same apparatus could be used to do highly sensitive and selective analyses of pollutants in air if the air samples were seeded into He in about 1% quantities. In many cases the first electronic transition would be driven by a two-photon process. The I4CI6Odetection could also work with the A = 1594.8 8, radiation being replaced by the absorption of two photons at A = 3189.6 8, (Bernheim et al., 1977). However, high-power densities at this relatively short wavelength may lead to somewhat higher probabilities of ionization for I2CI6Oand 12C'x0.The apparatus in Fig. 11 can also be equipped with photon detectors and used in spectroscopic studies where one wants not only accurate energy differences between levels but also lifetimes of the states and Franck-Condon factors for the transitions.

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SEEDED AND PULSED NOZZLE JET\

CHANNELTRON

FIG.11. Schematic diagram for the isotopically selective detection of I4Cl6O.

c.

CROSSED-BEAM STUDIES OF LASER-INDUCED COLLISIONS

By laser-induced collisions we refer here to processes of the type A(i) + B(i) + h w - , A(F) + B(F) That is, two atoms (or molecules) collide in the presence of a relatively powerful laser field, and the energy of the laser photons very nearly coincides with .@ + €; - €? - @. Clearly, energy is conserved if a photon is absorbed and a second-order mechanism for a process where the electronic state of each atom changes is provided by the possibility of photon absorption into a virtual state, followed by energy transfer of a portion of the absorbed energy to the other atom. A two-state theory for such processes has been given by Gudzenko and Yakovlenko (1972). Several experiments have been carried out which verify that a two-state theory is capable of predicting thermal rates for the process reasonably well (Harris rt d.,1976; Falcone rt a/., 1977; Lynch et al., 1978; Cahuzak and Toschek, 1978). A rather general solution to the two-state model has been given by Payne rt cil. (1979). The two-state model ignores (or at best, averages out) orientation effects in dealing with the collision, and all of the past experiments have determined a thermal rate constant. Far more detail could be obtained if one could carry out such experiments in a crossed-beam study where results could be obtained at fixed relative velocities and the products of the collision subjected to state-selective detection. The first elements of a theory which deals in detail with orientation effects have been given by

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Berman (1980). In addition, by studying angular distributions of the products, one could go a long way toward separating the relative importance of the long-range collisions mediated by dipole-dipole interactions from close collisions where valence forces come into play. Crossed-beam collision studies have usually been carried out with continuous beams having -10"/cm3 particles in each beam. With crossedpulsed nozzle jet beams, one can have as many as 10'5/cm3particles in each beam, and the spread in speeds of the particles is extremely small (about a few percentage points of the mean speed). In addition, with molecular species there is extreme cooling of the internal degrees of freedom, leading to a far cleaner situation where it is possible to do selectivestate preparation and detection with a pulsed laser. We shall see that the increased concentrations easily make up for the duty cycle effects encountered when one tries to use pulsed lasers with continuous beams, and make possible data collection on experiments involving lasers for product detection or state preparation on the same time scales as one encounters in experiments on continuous beams without pulsed lasers. We believe that an enormous number of studies which would have been impossible a few years ago can now be made with a combination of techniques similar to that which we shall describe in connection with the study of laserinduced collisions. Among the possibilities is the state-selective detection of neutral products and the preparation of populations of excited species for study. We consider crossed-pulsed nozzle jet beams of Ar and Xe. Each beam can be profiled by using RIS to selectively ionize the Ar or Xe and studying the ion signal as a function of where the lasers are focused and the time delay between triggering the pulsed jet and firing the laser. Accurate information on the velocity distribution can be obtained by two-photon excitation of [np"('P3,,)(n + l)p[3/2],), which, through spontaneous emission, produces a T, metastable population with high efficiency. Time-of-flight information on the metastables is a very accurate indicator of the velocity distribution. We suggest making a population of about lo8 atoms in the Xe excited state 15p5(('P3,2)6p[ I /?I0 ) by two-photon excitation as described in Section II1,B. After a 10 nsec delay, a laser is fired with photons having an energy equal to the difference between the excitation energies of the 'PI state of Ar and the 15p5(2P3,2)6p[1/2]o) state of Xe. These photons have energies -1.9 eV and are relatively near one-photon resonance with some of the higher ns states of Xe. Thus, the excited Xe atoms absorb a photon to virtually excite higher ns or nd states, with energy transfers occurring simultaneously to return Xe to its ground state, with the Ar atom being left in the 'P,state. An energy level diagram of the relevant states is shown in Fig. 12. Even at relatively low-power densities,

APPLICATIONS OF RIS 96,00(

-

27 1

Arl

95,50(

-

I

"E I

> 0 a W w z

95,OOC f i w ( X=6544.3A) c

80,OOC

Fiti. 12.

-

Energy level diagram for the laser-induced collision

1/210) + fiw (-6544 Xe(5p5CP3,2)6p[

A) + Ar(3p6)

Xe(5p6) + Ar(3p5eP1,*)4s[1/211)

the linewidth of the laser-induced process is -IO"/sec because of timeof-collision effects. Correspondingly, the probability that the simultaneous presence of a second pulsed laser to selectively ionize the Ar('P3 reaction product will affect its production is negligible if the power density is less than lo9W/cm2. With such a scheme, the probability of the process occurring for one of the Xe* atoms is P = n.,va~, where n A ris the argon concentration, vis the relative velocity, a i s the cross section, and T ~ the S pulse length. To study the process, one could operate with only one detected ion every lo4pulses with properly gated ion detectors. Thus, with lox excited Xe atoms, P could be as small as lo-'*. If n A r= 10'5/cm3, sec, we must have v -- 10; cm/sec, and T = 5 x

M. G. Payne et al.

272 10-12

u>-=

nArw

10-12

x lo5 x 5 x

=

2x

cmz

Since essentially all of the product can be detected and time-of-flight information enables one to deal with scattered light and any background ionization, the process can be studied very easily, even at relatively low power densities and greatly reduced concentrations. Since cross sections as large as cm2are expected for large power densities, it should be relatively easy to study angular distributions and velocity distributions for the products. By using Doppler-free excitation of the Xe and very narrow bandwidth light for detection, it is also possible to analyze populations of fine and hyperfine levels as a function of the state of polarization of the laser beams. Good absolute cross sections should be obtainable because the production of the Xe excited state can be measured by simply choosing the time-delayed laser so as to selectively ionize the population. Beam overlap and power densities can be chosen so as to yield absolute measurements of either reactants or products (Hurst et d ,1979). ACKNOWLEDGMENTS The research for this article was sponsored by the Office of Health and Environmental Research, U.S. Department of Energy under Contract W-7405-eng-26 with the Union Carbide Corporation.

REFERENCES Allen, L., and Eberly, J. H. (1975). “Optical Resonance and Two-Level Atoms.” Wiley, New York. Bardsley, J. N., and Biondi, M. A. (1970). Adv. At. Mol. Phys. 6, 53. Berman, P. R. (1980). Phys. Rev. A 22, 1848. Berman, P. R., and Lamb, W. E., Jr. (1%9). Phys. Rev. 187, 221. Bernheim, R. A., Kittrell, C., and Veirs, D. K. (1977). Chem. Phys. Lett. 51, 325. Bokor, J., Zavelovich, J., and Rhodes, C. K. (1980). Phys. Rev. A 21, 1453. Cahuzak, P., and Toschek, P. E. (1978). Phys. Rev. Lett. 40, 1087. Chen, C. H., Kramer, S. D., Clark, D. W., and Payne, M. G. (1979). Chem. Phys. Lett. 65, 419. Chen, C. H., Hurst, G. S., and Payne, M. G. (1980). Chem. Phys. Lett. 75, 473. Chen, C. H., Hurst, G. S., and Payne, M. G. (1981). To be published in “Progress in Atomic Spectroscopy-Part C” (H. F. Beyer and H. Kleinpoppen, eds.). Plenum, New York and London. Coggiola, M. J., Schulz, P. A., Lee, Y.T., and Shen, Y . R. (1977). Phys. Rev. Lett. 38, 17. Dunning, F. B., and Stebbings, R. F. (1974a). Phys. Rev. Lett. 32, 1286. Dunning, F. B., and Stebbings, R. F. (1974b). Phys. Rev. A 9, 2378. Duzy, C., and Hyman, H. A. (1980). Phys. Rev. A 22, 1878.

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Falcone, R. W., Green, W. R., White, J. C., Young, J. F., and Harris, S. E. (1977). Phys. R E V .A 15, 1333. Gentry, W. R., and Giese, C. F. (1977a). J. Chem. Phys. 67, 5389. Gentry, W. R., and Giese, C. F. (1977b). Phys. Rev. Lett. 39, 1259. Gentry, W. R., and Giese, C. F. (1978). Rev. Sci. Instrum. 49, 595. Grant, E . R., Coggiola, M. J., Lee, Y. T., Schulz, P. A. Sudbo, Aa. S., and Shen, Y. R. (1977). Chem. Phys. Lett. 52, 595. Grynberg, G., Biraben, F., Massini, M., and Cagnac, B. (1976). Phys. Rev. Lett. 37, 283. Gudzenko, L. I., and Yakovlenko, S. I. (1972). Zh. Eksp. Teor. Fiz. 62, 1686; Sov. Ph,ys.JETP (Engl. Trans/.) 35, 877 (1972). Harris, S. E., Falcone, R. W.,Green, W. R., Lidow, D. B., White, J. C., and Young, J. F. (1976). I n “Tunable Lasers and Applications” (A. Mooradian, T. Jaeger, and P. Stoketh, eds.), p. 193. Springer-Verlag, Berlin and New York. Hawkins, R. T., Egger, H., Bokor, J., and Rhodes, C. K. (1980). Appl. Phys. Lerr. 36, 391. Herzberg, G. (1950). “Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules.” Van Nostrand-Reinhold, Princeton, New Jersey. Holstein, T. (1947). Phys. Rev. 72, 1212. Holstein, T. (1951). Phys. Rev. 83, 1159. Holstein, T., Alpert, D., and McCoubrey, A. 0. (1952). Phys. Rev. 85, 985. Hurst, G. S., Payne, M. G., Nayfeh, M. H., Judish, J. P., and Wagner, E. B. (1975) Phys. R E V .Lett. 35, 82. Hurst, G . S., Allman, S . L., Payne, M. G., and Whitaker, T. J. (1978). Chem. Phys. Lett. 60, 150. Hurst, G. S., Payne, M. G., Kramer, S. D., and Young, J. P. (1979). Rev. Mod. Phys. 51, 767. Hurst, G . S., Payne, M. G., Kramer, S. D., and Chen, C. H. (1980). Phys. Toddy 33, No. 9, 24. Lambropoulos, P. (1976). Adv. At. Mol. Phys. 12, 87. Lecompte, C., Mainfray, G., Manus, C., and Sanchez, F. (1974). Phys. Rev. Lett. 32. 265. Lidow, D. B., Falcone, R. W., Young, J. F., and Harris, S. E. (1976). Phys. Rev. Lett. 36, 462. Lynch, L. J . , Lukasik, J., Young, J. F., and Harris, S. E. (1978). Phys. Rev. Lett. 40, 1493. Mainfray, G. (1978).In “Multiphoton Processes” (J. H. Eberly and P. Lambropoulos, eds.), p. 253. Wiley, New York. Messiah, A. (1966). “Quantum Mechanics,” p. 722. North-Holland Publ., Amsterdam. Mitchell, A. C. G., and Zemansky, M. W. (1934). “Resonance Radiation and Excited Atoms.” Macmillan, New York. Payne, M. G., and Cook, J. D. (1970). Phys. Rev. A 2, 1238. Payne, M.G., Talmage, J. E., Hurst, G. S., and Wagner, E. B. (1974). Phys. Rev. A 9, 1050. Payne, M. G., Hurst, G. S., Nayfeh, M. H., Judish, J. P., Chen, C. H., Wagner, E. B., and Young, J. P. (1975). Phys. Rev. Lett. 35, 1154. Payne, M. G., Anderson, V. E., and Turner, J. E. (1979). Phys. Rev. A 20, 1032. Payne, M. G., Chen, C. H., Hurst, G. S., Kramer, S. D., Garrett, W. R., and Pindzola, M. (1981). Chc,m. Phps. Lett. 79, 142. Phelps, A. V. (1955). Phys. Retz. 99, 1307. Phelps, A. V. (1959). Phys. Re\*. 114, 1011. Phelps, A. V., and Molnar, J. P. (1953). Phys. Rev. 89, 1202. Pindzola, M. (1981). Phys. Rev. A 23, 201. Purser, K. H., Litherland, A. E., and Gove, H. E. (1979). Nucl. Instrum. Methods 162,637. Rundel, R. D., Dunning, F. B., Goldwire, H. C., and Stebbings, R. F. (1975). J. Opt. Soc. A m . 65, 628.

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Sargent, M., 111, Scully, M. O., and Lamb, W. E., Jr. (1974). “Laser Physics.” AddisonWesley, Reading, Massachusetts. Smalley, R. E., Rarnakrishna, B. L., Levy, D. H., and Wharton, L. (1974).J. Chem. Phys. 61, 4363. Smalley, R. E., Wharton, L., and Levy, D. H. (1975). J . Chem. Phys. 63, 4977. Stebbings, R. F., Dunning, F. B., Tittel, F. K., and Rundel, R. D. (1973). Phys. Rev. Lett. 30, 815. Teets, R., Eckstein, J., and Hansch, T. W. (1977). Phys. Rev. Lett. 38, 760. Thonnard, N., and Hurst, G . S. (1972). Phys. Rev. A 5 , 1 1 10. Vasilenko, L. S., Chebotaev, V. P., and Shishaev, A. V. (1970). JETP Lett. (Engl. Trans/.) 12, 113. Wallenstein, R. (1980). “Generation of Narrow-band Tunable Coherent VUV Radiation.” Presented at the Eleventh International Quantum Electronics Conference, Boston, Massachusetts, June 23-26. Zoller, P. (1979). Phys. Rev. A 19, 1151. Zoller, P., and Lambropoulos, P. (1980). J . Phys. B 13, 69.

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 17

INNER-SHELL VACANCY PRODUCTION IN ION-ATOM COLLISIONS C . D . LIN and PATRICK RICHARD Department of Physics Kansas State University Manhattan. Kansas 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Measurements of Inelastic Ion-Atom Collisions A. General Considerations . . . . . . , , . , , , , , . , B . Single K-Vacancy Production . . . , , . . . . . . . . C. Double K-Vacancy Production . . . . . . . . . . . . 111. Theory of Inelastic Ion-Atom Collisions , . . . . . . . . A. General Considerations . , . , . . . . . . . . . , . . B. Low-Velocity Region: MO Model . . . . . . . . . . . C. Direct Excitation and Ionization at High Velocities . . . D. Electron Capture at Intermediate Velocities . . . . . . E. High-Energy Charge Transfer . . . . . . . . . . . . . F. Atomic Models . . . . . . . . . . . . . . . . . . . I v. Comparison of Theories and Experiments . . , , , , . . . A. Total Single-Electron Transfer Cross Sections . . . . . . B . Differential Cross Sections . . . . . . . . . . . . . . C. Double K-Electron Transfer . . . . . . . . . . . . . V. Concluding Remarks . . . . . . . . , . . . , . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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

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

.

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

275 277 277 281 299 303 303 305 307 3 10 319 324 326 327 336 343 347 348

I. Introduction This work is devoted to a review and summary of the understanding of the process of K-shell vacancy production in ion-atom collisions. In order that this subject be put in perspective, it is necessary to discuss the role of K-shell electron transfer along with excitation and ionization processes as the mechanisms of removing the K-shell electrons. Recent experimental data and theoretical models are summarized and evaluated over a broad range of collision parameters to assess our understanding of this subject. The subject of ionization has been reviewed in several works. Electron capture at high velocities has also been reviewed recently from the theo275 Copyright 01981 by Academic Press, Inc.

AU rights of reproduction in any form reserved. ISBN 0-12003817-X

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retical viewpoint. In this article we present ionization in a one-to-one context with the K-shell electron transfer process for collisions from the very asymmetric systems, where ionization is the dominant K-vacancy production mechanism, to near-symmetric systems, where K-electron transfer is the dominant mechanism. Experimental data from heavy-ion collisions are presented in parallel to selective data from the simple collisions such as H + + H and H + + He. This is important in view that many theoretical models have been developed originally for simple collision systems. Because of the range of parameters that can be controlled in heavy-ion collisions, heavy-ion collision data provide new grounds for testing these and some new theoretical models. Only ab initio theories and theories which treat electrons quantum mechanically will be discussed. Several semiempirical and classical theories have had definite impact on this subject over the years, but our understanding in this subject acquired in recent years has made some of these semiempirical and classical theories obsolete. The second section presents a review of the experimental methods of measuring K-shell vacancy production cross sections and measurements, where excitation, ionization, and charge transfer cross sections can be uniquely or reasonably identified. The results of these measurements over the variation of projectile charge q, projectile velocity v, projectile nuclear charge Z p, and target nuclear charge ZTare presented for the systematics where available. Some recent results for the impact parameter dependence and the double K-shell electron transfer cross sections are also presented. In the third section the status of the theory of electron transfer in ion-atom collisions is reviewed. In the last few years we have witnessed many theoretical models proposed for electron transfer processes. Our review emphasizes the evaluation of the basic assumptions and the assessment of the region of validity of these models. Little attention is paid to excitation and ionization theories except the general formulations. This area in general lacks new theories, especially in the regime of nearsymmetric collisions. The fourth section presents comparisons between theories discussed in the third section and experiments discussed in the second section for the case of target K shell to projectile K-shell electron transfer. As the collision parameters are varied, we can see the limitation of various theoretical models. Total and differential cross sections are compared, and in general we can conclude that existing theories explain most of the experimental data despite some discrepancies in finer details. The last section is a summary of the status of our understanding of this subject and the direction of some future theoretical and experimental

INNER-SHELL VACANCY PRODUCTION

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work, which will require more detailed knowledge of the collision dynamics.

11. Experimental Measurements of Inelastic Ion-Atom Collisions A. GENERAL CONSIDERATIONS Inelastic ion-atom collisions are studied experimentally through the measurement of the inelastic processes of ionization, excitation and electron transfer. In this work we shall restrict ourselves to those inelastic collisions in which K-shell vacancies are produced. Measurements of total and differential cross sections for K-vacancy production as a function of the available parameters of projectile charge Z,, charge state q , and velocity u , and of target charge Zr have shed much light on the competing K-vacancy production processes as well as the models or theories for describing these processes. Some of the interesting problems under study at present are the relative importance of target K-shell to projectile K-shell, L-shell, etc., electron transfer and direct Coulomb ionization as the projectile q and projectile Z , to target Z, are varied; the energy dependence of target K-shell to projectile K-shell electron transfer from lowscaled velocities (A = u/v, ratio of projectile velocity to average orbital electron velocity) to high-scaled velocities; the impact parameter, b , dependence of K-vacancy production by K-shell to K-shell (K-K) electron transfer and the importance of double K-vacancy production as a function of projectile Z, and velocity. In this section, the experimental results from these types of studies will be reviewed with the idea in mind of giving some of the general features of K-vacancy production, with an emphasis on light collision systems and systems for which the initial atomic states of the collision are specified with precision and for which there is at least partial knowledge of the final state. Our sources of knowledge about inelastic collisions can be classified into a few general categories: (1) Detection of the K X-rays or K-Auger electrons emitted by the excited or ion. This gives a measure of the primary excitation cross section # in collisions where there is only single K-vacancy production assuming that the fluorescence yield, o,,,is known (e.g., uK=crk/%). In cases

atom

where multiple ionization is important, then neither the exact final state nor the exact final charge state can be specified in general. In this case

278

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some knowledge of the final distribution of states and the effective fluorescence yield must be obtained as will be discussed in the next section. (2) Direct detection of the ejected electrons. This gives an unambiguous measure of K-shell processes only for collision systems which have no electrons except in the K shell. ( 3 ) Charge-state analysis of collision products. This measures electron gain or loss but gives no information on the final state of excitation. This method can be pertinent to K-vacancy production for projectiles with zero, one, or two K-shell electrons. For example, final-state charge analysis of the collision Nee+(ls)ions plus H going to Ne'"+plus H can yield the K-shell ionization cross section. (4) Charge-state analysis of collision products plus coincidence with emitted K-shell X rays or Auger electrons. This type of experiment can select inner-shell charge-transfer processes. For example, charge transfer from target K-shell to all bound states of the projectile can be measured by observing the one electron gain beam in coincidence with the target K-shell deexcitation. ( 5 ) Inelastic energy loss of the projectile. It is worthwhile to review briefly the methods of measuring various types of collision systems in which the inelastic processes of K-shell ionization, K-shell excitation, and K-shell electron transfer are observed and to examine to what extent the processes can be identified. I . Light Symmetric and Near-Symmetric Collision Systems

K-vacancy production in light symmetric and near-symmetric collision systems are fundamental to all of atomic collision physics since it addresses the basic three-body (ion-electron-ion) scattering problem. These systems are defined as those containing only K-shell electrons in the initial channel (e.g., H+ + H, H+ + He, He+ + H, Li+ + H). All inelastic atomic scattering processes are thus K-shell-vacancy producing events. The three K-vacancy production cross sections can be measured uniquely for these systems as illustrated in the following example: (1) K-shell ionization of He in H + + He + H + + He+ + e collisions can be measured by observation of the He recoils, the ejected electrons (Rudd and Madison, 1976; Selov'eret al., 1962; deHeer et al., 1%6; Toburen, 1979), or the H + energy loss spectrum (Park et al., 1976, 1978a, b); (2) K-shell excitation of He in Ht + He + Hf + He(lsn1) collisions can be measured by observing either the H+ energy loss spectrum or the deexcitation photons from excited He; and (3) K-shell electron transfer from the K-shell of He to some level of H+ in H+ + He -+ H + Het collisions can be measured by observing the neutralized beam with a charge state selector (Allision,

INNER-SHELL VACANCY PRODUCTION

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1958; Bratton et al., 1977). The projectile velocity dependence of the total cross sections and the angular distribution of reactions are the systematics which provide crucial tests of the theoretical scattering models for light symmetric collisions systems (see Sections 111 and IV). 2 . Light Asymmetric Collision Systems

Since an exact theory for describing inelastic K-shell excitations does not exist, it is not sufficient to critique the theory by comparing with only H + + H measurements. The logical extension of the study of light symmetric collision systems is the study of the Z-dependence of the cross sections for light asymmetric collision systems. These systematics can be accomplished by studying either the target Z-dependence in light projectile asymmetric systems (e.g., H + + Ne, Mg, . . . , Ar) or the projectile Z-dependence in light target asymmetric systems (e.g., A;+ + He for q = Z , Z - 1, and Z - 2). In the former case of light projectile asymmetric systems, referred to as light on heavy collisions, the K-vacancy production mechanisms can be viewed as an effective three-body problem. The excitation of the K-electron is caused by the motion of the two nuclear charges moving along a semiclassical trajectory. By assuming that the other electrons in the atom or ions are to shield the nuclei only, the excitation of the electron is reduced to solving the motion of the single active electron in a time-varying potential field quantum mechanically. In these collision systems K X rays or K-Auger electrons are emitted in all three inelastic channels due to the presence of the outer-shell electrons in the target. For example, (1) H + + Ne + H + + Ne(1s-') + e (K-shell ionization) gives rise to K, X rays and KLL-Auger electrons; (2) H + + Ne +. H + + Ne(ls-'nl) n > 2 (K-shell excitation) gives rise to K,, K,, etc., X rays and KLL- and KLM-Auger electrons; and (3) H + + Ne +. Ne( ls-') + H (K-shell electron capture) gives rise to K, X rays and KLLAuger electrons. This latter process of K-shell electron transfer can be measured by performing a coincidence between the H projectiles and the emitted K, X-ray or KLL-Auger electrons, as demonstrated in a series of experiments by Macdonald et al. (19741, Cocke et al. (1976, 19771, and Rodbroet al. (1979). The main contribution to the target K X-ray or Auger electron production is K-shell ionization with very little K-shell excitation due to the filled outer shells of the neutral atom and very little K-shell electron capture. K-shell ionization for light projectile asymmetric systems has been thoroughly investigated and found to yield cross sections in good agreement with direct K-shell Coulomb ionization such as the plane wave Born approximation, PWBA (Merzbacher and Lewis, 1958), semiclassical approximation, SCA (Bang and Hansteen, 1959), and the binary encounter approximation, BEA (Garcia, 1971, and references therein).

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Light target asymmetric systems, referred to as heavy on light collisions, allows one to study the effect of outer-shell electrons on K-shell ionization and K-shell excitation by observing the K-shell vacancy production as a function of its incident charge state. These types of systems have recently been investigated by three separate methods: (1) Total X-ray production. For example, by observing the projectile K X rays in AQzf+ He collisions, one can obtain the cross section for K-shell excitation plus K-shell ionization for q = 1 to Z - 2; the cross section for K-shell excitation plus capture to excited states of the projectile for q = 2 - 1 (one-electron ion); and the cross section for capture to excited states of the projectile for q = Z (bare ion) (Hopkins et al., 1976b; Guffey et al., 1977; Schiebel et al., 1977; Doyle er al., 1978). By comparing the cross sections for various incident projectile charge states it is possible in some instances to deduce separate cross sections for K-shell ionization and K-shell excitation. (2) High-resolutionK X-ray studies. One can obtain the relative cross sections for capture, ionization, and excitation, by observing the different final states in high-resolution X-ray spectra for each incident projectile charge state as demonstrated in a series of experiments by Macdonald et al. (1973),Hopkinset ul. (1974), and Tawaraet al. (1978a, 1979a). Combining the results of this method with the one above gives the unique cross sections for the three possible processes. ( 3 ) Electron gain and electron loss. The total charge-exchange cross sections for bare ions on helium gives the total capture to all states of the bare projectile. These can be compared to the total X-ray production cross section, which gives the total capture to all but the 1s and 2s states and states which cascade to the 2s state of the projectile (Dillingham, 1980). The K-shell ionization cross section of the projectile by helium atoms can be determined for the case of one-electron ions by observing the electron loss (Dillingham, 1980). These measurements avoid the question of fluorescence yield as required in the X-ray production from light on heavy collisions discussed above. The ideal experiment that also avoids the question of electron screening for heavy on light collisions would be, for example, one-electron ions on bare H or He, which to date has not been successfully negotiated.

3 . Heavy Asymmetric to Near-Symmetric Collision Systems

The final category of collisions is heavy ion-atom systems. These systems vary from the heavy near-symmetric collisions to the heavy asymmetric collisions and can be identified as those systems in which the target

INNER-SHELL VACANCY PRODUCTION

28 1

has K, L, and possibly higher shells of electrons and the projectile has a Zp greater than 2 and q from 1 for the singly charged ion to Z,for the bare ions. Nearly all of the information about inelastic K-shell processes in these collisions comes from measurement of the K X-ray or K-Auger electrons. Inelastic energy loss of the projectile has been used to determine the probability for K-shell excitation for a few selected systems (Fastrup, 1975; Fastruper al., 1971, 1974). In this general type of collision it is difficult if not impossible to keep track of all electrons or of all the final excited ionic states. For this reason it is difficult to go much beyond identifying from which collision partner the K X rays and the K-Auger electrons originate. A large number of measurements has been made using these collision systems and compared with one or more proposed theoretical models. Direct Coulomb ionization of the K shell has been identified as the basic mechanism for the production of K-shell vacancies for asymmetric collisions (Z, << Z,).At high velocities, the magnitude and energy dependences of the measured cross sections are in good accord with the PWBA and SCA calculations; in the low-velocity region, a semiempirical model based upon corrections to the PWBA and SCA theories (Basbas et al., 1973, 1978; Brandt and Lapicki, 1974; Andersen et al., 1976) has been applied in an attempt to extend the region of validity of the model. If the collision partners have Z, ZTand the projectiles carry no K-shell vacancy into the collision, the production of K-shell vacancies in the projectile and target atoms are understood in terms of the idea of electron promotion in the molecular orbital model (Fano and Lichten, 1965) for collisions at small velocities. Briggs and Macek (1972) were able to show that the K vacancies made in symmetric collisions could be explained by electron promotion via rotational coupling between 2pa and 2 p r molecular orbitals at low-scaled velocities. Briggs and Taulbjerg (1975, 1976) and Taulbjerg et al. (1976) extended the calculations to asymmetric systems. At medium- and high-scaled velocities the target K-vacancy production can be thought of as target K-shell ionization and target K-shell electron transfer to projectile K, L, and higher shells.

-

B. SINGLE K-VACANCY PRODUCTION

In this section a presentation of sample data will be given to describe the important aspects of the total and differential cross sections for K-shell electron ionization, excitation, and charge transfer as a function of the projectiles’ v ,q , Zpand the targets’ Z,. These parameter variations form a basis for evaluating theoretical models for K-vacancy production.

282

C . D . Lin and Patrick Richard

I . Projectile Charge-State Dependence of K X-Ray and K-Auger Electron Production Cross Sections One of the most revealing, unique, and sometimes bothersome aspects of ion-atom collisions is the projectile charge state dependence of K X-ray and K-Auger electron production cross sections as observed in early experiments by Macdonald et al. (1972), Winters et al. (1973), Mowat et al. (1972, 1973, 19741, and Hopkins et al. (1976a). The variation of the K X-ray or K-Auger electron production cross section with incident projectile charge state is due to the dependence of the target and projectile final-state distribution on the electron configuration of the incident projectile. In order to fully appreciate this problem, it is necessary to discuss K-fluorescence yields. K-vacancy production cross sections, uK,that are determined from K ! , respecX-ray or K-Auger electron production cross sections, a# and a tively, require a knowledge of the K-shell fluorescence yield %(j ) of the final state, j . The K-vacancy production cross section is given by a" = a # / w K (j ) = a:/(1

- w"(

j ) ) = a # + at

The state fluorescence yield w K ( j )is given by the ratio r#(j)/FfoTAL( j), where r#(j ) and Go,,, ( j ) are the X-ray and total decay rates, respectively, of the K vacancy of the state j . These rates can be calculated with some degree of accuracy using an appropriate atomic model. If more than one final state with a K vacancy is formed in the collision, then the K-vacancy production cross section is still given by g K

= ax" + (TK A

(2)

But it is now necessary to define an average fluorescence yield as &.c

+

= a$/(a$

( r : ,

= a#/aK

(3)

which is a measurable quantity. If a#varies with projectile charge, 4 , we then must write 6 x 4 ) = 4\.(q)aK(4)

(4)

The projectile charge-state dependence of ak(q) can thus have two extreme situations. In one extreme aK(q)is nearly independent of q and the variation is due primarily to wX,(q). This situation will be dominant when wKis much less than one and when multiple outer-shell ionization is large. The other extreme is when wIv(q) is nearly independent of 4 and the variation of a#(q)is due primarily to aK(q).An example of an extreme variation of wXv(q) with q is the projectile charge-state dependence of the Clq++ Ne collision system depicted in Fig. 1. The measured a# and (r!

283

INNER-SHELL VACANCY PRODUCTION

L

_1_ _1 _, _ -I- - -l -

1

0

l

I

I

5

I

I

10

I

I

1

1

I

15

9 FIG.1. The Ne K X-ray production cross sections, u k , the Ne K-shell Auger production cross sections, a f i ,and the Ne fluorescence yields (w.!, /w$) as afunction of incident projectile charge state, q , for Cia+ + Ne collisions: (a) CP++ Ne; (b) Ne fluorescence yield. The X-ray data at 1.4 MeV/amu (solid points) and the Auger data are from Burchet a / . (1974); the X-ray data at 1.2 MeV/amu (open points) are from Mowat el a / . (1972, 1974).

cross sections for 4 between 5 and 15 from Burch et al. (1974) and Mowat et al. (1972, 1974) are shown in Fig. la. One can see that m K changes by a factor of 21, whereas w i v , obtained from use of Eq. (4), changes by a factor of 7. Two other examples of the variation of cr; with 4 are given in Fig. 2. In Fig. 2a the Fq++ Ar cross section taken from Macdonalder al. (1972) shows a sharp rise for q = 8 and 9. This sharp rise occurs at the onset of K-shell vacancies in the projectile and is interpreted as the onset of the target K-shell to projectile K-shell charge transfer channel (Halpern and Law, 1973), which is closed for 4 = 4-6, where the cross section is

C.D . Lin and Patrick Richard

284

Fq' + A r 6

z 0

-t

0

W

cn cn cn

0 LT 0

1I N

E

9 I

I

I

Cuq'

+

I

I

Ar

V

0

I 1

/

FIG.2. The Ar K X-ray production cross sections for (a) 1.9 MeV Fa++ Ar (Macdonald 1972) and (b) 1 MeV/amu Cuq++ Ar (Schiebel and Doyle, 1978) as a function of incident projectile charge state, q . er a / . ,

relatively constant. The slight rise for q = 7 has been explained by the presence of a small fraction of metastable (ls2s) 3S ions in the beam (Schiebel et al., 1977). In Fig. 2b the Cuq++ Ar cross section taken from Schiebel and Doyle (1978) shows a gradual rise forq = 10-18 and a sharp rise for q = 19-22. This change of slope occurs at the onset of L-shell vacancies in the projectile and is thus explained as the onset of target K-shell to projectile L-shell charge transfer. In the two types of examples given here the mechanisms could be interpreted either in terms of direct K- to K- and K- to L-shell charge transfer or in terms of the molecular orbital model through the 2 p . l r - 2 ~rotational ~ coupling andor 2pu-lsu vacancy sharing; both will explain qualitatively the observed variation with q . The projectile charge-state dependences of target K X-ray cross sections, u!, discussed so far are for gas targets (see Figs. 1 and 2). Similar

INNER-SHELL VACANCY PRODUCTION

285

30

0

q FIG.3. (a) The Si K X-ray production cross sections for 19 MeV Fq++ SiH, (gas) and Fq++ Si (foil). The fluorescence yields for the gas and the foil as a function of projectile q obtained from Eq. (6) using the relative K, satellite intensities from high-resolution X-ray measurements. (c) The resultant K-vacancy cross sections using Eq. (4) (Tawara et al., 1979b).

dependences have been observed for solid targets (Gray et af., 1976, 1979; McDaniel et af., 1977; Gardner et af., 1977, 1979; Schmiedekamp et af., 1979; Tawara et af., 1978b). For sufficiently thin targets K X-ray cross sections can be related to K-vacancy prcduction cross sections using Eq. (4). The reader is referred to a review by Gray (1980), where the target thickness dependence of K X-ray yields and their relation to target and projectile vacancy production cross sections are described in detail. In only one case has there been a direct comparison of and ogv for one element in gaseous and solid form (Tawara et af., 1979b). Figure 3 shows this comparison for Fq++ Si. The interesting result is that the oxvvalues are constant for the solid target and not for the gas target. This result is presumably due to the rapid relaxation of the outer-shell vacancies prior ~

286

C . D . Lin and Patrick Richard

to filling of the K-vacancy in the solid target. The quantities C T ~and wk\ vary in such a way as to keep the vacancy production cross sections the same within errors for the gas and solid. This result adds a great deal of confidence to the use of thin solid target data as well as gas target data in comparisons with single-collision theory treatments of K-vacancy production. The quantity wk, defined in eq. (4) can be obtained by other methods. A theoretical expression for w i \ using Eq. (4) is wk\. = P "(j)W"(J?

(5)

where the P"(J) are the probabilities of forming the K vacancy in atomic statesj. If a total of N atoms result in a K vacancy and nj are in statej, then P"(J) = nJN. The P h ( j ) values cannot be calculated in general without prior knowledge of the reaction mechanism. The distribution of j states usually consists of one or more electron configurations of the form (ls)(2s)'(2p)'(3s)' * . . . Each configuration consists of several states called terms, and their X-ray decay forms multiplets. If the terms are formed statistically, then the values ofPKG)within the term are known, and a multican be calculated theoretically, using Eq. plet fluorescence yield, GULT, (4). Bhalla (1975a, b), Chen and Crasemann (1974, 1975), Ahmed er al. (1976), and Tunnel1 er al. (1979) have performed several calculations of whG) and q$ulfor several elements over a large range of the parameters i, j , k , etc. In order to use these calculations to obtain wk\. ,one must either be able to measure in high resolution the relative strengths of all electron configurations or to calculate the relative strengths. If the relative strengths are also assumed to be statistical, then an average multiplet yield, w h ( n ) , corresponding to the average over all (ls)(2s)'(2p) configurations with equal i +j = n values (configurations with an equal number of L electrons) can be calculated. The latter groups of electron configurations can be resolved and are usually referred to as the I& satellite X rays. A semiempirical value of wk\. can thus be obtained from the relation l/wi,=

2n [R,/wK(n)]

where Rn are the relative K, X-ray satellite intensities. If the relative Auger electron satellites A , are measured, then the expression for obtaining w i v is

2n A,d'(n)

(7)

The Si w i v values given in Fig. 3 for SiH,gas and Si foil targets were obtained by the use of Eq. (6). The R, values were obtained from highresolution X-ray spectra and the multiplet fluorescence yields taken from

INNER-SHELL VACANCY PRODUCTION

287

FIG.4. The average K fluorescence yield, wfi, , for Ti obtained from the method of Eq. (6). The open and closed symbols indicate calculations using the measured R. values for Sc and Ti, respectively. Measurements for H , He, Li, C, and CI beams at energies between 0.5 and 4.5 MeV/amu are indicated. The solid curves are the results of scaling R , with energy. The dashed line is an average value of wl;, for projectiles C through CI (Hall et d.,1981).

Tunnel1 et d.(1979). Another example of ok\ values obtained for solid targets by this technique using Eq. (6) is given in Fig. 4.The fluorescence yields of Ti are calculated for projectiles of H through C1 at high energies between 0.5 and 4.5 MeV/amu, using R, values obtained from highresolution X-ray spectra. The w h ( n ) values for Ti are used together with the R, values of Ti and Sc, indicated by the closed and open circles, respectively. The energy dependences given by the solid lines are obtained by scaling the values of R, with energy. The maximum variation in wh\ from H to C1 projectiles is only about 20% for this collision system. The results are taken from Hall er ul. (1981). In the projectile charge-state dependence studies of K-vacancy production cross sections obtained from mk or m k , it is important to be able to remove the q-dependence of the fluorescence yield in order to compare the results with theoretical calculations. Even though highly accurate values of w k , cannot always be obtained, it is relatively straightforward to follow the considerations given in this section to estimate the values of wh, and the uncertainties in ahintroduced by the q-dependence of wk, . The values wk\ are also projectile Z , and E dependent, as illustrated in the above example.

288

C . D . Lin and Patrick Richard K-SHELL IONIZATION

,/'

F"+He 4

, I

I

si K X-RAY PRODUCTION ( b ) AND FLUORESCENCE YIELDS

t ,

Si'q+He

0.5

I

I

3.4

I

0.3

3 3.2 0. I

0

2

4

6

8

10

12

q

FIG.5. (a) F K-shell ionization, OF, and K-shell to L-shell excitation per electron, a&, for I5 MeV Fq++ He as a function of 4. ut increases one order of magnitude, while up decreases by a factor of 3 in going from 4 = 2 to 9 = 8 (Tawara et a / ., 1979a).(b) Si K X-ray production cross section, at and u h for 15 MeV Si" + He as a function of4. A theoretical estimate of utxis subtracted from uh in order to obtain w h (Doyle et al., 1978).

Projectile charge-state-dependent data have also been obtained in heavy on light collisions, where the projectile X rays rather than the target X rays are measured. Total X-ray production cross sections, a! (solid circles) versus q are shown in Fig. 5b for SiQ++ He collisions at 15 MeV (Doyle el al., 1978). The variation of ak with q in this case is due primarily to the variation in wk, withg. It is also expected that part of the increase is due to the increase in the K-shell to L-shell excitation cross section, akx. The contribution due to excitation was calculated and subtracted in order to obtain the fluorescence yields (indicated by solid triangles). A direct measurement of the projectile charge-state dependence of K ionization,

INNER-SHELL VACANCY PRODUCTION

289

a:, and K-shell to L-shell excitation, g 2 h , has been made for 15 MeV Fq++ He collisions by Tawara et al. (1979a) and are presented in Fig. 5a. These cross sections were obtained by high-resolution X-ray measurements in which the K-ionization and K-shell excitation peaks are energetically separated. The K-shell excitation cross section per electron (solid circles) increases by an order of magnitude in going from q = 2 to q = 8. This is the only directly measured charge-state dependence of K-shell excitation known by the authors.

2. Velocity Dependence of K-Vacancy Production

Inelastic ion-atom collisions producing K-shell vacancies have a very pronounced dependence on projectile velocity. The projectile velocity dependence of K-vacancy production gives insight into the regions where different mechanisms may be in effect and delineates regions where processes may be dominant. In this section we shall compare the observed velocity dependences of cross sections for K-shell ionization, target K-shell to projectile electron transfer, target K-shell excitation, or any other possible distinct K-shell process that can be identified. The first sample result is for the light projectile asymmetric system H+ + Ne, which has a Z , / Z , ratio of 0.1. As can be seen in Fig. 6, K-shell ionization of Ne (Woods et al., 1976; Stolterfoht et al., 1973; Schneider and Stolterfoht, 1975) as deduced from total Auger electron measurements is orders of magnitude larger than the target K-shell to projectile charge transfer (Rgdbro et af., 1979) measured by a K-Auger-neutral-H coincidence. The charge transfer is assumed to be primarily to the K shell of H + from binding energy considerations. K-shell ionization has a peak near a scaled velocity of 1.0, while K-shell to K-shell transfer has a peak near a scaled velocity of 0.7. The electron transfer cross section falls off much more rapidly with energy than the K-ionization cross section. The K-shell ionization cross section is described exceedingly well by a PWBA or SCA direct Coulomb ionization calculation. The comparison of the electron transfer data with theory will be given in Section IV. Figure 7 shows the velocity dependence of K-shell ionization and K-K charge transfer over a large energy range for the asymmetric system Fq+ + Ti, which as a Z , / Z , ratio of -0.4(Hallet a/., 1981). K-K charge transfer is taken as the difference in the cross section for bare ion projectiles (Fy+), (Th2, and for projectiles with no K-shell vacancies (e.g., F"), v k Oi.e., , alili = c K 2- ulio.In this asymmetric system the K-shell ionization and K-K charge transfer are comparable in magnitude over the entire energy range studied. The K-K charge transfer reaches a plateau at a scaled velocity of -0.6, whereas the K-shell ionization is still rising at a scaled velocity of 0.8. The K-shell ionization is no longer fit by a simple PWBA or

C . D . Lin and Patrick Richard

290

-

N

5

c

Z

F

0 W

cn cn cn

0 LT 0

5.0

I0.0

10 .

21)

3.0 4.0 5.0

E /A ( MeVIarnu

(b)

(a) The total K-shell vacancy production cross section for Ht + Ne collision (Z,/Z., = 0.1) as a function of energy. Data above 0.4 MeV from Woods ef n / . (1974, 1976) and data below 0.5 MeV from Stolterfoht ef ul. (1973) and Schneider and Stolterfoht (1975). This cross section is essentially the K-shell ionization cross section. (b) The K-K charge transfer obtained from Ne K- Auger-neutral-H coincidences (Rodbro ef d.,1979). Note the different energy scales. The scaled velocities are indicated at the top of each figure. The solid lines are eye guides. FIG.6.

SCA calculation, labeled by (T in Fig. 7. Various theoretical calculations of direct Coulomb ionization to account for Coulomb deflection of the projectile and increased binding of the target K shell by the projectile, labeled a,"" (Basbas et d . , 1973) and ak,,(Basbas et nl., 1978). are given. The calculation attllis the original formulation given by Basbas et nl. (1973), and (T:,~~ is the improved formulation for treating increased binding in close collisions by Basbas et a/. (1978). The curve c ~ l \ includes , ~ ~ the two above effects and the so-called polarization effect due to K ionization in distant collisions, which becomes important at relatively high velocities (Basbas et i l l . , 1978). The data and the latter theory give amazingly good agreement, especially considering the size of the corrections. The increased binding effect brings the theory into accord with the data at the low-scaled velocities, whereas the additional polarization effect brings the theory into accord at the higher scaled velocities. The numerical results of the calculations were obtained from a program provided by G . Basbas (private communication, 1979).

29 1

INNER-SHELL VACANCY PRODUCTION v/v,

v/v,

z

0

I0-

0.0

1.0

2.0

3.0

4.0

5.0 6.0 20

E/A (MeVIamu 1 ( b)

F I G .7. The K-shell ionization (a) and K-K transfer cross sections (b) for F + Ti(Z,,/ = 0.41 as a function of projectile energy (Hall ct t r l . , 1981). The cross sections are cornparable over the entire measured energy range. The calculations are o K= PWBA, UPM = PWBA with bindingandCoulombdeflectioncorrections(Basbas p f c r / . , 1973);ir1\112= PWBA with binding, Coulomb deflection correction, and radial cutoffs (Basbas ct u / . , 1978); and tr:,,,. = PWBA with binding, Coulomb deflection correction and polarization corrections (Basbas C I crl. , 1978). The solid line through the K-K transfer data is an eye guide. The scaled velocity is indicated at the top of the figure.

ZI

Figure 8 depicts the velocity dependence of K-shell ionization and K-K transfer for Siq++ Ti, which has a Z,,/Z.,.ratio of -0.64 (Hallcf ( I / . , 1981). In this more symmetric collision system the K-K transfer is larger than K-shell ionization over the energy range studied. The K-K transfer is relatively flat compared to K-shell ionization. The K-shell ionization curve can no longer be fit very well by the at,5l,calculation. Other processes such as target K-shell to projectile L-shell charge transfer are possibly contributing to the K-shell ionization data. This will be seen more clearly in Section II,B,3, where the Z , dependence of these cross sections will be discussed and a comparison with theory presented. The theory for K-shell to K-shell charge transfer will be discussed in Section 111, and comparison of data will be reserved for Section IV.

292

C . D . Lin and Patrick Richard v/v, Q l2,

1617

,

0;4 9.5

9.6

0:

9.8

~

K -VACANCY PRODUCTION

4

t

A

Si + T i K - K TRANSFER

4

K IONIZATION

'""0

1.0

21)

3.0 4.0 5.0 E(MeV/amu)

6.0 7.0

FIG.8. The K-shell ionization and K-K transfer cross sections for Si + Ti(Z,/Z, = 0.64)as a function of projectile energy (Hall ef a / . , 1981). The K-K transfer dominates over K-shell ionization over the entire measured energy range. The solid lines are eye guides to the data. The scaled velocity is indicated at the top of the figure.

The energy dependence of K-shell ionization has been studied by , electron-loss measurements for one-electron projectiles of C j+, N , 0 '+ and FS+onHe by Dillingham (1980), where the ratios Z,/ZT (here ZTtaken as projectile) are 0.33, 0.29, 0.25, and 0.22, respectively. At the present time this is the only such measurement of the ionization of one-electron heavy ions. Figure 9 depicts the measured energy dependences and the PWBA predictions. Excellent agreement is obtained. The PWBA prediction with Coulomb deflection, increased binding, and polarization correcgive almost the same prediction as the PWBA result c Kfor this tions aFB,, case over the measured energy range. More systematics of this type would be useful in assessing the effects of electron screening, increased binding, etc., in Coulomb ionization. The energy dependence of projectile K-shell excitation in light target asymmetric collisions of F'+ + He and F8++ He have also been measured

293

INNER-SHELL VACANCY PRODUCTION

h

E

0 c b

I"

L

N '

E

0

v

'

t

,

/

/

' , 10

I

20

l

30

l

I

40

1

1

1 50

Projectile Energy (MeV) FIG.9. The K-shell ionization of the projectile ions C5+,N", O", and FXfby neutral He in the energy range 5-40 MeV using electron loss measurements (Dillingham, 1980). The solid curves are the PWBA predictions.

(Tawara et al., 1979c) by the method described at the end of Section II,B, 1, i.e., high-resolution X-ray spectroscopy. The measurements are in good agreement with PWBA calculations, as shown in Fig. 10, by neglecting the effects of the electrons in the helium atom. Notice that the electrons in He can screen the target nucleus as well as ionize or excite the electrons in the projectile. 3. Z,-Dependence of K-Vacancy Production

A very large body of data exists on the Z,-dependence of cross sections for very heavy collision systems ( Z , , ZT > 30) at fairly low-scaled velocities. See, for example, the reviews by Mokler and Folkmann (1978), Leisenet (11. (1980), and Meyerhof and Taulbjerg (1977). This discussion is restricted to low Z systems, where both K-shell ionization and K-K charge transfer have been measured. These types of studies require measurements with at least one-electron heavy-ion beams, and better yet bare ions. For this reason these studies are limited by accelerator consid-

C . D . Lin and Patrick Richard

294

-

1 -

lu

E

F'6, F"-

He K - SHELL EXCITATION CROSS SECTION

-1 -

PWBA, NO SCREENIN

0

1

'"'"0

10

5

15

20

25

30

35

40

ENERGY (MeV) FIG. 10. The K-shell to L-shell excitation cross sections of the projectile ions F6+and Fn+by neutral He in the energy range of 10-40 MeV. PWBA calculations with and without screening of He by its electrons are given for F"+and without screening for F'+ (Tawara et d., 1979~).

1020

'0''

i 2

4

6

8

10

ZP (a)

FIG.11. (a) K-shell ionization of Ne obtained from total K-Auger electron cross sections for projectiles H+,He'+, Li3+(Rodbroet a / . , 1979); N"', F3+(Woodset a/.. 1976): and Ne3+ (K. Holzer and R. Brenn, private communication, 1980) at 0.5 MeVlamu (scaled velocity of (Woodset 0.56). (b) K-K transfer for H', He'+, Li'+ (Rodbroet a / . , 1979); N7+,OH+,and FY+ a/.. 1976) on Ne at 1.7 Meviamu (scaled velocity of 1.03). The three low Z , K-K transfer data are obtained from K-Auger electron-gain beam coincidences, whereas the three high Z,, data are obtained from charge-state dependence measurements. The arrow indicates the position of symmetric collisions. The solid lines are eye guides to the data.

295

INNER-SHELL VACANCY PRODUCTION

erations to projectiles with Z 5 20. For a review of the low-energy lightion collision studies the reader is referred to Stolterfoht (1978). Figure 11 depicts the Zp dependence of K-shell ionization of Ne at 0.5 MeV/amu and K-K charge transfer at 1.7 MeV/amu. A complete set of data at the same velocity does not exist at present for Ne. The K-shell ionization data are taken from total Auger electron production cross sections, whereas the K-K charge transfer is obtained by KAuger-electron-neutral H beam coincidences for Z , = 1, 2, and 3 and by subtracting cross sections for three-electron projectile ions from those for bare projectile ions for Z , , = 7, 8, and 9 (i.e., ml,, = ml,z - m,,o). The K-shell ionization curve is essentially constant from ZI, = 3 (very asymmetric) to Zp = 10 (symmetric). The K-K transfer, on the other hand, rises very steeply over the range of Z,. One can see that this is a general result for scaled velocities 5 1 . 0 by the comparisons of another system ( A ; + + Ti).where both K-shell ionization and K-K charge transfer are taken at the same scaled velocity (Fig. 12). The steeply rising K-K transfer curve crosses the K-shell ionization curve at ZI,/ZT 0.36 (Z,, = 8). From the energy dependences given in Figs. 11 and 12 one can see that the crossover point will be nearly scaled velocity independent. At 4 MeV/amu it occurs at ZIB/ZT 0.41 (Z, = 9) (Hall et NI., 1981).

-

-

loi7 K-SHELL VACANCY PRODUCTION

h

I O2Zo

5

10

z,

15

20

FIG.12. The Z,,-dependence of K - K transfer and K-shell ionization of Ti at 1.75 MeV/ arnu (scaled velocity of0.43) (HaUrt o/., 1981). The two cross sections cross at a Z,,/ZTratio of 0.36. The arrow indicates the position of svrnrnetric collisions.

C . D . Lin and Patrick Richard

296

I O'* K-SHELL VACANCY PRODUCTION

41

10-191

1

0.5 MeVIamu

I

0

t0 W

cn cnl

cn

0 [r 0

0

5

10

15

ZP FIG. 13. The Z,-dependence for K-shell vacancy production in Ti at 0.5 MeV/amu (scaled velocity of 0.24). The K-shell ionization calculations are the same as in Fig. 7. The one additional curve labeled uOBhS is an OBK calculation for target K-shell to projectile L-shell charge transfer scaled by a factor of -15. The data are from Ball er a / . (1981).

The Z,-dependence of K-shell ionization at low velocity has many interesting features, which have been pointed out in several measurements for heavy ions. Figure 13 depicts the situation for 0.5 MeV/amu (scaled velocity 0.24) for A:+ + Ti collisions (Hallet al., 1981). This behavior with Zp is in large contrast to that in Fig. 12 for a scaled velocity 0.43. The low-velocity cross section shows a dramatic upturn above Z, = 7. The Z,-dependences of the theoretical calculations of aK,atB1, atBz,and atBpas defined in the discussion of Fig. 7 are also given in Fig. 13. At these low velocities the upB,calculation is accurate only up to a Zp/Z, of -0.32 (Z, = 7). The large increase in cross section above Z, = 7 is attributed to the onset of target K-shell to projectile L-shell charge transfer. This process may be due to direct K-shell to L-shell charge transfer (solid line in Fig. 13 labeled ao131,s is a scaled first Born approximation for electron capture, or OBK, calculation) or multistep molecular orbital processes in ~ which projectile-L to target-L transfer is followed by 2 p r - 2 ~rotational coupling and followed by 2pa-lsa vacancy sharing (see, for example, Briggs, 1976; Schneider and Stolterfoht, 1979).

INNER-SHELL VACANCY PRODUCTION

1

K-SHELL TO K-SHELL CHARGE TRANSFER

t

\

'0"b

297

1

I

I 02c

Id" 0

5

10

15

20

25

30

35

Z T

FIG. 14. The Z , dependence of K-K transfer and K ionization for F projectiles at 1.7 MeV/amu. The data are from several sources: Ne, Woods er a / . (1976); Si, Tawara et a / . (1978b); Ar, Macdonalder a / . (1972); Ti, V, Cr, Fe, and Co, Schmiedekamper a / . (1979); Cu, Gardner er a / . (1977).

4. Z,-Dependence of K-Vucuncy Production

The target Z-dependence of K-vacancy production where both K-shell ionization and K-K transfer have been measured is scarce in the literature. One such case gathered from several literature sources is for fluorine projectiles at 1.7 MeV/amu. Figure 14 shows the systematics for Z , between 10 and 29 corresponding to a ZP/ZTrange of 0.9-0.31. The cross sections slowly approach a crossover, with K-K transfer being larger than K ionization over the whole range. Both cross sections are monotonically decreasing, with Z, over 4 orders of magnitude. The Z,-dependence for a fixed scaled velocity would be of interest, but requires a very large range of projectile velocities. 5. b-Dependence qf K-Vclccincy Production

One of the most crucial tests of the theories of vacancy production is the comparison of impact parameter dependence predictions with experi-

298

C . D . Lin and Patrick Richard

b (lO-"cm)

FIG.IS. The K-shell ionization of Cu at 1.414 MeV/amu, giving the measured impact parameter dependence for H+, O"+,and On+on Cu. The theory curves are Z z times P,(b ), as predicted by SCA, and P,,.,4,,,,(h),a s predicted by SCA, with a correction for increased binding effects. Data and calculations are from Andersen c,t rrl. (1976).

ment. Several studies of the impact parameter dependence of K-vacancy production have been performed to date. In this discussion only cases where both K ionization and K-K transfer have been investigated will be emphasized. Light projectile asymmetric collision systems have ionization probabilities, P ( b ) ,described fairly well by the SCA theory. Figure 15 contains such a typical result for 1.414 MeV/amu H + + Cu collisions from Andersen et cil. (1976). As the projectile charge and charge state are increased, one would expect deviations in the P(b)function due to increased binding effects in the target, as exhibited in the total cross sections (see Figs. 7 and 13) and the growing in of the K-K electron transfer process (see Fig. 12). The idea of an increased binding effect in K-vacancy production by Coulomb ionization has been extended to the SCA model by Andersen et al. (1976). It is more straightforward to establish an impact parameter dependence of the increased binding effect in this formulation than in a PWBA formulation. A scaling of the impact parameter dependent probability is given based on the SCA results of Hansteen ef al. (1975). The resultant scaling takes the form

INNER-SHELL VACANCY PRODUCTION

299

where the variables with primes refer to H projectile parameters and the variables without primes refer to heavy-ion projectile parameters. E , is the binding energy of a 1s electron wave function due to a projectile at the distance of closest approach assumed to be h . The I functions are taken from Hansteen et a/. (1973, and the scaling is affected by letting b E , / v = h ' E 1v and u 1r,,E , = v / r E k . E , and rl\ are Z-dependent when increased binding is included according to their prescription and thus leads to a deviation from a Z $ scaling and an altered P(b). A sample of the results is given in Fig. 15. The data are for 06+ and Ox+ plus Cu, which has a Z,/Z, = 0.28, where one'expects the deviations from a Z: dependence. The SCA results without increasing binding are taken as the curve Z2, x P,(b), where P H ( h is ) the ionization probability for 1.414 MeV/amu H + + Cu. The calculated curve including increased binding is labeled PhcA(B)(b ). The magnitudes of the two calculations differ by about a factor of four and have a slightly different shape. The P,,,,,,(b) fits the data fairly well. Recent results by W. Schadt et N / . (1980) for Ne-Ni (ZL,/Z,,.= 0.36) collisions at 8 MeV, 12 MeV, and 18 MeV, which corresponds to scaled velocities of 0.16, 0.20, and 0.24, respectively, demonstrate the good agreement between experiment and data over a very large h range. One of the very few sets of data for P(b)for K-vacancy production with and without the effects of K-K transfer is given in Fig. 16 for FH++ Ne and F6++ Ne at 4.4 and 10 MeV (Hagmann et a / . , 1981). The F6++ Ne data contains only Ne K vacancies produced by K ionization and possibly target K-shell to projectile L-shell transfer (open triangles in Fig. 16). The Fx++ Ne contains in addition Ne K vacancies produced by K-K electron transfer. Similar to the case of the total cross sections, the P(b) for K-K transfer is obtained by subtracting the P(b) for F6+from the P(b)for FH+, Pl,l,(b)= PB+(b) - P,+(b).The results of this subtraction will be presented in Section IV, together with theoretical calculations for K-K transfer. The important observation of the quantum mechmical interference effects at the low velocity will provide a much more stringent test of theory than the total cross section alone, which is relatively energy independent (see Fig. 25 in Section IV). +

C. DOUBLE K-VACANCY PRODUCTION Many types of multielectron excitations occur in ion-atom collisions, as discussed in Section II,B,l. One of these processes that can be readily identified in K X-ray (Richard et ul., 1972) and to some extent in K-Auger

300

C . D . Lin and Patrick Richard 1.2 I.o

F"'I"""""'""'""'""'

1

1

ai

Oo

a2

(a) 1.1

~3

0.4

0.5

a

b [a.u.]

F"""1''""'"""'""'""'" 4A MeV

-a

F* + NO

n 0.6 0.5-

I

0.4

F"+Ne

a -

a3 -

-

02 ai 0

0

(b)

0.1

02

03 0.4 b [au.]

0.5

0.6

FIG.16. The impact parameter dependence for Ne K-vacancy production in FH++ Ne and F6++ Ne: (a) at 10 MeV; (b) at 4.4 MeV (Hagmannet al., 1981). The solidcurves are eye guides to the data. The F'+ data contain the additional contribution of K-K charge transfer.

spectra (Woods et al., 1975b) is double K-vacancy production. This process is also of theoretical interest because of its relation to single K-vacancy processes and because there are only two K-shell electrons to consider. Double K-vacancy production can occur through double K-ionization, double K-shell excitation, double K-shell electron transfer, and the three combinations of K-ionization-K-excitation, K-ionization-K-electron transfer, and K-electron transfer-K-excitation. The situation is greatly simplified for target K-vacancy production, where K-shell excitation is known to be very small, thus leaving only three possible double Kvacancy production processes. The projectile q-dependence, Z,-dependence, and E-dependence of double K-vacancy production are important to assess the role of the dif-

INNER-SHELL VACANCY PRODUCTION

301

ferent possible processes. The first study of the q-dependence of double K-vacancy production was performed by Woods et al. (1975b) for Nq+, Oq+, and Fq+on Ne. These authors observed a very large enhancement of the KLL-Auger hypersatellites of Ne for the bare projectiles compared to projectiles with one or more electrons. The enhancement is interpreted in terms of the opening of the double K-K electron transfer channel. Lin (1979) performed calculations of the double K-transfer cross section, which are substantially in agreement with these experimental results. In analogy to single K-vacancy production, the following expressions can be used to understand the q -dependence of double K-vacancy production: (urnrefers to a cross section due to projectiles with n K vacancies) Single K:

uK0

= ur

c(q s

+ 4uK-K = C T ~+ u K - K

z - 2)

( T K ~ = (T?

(4= Z

vKZ

(4= z)

Double K: cr% = utK

1)

(9a)

z - 2) (4= Z - I )

(9b)

-

(q

uZK K1

- r f K + t't,E-K

&;

= uf" + uf,"K-~ (TtK-ZK

+

~

(4= z)

The difference in electron screening effects of the projectile with different q on ionization and K-K transfer is neglected in these expressions. We see

from these equations that (+9,K-2K

=

uK-K

(at3 - &):

=

-

uK2 - uK0 and

(mi!

-

&i)=

- 2uZK K 1 + &$

(9c)

The fluorescence yields of the double K-vacancy states are not well known but, for similar outer-shell configurations as single K-vacancy states, the fluorescence yields of double and single K-vacancy states are approximately equal (Bhalla, 1975b). Recent extensive studies of the K hypersatellite X rays of Ti by Hall et al. (1981) demonstrate the q-dependence expected by Eq. (9). The K, hypersatellites of Ti have been studied with projectiles between Z , = 1 and Z p = 17 and demonstrate very convincingly the double K-K transfer mechanism. Figure 17a depicts the Zp dependence of u2KK2/uK2(circles) and ( T Z K K O / ( T ~ ~ (squares), which are the ratios of double K- to single Kvacancy production for bare ions and for few electron ions, respectively. For Zp < 7, the two ratios are the same. The low-Z data (Z s 7) (open circles) are from Awaya et al. (1979). If Coulomb ionization is the dominant mechanism for producing both double and single K vacancies, then the ratio should follow a Z2, dependence for small ionization probabilities. The solid curve in the figure demonstrates the Z $ behavior. The falloff of the few-electron projectile data below the ZZ, curve is presuma-

302

C . D . Lin and Patrick Richard

0.4

0. I

€ / A (MeV/omu)

FIG.17. (a) The Z,-dependence of the ratios of double K-vacancy to single K-vacancy production cross sections of Ti by bare ions (circles) and few electron ions (squares). The open circles are 6 MeV/amu measurements from Awaya et a / . (1979); the solid symbols are 5 MeV/amu measurements from Hall et a/. (1981). The solid curve is a Z $ curve. The rise in the data at high Z , , for bare ions is attributed to double K-K charge transfer. (b) The energy dependence for Siq++ Ti is from Hall et u / . (1981). The triangles are the double K- to single K-vacancy ratio for one-electron ions.

bly due to the increased binding effect, whereas the rise of the bare-ion projectile data above the ZZ, curve is due to the increase of the double K-K transfer as one approaches the resonant charge transfer. Figure 17b depicts the energy dependence of the ratios defined above for the case of Si + Ti from Hall et al. (1981). The triangles represent the double K- to single K-vacancy ratio for one-electron ions.

INNER-SHELL VACANCY PRODUCTION

303

The double K-K transfer cross sections obtained by using Eq. (9) will be compared with theory in Section IV. Both the projectile E-dependence and Z,-dependence of the cross sections are available for the case of Ti.

111. Theory of Inelastic Ion- Atom Collisions A. GENERAL CONSIDERATIONS The motion of heavy particles in ion-atom collisions at moderate energies (E >> 1 eV) is described adequately by well-defined classical trajectories, because the de Broglie wavelength associated with the relative motion of the colliding heavy particles is very small compared with atomic dimensions. In contrast, the electronic motion, which is associated with the discrete nature of atomic states, must be described by quantum mechanics. In this semiclassical approximation, the trajectories of heavy particles are determined classically by an assumed potential W ( R ) ,often chosen intuitively or deduced from experimental elastic scattering data, while the electronic motion is obtained by solving the time-dependent Schrodinger equation (or Dirac-Breit equation for relativistic inner-shell electrons), where the time-varying potential arises from the changing internuclear separation. Consider first a one-electron collision system consisting of projectile nucleus P, target nucleus T, and electron e - . We choose a coordinate system as depicted in Fig. 18. The incident projectile has initial velocity v and impact parameter b, the internuclear separation is denoted by R =

FIG. 18.

A coordinate system consisting of one electron and two nuclei.

C . D . Lin and Patrick Richard

304

R ( t ) , which is obtained by solving the classical equation of motion. The theoretical problem then reduces to finding solutions of the timedependent Schrodinger equation

[-iV +

V&T)

+ Vp(rp) +

a

W(Z?)J*(r, t ) = i - Y ( r , at

t)

(10)

Atomic units are used in Eq. (10) and throughout the rest of this article except where otherwise stated. In Eq. (lo), we assume that the electron experiences a potential VT from the target nucleus and a potential V, from the projectile. With this somewhat general choice of V, and V,, Eq. (10) describes many-electron systems in a single-electron approximation in that the passive electrons are treated as only providing screening to the active electron considered in Eq. (10). The position vectors rT, r,, and r of the active electron are defined in Fig. 18. We choose the origin C of the coordinate system at the midpoint of the internuclear axis. Our discussion of inelastic inner-shell processes in ion-atom collisions will be based primarily upon Eq. (10). The limitation of the single-electron model for many-electron systems as given by Eq. (10) will be discussed in Section II1,F. In the following, we shall first consider different collision models used in the solution of Eq. (10) for various inelastic processes, for different collision velocities, and for different collision conditions. For true one-electron systems, VT = - Z T / r T and V, = - Z p / r p ,where Z , and ZT are, respectively, the nuclear charges of the target and the projectile. There are many existing elaborate one-electron system calculations in the literature, which allow us to assess the theoretical models adopted in the many-electron ion-atom collision systems. On the other hand, the greater variety of effective ZT and Z p provided by heavy-ion-atom collisions often allows us to test theoretical models not available from one-electron systems. The precise nature of the effective internuclear potential, W(R),does not arise in the computation of transition probabilities and total cross sections since the internuclear potential can be eliminated from Eq. (10) by a phase transformation

[

q ( r , t) = +(r, t) exp -i j ' ~ ( ~ ( t ' ) ) d t ' ]

(11)

such that Eq. (10) reduces to

a

[ - 4 V + vT(rT) + Vp(rp)lW, t ) = i W, t) at

(12)

The potential W(R) does enter in determining the trajectory R = R ( t ) . However, in many calculations at higher energies, straight-line trajectories are often used. [A proper treatment of W(R) is important in describing differential cross sections; see Section IV,B.]

INNER-SHELL VACANCY PRODUCTION

305

The dynamic solution of Eq. (12) and the dominant inelastic processes depend critically upon the parameter A = u/u, (where u is the collision velocity and u, the orbital velocity of the electron). Existing theoretical models solve Eq. (12) based primarily upon close-coupling methods (or eigenfunction expansion) or its variations. In the limit u<< v,, the electron responds to the slowly varying Coulomb fields from the two nuclei, and a transient molecule is formed during the collision such that a description based upon molecular orbitals (MO) is appropriate. In the high-velocity limit u >> u . , the projectile imparts, primarily, a transverse momentum to the electron (Jackson, 1975, Chapter 13), and the major inelasti: process is the direct Coulomb ionization. Recent work in inner-shell vacancy production by bare projectiles is I , even though it covers a large range of ZTand mostly in the region A Zp.In Section I1 experimental results were presented showing that charge transfer plays a dominant role in the inner-shell vacancy production, particularly for ZT Zp. Responding to this new situation, many theoretical models have been proposed in the last few years. In the next few sections, we shall discuss these physical models and their limitations. Comparison with experimental results will be presented in Section IV.

-

-

REGION:MO MODEL B. LOW-VELOCITY In the low-velocity region, it is convenient to expand the timedependent electronic wave function in terms of trmveling adiabatic molecular orbitals (MO): Icr(r9

1) =

c 4 f ) d d R ( t ) ;rl I/,[R(f' dt ' + jJr, R)v - r x exp{ i [ I1 f

-

(13)

)]

where &is the adiabatic MO and Uj(R)is the MO energy. For transitions to low-lying states, expansion (13) is often truncated to only a few terms. In the context of ion-atom collisions, the MO model based upon Eq. (13) has been reviewed recently by Briggs (1976). In this article we are interested in the velocity region which is higher than those generally considered in the MO model, and we will mention below only those developments in the MO calculations which are relevant to later sections. In Eq. (13) the velocity-dependent phase factor, generally called the electron translational factor (ETF), was originally introduced by Bates and McCarroll (1958) to account for the linear momentum of the electron

306

C . D . Lin and Patrick Richard

associated with moving centers. These factors are needed to ensure that the calculation is independent of the choice of the origin and to avoid spurious couplings. However, the functions A(r, R) are only rigorously defined in the limit, R -+ x wheref, = + $ ( - 4) if the orbital& dissociates to an atomic orbital centered at P ( T ) . Very often one also imposes& = 0 at R = 0. This condition does not follow from the translational invariance principle and thus should be viewed as an auxiliary condition imposed by the individual physical model. Except for these two limits, the functions f;(r, R) are not rigorously defined. In recent years, various forms of electron translational factors have been proposed. The simplest one is the plane wave translational factor originally proposed by Bates and McCarroll (1958). More sophisticated but channel-independent ETFs have also been proposed by Schneiderman and Russek (1969) and have been adopted in recent MO calculations for heavy ion-atom collisions (Briggs and Macek, 1972; Taulbjerg and Briggs, 1975; Fritsch and Wille, 1978). More recently, Thorson and Delos (1978a, b) and Crothers and Hughes (1978, 1979) have proposed ETFs derived from variational methods. These are channel dependent and in general more complicated in form. Because of the different forms of ETFs used by different workers, the dynamic MO calculations for a given set of MO orbitals may still differ. The importance of ETFs has been recognized recently. Molecular orbital calculations without ETFs have been shown to converge very slowly (Winter and Lane, 1978) with respect to the increase in the number of channels included. However, the sensitivity of dynamic calculations with respect to the various forms of ETFs for a given basis set has not been widely studied. In Table I, we show the charge transfer cross sections for H2++ H( 1 s) at several energies calculated by Winter and Hatton (1980) and by Vaaben and Taulbjerg (1979). Notice that the former used plane wave ETFs, while the latter used the Schneiderman-Russek ETFs. Both groups showed the convergence with respect to the size of the basis set except at the highest energy point. The discrepancy between the two sets of calculations is about 20%, which is probably due to the different E T F a (The difference at E = 1 keV is presumably due to different trajectories used in the two calculations.) Substitution of Eq. (13) into Eq. (12) results in a set of first-order coupled differential equations for { a j ( t ) } , which are to be solved with suitable initial conditions { aj(- 00)) to obtain transition amplitudes for each impact parameter b and each impact energy E. Coupling terms between the channels are usually classified in terms of two “mechanisms”: radial couplings and rotational couplings. According to the perturbed stationary state (PSS) model, where the ETFs in the expansion (13) are neglected, radial coupling occurs between states with identical projected angular momenta along the internuclear axis, while rotational coupling

307

INNER-SHELL VACANCY PRODUCTION TABLE I

TOTALCROSS SECTIONS (IN UNITS OF lo-'' Cm') FOR ELECTRON TRANSFER INTO VARIOUS STATES OF He+ B Y He2+IMPACTON H(1s) CALCULATED IN T H E MOLECULAR STATE EXPANSION^ Vaaben and Taulbjerg

1 3 8 20

0.1 0.174 0.284 0.449

Winter and Hatton

36

206

3'

4'

1Or

0.12 1.24 5.37 10.32

0.13 1.25 5.17 9.58

0.25 1.45 5.98 10.2

0.268 1.49 6.12 10.8

0.268 1.50 6.35 12.3

" The calculations of Vaaben and Taulbjerg (1979) used Schneiderman-Russek type translational factors and the calculations of Winter and Hatton (1980)used plane wave translational factors. * A 2pu trajectory was used. ' A rectlinear trajectory was used.

occurs between states where these projected angular momenta differ by one unit. The introduction of ETFs negates this simple result, but under favorable conditions the collision dynamics can still be approximated as the operation of a single (or several) rotational and/or radial couplings. Our understanding of heavy-ion collisions at lower energies ( A << 1) owes its advance to the prevailing of these favorable conditions. A great deal of progress has been made since the original model of Fano and Lichten (1965). Dynamic calculations based upon the more refined MO models have since been studied for several systems (see Briggs, 1976). However when favorable conditions do not apply, we find that our understanding of collisions at lower energies is still very limited. Since ionizations in the MO regime are still evasive of theoretical calculations, even for simple systems, the interpretation of inner-shell vacancy production cross sections by ionizations at low energies still has to rely on either semiempirical formula (Basbas et al., 1973; Lennarder al., 1978) or simplified theoretical models (Briggs, 1975; Meyerhof, 1978).

c. DIRECTEXCITATION A N D IONIZATION A T HIGHVELOCITIES 1. General Remarks

For faster collisions where the orbiting velocity of the electron is not large compared to the velocity of the projectile, the quasi-molecular com-

C. D. Lin and Patrick Richard

308

plex is not formed. It is better to replace Eq. (13)with a two-center atomic expansion (Bates, 1958):

+(r, t ) =

2n an(t)4n(rT)exp[-i(r,t + tv r + BuV] + 2 bm(t)&(rp) exp[-i(rmt - t v r + Qu'~)] m

(14)

In Eq. (14), c#Jn(rT) and &,(rp), respectively, are atomic eigenstates or pseudostates centered on T and P, and en and E , are the corresponding eigenenergies. The velocity-dependent phase factors are the plane wave ETFs. Unlike the MO expansion in Eq. (13), the ETFs for atomic expansion with true atomic eigenstates are well defined. In Eq. (14), the electronic wavefunction $(r, t ) is expanded over a basis set that is overcomplete. The usefulness of the expansion, however, often depends on whether the expansion can be truncated to only a few terms. For fast collisions, the projectile imparts primarily a transverse momentum to the electron, and there is little electron density concentration near the projectile. Therefore, the expansion around the projectile can be neglected if direct excitation and ionization are the subject of interest. On the other hand, if v v,, then the electron will have substantial probability in the vicinity of the projectile, especially when Zp ZT. In this case, a two-center expansion is more appropriate.

-

-

2. One-Center Atomic Expansion In a one-center expansion for excitation and ionization at high velocities, we can neglect the ETFs, and thus Eq. (14) reduces to

Substitution of Eq. (15) into Eq. (12) gives a set of coupled first-order equations ib,, =

2 Vn,(R) m

exp[i(r,

-

rm)t]am(t)

(16)

where vP(rP.)l+m(rT)) If the inelastic transition probability is small, it is possible to obtain the first-order solutions by setting a&) = 1 for the elastic channel and am(t)= 0 for m # 0 on the right-hand side of Eq. (16). The resulting equation is integrated to give Vnm = ( &(rT)I

309

INNER-SHELL VACANCY PRODUCTION

a,(t)

=

-i

If

v,,exp[-i(t-,

(n z 0)

-En)r’]dt‘

(17)

By letting t + m, Eq. (17) gives the first-order approximation for the inelastic transition probability amplitude from initial state 0 to a final state n . This corresponds to the semiclassical approximation (SCA) for excitation and ionization. If the trajectory R = R ( t ) is assumed to be rectilinear, then this approximation is equivalent to the plane wave Born approximation (PWBA) in the wave picture (McDowell and Coleman, 1970; Taulbjerg, 1977; Bethe and Jackiw, 1968). Higher order approximations can be obtained from Eq. (16) by the substitution

4) = 4 0 ) exp (-

1‘

v n n ( t f ) dt’>

(18)

The equation for dn(t)becomes i&t)

=

C

m#n

d , ( t ) ~ , , ( ~exp[-i(a, )

-

a,)t]

(19)

where

and the first-order two-state solution of Eq. (19) becomes

dn(t)= - i

1‘v,, exp[-i(6,

-

a , ) t ’ ] dt’

(21)

--r

This last expression corresponds to the distortion approximation. By comparing the transition amplitudes from Eq. (21) and from Eq. (17) we notice that the distortion approximation differs from the SCA (or the PWBA) only in the phase term in the integrand by an amount V, shown in Eq. (20). Higher order Born approximations based upon Eq. (16) have also been proposed by several authors. Interested readers are referred to the recent review by Bransden (1979). Extension beyond the first Born or SCA calculations can also be achieved by including a few more states in the expansion (15). For excitations to low-lying bound states, the one-center multistate expansion has been adopted in the close-coupling calculations (Flannery and McCann, 1974; Briggs and Roberts, 1974). For direct ionization Reading, Ford, and co-workers (Reading et ul., 1976; Ford et al.. 1977; Fitchard ef ul., 1977) have employed a one-center multistate expansion including discrete as well as discretized continuum states (pseudostates). Target-centered atomic and pseudostates with angular momentum up to 1 = 2 are included in the works by the latter group, and the method has been applied to

310

C . D . Lin and Patrick Richard

vacancy production of inner-shell electrons by light ions. It might be remarked that in the calculations of Reading, Ford, and collaborators, U-matrix approximation is generated within the finite basis set. They show that this technique is often numerically preferable to solving the coupled differential equations spanned by the same basis set. The results of these one-center, multistate, close-coupling calculations are in general in better agreement with experimental data than the first Born calculations, but discrepancies still appear at lower energies. The advantage of the one-center expansion is that coupling matrix elements are functions of R only. They need not be recomputed for different impact parameters and different collision velocities. But as the collision velocity decreases, the orbiting electron becomes more likely to be pulled to the vicinity of the projectile. Such charge distribution near the projectile cannot be easily represented by atomic orbitals centered on the target unless states with high orbital angular momenta are included in the expansion. In this case, a two-center expansion provides a more direct and convenient way of representing such a charge distribution. In actual physical processes, charge transfer often dominates direct excitation and ionization processes in the lower velocity region ( u s 0,). Thus, coupling with charge transfer channels is often important in the calculation of direct processes.

D. ELECTRON CAPTURE

A T INTERMEDIATE VELOCITIES

h’e define the intermediate-velocity region for collisions as that region where the collision velocity u is comparable to the characteristic orbital velocity u, of the electron in the initial state. This is the region where the total capture cross section peaks (except for symmetric collisions). For near-symmetric collisions, the capture probability is not small in this velocity region, and the perturbation approach is not suitable. On the other hand, since the collision is too fast for an MO description, the molecular orbital expansion (13) is not suitable. In this energy region, the two-center atomic expansion (14) provides a convenient basis for describing the capture process. The usefulness of expansion (14) depends on whether the actual physical problems can be approximated by a small truncated basis expansion. In this section we first examine the simple two-state, two-center atomic expansion method; later the multistate expansion and the three-center expansion will be described. Some of these methods have been applied to the heavy-ion collision studies; others are limited to simple collision systems only.

INNER-SHELL VACANCY PRODUCTION

311

I . Two-State, Two-Center Atomic Expansion a. Coupled equations. capture, we write $(r, t )

=

In a truncated two-state expansion for electron

AT(f)&(rT)exp[-i(hV*r

+

+ ETf)] + Bv't + epf)]

f QV2t

Ap(f)c#Jp(rp)exp[-i(-hv-r

(22)

where all quantities with subscript T refer to the initial state centered on T and those with subscript P to final states centered on P. Substitution of Eq. (22) into the time-dependent Schrodinger Eq. (12) gives a set of coupled equations for AT(f) and Ap(f). However, we follow a similar procedure in going from Eq. (17) to Eq. (21), which leads to the distortion approximation. After a suitable phase transformation from [ A T ( f )A,(t)] , to [dT(t),d , ( t ) ] ,one gets the coupled equations for dT(t) and d p ( t ) :

The derivation of Eqs. (23) is given by Lin et al. (1978) as well as McDowell and Coleman (1970) using slightly different notations. The definition of the matrix elements hij and Sij (i, j = T, P) is given in Eq. (6) of Lin et al. (1978) (with some obvious changes), and UT and Up are defined as

Equations (23) are expressed in a form similar to the coupled equations derived from the two-state MO expansion. The second term on the righthand side of Eqs. (24) describes the distortion potential of each electronic state due to the other heavy particle, but UT and Up are nor the MO adiabatic potentials. Nevertheless, the coupling terms in Eqs. (23) consist of two multiplicative components similar to the MO expansion: one part from the oscillating energy difference and the other part from the dynamic potential coupling (corresponding to radial or rotational couplings in the MO model). Since electron translational factors are included in the definition of off-diagonal matrix elements hi, and Si,(i # j ) , the potential coupling terms also have oscillatory behavior as a function of time at higher velocities. Thus, the coupling terms in Eqs. (23) are determined by the product of two oscillatory functions. The distortion term exp[ - i s' ( UT Up) dt] = exp[ - ( i / v ) s" ( UT - Up) dR] oscillates faster with decreasing

C.D. Lin and Patrick Richard

3 12

velocity u and dominates the energy dependence of electron transfer cross sections at lower velocities. On the other hand, the dynamic potential couplings [(hPT- SpThTT)/(l - S2)and its counterpart in the first equation in (23)] can be shown to have faster oscillations at higher velocities and to determine the energy dependence of electron capture cross sections at higher energies. Capture cross sections peak when the two oscillating components have the same amplitude but opposite phase, which occurs when u u,.

-

b. Connection with various perturbation theories. If the capture probability dp is small, we can set dT(t)= 1 and integrate Eq. (23b) to give

This expression is the correct expression for the distortion approximation where initial and final states are not orthogonal at finite R . An expression similar to this one, generally referred to as the distorted wave approximation, was derived by Bassel and Gejuoy (1960) in the wave picture. Equation (25) reduces to the standard distortion approximation (21) when all the nonorthogonality terms (STP= S*,, and S 2 = STPSm)are set at zero. If we further set Ui= E i (i = T, P) and use the explicit definition of hPT, then we obtain the first-order approximation for electron capture: dp(+m) = -i

I_mm


exp[-i(eT -

ep)fI

dt

(26)

where c$p =

c#Ip exp[(i/2)v*r]

and

6T= c#IT

exp[(-i/2)v*r]

This is the semiclassical version of the first Born approximation for electron capture, known as the OBK approximation, named after Oppenheimer (1928) and Brinkman and Kramers (1930). In the wave formulation, the OBK approximation gives simple formulas for total capture cross sections between hydrogenic initial and final states (see McDowell and Coleman, 1970, Chapter 8). These formulas have been widely used (and abused). We shall see in the next few subsections that there are no collision systems at any velocity where electron capture cross sections are predicted by the simple OBK approximation. In passing, it might be noted that if we start with the time-dependent Schrodinper Eq. (lo), the first-order expression for the capture amplitude will be dp(+m) = -i

m

(&IVp

+ WR)I&T) exp[-i(ET

-

eP)fl dt

(27)

INNER-SHELL VACANCY PRODUCTION

313

This corresponds to the semiclassical version of the Jackson-Schiff (1953) approximation. It differs from the OBK result (26) because of the inclusion of the internuclear potential W ( R ) .Deficiency of this approximation and its variants was discussed by Lin et al. (1978). There are other forms of charge transfer theories in the literature which can be visualized as a further approximation to Eq. (25). If we replace Uiby E t (i = T, P), then Eq. (25) reduces to

Equation (28) can be obtained from the OBK formula (26) by requiring that the final state dr be orthonormalized to 4Tat all R (Lin and Soong, 1978). If we set S = 0, then A,,., - SYl.hrT in Eq. (28) can be written as C&~lVl. - ( ~ J T I V ~ I ~ T ) J $ TThe ). operator VrI - (4TlVp14T)= Vp - ( Vll) has been referred to as the Bates-Born potential (Theisen and McGuire, 1979; Junker, 1980). c. Region q f vcilidity for the titv-state, twv-center irtoinic expiinsion method. All of the first-order approximations discussed in Section III,D,l,b above can be derived from Eq. (25) with additional assumptions (although some are unjustified), while Eq. (25) is the approximate solution to the two-state coupled equations (23) if the capture probability is small. The two-state models (17) and (21) for direct processes are known to be valid at higher velocities, but this does not guarantee that Eqs. (23) or (25) are valid for charge exchange at higher velocities. In fact, the two-state model (23) is not valid at either very high or very /OH, velocities, but there are circumstances in which it is possible to obtain accurate results for electron capture cross sections in a simple two-state model, as emphasized recently by Lin (1978) and Lin et d.(1978). As pointed out earlier, except in the case of resonance capture, the u,. At these velocities, cross section for capture often peaks near u capture occurs primarily at large impact parameters, i.e., at impact parameters comparable to or greater than the typical orbital radius of the electron in the initial state, for which the wave functions can be easily approximated by a two-center, two-state atomic expansion. However, as the velocity increases or decreases away from u,, total electron capture becomes dominated by contributions from small impact parameters. (For symmetric resonance capture, capture is still dominated by large impact parameter collisions even for u < u, .) Therefore, in terms of total capture cross sections, the two-state model is useful in the intermediate velocity u,. For example, in the case of protons on helium atoms, region u experimental data and two-state calculations agree reasonably well up to

-

-

C . D . Lin and Patrick Richard

3 14

TABLE I1 COMPARISON OF T H E TWO-STATE CALCULATIONS~ A N D T H E PSEUDOSTATE CALCULATIONS~ FOR THE TOTAL CAPTURE CROSSSECTIONS FOR p + H(1s) * H(1s) + p REACTIONS' Energy (keV) 4 10 15 20 25 40

60 100 300 1000 'I

Two-state

Pseudostate

1.15 (-15)" 7.79 (-16)" 5.60 (-16) 4.18 (-16Y' 2.76 (-16) 1.51 (-16)" 4.98 (-17Y' 1.012 (-17) 1.711 (-19) 5.12 (-22)

1.13 (-15) 7.77 (- 16) 5.81 (-16) 4.14 (-16) 2.93 (- 16) 1.13 (-16) 4.20 (-17) 8.89 (-18) 8.51 (-20) 2.63 (-22)

Two-state calculations from McCarroll (1961).

* Pseudostate calculations from Cheshire et ui. (1970). The cross sections are given in cmz; A ( - @ = A x 10-8. Interpolated from MCCZUTOU (1961).

100 keV, but by 500 keV the two-state results are about twice those given by the data. In Table 11, we show a similar comparison between the two-state and the more accurate multistate calculations of Cheshire et al. (1970) for H + + H( 1s) +. H( 1s) + H +.Notice that the deviation starts to appear for E 2 300 keV. A comparison of two-state calculations with experimental data for inner-shell vacancy production will be presented in Section IV. 2 . Two-Center, Multistate Expansion, and Three-Center Expansion

The simple two-state model becomes inadequate for collisions occurring at small impact parameters, as well as for transitions to excited states. In the latter case, the failure can result from the neglect of coupling through intermediate states or from the neglect of final-state coupling with other near-degenerate states. For the simple H + + H, H + + He+, and + H collisions, two-center multistate calculations including lowHe '+ lying excited states of both target and projectile have been employed in the past to obtain excitation and capture cross sections, but the results are often unsatisfactory, especially for small impact parameters. The major difficulty is related to the deficiency in trying to expand an electronic wave

INNER-SHELL VACANCY PRODUCTION

3 15

function at small internuclear separation in terms of two-center atomic orbitals. Consider, for example, the H + H( 1s) collision for u < u, . In the united atom limit the electronic wave function is better represented by the He+(ls) wave function. This wave function cannot be adequately expanded in terms of H(1s) wave functions on the two centers. Since He+(ls) overlaps strongly with the continuum spectrum of the Hamiltonian of the hydrogen atom, it was concluded that continuum states or pseudostates simulating continuum states should be incorporated in the expansion. In the past, pseudostates and Sturmian basis functions have been used in the two-center expansion (Gallaher and Wilets, 1968; Cheshire er al., 1970). Recent calculations of Shakeshaft (1976, 1978b) include 12 states and 35 states on each center, respectively. To illustrate directly the importance of the He +(1s) wave function in the united atom limit for collisions at small impact parameters without resorting to a large basis set, it is intuitively attractive to expand the wave function about three centers, T, and P, and the center of charge C, rather than two centers. For the collision, p + H(1s) + H(1s) + p, Anderson et a/. (1974) and, more recently, Lin et d.(1981) have applied the idea of the three-center expansion. In the work of Lin et al. (1981) a three-center, three-state expansion is adopted: +

+(r, t ) = AT(f)& exp[-i(4v*r

+ Qu2t +

+Ap(f)c$p exp[-i(-iv*r

eTt)l

+ QuZt + e p t ) ]

where & and Ec are the He+(ls)wave function and energy, respectively. The result of the three-center expansion shows that capture probabilities differ from the two-center, two-state results only for small impact parameters, but not for large impact parameters. Thus, the total capture cross sections are not changed significantly. In Fig. 19, we show the capture probability P(b) at a fixed angle 8 = 3” with various collision energies. These collisions correspond to scattering at very small impact parameters, as indicated on the top scale of the graph. The experimental data are taken from Lockwood and Everhart (1962) and Helbig and Everhart (1964). Notice that the two-center, two-state calculation does not reproduce the oscillation (the predicted maxima and minima of the two-state model are indicated by arrows), while the pseudostate expansion of Cheshire et al. (1970) (shown by dashed-dot lines) and the three-center expansion (shown by a solid line) both reproduce the oscillation and the maxima and minima well. The three-center expansion suffers from the disadvantage that threecenter matrix elements are very difficult to calculate. An alternative ap-

316

C . D . Lin and Patrick Richard

E(keV) FIG.19. Total charge transfer probability P ( b )for H+ + H(1s) -+ H(ls) + H+ at the fixed scattering angle of 3". Experimental data are taken from Lockwood and Everhart (1962)and from Helbig and Everkart 11964). Theoretical results: (- . .-) two-center atomic expansion three-center, three-state including Is, 2s, 2p, 3s, 3p states (Cheshire er al., 1970); (-) calculations (Lin et u l . , 1981);(---) two-center 1s and 1s atomic expansion (Fritsch and Lin, 1981), where 1s is the He+(ls) wavefunction orthonormalized to H(ls) wavefunction. The predicted positions of maxima and minima of the oscillation from the two-state, two-center expansion are indicated by arrows.

proach is to put the united-atom (UA) basis functions on the two centers T and P. However, to satisfy boundary conditions properly, these UA functions are orthogonalized to the true physical states in the separated-atom limits, resulting in pseudostates having energies in the continua. This method provides a systematic way to introduce pseudostates. Recent calculations by Fritsch and Lin (1981) showed that the method reproduces the P(b)oscillation at 3" for H + H( 1s) + H( 1s) + H (shown by dashed lines in Fig. 19) and agrees well with the molecular coupled-state calculations of Winteret al. (1980) for H + + He+(ls)-+ H(ls) + He2+in the u < u, region. The latter calculation shows that there is an energy region where the atomic and molecular basis expansion methods overlap. The two-center and three-center expansions are often necessary in calculating excitation and charge transfer to low-lying states for nearsymmetric collisions. For asymmetric collisions where the projectile charge 2, is much smaller than the target charge &, simplification of the two-center expansion (14) can be made by retaining only a few projectilecentered states and keeping many target-centered states. In this type of +

+

INNER-SHELL VACANCY PRODUCTION

3 17

asymmetric collision, the electron transfer probability is small and the time-dependent wavefunction +(r, t ) is well represented by a one-center expansion about the target. If we denote the sum in the first line of Eq. (14) by $rr and truncate the sum over projectile-centered states to only one state, +$, the final state for which the charge transfer probability is to be calculated, then one can rewrite Eq. (14) approximately as +(r, t ) =

JIT +

bA(t)+\J

(30)

where 9%is the traveling atomic orbital with velocity v if the origin of the coordinate system is chosen at the target center. Equation (30) takes a form similar to the two-state model (22) except that the “target” state is no longer the atomic target state in the separated-atom limit and that the unitarity condition is not imposed. Substitution of Eq. (30) inLo the timedependent Schrodinger equation and solving for bh (+ x ) , an equation similar to the distortion approximation is obtained. This method has been used recently by Ford, Reading, and Becker (Reading and Ford, 1979; Ford et al., 1979, 1981) to calculate electron capture cross sections from the inner shells of atoms by light ions. Notice that this method is valid only when the capture probability is small. In the opposite type of asymmetric collision where the projectile charge Z, is much greater than the target charge Z,, the basic physical process is different and different physical models are needed. Except at very high velocities, the electron will be captured to the excited states of the projectile. Since many states are populated, the coupled-channel expansion (14) is impractical. At higher collision velocities, the coupling between final excited channels might be neglected and capture to an individual final state might be approximated by a two-state model in the distorted wave approximation (28). This method has been used by Rufuku and Watanabe (1978, 1979) for calculating total capture cross sections by a variety of bare projectiles from hydrogen atoms. However, this method should be used with caution. Even though the capture probability to individual channels is small, the total capture probability is close to unity, especially u,. The validity of the distorted wave approximation becomes for u questionable since the elastic scattering amplitude $ ( t ) in Eq. (23b) can no longer be approximated by & ( t ) = 1, a condition required for Eq. (25) to be true. At still lower energies, u s u,, because of the significant unbalance between the charges Z, and Z,, it is expected that the total charge transfer probability nearly reaches unity at small b , followed by strong oscillations at large b before the cutoff impact parameter b, where capture is no longer likely. In this case, the total capture cross section is not likely to depend sensitively upon the collision velocity until it reaches the very low energy regime. For these collisions, Ryufuku and Watanabe

-

C. D. Lin and Patrick Richard

3 18

(1978, 1979) had developed the so-called “unitarized distorted wave approximation (UDWA)” for calculating total electron capture cross sections for highly charged bare ions on hydrogen atoms. The basic idea is to include in the expansion (14) the initial state, H( Is), and all the important final states in the projectile. Instead of solving the resulting large coupled-state equations, they rewrite the time-dependent Schrodinger equation in the interaction picture. By dropping the chronological ordering operator T in the expansion of the S matrix and neglecting coupling between final states, they obtain a simpler formula for calculating transition probabilities to individual states. While the validity of these approximations is not clear, the method still requires calculations of P(b) to a few hundred final states for systems such as Si 1 4 + + H( 1s) -+Sit3+(nlm)+ H in order to obtain total capture cross sections. It is interesting to comment that for these systems charge transfer to individual final states is likely determined only after the charge clouds on the two centers separate, i.e., when the electron no longer oscillates from one center to the other. This situation should be built directly into the theoretical model in future studies where individual final states are not explicitly included if only the total capture cross sections are needed.

+

3. Eikonal Approximation

In the eikonal approximation for charge transfer between initial, & , and final, c#+, states, the transition amplitude does not follow from the distortion approximation (25). Instead, in the “post” form, it is given by

I

A ( + m ) = -i S_:dt ( ~ l p ( r p ) --G. /rT

I

t ))

(3 1)

for V , = -&/rT, if we limit ourselves to pure Coulombic three-body problems. The traveling atomic orbitals in the eikonal approximation are given by Ijlp(rp)= +P(rp)exp[-i(+t - $V r + Bv2t)] (324 +&TI

t)=

&(rT)

x exp

exp[-i(ETt

[ -i

+ $v * r + t u 2 t ) ]

(--Zp/rp)dt’]

The last exponential factor in Eq. (32b) describes the additional phase on the initial state due to the Coulomb interaction between the electron and the projectile. Without this last factor, Eq. (31) is reduced to the OBK approximation. If the “prior” form is adopted in Eqs. (31) and (32), then the final state will be multiplied by the additional Coulomb phase due to the interaction between the electron and the target nucleus.

INNER-SHELL VACANCY PRODUCTION

319

The eikonal approximation was first applied by Dewangan (1975, 1977) and by Tsuji and Narumi (1976) to electron capture for H + + H(1s) -+ H(n = 1,2) + H+. The method has been generalized recently by Eichler and Chan (1979a, b) and Chan and Eichler (1979) for charge transfer to arbitrary hydrogenic states of the projectile, in connection primarily with the study of electron transfer from hydrogen atoms to incident bare projectiles. Using a sum rule first given by Fock (1935) to carry out the subshell summations, a closed-form expression for capture to a given n th principal shell can be obtained in this approximation. (A similar sum rule is applicable to the OBK approximation; see McDowell and Coleman, 1970, Chapter 8.) The main attraction of the eikonal approximation (22) is that the total capture cross section to a given n th shell can be expressed in a closed form and often improves the agreement with experiment over the simple OBK approximation. However, there are still a number of shortcomings in this model: ( 1 ) It is useful, intrinsically, only when the capture probability is small since it is a perturbation approach. (2) There exists the discrepancy in this model between the post and the prior form for the capture probability. The problem associated with nonorthogonal states is not treated in this approximation. (3) The model does not approach the correct high energy limit (the relation of the eikonal approximation to the Born series expansion is discussed recently by Eichler and Narumi, 1980), nor is it valid in the intermediate energy region if back-coupling is important. Nevertheless, the model predicts reasonable agreement for total capture cross sections with experimental data and is particularly useful when capture to excited states dominates. It provides guidance to estimate charge transfer cross sections for experiments where initial or final states of the process are not defined. On the other hand, the agreement or predictions should always be judged critically in view of the above shortcomings. E. HIGH-ENERGY CHARGE TRANSFER Two recent reviews have addressed the subject of charge transfer at high velocity. Shakeshaft and Spruch (1979) reviewed the various mechanisms of charge transfer at asymptotically high impact velocities. In particular, the dominance of the second Born term over the first Born term (the OBK approximation) and the relation of the second Born to the classical double-scattering model suggested by Thomas (1927) are thoroughly examined in many examples. The other review by Belkic et al.

C . D . Lin and Patrick Richard

320

(1979) discussed the various second-order theories for charge transfer and the comparison with second Born results at asymptotically high velocities. In particular, the so-called continuous distorted wave (CDW) approximation has been applied by these authors and by others to calculate electron capture cross sections for a wide variety of collision systems, including some heavy-ion collisions. I . Second Born Approximation

The fact that the charge exchange process at the asymptotically highenergy limit is not given by the first Born approximation but by the second Born approximation was first proved by Drisko (1955) and subsequently studied by a number of workers (see the review by Shakeshaft and Spruch, 1979, for full references). The second Born cross section for K-K charge transfer in the high-velocity limit is u,32=

-

aOwh[0.295+ ( ~ T u / ~ " ) / ( + Z ZT)I P

(33)

while uosh u-I2 in the high-velocity limit. Thus, the second Born predicts a u - l l dependence at asymptotic energies. However, the approach to the high-energy limit is very slow. Some other recent developments in this area are worth mentioning. It has been shown (Shakeshaft, 1978a) that the third Born does not change the coefficient of the u-" term at asymptotic energies, but the coefficient of the u-12 term is modified. It also has been shown (Briggs and Dube, 1980) that capture from the ground state to the excited states also has the U-" dependence at asymptotic high volocities. However, it must be pointed out that for collision energies of a few rnegaelectron volts per nucleon, not only is the nonradiative capture cross section very small, but also charge exchange takes place predominantly by a radiative process (Briggs and Dettrnann, 1974, 1977; Lee, 1978). In evaluating the second Born amplitude, summation over intermediate states (in terms of Green's functions) are often carried out using some type of "peaking approximation." For hu/e2 >> Z,, Z,, free-particle Green's functions are often used in the evaluation of the second Born amplitude. With the help of peaking approximations, this gives the well-known second Born predictions for charge transfer at high velocities. The accuracy of the peaking approximation has been examined recently by Simony and McGuire (1981), where they evaluated the second Born term for H + + H( 1s) + H(1s) + H numerically without resorting to a peaking approximation at energies ranging from 5 MeV to 50 MeV. At higher energies, they showed that the error introduced by the peaking approximation is less than 15%. +

INNER-SHELL VACANCY PRODUCTION

32 1

Although it is generally agreed that higher Born terms do not alter the second Born prediction of u-” at asymptotically high energies, the importance of these higher Born terms in the Born series at slightly lower energies is not clear. In other words, the usefulness of the second Born calculations at slightly lower energies is not known, especially since there are little high-energy data for comparison. In a recent article, Shakeshaft (1980) discussed a new peaking approximation, which is valid for asymmetric (Z, << Z,) systems and for collision velocities hu/e2 >> Z , but arbitrary Z,. By using free-particle Green’s functions, he calculated K-K capture cross sections for Z , = 1 and Z, = 1-18. The resulting second Born cross sections are much higher than the first Born (OBK) cross sections, which are known to overestimate experimental capture cross sections for comparable systems (Cocke et ul., 1977; Radbro et a/., 1979). This result would imply that higher Born terms are important and have to be computed (an almost impossible task). Since the intermediate states included in the second Born amplitude are supposed to represent the direct Coulomb ionization channels, which are the major inelastic channels at high collision velocities, the free Green’s functions used in Shakeshaft’s calculation are obviously inappropriate for hole2 5 Z , . A calculation using Coulomb Green’s functions in the second Born amplitude has been carried out recently by Macek and Shakeshaft (1980). It was found that by using Coulomb Green’s functions, rather than the freeparticle Green’s functions, the second Born cross section is greatly reduced and becomes comparable in size to the first Born cross section in the region hu/e2 s Z,. We show in Table I11 the results of these two different second Born calculations and comparison with the OBK (first Born) results.

2. Other High-Energy Theories Since the second Born term does not become dominant until at very high velocities, other charge transfer theories are called for at high (but not asymptotically high) energies. There are several of these “highenergy” charge transfer theories proposed over the years which have been applied to simple collision systems. In the last few years we witnessed the application and suitable modification of some of these methods to heavy ion-atom collisions. Three such theories are singled out for discussion here. All three theories do give the u-I1 dependence at asymptotic velocities, but some do not have the same coefficients as the second Born predictions. For details of the asymptotic predictions, we refer the readers to Belkic et al. (1979).

C . D . Lin and Patrick Richard

322

TABLE III ELECTRON CAPTURECROSSSECTIONS ( I N UNITS OF TO’) BORNAPPROXIMATION A N D THE COMPARISON W I T H T H E FIRSTBORN(OBK) RESULTS”

C A L C U L A T E D IN T H E S E C O N D

ELah

Z,

(MeV/amu)

10

2.5

u?c

U2f

10.0

5.1 ( - 6 ) b 1.4 (-6) 1.4 (-7)

8.4 ( - 5 ) 1.4 (-5) 8.3 (-7)

3.4 (-6) 8.4 (-7) 6.4 (-8)

2.5 5.0 10.0

1.7 (-8) 4.5 (-8) 4.0 (-8)

1.4 (-6) 2.0 (-6)

3.2 (-8) 5.3 (-8) 2.8 (-8)

5.0

20

uOHh

1.0 (-6)

“ Second Born cross sections calculated using free-particle Green’s function are given in the column under uZfand those using Coulomb Green’s function are under u ~ The . projectile has Zr = 1 and the target has Z , = 10 and 20 and &ah is the projectile energy in the laboratory frame. The initial and final states are both hydrogenic Is states. Taken from Macek and Shakeshaft (1980). A(-B) = A

X

The impulse approximation has been applied without further approximations to proton-hydrogen and proton-helium (Bransden and Cheshire, 1963; Coleman and Trelease, 1968) collisions in a full quantum mechanical description. However, it is more convenient to describe the theory in the semiclassical approximation. This theory can be derived by a systematic expansion of the Green’s functions in terms of the target or the projectile potential (Briggs, 1977). For asymmetric collisions such as the capture of neon K-shell electrons by protons, the impulse approximation can be derived by assuming a single collision with the projectile proton and multiple scatterings with the stronger target neon potential. By adopting a “prior” form for the scattering amplitude and assuming straight-line trajectories, Jakubassa-Amundsen and Amundsen (1980) and Kocbach (1980) have derived the amplitude for charge transfer in the impulse approximation to be

a. Impulse approximation.

x

/ dr exp[-i(q

* v + &v’

+ eT

- ep)f1

INNER-SHELL VACANCY PRODUCTION

323

where & is the initial state wave function, &(q) is the final state wave function in momentum space, and +k)is the continuum Coulomb function in the field of the target nucleus. Equation (34) illustrates explicitly that charge transfer is directly related to the Coulomb ionization process (through the matrix element ($k)lVp(rp)I+?)I. Multiple scattering of the electron by the target nucleus is considered in this approximation, and thus second and some higher Born terms are included. (The second Born includes only a single collision with the target nucleus.) It predicts the same high-velocity limit as the second Born theory but is applicable to lower energies because multiple scattering is included (see Section IV). If the “post” form of the transition amplitude is adopted (Briggs, 1977), then the impulse approximation includes the multiple scattering with the projectile but only a single collision with the target. Such a theory was proposed (Briggs, 1977) to study electron transfer from light atoms by highly charged heavy ions, but its validity has been questioned (Belkic et al., 1979). It might be noted that various peaking approximations are often used in evaluating the transition amplitude in the impulse approximation, but a recent calculation by Jakubassa- Amundsen and Amundsen ( 1980) carried out the calculation numerically without any peaking approximation.

b. Continuum intermediate-state approximation and continuum distorted urave approximation. In evaluating the transition amplitude (34), various peaking approximations are often applied. Since the Coulomb function $hi)v is sharply peaked at q = 0, and if we neglect the q-dependence of the Coulomb function in (34), we obtain the transition amplitude A:!:)

=

-i

J-1dt m

(&rp)N(vT)

x exp{ -i[v

r

+ (eT

IFdivT, -

ep)t]}

1 , iv

‘rT

+

iurT)IVP(rP)t4T(rT))

(35)

where we have expressed the Coulomb function explicitly. In Eq. (3% v T = ZT / v and N ( v ) = I‘( 1 - i u ) e n u i is 2 the normalization for the Coulomb function. Equation (35) in its present form iE called by Belkic (1977), the continuum intermediate-state (CIS) approximation. Notice that if lFl(i uT , 1 , iv * r T + iur,) in Eq. (35)is replaced by its asymptotic form, then Eq. (35) reduces to the “prior” form of the eikonal approximation [cf. Eq. (31)l. The impulse and CIS approximation do not treat the scattering processes symmetrically. The continuum distorted wave approximation (CDW) introduced by Cheshire (1964) treats both the scattering of the electron by the projectile and by the target. In the semiclassical impact parameter approximation, CDW describes the initial state as $( t+,

324

C . D . Lin and Patrick Richard

r,,)+.,(rT),the product of a Coulomb function describing the Coulomb scattering of the electron by the projectile and an initial target atomic wave function 4Tcentered on the target. A similar expression is chosen for the final state. Thus, instead of Eq. (3% the transition amplitude takes the form

x exp{-i[v

r

+ (eT

-

ep)t]}) dt

(36)

where 0 is a certain differential operator which can be deduced from the time-dependent Schrodinger equation. These perturbative approaches can be regarded as the first-order approximation to some multiple-scattering theories, similar to that of the OBK approximation, which is the first-order term of the Born series. In particular, Gayet (1972) has shown that CDW is the first term in the multiple-scattering series of Dodd and Greider (1966). Notice that none of these theories consider the nonorthogonality of the initial and final states. These high-energies theories, unlike the second Born approximation, have been applied to the intermediate- and high-energy charge transfer collisions. The impulse approximation has been shown to apply in the v v, region for very asymmetric systems. Comparison of these calculations with experimental data will be discussed in Section IV. Notice that CDW has been applied extensively to a number of collisions in heavy-ion-atom collisions. The readers are referred to the review by Belkic et al. (1979).

-

F. ATOMICMODELS

Our theoretical discussions so far have been limited to one-electron, two-center problems. We have written Eq. (10) in the form which is valid for multielectron systems if the collision can be simplified by an active electron approximation. In this approximation, we deal with only the active electron which participates in the inelastic transitions. What happens to the other passive electrons is not considered. Such calculations are to be compared with inclusive experimental measurements, which do not provide details about transitions of passive electrons, or with measurements where such detail information are integrated. This is particularly important for heavy-ion collisions since multiple transitions involving more than one electron often are very prominent. Although it is straightforward to generalize the one-electron models presented in the previous sections to many-electron systems in terms of

INNER-SHELL VACANCY PRODUCTION

325

many-electron Hamiltonians and wave functions, this procedure is definitely inappropriate for most heavy-ion collisions since the many-electron final states are not specified in these measurements. It is desirable to see what kinds of measurements can be interpreted in terms of single-electron models. This question has been undertaken by Reading and Ford (1980) recently, following an earlier article by Reading (1973). Consider, for example, the K-shell-hole production of a n atom by heavy ions. The K-shell electron can be lifted to the unoccupied orbitals of the composite system directly without disturbing any other passive electrons, or it can proceed through a two-step process in which an L-shell electron is lifted, followed by an excitation of the K-shell electron to the L-shell vacancy. This K-shell vacancy formed in the first step can also be filled by the knockdown of an L-shell electron. All these single-electron processes can also be accompanied by the simultaneous hole production of other electrons. Starting with an independent electron approximation where initial and final electronic wave functions are represented by determinants consisting of single-electron orbitals, Reading and Ford (1980) showed that the probability for the K-hole production in an inclusive experiment, i.e., an experiment where a K hole is observed but the final states of other electrons are not determined, can be calculated in the single-particle approximation directly. To be more precise, letting uLibe the transition amplitude from orbitalj to orbital i, these authors showed that, in an inclusive experiment, the K-hole production probability is

where the K-shell orbital has the index 1, the first summation is over all unoccupied orbitals, and the second summation is over the occupied orbitals. This simplification is possible because closure relations can be applied in an inclusive experiment. The probability p1in Eq. (37) includes single and double K-shell vacancy production, as well as K-hole production accompanied by other multielectron transitions. However, for experiments like multihole production or like charge transfer, accompanied by hole production, interference terms appear. For example, the two-hole production probability is given by the determinant

The equality p12 = plp2 occurs when the two orbitals have opposite spins since for spin-independent forces, for a given k , either [ i l k or f h k is

326

C . D. Lin and Patrick Richard

zero if the orbitals 1 and 2 have opposite spins. This result (38) alters the simple binomial form discussed earlier by several authors for multiplehole productions (McGuire and Richard, 1973; Hansen, 1973; McGuire and Weaver, 1977). These earlier works, although they also start with an independent electron approximation, fail to incorporate the antisymmetrization of the product functions. The importance of these interference terms will be discussed further in connection with the comparison with experimental data in Section IV,A. The validity of these results hinges upon the independent electron model where electron correlation is completely neglected. Even within the single-electron model from which ark’s are computed, there are intrinsic complications in the assumptions of suitable oneelectron potentials [e.g., V, + V p in Eq. (lo)]. For lower collision velocities, particularly in the MO region, the active electron is often assumed to be under the screening potential due to the fully relaxed passive electrons, i.e., the electronic orbitals of passive electrons are assumed to be the molecular orbitals at that particular internuclear separation. For collisions at higher velocities, on the other hand, one often assumes that the passive electrons remain at their respective initial atomic orbitals throughout the collision, i.e., the screening is by the passive electrons in the unrelaxed orbitals. For inner-shell excitation at intermediate velocities, the outer electrons neither relax completely nor stay at the atomic orbitals of the initial state. In fact, most likely these outer electrons will be ionized. Therefore, even with an atomic potential, this potential is not necessarily the potential of the neutral atom. Fortunately, the inner-shell transition probability often does not depend upon the degree of outer-shell ionization significantly (Lin and Tunnell, 1980). However, if full relaxation of the molecular orbitals for passive electrons is assumed, the calculated K-K charge transfer cross section are found to deviate significantly from those calculated from the atomic models where the orbitals of passive electrons are frozen (Fritsch el al., 1981).

IV. Comparison of Theories and Experiments In this section selective experimental data are displayed in comparison with calculations of theoretical models. The data are chosen for those cases where calculations are available and thus provide a testing ground for examining the theoretical models discussed in the previous section. We shall limit the comparison primarily to experimental charge transfer cross sections over a wide range of collision partners in the intermediateand high-energy region and mostly to inner-shell processes, where the

327

INNER-SHELL VACANCY PRODUCTION

single-electron approximation is more likely to be valid and the dynamic models are not clouded by many-electron effects.

A. TOTALSINGLE-ELECTRON TRANSFER CROSSSECTIONS I . H + + He

In Fig. 20 we show the total electron capture cross sections for protons colliding with helium atoms in the energy range from 2 to 1000 keV. Experimental data are adopted from the compilation of Tawara and Rus,

I

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FIG.20. Total charge transfer cross sections for H' on He collisions. Experimental results are taken from the compilation of Tawara and Russek (1973). Theoretical calculations: curve 1, two-state, two-center atomic model calculations (Green rt a / .. 1965; Bransden and Sin Fai Lam, 1966); curve 2, CDW calculations (Belkic et a\., 1979); curve 3, impulse approximation (Bransden and Cheshire, 1963).

328

C. D . Lin and Patrick Richard

sek (1973), where references to the original sources can be found. The total electron capture cross section peaks at E 25 keV and drops sharply as the collision energy decreases or increases away from this value. Three theoretical calculations are shown in Fig. 20. The results of the coupled two-state atomic expansion method by Bransden and Sin Fai Lam (1966) and Green et af. (1965) are shown as curve 1 . The calculation reproduces the overall shape and magnitude of the experimental data very well but tends to overestimate the cross sections for collision energies away from the peak. Sin Fai Lam (1967) has performed a five-state calculation by including the 2s and 2p states of the hydrogen atom in addition to the 1s states of H and He, but the result is essentially identical to the earlier two-state calculation. This is not surprising since capture in this case is dominated by the transfer to the ground state of the hydrogen atom. In fact, when the capture is primarily to the 1s ground state, the capture cross sections to the excited bound orbitals with principal quantum number n are often approximated by v,,= al,/n3 where vlsis the 1s capture cross section. (This rule is usually valid for high-energy collisions and originates from the n -3’2 dependence of the hydrogenic radial wave functions near the origin.) Thus, in comparing with experimental total capture cross sections, theoretical results for ulsare calculated and the total theoretical capture cross section is approximated by vto,= 1.2 vlsby summing over the excited-state contributions and assuming the l / n 3 dependence for excited states. In Fig. 20, results from two other high-energy theories are also shown. In particular, curve 2 gives the CDW results (see Belkic et af., 1979) and curve 3 gives the results from the impulse approximation (Brandsen and Cheshire, 1%3). The CDW approximation, when applied to the intermediate-energy region, often overestimates the total cross section. This is due to the fact that CDW is a perturbation theory and fails when back-coupling becomes important (McCarroll, 1961). This back-coupling is considered in the coupled-state calculations. In the high-energy region shown (-400-1000 keV), there are some discrepancies among the three theories, with the coupled two-state results tending to be largest. The discrepancies become more pronounced at increasing energies since the coupled-state method is not suitable for high-energy charge transfer processes. The calculations shown are obtained using uncorrelated helium wave functions. Correlation usually does not play any significant role for collisions at intermediate energies. Baynard and Szuster (1977) have used correlated helium wave functions in their CDW calculations, and have obtained less than 20% effects compared with uncorrelated wave function calculations.

-

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E (MeV) FIG.21. Total K-shell charge transfer cross sections for H’ on Ne collisions. Experimental data: ( 0 )Rodbro et a / . (1979); (W) Cocke et ml., (1977). Theoretical calculations: curve 1, two-state atomic expansion (Lin el a/., 1978); curve 2, target-centered multistate calculations (Ford et a / . . 1981); curve 3, CDW (Belkic P I al., 1979); curve 4, impulse approximation (Jakubassa-Amundsen and Amundsen, 1980).

2. H + + Ne(K)

Electron transfer from the outer shells of neon by protons is the major inelastic process for E 10-50 keV (Lin and Tunnel], 1980). As far as the electron capture mechanism is concerned, outer-shell capture in this system is similar to the symmetric H f + H collision. On the other hand, capture of K-shell electrons by protons is a very asymmetric collision in that the electron is originally bound by Z,. 10, while the incoming projectile has only Z,, = 1. Experimental cross sections for K-shell electron transfer from Ne to protons were measured recently by Cocke et al. (1977) and by Radbro et al. (1979) by detecting the K-Auger electrons in coincidence with charge-changing projectiles (see Fig. 6 in Section 11). Their results are shown in Fig. 21, together with theoretical calculations.

-

-

330

C . D.Lin nnd Patrick Richurd

Curve 1 is from the two-state atomic expansion (TSAE) model (Lin et al., 1978). This model predicts reasonable cross sections near the high-energy side of the peak but fails badly on the lower energy side. The failure of the simple TSAE model in the low-velocity region is not unexpected since transitions to the final states through intermediate states are important at small collision velocities. Near the peak, the one-step distant collision dominates, and the TSAE model is useful. At higher energies close collisions dominate the charge transfer process, and again the TSAE model fails. To include the effects of intermediate states, which in this case are primarily target centered, the multistate atomic expansion and the impulse approximation have been adopted. In curve 2, the results of the multistate calculations of Ford et a/. (1981) are displayed. In this calculation, targetcentered excited states and pseudostates (representing the continuum) are used in a one-center expansion to represent the “exact” wave function of the system. Charge transfer probability is then projected from this “exact” wave function. In their calculation target-centered wave functions with I = 0, 1, 2 are included. Curve 4 is the result of the impulse approximation (Jakubassa-Amundsen and Amundsen, 1980). This special type of impulse approximation used by these authors is designed primarily for the asymmetric systems. Multiple scattering with the strong target potential is included to high orders, whereas the weaker interaction with the projectile potential is treated to first order. The agreement of this theory with data is very good. This indicates that a second Born calculation alone will not be adequate since higher order terms are important, as can be concluded from this theory and the negative result of the second Born calculation (Shakeshaft, 1980). In Fig. 21 we also show the CDW results, curve 3, which agree with experimental data quite well at higher velocities but not at lower velocities. 3. Li3++ Ne(K)

Rsdbro et ul. (1979) also measured the total electron transfer cross sections from the K-shell of neon to bare Li ions. Their results are shown in Fig. 22 for incident energies between 1 and 3 MeV/A. Theoretical calculations from the TSAE method are shown as curve 1 and from the CDW method are shown as curve 3. Two curves from the work of Ford et al. (1981) are shown. Curve 2a corresponds to the prediction from the single-electron model; curve 2b gives the results when interference with other inelastic processes is included (see Section 111,F). There is a noticeable discrepancy at lower energies between curves 2a and 2b, signaling the importance of many-electron processes. No experimental data are available to test this conclusion. Since electron capture from the outer

33 1

INNER-SHELL VACANCY PRODUCTION

Li ”-Ne

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0.5

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2

3

4

5

E (MeV / amu) FIG.22. Total K-shell charge-transfer cross section for Li3+on Ne collisions. Experimental data are taken from Rodbro et a / . (1979). Theoretical calculations: curve 1 is from the TSAE model; curves 2a and 2b are from Ford et a / . (1981). Curve 2a includes the manyelectron effects; curve 3 is from the CDW model (Belkic et a/.. 1979).

shells of neon is very large for E < 1 MeV/A, the purity of Li3+ions is difficult to maintain to allow the data in Fig. 22 to be extended to lower energies. However, it must be pointed out that the conclusion in the work of Ford et al. (1981) depends upon the accuracy of the transition amplitude for other inelastic processes. If the L-shell capture probability (which is large for E < 1 MeV/amu) is not accurately calculated, then the manyelectron effects will be overestimated. Notice that their method for charge transfer is valid only when the capture probability is small and probably not very accurate for the L-shell capture of neon by Li3+.Their calculation for H + Ne(K) shows a very small contribution from many-electron effects. +

4. F 9 + + Si(K)

The cross sections for charge transfer from the target inner shell by heavy projectiles can also be deduced by studying the projectile charge-

332

C. D . Lin and Patrick Richard

state dependence of the production of inner-shell vacancies. By assuming that inner-shell vacancies are created by the direct Coulomb ionization mechanism and by the charge transfer processes, and further assuming that the cross sections due to Coulomb ionization are independent of the charge state of the projectiles, then experimental K-K charge transfer cross sections can be calculated [see Section 11, Eq. (9)]. For example, the K-shell vacancy production cross section of Si due to bare F9+ions consists of direct ionization plus capture into the K-shell and the higher shells of the fluorine ions, whereas in the K-shell vacancy production cross section of Si due to helium-like and other lower charge-state fluorine ions, the K-K charge transfer mechanism is not allowed because the K-shell is occupied. If we assume that the K-K charge transfer channel is the only difference between the two systems in producing K vacancies in the target, then the K-K charge transfer cross sections for the collision system F9++ Si(K), for example, can be calculated by subtracting the vacancy production cross sections of target atoms by helium-like and other lower charged ions from those of the bare projectiles. In these types of experiments, K X-ray-ion coincidence is not needed but the charge-state dependence of the cross section has to be measured. With the assumptions in the previous paragraph in mind, we show the K-K charge transfer cross sections of F9++ Si(K) in Fig. 23. The experimental data are from Tawara er al. (1978b), while the theoretical results are from the various coupled-state calculations. Notice that the data displayed in Fig. 23 cover the lower energy side of the a,, peak. It was explained in Section IV,D, 1,a that in this region the coupled-state calculation is more sensitive to the potential U, and Up and thus to the atomic potential used in the model calculation. Curve 2 in Fig. 23 is the TSAE result using screened Coulombic target potential, while curve 1 is from the TSAE calculation using a Herman-Skillman atomic potential (Herman and Skillman, 1963) for the target (Lin and Tunnell, 1980). The difference originates primarily from the potential Up(R)used in the two atomic models. We conclude that a more realistic atomic potential is important in calculating aKKin the lower velocity region. In Fig. 23 we also notice that there is still some discrepancy between curve 2 and experimental data. It has been questioned whether this is an indication of the breakdown of the atomic expansion model in this velocity region and whether a molecular coupled state expansion is preferable. In an attempt to answer this question, Fritsch et al. (1981) used a variable-screening model (Eichler and Wille, 1974, 1975; Fritsch and Wille, 1978) to calculate a,, according to a two-state MO expansion. Two types of MO calculations are performed. In the usual M O calculation, all the passive electrons are assumed to be completely relaxed during the collision. A two-state MO calculation based upon

INNER-SHELL VACANCY PRODUCTION I

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E( MeV/amu ) FIG.23. Total K-K electron transfer for FS+on Si collisions. Experimental data are taken from Tawara et a / . (1978b). Theoretical calculations: curves 1 and 2 (Lin and Tunnell, 1980). TSAE calculations using Herman-Skillman target potential (curve 1 ) and screened hydrogenic target potential (curve 2); curves 3 and 4, the two-state molecular orbital model calculations (Fntsch et a / . , 1981) using a fully relaxed Hamiltonian (curve 3) or a frozen Hamiltonian (curve 4).

this model is shown in curve 3. In this calculation the electronic Hamiltonian is different from the one used in the TSAE calculation (from which curve 1 is based), where the passive electrons are assumed to be frozen. Thus, the comparison between curve 1 and curve 3 is more than just the comparison between the two-state A 0 and MO models. Since in the energy region studied, the projectile velocity is not slow compared with the orbital velocities of outer-shell electrons, full relaxation of passive electrons seems to be an undesirable assumption. Thus, in curve 4 a two-state MO calculation based upon a frozen model Hamiltonian (identical to the one used for curve 1) is also displayed. Neither curve 3 nor curve 4 shows improvement with experimental data in the energy region shown. Thus, we conclude that a two-state A 0 or MO expansion is not quite adequate in the lower energy end of Fig. 23, and a multistate calculation is needed. This conclusion is consistent with the recent work of Winter and Hatton (1980), where they

334

C . D . Lin and Patrick Richard 1

~

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1

- SINGLE K-K CHARGE TRANSFER

~

1

[

l

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t 8

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ZP FIG.24. Projectile Z,-dependence of K-K transfer cross sections with Ti target at 5 Meviamu. Experimental data are taken from Hall et a / . (1981); the solid line is from TSAE calculations.

studied H + + He+(ls)+ H(1s) + He2+,using a multistate MO expansion, and found that ukkis reduced substantially from the two-state result when additional states were included in the expansion. 5 . Projectile Z,-Dependence of K-K Transfer Cross Sections

The feasibility of choosing the nuclear charge Z, and the charge state q of the projectile permits experimentalists to measure the Z,-dependence of the K-K charge transfer cross sections for a given target, as discussed in Section 11. In Fig. 24 we show the Z,-dependence of crKh for a Ti target at a fixed projectile velocity of 5 MeV/amu (Hall et a l . , 1981). The cross section rises rapidly (but less than the Z,? prediction of the OBK theory) at small Z, &. The solid curve gives the TSAE and tapers off near symmetry Z, prediction, which shows quite reasonable agreement with experimental data. No other theoretical calculations are available for comparison, but the perturbation theories are not expected to be valid for some near-symmetric collisions considered here.

-

335

INNER-SHELL VACANCY PRODUCTION

‘ 7 NEON K-AUGER CROSS SECTION

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In the last example for total capture cross sections we consider the E-dependence and q -dependence of the neon K-vacancy production cross sections for Fq++ Ne in the energy range of 4-35 MeV. In Fig. 25 the neon K-Auger cross sections of Woods et al. (1976) and Hagmann el al. (1981) are displayed for various F p +ions. These cross sections are essentially equal to the vacancy production cross sections CT; of Ne K-shell electrons since the fluorescence yield w for Ne is very small. We notice from Fig. 25 that crQv for y G 6 are essentially independent of y at these velocities. [Large charge-state dependence effects are observed at considerably lower velocities (0.1-0.5 MeV) by Hoogkamer er al. (1977) and less pronounced effects at intermediate velocities (1-4 MeV) by K. Holzer and R. Brenn (private communication, 1980) for Ne-Ne collisions.] The large value of aP,for q = 8 is attributed to the opening of K-shell vacancies in the projectile and the opening of the K-K charge transfer channel as an efficient mechanism for Ne K-shell electron removal. By assuming CT$ = wFS6 + C l i h , where (Tkii is the K-K charge transfer cross sections for F H ++ Ne(K) calculated from the TSAE model, and u $ ~ ”

336

C. D. Lin and Patrick Richard

is from experimental results, we compare a$thus calculated (solid curve) with experimental a$ in Fig. 25. The “theoretical” at shows reasonably good agreement with the experimental a$.

B. DIFFERENTIAL CROSS SECTIONS I . Introduction

Differential cross sections for electron capture in ion-atom collisions often exhibit detailed information not evident in the integrated total cross sections. In Section I11 we showed that for the proton-hydrogen atom resonant capture, the capture probability at small impact parameters, as displayed by the P(b)oscillation with E in Fig. 19, exhibits very different behavior from the different theoretical models, although the total capture cross sections predicted by these models do not show any significant discrepancies. From the experimental standpoint, there has been a lack of data for d a l d f l for charge transfer. This is mainly due to the fact that angular distributions for electron capture are sharply peaked in the forward direction at high impact energies. Since the first daldfl measurement of K-charge transfer in heavy-ion collisions by Cocke et al. (1976) for H+-Ar at an impact energy of 6 MeV and the subsequent measurement by Bratton et al. (1977) for H+-He at 293 keV, many more such measurements for different systems and different energies are being carried out at various laboratories. 2 . Theoretical Consideration In discussing differential cross sections, experimentalists measure duldfl directly, while all the semiclassical theoretical models adopt the impact parameter approximation, where the electron transfer probability P(b) at a given b is directly calculated. If the scattering of the heavy particles can be unambiguously described by a hyperbolic orbit calculated from classical mechanics, then there is a one-to-one correspondence between the scattering angle 19 and the impact parameter. Assuming, for example, that the projectile is scattered solely by the Coulomb field of the target nucleus, then the relation

d a / d f l = P(b)(da/dfl)Rutherford

(39)

holds, and the value of P(b) can be calculated from experimental measurements, However, the simple relation (39) is valid only if the quantum diffraction effect is small compared with the deflection of the particle due to the field.

INNER-SHELL VACANCY PRODUCTION

337

The circumstance where Eq. (39) is valid has been derived by Bohr (1948). Assuming Coulomb scattering between charges Z,,e and 4 . e and defining the parameter K

=

(Z,J+')/hv

(40)

Bohr showed that K >> 1 is the necessary and sufficient condition for the justification of the classical consideration (39). For inner-shell processes, Eq. (39) can be easily extended to include the screening effect. By assuming a screened potential of the form V ( r )= ( b/r)ePr'"(the parameters n and b can be fitted from the elastic scattering data), an equation similar to Eq. (39) can be derived from which the experimental P(b)is deduced. If the condition K >> 1 is not valid, then the experimental d d d R should be compared with the theoretical d d d R directly. This procedure is generally true for collisions at high velocities [cf. Eq. (40)l. In principle, it is possible to formulate the collision problem in a quantum approach where differential cross sections are calculated directly. On the other hand, one would like to have the simplification introduced by the impact parameter approximation. It is desirable to have a theory of d d d R for heavyparticle collisions which gives differential cross sections identical to the quantum treatment to first order in the ratio of the electron mass to the mass of the heavy nuclei, since this is the limit where the impact parameter method is expected to be valid. This is achieved by employing the eikonal approximation, as formulated by Schiff (1956), Glauber (19591, and McGuire and Weaver (1977). By employing a partial wave analysis in which the summation over partial waves is replaced by an integral and the Legendre polynomials by their asymptotic form, these authors derived an expression for dddR. In the context of atomic collisions, McCarroll and Salin (1968), Wilets and Wallace (1968), and Chen and Watson (1968) also derived the desired equation from the full quantum formalism in which all quantities are replaced by their limits as the masses of the atomic nuclei become infinite. The results for the differential cross sections for transition from K-shell to K-shell in the center-of-mass system is

where p is the reduced mass and 6(b) = - ( i / v )

IoffiW ( R )dZ

(42)

is the eikonal phase calculated with an effective internuclear scattering potential W ( R )and Z = vt. Thus the internuclear potential W ( R )plays a definite role in the calculation of d a / d R . although the precise nature of W ( R )is not important in the calculation of the total cross sections, espe-

338

C . P . Lin and Patrick Richard

cially for high-velocity collisions where straight-line trajectories are often assumed in the theoretical models. This dependence of daldfl on W ( R ) causes additional complications in comparing theoretical and experimental differential cross sections, because the precise nature of W(R)for a given process is not accurately known. For multielectron ion-atom collisions, the screening of passive electrons has to be considered, but a precise description of the relaxation of the electrons and the degree of screening depends upon the dynamic scattering process. For electron transfer problems, a consistent choice of internuclear potential is especially important for those first-order theories where the nonorthogonality of initial and final states are not considered (Belkic et al., 1979, p. 357) since the internuclear potential contributes to the transition amplitude in these models. 3 . d a l d f l at Medium and High Velocities

The differential cross section for electron transfer for protons on helium atoms at 293 keV is shown in Fig. 26. Experimental data are taken from Bratton e f al. (1977) and are compared with three theoretical calculations. The solid curve (curve 1) is from Lin and Soong (19781, where the TSAE model was used in calculating the capture probability amplitude A(b)[cf. Eq. (41)] and a static potential was used in calculating the eikonal phase q b ) . This static potential was chosen identical to the early second Born approximation (SBK) work of Rogers and McGuire (1977) in which the OBK approximation was used in calculating A(b). This latter work is not shown in Fig. 26 since the OBK overestimates aKKby a factor of 3.6 at this energy. The static potential model used in this work has the undesirable feature that the target nucleus is screened by the passive electrons as well as by the active electron. The inclusion of the active electron in screening the target nucleus results in double counting of the active electron-target nucleus interaction since this interaction is treated already in calculating A ( b ) .The dashed lines are from the recent work of Rivarola et al. (1980), where CDW was used to calculate A(b)and a different static potential is used to compute 6(b).[To be more precise, in the CDW calculation d d d f l is computed in the wave formulation assuming no internuclear interaction between the heavy particles. The A(b) is then obtained from the inverse transformation of Eq. (41).1 The screening potential used by these authors excludes the screening of the target nucleus by the active electron and frozen orbitals are assumed for passive electrons to obtain the screening potential. If this static potential is used in calculating 6(b),together with the TSAE result for A(b), then curve 3 is obtained. In Fig. 26 the experimental data pertain to capture to all bound states of the projectile, while theoretical calculations include the dominant K-K

339

INNER-SHELL VACANCY PRODUCTION

Io5 i

0.01 0.02

0.03

0.04

0.05 0.06

0.07

FIG.26.

Differential cross sections for total electron capture cross sections for protons a / . (1977). Theoretical calculations: curves 1 and 3 are obtained using the same capture transition amplitude calculated from the TSAE model but different eikonal potentials for the internuclear motion (see text); curve 2 is from the CDW model (Rivarolaet d.,1980). The eikonal potentials used in calculating curves 2 and 3 are essentially identical.

on He at 293 keV. Experimental data are taken from Bratton et

charge transfer only. The overall agreement between theory and experiment is good, although the work of Rivarola et al. (1980) showed some structure not evident in the experimental data. It is not clear if the structure can be washed away by including explicitly the capture to excited states or if it is the result of the approximations used in calculating A(b) and/or q b ) . The role of the internuclear potential in determining d d d f l becomes more important with decreasing collision velocities since the eikonal phase (42) is inversely proportional to the projectile velocity and since the range of impact parameters contributing to (T also increases with decreasing velocities. Thus, the measurement of differential cross sections at lower energies will provide a more sensitive test of the model for different internuclear potentials. In Fig. 27 we show dddfl for the capture of

340

C . D . Lin und Patrick Richard

FIG.27. Same as Fig. 26 except at 30 keV. Experimental data are from Martin et a / . (1981). The two theoretical curves differ in the eikonal potentials used; both use the same capture amplitude calculated from the TSAE model.

electrons from helium atoms by protons at E = 30 keV (Martin et al., 1981). This corresponds to the energy where the cross section peaks, and thus capture contributions from large impact parameters are important. In Fig. 27 the experimental data are compared with two theoretical calculations in which A(6)is obtained from the TSAE calculation but with two different static potentials for calculating 6(b)[cf. Eq. (41)l. If the static potential includes the screening of the target nucleus by the active electron, then the resulting d u / d Q as displayed by the solid line, shows a dip in the angular distributions not observed in the experimental data. On the other hand, if the screening of the target nucleus by the active electron is then we obtain the curve shown in dashed lines. not included in W(R), This curve does not predict any dip and is in better overall agreement with experimental data. It must be pointed out that this later screening potential, as described by Rivarola et al. (1980), is the consistent internuclear potential to be used in calculating differential cross sections if the many-electron ion-atom collision is described by the active-electron approximation and the passive electrons are treated as frozen during the collision.

INNER-SHELL VACANCY PRODUCTION

1

'0

3

.01

.02

.03

34 1

3

.04

.05

-06

.07

8,,.(degrees) Fic;. 28. Differential cross sections for H + + H(ls) + H(1s) + H + at 50 MeV. The solid line labeled OBK is the first Born (OBK) calculation. The other three curves are from the second Born calculations, with peaking approximation(s) used in obtaining curves 2 and 3. Curve I is obtained by evaluating the second Born term numerically (Simony and McGuire, 1981).

4.

d u l d n ut Asyrnptoticdly High Velocities

The mechanism of charge transfer at asymptotically high velocities differs from the corresponding process at lower energies. Several theories for charge transfer at asymptotic velocities predict a peaked structure in the differential cross sections. In Fig. 28 we show the theoretical predictions of duldRfor H + + H ( l s + 1s) at 50 MeV from the first Born (OBK) and the second Born approximation. The second Born prediction calculated numerically (Simony and McGuire, 1981) is shown in curve 1, while the two other curves are the approximate second Born predictions resulting from two peaking approximations (curve 3) or one peaking approximation (curve 2) in the evaluation of the second Born amplitude. The second Born prediction is characterized by a minimum near 8 = 0.03' and a second peak at 6 -0.05'. The minimum originates from the interference of the first and second Born amplitudes, while the second peak is the con-

C. D. Lin and Putrick Richard

342

tribution from the double scattering in the second Born theory of charge transfer. The prediction shown in Fig. 28 has not yet been confirmed experimentally because of the small cross sections and the small angles involved. According to the calculations of Simony and McGuire (1981), the structure shown in Fig. 28 disappears for E d 5 MeV. More importantly, it is not clear whether the structure shown in Fig. 28 will be washed away by the contribution from higher Born terms. Experimental measurements are desirable to check these predictions. 5 . Iinpuct Parumeter Dependence for K-K Charge Transfer

For collisions involving heavy ions, the quantum diffraction effect is less important for a given collision velocity u compared with lighter ions, and the condition K >> 1 is satisfied so that Eq. (39) can be used to obtain an experimentally determined P(b). In Fig. 16 given in Section 11, we notice that the Ne K-Auger electron vacancy production probabilities at different impact parameters showed quite different magnitudes and behaviors for F'lt and F x +ions. If we assume that the difference is due to the additional K-K charge-transfer channel for Fx+ions in creating Ne K vacancies, then by subtracting the vacancy production probability for F" ions from that for Fx+ions at each impart parameter, we obtain the experimental P(b)for K-K charge transfer for Fx++ Ne collisions. These experimental results at 10 MeV and 4.4 MeV are shown in Fig. 29 in comparison with the TSAE calculations. In Fig. 29b, the experimental probability P(b) at 10 MeV is very small at small impact parameters and peaks at b = 0.23 a.u. Both the behavior and the magnitude are well reproduced by the TSAE calculations. In Fig. 29a, the P(b) for the same system but at a lower energy of 4.4 MeV is shown together with the TSAE prediction. The P ( b )shows oscillatory behavior with a minimum occurring at b = 0.23 a.u. which is reasonably well reproduced by the TSAE calculation. The two peaks at h 0.1 a.u. and h 0.35 a.u. are also well predicted by the calculation. The only other P(b) measurement for K-K charge transfer involving heavy ions of which we are aware is the work of Schuch ef af. (1979). In this work, a hydrogen-like S'"+ beam at 32 MeV was collimated to collide with an Ar gas, and the scattered particles were detected in coincidence with emitted characteristic X rays of S and Ar. From the measured PP\'(h)/P:(b) ratio, they compared the data with the semiempirical twostate model predictions of Briggs (1974). The comparison, as presented in Fig. 2 of Schuch et al. (1979), shows large discrepancies as opposed to the good agreement shown in Fig. 29 for Fxt on Ne. The experimental ratio is

-

-

INNER-SHELL VACANCY PRODUCTION I

I

I

I

0.2

0.3

1

I

I

0.5

0.6

343

1

1.0

-an

0.6

0.4 0.2

0

0.1

0.4

b ( 0.u. 1

F I G 29. . Impact parameter dependence of K-K charge transfer probability: (a) FXf+ N e at 4.4 MeV; (b) FK++ N e at 10 MeV. Experimental data are taken from Hagmann et d. (1981), and the theoretical calculations are from the TSAE model.

much less (by a factor of -10) than the theoretical prediction and the data showed two minima, while the minimum at larger b is not predicted by the theory. The two-state model of Briggs is essentially a simplified TSAE model. We have made a TSAE calculation for this system, and the result does not agree with experimental data. It appears that a new measurement is called for.

C. DOUBLE K-ELECTRON TRANSFER Double- and multiple-electron transfer processes are common phenomena in heavy-ion collisions. A great number of experimental data have been accumulated over the years (Betz, 1972; Tawara and Russek, 1973; Richard, 19751, but there is a lack of theoretical analysis because the final states of the multiple processes in these experiments are often not determined, and because of the difficulties theorists have in dealing with

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single-electron transfer processes much less multielectron transfer processes. However, there are some recent experimental data which can be interpreted as resulting from the transfer for two K-shell electrons to the K shell of the bare projectiles. In an earlier study, Woods et al. (1976) bombarded neon atoms with highly ionized fluorine, oxygen, and nitrogen ions and observed a large enhancement of the hypersatellite structure in the neon Auger electron spectra when and only when bare projectiles were used. These hypersatellites, lying on the high-energy side of the satellite lines, are due to the filling of double K-shell vacancies by outer electrons. Starting with an independent electron approximation, i.e., by neglecting the correlations between the electrons, McGuire and Weaver ( 1977) as well as Reading and Ford (1980) derived an expression for the double K-K charge transfer probability in terms of the single K-K charge transfer probability P( 6 ) calculated in the single-electron approximation. In terms of P(b), the total cross sections for the single-electron transfer is uKK = 2 * 2.rr

Ix P(l

-

P ) b dh

(43)

0

where the first factor 2 comes from the two K-shell electrons; the double K-K electron transfer is uZtiaK = 2n

J m P 2 bdb 0

(44)

This result is independent of how P(b) is calculated in the single-electron approximation as long as the electron-electron correlation is neglected. By using the P(b)calculated using the TSAE model, Eq. (44) allows us to calculate double K-K electron transfer cross sections. To show the validity of this simple model, we display in Fig. 30 the symmetric resonant double-electron transfer cross sections for He2++ He + He + He2+.The experimental results, shown as a solid line, are due to Pivovarer a/.(1962); the circles are from Allison (1958); and the triangles are from Bayfield and Khayrallah (1975) and Shah and Gilbody (1974). The dashed-double dotted lines are from the work of Fulton and Mittleman (1966), using coupled-state calculations in terms of atomic orbital expansion. The result of using Eq. (44) is shown in dashed lines where the TSAE is used to compute P(b). The agreement between this simple calculation and experimental data is quite reasonable. To compare the double K-K electron transfer data from heavy-ion exfor SiI4++ periments, we show in Fig. 31 the energy dependence of uzti.2K Ti and in Fig. 32 the projectile Z,-dependence of uZKaK for 5 MeV/amu projectiles on Ti taken from Hall et al. (1981) where the cross sections are obtained according to Eq. (9b). The results of TSAE calculations in con-

- 9 ' 01

200

400

600

800 1000

E (keV)

FIG. 30. Double K-K electron transfer in He'+ on He collisions. Experimental data Pivovarer (11. (1962);(0) Allison (1958);(A) Shah and Gilbody (1974) and Bayfield and Khayrallah (1975). Theoretical calculations: (I- . -) Fulton and Mittleman (1966); (-. -) TSAE model [together with Eq. (44)l.

(--)

01 Q3 0.4 0.5 I I n i I

v/v, 0.6 I

0.7

0.8

I

I

I

1111111111/IIII

Ioz'b,o

10 .

2.0

30

4.0

5.0

6.0

7.0

E/M (MeV/omu) + FIG. 31. Energy dependence of double K-K charge-transfer cross sections for . % I 4on Ti. Experimental data are from Hall ef a / . (1981); theoretical calculations are from TSAE model [together with Eq. (44)l. The x are from A. L. Ford and J. F. Reading (private communication, 1980).

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0

5

10

15

ZQ

FIG.32. Projectile Z,,-dependence of double K-K charge transfer cross sections for Ti target for projectiles with E = 5 MeV/amu. Experimental data are from Hall et a / . (1981); theoretical calculations are from TSAE model [together with Eq. (44)j.

junction with Eq. (44) (given by the solid line) are also shown for comparison. The calculations agree with the E-dependence and Z,-dependence in support of the validity of the independent electron model and the TSAE model. The results of the multistate calculations of A. L. Ford and J. F. Reading (private communication, 1980) for the E-dependence are indicated by x in Fig. 31 and are also in reasonable agreement with the experimental data. A critical test of the independent electron approximation used in calculating crhh and (72h& given in Eqs. (43) and (44) can be performed by measuring the impact parameter dependence of the single and double K-K charge transfer probabilities by bare projectiles for near-symmetric collisions. From Eq. (43), it is clear that the single K-K capture probability 2P( I - P ) will show a minimum for impact parameters b, where P(b) 1.

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INNER-SHELL VACANCY PRODUCTION

347

At these impact parameters the corresponding double K-K capture probability P' will show a maximum. A direct experimental test of these predictions in heavy-ion collisions is desirable to assess the validity of the independent electron approximation and the importance of electron correlation in inner-shell processes. The resonant He2++ He case has been measured by Keever and Everhart (1966) at three impact parameters over the incident ion energy range between 3 and 200 keV.

V. Concluding Remarks In this article we have reviewed the mechanisms of inner-shell vacancy production in heavy-atom collisions. We have compared the relative importance of direct Coulomb ionization and charge transfer in the vacancy production by varying the projectile charge state q , projectile nuclear charge Z,, target nuclear charge Z, , and collision velocity u. It was found that K-K charge transfer plays a dominant role in the target K-shell vacancy production for near-symmetric collisions. Experimental K-K charge transfer cross sections are shown to be accurately predicted by the theories developed in the last few years. The study in heavy ion-atom collisions provides new tests of these theories and establishes the region of validity of these theories. The experimental data and theories discussed in this review concern primarily the intermediate velocity region, typically for 0.3 G u/u, s 3. At higher and lower collision velocities, the validity of various theoretical models is less clear. For charge transfer at high velocities, the capture of the target electron has to occur at small impact parameters. Perturbation theories based upon single-collision models are not likely to be valid. For collisions at smaller velocities, the situation is clouded by the increasing importance of many-electron effects, including the modeling of the relaxation of passive electrons. Notice that in the usual MO model the full relaxation of passive electrons is assumed (Briggs, 19761, while in the atomic model or other charge transfer theories for u 2 u,, the passive electrons are assumed to be frozen in the collision. How the passive electrons relax in a many-electron ion-atom collision remains to be investigated. The situation of ionization at low collision velocities is not quite clear. There is a lack of studies of Coulomb ionization in the simple systems at low velocities. Initial calculations based upon the MO model showed the sensitivity of coupling terms with respect to ETFs (Rankin and Thorson, 1978). Simpler models based upon modified PWBA (Basbas et a / . , 1978)

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and SCA (Andersen ef al., 1976) models show some agreement with experimental data, but the theoretical foundations of these models remain to be examined. Many subjects have to be left out of this review due to the limited space. For example, charge transfer to heavy bare ions from simple atoms, which is important for thermonuclear fusion modeling, is not fully discussed. These problems are characterized by the population of many excited levels in the final states, which are often not identified. In Section I11 we discussed the application of some theoretical models to these problems, but critical tests of the theory are not possible because only total charge transfer cross sections are available. We have also not discussed excitations (except in limited areas) and capture to excited states. This work indicates a few of the interesting questions of theory and experiment which remain to be investigated in this area of ion-atom collisions. The general problem of total and differential cross sections for K-shell radiative and nonradiative electron capture at high velocities needs further investigation as does the double K-capture process. Additional unique investigations of the effects of spectator electrons in all the atomic K-shell processes are still needed. By measuring polarization as well as orientation of the target and projectile states, information about detailed transition amplitudes could be obtained. These measurements would provide more critical tests of the theoretical model. ACKNOWLEDGMENTS The authors wish to acknowledge the Department of Energy, Division of Chemical Sciences for partial support during the time this manuscript was prepared. CDL is also supported in part by Alfred P. Sloan Foundation. Critical comments of the manuscript by Dr. A. Salin are greatly appreciated.

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Schneider, D., and Stolterfoht, N. (1975). Phys. Rev. A 11, 721. Schneider, D., and Stolterfoht, N. (1979). Phys. Rev. A 19, 55. Schneiderman, S. B., and Russek, A. (1969). Phys. Rev. 181, 256. Schuch, R., Nolte, G., Schmidt-Bocking, H., and Lichtenberg, W. (1979). Phys. Rev. Lett. 43, 1104.

Selov’er, E. S., I l k , R. N., Oparin, V. A., and Fedorenko, N. V. (1962). Sov. Phys. -JETP (Engl. Transl.) 15, 459. Shah, M. B., and Gilbody, H. B. (1974). J . Phys. B 7 , 256. Shakeshaft, R. (1976). Phys. Rev. A 14, 1626. Shakeshaft, R. (1978a). Phys. Rev. A 17, 1011. Shakeshaft, R. (1978b). Phys. Rev. A 18, 1930. Shakeshaft, R. (1980). Phys. Rev. Lett. 44, 442. Shakeshaft, R., and Spruch, L. (1979). Rev. Mod. Phys. 51, 369. Simony, P., and McGuire, J. H. (1981). To be published. Sin Fai Lam, L. T. (1967). Proc. Phys. Soc., London 92, 67. Stolterfoht, N. (1978). Top. Curr. Phys. 5, 155. Stolterfoht, N., Schneider, D., and Harrison, H. G. (1973). Phys. Rev. A 8, 2363. Taulbjerg, K. (1977). J. Phys. B 10, L341. Taulbjerg, K., and Briggs, J. S. (1975). J. Phys. B 8, 1895. Taulbjerg, K., Briggs, J. S., and Vaaben, J. (1976). J. Phys. B 9, 1351. Tawara, H., and Russek, A. (1973). Rev. Mod. Phys. 45, 178. Tawara, H., Richard, P., Jamison, K. A., and Gray, T. J. (1978a). J. Phys. B 11, L615.

INNER-SHELL VACANCY PRODUCTION

353

Tawara, H., Richard, P., Gray, T. J., Newcomb, J., Jamison, K. A., Schmiedekamp, C., and Hall, J. M. (1978b). Phys. Rev. A 18, 1373. Tawara, H., Richard, P., Jamison, K. A., Gray, T. J., Newcomb, J., and Schmiedekamp, C. (1979a).Phys. Rev. A 19, 1960. Tawara, H., Richard, P., Gray, T. J., Macdonald, J. R., and Dillingham, R. (1979b). Phys. Rev. A 19, 2131. Tawara, H., Terasawa, M., Richard, P., Gray, T. J., Pepmiller, P., Hall, J., and Newcomb, J. (1979~).Phys. Rev. A 20, 2340. Theisen, T. C., and McGuire, J. H. (1979). Phys. Rev. A 20, 1406. Thomas, L. H. (1927). Proc. R . SOC. London 114, 561. Thorson, W. R., and Delos, J. B. (1978a). Phys. Rev. A 18, 117. Thorson, W. R., and Delos, J. B. (1978b). Phys. Rev. A 18, 135. Toburen, L. H. (1979). IEEE Trans. Nucl. Sci. NS-26, 1056. Tsuji, A,, and Narumi, H. (1976). J. Phys. Soc. Jpn. 41, 357. Tunnell, T. W., Can, C., and Bhalla, C. P. (1979). IEEE Trans. Nucl. Sci. NS-26, 1124. Vaaben, J., and Taulbjerg, K. (1979). Abstr. Contrib. Pup. Int. C'onf. At Phys.. 11th. 1979 p. 566. Wilets, L., and Wallace, S . J. (1968). Phys. Rev. 169, 84. Winter, T. G., and Hatton, G. J. (1980). Phys. Rev. A 21, 793. Winter, T. G., and Lane, N. F. (1978). Phys. Rev. A 17, 66. Winter, T. G., Hatton, G. J., and Lane, N. F. (1980). Phys. Rev. A 22, 930. Winters, L. M., Macdonald, J. R., Brown, M. D., Chiao, T., Ellsworth, L. D., and Pettus, E. W. (1973). Phys. Rev. A 8, 1835. Woods, C. W.,Kadman, R. L., Jamison, K. A., Cocke, C. L., and Richard, P. (1974).J. Phys. B 7, L474. Woods, C. W., Kauffman, R. L., Jamison, K. A,, Stolterfoht, N., and Richard, P. (1975a).J. Phys. B 8, L61. Woods, C. W., Kauffman, R. L., Jamison, K. A., Stolterfoht, N., and Richard, P. (1975b). Phys. Rev. A 12, 1393. Woods, C. W., Kauffman, R. L., Jamison, K. A., Stolterfoht, N., and Richard, P. (1976). Phys. Rev. A 13, 1358.

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II

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,VOL. 17

ATOMIC PROCESSES IN THE SUN P. L . DUFTON* and A . E. KINGSTON Department of Applied Mathemutics and Theoretical Physics The Queen's Unitwsity c?f'Belfust Belfast, Northern Ireland

I. Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . 355 359 361 . . . . . . . . . . . . . . . 370

11. Atomic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 111. Bound-State Wave Functions . . . . . . . . . . . . . . . . . . .

IV. Spontaneous Decay of Bound States V. Electron Excitation . . . . . . . . . . . . . . . A. Electron Excitation of Be-Like Ions . . . . . . B. Electron Excitation of Many-Electron Systems . V1. Proton Excitation . . . . . . . . . . . . . . . . VII. Applications of Atomic Data to Solar Plasmas . . . VIII. Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

. . . . . . . . 381 . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

382 401 403 406 414 415

I. Introduction The sun is a unique astronomical object in that its proximity allows its electromagnetic spectrum to be observed from the y ray to the radio regions, corresponding to 18 decades in energy. Additionally, for significant portions of this range, it is possible to combine high spectral, temporal, and spatial resolutions. For example, the ultraviolet spectrometer (Woodgate et al., 1980) on the Solar Maximum Mission Satellite has a resolving power of typically lo4 and is capable of obtaining a significant signal in an emission line in 64 msec from a spatial area of 3 x 3" on the solar disk. Hence, the spectroscopic information available for the sun is more detailed than that for any other astronomical plasma. * Present address: Department of Pure and Applied Physics, The Queen's University, of Belfast, Belfast BT7 INN, Northern Ireland. 355 Copynghl (c) 1981 by Academic Press, Inc All nghts ot reproduction in any form reserved. ISBN 0-12-003n17-x

LogN, LogT,

I

I

I

I

11

12

10

PHOTOSPHERE-

8

I

0

-

CHROMOSPHERE

--I

2000

CORONA

I 1000

h lkml

FIG. ,I. A typical model of the electron density and temperature structure in the solar atmosphere. The photospheric region results are from Vernazza cr a / . (1976) and transition region and corona from Dupree (1972).

ATOMIC PROCESSES IN THE SUN

357

In order to interpret these observational results, it is necessary to have high-quality atomic physics data, the type of data depending on the region of the solar atmosphere being studied. For example, the visible and UV continuum radiation arise mainly from deep layers of the solar atmosphere, known as the photosphere (see Fig. 1). This region has typically an electron temperature of 5000 K and a gas density of lOI7particles cmP3, with the most abundant element hydrogen being principally neutral or in the form of the negative hydrogen ion. Here the most important processes for determining the continuum fluxes are the photoionization and freefree rates (and the inverse radiative rates) for hydrogen, H-and, in the UV region, the more abundant heavier ions such as C I, Si I, etc. Many of the weaker absorption lines due to atomic and molecular species are also produced in the photospheric region, and to predict their strengths it is necessary to have accurate values for their oscillator strengths and their radiative and impact damping constants. For the former, recent advances in experimental techniques mean that for some ions, such as a neutral iron, accuracies of better than 1% are now possible (see, for example, Blackwell et al., 1979). A good description of the variety of data required for predicting the photospheric spectrum is given by Lambert (1978). The cores of stronger lines, where the line opacity is large, are formed higher in the atmosphere in the chromospheric region (see Fig. 1). Here electron temperatures range from 5000 to 30,000 K, and hydrogen now starts to ionize. Also the population of ionic levels now differs significantly from that predicted by the Boltzmann and Saha equations, and it is necessary to consider the interaction of the radiation field and the ions on a microscopic level. Hence, all atomic processes that populate or depopulate ionic levels such as photoexcitation and ionization and electron collisional excitation and ionization, together with their inverse processes, are potentially important in determining the population of ionic levels. Also when considering the radiation field it is necessary to consider both absorption (including free-free transitions) and scattering processes. For the latter, line scattering can lead to a significant frequency redistribution in the cores of the profiles due to both Doppler effects from the thermal motion of the scattering ion and to the energy uncertainty due to the finite lifetime of the excited state. The extreme complexity of this problem has limited progress, although in the last few years, realistic calculations of the Ca I1 and Mg I1 H- and K-line cores have been made by Linsky and his co-workers (see, for example, Basri et al., 1979). In the outermost layers of the atmosphere, called the corona, extremely high electron temperatures in the range lo7 to 5 x lo7 K exist, together with lower electron densities of the order of lo7cm+. Under such circumstances, hydrogen, helium, and the lighter elements are fully ionized with

358

P. L. Dufton

and A .

E . Kingston

heavier elements, such as the cosmically abundant element iron, being either fully stripped or in high ionization stages. The transfer from the chromosphere to coronal conditions occurs over a small height range of approximately 100 km, known appropriately as the transition region. Here the degree of ionization of elements heavier than helium (hydrogen and helium are effectively fully ionized) varies markedly with height, and together with the corona produces a rich emission line spectra by electron collisional excitation followed by spontaneous emission. These emission spectra lie predominately in the UV and X-ray regions, with the more highly ionized species producing the lines at the shorter wavelengths. The above outline assumes that the sun is spatial homogeneous and hence that there is a unique atmospheric structure. Observations show that this is not the case. For example, solar photographs taken in the core of the hydrogen Balmer a line show a complicated network structure associated with the local magnetic fields, while in the transition region there are areas characterized as quiet or active whose emission line intensities can differ by an order of magnitude or more. In the corona, there are also marked differences with regions of low intensity called coronal holes. Additionally there are time variations in the solar atmosphere, the most spectacular of these being associated with flares. These explosive events stretch down from the corona, through the transition region, and into the chromosphere and are associated with active regions. The mechanisms controlling their production and evolution are at present of considerable interest and the Solar Maximum Mission Satellite launched in February 1980 has as its primary aim the study of solar flares from the y ray to the visible spectral regions. The availability in the last decade of rockets and satellites as platforms for quantitative spectroscopy at wavelengths of less than 3000 A has led to extensive observation of the solar emission line spectrum. Therefore, the scope of this review will be limited to the atomic physics relevant to the study of the regions producing this spectra, i.e., the transition region and the corona and transient phenomena such as flares. However, the variety of ionization stages and atomic processes considered is such that many of the data discussed will also be relevant to studies of the solar photosphere and chromosphere and to other astronomical objects, such as flare stars, H I1 regions, and planetary nebulae, which have emission line spectra. Although the atomic physics data and their application will be discussed generally we will, in many cases, use the four-electron Be-like sequence as a representative example. This sequence is chosen as one both with a rich solar line spectrum and because the relevant atomic data have been intensively studied.

ATOMIC PROCESSES IN THE SUN

359

11. Atomic Spectra When attempting to interpret the UV and X-ray region of the solar spectrum it is essential to have a comprehensive knowledge of the wavelengths of the spectral lines of each ion. Primarily the energy structure is used to identify the rich emission line spectra, and hence it is important to know the energy structure to very high accuracy. As we shall see in the following section, ab initio theoretical calculations of atomic energy levels may give wavelengths which are accurate to better than 1%, but in general this is not of sufficient accuracy to be used for line identification. In many cases observations of the radiation from laboratory plasmas, produced, for example, by theta pinches, intense laser beams, or beam foil stripping, can supply the ionic energy levels. However, in other cases, particularly for ions with even numbers of electrons which have low lying metastable levels, laboratory plasmas are inadequate because the high laboratory densities coupled with the low ionic column densities, compared with those observed in the sum, tend to mitigate against spin-forbidden transitions. (2s') IS8 transitions, which are Hence, for example, the (2s2p) "Pp strong features for all abundant ions in the sun, are not usually observable in laboratory plasmas. To determine the energy structure of these ions it is necessary to use semiempirical fits to extrapolate along isoelectronic sequences (Edlen, 19791, or indeed to use the sun as a laboratory source. The achievement of high wavelength stability in UV and X-ray spectrographs flown on earth-orbiting satellites has resulted in the latter method being extensively used (Edlen, 1979). A comprehensive review of the classification of the spectra of ionized atoms has been given by Fawcett (1974, 1973, and a useful list of the wavelengths of lines of Be-like ions is given by Edlen (1979). A knowledge of the ionic structure is not only important for line identification but it is also required as a guide to the atomic processes which may be important in determining the level populations of excited states and hence the strength of the emission line spectra. For example, the energy separation of LS degenerate fine structure levels may well determine whether electron or proton collision rates are sufficient to ensure that the corresponding ionic populations follow Boltzmann statistics. Since a very wide range of ionization stages are observed in the solar spectrum, it is found that there are significant quantitative differences among different ions even if they are in the same isoelectronic sequence. This is illustrated in Fig. 2, which shows the energy diagrams for the 46 lowest levels of three ions in the Be sequence, C 111, Si XI, and Fe XXIII. In the figure the energy scale E' is normalized to the ionization potential of the ion.

-

FIG. 2. Energy levels for (2/2/’) and (2/3/’) states in the &-like ions C 111, S i x , and Fe XXIII. The energies have been normalized to the ionization potential of the relevant ion.

ATOMIC PROCESSES IN THE SUN

36 I

A number of trends are observed as the nuclear charge Z of the ion increases. For C I11 the (2121') states are well separated, but as Z increases, these states become closer together, and a similar pattern is observed for the (2131') states. This is due to the fact that a s Z increases, the system becomes hydrogenic as the electron-nuclear interaction becomes greater than the electron-electron interaction. The energy difference for A n # 0 transitions increases as Z ' , while the difference for A n = 0 transitions increases as Z. It is also seen that as Z increases, the spin-orbit interaction increases and the splitting of LS levels into their separateJ components becomes more pronounced. For example, in C 111 the splitting in the (2s2p) :3Pg=o+ (2s2p) 3Pg=2 levels is only 0.15% of the (2S2)'S + (2s2p) 3P"energy difference, compared with 34% for Fe XXIII. The energy level diagram also provides a useful guide to the approximations which are appropriate in a particular theoretical calculation. For example, the very large energy differences in the (2121') levels in C I11 would suggest that in C 111 calculations it is important to take careful account of electron-electron interaction, while the large energy splitting in the IP levels of Fe XXIII would suggest that it is necessary to include the spin-orbit interaction in Fe XXIII calculations. The diagram also indicates that for C I11 a close-coupling calculation would give good results by including (2121') and (2~31)states as these states are well separated from the (2~31)states, but for Fe XXIII the (2~31)states lie close to the (2~31) states and would have to be included in a calculation.

111.

Bound-State Wave Functions

In order to interpret the solar spectrum it is important to have an accurate knowledge of the basic atomic processes producing the radiation. For example, in the case of emission lines it is usually necessary to know the rates of spontaneous decay and the rate of electron excitation to a state. Unfortunately, it is difficult to study multiply charged ions in laboratory plasmas, and it is often necessary to use ub initio quantum mechanical calculations to obtain estimates for the rates of the basic atomic processes. For such calculations it is necessary to have accurate wave functions for the states which are being considered. Over the last ten years there have been considerable theoretical advances in the calculation of nonrelativistic wave functions and their use in calculating atomic rates (Oksiiz and Sinanoglu, 1969; Sims and Whitten, 1973; Eissnerrt d., 1974; Laughlin and Victor, 1974; Nicolaides and Beck, 1975; Watson and O'Neil, 1975: Nussbaumer and Storey, 1978, 1979b; Hibbert, 1980).

362

P . L . Dufton and A . E. Kingston

Progress has also been made by Kim and Desclaux (1976) and Armstrong et al. (1976) in calculating atomic data using the relativistic multiconfigurational Hartree-Fock method. The relativistic randomphase approximation has also been used to calculate atomic lifetimes (Lin and Johnson, 1977; Cheng and Johnson, 1977). In this review we shall consider in detail the configuration interaction wave functions generated by the computer program CIV3 (Hibbert, 1970; Burke et al., 1972; Hibbert, 1975; Glass and Hibbert, 1976, 1978b). These wave functions have been used extensively in both atomic lifetime calculations and electron excitation calculations. For light atoms relativistic effects are small, and the Hamiltonian for an atom or ion may be written as

where N is the total number of electrons and 2 is the nuclear charge. Since this Hamiltonian commutes with the total spin and orbital angular momentum operators S' and L' and their z components S, and L, ,we seek wave functions of the form q ( L S M L M s ) . A configuration interaction (CI) atomic state wave function is represented by a configuration interaction expansion of the form .w

* i ( W

=

2

(2)

Uij+jbiLS)

j= 1

where L and S are the total spin and orbital angular quantum numbers of the state and CY defines the coupling scheme. Each of the configuration wave functions rC15(aiLS),which is, in effect, a linear combination of determinants, is defined with respect to its subshells: +j(CY&S) =

{(ls)"(CYlL1S1)(2S)hs(CY2L2S2)

.

* *

:aLS}

(3)

so that each subshell, with A, electrons, has a set of quantum numbers ajLJS5defining the seniority and the orbit and spin angular momenta of the subshell. The subshells are coupled to form configuration wave functions +,(a&S), which are eigenfunctions of L' and S'. The subshells are themselves constructed from one electron orbitals, which are products of a radial function, a spherical harmonic, and a spin function: Vnlrni

= (1 /r)f'nl(r) Y $ / ( ~ , +x(m J ) A.

(4)

In his work Hibbert writes the radial functions in terms of Slater-type orbita1s : k pnl(r)= cjnlr'jn/e-Ljii/r (5)

2 I=

I

363

ATOMIC PROCESSES IN THE SUN

and orthonormalizes his orbitals to satisfy

L"

pnl(r)Pn*l(r) dr =

an,

for

1+ 1

5

n'

In

(6)

With an initial estimate for the parameters in P n l ( r ) a, Hamiltonian matrix can be constructed with matrix elements given by

HU

= (W ~ N F l I W

(7)

This matrix can then be diagonalized. The resulting eigenvalues are upper bounds to the corresponding exact energies, and the associated eigenvectors provide the coefficients aLi in Eq. (2) (Perkins, 1965). A different choice of the initial estimate for the parameters in Pnl will, of course, lead to different eigenvalues, and so because of the upper-bound property the eigenvalue can be treated as a function to be minimized with respect to variations of the parameters in Pnl(r).This procedure has been computerized and is available in a program called CIV3 (Hibbert, 1970, 1975; Glass and Hibbert, 1978b). As an example, in calculations of the wave functions of Be-like ions the Is, 2s, and 2p orbitals are normally taken from Hartree-Fock calculations, these being calculations which only take one term in the expansion of the wave function in Eq. (2), and hence do not take account of configuration interaction. These three orbitals could be used to construct the simplest configuration interaction wave function for the six lowest levels of a Belike ion in which the ISe states would use two configurations, viz., ( l s ) * ( 2 ~ )and ~ , ( 1 ~ ) ~ ( 2 pthe ) ~ ,other four states being represented by a single configuration of the form (ls)*(2s2p)or ( l s ) * ( 2 ~ )The ~ . accuracy of such CI wave functions can be increased by including more one-electron orbitals and hence increasing the number of configurations in Eq. (2). For example, Burke el af. (1972) included 3s, 3p, and 3d orbitals in their calculations on the three lowest levels of Be-like ions Be I to Ne VII. They varied the parameters in the radial functions to optimize the lowest eigenvalue of the Hamiltonian matrix formed from the following seven configurations of the ground state, ISe: (ls2)(2sz), (ls2)(2s 3s), (ls2)(3s2), (1s*)(2p2) (ls2)(2p3p), (ls2)(3p2), (ls2)(3d2)

(8)

These optimized radial functions were then used to calculate the Hamiltonian matrix, which in turn gave the lowest eigenvalues and corresponding eigenvectors of each of the states IS, 3P, and 'P. The following six configurations were used for the 3P and 'P states:

364

P . L. Dufton and A . E . Kingston (1S2)(2s2p), (lS2)(3s2p), (ls2)(2s3p), (ls2)(3s3p) ( ls2)(3p3d), (ls2)(2p3d)

(9)

while the seven configurations listed in Eq. (8) were used for the ' S state. In Table I we list the energies of the ground (2s') IS state and lowest (2s2p) 'P state of some Be-like ions which were calculated using CI wave functions with an increasing number of configurations. It is clear that for the total energy of the ground state the simplest CI function gives a significant improvement compared to the Hartree-Fock function, but the addition of further CI terms produces only small improvements in the total energy. The Hartree-Fock energies of the excited 'P states are more accurate than Hartree-Fock energies of the ground ' S states, but the CI functions again produce small improvements. In the two cases for which we have a total energy from experiment, the most elaborate CI functions give energies which are considerably higher than the experimental energy. This is chiefly due to the fact that in these calculations the two inner 1s core electrons are "frozen" and do not interact with the outer electrons. However, when we consider the energy difference between two states, the error introduced by this approximation cancels and we obtain quite reliable energy differences. Table I also shows that as the nuclear charge increases, the difference between the experimental and theoretical excitation energy increases considerably. This divergence of theory from experiment can be chiefly attributed to the neglect of relativistic effects in these theoretical calculations. These effects are particularly important in the (Is') core. Full relativistic calculations for Be-like ions have been carried out by several groups. Kim and Desclaux (1976) carried out relativistic Hartree-Fock calculations, and Armstrong et al. (1976) used multiconfiguration Dirac-Hartree-Fock calculations, while Lin and Johnson ( 1977) and Cheng and Johnson (1977) used the relativistic random-phase approximation. In Fig. 3 we compare the theoretical energy differences of the (2s2p) 3Pand (2s') IS + (2s2p) 'P calcutwo lowest transitions (2s') IS lated by Armstrong et al. (1976) using both relativistic and nonrelativistic wave functions. In order to make the comparison simpler, we have, in fact, plotted the nuclear charge, Z, against the excitation energy divided by Z. For large Z the nonrelativistic energy difference for An = 0 transitions increases as Z, and our plots tend to a constant at large Z. The relativistic calculations of the energy difference increase much more rapidly than Z for large Z. When Z is small, relativistic effects are small, for example, for the ' S + 'Ptransition in F VI relativistic effects increase the energy difference by about 0.5%. However, as Z increases, the effect is much larger, the increases for Ar XV and Kr XXXIII being 5 and 50%, --+

TABLE I THEORETICAL NONRELATIVISTIC ENERGIES E A h D ENERGY DIFFERENCES (IN a.U.) OF T H E (lS-3(2S2)'s (lS2)(2S2p)'P STATES OF c 111, si XI 4 N D Fe XXIIl C A L C U L A T E D U S I N G C O N F I G U R A T I O N INTER4CTION (c1)W A V E F U N C T I O N S WITH DIFFERENT N U M B E ROF S ORBITALS

AND

Orbitals used in CI expansion

Hartree -Foc k Is, 2s. 2p

CI Is, 2s, 2p

2p (-1. 1s 2s ' 3s, 3p, 3d -1

-1

1s 2s -' 2p CI: -' -1

3s, 3p, 3d, 4f

Experiment

c 111 E('S) E( ' P ) AE = E('P) - E ( ' S )

-36.47742b -36.0069gb 0.47044b

- 36.4087''

- 36.4770"

"

0.4484"

-35.9603" 0.5168"

-223.8149" - 222.4599" 1.3550"

-223.9760d -222.4599" 1.5161"

-223.991 1" - 222.5064" 1.4847"

-223.9911" -222.5105" 1.4806d

-804.9571" -802.2565" 2.7006"

- 805.2595"

-805.2760" - 802.3032" 2.9728"

-805.2760" -802.3079" 2.%81"

- 35.9603

- 36.54747' -36.08110' 0.46637'

Si XI

AE

1

E('S) E('P) E('P) - E('S)

- 224.56 1 19' - 223.0604' 1.5008'

Fe XXIII

E('S) E( ' P ) 1E = E ( ' P ) - E ( ' S )

Burke et a / . (1972). Bemngton et ul. (1977). Kelly and Palumbo (1973). Glass (1979a) and private communication.

- 802.2565

3.0030"

3.4302''

P . L . Dufton and A . E . Kingston

366

'i.

c

:0 1

c

.c _

/

n

Y x

-I-

Cr 5

Ne 10

Ne

Ar

20

50 Nuclear charge Z

-

10

Ar

20

Fe

Xe

100

50

-

FIG. 3 . Excitation energy divided by the nuclear charge for the ( 2 ~ ' ) ' s ~ (2s2p):'PI and (2s2)'So (2s2p) 'PItransitions in the Be-like ions. Theoretical results: (-) nonrelativistic calculations: (--) relativistic Dirac-Hartree-Fock calculations (Armstrong t't t i / . , 1976).

respectively, while for Be-like Pb, the increase is more than a factor of 10. There is no doubt that for very heavy ions it is necessary to carry out full relativistic calculations. These calculations are much more difficult to perform than nonrelativistic calculations, and to increase the accuracy of the calculations requires considerable labor. Fortunately, the abundance of heavy ions in the sun is very small, and in general we only need to consider ions up to Fe XXIII. For these ions relativistic effects are important enough to be included in a calculation, but it is not necessary to carry out a full relativistic treatment. If, for example, we consider the full relativistic calculations of Armstrong ef a/.(1976), using the equivalent of a Is, 2s, 2p CI wave function, their relativistic results for the excitation energy of the ' S + 'P transitions in C I11 and Fe XXIII differ by 12 and 2%, respectively, from the experimental values, the difference being mainly due to the simplicity o f the wave function. For the same transitions more complicated non-

ATOMIC PROCESSES IN THE SUN

367

relativistic CI wave functions (Table I) gave results which were in error of 0.9% for C I11 and 15% for Fe XXIII, the difference here being due chiefly to the neglect of relativistic effects. Clearly it is desirable to use a method which will use accurate CI wave functions but which also allow us to include relativistic effects. One method for doing this has been given by Hibbert and Glass (Glass and Hibbert, 1978a,b). They use the Breit-Pauli approximation, which is valid for low Z (Z << 137), and write the Hamiltonian in the form =

HBP

HNR

+ HRel

(10)

where HNRis the nonrelativistic Hamiltonian and HRelis the relativistic part of the Hamiltonian, which may be written as HRel

= Hso

+

Hmass

+

HI11

+

Hsoo

+

Hss

+

+ HDZ +

Hssc

( ll)

H,, is the one-body spin-orbit interaction of each electron magnetic moment with the magnetic field which it produces by its motion in the field of the nucleus; Hmassis the relativistic mass correction; H,, the one-body Darwin term, the relativistic correction to the potential energy; H,, the spin-other orbit interaction is the sum of the spin-orbit coupling of an electron in the Coulomb field of another electron and the interaction of the spin-magnetic moment of an electron with the orbital current of another electron; H,, is the dipole interaction of the spin-magnetic moments of two electrons; H,, the orbit-orbit interaction; HI,, the two-body Darwin term, the relativistic correction to the potential energy; and H,,, the electron spin contact term. It is convenient to rewrite HBp= nonfine structure

+ fine structure

(12)

where nonfine structure fine structure

= =

+

HyR H,,,,,, + H,,, H,, + H,,, + Hbs

+ H,, + HI,2+ H,,,

(13) (14)

the nonfine structure terms commute with L2, S,, L2,and L, and the fine structure terms commute with J' and J,, where J is the total angular momentum and J, is its azimuthal component. The J-dependent configuration interaction (CI) expansion takes the form M

QdJ, MJ) =

~,+,((-w,Z-,S,JMJ)

(15)

j= I

Here we allow for the breakdown of LS coupling by adopting an intermediate coupling scheme in which the sum overj includes all configurations

P. L . Dufton and A . E. Kingston

368

for which

Lj

+ Sj = J

(16)

The configurations are constructed from the one-electron orbitals given by Eqs. (4) and (5). For very heavy systems relativistic effects produce changes in the radial functions, the inner orbitals are contracted, and the valence orbitals are expanded (Burke and Grant, 1967; Grant, 1970, 1979). However, we will assume that these effects are negligible for systems in which the Breit-Pauli Hamiltonian is valid (Z << 137), and we therefore determine the parameters in the one-electron orbitals by using only the nonrelativistic Hamiltonian in the optimization process. The coefficients { b,} are then determined as the eigenvectors of the Hamiltonian matrix with basis { Q i } , where the Hamiltonian is now the full nonrelativistic Hamiltonian plus relativistic corrections. More formally for a given parity n we can write Eq. (15) as

where (AJ&ljin) is the atomic state function at,Ic](Ljs])JiMJ(n)the single configuration 49, and ( AiJinlCILjSjn) = bUis called the mixing coefficient. 4 and S1are, respectively, the total orbital and spin angular momentum of the configuration, while At distinguishes states with the same Ji but different energies, and Cj distinguishes configurations with the same LJ1. We can also write Eq. (2) as IrjL]s]MLJMsJn)

=

2

( r]L]sjnlc]L]s]n)Ic]L$jMLjMSjn)

( 18)

CJ

rl distinguishes states with the same LISl but different energies. As this is a unitary transformation, we may write IclLjs.@L]MsJn)

=

2(

C]L]s]nlr]LjS].rr>

lr]L]sjMLJMsjn)

( 19)

nJ

which with ICj(Ljsj)JiMJjn) =

2 ML,MS,

gives Eq. (17) as

1AiJIMJin we write

)

c(Ljsdi:

M~JMsJMJI)(cjLjs]ML~Ms~n) (20)

ATOMIC PROCESSES IN THE SUN

369

This is called a "term-coupling'' coefficient (Jones, 1975; Saraph, 1972). may be expanded in terms of the The atomic state wave functions {Qi} single configuration basis {4J and mixing coefficients (6,) or in terms of the CI basis {*J and term-coupling coefficients &}. As an example of a nonrelativistic configuration interaction wave function and a relativistic configuration interaction wave function we give the wave functions for Ca XVII using only Is, 2s, and 2p orbitals, and with the (ls2) core "frozen," we have from Eq. (2)

W ( ~ S *IS"] ) = -0.972+[(2s2) ISe]+ 0.236$[(2p2) ISe; W(2s2p) 3P"] = 1.00$[(2s2p) 3P"] W(2s2p)

I",]

=

1.00+[(2s2p) 'P"] (23)

~ ( 2 ~ 3 2 ~) =1 1.00+[(2~*):]pel 'l"(2p2) IDe] = 1.00+[(2p2)ID"] '4"(2p2) Isel = 0.236$[(2s') ISe] + 0.972+[(2p2) ISe]

If we now take the Breit-Pauli Hamiltonian with the spin-orbit mass correction and Darwin terms in addition to the nonrelativistic operators and diagonalize the Hamiltonian matrix to obtain the mixing coefficients { b , } , then Eq. (15) becomes @['Sg] = -0.9754[(2S2) 'Sg] + 0.0124[(2p')3B]

@["pg]

=

+ O.2224[(2p2)'SGI

I .004[( 2s2p) "pl]

@[3Py] = 0.9974[(2s2p)''P'J

+ 0.0724[(2~2p)'P:]

@["px]

=

I .004[(2s2p)"Ppx]

@['P!]

=

O.O724[(2~2p)~P!]- 0.997+[(2~2p)'PP]

@[3peO]

=

@[!El

=

-O.O394[(2~')~Sg]- 0.9924[(2~')"@]- 0.120~$[(2p')'S~] (34) I .004[(2p"3Fq

@["P!j] = -0.970~#1[(2p'):"P$]+ 0.2414[(2p2)ID;] @[ID;]

=

0.2414[(2p2)

+ 0 . 9 7 0 4 [ ( 2 ~ ID;] )

- 0.1264[(2p2))"Pg]+ 0.9684[(2p2)'S8] @['Sg] = 0.2194[(2~')~S5]

We note the large mixing of "& and ID; and also l3peO and (2p') 'S", There is also smaller mixing of (2s') IS: and Tf,, as well as 'jPP and IPY. Combining Eqs. (23) and (24), for example, for J = 0 even, we obtain the term-coupling coefficients f &to give Eq. (21):

370

P . L. Dufton and A . E . Kingston (2s')'Sg = O.9998(2s2)'Sg+ 0.01 18(2~~)~peO - O.O141(2p')'S~ ( 2 ~ ~ ) ~ p=e O 0.0099(2~~)'S5 - 0.9920(2~')~E - 0. 1261(2p')'Sg

(25)

- 0.9919(2p')'Sg (2p2)'S8 = O.O155(2~')'S~- O.I260(2~~)~peO

An interesting set of calculations has been carried out by Glass (1979a) on the Be-like ions using a CI wave function with seven orbitals in the Breit-Pauli approximation, with different terms in Eq. (1 1) being used in the Breit-Pauli Hamiltonian. Table I1 is a comparison of his results for the excitation energy of (2s') ' S 4 (2s2p) 'P for Si XI and Fe XXIII with experimental results, with Hartree-Fock calculations, and with the full relativistic calculations of Armstrong et al. (1976). The terms H,, , H,, , and H,, only change the energy difference slightly; a much more marked effect is produced by the inclusion of the H,,,, and H,, terms only. The H,, term added to these produces a small but significant effect, and adding on the H,, and H,, produces a very small change. From Table I1 it is clear that by using a CI wave function with the above five terms in the BreitPauli Hamiltonian we can produce theoretical energy differences which differ from the experimental value by about 1% for Be-like ions from C I11 to Fe XXIII, the CI wave function being mainly responsible for the good agreement at low Z and the Breit-Pauli approximation being mainly responsible for the agreement at large values of Z.

IV. Spontaneous Decay of Bound States Over the last decade considerable advances have been made in the measurement and calculation of lifetimes of excited states of positive ions. Beam foil experiments have provided useful data for a large number of transitions in a variety of ions (see, for example, Pegg et al., 1979, and other papers in that journal). Accurate theoretical calculations by a number of groups have also provided a large amount of useful data (Glass, 1979a; Nussbaumer and Storey, 1979a,b; Laughlin et al., 1978; Victorov and Safronova, 1977; Hibbert, 1980). At present the best theoretical results are in general more reliable than the data obtained from experiment, and we shall concentrate on the theoretical results. If one neglects both the spatial variation of the electric vector of the radiation field in the vicinity of an atom and the interaction between the atom and the magnetic vector of the radiation field, then the electric dipole transition probability of a spontaneous radiative transition from an excited atomic state a to a

TABLE I1 RELATIVISTIC (Rel) A N D N O N R E L A T I V I S(N T IRel) C C A L C U L A T I OON FS THE ENERGY OF (2s)' ' S + (2s2p) 'PT R A N S I T I O (inN a.u.)

_ _ _

Orbitals used in CI expansion

c 111 Si XI Fe XXIII

-

CI: Is, Zs, 2p, 3s, 3p, 3d, 4f

CI: Is, 2s, 2p

Hartree-Fock Is, 2s. 2p N Re1

N Re1

Rel"

N Re1

Relb

Rel'

Reld

Rel'

Experiment

0.4484 1.3550 2.7006

0.5168 1.5161 3.0030

0.5204

0.47044 1.4806 2.%81

1.4809 3.0124

1.5136 3.4251

1.5141 3.4790

1.5140 3.4699

0.46637 1.5008 3.4302

3.504

Full relativistic calculations Armstrong et al. (1976). Relativistic calculations of Glass (1979a). with H\n + H, + H,,,, + H,, ' With H V R + H,,,, + H D ~ . 'With HhR+ H, + HmaS+ HD,. With HNR+ H,, + H,, + H,, + Hma,,+ H D ~ .

a

THE

312

P . L. Dufton and A . E . Kingston

lower state b is AEl(a + b) = ( 6 4 + / 3 h A 3 ) S ( b , a)

(26)

where

= S(a, b)

and the wavelength of the transition A

=

c/u

(29)

with h u = (Eb - Ea) being the energy difference of the transition. Electric dipole transitions are often called E 1 transitions to distinguish them from electric quadrupole (E2) and high-order transitions. If we write the normalized spherical harmonic as Yl,(i) and define C,, = ( 4 n / 3 )”’Y I P ,then it may be shown that I(blrila)12 =

c.

l(~l~tClP(~i)I~)l2

(30)

Y

This matrix element may also be written in terms of the oscillator strength for the transition

This is the “length” form of the oscillator strength. If the wave functions used are exact, then the “length” formula is equivalent to the “velocity” form

Hence, for a given calculation the difference between length and velocity results is often a useful indication of the error introduced by the use of approximate wave functions. For transitions between two energy levels a with w, states with wave functions +(ai) and b with wb states with wave functions $(bI),the transition probability is obtained by summing over all final states and averaging over all upper states and is given by AEl(a + b) = (64.rr4/3hA3)1(1/wa)S(b, U )

where

(34)

373

ATOMIC PROCESSES IN THE SUN

For LS coupling we can derive selection rules for allowed transitions. For example, the operator for these electric dipole transitions, T i , is spin independent, and the matrix element is zero if

xi

AS # 0

(36)

and as the operator is also an odd function the matrix element for electric dipole transitions is also zero if the initial and final states have the same parity. Using similar arguments, we can derive other selection rules for electric dipole radiation. For these transition probabilities to be nonzero we must have AS

=

0; A L

=

0,

kI

(not 0 + 0);

AJ = 0,

(not 0 + 0)

kI

(37)

and the parity of the state must change. In general the electric dipole transitions give the strongest lines. However, if a transition is forbidden by the electric dipole selection rules, it may not be forbidden if one includes in the radiation field either the variation of the field in the vicinity of the atom or the magnetic vector of the radiation field. The former gives rise to electric quadrupole radiation and the latter to magnetic dipole radiation. The electric quadrupole transition probability is AEz(a-+ b) = (32n6/3hA')SEz(b,a )

(38)

where

-

In LS coupling this transition probability is nonzero if AS

=

AL

0;

=

0,

AJ = 0,

kI,

5 2 (not 0 + 0, 0

51,

&2

(not 0 -

0, & +

1)

t, 0 ++ 1)

(40)

and the panty of the states must not change. Electric quadrupole transitions are often called E 2 transitions. The magnetic dipole transition probability is Aul(a -+ b)

=

(64n4/3hA")SMM,(b, a)

(41)

~ , , ( b , a)

I(-e/2rnc)(blL + 2Sla)I2

(42)

where =

For this transition probability to be nonzero in LS coupling, we must have AS

=

0;

AL

=

0;

AJ

=

0,

-t I

(not 0 + 0)

(43)

and the parity of the states must not change. Magnetic dipole transitions are often called M 1 transitions. In general electric quadrupole and magne-

P . L . Dufton and A . E . Kingston

374

TABLE 111

-

NONRELATIVISTIC CALCULATIONS OF THE DIPOLE LENGTH A N D VELOCITY F O R M U L A S , EQS.(31) A N D (33). FOR T H E OSCILLATOR STRENGTH OF T H E ( 2 . ~ 2'PI ~ ) (2s2) 'So TRANSITION I N C 111, Si XI, A N D Fe XXIII OBTAINED USING CONFIGURATION INTERACTION (CI) W A VE FUNCTIONS WITH DIFFERENT NUMBERS OF ORBITALS

Orbitals used in CI expansion

CI: _ Is,_2s._2p 3s. 3p, 3d

Hartree-Fock Is, 2s, 2p

C1 Is, 2s. 2p

CI: Is, 3s, 2p -

1.0711" 0.5634"

0.7%7" 0.6140"

Si XI Length Velocity

0.265' 0.161'

0.2652' 0.1583'

0.2607' 0.2590'

0.2599' 0.2597'

Fe XXIIl Length Velocity

0.132' 0.080"

0.1323' 0.0813'

0.1310' 0.1328'

0.1308' 0.1330'

3s, 3p, 3d, 4f

c I11 Length Velocity

0.7772b 0.783gb

Burke ef a / . (1972). Berrington el 01. (1977). ' Glass (1979a), and private communication.

tic dipole transition probabilities are of the same order of magnitude, but they are both about smaller than electric dipole transition probabilities. However, if the electric dipole transition is zero, then the electric quadrupole and magnetic dipole transitions are important. As an example, transitions between the (2s2),(2s2p), and (2p2) configurations of the Be-like ions produce the strongest lines for these ions in the solar spectrum. We shall consider first the recent advances which have been made in these transitions before considering An # 0 transitions. The oscillator strengths of the electric dipole transitions between the n = 2 levels of the Be-like ions have been calculated using the nonrelativistic CI wave functions discussed in the previous section (Burke et nl., 1972; Berrington et ul., 1977, Glass, 1979a). Table I11 is a comparison of the electric dipole oscillator strength of the (2s2p) 'PI -+= (2$) 'Sotransition in C 111, Si XI, and Fe XIIII, calculated using the length and velocity formulas, Eqs. (31) and (33), with a variety of CI wave functions. For low values of Z and the length formula, elaborate CI wave functions are required to give an accurate result, but for large Z quite accurate results are obtained using simple CI functions or even Hartree-Fock functions. To obtain accurate results with the velocity formula it is necessary to use elaborate CI wave functions for all values of 2.

375

ATOMIC PROCESSES IN THE SUN

,./- 4

u 0.01

C 0 Ne

0.001

I I I I III

Ar

I

Fe

Mo

Pb

I I I I Ill

FIG.4. Nonrelativistic and relativistic multiconfiguration Dirac-Hartree-Fock calculations of the electric dipole oscillator strengths of the transitions (2s2p)'P1--* (2s2)'S0 and ( 2 ~ 2 p ) ~4P(~2 ~ ' ) ' s ~Theoretical . results: (-) nonrelativistic results; (---) relativistic results (Armstrong et al.. 1976).

In the previous section we have seen that relativistic effects are important in calculating atomic energy levels for large 2, and they are also important in calculating oscillator strengths for large Z. Using a relativistic, multiconfiguration, Dirac-Hartree-Fock treatment, Armstrong et al. (1976) have calculated the electric dipole oscillator strengths for transitions between the (2s2) and (2s2p) levels of ions in the Be-like series. These are plotted in Fig. 4, where they are compared with their nonrelativistic calculations. For large Z a full relativistic treatment is essential, but for ions up to about Fe XXIII in the Be sequence, relativistic

TABLE IV RELATIVISTIC (Rel) A N D NONRELATIVISTIC (N Rel) CALCULATIONS OF T H E ELECTRIC DIPOLE, E l , OSCILLATOR STRENGTH FOR THE TRANSITIONS (2S2p)'Pl + (2S2)'s0 A N D (2S2p)3P1-+ (2S2)ls0

Orbitals used in CI expansion

Hartree-Fock Is, 2s, 2p N Re1

N Re1

0.319 0.199 0.132

0.198 0.132

-

-

_ _ _ _

CI: Is, 2s. 2p, 3s, 3p, 3d, 4f

CI: Is, 2s, 2p ReP

N Re1

Relb

Rel'

Reld

Rel'

(2s2p)'P1 + (2S')lSO

Mg IX Ar XV Fe XXIII

Mg IX Ar XV Fe XXIII

0.209 0.156

2.2 (-4)+ 1.5 (-3)

0.311 0.1% 0.131

-

0.311 0.195 0.130

0.316 0.207 0.156

1.94 ( - 5 ) 1.97 (-4) 1.02 (-3)

Full relativistic calculations of Armstrong et al. (1976). Relativistic calculations of Glass (1979a, b) with H,R + H,, + H,, + H,,. ' With HNR + H,,,, + HD1. With HhR + H , + HmaS + HD1. ' With H N R + H,, + H , + H, + H,,,,,, +HD~. 'The number in brackets denotes the power of ten by which the number should be multiplied.

0.316 0.206 0.154

0.3 16 0.207 0.154

3.17 ( - 5 ) 2.98 (-4) 1.87 (-3)

2.02 ( - 5 ) 2.29 (-4) 1.64 (-3)

377

ATOMIC PROCESSES IN THE SUN

effects do not dominate, and it is appropriate to use the Breit-Pauli approximation (Glass and Hibbert, 1978b) discussed in the previous section. Table IV gives the results for the Breit-Pauli approximation for the (2s2p) 'PI += (2s') ' S o transition in Mg IX, Ar XV, and _ Fe XXIII obtained _ _ using the CI wave function with Is, 2s, 2p, 3s, 3p, 3d, 4f orbitals and including different relativistic terms in the Hamiltonian (Glass, 1979a). The inclusion of the five relativistic terms H,, ,H,,, ,H,, ,H,,,,,, ,and H,, in Eq. (11) increases this electric dipole oscillator strength by 2 , 6 , and 18% for Mg IX, Ar XV, and Fe XXIII, respectively. The major increase is brought about by the inclusion of the non-fine-structure terms H,,,, and HDl, the fine structure terms H,, + H,, + H,, having little effect. Figure 5 is a comparison of the nonrelativistic and relativistic calculations of Glass (1979a) for this electric dipole oscillator strength for other ions in the Be-like sequence. As the nonrelativistic oscillator strength decreases as I/Z for large Z, it is convenient to plot the product Zfagainst 2. The small difference between the relativistic calculations of Armstrong et cil. (1976) and those of Glass is due to the more elaborate CI wave function used by Glass. Also plotted in the figure are the results of Nussbaumer and Storey (1978, 1979a,b). The agreement between the three calculations is excellent, but they do not agree with the beam foil measurements, which are also plotted (Pegg et al., 1977, 1979).

5

+

1

FIG.5 . Product of the nuclear charge Z of the ion and the oscillator strengthf for the (2s2p)lP, (2s2)'So transition for Be-like ions. Theoretical results: (-) relativistic; (---) nonrelativistic results (Glass, 1979a. Hibbert, 1974);( 0 )relativistic results (Armstrong et u / . , 1976); ( + ) relativistic results (Nussbaumer and Storey, 1978, 1979a, b). Beam foil measurements: ( A ) (Pegg er al.. 1977); (0) (Pegg ct d., 1979). +

P. L. Dufton and A . E . Kingston

378

Calculations have also been carried out on the other electric dipole transitions between the n = 2 states of the Be-like ions (Nussbaumer and Storey, 1978, 1979a,b; Hibbert, 1974; Glass, 1979a). The agreement between the results obtained by these two groups is very satisfactory. In the relativistic multiconfiguration Dirac-Hartree-Fock treatment of transitions between (2s2)and (2s2p) states, Armstrong et al. (1976) automatically obtained a nonzero electric dipole oscillator strength for the transition (2s2p) 3P, + (2s2) ' S o . As this transition involves a change of spin, it is strictly forbidden in LS coupling. However, in a full relativistic treatment this transition is allowed; it,also becomes allowed in the BreitPauli approximation, where the wave functions for the (2s2) 'So and (2s2p) 3P1state$ are written [Eq. (2411 as

@[W) IS81 = 61,14[(2s2)IS.,] + b1,64[(2P2)3pe,l+ b,,10"P2) @[(2s2p)sPY]= b,,+[(2s2p)"Y]

'SEl

+ 63.d "2s2p) 'PYI

(44) (45)

The products of the singlet terms in the two equations, together with the product of the triplet teims, give rise to a nonzero result for the electric dipole oscillator strength. Table IV gives the results in the Breit-Pauli approximation for the electric dipole oscillator strength for the ( 2 ~ 2 p )+ ~ P(2s2)'S0 ~ transition in Mg IX,Ar XV, and Fe XXIII obtained by Glass (1979b). Clearly both the fine structure terms H,, + H,, + H,, and the non-fine-structure terms H,,,,,, + HDlare important for this type of transition. The Breit-Pauli results of Glass are in good agreement with the relativistic calculation of Armstrong et al. (1976). Table V is a comparison of the Briet-Pauli approximation results for the electric dipole probabilities for the transition (2s2p) 3P7 + (2.9) ' S o obtained by Glass and Hibbert (1978a) and Glass (1979b), with results obtained by Nussbaumer and Storey (1978, 1979a,b) and Laughlin er al. TABLE V TRANSITION ~ O B A B l L l T l E S(SeC-') O F T H E ELECTRIC DIPOLE, E l , SPIN-FORBIDDEN TRANSITION (2S2p)3e + (2S')'so IN THE Be-LIKE IONS CIII

N IV

OV

NeVII

CaXVII

FeXXIII

References

8.60'"

4.95'

1.9

1.744

6.766

5.45'

9.59'

5.77

2.253

i w

6 .77s

5.01'

1.102

6.04*

2.373

2.054

Glass and Hibbert (1978), Glass (1979b) Nussbaumer and Storey (1978, 1979a, b) Laughlin, Constantinides, and Victor (1978)

a

Superscript denotes the power of ten by which the number should be multiplied.

ATOMIC PROCESSES IN THE S U N

379

(1978). The calculations are in quite good agreement; however, we would expect that the results of Nussbaumer and Storey to be somewhat more accurate than the other calculations because of the complexity of their wave functions. For the range of nuclear charge, Z, considered in Table IV, the spinforbidden electron dipole transition probabilities increase approximately as Zx.This very rapid increase is to be compared with a linear increase in the spin allowed electric dipole transitions probabilities for An = 0 transitions. A study of the wave function in Eq. (24) shows that spin-forbidden ' pP from electric dipole (El) transitions also take place to (2s2p) P (2p') 3Po,l.z,to (2s2p) 3P:2 from (2p') ID,, and to (2s2p) 'P!: from (2p2) ' S o . The transition probabilities for these transitions in the Be-like ions have been calculated by Nussbaumer and Storey (1978, 1979a,b). They all increase very rapidly as Z increases. Magnetic dipole ( M 1) transition probabilities have also been calculated for the spin-allowed transitions, (2s2p) "p:' to (2s2p) 3P8 and (2s2p) P ' pz" to (2s2p) P'pP (Nussbaumer and Storey, 1978, 1979a,b; Tunnel1 and Bhalla, 1979; Naqvi, 1951). These are much smaller than the magnetic dipole (Ml) transition probabilities for the spin-forbidden transitions from (2s2p) 'P! to (2s2p) 'P8,1,zand (2p2)3P1to (2s') ISo (Nussbaumer and Storey, 1978, 1979a,b; Naqvi, 1951). Electric quadrupole (E2) transition probabilities have been calculated for the spin-allowed transitions (2p7 ID, to (2s') ' S o and (2p2) ' S o to (2p2) ID, and the spin-forbidden transition (2p2) 3pz to (2s2) ' S o (Nussbaumer and Storey, 1978, 1979a,b). The magnetic quadrupole (M2) transition (2s2p) "pz" to (2.9) ' S o has been considered recently by several groups (Nussbaumer and Storey, 1978, 1979a,b; Shorer and Lin, 1977; C. D. Lin et af., 1978; D. L. Lin et af., 1978). There is good agreement among all the recent calculations, but they all differ significantly from earlier calculations of Osterbrock (1970). For N IV the transition probability of this magnetic quadrupole transition is approximately 5 x l(r times smaller than the transition probability for the (2s2p) 'P! to (2s2) ' S o transition and may be detectable in the solar spectrum. The M2, transition probability for the transition has also been calculated by Shorer and Lin (1977). (2p2)'SO+ (2~2p)~Ppz" Table VI is a summary of the theoretical results which are available for transitions between ( 2 ~ 7(2s2p), , and (2p2)configurations in C I11 and Fe XXIII. In general, calculations which have been carried out for C I11 have also been carried out for all low Z values up to Ne VII; for larger values of Z much less effort has been put into calculating E2, M1, and M2 transitions. For low values of Z the spin-allowed electric dipole transition prob-

TABLE VI THEORETICAL TRANSITION PROBABILITIES (A) I N sec-'

FOR

= 0

TRANSITIONS I N Be-LIKE

c 111 A N D Fe XIII"'b

c I11 0

5-11c.d

1

9.592' 5 . W 3 (M2) 1 .78S9

2.39-7 ( M 1)

9.23-' ( M I ) 3.83-2 (E2) 2.373 (E2)

4.443*

2 1 0

1

2 2

1.453(MI)

0

2.41P(M 1) 1.09 3 ( M l ) 1 .3309 3.32Y 3.336n 4.6W3 2.0373

1.81 3 ( M l )

5.5378 9.985" 2 .927' 4.57-YM2)

2.439 1 SI2' I .3183 1.407* 2.1149

4.34' (E2)

Fe XXIII 3-5c 5.0147 7.51 (M2) 1.974'"

6.50-' (E2)

6.4329

9.63VM I ) 1.225Iu 4.0W9 5.2069 4.84or 1 .I3648

4.4549 7.5029 4.7339

2 .2507 7.5946 3.6178 4.3539 3.037Iu

" The results tabulated are mainly taken from the work of Nussbaumer and Storey (1978, 1979a. b). The calculations are all electric dipole transitions ( E 1) except those denoted by (E2) electric quadrupole transitions, ( M 1) magnetic dipole transitions, and ( M 2) magnetic quadrupole transitions. Estimate of two-photon ( E I + M 1) transition rate (Laughlin, 1980). * The superscript denotes the power of ten by which the number should be multiplied. Shorer and Lin (1977).

ATOMIC PROCESSES IN THE SUN

381

abilities are very much larger than all other transition probabilities, but for large Z the spin-forbidden E 1 transition probabilities are comparable with the spin-allowed probabilities. The E2, M1, and M2 transition probabilities all increase rapidly with Z, but even for Fe XXIII they are much smaller than the E 1 probabilities. Most of the spectral lines of the Be-like ions which have been observed in the solar spectrum have been from An = 0 transitions. A small number of lines have been seen from transitions between states with (2s2),(2s2p), ( 2 ~ 9and , ( 2 ~ 3 ~(12, ~ 3 ~(2s3d), 1, (2p3s), ( 2 ~ 3 ~and 1 , (2p3d) configurations (Malinovsky, 1975; K. G Widing, personal communication, 1980). A very extensive series of CI calculations has been carried out by Hibbert (1979, 1980) on the transition probabilities between all states of 0 V having the form 1s%/n'I, n = 2, n' = 2, 3. In all, approximately 400 electric dipole spin-allowed and spin-forbidden transitions were considered, and about 50 electric quadrupole spin-allowed transitions were also included. The lifetimes of the states were compared with beam foil experiments, and, in general, theory and experiment were in good agreement (Martinson et al., 1971; Irwinet a/., 1973; Knystautas and Drouin, 1975; Buchet et nl., 1976, Engstrom et al., 1979). M1 and M2 transitions were also considered, but they did not affect the lifetimes significantly. Accurate CI results have also been obtained by Glass (1979~)for the transition probabilities for the spin-allowed electric dipole between the configurations ( 2 ~ 7(2s2p), , (2p2),(2s3s), (2s3p), and (2s3d) in C 111, N IV, 0 V, and Ne VII. His results are in good agreement with beam foil measurements (Buchet-Poulizac and Buchet, 1973; Barette and Drouin, 1974; Buchet and Buchet-Poulizac, 1974). Finally we note the extensive set of transition probabilities obtained by Czyzak and Krueger (1979) for N IV and 0 V for transition of the form n11,n212+ n i l i n i l i , where n,ll and n i l ; were either 2s or 2p and n212 went up to 6p and nh/i went up to 7f.

V. Electron Excitation In this review we have attempted to discuss recent advances in calculations for atomic processes which are important in the sun. We have considered the Be-like ions in detail because they have many lines in the solar spectrum and also because the theoretical methods which have been used for Be-like ions can be readily extended to other isoelectronic series. The theoretical results which we obtained for the excitation energies and transition probahilities in the Be-like ions are typical of the results that we would expect from any ion which is observed in the sun.

382

P. L . Dufton and A . E . Kingston

Recent calculations on the electron excitation of Si 111 (Baluja et al., 1980) and S IV (Dufton and Kingston, 1980) suggest that for electron excitation, ions in the third row of the periodic table are quite different from ions in the second row. Hence, in this section we first consider in detail electron excitation of Be-like ions and then consider electron excitation of Si I11 and S IV. The results for the Be-like ions are typical of the results that we would expect for ions in the first row of the periodic table. It is not yet known if the results for Si 111 and S IV are representative of heavier ions. A. ELECTRON EXCITATION OF Be-LIKE IONS

Sections I1 and IV show that for Be-like ions the observed energy levels of ions are in general more reliable than the calculated values, and, although the measured transition probabilities are obtained quite accurately at present, the theoretical values are normally to be preferred. Some work has been carried out on the measurement of electron excitation rates for ions using theta pinches. Excitation rates have been obtained for transitions in Ne VII by Tondello and McWhirter (1971) and for transitions in N IV, 0 V, Ne VII, and Si XI by Johnston and Kunze (1971). However, chiefly because of the uncertainty of determining the populations of the ground (2s') 'Sostate and the metastable (2s2p) 3P1state these rate coefficients were estimated to have uncertainties of a factor of three. Because of this large uncertainty in the measured electron excitation rate coefficients, it is important to obtain accurate theoretical rate coefficients for ions which are important in the sun. When electron excitation processes for a particular ion are important, the thermal energy of the free electron is usually approximately equal to or less than the electron excitation energies of the ion. Hence, in our theoretical calculations we will be chiefly interested in the excitation cross section at low energies, i.e., at electron energies up to approximately ten times the excitation energy. The nonrelativistic Hamiltonian for the ( N + 1) electron system, comprising an ion with N electrons and a free electron, is

c (-gv;

N+l

N+1

W+'=

i=l

- Z/rJ

+ 2

I/ru

irj=I

where Z is the nuclear charge of the ion. With this Hamiltonian the total orbital, L, and spin, S, angular momentum of the electron-ion system are conserved as well as their azimuthal components ML and Ms. Because of this we construct the total wave function for the system in terms of channel functions, which are eigenfunctions of L', S', Lz, and Sz. We consider

383

ATOMIC PROCESSES IN THE SUN

a given target state which has quantum numbers Li, Si, MLi, M s i , and which is distinguished by Ti from other states with the same quantum numbers. Denote the wave function for this target state by *i$fs,@l,

. .

x 2 ,

9

(47)

XN)

where X denotes the r radial, i angular, and CT spin variables. The free electron is denoted by quantum numbers f i , si,mli, m,,, and also ki, where fk: is the energy, in atomic units, of the free electron incident on the state T i . The total energy E is given by E = &k: + E i , where Ei is the total energy of State Ti. The free electron is described by the function Y'Tt ( 8,

d) X Y f t ( CT)[ F&-)/ r ]

(48)

and we now couple Eqs. (47) and (48) to form a channel function, which represents an atomic eigenstate, coupled with the spin and angular functions of the incident electron to form an eigenstate of the total orbital, L, and total spin S angular momenta and their z components. In this representation we will use i to denote the quantum numbers

Ti, Li, S i , l i , s i , ki, L, S,

ML,

M,, and the parityp

(49)

and the channel functions are @&I,

X29

.

@$s,

-

.

9

Xn,

( x l y

i ~ + Vi N, + ~ )

x2,

*

*

.

?

xn)Y/irnl,(~N+l, dNfl)X?'(~N+l)

(50)

where the C's are Clebsch-Gordan coefficients. For each L , S , and parity p (odd or even) we write the total wave function for a particular energy E as *,AX],

x p ,

9

XN+J

=

2 i

@i[~i(rN+i)/rN+i~

+ 2 dj4j

(51)

1

where the operator d ensures that the wave function is antisymmetric in all the coordinates of the ( N + I ) electrons. The functions d1are ( N + I ) electron functions, which are constructed from one-electron orbitals to have the same parityp and orbital, L, and spin, S, angular momenta as V E .They are included partially to remove constraints on the total wave function which are imposed by orthogonalizing the one-electron orbitals and partially to take account of short-range electron correlation. Substituting V Einto the Kohn variational principle (Kohn, 1948) and taking arbitrary variations of the functions Fi and coefficients d l , we

384

P.L. Dufton and A . E . Kingston

obtain

where n is the number of channel functions retained in the expansion and (N + I ) electron bound state functions in Eq. (51). In Eq. (52b) the integration is over all electronic coordinates, and in Eq. (52a) the integration is over all electronic coordinates except the radial coordinates of the scattered electron. The resulting equations are satisfied subject to the boundary conditions

m is the total number of

where Bi = kir - 4Lin - q i ln(2kir) Ti = - ( Z

-

N)/ki,

+ crli

vli =

arg r(ij+ I

+ ;Ti)

(54)

The second subscript j on Fu is introduced to label the n independent solutions. For a given E , L , S. and p the R i j matrix is a function of the incident channel TiLiSilisiand the outgoing channel T;L;S;l;s;, and we write Rij =

R”(TiLiSilisiLSki;T ; L ; S i l j s ; L S k ; )

(55)

or in a more compact form Rij

=

RLSli(riLiSils; r ; L ~ S ; l ’ s ’kg) :

(56)

When we have calculated the RLSiCiLiSils: LISIl‘s’;k : ) matrix for a given incident electron energy relative to the ground state of +kq, and for all values of L and S and panty p (odd or even), we can obtain the total cross section (in of) for a transition from one state riLiSi to another r ; L ; S ; ,using e ( r i L i s i+ r ; L ; s ; )

where the TLSI~(TiLiSils: r i L i S i l ’ s ’ ;k : ) matrix in Eq. (57) was obtained from the RL“4TiLiSils;r; L i S i l ’ s ’ ;k ; ) matrix using the relation T

=

The total collision strength is

-2iR/(I - iR)

(58)

ATOMIC PROCESSES IN THE SUN

385

and hence

For an atom in which the term splitting is small compared with the term separation, we can use the RL"P(TiLiSils; T ; L ; S ; l ' s ' ;k:)matrices to obtain transitions between different fine structure levels by transforming to pair coupling, where

Li

+ Si = J i ,

Ji + 1 = K,

and

K

+s =J

(61)

Using Racah recoupling coefficients (Racah, 1943), Saraph (1972, 1978) has shown that for a given J and parity, p, RJp(TiLiSiJi, IK; T;L;SiJi/'K'; k q ) =

2 ~ ( L S JL,siJi, , i K ) w w i L i s , i s ;I - ; L ~ ; W ;k : ) LS x C(LSJ, L:S:J;, / ' K ' )

(62)

where the C's are products of two Racah coefficients. Collision strengths between the fine structure levels riLiSiJi+ T;LIS;J; are then given by

a(riLis,Ji r;L;s;m = 2 (25 + l)IF'l,(TiLiSiJi/K; r i L ; S : J ' / ' K ' ;ki)12 -+

(63)

ll'kk' JP

where the matrix T J l )is obtained by substituting RJ1)into Eq. (58). The accuracy of any electron-scattering calculations depends on a t least two things, namely, the accuracy of the atomic target states used in the expansion of the total (N + I ) electron wave function [Eq. (5111 and the number of target states used in this expansion. If we use a large number of target states in Eq. (51) which have poor wave functions o r only a small number of accurate target states in Eq. (51), we obtain inaccurate cross sections. Over the last ten years several groups have produced useful computer packages for obtaining accurate electron excitation cross sections for ions. A method which has been used extensively for the electron excitation of Be-like ions is the R-matrix method. This method was developed by Burke and his co-workers (Burke, 1973; Burke and Robb, 1975; Berrington et ul., 1974, 1978) and uses the CI wave functions discussed in the

386

P . L. Dufton and A . E . Kingston

previous sections. In the R-matrix method the ion is enclosed in a sphere of radius a. In the internal region r < a , exchange effects are important, and the electron-electron interaction is large. Here, the ( N + I ) electron wave function is expanded in a finite set of energy-independent basis functions. Logarithmic boundary conditions are imposed on the wave function on the surface of this sphere. In the outer region r > a, the interaction is weak, and we can neglect exchange effects and can approximate the direct potentials by a few terms in their asymptotic expansion. For r > a, we solve a set of coupled ordinary differential equations to obtain the wave functions, and these are matched to the inner region solutions on the boundary r = a to give the radial functions and the R matrix. Another powerful computer code, I M P A C T , has been developed by Seaton and his co-workers (Eissner and Seaton, 1972; Seaton, 1974a,b; Crees et al., 1978) for electron ion scattering. In this method the coupled integrodifferential equations are solved using radial functions, which are tabulated at a small number of points. As in the R-matrix method, these solutions are obtained out to some point r = a, for r > a exchange effects are neglected and the direct potentials are represented by their asymptotic forms. Hence, for r > a, the asymptotic solutions are obtained by solving a set of coupled ordinary differential equations. Matching these asymptotic solutions to the inner region solutions at r = a gives the radial functions and the R matrix. Henry and his co-workers (Smith and Henry, 1973; Henry ef al., 1981) have also developed a practical method of obtaining accurate electron excitation cross sections. It is called NIEM, noninteractive integral equation method, and is based on the work of Sams and Kouri (1969). The three methods discussed above all solve the scattering problem using a moderate number of target states in the expansion of the total wave function Eq. (51). Mott and Massey (1933) suggested that if the coupling between the initial and final states is weak and the coupling to all other states is weak, then we could use only two target states, the initial state and the final state, in Eq. (51). They also assumed that the coupling potential between the two states could be ignored in the incoming channel. The solutions of the two integrodifferential equations allowed for the distortion of the incident and outgoing waves of the electrons by the target atoms. The method was called the distorted wave approximation. Since that time there have been many variants on the distorted wave approximation. At least six of these variants have been discussed by Henry (1981) in a recent review. As the distorted wave methods are normally only approximations to the collision problem with two target states in V E , results from these methods should not be as accurate as R-matrix calculations, which include two or more target states.

387

ATOMIC PROCESSES IN THE SUN

Another approximate method which has been used extensively at large electron impact energies or large values of L is the Coulomb-Born approximation. In this approximation exchange effects are ignored and the initial, i, and final, f, state wave functions are taken to have the form *,(Xi

?.

..

= Q,@i , . . . , X N , i N + i ,

aN+l)(kirN+l)-'F~,(kirN+l) (64)

qAX1,.

. . , XN+A =

Q f W l , ..

., X N ,

r N f l , (TN+l)(kfrN+l)-'Flf(~r,r,+l)

respectively, where the channel quantum numbers of Q, are TiLiSil,sikiLSMLMs,

and the parity p

I',L$,lfsf~LSMLMS,

and the parity p

and Qf are In Eq. (64) Fl(kr) is the regular Coulomb function for charge z = Z - N, which satisfies the boundary condition

Fl(kr))Zrn sin(kr - bln

-

-q In(2kr)

+ al)

(65)

with q = -(Z - N ) / k

and

a! = arg r(l + I

+ iq)

(66)

The R matrix is then given by R;iB= -2(k,kf)"2('PllVl'Pf)

(67)

where

v = zN - -1 t=1 rt.N+i

N rNfl

(68)

Using CI bound state wave functions [Eq. (2)], this reduces to R 5" = -2(k,k$"2

jmV,,(r)F~fkfr)FII(kir) dr 0

(69)

where Vf, is the direct potential between the initial and final channel. As an example of recent calculations on electron excitation of ions we will first consider the Be-like ion C 111. There have been many theoretical estimates of the electron excitation collision strength for the transition from the ground (2s') IS state to the lowest metastable level (2s2p) 3Pof C 111. This transition is not only important in solar applications, but is also important theoretically as a test calculation for spin-forbidden transitions. Calculations have been carried out by Osterbrock ( 1970), Eissner (1972), Flower and Launay (19731, Hershkowitz and Seaton (1973), Berrington et al. (1977, 1980), Mann (1980), and Robb (1980). Some of these

P . L . Dufton and A . E . Kingston

FIG.6. Collision strength for the (2s2)'S+ ( 2 ~ 2 p ) ~transition P in C 111. Theoretical results: (-) six-state, seven-orbital R -matrix calculations (Berringtonet a / ., 1977); (---) six-state, three-orbital R-matrix calculations (Berrington et d., 1977); (+) distorted wave calculations (Mann, 1980); (A) distorted wave calculations (Flower and Launay, 1973); (0) distorted wave calculations (Eissner, 1972).

calculations are compared in Fig. 6. The most accurate calculations are the R-matrix results of Berrington et al. (1977) (see also Berringtonet al., 1981). They used the six lowest states (2s') IS, (2s2p) 3*lP,and (2p9 ", ID, and IS, of the ion in their expansion of the total wave function, and in the CI wave functions for the target states they employed seven one-electron orbitals. Their calculations show that at low electron energies the collision strength is dominated by the series of resonances which converge to the excitation thresholds of the other four states. A more detailed graph of this resonance structure is given in Fig. 7. Since this collision strength decreases rapidly as the energy increases, this resonance region makes a significant contribution to the electron excitation rate coefficient at low temperatures. In order to judge the accuracy of their calculations Berrington et al. (1977) also carried out R-matrix calculations in which they also used six target states, which were obtained with only three one-electron orbitals, Is, 2s, and 2p. The accuracy of these wave functions was discussed in Sections I11 and IV. At an excitation energy of 10 Ry these less accurate

ATOMIC PROCESSES IN THE SUN

389

i

I

+

(2S21'S-(2S P)'PO

3P

'PO I

0.6

I

I

ae 1 ELECTRON ENERGY [fly,)

'

I

1.2

-

FIG.7 . Collision strength for the (2s')'S (2s2p):'Pand (2s')'S -+ (2s2p)'P transitions in C 111 in the resonance region. Theoretical results: (-) six-state, seven-orbital R-matrix calculations (Berrington et u / . , 1977); ( + ) distorted wave calculations (Mann, 1980); (0) distorted wave calculations (Eissner, 1972).

calculations were about 20% too small, but at 2 Ry they were 20% too large. These Is, 2s, 2p results are in good agreement with the closecoupling results of Robb (1980). Calculations were also carried out using only the three lowest target states constructed from the seven one-electron orbitals discussed above. The results of these calculations, for one partial wave, are compared with the six-state, seven-orbital calculations in the resonance region (Fig. 8). It is clear that the introduction of the three higher states in the six-state calculation lowers the position of the resonances and also alters their structure. Also plotted in Fig. 6 are the distorted wave results of Flower and

P. L. Dufton and A . E. Kingston

390

F-l-

0.8

I I

I /

\I

I

\I

1

i 0.5

0.6

I

0.7 ELECTRON ENERGY (Ryl

0.8

I

I9

FIG.8 . 'P" partial collision strength for the (2s2)'S + ( 2 ~ 2 p ) ~transition P for C I11 in the resonance region. Theoretical results: (-) six-state, seven-orbital R-matrix calculations (Bemngton cr ( I / . , 1977); (---) three-state, seven-orbital R -matrix calculations (Berrington P I ctl.. 197).

Launay (1973); their results are in quite good agreement with the threeorbital, six-state results of Berrington et af. (1977). These three-orbital, six-state R-matrix results are also in good agreement with the distorted wave calculations of Mann (1980) at energies above about 2.5 Ry and with the distorted wave results of Eissner (1972). The distorted wave calculations of Mann (1980) do not give any resonance structure, and in Fig. 7 we compare his results in the resonance region with the accurate results of Berrington et af. (1977). We shall show in the case of 0 V that this resonance structure leads to a large increase in the electron excitation rate coefficient. The electron excitation transition from the ground ( 2 s Z ) l Sstate of C 111 to the (2s2p) 'P state is important for solar studies. It is also important theoretically as it is typical of a spin-allowed optical transition. The most accurate calculations available at present are again the six-state, sevenorbital R -matrix results of Berrington et af. (1977). In these calculations the first eight partial waves were calculated using the R-matrix method, and the remaining partial waves were obtained using the Coulomb-Born

ATOMIC PROCESSES IN THE SUN

391

FIG.9. Collision strength for the (2s')'S + (2s2p)'P transition in C III. Theoretical six-state, seven-orbital R-matrix calculations (Berrington et al.. 1977); (---) results: (-1 six-state, three-orbital R-matrix calculations (Berrington et ( I / . . 1977); (V) close-coupling calculations (Flower and Launay, 1973); ( + ) distorted wave calculations (Mann, 1980); (0) seven-orbital Coulomb-Born calculations (Berrington et al.. 1977).

approximation with the same seven-orbital target states. Figure 9 presents the total collision strength as well as the contributions from L s 7 and from L > 7. At low energies there is a small amount of structure in the collision strength; this structure is shown in more detail in Fig. 7. Since the collision strength for optically allowed transitions increases with increasing energy, the contribution from the resonance region to the electron excitation rate coefficient does not dominate, in contrast to the spinforbidden transition ( 2 ~ ~+) (' 2s~ 2 ~ ) " . R -matrix calculations were also carried out, with six target states generated from only three orbitals, for electron incident energies above the resonance region. These results are only about 5% lower than the accurate results and are close to the R-matrix calculations of Robb (1980). The close-coupling results of Osterbrock (1970) are, however, about 50% higher than the accurate results at low energies, while the close-coupling results of Flower and Launay are about 20% lower. The distorted wave results of Mann ( 1980) are in good agreement with the accurate R -matrix

392

P . L. Dufton and A . E . Kingston

calculations; only in the resonance region do the distorted wave results differ by more than 10% from the R-matrix results. Also plotted in Fig. 9 are the Coulomb-Born results obtained with the seven-orbital CI wave functions. These Coulomb-Born results are much closer to the accurate results than the Coulomb-Born results of Nakazaki and Hashino (1977). However, the Coulomb-Born results obtained from the seven-orbital CI wave functions are in good agreement with the distorted wave results of Mann (1980), except in the resonance region, where they are 20% higher. This reasonable agreement between the R -matrix, distorted wave, and Coulomb-Born results is to be expected for an optically allowed transition. For these transitions high partial waves are important, and these partial waves are insensitive to the procedure which is adopted to calculate them. Six-state, seven-orbital R -matrix calculations have also been carried out for 0 V (Berrington et al., 1977). The results of these calculations for transitions from the ground to the (2s2p) 3Pand (2s2p) 'P states are given in Figs. 10 and 11, where they are compared with other calculations. As in

----+-I

T

I

Electron energy iRy

I

1

FIG.10. Collision strength for the (2S')'S -P (2s2p)'P transition in 0 V. Theoretical results: (-) six-state, R-matrix calculations (Berringtonel al., 1979); (+) distorted wave distorted calculations (Mann, 1980);(0)distorted wave calculations (Malinovsky, 1975); (0) wave calculations (Saraph, 1972).

ATOMIC PROCESSES IN THE SUN

4 3P0

-

(2s2p):'Pand ( 2 ~ ~ ) )(2s2p)lP 's transitions FIG.11. Collision strength for the (2S')'S for 0 V in the resonance region. Theoretical results: (-) six-state R-matrix calculations (Berringtonef ul., 1977); (+) distorted wave calculations (Mann, 1980); (0) distorted wave calculations (Malinovsky, 1975). --f

the case of C I11 the distorted wave results for 0 V (Saraph, 1972; Malinovsky, 1975; Mann, 1980) are all in reasonable agreement with each other, and they are also ,in agreement with the more accurate R-matrix results, except in the resonance region, where the R -matrix results are transition. P much larger for the (2s')'S -+ ( 2 ~ 2 p ) ~ In their calculations of the electron excitation of C 111, Berringtonet al. (1977) used the six target states (2s2)'S,( 2 ~ 2 p ) ~'P, P , and (2p2I3P,ID, IS, and seven orbitals. These calculations automatically give the excitation cross sections for all of the transitions between these six states. It is difficult to compare these cross sections directly; it is much more convenient and useful to compare the electron excitation rates. For a transition from a state i to f with cross section Q(i -+f ) (in a ; ) the rate coefficient C(i + f ) in units of cm3 sec-I at a temperature of T K is E&(i-+ f ) exp[-(3.158 x 10; E ) / T ] dE

(70)

394

P. L . Dufton and A . E . Kingston

1

tog [Electron temperature, Te] Fic;. 12. Rates for electron excitation of the ground (25')'s state of C 111 to the ( 2 ~ 2 p ) " P ~(2s2p)'P, , ~ , ~ , (2p*)3PPo,1,2, (2p2)'D,and (2p2)'Sexcited states (Dufton et a ) . , 1978).

where E is the energy of the incident electron and AEi, is the excitation energy in atomic units. Figure 12 is a plot of the electron excitation rate coefficients for excitation of the ground state, (2s2)IS, of C I11 to the first five excited states. At low temperatures the excitation to the ( 2 ~ 2 p ) ~state P is the dominant process, but as the temperature is increased, excitation to the (2s2p) 'P state becomes important. Excitation to the (2p2)'D state is always at least an order of magnitude slower than excitation to the (2s2p) 'P state, and excitation to the (2p2)3Por the (2p9 'S states is about two orders of magnitude slower. As the ( 2 ~ 2 p ) ~states P are metastable, their populations may be quite large, and electron excitation from them is important. We compare in Fig.

ATOMIC PROCESSES IN THE SUN

395

c l

Log[Electron temperature 61 F IG. 13. Rates for electron induced transitions from the metastable (2s2p)'P0 state of C 111 to the (2s')'S, ( 2 ~ 2 p ) ~ P (2s2p)'P, ,,~, (2p')3Ppo.1,2 (2p')lD, and (2p')lS states (Dufton o f a / . , 1978). Also plotted on the graph as a dashed line is the rate for the proton induced fine structure transitions (2s2pPP0 + ( 2 ~ 2 p ) [ ~ + P , 3P2](Doyle ef a / . . 1980).

13 the rate coefficients for electron excitation and deexcitation from the ( 2 ~ 2 p ) fine ~ P ~structure level. For low temperatures the electrons are excited to the other (2s2p) fine structure levels, and deexcitation to the ground (2s2)'S state is also important. However at 200,000 K excitation to the (2p2)3P0,1,zlevels is as important as excitation to the ( 2 . ~ 23P1,2 ~ ) levels, with excitation to the (2s2p) 'P and (2p2)ID levels being an order of magnitude smaller and excitation to the (2p2)IS level two orders of magnitude smaller. A comprehensive tabulation of electron excitation rates for other transitions between the six lowest levels of C I11 has been given by Dufton et af. (1978). Rate coefficients for transitions between the six lowest levels of N

P . L . Dufton and A . E . Kingston

396

IV, 0 V, and Ne VII are also available (Dufton et a/., 1978, 1979). As the nuclear charge increases, the rate coefficient for a given transition decreases rapidly. If we consider an optically allowed transition, the cross section (in nu;) varies for large values of E as (2.303A,/AEi,)(log E)IE

(71)

where A, is the optical oscillator strength for the transition i + f. For An = 0 transitions, the oscillator strength for a given transition varies approximately Xf = C / A E , ,as we go dong an isoelectronic series. Hence, at high energies we expect the cross section for a particular optically allowed An = 0 transition to vary as AE1;' in an isoelectronic series. It is found that the low-energy cross section also varies as A E i 2 . Changing the variable in Eq. (70) to X = E/AE, it is found that the rate coefficient, written as a function of T / A E , varies as AE-"' for a particular optically allowed An = 0 transition in an isoelectronic series. As an example of this variation we plot in Fig. 14 the electron excitation rate coefficient for the transition (2s') ' S o (2.52~)'PI, in C 111, 0 V, and Ne VII. It would appear that this procedure may provide a reliable way of obtaining rate coefficients for higher members of the isoelectronic series.

30.000

100.000

I

1 I11111 500.000 30,000

100,000 IT'KI / [AE(Ry)l

500,000

Ftc. 14. Electron excitation ratesfor the(2s2)IS0-+( 2 ~ 2 p ) ~ Pand(2s2)'So+ ~,,~ (2s2p)'PI transitions in C 111, 0 V, and Ne VII (Dufton ef al., 1978, 1979). The rate for the (2s2)'S0+ (2~2p)~P transition ~ , , ~ has been multiplied by (AE)3.5and the rate for the (2s2)'S0 (2s2p)IP1 transition has been multiplied by (AE)2.5. -+

ATOMIC PROCESSES IN THE SUN

397

-

FIG. 15. Collision strength for the (2s')lS ( 2 ~ 2 p ) ~transition P in 0 V. Theoretical R-matrix 12-state calculations (Berrington et a / . , 1979); (---) R-matrix sixresults: (-) state calculations (Berringtonet a / . , 1977).

Because of the complex resonance structure which exists in the cross section for the (2s') IS + (2s2p)3Ptransition it is difficult to judge how this cross section varies as we move up the isoelectronic series. Empirically we find that if we again plot the temperature in terms of T / A E , then the rate coefficient appears to vary as AE-3.5.The plot of the rate coefficient multiplied by AE3." is given in Fig. 14. To test the accuracy of their six-state R -matrix calculations for electron excitation of 0 V, Berrington et a / . (1979) repeated their calculations with 12 target state functions consisting of the six n = 2 states with configurations (2s2),(2s2p), and (2p9 and also the six n = 3 states with configurations (2s3s), (2s3p), and (2s3d). The collision strength which they obtained ~P is given in Fig. 15. It is seen that the for the (2s') IS + ( 2 ~ 2 p ) transition introduction of the six new ( 2 ~ 3 sIg3S, ) (2s3p) Is3P, and (2s3d) 1*3Dstates introduces new series of resonances converging to their thresholds. Similar resonance series were found in the (2s') 'S + (2s2p) 'P collision strength. The effect of these resonances on the rate coefficient for the (2s') IS + (2s2p) 3P transition is large [unlike the (2s2)ISe + (2s2p) 'P transition, where it is less than 2%] and is demonstrated in Fig. 16. As the new resonance structure occurs at incident electron energies between 3 and 6 Ry,it is found that at the temperature of maximum abundance of 0

P. L. Dufton and A . E. Kingston

398 5

I

I

I

m 1

0

c

X

T

L

al m

E

U

0,

c

0

L

I

=:!

0

c

0

-U

c

X

al C 0 L

c

U

22

w

\

1

L 7

5

53

59

56 l o g Te

FIG.16. Electron excitation rate for the (2s')'S + ( 2 ~ 2 p ) transition ~P in 0 V. Theoretical results: (-) R-matrix 12-state calculations (Berrington et a l . . 1979); (---) R-matrix sixstate calculations (Berringtonet a / . , 1977);(------) distorted wave calculations (Mann, 1980).

V in the sun, T = 200,000 K, the effect of these new resonance structures is about 10%. However, at higher temperatures we would expect a much larger increase in the cross section. An interesting calculation by Cowan (1980) on the rate coefficient for the (2s2)' S + (2s2p) 3P transition in 0 V suggests that for rate coefficient calculations it is possible to include the resonance effects using simple perturbation methods. His calculations also

ATOMIC PROCESSES IN THE SUN

399

suggest that in the region of 200,000 K this rate coefficient should not be greatly affected by the inclusion of more target state functions. A small number of calculations have also been carried out in Be-like ions on the electron excitation of transitions in which An changes by one or more. Distorted wave results for 0 V have been reported for a small number of energies by Malinovsky (1975) for the transitions from the (2s2)' S and ( 2 ~ 2 p ) ~states P to the ( 2 ~ 3 sI%, ) (2s3p) '*3P(2s3d)Is3D states. These calculations have been corrected and extended to include excitations from the (2s2p) 'P state (W. Eissner, private communication, 1980). Scaled electron impact collision strengths, Z 2 Q for transitions between the (2~21)states and (2131') states ( l = s or p andl' = s, p, or d) have been obtained by Parks and Sampson (1977) using the method suggested by Burgess et al. (1970). The Born approximation has been used to calculate excitation cross sections for transitions from the ground state of C 111, N IV, and 0 V to the (2sns) IS, (2snp) 'P, and (2pnd) 'Dstates for n = 3 , 4 , and 5 (Ganas and Green, 1979). For Si XI nonexchange distorted wave approximation has been used to calculate the electron excitation rate from the ground state to the (2s4p) 'P, (2sSp) 'P, and (2s4d) ID states (Davis et al., 1977). We have seen in Sections I11 and IV that for heavy ions it is important to take account of relativistic effects when calculating atomic energies and oscillator strengths. It is evident that it is also important to consider relativistic effects in the target wave functions in calculations on the electron excitation cross sections of heavy ions. We have already discussed (Section 111) a method of obtaining quite reliable CI relativistic wave functions for ions in which relativistic effects were not very large (Glass and Hibbert, 1978b). In this method the parameters of the one-electron orbitals [ Eq. (4)] are obtained by diagonalizing the nonrelativistic Hamiltonian matrix, but the coefficients bii in theJ-dependent CI wave function expansion [Eq. ( 1 3 1 are determined by diagonalizing the Breit-Pauli Hamiltonian. In intermediate coupling the wave function for a fine structure state with a given Jiand MJi and parityp is [Eq. (2111 IAiJiMJiP)

=

L.S. 1 1 1 .. 1

f(AiJi

9

r i L i S i p ) / r i ( L i S i ) J i M J , )p

(72)

where f [Eq. (22)] are called term-coupling coefficients (Jones, 1975; Saraph, 1972). Here A+ distinguishes between two states with the same values of J i and M ibut different energies, and T idistinguishes between two states with the same List but different energies. Using similar arguments (Jones, 1975)we can transform theRJWiLiSiJilK;T : L ; S ; J ; l ' K 'k; ? ) matrix to intermediate coupling using

P. L. Dufon and A. E. Kingston

400

RJD(AiJi1K; AiJil'K'; k : ) =

I: f(AiJt, T i L i S i ) R J ~ ~ ( T i L I S f JTtil LK,;' S ; J ; I ' K 'k; : )

I'lLiSt

x f ( A ; J ; , T;L;S;)

(73)

With this term-coupled R matrix in Eq. (58) we obtain the term-coupled TJi,matrix. This can then be used in R(AJi

+

AiJi)

to give the collision strength from one fine structure level AJi to another AiJ;. The effect of including relativistic effects in electron excitation calculations is most marked in spin forbidden transitions. Some very interesting term-coupling calculations have already been carried out for the electron excitation of Fe XXIII from the (2s2)'Sostate to the (2s2p)3P1state. These are displayed in Fig. 17. The nonrelativistic close-coupling calculations of Henry and Bhadra (1980)only differ by about 20% from the nonrelativistic distorted wave calculations of Mann (1980). The shape of the two collision strengths is also similar; they both decrease rapidly as the incident electron energy increases. This is the typical behavior of a spin-forbidden transition. The relativistic close-coupling (Robb, 1980)and distorted wave calculations (Mann, 1980) are also in close agreement, but they differ significantly from the nonrelativistic results. For example, at the excitation threshold the relativistic calculations are a factor of two larger than the nonrelativistic calculations, and at an energy of one hundred times threshold the factor is greater than ten. The inclusion of relativistic effects in heavy ions makes the collision strengths for spin-forbidden transitions similar in size and shape to the collision strengths for spin-allowed transitions. This is similar to the effect of including relativistic effects in transition probability calculations for heavy ions, where spin-forbidden transitions can have quite large transition probabilities when relativistic effects are included. For Fe XXIII relativistic effects are also important for the transitions (2s2)'S+ ( ~ P ~ and ) ~ (2s')'S P + (2p')'D. Collision strengths for these transitions in Fe XXIII and other transitions between ( 2 ~ 9(2s2p), , and (2p7 configurations have been obtained by Mann (1980), Robb (1980), Younger (1980), Feldman et al. (1980), Bhadra and Henry (1980), and Parks and Sampson (1977).

40 1

ATOMIC PROCESSES IN THE SUN

7

A

-------- --------------------

i

-3

10

I

I

I

I

I

l

l

0

I

I

I

I

l

l

10 X = [Energy of e l e c t r o n ] I [Excitation energy]

1

I 100

~P in FeXXIII. TheoretFic. 17. Collision strength, a, for the (2s')'S + ( 2 ~ 2 p ) transition six-state close-coupling calculations (Robb, ical results: (A) relativistic calculations, (-) 1980);(---)distorted wave calculations (Mann, 1980);(B) nonrelativistic calculations, (-) six-state close-coupling calculations (Henry and B hadra, 1980);(---) distorted wave calculations (Mann, 1980).

B. ELECTRON EXCITATION OF MANY-ELECTRON SYSTEMS In the previous section we have discussed the accuracy of electron impact excitation data for light ions using the Be sequence as a representative example. Here we discuss briefly results for ions in the third row of the periodic table and in particular new results for the S IV and Si 111 ions, which are both observed in the solar emission line spectrum. Electron impact collision strengths for S IV have been published by Bhatia rt (11. (1980) and Bhadra and Henry (1980). The former used a distorted wave approximation and relatively simple configuration interaction target wave functions; the latter chose a close-coupling calculation with more complex wave functions. For the (3S33p)rPo + (3s3p2)'P' transitions (the most important transition for the interpretation of solar UV

P. L . Duflon and A . E . Kingston

402

I

I

1

1

1

I

1

I

P*

A

I

0.7

I

0.8

Y

2

L

6

1

E IRy) FIG. 18. Collision strength, R, for the intercornbinationtransition in S IV from Dufton and Kingston (1980). Also shown are the close coupling results ( x ) of B hadra and Henry (1980) and the distorted wave results (0)of Bhatiaer 01. (1980). Note the change of scale on the energy axis between 0.95 and 1.5 Ry.

observations) the results of Bhadra and Henry are approximately 50% higher than those of Bhatia et af., which Bhadra and Henry attribute principally to the difference in target wave functions. Both sets of authors limit themselves to moderately high electron impact energies away from the region where resonances effects could be important. Recently Dufton and Kingston (1980) carried out R -matrix calculations using target wave functions similar to but somewhat more complicated than those of Bhadra and Henry. For the common energies points agreement between the two calculations is good, with, for example, the differences for the 2Po+- 4Petransition being less than 20%. However, in the low-energy region the collision strengths for many transitions are dominated by the resonance structure. This structure is shown in Fig. 18 for the sP"+ 4Ptransition, and is particularly complex due to the many partial waves which contribute to it. The resonances have the effect of increasing the collision rate, and for an electron temperature of 70,000 K (corresponding to the maximum ionization fraction for S IV) the R-matrix results give a rate a factor of three higher than that deduced from the values of Bhadra and Henry. A similar effect is found for Si 111, for which Baluja et al. (1980) pub-

403

ATOMIC PROCESSES IN THE SUN

$2 15

1c

?;

5

_L

05

06

07

Energy

0755,

-

lRyl

FIG.19. Collision strength, a, for the (3s2)'S (3s3pI3Ptransition in Si 111. Theoretical results: (-)R-matrix calculation (Balujaet a / ., 1980);(---)distorted wave calculation (see Nicolas, 1977).

lished results from a major 12-state calculation. Again for many transitions, the low-energy cross sections are dominated by resonances, and in Fig. 19, the structure for the (3s') ISe + (3s3p)3Putransition is shown. This ion has a structure analogous to that of the Be sequence, but with additional closed shells. However, while in the Be sequence the effect of resonances was to increase the IS" .+ 3P"collision rate by less than 50%, in the case of Si 111, the rate is increased by factors of between two and six compared with the calculations of Nicolas (1977), which did not include resonances. Hence, there is some evidence that for some transitions in heavier ions, the collision rates are dominated by resonances, and that calculations which fail to take such effects into account will give incorrect results by up to an order of magnitude.

VI. Proton Excitation For most transitions, excitation by electrons is very much more effective than excitation by protons. However, it was pointed out by Seaton (1964) that excitation by protons can be important for excitation between fine structure levels. If we consider a plasma at a given temperature, then the kinetic energy of the electrons, tm,u,2, is equal to the kinetic energy of

404

P. L . Dufon and A . E. Kingston

the protons, tmpug, where me and ue are the electron mass and velocity and mp and up are the proton mass and velocity, where &/me = 1836. The cross sections for electron and proton excitation are approximately equal when their velocities are equal. As the electron excitation cross section is largest for electron energies from one to about ten times the excitation energy of the transition, the proton excitation cross section will be largest at approximately 2000-20,000 times the excitation energy. Proton collisions will therefore be important for transitions which have excitation energies very much smaller than the thermal energy of the plasma. Excitation by other ions such as He+ will be much less important due to the higher abundance of the protons. There have been many estimates of the cross section for fine structure transitions induced by proton impact (Seaton, 1964; Bahcall and Wolf, 1968; Reid and Schwartz, 1969; Bely and Faucher, 1970; Masnou-Seeuws and McCarroll, 1972; Sahal-Brechot, 1974; Faucher, 1977; Faucher and Landman, 1977; Kastner, 1977). The close-coupled impact parameter approximation of Reid and Schwartz (1969) and Masnou-Seeuws and McCarroll (1972) has been used by Doyle et af. (1980) to obtain proton excitation rates for the fine structure transitions between the levels of the Be-like ions C 111,O (2s2p)3Po,1,2and between the levels (2p2)3P0,1,2 V, and Ne VII. In this approximation the perturbing proton follows a classical Coulombic trajectory and the intermultiplet transitions are caused by the quadrupole component of the electrostatic interaction of the proton with the electrons of the ion. This arises because, in the expansion of the electrostatic interaction in terms of R , the distance between the proton and the nucleus, the first nonzero term in this expansion is the , ( r 2 ) is the expectation value o f r 2 for the quadrupole term ( r * ) / R 3 where p electron. Faucher and Landman (1977) have carried out a more exact quanta1 treatment of these proton-ion collisions, and they have shown that the use of semiclassical methods does not give rise to significant errors. They suggest that the major error in the use of semiclassical methods arises from an inexact treatment of the penetration by the proton of the electron cloud of the ion. In their work Doyle et al. (1980) took account of this penetration by using several different short-range potentials and concluded that this penetration gave an error of about 1% in the cross sections below 200 eV but gave larger errors at higher energies. Figure 20 gives the proton excitation rates for the (2s2p) "Pp + (2s2p) g for C 111, 0 V, and Ne VII, which and ( 2 ~ 2 p ) ~ P+g ( 2 ~ 2 p ) ~ Ptransitions were calculated by Doyle et al. (1980). It was found convenient to plot the proton excitation rate multiplied by ( Z ' )' against a reduced temperature T K divided by ( Z ' ) 3 ,where Z ' is the nuclear charge minus two. The authors quote an accuracy of 1% for these results for temperatures below lo6K

ATOMIC PROCESSES IN THE SUN

I

405

1 -

FIG.20. (2s2p):’PPand

and an error of 3% at a temperature of lo7 K. Their results for the transitions (2p2)3P1+ (2p’) 3Pzand (2p7 3P0+ (2p’) 3Ppare only slightly different from those for the ( 2 ~ 2 p ) ~transitions. P state to the ( 2 ~ 2 ~ ) ” : The rate for proton excitation of the (2~2p)~Pp,O state for C 111, 0 V, and Ne VII was also calculated by Doyle et al. (1980); their results are shown in Fig. 21. They also found that the rate

Flci. 21. Fine structure proton excitation rate multiplied by (2’)’ for the (2s2p)’P: (2s2p):’Pptransition in C 111, 0 V, and Ne VII (Doyle el (I/.. 1980).

-+

406

P. L . Dufton oiid A . E . Kingston

for the (2pz)3P0+ (2p')3P1 transition was close to that for the "~ ( 2 ~ 2 p )+ ~ P( ~2 ~ 2 ~ )transition. Similar calculations were carried out by Malinovsky (1975) for 0 V using the formulas of Sahal-Brechot (1974). For J = 0 + 2 and J = I + 2 transitions, her results differ by less than 20% from the results of Doyle et rrl. (19801, but for J = 0 + 1 transitions her results differ by about 30%. The proton excitation rate for excitation of the (2~2p)~Pb\ level to the (2s2p):'P:' and 3PBlevels in C 111 is plotted in Fig, 13, where it is compared with the rates of electron excitation of that level. At low temperatures the proton excitation rate is much smaller than the equivalent electron rate, but as the temperature increases, the proton excitation rate increases rapidly while the electron excitation rate decreases rapidly, at higher temperatures the proton rate being larger than the electron rate.

VII. Applications of Atomic Data to Solar Plasmas There are a number of excellent reviews (Gabriel and Jordan, 1972; Dupree, 1978; Dere et (11.. 1979; Feldmanet al., 1978)on the interpretation of the solar emission line spectra. Hence, here we shall limit ourselves to a description of how the observed emission line spectra can be used to derive the parameters of the emitting plasma. Given the atomic data discussed in the previous sections, it is possible to predict intensities for the solar emission line spectrum. The line strengths will depend upon the population of the ionic upper level of the transitions along the line of sight. In deriving such level populations it is often possible to consider each ionization stage independently. This is equivalent to assuming that the processes between different ionization stages are slow compared with those within each stage. Then for a set of n levels of a given ion, the change in the population N i , of level i is given by n

+

2 N,Aji

J= i

-

Ni

I=

A,

(75)

1

where C , is the electron collision rate for a transition between levels i andj and unit electron density with Cii = 0 and Aij is the spontaneous radiative deexcitation rate from level i to levelj with Aii = 0. The first two terms are the collisional rates into and out of level i, respectively, while the second two terms give the corresponding radiative rates. As only spontaneous radiative deexcitation is considered, the sums

ATOMIC PROCESSES IN THE SUN

407

in the latter terms are limited to downward transitions. For some ions, e.g., He-like species, stimulated processes may also be important, and this would lead to additional terms, including the local radiation field in Eq. (75) (see, for example, Doyle, 1980). This formulation also assumes that transitions via ionic impacts are negligible. As discussed above, the solar plasma parameters are such that this process will only be significant for mixing between fine structure levels. These additional rates can be easily included via expressions analogous to the first two terms in Eq. (75). For a stationary plasma, the time derivatives will be zero, and Eq. (75) reduces to n simultaneous equations of the form

Cij +

N,

Aij 5= 1

j=l

together with the constraint that the level populations are related to the by volume density of the ionization stage, NION, Ni

=

NION

(77)

i= 1

(the neglect of highly excited levels, which are not included in the rate equations, from this normalization is not likely to be serious as for solar plasma conditions they have minute populations). The set of equations (76) has two limiting solutions corresponding to low- and high-density regions. For cases with low electron densities, where level i has an allowed transition to the ground state, the first term in the denominator is negligible. Also under such conditions the values of the excited level populations are very small (by typically eight orders of magnitude for N, = lo9 cm-3) compared with the ground state. The first term in the numerator is then the more significant, and the coronal approximation of Elwert (1952) is found, viz.,

Ni

=

(78)

N,NICIi/Ail

where the subscript 1 refers to the ground state. In contrast at high electron densities the radiative terms become insignificant, and Eqs. (76) reduce to N~ =

5= 1

cij,

i

=

I ton

(79)

408

P. L. Dufon and A . E. Kingston

The relation between inverse collisional rates (see, for example, Mihalas, 1978) then leads directly to the thermodynamic equilibrium population distribution, namely, NjINi = gdgi exp( - 4 i I k T J

(80)

where gj,gi are the level degeneracies and Eliis the energy difference. For El transitions with spontaneous deexcitation rates of more than lo8 sec-', the high-density limit is only achieved for electron densities of the order of 10'' ~ m - much ~ , higher than those found in the solar transition region or corona. However, for groups of levels which have no large allowed transitions (for example, fine structure levels), the high-density limit can often be obtained at solar densities. The line emissivity, eu, for a transition between levels i a n d j will be given by (see, for example, Mihalas, 1978) =

(h ~ / 4 ~ ) N j A j i

(81)

and for an optically thin plasma the observed surface flux will be simply an integral over the line-of-sight region, where the line-emitting ion is abundant In the idealized case of a constant density and temperature with a line-of-sight dimension, 1, the observed flux will be FU

=

(h~/4n)AjilNj

(83)

and we note that for the coronal approximation the line strength becomes independent of the Einstein A coefficient and proportional to the collisional excitation rate. Equations (76) and (82) provide a general solution for a stationary optically thin plasma. In a typical case the lowest 10-20 levels (including each fine structure level separately) would be included in the statistical equilibrium calculation, requiring the solution of the same number of simultaneous equations. This procedure would be undertaken at various positions with different electron densities and temperatures along the line of sight and the line flux calculated by direct numerical integration of Eq. (82). Several programs exist (see, for example, Dufton, 1978) which solve this relatively simple set of equations. Such calculations are normally used to deduce plasma parameters, such as the electron density, from the observed relative line strengths. As an illustrative example we shall consider an isothermal, constant-density plasma (because a given ion has normally a significant population over only a relative small region of plasma, this approximation is useful for many solar features). Then the electron temperature can be deduced from

ATOMIC PROCESSES IN THE SUN

409

the ratio of the intensities of emission line originating from levels with different excitation energies of the same ion. The method is based on the different temperature dependence of the excitation rates populating the two levels. If we consider two levels i and j for which the principal rates are spontaneous radiative deexcitation and electron impact excitation from the ground state (i.e., excitation rates from metastable levels are not significant), then the coronal approximation gives

where ljl is the intensity of the transition from level j to the ground state, E j is the energy of level j relative to the ground state and Clj is the excitation rate to levelj, which may be written as (see, for example, Gabriel and Jordan, 19721,

where g , is the degeneracy of the ground state and Ru the effective collision strength is a slowly varying function of temperature. Then

Hence, given the observed line ratio, R, the electron temperature, T, can be derived. Unfortunately for the line ratio to be sensitive to temperature the value of (Ed- Ei)must be large. In such cases the transitions from the two levels to the ground state will be at well-separated wavelengths, leading to problems in determining the observational ratio accurately. For example, in 0 V the (2s3p)'P + (2s')'S and (2s2p)lP + (2s2)IS lines used by Malinovsky (1975) to derive temperatures lie at 172 and 630 A, respectively. The sensitivity of this ratio to electron temperature is illustrated in Fig. 22. This problem may be circumvented by considering a transition from level j to some excited level k. Then, provided that this spontaneous radiative transition is the main depopulation mechanism for level j , Eq. (86) becomes

and the wide wavelength separation need no longer necessarily exist. An example of such a situation for 0 V is the (2p7 IDe + (2s2p) IP0transition P "(2s7 ISe transiat 1371 A, which may be compared with the ( 2 ~ 2 p ) ~+

P. L. Dirfron and A . E. Kingston

410

0.2 I(13711 111218)

0.1

I

I

5

5.5

Log Te

6

FIG.22. Temperature-sensitive line ratios in 0 V. The solid curve is the ratio ,Of the intensitiesofthe(2~3p)~P"(2s2)'Selineat172A tothe(2s2p)'P:-(2s2)lSelineat630A.The dashed curve is the ratio of the ( 2 ~ 2 p ) ~ + P "(2s2)lSeline at 1218 A to the (2p')ID'- ( 2 ~ 2 p ) ~ P " line at 1371 A.

tion at 1218 A. These lines are at similar wavelengths, and for electron densities less than 3 x 10'O cm3both obey the coronal approximation. The temperature sensitivity of the ratio is again shown in Fig. 22 (Duftonet al., 1978). Although this is not as large as for the other 0 V ratio, this is probably more than compensated by the greater accuracy with which the observational ratio can be measured. Although a number of different techniques exist for estimating the electron density, these all ultimately depend on comparing the population of a level whose population mechanisms are dominated by collisional processes with that of a level having a spontaneous deexcitation rate, which is greater than any of the collisional rates. The relative population of the former is given by Eq. (80) and is independent of electron density, while that of the latter is proportional to electron density. Hence, any method measuring the ratio of these populations will yield an estimate of electron density. One approach consists of comparing the intensity of an intercombination spin-forbidden transition with that of an allowed transition. As an

ATOMIC PROCESSES IN THE SUN

41 1

example, the 0 IV intercombination lines ( 2 ~ 2 ~4Pe ' ) + ( 2 ~ ~2P" 2 can ~ ) be ~ ) ' s ' lines, as has been compared with the C IV resonance ( ~ P ) ~ P ( 2' + discussed by Feldman and Doschek (1978). For electron densities N , 3 3 x 10"' ~ m - the ~ , upper levels of the 0 IV transitions have a Boltzmann population compared with the ground state, while the C IV upper levels are still in the coronal approximation due to their large spontaneous deexcitation rates. Hence, for large electron densities the ratio of the C IV line intensities to those of 0 IV are proportional to electron density. To use this method to derive absolute densities (compared with density differences between different solar regions) it is necessary t o know the carbon-to-oxygen abundance ratio in the emitting plasma (usually assumed to be the photospheric value) and also to choose ions which will have similar spatial distributions. In ionization equilibrium 0 IV has a maximum ionization fraction at approximately 1.2 x lo5 K (Jordan, 1969), while C IV is formed at a lower temperature of 9 x lo4 K in presumably an overlapping but not spatially coincident region of the atmosphere. This difficulty can be partially eliminated by considering a second optically allowed transition in an ion, such as N V, which has a temperature of maximum ionization fraction higher than that of 0 IV, and hence in effect bracketing the 0 IV ion. Such methods have been widely used by Doschek, Feldman, and their co-workers (Bhatiaet al., 1980; Feldman et ul., 1978). The problem of spatial coincidence can be considerably reduced by using two lines from the same ion. For example, in 0 V the resonance (2s2p)'P0 + (2s2)ISetransition at 630 A can be compared with the intercombination ( 2 ~ 2 p ) ~ + P " (2s2)ISetransition at 1371 A. The sensitivity of this ratio to electron density is shown in Fig. 23 (it also has a small variation with temperature), the diagnostic being useful for N , 2 3 x 10'" cmP3,where the collision rate for the intercombination line is comparable with or greater than the radiative rate. Although this is theoretically more reliable, the constraint of using only one ion often results in suitable line pairs being widely separated in wavelength, with the observational value of the ratio being therefore more difficult t o determine. Additionally, this method requires accurate atomic data. This is illustrated in Fig. 23, where the effect is shown of increasing the ISe + 3Purate by 25% and decreasing the ISe -+ 'PI' rate by a similar amount. Other more indirect methods exist for estimating the electron density. For example, in the beryllium sequence the ratio of the two optically allowed transitions, ( 2 . ~ 2'PU ~ ) + (2s') ISe and (2p2)3Pe+ (2s2p) 3P0, has been extensively used (Loulergue and Nussbaumer, 1974, 1976; Jordan, 1974; Dufton et al., 1978; Dupree et al., 1976; Malinovsky, 1975). The intensity of the former depends on the population of the upper level, while

P. L. Dufron and A . E . Kingston

412 I

I

I

I,'

I

I

Fic;. 23. Density-sensitive line ratios in 0 V. The solid curve is the ratio of the intensities of the ( 2 ~ 2 p ) ~+ P "(2s2)'Seline at 1218 A to the (2s2p)'P"+ (2s*)lSeline at 630 A. The heavily ~ P( 2e ~ 2 ~ ) " "multiplet at 760 A to the (2s2p)'P" + dashed curve is the ratio of the ( 2 ~ ~ ) .+ (2s2)'Seline at 630 A. Also shown are the effects of varying selected collision rates by 25%.

that of the latter is a measure of the populations of both its upper and lower metastable 3P0levels, because the principal population mechanism of the 3Pe states is electron excitation from the 3Pustates. The density dependence of this ratio is illustrated in Fig. 23 for 0 V, the results again being taken from Dufton et al. (1978). This indirect measurement of the population of the 3P0states has the advantage over the more direct measurement discussed above, in that the lines have a smaller wavelength separation (63&760 A for 0 V). However, the theoretical results are again sensitive to the atomic data, particularly for the collision rates. This is shown in Fig. 23 by plotting the theoretical ratio with the collision rates changed as discussed above. As can be seen, the electron density deduced from a given ratio is changed by more than a factor of two by these modest changes in the atomic data. Another method of determining electron densities utilizes lines within P e( 2 ~ 2 p ) ~ components P" in the same multiplet, for example, the ( 2 ~ * ) ~+ ~P" in the beryllium sequence and the ( 2 ~ 2 p ~+ ) ~( P2 ~ ~ 2 p ) components the boron sequence. For the latter, the lines of 0 IV (Feldman and Dos-

413

ATOMIC PROCESSES I N THE SUN

I

J

FIG.24. Ratio of the intensities ofthe ( J = 3 / ? , J ' = 3/2) line to the ( J = 5/Z,J ' = 3/2) line in the S IV (3s3p')'R- (3s23p)'F'!!. rnultiplet. The heavily dashed curve uses the collision rate data of Dufton and Kingston (1980). and the lightly dashed curve the data of Bhatiaet ti/. (1980).

chek, 1979) are of particular interest as they are well separated, while lying at a wavelength (1400 A) accessible to conventional UV grating spectrographs. They have been observed by many solar experiments, including the recently launched Solar Maximum Mission Satellite. All the lines in the multiplet are spin forbidden and have small EinsteinA coefficients. Hence, as the electron density increases, the population of the upper levels relative to the ground state changes from the coronal approximation to Boltzmann statistics. However, due to variation in the spontaneous radiative rates, this occurs at different electron densities for the different 'Pe fine-structure levels, leading to the density sensitivity. This method has many advantages, including the use of a single-ion, small-wavelength separation of the components and line ratios near to unity, which are important for detectors such as photographic film, which suffer from saturation effects. However, there is again the need for accurate atomic data. This can be seen in the case of the analogous (3~3p')~Pe, + (3s23p)'Po, transitions in S IV. In Fig. 24 the ratio of the intensities of the J = 3/2 t o J ' = 3/2 transition at 1416.9 A to the J = S / 2 to J' = 3 / 2 transition at 1406.0 A is plotted as a function of electron density.

414

P. L. Di!fton und A . E. Kingston

The dashed curve refers to the atomic data of Bhatia ef al. (1980) based on relative simple S IV wave functions and ignoring the effect of resonances in the electron excitation rates. The heavily dashed curve incorporates more sophisticated wave functions for both the spontaneous radiative and R -matrix collision strength calculations and explicitly includes the effect of resonances (Dufton and Kingston, 1980). The Einstein A coefficients are changed by typically 10-20%, but the collision rates are increased by factors of up to five (see the section on electron excitation calculations), and this, in turn, leads to major changes in the electron densities deduced from a given ratio. Besides being used as probes of electron density and temperature, line ratios can provide information of the dimensions of the emitting plasma through optical depth effects. The above discussion has assumed that the plasma is optically thin, i.e., any photon emitted has a negligible probability of being reabsorbed. In practice for strong allowed transitions there is a significant chance that a photon will be lost by the inverse process; the greater the extent or density of the plasma, the greater is the probability. Hence, by comparing the intensity of different lines, usually in the same multiplet, it is possible to deduce information about the extent of plasma where the relevant ion has a significant population. Although atomic physics data are required for such analyses, they do not have the crucial importance as, for example, in electron density diagnostics. For further details the reader is referred to Doyle and McWhirter (1980) and Nicolas (1977).

VIII. Conclusions In this :eview we have discussed recent advances which have been made in theoretical calculations of basic atomic data which are relevant in the analysis of solar emission line spectra. The best theoretical calculations are now estimated to be accurate to better than 5%, although such results are only available at present for a small number of ions. It is envisaged that in the future precise theoretical data for many other ions will become available and will be used in interpreting the large amount of observational data obtained from rocket- and satellite-borne solar instruments. Additionally, high-quality observations of the solar emission spectrum should provide some of the most rigorous checks on the accuracy of the theoretical calculations.

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ACKNOWLEDGMENTS We are grateful to our colleagues in the Department of Applied Mathematics and Theoretical Physics at Queen’s University, Belfast, in particular, Professor P. G. Burke and Drs. K. A. Berrington, J. G. Doyle, and A. Hibbert, for their help and advice. We would also like to thank the staff of the Harvard College Observatory, Boston, the Naval Research Laboratory, Washington, and the SMM at the Goddard Space Flight Center, Washington for their hospitality and assistance during our visits. We acknowledge useful discussions with o w collaborators in the QUACS consortium organised by Dr. R. W. P. McWhirter. Most of our work discussed in this review was supported by grants from the British Science Research Council.

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A

ac Stark shift, see Stark shift Adiabatic eigenergies, 64 Alkali atoms. quantum defect theory for, 123-124 Alkali metals, optical spectra of, 120-126 Alkali Rydberg states, see also Rydberg states quantum defect and, 123-124 relativistic effects in, 124-126 Alkalis ionization thresholds in, 152-156 Stark states in. 142-146 Amplified spontaneous emission. 193 SF transition to, 196-200 Anisotropy parameter, for valence photoelectrons. 21 Argon, see also Inert gases crossed-pulsed nozzle jet beams and, 270 two-photon excitation of. 240-249 Argon autoionization level, parameters of. 50 Argon K X-ray production cross sections, 284 ASE. see Amplified spontaneous emission Atomic expansion one-center, 308-310 two-state. two-center, 31 1-312 Atomic models. of ion-atomic collisions, 324-326 Atomic physics inert gases in, 239 resonance ionization spectroscopy in, 229-272 Atomic Rydberg states, 99-161, see also Rydberg states

electric field ionization and, 151-156 optical excitation in, 107-1 12 preparation of, 103,l 12 two-photon excitation of, 109 Atomic spectra, in sun, 359-361 Atomic structure, in presence of uniform electric field, 138-146 Atoms, photoionization of, see Photoionization Auger electron anisotropy, 24-25 Autoionizing level, photoionization cross section near, 49

B Barium, high-resolution Rydberg spectroscopy and, 129-130 Bates-Crothers classical trajectory. 60 Bates-McCarroll plane waves. 89 Be-like ions, electron excitation of. 374. 377. 382-400 Bloch sphere. 185. 187. 199 Bloch-type vibrations. 31 Bloch vector, 184-185, 197 collective, 190 Born approximations first and second, 320-321, 338 higher order, 309 Bose operator, 178 Bound states, spontaneous decay of, 3703 80 Bound-state wave functions, in solar spectrum, 361-370 Branching ratios, in intershell interactions, 22 Breit-Pauli Hamiltonian, 368-370. 378 Broad bandwidth lasers. multiphon excitation with. 231-249 419

INDEX

420 C

Callaway-Bartling model, 75, 80 Carbon monoxide energy levels in different isotopic species of, 268 isotopically selective detection of, 265268 I4C detection, laser spectroscopic techniques in, 265 CDW, see Continuum distorted waves Cesium experiment, in superfluorescence, 208-210 Charge transfer at asymptotically high velocities, 341342 at high velocity, 319-324 nonadiabatic, 55-93 specific background of, 56-63 as three-body process, 55 Chlorine 3ps shell, photoionization cross section for, 27 CIS, see Continuum intermediate-state approximation CIV3 computer program, 363 C1 (configuration interaction) wave functions, in solar spectrum, 359, 362-364, 369, 374-377 Classical ionization limit, stark levels and, 140 Classical trajectory Monte Carlo, 92 Closed-shell atoms, one-electron wave functions and, 1 I Collective effects in atoms with open shells, 25-30 collective oscillations and, 31 defined, 3 near inner-shell thresholds, 32-40 in photoionization of atoms, 1-52 relativistic effects of, 20-25 static rearrangement and, 33-36 Collective oscillations, 3 1-32 Collectivekelativistic effects Auger electron anisotropy and, 24-25 branching ratios and, 22 combination of, 20-25 Collectivization, of 4p shell in xenon, 4546 Collectivization of vacancies, 40-50 “shadow” levels in, 46-48

Collisional ionization, Rydberg state detection by, 116-117 Collision channels, wave function of, 128 Collision rates, of inert gases, 249-252 Collision systems heavy asymmetric to near-symmetric, 280-28 1 light symmetric and near-symmetric, 278-279 Comparison equations, phase integrals and, 63-83 Complex atoms, MQDT analysis of, 130 Configuration interaction wave functions, in solar spectrum, 359, 362-364, 369, 374-377 Continuum distorted waves, 91 approximation of, 323-324, 328 Continuum intermediate-state approximation, 323-324 Cooper minimum value, 3, 20 Copper, K-shell ionization of, 298 Correlation functions, superfluorescence and, 181-182 Coulomb function, in field of target nucleus, 323 Coulombic electron-hole interactions, 48 Coulombic field, Stark levels and, 148 Coulombic interaction matrix element, 44 Coulombic interactions, multiple series of, 255 Coulombic phase shift, 83 Coulombic potential, quantum defect theory and, 123 Coulombic wave function, 83 Crossed fields, Rydbert atoms in, 157 CTMC, see Classical trajectory Monte Carlo Cylindrical coordinates, saddle-point limit in, 139 D

DeBroglie wavelength, ion-atom collisions and, 303 Delos-Thorson variable, 80-8 1 DEP, see Double-electron photoionization Diabatic JWKB functions, 58 “Diabatic” notation, Stueckelberg model and, 77

42 1

INDEX Diabatic representation, defined, 57 Dicke superradiance, 159-161 Differential cross sections and duldR at asymptotically high velocities, 341-342 and duldR at medium and high velocities, 338-340 in inner-shell vacancy production, 336342 Dipole-dipole interaction, 168-169 Dipole polarizability, I5 Dirac-Breit equation, 303 Discretized continuum states, 309 Distorted-wave approximation, 56 Be-like election excitations and, 386 two-state model in, 317 Doppler broadening, line shape and width in, 257 Double-electron photoionization, 3, 30, 39, see also Photoionization Double K-electron transfer, 343-347 Double K-vacancy production, 299-303 Double resonance microwave spectrosCOPY,122-123 Dye lasers in quantum beat spectroscopy, 121-122 in Rydberg state excitation, I 11-1 12

E

Eikonal approximation, in electron capture, 318-319 Electric field ionization, 151-156 Electric fields Rydberg atoms and, 136-156 and stable and autoionizing states in alkaline earths, 147 Electromagnetic field interaction of with atoms, 2-3 Maxwell-Bloch description of, 216 Electron, potential energy surface of, 137 Electron bombardment, atomic Rydberg states preparation by, 105-106 Electron capture eikonal approximation and, 3 18-3 19 at intermediate velocities, 3 10-3 19 three-center expansion and, 314-3 18 two-center multistate expansion and, 314-318

Electron excitation of many-electron systems, 401-403 six-state R-matrix calculations of, 397 Electron excitation rate coefficients, for ground-state excitation, 394 Electronic transitions, laser drives for, 262 Electronic translation, perturbed stationary states and, 83-91 Electrons “fast” and “slow,” 37-38 “up” and “down,” 28 Electron-scattering calculations, accuracy of, 385 Electron transfer, atomic Rydberg state preparation by, 104-105 Electron translation factor, 305-308 Exponential model, advantages and applications of, 78-83 External fields, Rydberg atoms in, 131157

F

FY+ Ne(K) transfer cross sections, 335 F’ + Si(K) transfer cross sections, 332 Fabry-Perot cavity, 161 Fano formula, 49 Few-electron shells collectivization, 15-17 Field ionization Rydberg atoms and, 136-156 in Rydberg state detection, 114-1 16 Fluorescence, in Rydberg state detection, 113-1 14, see also Superfluorescence Fritsch-Wille model, Stueckelberg transition probabilities for, 79

G

Generalized random phase approximation with exchange, 33-36, see also Random phase approximation with exchange Gerade wave functions, 84 Green’s function, 180 GRPAE, see Generalized random phase approximation with exchange

422

INDEX

H H+ + He transfer cross sections, 327-329 H+ + Ne(K) transfer cross sections, 329330 Hartree-Fock function, 364, 374 Heisenberg operator deviated spins and, 225 equation of motion for, 178-180 Helium collisional effects on, 251 three-photon exciation in, 261-262 Heteronuclear collisions, 88-91 High-energy charge transfer, 3 19-324 impulse approximation and, 322-323 Holstein-Primakoff transformation, 225 Holstein theory, in resonance line broadening, 255-258 Homogeneous and inhomogeneous broadening, in superfluorescence, 193-202 Homogeneous broadening, vs. inhomogeneous Lorentian broadening, 201-202 Homonuclear collisions, 83-87 Hydrogen atom, theoretical and experimental investigations for, 138-142

I

IMPACT computer code, 386 Impact energies, cross sections for formation of H(2p) and H(2s) atoms in helium-hydrogen collisions, 87-91 Inelastic ion-atom collisions, see also Ionatom collisions experimental measurements of, 277-303 theory of, 303-326 Inert gases in atomic and molecular physics, 239 collision rates of, 249-252 crossed-beam studies of laser-induced collisions in, 269-272 lifetimes for excited states of, 249 photoionization cross sections of, 249252 resonance ionization spectroscopy of, 239-262

resonance radiation trapping in, 252-260 two-photon excitation of, 240-249 Inhomogeneous broadening delay time and, 200-201 SF emission damped by, 198 SF pulse and, 210-211 SF theory and, 193-194 Initiation time regimens, equations of motion in, 176-180 Inner-sheU thresholds, collective effects near, 32-40 Inner vacancy decay, 36-40 Inner-shell vacancy production differential cross sections in, 336-342 double K-electron transfer in, 343-347 theories vs. experiments in, 326-347 Interaction matrix element, 14 Interelectron Coulomb interaction, 2, 26 Intermediate velocities, electron capture at, 310-319 Ion-atom collisions atomic model of, 324-326 and direct excitation and ionization at high velocities, 307-3 10 inelastic, 277-326 inner-shell vacancy production in, 275348 low-velocity region in, 305-307 molecular orbitals and, 305-307 multielectron excitations in, 299-300 Ionization, by static electric fields, 151156 Ionization fields, timing sequences in measurement of, 153 Isotopically selective detection, of ' ' C ' 6 0 , 267

J

JWKB semiclassical phase integral, 57-58, 63

K K-Auger electron, detection of, 277 K-Auger electron production cross sec-

423

INDEX tions, projectile charge-state dependence of, 282-289 K fluorescence yield, for titanium, 287 K-K charge transfer impact parameter dependence for, 342343 in target K-shell vacancy production, 347 K-K charge transfer cross sections, 2, dependence of, 334 Krypton, two photon excitation of, 240249, see ulso Inert gases K-shell electron transfer, 275-276 K-shell excitation cross sections, of projectile ions F"+ and F", 294 K-shell ionization energy dependence of, 292 polarization effect due to, 290-291 K-shell vacancy, L-shell electron knockdown and, 325 K-shell vacancy production, 276 b-dependence of, 298 Coulomb ionization in, 301 double, 299-303 increased binding effect in, 298 in light symmetric or near-symmetric collision systems, 278-279 theoretical models of, 281-299 velocity dependence of, 289-293 Zp dependence for, 293-296 2, dependence of, 297-299 K-shell vacancy production cross section, for H ' and Ne collisions, 290 K X-ray electron production cross sections, projectile charge-state dependence of, 282-289 K X-rays detection of, 277 high-resolution, 280

L

Landau-Zener linear model unit, 67-68 Laser excitation, of atomic Rydberg states, 107- 108 Laser-induced collisions, crossed beam studies of, 269-272 Lasers, in Rydberg state excitation, 187112

Laser systems, pulse lengths and pulse energies in, 264 LCAO, see Linear combination of atomic orbitals Li3 + Ne(K), transfer cross sections for, 330-33 1 Light asymmetric collision systems, 279280 Light projectile asymmetric conditions, 279 Light symmetric collision systems, 278279 Light target asymmetric systems, 280 Linear combination of atomic orbitals, 56 approximations in, 81 diabetic formulation of, 66 Low velocity region, in ion-atom collisions, 305-307 L-shell electron, in K-shell vacancy, 325

M

Magnetic fields, Rydberg atoms in, 131136 Magnetic quadrupole transitions, in solar atomic processes, 379 Magnus approximation, 81-82 Manganese, photoionization cross sections near 3p threshold of, 29 Many-body perturbation theory, 4- 13 Many-electron systems, electron excitation of, 40 1-403 Maser oscillation, superradiance and, 159161 Maxwell-Bloch equations, 183 derivation of, 223-224 in superfluorescence, 193, 196-197 Maxwell-Bloch theory, superfluorescence and, 189, 207, 216 MBPT, see Many-body perturbation theory Mixed diabatic-adiabatic formulation, 66 MO, see Molecular orbitals Molecular orbital adiabatic potentials, 3 11 Molecular orbital calculations, two-state. 332-333 Molecular orbitals expansion of, 308, 31 I model of. 305-307

424

INDEX

Molecular physics inert gases in, 239 resonance ionization spectroscopy in, 229-272 Molecule, absorption spectrum of, 263 MQDT analysis of complex atoms, 130-131 in Rydberg state spectroscopy, 128-129 Multiphon excitation, with broad bandwidth lasers, 231-239

N

Near-symmetric collision systems, 278-279 Neon, three-photon excitation rates in, 261-262 Neon K X-ray production cross sections, 283 NIEM, see Noninteractive integral equation method Nonadiabatic charge transfer, 55-93 Noninteractive integral equation method, 386 Nonmolecular three-body analysis, 91-93 Nozzle beam, spectroscopic selectivity of excitation in, 264

0

OBK method, see Oppenheimer-Brinkman-Kramer approximation One-electron collision system, 303-304 Open-shell atoms, collective effects in, 2530 Oppenheimer-Brinkman-Kramer approximation, 56, 296, 312-313, 319 Optical excitation, of atomic Rydberg states, 107-1 12 Optical spectroscopy, high-resolution, 120121, see also Resonance ionization spectroscopy Optogalvanic detection, of Rydberg states, 1 I8

P Parabolic comparison equation methods, 74-76 Partial cross sections, 15-17 PCI, see Postcollisional interaction Perturbation theory correspondence rules of, 8-10 many-body, 4-13 Perturbed stationary states, 83-91 model of, 306 Phase integral method, 63-83 advantages of, 64-65 Photoelectrons angular distribution of, 18-20 polarization of, 22-24 Photoionization characteristics of, 13-32 collective effects in, 1-52 double-electron, 3, 30 total cross-section calculations in, 1315

Photoionization amplitude, 13 Photoionization cross sections of inert gases, 249-252 near autoionizing level, 49 Photoionization detection, of Rydberg states, 118-119 Photoionization spectra, in presence of electric field, 147-151 Planck distribution, 254 Planck function, 253 Plane wave Born approximation, 279-28 I , 292-294, 298, 309 Polarizability , autocorrelation function and, 256 Polarization Bose operator, 178 Polarization effect, K-shell ionization and, 290-291 Postcollision interaction, 36-40 Proton excitation, in solar atomic processes, 403-406 Proton-ion collisions, in solar atomic processes, 404 Pulsed supersonic nozzle jet beams, resonance ionization spectroscopy and, 262-272 PWBA, see Plane wave Born approximation

INDEX

Q Quantum beat spectroscopy, laser in, 121122 Quantum defect theory, for alkali atoms, I 23- 124

R Radiation transport, onset of, 257 Radiative decay, metastable state and, 260 Random phase approximation with exchange, 4-13 collective/relativistic effects and, 21 cross-section calculations in, 13 derivation of equations for, 5-6 generalization of, 33-36 in 1s argon photoionization, 39 for open-shell atoms, 26 radioactive decay probability and, 44 significant formula and general relations in, 11-13 technique of diagrams in, 6-9 time-forward diagrams for, 10 for open-shell atoms, 27 Rayleigh-Schrodinger perturbation theory, 140 Relativistic effects, in alkali Rydberg states, 124-126 Resonance ionization spectroscopy in atomic and molecular physics, 229272 Doppler-free excitation and, 239 high-resolution, 238-239 of inert gases, 239-262 multiphon excitation in, 231-239 and pulsed supersonic nozzle jet beams, 262-272 Holstein theory and, 255-258 trapping of in inert gases, 252-260 RIS, see Resonance ionization spectrosCOPY Rosen-Zener model, 75, 80 Rotation-matrix transformation, 59 RPAE, see Random phase approximation with exchange Rydberg atoms

425

in crossed fields, 157 in electric fields, 136-156 in external fields, 131-157 in ground state, 131 in magnetic fields, 131-136 properties of, 101 Rydberg electron, interaction with core, 1 I9 Rydberg formula, original proposal of, 100, 119 Rydberg ionization spectrometer, 115 Rydberg orbital overlap, 106 Rydberg state lifetimes, measurements of, 158-159 Rydberg states, 101-1 19 atomic, see Atomic Rydberg states atom preparation by electron bombardment in, 105-107 atom preparation by electron transfer in, 104-105

defined, 100 detection of, 112-119 double resonance microwave spectroscopy in, 122-123 energy positions of, 102 field ionization detection of, 114-1 16 fluorescence detection and, 113-1 14 high-resolution optical spectroscopy of, 120-12 1 hydrogenic, 157 light intensities necessary to populate, 11 1 maser effects in, 101 optogalvanic detection in, I18 photoionization detection in, 118-1 19 preparation of atoms in, 103-112 radiative properties of, 157-161 spectroscopy of, 100, 119-131 stability of, 102 superradiance and, 157 three-phase experiment in, 101-102 time-resolved spectroscopy and, 121 122 transition probabilities to lower states in, I03 transitions between, 159-161 Rydberg state spectroscopy alkali metals and, 119-126 complex atoms and, 130-131 high-resolution studies in, 129-130 two-electron spectra in, 126-130

426

INDEX S

Saddle point model, ionization thresholds and, 154 SCA, see Semiclassical approximation Schrodinger collective variable, 225 Schrodinger equation, time-dependent, 232, 304, 311 Schrodinger field, end-fire modes of, 178 Schrodinger operator, 178 Second Bonn approximation, 320, 338 Second-order potential method, 56 s electron angular distribution, 20-22 Self-consistent field, 2 choice of, 9-10 Semiclassical approximation, for excitation and ionization, 309 Semiclassical JWKB functions, 70 SF, see Superfluorescence Single charged-ion formation, 17- 18 Single-electron transfer cross sections, total, 327-336 Solar atmosphere electron density and temperature structure in, 356 time variations in, 358 Solar atomic processes atomic spectra and, 359-361 bound state wave functions and, 361370 electron excitation and, 381-403 proton excitation and, 403-406 solar plasmas and, 406-414 and spontaneous decay of bound states, 370-381 Solar corona, electron temperatures in. 357 Solar Maximum Mission Satellite, 355, 358, 413 Solar plasmas. atomic data and. 406-414 Solar spectrum, bound-state wave functions and, 361-370 and spontaneous decay of bound states, 370-38 I Stark effect, quadratic, 143 Stark levels, Rayleigh-Schrodinger perturbation theory and, 140-141 Stark shift, 234-235, 243 Stark spectra, of stable states. 150 Stark states

continuum and, 148 density of, 146 spectroscopy of in alkalis, 142-146 Static rearrangements, inner-shell thresholds and, 33-34 Stokes line, 64-65 bending of, 72 Stokes phenomenon, 63-64 Stueckelberg matrix, 63-78 Stueckelberg model, 70-71 Stueckelberg “nonphysical” branch cut, 66 Stueckelberg phase integral approximation, 64 Stueckelberg transition probabilities, for Fritsch-Wille model, 79 Sun, see d s o Solar (rrdj.) atomic processes in, 355-414 atomic spectra of, 359-361 Superfluorescence, 167-226 average behavior in 188-189 beam intensity in. 212 cesium experiment and, 208-210 classical behavior in, 185-188 collective Bloch vector in, 190 collective delay time and, 170 correlation functions and. 181-182 defined 168 delayeo pulse in, 213 dipole-dipole coupling and, I70 effective initial tipping angle in, 188-189 equations of motion and. 194-196 experimental results in. 213-222 experimental techniques in. 206-2 12 first observation of, 169, 206-207 fluctuating behavior in. 190-193. 219222 homogeneous and nonhomogeneous broadening in, 193-202 initiation of, 184-185. 219-222 initiation time regime and, 176-180, 225 inversion profile and, 211-212 Maxwell-Bloch equations and. 193. 196I97 Maxwell-Bloch theory and. 189 one-dimensional, 171, 202 “point” sample and, 223 pulse intensity in, 214 quantum-mechanical description of. 176 semiclassical theorv of. 172-176

427

INDEX single-shot outputs in. 185-188 spatial and temporal character of, 169 stochastic variables description of, 183I84 SVEA plane-wave end-fire modes in. 202 three-dimensional and multimode effects in, 202-205 transition to amplified spontaneous emission. 196-200 Superfluorescence experiments conditions for, 207-208 initiation and fluctuation of SF in, 219222 pulse parameters and, 213 spatial and temporal coherence in, 216219 superfluorescence output power, 204 Superfluorescence pulse, measurement of, 186 Superfluorescence pulse shapes, diffraction and, 205 Superradiance master oscillation and, 159-161 Rydberg states and. 157 Superradiant microwave transitions, 160 Supersonic nozzle jet beams, resonance ionization spectroscopy and, 262-272 SVEA Maxwell equation, 203 SVEA plane wave end-fire modes, in superfluorescence, 202-203 Symmetric resonance model, 80

Titanium, K hypersatellite X rays of, 301 Total single-electron transfer cross sections, 327-336 Transition probabilities, comparison of by various methods. 71 TSAE model, see Two-state atomic expansion model Two-center multistate expansion, electron capture and, 314-318 Two-electron spectra, in Rydberg state spectroscopy, 126-130 Two electrons-two holes, interaction with. 48-SO Two-photon excitation, of inert gases, 240-249 Two-state atomic expansion model, 330, 338. 340

U

Ungerade wave function, 84 Uniform electric field, atomic structure in presence of, 138-146 Uniform parabolic cylinder, function formula for, 74 United-atom basis functions, 3 16 “Up” and “down” electrons, 28

V T

Target nucleus, multiple scattering of nucleus of, 323 Term-coupling coefficients, 399 Three-body analysis, nonmolecular, 91 -93 Three-center expansion, electron capture and, 314-318 Three-photon excitation rates, in helium and neon, 261-262 Three-step processes, in strontium and other metals, 127 “Time-foward” diagrams, 10-1 I , 15-16 Time-resolved spectroscopy, 12 1-1 22 “Time-reverse” diagrams, 15- 16

Vacancies, collectivization of, 40-50, see also K-shell vacancy Vacancy energy and width, 43-44 Vacancy wave function, 41-43 Valence photoelectrons, anisotropy parameter for, 21

x Xenon, see also Inert gases crossed-pulsed nozzle jet beams of, 270 excited state of, 2.5 I

428

INDEX

ionization signal vs. energy squared for, 248 two-photon excitation of, 246-248 Xenon 4p shell, collectivization of, 45-46 Xenon 5s electrons, photoionization of, 17-18

Z

Zwaan-Stueckelberg method, 63, 74, 76 Zwaan-Stueckelberg phase integral, 64, 78

Contents of Previous Volumes

Volume 1 Molecular Orbital Theory Of the Spin Properties of Conjugated 1 A.1 Molecules, G. G. ~ ~ and A mos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational ~ ~ in E ~ counters between Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams a t Thermal Energies, H . Pauk and J. P. Toennies High Intensity and High Energy Molecular Beams, J. B. Anderson, R. P. Andres, and J. B. Fenn AUTHORINDEX-SUBJECTINDEX

Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner AUTHORINDEX-SUBJECT INDEX

volume 3 The Q'antal Calculation of Photo-~ ionization~ Cross Sections, A. L. ~ i Stewart Radiofrequency Spectroscopy of Stored Ions* H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering*F. Chanoch Bedo Reactive Collisions between Gas and Surface Wise and Bernard J . Wood . AUTHORINDEX-SUBJECTINDEX

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W. D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J. Mum, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W . R. S. Carton Volume 4 The Measurement of the Photoionization Cross Sections of the H. s. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Atomic Gases, James A. R. Samson Electronic Eigenenergies of the HyThe Theory of Electron-Atom Collidrogen Molecular Ion, D. R. sions, R. Peterkop and V. Veldre Bates and R. H. G'.Reid 429

~

430

CONTENTS OF PREVIOUS VOLUMES

Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L. Moiseiwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionization, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R. G. W. Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J . Seaton Collisions in the Ionosphere, A . Dalgarno The Direct Study of Ionization in Space, R. L. F. Boyd AUTHORINDEX-SUBJECTINDEX

Volume 5

The Meaning of Collision Broadening of Spectral Lines: T h e Classical-Oscillator Analog, A . Ben-Reuven The Calculation of Atomic Transition Probabilities, R. J. S. Crossley Tables of One- and Two-Particle ParentCoefficients of age for Configurations s s'"pq, C. D. H. Chisholm, A. Dalgarno, and F. R. Innes Relativistic Z-Dependent Corrections to Aomic Energy Levels, Holly Thomis Doyle AUTHORINDEX-SUBJECTINDEX

Fractions!

Volume 6

Dissociative Recombination, J . N. Bardsley and M. A. Biondi Analysis of the Velocity Field in Plasma from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo T a k a y a n a g i a n d Y uk i k a z u Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and R. T. Marrero T h e o r y a n d A p p l i c a t i o n of Sturmain Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston AUTHORINDEX-SUBJECTINDEX

Flowing Afterglow Measurements of Ion-Neutral Reactions, E. E. Fer son, F. C. Fehsenfeld, and A. L. ghmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Volume 7 Stored Ions 11: Spectroscopy, H . G. Dehmelt Physics of the Hydrogen Master, C. Audoin, J . P. Schermann, and P. The Spectra of Molecular Solids, 0. Grivet Schnepp

CONTENTS OF PREVIOUS VOLUMES

Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J . C . Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Paunez, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of MoleculesQuasi-Stationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J. Greenfield AUTHORINDEX-SUBJECTINDEX

43 1

Volume 9 Correlation in Excited States of Atoms, A. W. Weiss The Calculation of Electron-Atom Excitation Cross Sections, M. R. H . Rudge Collision-Induced Transitions Between Rotational Levels, Takeshi Oka The Differential Cross Section of Low Energy Electron-Atom Collisions, D. Andrick Molecular Beam Electronic Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy AUTHORINDEX-SUBJECTINDEX Volume 10

Relativistic Effects in the ManyElectron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K. L. Bell and A. E. Kingston Volume 8 Photoelectron Spectroscopy, W. C. Price Interstellar Molecules: Their Formation and Destruction, D. Dye Lasers in Atomic Spectroscopy, McNally W . Lange, J. Luther, and A. Steudel Monte Carlo Trajectory Calculations of Atomic and Molecular Recent Progress in the ClassificaExcitation in Thermal Systems, tion of the Spectra of Highly James C. Keck Ionized Atoms, B. C. Fawcett Nonrelativistic Off-Shell Two-Body A Review of Jovian Ionospheric Coulomb Amplitudes, Joseph C. Chemistry, Wesley T. Huntress, Y. Chen and Augustine C. Chen Jr. Photoionization with Molecular SUBJECT INDEX Beams, R. B. Cairns, Halstead Harrison, and R. I. Schoen The Auger Effect, E. H . S. Burhop Volume 11 and W. N . Asaad The Theory of Collisions Between AUTHORINDEX-SUBJECTINDEX Charged Particles and Highly Ex-

432

CONTENTS OF PREVIOUS VOLUMES

cited Atoms, I. C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M. J. Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W. D. Robb Role of Energy in Reactive Molecular Scattering: An InformationTheoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M. F. Golde and B. A. Thrush AUTHORINDEX-SUBJECT INDEX

Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser Excited Atoms in Crossed Beams, I. V. Hertel and W. StoN Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W. B. Somerville AUTHORINDEX-SUBJECT INDEX

Volume 12

Resonances in Electron Atom and Molecule Scattering, D. E . Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald F, Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in HeavyParticle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobashev Rydberg Atoms, S. A. Edelstein and T. F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree AUTHORINDEX-SUBJECTINDEX

Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. G. Gouedard, J. C. r i i : m , and J. Vigue Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid AUTHORINDEX-SUBJECT INDEX

Volume 13 Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson

Volume 14

CONTENTS OF PREVIOUS VOLUMES

433

Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Negative Ions, H. S. W. Massey Theory of Low Energy ElectronAtomic Physics from Atmospheric Molecule Collisions, P. G. Burke and Astrophysical Studies, A. AUTHORINDEX-SUBJECT INDEX Dalgarno Collisions of Highly Excited Atoms, R. F. Stebbings Theoretical Aspects of Positron Volume 16 Collisions in Gases, J. W. Atomic Hartree-Fock Theory, M. Humberston Cohen and R . P. McEachran Experimental Aspects of Positron Experiments and Model CalculaCollisions in Gases, T. C. Griffith tions to Determine Interatomic Reactive Scattering: Recent AdPotentials, R. Duren vances in Theory and Experi- Sources of Polarized Electrons, R. ment, Richard B. Bernstein J. Celotta and D. T. Pierce Ion-Atom Charge Transfer Colli- Theory of Atomic Processes in sions a t Low Energies, J. B. Strong Resonant ElectromagHasted netic Fields, S . Swain Aspects of Recombination, D. R. Spectroscopy of Laser-Produced Bates Plasmas, M. H . Key and R . J . Hutcheon The Theory of Fast Heavy Particle Collisions, B. H. Bransden Relativistic Effects in Atomic Collisions Theory, Atomic Collision Processes in ConB. L. Moiseiwitsch trolled Thermonuclear Fusion Research, H. B. Gilbody Parity Nonconservation in Atoms: Status of Theory and ExperiInner-Shell Ionization, E. H . S. ment, E. N. Fortson and L . Burhop Wilets Excitation of Atoms by Electron INDEX Impact, D. W. 0. Heddle

Volume 15

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