Advances in Atomic and Molecular Physics, Volume 22

Advances in

ATOMIC AND MOLECULAR PHYSICS VOLUME 22

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ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

Sir David Bates DEPARTMENT OF APPLIED MATHEMATICS A N D THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 22 1986

@)

ACADEMIC PRESS, INC.

Harcourt Brace Jovanovich, Publishers

Orlando San Diego New York Austin Boston London Sydney Tokyo Toronto

COPYRIGHT 0 1986 BY ACADEMIC PRESS, INC ALL RIGHTS RESERVED NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS. ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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ISBN 0-12-003822-6 l ’ l ~ l Y I k I 1IN I H t L I N l r t l ) 5TArt\

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(alk. paper) Or

AMkRICA

Y 8 7 6 5 4 3 2 I

LTD.

NUMBER 65-18423

Contents

Positronium-Its Formation and Interaction with Simple Systems J. W. Humberston

I. 11. 111. IV.

Introduction Positronium Formation in Positron - Atom Scattering The Interaction of Positronium with Simple Systems Concluding Remarks References

1

2 22 33 34

Experimental Aspects of Positron and Positronium Physics T. C. Grifith I. 11.

111.

Introduction Annihilation Spectra for Positrons in Gases Cross-Section Measurements with Positron Beams References

37 39 52 72

Doubly Excited States, Including New Classification Schemes

C.D.Lin I. 11.

111. IV. V. VI. VII. VIII.

Introduction Analysis of Radial and Angular Correlations Classification of Doubly Excited States Solution of the Two-Electron Schrijdinger Equation in Hyperspherical Coordinates Body-Frame Analysis of Correlation Quantum Numbers Effects of Strong Electric Fields on Resonance Structures in H- Photodetachment Doubly Excited States of Multielectron Atoms Concluding Remarks and Perspectives References V

77 81 96 109 115 125 131 138 140

vi

CONTENTS

Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms I. 11. 111. IV. V.

H . B. Gilbody Introduction Outline of Experimental Methods Charge Transfer Ionization Cross Sections for Electron Removal from Hydrogen Atoms in Collisions with Positive Ions References

143 144 152 182 189 192

-

Electron - Ion and Ion Ion Collisions with Intersecting Beams K. Dolder and 3. Peart I. 11.

111. IV. V. VI . VII . VIII. IX. X.

Introduction Notes on Experimental Techniques Ionization of Positive Atomic Ions by Electron Impact Measurements of Dielectronic Recombination Measurements of Electron-Impact Excitation of Positive Ions Scattering of Electrons by Ions Collisions between Electrons and Negative Ions Collisions between Electrons and Molecular Ions Collisions between Positive and Negative Ions Collisions between Positive Ions References

197 199 20 1 213 215 222 224 226 228 232 237

Electron Capture by Simple Ions

Edward Pollack and Yukap Hahn I. 11. 111. IV . V. VI. VII.

Introduction Theory Experimental Background Typical Studies Ion- Molecule Charge-ExchangeCollisions: He+ Ion -Molecule Collisions: Other Systems Electron-Transfer-Excitation Collisions References

+ H2

243 244 26 1 266 273 293 305 3 10

Relativistic Heavy-Ion- Atom Collisions R. Anholi and Harvey Goufd I. 11.

Introduction Experiments

315 317

CONTENTS

111. IV. V. VI. VII. VIII.

IX.

Ionization Processes Electron-CaptureProcesses Collisions in Solid Targets X-Ray Continuum Processes Ultrarelativistic Collisions Relativistic Few-Electron Ions in Quantum Electrodynamics Experiments Conclusions References

vii 324 345 357 369 374 377 379 38 1

Continued-Fraction Methods in Atomic Physics S. Swain

I. 11. 111.

IV. V.

INDEX

Introduction Continued-FractionSolutions to Linear Equations Perturbation Theories The Density Matrix and Rate Equations Conclusions Appendix: Derivation of the Continued-Fraction Expansion of a Determinant References

387 389 398 415 427 427 428

433

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II

ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL 22

POSITRONI UM- ITS FORMIATION AND I N ~ ~ ~ C WITH T r ~ N SIMPLE SYSTEMS J. W. HUMBERSTON Department of Physics and Astronomy University College London London WCIE 6BT. England

I. Introduction During the past few years very significant developments have taken place in the study of the interactions of positrons with atoms and molecules. Progress has been particularly impressive on the experimental front where the availability of more intense positron beams and improved detectors have made possible much more sophisticated experiments, and this in turn has stimulated further associated theoretical investigations. Recent experimental developments are reviewed by Griffith in another chapter in this volume, and a more comprehensive review by Charlton ( 1985)has also recently been published. One of the most interesting processes to occur in positron - atom collisions is positronium formation, a simple example of a rearrangement collision in which the incident positron combines with one of the electrons in the target atom to produce a bound positron -electron system, positronium (Ps), and a residual ion; thus e+ A + Ps f A+

+

Positronium formation cross sections have recently been measured directly (Charlton et al., 1983; Fornari et al., 1983) over a wide range of incident positron energies for several target systems, and a significant discrepancy is found between the results of the two experiments which becomes increasingly pronounced as the positron energy is raised. The possibility ofresolving this discrepancy provides an added incentive to study positronium formation theoretically, and it is this topic which will form the major part of this 1 copyright 0 1986 by Academic Press, Inc. AU rights of reproduction in any form reserved.

2

J. W. Humberston

article. We shall also discuss the interaction of positronium with simple systems. Other theoretical aspects of positron collisions in gases, and particularly elastic scattering and annihilation, are treated in the review articles of Fraser (1 968), Bransden (1969), Massey (197 l), Drachman ( 1972a), Massey et al. (1974), Humberston (1979), and Ghosh et al. (1982).

11. Positronium Formation in

Positron -Atom Scattering Positronium has the structure of a hydrogenic atom with a reduced mass of half the electron mass. The ground-state energy is therefore - 6.8 eV and the dipole polarizability is 36 a:. The two spin states are referred to as parapositronium ( S = 0) and orthopositronium ( S = 1 ti),and annihilation is into two and three y rays respectively with lifetimes in the ground state of 1.251 X l0-lo s and 1.418 X lO-’s (Gidley et al., 1982). We shall neglect spin-dependent forces and generally make no distinction between ortho- and parapositronium. Consequently, of all the positronium formed in the collisions of positrons with the target system, one-quarter is assumed to be parapositronium and three-quarters orthopositronium. If the ionization energy of the target atom is Ei,the energy of the positron at the threshold for ground-state positronium formation is Eps= Ei- 6.8 eV

For atomic hydrogen and helium, the two target atoms with which we shall be most concerned, the positronium formation thresholds are 6.803 and 17.6 eV, respectively. If the ionization potential is less than 6.8 eV, as is the case for the alkali atoms, the positronium formation channel is open even at zero positron energy; indeed the reaction is exothermic. The energy interval between the positronium formation threshold and the first excitation threshold of the target atom, El (assuming El > &), is referred to as the Ore gap. Within this energy gap positrons can either be elastically scattered or form positronium. Direct annihilation ofthe positron with one of the electrons in the target atom is also possible, but the cross section for this process is very much smaller than that for either elastic scattering or positronium formation. The Ore gaps for atomic hydrogen and helium are 6.8- 10.2 and 17.8-20.6 eV, respectively. Positronium formation can of course occur at energies beyond the upper limit of the Ore gap, and it continues to make a significant contribution to the total cross section at positron energies as high as 100 eV, but it then

POSITRONIUM FORMATION AND INTERACTION

3

competes with many other inelastic processes and it becomes almost impossible to treat the process in a very precise manner. Within the Ore gap only two channels are open and very accurate results can be obtained, at least for simple target atoms, using rather similar techniques to those employed to calculate the essentially exact values of the scattering parameters for low-energy elastic scattering of positrons by hydrogen (Schwartz, 196la; Humberston and Wallace, 1972; Armstead, 1968; Register and Poe, 1975) and helium (Humberston, 1973;Campeanu and Humberston, 1975; Humberston and Campeanu, 1980). Detailed theoretical investigations of positronium formation in the Ore gap have so far only been made for atomic hydrogen. No experimental results are yet available for this system, although the development of much more intense positron beams may make such an experiment possible in the near future. Nevertheless, the study of the process in the relatively simple positron-hydrogen system is expected to lead to a better understanding of positronium formation in more complex systems. A. POSITRON-HYDROGEN SCATTERING Many different approximation methods have been used to calculate positronium formation cross sections, and we will not attempt to review them all here. Instead we shall concentrate mainly on the more elaborate methods which have yielded the most accurate results within the Ore gap. References to other simpler methods are given in the review of Ghosh et af. (1982). The Hamiltonian of the positron-hydrogen system can be written in atomic units (using the nomenclature in Fig. 1 and assuming an infinitely massive proton) as

which is appropriate when considering the system as positron- hydrogen, or as

which is appropriate when considering the system as positronium -proton. For each partial wave the total wave function can be written in the twocomponent form as

J. W. Hurnberston

4

e+

P FIG.1. The positron- hydrogen system.

where the first component, Y, , represents positron - hydrogen elastic scattering with positronium formation, and the second component, Y2,represents positronium - proton elastic scattering with electron attachment to the proton. The asymptotic forms of the two components are yl

-

I,+m

y/,0(i1)~1’2~H(r2)[j/(kr1)

- KIln/(krl)]

- y,,,(p^)(2K)L’2~R(r3)K2,n,(K~)

P-m

’% P-m Y , , 0 ( ~ ) ( 2 ~ ) 1 ’ 2 ~ ~ ((w) ~ 3 )[ j K,,n,(~p)l / r,+-

(3)

- Y/,0(i,)(k)”2~H(r2)K12n/(kY1)

where KO(i, j = 1, 2) are the elements of the K matrix (reaction matrix), and &(r2) and &(r3) are the wave functions of the hydrogen atom and the positronium. The wave numbers of the positron and positronium are k and K , respectively, and energy conservation gives

2 E = k 2 - l = f ( r ~ ~ 1)The cross section for scattering between channels u and v’ is

(4)

5

POSITRONIUM FORMATION AND INTERACTION

in units of nu;, where subscripts 1 and 2 refer to the positron- hydrogen and positronium-proton channels, respectively, and kl = k and k2 = K. Thus, the cross sections for positron - hydrogen elastic scattering and positronium respectively. formation are q l and o12, The most accurate results for the s-,p-, and d-wave contributions to the positronium formation cross section are believed to be those obtained by Humberston (1982, 1984) and Brown and Humberston (1984, 1985) using the Kohn variational method with very elaborate trial functions containing many variational parameters. Although it is not a bounded variational method, and very anomalous results can sometimes be obtained (Schwartz, 196 1a), the method is simple to use and has yielded very accurate results for the elastic scattering of positrons by hydrogen and helium atoms below the positronium formation threshold. For the present two-channel problem the stationary Kohn functional takes the matrix form

12

K12

=

G

I

K:2 - ('y,, LYl)

I

IKh K:,l

I

(TI, L'y2)

(6)

L'yd ( ' y 2 , L'yA where L = 2(H - E ) ,and Yl and Y2are suitably chosen trial functions with the asymptotic forms given by Eq. (3), namely 'y1=

K2J

('y2,

K:

y , , 0 ( ~ 1 ~ H ( ~ 2 ) k 1 ~ 2 ~-~ , ( ~In,@,)[ ~1)

-

+

1 - exP(-Jr,)R

y,O(i)&s(r3)(2K)1/2K'2 I n/(Kp)[ - exp(-pp)lq Y,,0(?1,?2)ci exp[-((ar, j?r2 yr3)]ryr$rp

x

+ +

(7)

i

'y2

=

y , , o ( i ~ ~ ( r ~ ) ( 2 K ) 1 / 2 (j , JG,n,(rcp)[ (~p) 1 - exp(-w)l4)

- y , * 0 ( ~ 1 ) ~ ~ ~ 2 ) ~ 1 ~ 21~-~ exp(-Ar,)lP 2 ~ / ( ~ ~ 1 ) [ Y,,o(?l,f2) di exp[-((ar, Pr2 yr3)]r7r$ry

+

C

+ +

i

+

For each value of1there are (1 1 ) differentangular functions Y,,o(?l,p2) with the parity of 1(Schwartz, 196 1b), and associated with each such function is a summation over the index i in Eq. (7), which includes all terms with ki li mi S o,where ki, l,, mi,and o are non-negative integers. Further details of the method of calculation are given by Humberston ( 1982). Very accurate results can be obtained if sufficient short-range correlation terms are included, but in order to establish the precision of a particular result it is necessary to investigatethe convergencewith respect to systematic improvements in the trial function. Increasingthe value of o provides such a systematic improvement, and the numbers of terms generated by the above scheme for o = 1,2, . . . , 7 are 4, 10,20, 35, 56,84, and 120, respectively.

+

+

J. W. Hurnberston

6

Although, as mentioned earlier, the Kohn variational method is not a bounded method, it is almost invariably found in practice to give lower bounds on the diagonal elements of the K matrix. As the trial function is improved by increasing o,these matrix elements increase monotonically and converge to the (presumably) exact values. Indeed the convergence is often sufficiently smooth that extrapolation to infinite o is possible, giving even more accurate results (Humberston, 1984). No such bound principle applies to the off-diagonal K-matrix element KI2(=K2J and accordingly the variation with o can be somewhat erratic, although for sufficiently large values of w the results do seem to converge. We will now consider the results obtained by the above procedure for each partial wave in turn. 1. s Wave

The variations of the elastic scattering and positronium formation cross sections with o at four positron energies within the Ore gap are shown in Figs. 2 and 3. The lower and upper boundaries of the Ore gap correspond to k = l / f i = 0.7071~;' and k = = 0.8660ai1,respectively. The con-

0 1L

0 12

-

N

0 10

k

0

0.8!

m

5

008

c

6

0.8( 0 06

0.7!

0 OL

0.71 0 02

I

0

0

1

I 2

I

I

3

L

I 5

I 6

1 7

W

FIG.2. Variation of the s-wave elastic cross section with w.

POSITRONIUM FORMATION AND INTERACTION

0 008

7

k = 071

-

0 OOL 0 .

I

-0

m

E

0

L

I

I I

1

1

t

t?

0.006

W

FIG.3. Variation of the s-wave positronium formation cross section with w .

vergence of the elastic scattering cross sections is particularly impressive, and even the positronium formation cross sections have probably converged to within 10%oftheir exact values. Details of the convergence of the individual K-matrix elements are given by Humberston ( 1984). The energy dependence of the s-wave contribution to the positronium formation cross section is plotted in Fig. 4. A conspicuous feature, which has been examined in detail by Humberston ( 1982), is the initial very rapid rise from zero at the threshold, but the magnitude of the cross section remains very small relative to the elastic cross section and, as we shall see, also to higher partial-wave contributions to the positronium formation cross section. Also plotted in Fig. 4 are the results from other calculations using a variety of approximation methods. The extraordinarily wide range of values, spanning several orders of magnitude, illustrates the sensitive dependence of the results on the quality of the trial function. Even some of the other elaborate variational calculations, such as those of Stein and Sternlicht ( 1972), Chan and Fraser (1973), and Winick and Reinhardt (1978), give significantly different results. Stein and Sternlicht also used the Kohn variational method with a similar,

J. W. Humberston

8 I

1 I 0.006

- I

0.005

0.004 I

N O

m

k

t?

0.003

0.002

0.001

0

0.5

0.6

0.7

k2[ai2) FIG.4. Results for the s-wave positronium formation cross section in positron- hydrogen scattering:(A) Humberston (1982); (B) Stein and Sternlicht(1972); (C) Chan and Fraser (1973); (D) Winick and Reinhardt X lo-' (1978); (E) Fels and Mittleman (1967); (F) Dirks and Hahn (1967); (I) (1971); (G) Wakid and La Bahn X 10 (1972); (H) Bransden and JundiX coupled static approximation; (J) Born approximation X lov2.

but slightly less flexible, trial function to that of Humberston. Up to 84 correlation terms (o= 6) were included and very satisfactory agreement with the results of Humberston was obtained except at energies close to the positronium formation threshold. Their values of the diagonal elements of the K matrix then fall slightly below those of Humberston, strongly suggesting that their results are less accurate. Chan and Fraser used a method based on the formulation of the coupled static approximation with the addition of several short-range correlation terms, &. The method amounts to using the Kohn variational method with

POSITRONIUM FORMATION AND INTERACTION

9

trial functions of the form

and allowing the variational method to determine the forms of the functions Fi( rl) and Gi(p)(i = 1,2). These emerge as numerical solutions to a set of coupled integrodifferential equations with boundary conditions at infinity given by Eq. (3). Although the method is more complicated than the purely algebraic Kohn method, it has the advantage of yielding rigorous lower bounds on the diagonal elements and eigenvalues of the K matrix provided the total energy of the system is below the lowest eigenvalue of the operator QHQ, where Q is the closed channel projection operator (Hahn, 1966); Q

=

( -I#t)(&II

and

The eigenvalues ofQHQ are related to the positions of Feshbach resonances and such resonances are known to exist just below the n = 2 excitation threshold of the hydrogen atom (Doolen et al., 1978), although none has yet been found in a direct calculation of the type being described here. The three resonances found by Seiler et af. ( 197 1) have been shown to be artifacts of their method of calculation, which neglected the open positronium channel. When this channel was included in the trial function the resonances disappeared (Drachman, 1975). Only 26 short-range correlation terms of a rather restricted form were used by Chan and Fraser, and, not surprisingly, the values of their diagonal K-matrix elements and eigenphases are somewhat less positive than those of Humberston. However, their results are significantly better than those obtained from earlier rigorous lower-bound calculations such as those of Dirks and Hahn ( 1 97 1) and the coupled static approximation. A somewhat less conventional method of calculation has been used by Winick and Reinhardt (1978). They calculated the various partial-wave elastic scattering amplitudes, t,, from the off-shell elastic scattering T matrix and hence the elastic scattering across sections 4

a;, = -(21+ k2

1)ltf

J. W. Humberston

10

Then, using the optical theorem, they also obtained the partial-wave total cross sections 4 at=-(21+

l)Imt,

k2

(10)

The difference between a , and ae,is the cross section due to all inelastic processes, and within the Ore gap the only such process is positronium formation. The matrix representation of the total Hamiltonian that is required for the calculation of the T matrix was generated using basis functions of the form, using the same nomenclature as in Fig. 1, (1 1) (ar, + &)lrFrlrT'Y,, ,,+(?I Again all functions with ki li mi S o were included with the exception of a few terms when o = 7, so that the maximum number of functions was 105. Given the similarity of these basis functions in form and number to those used by Humberston, one might have expected the two sets ofresults to +i = ewt-

0.8

-

0.7

-

+ +

k

0.71

0.75 0 80

-

0.6

0.85

"b 0.5-

-2

g

0.6

-

0.3 0.2

-

0.1

-

01

I

I

I

I

I

I

I

W

FIG.5. Variation of the pwave elastic scattering cross section with w.

POSITRONIUM FORMATION AND INTERACTION

11

be in good agreement. This is indeed so for the elastic scattering cross section, but the positronium formation cross section is approximately five times larger than Humberston’s. The discrepancy is most probably due to the smallness of the positronium formation cross section in relation to the elastic cross section. The elastic and total cross sections are therefore very similar in magnitude, and, as both cross sections are slightly in error, the subtraction procedure is very likely to introduce a large percentage error into the positronium formation cross section. 2. p Wuve

The convergence of the p-wave elastic scattering and positronium formation cross sections with respect to o,as obtained by Brown and Humberston, is shown in Figs. 5 and 6. These results are slightly different from those already published by Brown and Humberston ( 1984) and correspond to an improved choice of values of the nonlinear parameters in the trial function that gives much better convergence to well within 10%of the exact values. Except at energies very close to the threshold, thep-wave contribution to the 0 8 -

m

r

O L -

t?

0302-

01

-

Y

0

-

Y

Y

Y

1

1

I

y

071

J. W. Humberston

12

1.2 1 .o

0.L 0.2 0

0.5

0.6

0.7

k2(aiz)

FIG.7. Results for thepwave positronium formation cross section in positron- hydrogen scattering:(A) Brown and Humberston (see text); (B) Chan and McEachran (1976); (C) Winick and Reinhardt (1978); (D) coupled static approximation;(E) Born approximation.

positronium formation cross section is comparable to the elastic scattering cross section and is very much larger than the s-wave contribution. Figure 7 illustrates the energy dependence of the pwave contribution to the positronium formation obtained from various calculations. The differences between them are very much less than for s-wave scattering, with a particularly impressive measure of agreement between the results of the three elaborate calculations of Chan and McEachran (1976), Winick and Reinhardt (1978), and Brown and Humberston. This characteristic also applies to most of the simpler approximations, so that even the Born approximation results are little more than a factor of two too large compared to more than a factor of 100 for the s wave. Likewise the results of the coupled static approximation are less than 50% below the exact values, whereas for the s-wave contribution they are a factor of 10 too small. Chan and McEachran ( 1976) used a very similar method to that of Chan and Fraser for s-wave scattering with a set of up to 50 short-range correlation terms of the form

13

POSITRONIUM FORMATION AND INTERACTION

These terms have no explicit dependence on the electron -positron coordinate r, ,although implicit dependence on r, is introduced by the coupling of the two angular momenta. Considering that the trial function is significantly less flexible than that of Brown and Humberston, the results obtained for the positronium formation cross section are remarkably good. Rather less good agreement with the accurate values is found for the elastic scattering cross section, however, and the values of the diagonal K-matrix elements are also rather low. The method of Winick and Reinhardt clearly works much better for the p wave than the s wave, presumably because the magnitude of the total cross section is approximately double the elastic cross section and therefore very little precision is lost by the subtraction procedure described above. 3. d Wave The convergence of the d-wave elastic scattering the positronium formation cross sections with respect to w, as calculated by Brown and Humberston ( 1985), is displayed in Figs. 8 and 9. Optimization with respect to the nonlinear parameters in the trial function has not been completed and the

I

0.5

k

-1

b-

0.2

0 .l

0

0

1

2

3

4

5

6

7

w

FIG. 8. Variation of the d-wave elastic scattering cross section with Q.

J. W. Hiimberston

14

1 .L

c

1

0

I

I

I

1

I

I

I

1

2

3

4

5

6

1

W

FIG.9. Variation ofthe d-wave positronium formation cross section with w. The results for k = 0.7 la;' are too small to be plotted.

convergence is not quite as good as for the p wave. Nevertheless the most accurate results (o= 7) are probably within 10% of their exact values. The d-wave contribution to the positronium formation cross section is even larger than the p wave except, of course, just above the positronium formation threshold, and it is probably the dominant contribution within the Ore gap. Very few investigations have been made of the d-wave contribution to the positronium formation cross section, and the only other elaborate calculations are those of Winick and Reinhardt (1978). Their results, given in Fig. 10, are again in good agreement with those of Brown and Humberston, although rather less so than for p waves. The improvement in the Born approximation results in going to higher partial waves is maintained, and very reasonable agreement with the accurate results is now obtained. Rather surprisingly, no such improvement is found for the coupled static approximation (Abdel-Raouf et al., 1984), the results being still 40% too low.

POSITRONIUM FORMATION A N D INTERACTION

15

k 2 ( ai2)

FIG.10. Results for the d-wave positronium formation cross section in positron - hydrogen scattering: (A) Brown and Hurnberston (1985); (B) Winick and Reinhardt (1978); (C) Born approximation;(D) coupled static approximation (Abdel-Raouf, 1984).

4. Higher Partial Waves A complete description of positronium formation in positron - hydrogen scattering requires the contribution from all higher partial waves. The only very detailed investigations are those of Winick and Reinhardt (1978), who have calculated elastic and positronium formation cross sections for all partial waves with I S 5 . Given the good agreement with Brown and Humberston forp and d waves, it is likely that their results for higher partial waves are also quite accurate, provided that the positronium formation cross section is not too small relative to the elastic scattering cross section. According to their results the only other significant contribution to the positronium formation cross section within the Ore gap comes from the fwave. It is also very likely that the Born approximation gives quite accurate results for all higher partial waves, considering the dramatic improvement in accuracy in going from s to p to d wave. Drachman ez al. ( 1976) made this assumption in order to estimate the energy dependence of the positronium formation cross section over a wide energy range.

16

J, W. Humberston

5. Total Cross Sections

The sums of the first three partial-wave contributions to the positronium formation cross section are plotted in Fig. 1 1. Also plotted there is an estimate of the total positronium formation cross section, where the Born approximation has been used for all partial waves with I > 2. These results are consistent with the sum of the first five partial cross sections calculated by Winick and Reinhardt. It is instructive to examine the behavior of the total elastic scattering cross section in crossing the positroniurn formation threshold, and the sum of the first three partial-wave contributions below and above the threshold is plotted in Fig. 12. Above the threshold the positronium formation cross section is also added to give the total (elastic plus positronium formation) cross section. There appears to be a slight discontinuity in the magnitude of the elastic scattering cross section in crossing the threshold, but this is probably due to the existence of a narrow cusp in the cross section. The earliest experimental estimates of the magnitude of the positronium formation cross section in the Ore gap were based on the assumption that the elastic scattering cross section there is a linear (in energy) extrapolation of the elastic cross section below the threshold (Stein et al., 1978). The positronium formation

k2(o,,-2/

FIG.1 I . Sums ofthe partial-wavecontributionsto the positroniumformation crosssection in positron-hydrogen scattering. The contributions for 1 > 2 are obtained from the Born approximation (Drachman ef al., 1976).

POSITRONIUM FORMATION AND INTERACTION

I

17

/

2.5 I I

- 2.0

-z

- 0

t

1.5

0 .-*

I

rt

r/ I

/B

I

I/

C

I

U

VI 0 VI VI

1.0

V

A

I

A

L

0 .05 0.2

0. G

0.6

k2

FIG.12. Total cross sections for positron- hydrogen scattering: (A) elastic scattering; (B) elastic scattering plus positronium formation cross sections; (C) linear extrapolation of the elastic scattering cross section from below the positronium formation threshold (k2= 0.5~;~) into the Ore gap.

cross section was then determined from the difference between the total cross section and the extrapolated elastic cross section, also plotted in Fig. 12. This procedure leads to an underestimation of the positronium formation cross section in atomic hydrogen near the upper boundary of the Ore gap of nearly 20%. Adding the higher partial-wave contributions to the elastic scattering and total cross sections, as calculated by Winick and Reinhardt, modifies the situation slightly, but the elastic scattering cross section within the Ore gap still dips significantly below the extrapolation line to give an underestimation of the positronium formation cross section of more than 15%. If this behavior of the elastic scattering cross section within the Ore gap is found in other atoms, the experimental measurements of the positronium formation cross section by Charlton et al. (1983) are likely to be too low, because these authors had to normalize their relative results to the value as determined by the extrapolation - subtraction procedure described above at one energy close to the upper boundary of the Ore gap. No such normalization procedure was required in the other direct measurements of positronium formation cross sections by Fornari et al. ( 1983), and it may therefore be significant that even within the Ore gap these results are slightly larger than

J. W. Humberston

18

those of Charlton et al. (see Fig. 14 for the results for helium). As the positron energy is raised the discrepancy between the two experimental sets of results becomes much greater, but it cannot be attributed entirely to a normalization error. A likely explanation is provided in the next section. 6 . Angular Distribution of Positronium

The differential cross section for the formation of positronium is given in terms of the partial-wave K matrix elements by (in units of aaa)

and the results obtained by including only the first three partial waves are plotted in Fig. 13. At k = 0 . 7 1 a ~just ~ , above the positronium formation k (a;’)

I

0.71

0.75 I

I

I

0.85

0

45

90

135

180

e (deg) FIG.1 3. Angular distribution of positronium produced in positron- hydrogen scattering.

POSITRONIUM FORMATION AND INTERACTION

19

threshold, rather more than half the positronium is produced at angles greater than 90" relative to the incident positron direction, but as the positron energy is raised the angular distribution rapidly becomes forward peaked so that already at k = 0.85~1;' approximately 60% is produced at angles less than 45 '. The addition of higher partial waves is expected to enhance the forward peaking still further. Qualitatively similar angular distributions at this energy have also been obtained by Drachman et al. (1 976), using the Born approximation as modified in the manner described previously, and by Mandal et al. (1979) using the distorted-wave approximation. According to Drachman et al. the angular distribution becomes even more strongly forward peaked as the positron energy is raised beyond the Ore gap. Rather similar angular distributions, again becoming more strongly forward peaked with increasing positron energy, have also been obtained in the distorted-wave calculations of Mandal et al. ( 1979) for helium. The form and energy dependence of the differential cross section for positronium formation provide an explanation of why the positronium formation cross sections measured by Charlton are probably too low. The geometry of their apparatus is such that positronium emerging at small angles to the incident positron direction is likely to escape from the detection region before annihilating and will therefore not be detected as having been positronium. Clearly the probability of escaping detection increases with increasing positron energy, and this is consistent with the energy variation of the discrepancy between the two experimental measurements (see Fig. 14). B. POSITRONHELIUM SCATTERING We have discussed positronium formation in low-energy positron hydrogen scattering in some detail because very accurate results can be obtained against which to compare the results of simpler approximation methods for use with more complex atoms. Unfortunately, as we have seen, the simple methods are not very satisfactory for the first few partial waves at low positron energies, and any results obtained by their use in more complex systems should be treated with caution until very accurate theoretical results are available. There is the additional problem of the use of an inexact target wave function. This is conveniently circumvented in elastic scattering by using the method of models (Drachman, 1972b; Humberston, 1973), but no such self-consistent formulation is possible for a rearrangement process. All the calculations of positronium formation cross sections in positron -helium scattering have used very simple helium wave functions, and agreement

J. W. Humberston

20

20

LO

60

80

Positron energy [ eV 1

FIG.14. Results for the positronium formation cross section in positron-helium scattering. Experimental results: (+) Fornari et al. (1983); (0)Charleton et al. (1983). Theoretical results: (A) Mandal et al. (1979); (B) Khan and Ghosh (1983); (C) Mandel et al. (1975); (D) Mandal et al. ( 1976); (E) Kraidy and Fraser X 0.5 [asquoted by Costello et al. ( 1972).The curve is actually the total cross section, and an approximately constant value of0.35nae for the elastic scattering cross section should be subtracted from it to give the positronium formation cross section.] (F) Born approximation.

between the results of a simple method of approximation and experimental measurements is therefore no guarantee that the results are correct. With this warning in mind we examine the various approximation methods that have been used for calculating positronium formation cross sections in helium and compare the results with the experimental values. Nothing comparable to the elaborate variational calculations described earlier for hydrogen has yet been attempted for helium, although it is now

POSITRONIUM FORMATION AND INTERACTION

21

feasible to do so and it is to be hoped that accurate results will be obtained in the near future. The two sets of experimental results are plotted in Fig. 14. Because the measurements of Charlton et al. (1983) probably suffer from the previously mentioned systematic errors associated with incorrect normalization and the failure to detect positronium produced at small angles to the incident positron beam direction, it is likely that the results ofFornari et al. ( 1983)are the more accurate, particularly because no evidence has yet emerged of any significant systematic errors in them. Further support for this set comes from recent measurements of the cross sections for ionization by positron impact, both inclusive and exclusive of positronium formation (Sinapius et al., 1985). The results of several theoretical calculations are also plotted in Fig. 14. They all relate to the formation of positronium in its ground state, whereas the experimental measurements include the contributions from all energetically permitted states, although the cross section for formation into the positronium state with principal quantum number n is expected to vary as r r 3 . Again the Born approximation (Mandal et al., 1975) greatly overestimates the cross section as, to an even greater extent, does the variant of the coupled static approximation used by Kraidy and Fraser ( 1967). These authors added long-range polarization potentials to the standard coupled static formulation. In its standard form without the polarization potentials, however, the coupled static approximation underestimates the cross section by a substantial factor (Mandal et al., 1975). The best agreement with the experimental results ofFornari et al. has been obtained by Mandal et af. (1979) using the distorted-wave approximation. Also, in fair agreement with this experimental set, but in better agreement with the measurements of Sinapius et al., are the polarized orbital results of Khan and Ghosh ( 1983). In contrast the polarized orbital results of Mandal et al. (1976) greatly overestimate the positronium formation cross section just above the threshold, due to a resonant-like feature in the p wave, but then rapidly become too low. Most of these results exhibit the dominance of the d-wave contribution to the positronium formation cross section found also in positron- hydrogen scattering. Also, whenever differential cross sections for positronium formation have been calculated, they are found to be peaked in the forward direction and become even more so as the positron energy is raised. C. POSITRON SCATTERING BY OTHERATOMS Positronium formation cross sections have been measured by Charlton et al. (1983) and Fornari ef al. for numerous atoms, but very few theoretical

22

J. W. Humberston

investigations have been made for any target systems other than hydrogen and helium. Even than, as might be expected, only rather simple approximation methods have been used. The next simplest target system is lithium, where, as for all the alkali atoms, the positronium formation channel is open at zero positron energy. No experimental measurements of positronium formation cross sections have been made for any of the alkali atoms, and it is only very recently that the first total cross sections, for potassium, were measured by Stein et al. (1985). In a sense the alkali atoms are rather simple to treat theoretically because they have one electron outside a closed shell. They can therefore be considered as a modified hydrogen atom, and positron scattering from them could be investigated using the elaborate variational methods described in Section II,A, although this approach has not yet been tried. The only calculations of the positronium formation cross section that have been made for lithium are by Guha and Ghosh (198l), who used the Born approximation and the coupled static approximation both with and without additional polarization potentials, and by Mukhejee et al. ( 1989, who used a distorted-wave approximation. The most accurate results are probably those obtained from the polarization potential calculations of Guha and Ghosh, and they give a cross section of approximately 5 0 ~ at~ 8 positron energies close to zero, falling rapidly to 2 n d at an energy of 10 eV. Again the largest contributions to the positronium formation cross section come from the d wave, and the differential cross section also exhibits strong forward peaking. Positronium formation cross sections have been calculated for the rare gases argon and neon by Gillespie and Thompson (1 977). The Born and distorted-wave approximations were used, but neither method gives results in good agreement with the experimental values.

111. The Interaction of Positronium with Simple Systems Having discussed the formation of positronium in some detail, we shall now consider the interactions of positronium with simple systems. More precisely, we shall consider the interactions between a positron, an electron, and one other system, which may be a single particle or an atom or ion. Becausethe static interaction between a positron and an atom is repulsive, positrons are much less likely than electrons to be bound to atoms. It has

POSITRONIUM FORMATION AND INTERACTION

23

been rigorously established by Armour ( 1982, 1983)that a positron does not bind to a hydrogen atom. Earlier well-converged variational calculations of positron - hydrogen elastic scattering had already provided very convincing evidence of this fact by yielding a nodeless zero-energy wave function and a negative scattering length, u = -2.10u0(Humberston and Wallace, 1972).A negative value for the scattering length does not, of course, preclude the possibility of a bound state, but such a state would not then be particularly weakly bound and should therefore be readily found. Similar calculations for positron - helium scattering (Humberston, 1973) give a less negative value for the scattering length (a = -0.48 uo),suggesting that the positron-atom interaction is less attractive for helium than for hydrogen. This, coupled with the absence of any negative eigenvalues of the total Hamiltonian matrix, makes it highly improbable that a positron can bind to a helium atom. The total energy of a positron-atom system, &-A, may be more negative than that of the atom alone, as, for example, in lithium, but unless it is also more negative than the lowest energy ofthe corresponding positronium - ion system, Ps-A+, it does not imply a true bound state. There is no certain example of a positron being bound to an atom or positronium to a positive ion, although the calculations of McEachran et al. (1 980) lend some support for the possible existence of a positron - xenon bound state. A. THEPOSITRONIUM - ELECTRON SYSTEM

Positronium can bind to a third positively or negatively charged particle provided the mass ofthat particle is sufficiently low. Thus, the e+e-p+ system is still not bound (Armour, 1983), but the singlet e+e-e+ system, or its antisystem e-e+e- (the positronium negative ion, Ps-), is bound. This was established theoretically by Wheeler ( 1946)and first detected experimentally by Mills ( 1981a). The most accurate determination of the energy for breakup into positronium plus electron has been made by Bhatia and Drachman (1983) with the result 0.3266769 eV. These authors used a Hylleraas wave function of the form (YR- =

C ci(1 + Pl2){r+r?exp[-((yr, + Sr2)])ry2 i

where rl and r2 are the coordinates of the two electrons relative to the positron and r12= Ir, - r21. All terms with li mi niS o were included and the most elaborate trial function contained 220 terms (o= 9). The rate for annihilation of the positron with one of the electrons in Ps- is given in terms of the normalized bound-state wave function by

+ +

3, = 100.62(wld(r,)lw)

ns-'

(15)

24

J. W. Humberston

and the result for the most accurate wave function is ;Z = 2.0861 ns-*, or a mean of life of 4.7936 ns, consistent with the experimental rate of 2.09 f 0.09 ns-I measured by Mills (1983). This value is very close to the appropriately weighted average ( 1 to 3) of the annihilation rates for singlet and triplet free positronium and therefore implies that the configuration of Ps- is, as expected, an electron weakly bound to positronium. Bhatia and Drachman also investigatedthe possibility of a 'Pebound state of Ps-. Mills (1 98 lb) had speculated that the state, to be thought of as an electron weakly bound to positronium in the n = 2 state, might exist, although his subsequent theoretical investigations suggested that this was not so. Bhatia and Drachman carried out a much more detailed investigation using a suitably amended form oftheir ground-state wave function [Eq. ( 14)] with as many as 120 correlation terms, but no bound state was found. Such a state can exist, however, if the ratio of the mass of the positive particle to the mass of the negative particle is increased beyond 16.8 or reduced below 0.4047, and hence a 'Pebound state does exist for H-, e-,u+e-, and Hf. Very similar results to those quoted above for the binding energy and annihilation rate of Ps- have been obtained by Ho ( 1983).He also calculated the angular correlation function for the two y rays from the annihilation process, obtaining a full width at half-maximum of 1.4 mrad. As well as the ground state, Ps-, the positronium -electron system has a number of autoionizing states, or resonances in electron - positronium scattering, associated with the various thresholds for excitation of the positrone ~considerable detail, ium. Ho (1 984) has studied the ISe and 3 P r e s ~ n a n cin using the stabilization and complex rotation methods with appropriately symmetrized Hylleraas trial functions of the form given in Eq. ( 14), and has determined the positions and widths given in Tables I and 11. The energy width of a resonant state corresponds to the lifetime for autoionization, TA, according to T A = h /r,and Ho ( 1985) has established that the lifetimes for autoionizion of the five lowest resonances are several orders of magnitude smaller than the lifetimes for annihilation (see Table I). Autoionization is therefore much more probable than annihilation. Nonresonant elastic scattering of electrons by positronium has recently been investigated by Ward et al. ( 1985). Very well converged values of the singlet and triplet s-wave phase shifts were obtained using the Kohn variational method with scattering trial functions containing rhany Hylleraastype short-range correlation terms. The singlet and triplet scattering lengths were also obtained from the effective range formula with the values a+=(12.38-t0.07)%,

a-=(5.8+0.2)a0

and hence the zero-energy cross section is 2 2 8 ~ ~ :Calculations . were also

POSITRONIUM FORMATION AND INTERACTION

25

TABLE I

‘st RESONANCES OF PS- ASSOCIATED WITH EXCITATION THRESHOLDS OF POSITRONIUM

n

-E,(Ry)

r (RY)

2

0.1520608 0.12730 0.070683 0.05969 0.04045 0.0350 0.03463 0.0258 0.02343

0.000086 0.00002

3 4 5

Autoionization lifetime (10-9 s)

Annihilation lifetime (10-9 s)

0.00056 0.00242 0.000323 0.000440 0.000202 0.000 I6 0.00014 0.000I0 0.00034

5.028 6.920 23.474 32.895 80.000

o.Oo015

0.000 11 0.00024 0.0003 0.00034 0.00045 0.00014

made using the static exchangeapproximation. The results, shown in Fig. 15, are qualitatively similar to those for electron - hydrogen scattering, and, as there, the agreement between the two methods is very good for triplet scattering but considerably less good for singlet scattering. In a search for resonancesjust below the n = 2 excitation threshold of the positronium, the lowest ISe resonance was found in the results from the Kohn variational method at a positron energy of 4.767 eV, and with a width of 1.2 meV, in good agreement with the values calculated by Ho ( 1984). However, no evidence was found of the much narrower second resonance also found by Ho below the threshold. Presumably an even more flexible trial function than that used by Ward et al., with 70 correlation terms, is required before this resonance is revealed in a scattering calculation. Ward et al. ( 1986a) have also carried out similar calculations for p-wave scattering and again there are qualitative similarities to electron - hydrogen scattering. In addition to its intrinsic value, an accurate wave function for TABLE I1

IScRESONANCES OF Ps- ASSOCIATED WITH EXCITATION THRESHOLDS OF POSITRONIUM

2 3 4

0.12706 0.05873 0.034 I5

o.ooO0 1 o.ooo02 0.00002

J. W. Humberston

26 3.0

0

0.2

0.1

0.3

0.4

0.5

0.6

k (a;')

FIG. 15. Singlet and triplet s-wave phase shifts for electron-positronium scattering. The upper set of curves are the triplet phase shifts, to which R radians have been added, and the lower , Accurate variational results; ---, static exchange approximation. set are the singlet. -

p-wave electron-positronium scattering is required for an accurate calculation of the photodetachment cross section of Ps-. This can be written in several formally equivalentways (Bhatiaand Drachman, 1986), the simplest of which is (in units of aaf)

where the dipole transition operator is QL

=

* (ri

+W 2 )

27

POSITRONIUM FORMATION AND INTERACTION

and R2 = r2 - frl , v/k is the p-wave electron - positronium wave function with electron wave numberk, o is the angular frequency, or, in our units, the energy of the photon, and a is the fine structure constant. Bhatia and Drachman ( 1986) have calculated this cross section using a simple approximation in which the final state is represented by the plane wave

and the Ps wave function is taken to be VPS- =

c(1 -k pi

2>#dri

)e-yR2 /R2

(19)

with y = 0.1265. This is the form of the exact Ps- wave function for R2 >> rl, and it is expected to be a reasonably accurate representation of the wave function over most of the relevant region of configuration space because the electron is rather weakly bound to positronium. For the same reason the further approximation can be made of dropping the exchange terms in V/R- and v/k and only retaining the term R R2 in QL. The final expression for the photodetachment cross section is then, in terms ofthe wavelength 3, of the incident

1 .o

'\ \

C

.-0

L

V

/

Ln

. d

/

C

/

/

/

\

/

\

\

\ \

/

0

5

10

15

20

Wavelength h

25 (

30

35

LO

lo3 A )

FIG. 16. Photodetachment cross section for Ps- according to Bhatia and Drachman ( 1986).

J. W. Humberston

28

radiation,

where & = 37953.46 A. This function is plotted in Fig. 16. A similar approximation applied to the photodetachment of H- (Ohmura and Ohmura, 1965; Bell and Kingston, 1967) was found to give quite accurate results, and it is expected to be equally satisfactory here. This has recently been confirmed at all but very short wavelengths by the calculations of Ward et al. (1986b), in which accurate variational wave functions were used for the initial and final states. For wavelengths close to the threshold for photodetachment of Ps- into an electron and positronium in the n = 2 state ( A = 2.283 X lo3 A), however, the photodetachment cross section exhibits a sharp rise due to a cusp-like feature in the electron-positronium singlet p-wave phase shift. Photodetachment of Ps- has been suggested as a mechanism for the production of a monoenergetic beam of positronium. The ps-, having been formed in the manner described by Mills ( 198 1 a), would be accelerated electrostaticallyto the required energy and then undergo photodetachment to produce a beam of positronium.

B. THEPOSITRONIUM-HYDROGEN SYSTEM Positronium can bind to a hydrogen atom, provided the two electrons are in a spatially symmetric state (spin singlet),to form the molecule positronium hydride, PsH. The most accurate value of its energy of dissociation into positronium and hydrogen has been calculated by Ho (1978) as Ed = 1.021 12 eV. This result was obtained using a generalized Hylleraas type of trial function containing all six interparticle distances of the form

C cie-qrpr$r7$(1 + Plz)(e-("I+Yr+pp) n

ry =

i

(21)

+ +

where the nomenclature is as defined in Fig. 17. All terms with hi qi mi n, s o were included, and a very well converged result was obtained for w = 6 (210 terms). The rate of annihilation of the positron with one of the electrons into two y rays is given by

+

FQSITRONIUM FORMATION AND INTERACTION

29

P

FIG.17. The positronium- hydrogen system.

This parameter has not been calculated by Ho, but Page and Fraser (1974), using a similar, but slightly Iess flexible, trial function which gave a binding energy of 1.0014 eV, obtained the value A = 2.327 ns-' or a mean life of 4.30 X s. Again, as for Ps-,the closenessofthis annihilation rate to the average of the singlet and triplet annihilation rates for free positronium implies a configuration for PsH of positronium rather weakly bound to the hydrogen atom. Positronium hydride has no bound excited states, but Ho (1978) and Drachman and Houston (1975) have found a resonance in positroniumhydrogen scattering in the electron spin singlet state. Ho, using the complex rotation method with the trial function described above, found an s-wave resonance with a total energy E, = - 16.40 f 0.014 eV (a scattering energy of 4.20 eV) and width r = 0.075 eV. Similar results (scattering energy 4.5 eV, r = 0.06 eV) were obtained by Drachman and Houston (1975) using the stabilization and complex rotation methods. Their wave function was slightly less flexible than Ho's, with no dependence on the electronelectron coordinate r,2. Nonresonant positronium- hydrogen elastic scattering has been investigated by Houston and Drachman (1 979, Hara and Fraser (1975),and Page (1976). Houston and Drachman extracted s-wave phase shifts from eigenvectors of the total Hamiltonian matrix generated with the set of normalizable basis functions used in the calculation of the binding energy of Ps- H. The eigenvectorcorrespondingto the lowest eigenvalue gives an approximation to the wave function of the bound state, but the eigenvectors corresponding to higher eigenvalues give approximations to the wave functions

J. W. Humberston

30

for singlet positronium - hydrogen scattering at various energies for small to intermediate values of the separation coordinate R (see Fig. 17). Clearly they cannot represent the forms of the scattering functions in the asymptotic region R 4 correctly because they are normalizable, but if the basis functions are suitably chosen the asymptotic form of the solution will be reasonably accurately established for intermediate values of R. The phase shift can then be obtained by fitting the wave function at such values of R to = $H(TZ)$Ps(PI)F(R)

(23)

where

and the wave number of the positronium, k, is related to the total energy of the system, E (in Ry), by (k2/4) - 1.5 = E. Using these phase shifts and the binding energy of Ps- H in the usual effective-rangeformula, Drachman and Houston also obtained the singlet scattering length, a+ 5.3U0, and effective range r: = 2 . 5 ~ ~ . Singlet and triplet (spatially symmetric and antisymmetric in the coordinates of the two electrons) phase shifts, 6t and 67,respectively, have been calculated by Hara and Fraser (1 975) in the static exchange approximation for several partial waves. In addition, the scattering lengths were determined with the values a+ = 7.275% and a- = 2.476%. The total elastic scattering cross section is, in units of naf, 1

2 +

+

(21 l)(sin2 S,+ 3 sin2S r ) (26) k l and the cross section for conversion of ortho- to parapositronium is gel= 7

1 can = - (21 4k2 I

C + 1) sin2(6t - ):6

Both cross sections are plotted in Fig. 18. The most accurate values of the singlet and triplet scattering lengths are those calculated by Page (1976). He used the Kohn variational method with trial functions of the form (using the nomenclature of Fig. 17)

POSITRONIUM FORMATION AND IhTERACTION

31

80

6C

-

NO

Y

C

.-0 t

Q u)

LO

u) v)

e

0

20

n

I

0

02

QL

I

I

I

I

06

08

I

10

12

k (a,’ )

FIG.18. Cross sections for positronium-hydrogen scattering: (A) elastic scattering; (B) ortho- to parapositroniumconversion.

+ +

where the second summation includes all terms with qi hi mi S o.The most elaborate trial functions, with o = 4 (N = 35) give the results a+ = 5.844a0, a- = 2 . 3 1 9 ~corresponding ~ to a,, = 49.57ra; and awn= 3.18naZ. For the triplet state, where there is no bound state of the positroniumhydrogen system, the Kohn variational method immediately gives a rigorous upper bound on the scattering length. An upper bound can also be obtained for the singlet state, where there is a bound state, PsH, provided the shortrange correlation terms in the trial function are sufficiently flexible to represent the bound state. Such is the case here with Page’s trial function, and so both the values given above are upper bounds. They are lower, and therefore more accurate, than the values of Hara and Fraser. The triplet scattering length of Page is larger than that of Drachman and Houston, but, because this latter result is not an upper bound, it cannot be assumed to be the more accurate.

J. W. Humberston

32

For sufficiently large values of the coordinate R the interaction potential between the positronium and the hydrogen atom simplifies to the van der Waals form C/R6,where C = 64 (Hara and Fraser, 1975). The short-range correlation functions used by Page are not very well suited to representing the rather long-range distortions produced by this potential, and Page’s results could probably be improved by the addition of longer-range correlation terms to the trial functions. Page also determined the singlet effective range from the PsH binding energy and the scattering length using the negative energy form of the effective range formula, and obtained the value r$ = 2.904. He was unable to calculate the triplet effective range in this way, however, because there is no equivalent bound state.

C . OTHERSYSTEMS Positronium can bind to itselfto form the positronium molecule, Ps,. This was first established by Hylleraas and Ore ( 1947) after previous unsuccessful attempts by Wheeler (1946) and Ore (1946). They calculated the binding energy with respect to breakup into two positronium “atoms” as 0.11 eV, but this was soon increased to 0.135 eV by Ore ( 1 947). A study of positronelectron annihilation in Ps, has been made quite recently by Tisenko (1 98 l), who, using the wave function of Hylleraas and Ore, calculated the rates of decay into two and three y rays as 1.6 X 1Olo and 0.43 X lo8 s-l, respectively. Positronium does not bind to helium, but positronium scattering by helium is of considerable interest because the rate of quenching of orthopositronium in the collision process can be calculated from the scattering wave function, and this quantity can also be measured experimentally.It therefore provides a further opportunity for comparing theoretical and experimental results. The quenching rate is usually expressed in terms of the parameter ,2&, an effective number of electrons with which the positron can form a singlet spin state. For the positronium -helium system, containing three electrons,

I

Jer= 3

-

l(x$(sp9sl), W I 2 4 r p- r,) hpCfr, & 4

(29)

where Y is the total wave function normalized to unit positronium density as rp m, and &, sl) is the singlet spin function for the positron and electron 1 . Barker and Bransden (1 968) and Fraser and Kraidy (1 966) have investigated positronium -helium scattering in the static exchangeapproximation,

POSITRONIUM FORMATION AND INTERACTION

33

using a very simple uncorrelated helium wave function. At zero energy the scattering cross section is 13nai, dropping to 7.7na;k at 13.6 eV. The zeroenergy value of , Z , is 0.0347 compared with the experimental value of 0.125 0.002, obtained at room temperature (Coleman et al., 1975). In an attempt to improve on these calculations, Barker and Bransden added a long-range van der Waals potential to the static exchange equation and obtained the correspondingcross sections as 1 6 . 9 ~and ~ 6 7 . 6 ~ and ~ ; ,Zeff= 0.0445. The results for the cross sections are almost certainly much closer to the exact values than might be inferred from the rather poor agreement between the theoretical and experimental values of ,Zeff,for it is well known that the calculation of the annihilation rate, where only a very restricted region of configuration space is sampled, provides a particularly stringent test of the accuracy of the wave function. More elaborate trial functions containing very flexible electron- positron correlationterms, together with a more accurate correlated helium wave function, must be used if accurate values of ,ZeEare to be obtained.

+

IV. Concluding Remarks We have studied the formation of positronium and its interaction with various simple systems in considerable detail, and have found a wealth of interesting features. Many of these are still not amenable to experimental investigation, although the development of much more intense positron beams of more precisely defined energy (Howell et al., 1984; Lynn and Frieze, 1984; Hulett et al., 1984) is likely to transform the experimental situation within the next few years. This in turn will, no doubt, stimulate further theoretical developments. Another important stimulus to further theoretical investigations of such systems has come from the attempts made to understand the detailed features of the spectrum of the electron -positron annihilation radiation observed to be coming from the direction of the galactic center and also from solar flares. The experimentalobservationshave been reviewed by Leventhal et al. (1 978) and Leventhal ( 1986),and attempts to explain the observations have been described by Bussard ef al.( 1979), Drachman (1 982, 1984), and McKinley ( 1986). Particular attention has been given here to the three-body systems e+e-p and e+e-e-, and very accurate results have been obtained for bound-state, resonance, and scattering parameters, but only in a rather narrow energy range. Much still needs to be done at somewhat higher positron energies

34

J. W. Humberston

where more inelastic channels are open and yet the high-energy approximations are still not valid. Mention has also been made of some four- and five-body systems involving positrons and positronium, but the results are generally of an inferior quality. Attention will increasingly be given to these more complex systems, particularly where comparisons between theoretical and experimental results can be made. It should be possible, with the computing power currently available, to obtain very accurate results for the four-body systems, and one of the more urgent tasks here is to calculate accurate values of the positronium formation cross section in positron - helium scattering where, as we have seen, a significant discrepancyexistsbetween the two sets of experimental results. The availability of accurate theoretical results was a significant factor in the successfulresolution of similar discrepanciesin the expenmental measurementsof the cross sections for the elastic scatteringof low-energy positrons by helium (Kauppila and Stein, 1982; Humberston, 1978), and accurate theoretical values of the positronium formation cross section should be equally valuable.

ACKNOWLEDGMENTS Most ofthis review article was written during a recent visit to York University, Toronto, and the author gratefully acknowledgesthe financial support provided there by the Natural Sciences and Engineering Research Council ofcanada. He is also indebted to ProfessorR. P. McEachran and his colleagues,Professors J. W. Darewych and A. D. Stauffer, for many fruitful discussions and their hospitality at York University. Finally, he wishes to thank Mrs. Virginia Peteherych for typing most of the manuscript.

REFERENCES Abdel-Raouf,M. A., Darewych, J. W., McEachran, R. P., andStauffer, A. D. (1984). Phys. Lett. 100A, 353. Armour, E. A. G. (1982). Phys. Rev. Left.48, 1578. Armour, E. A. G. (1983). J. Phys. B 16, 1295. Armstead, R. L. (1968). Phys. Rev. 171,91. Barker, M . I., and Bransden, B. H. ( I 968). J. Phys. B I, 1 109. Bell, K. L., and Kingston, A. E. (1967). Proc. Phys. SOC.90,895. Bhatia, A. K., and Drachman, R. J. (1983). Phys. Rev. A 28, 2523. Bhatia, A. K., and Drachman, R. J. (1986). In “Positron (Electron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Bransden, B. H. (1969). Case Stud. At. Collision Phys. 1, 171.

POSITRONIUM FORMATION AND INTERACTION

35

Brown, C. J., and Humberston, J. W. (1984). J. Phys. B 17, L423. Brown, C. J., and Humberston, J. W. (1985). J. Phys. B 18, L401. Bussard, R. W., Ramaty, R., and Drachman, R. J. (1979). Astrophys. J. 228,928. Campeanu, R. I., and Humberston, J. W. (1975). J. Phys. B 8, L244. Chan, Y. F., and Fraser, P. A. (1973). J. Phys. B 6,2504. Chan Y. F., and McEachran, R. P. (1976). J. Phys. B 9,2869. Charlton, M. (1985). Rep. Prog. Phys. 48, 737. Charlton, M., Clark, G., Griffith, T. C., and Heyland, G. R. (1983). J. Phys. B 16, L465. Coleman, P. G., Griffith, T. C., Heyland, G. R., and Killeen, T. L. (1975). J. Phys. B 8 , 1734. Costello, D. G., Groce, D. E., Herring, D. F., and McGowan, J. W. ( 1 972). Can. J. Phys. 50,23. Dirks, J. F., and Hahn, Y. (1971). Phys. Rev. A 3 , 310. Doolen, G. D., Nuttall, J., and Wherry, C. J. ( I 978). Phys. Rev. Lett. 40, 3 13. Drachman. R. J. (1972a). Proc. Int. Conf: Phys. Electron. At. Collisions, 7th, 1971 p. 277. Drachman, R. J. (1972b). J. Phys. B 5, L30. Drachman, R. J. (1975). Phys. Rev. A 12, 340. Drachman, R. J. ( 1 982). Can. J. Phys. 60,494. Drachman, R. J. (1984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), p. 203. Plenum, New York. Drachman, R. J., and Houston, S. K. (1975). Phys. Rev. A 12,885. Drachman, R. J., Omidvar, K., and McGuire, J. H. (1976). Phys. Rev. A 14, 100. Fornari, L. S., Diana, L. M., and Coleman, P. G. (1983). Phys. Rev. Lett. 51,2276. Fraser, P. A. ( 1 968). Adv. At. Mol. Phys. 4,63. Fraser, P. A., and Kraidy, M. (1966). Proc. Phys. SOC.89, 533. Ghosh, A. S., Sil, N. C., and Mandal, P. (1982). Phys. Rep. 87, 313. Gidley, D. W., Rich, A., Sweetman, E., and West, D. (1982). Phys. Rev. Lett. 49, 525. Gillespie, E. S., and Thompson, D. G. (1977). J. Phys. B 10, 3543. Guha, S., and Ghosh, A. S. (1981). Phys. Rev. 23, 743. Hahn, Y. (1966). Phys. Rev. 142,603. Hara, S., and Fraser, P.A. (1975). J. Phys. B 8, L472. Ho, Y. K. (1978). Phys. Rev. A 17, 1675. Ho. Y. K. (1983). J. Phys. B 16, 1503. Ho, Y. K. ( 1 984). Phys. Lett. A 102A, 348. Ho, Y. K. (1985). Phys. Rev. A 32,2501. Howell, R. H., Alvarez, R. A., Woodle, K. A., Dhawan, S., Egan, P. O., Hughes, V. W., and Ritter, M. W. ( 1 984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), p. 155. Plenum, New York. Hulett, L. D., Dale, J. M., Miller, P. D., Moak, C. D., Pendyala, S., Triftshhser, R. H., Howell, R. A,, and Alvarez, R. A. (1 984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), p. 195. Plenum, New York. Humberston, J. W. (1973). J. Phys. B 6 , L305. Humberston, J. W. (1979). Adv. At. Mol. Phys. 15, 101. Humberston, J. W. (1982). Can. J. Phys. 60, 591. Humberston, J. W. (1984). J. Phys. B 17,2353. Humberston, J. W., and Campeanu, R. I. ( 1 980). J. Phys. B 13,4907. Humberston, J. W., and Wallace, J. B. G. (1972). J. Phys. B 5, 1138. Hylleraas, E. A., and Ore, A. ( 1 947). Phys. Rev. 71,493. Khan, P., and Ghosh, A. S. (1983). Phys. Rev. A 28,2 I8 1 . Kraidy, M., and Fraser, P. A. (1967). Proc. lnt. Conf:Phys. Electron. At. Collisions,5th p. 110. Leventhal, M. (1986). In “Positron (Electron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press.

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Leventhal, M., McCallum, C. J., and Stang, P. D. (1978). Astrophys. J. Lett. 225, L11. Lynn, K. G., and Frieze, W. E. (1984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), p. 165, Plenum, New York. McEachran, R. P., Stauffer, A. D., and Campbell, L. E. M. (1980). J. Phys. B 13, 1281. McKinley, J. M. (1986). In “Positron (Electron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Mandal, P., Ghosh, A. S., and Sil, N. C. (1975). J. Phys. B 8,2377. Mandal, P., Basu, D., and Ghosh, A. S . (1 976). J. Phys. B 9, 2633. Mandal, P., Guha, S., and Sil, N. C. (1979). J. Phys. B 12,2913. Massey, H. S. W. (1971). Con$ At. Phys., 2nd, 1970p. 307. Massey, H. S. W., Burhop, E. H. S., and Gilbody, H. B. (1 974). “Electronic and Ionic Impact Phenomena,” Vol. 5 . Oxford Univ. Press, London and New York. Mills, A. P. (1981a). Phys. Rev. Left 46, 717. Mills, A. P. (1 98 1b). Phys. Rev. A 24, 3242. Mills, A. P. (1983). Phys. Rev. Lett. 50, 671. Mukhejee, S. C., Majumdar, P. S., Datta, S., and Ghosh, A. S. ( 1 986). In “Positron (Electron)Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Ohmura, T., and Ohmura, H. (1965). Phys. Rev. 118, 154. Ore, A. (1946). Phys. Rev. 70,90. Ore, A. (1947). Phys. Rev. 71,913. Page, B. A. P. (1976). J. Phys. B 9, 11 1 I. Page, B. A. P., and Fraser, P. A. (1 974). J. Phys. B 7, L389. Register, D., and Poe, R. T. (1975). Phys. Leff.51A, 431. Schwartz,C. (1961a). Phys. Rev. 124, 1468. Schwartz, C. (1 96 I b). Phys. Rev. 123, 1700. Seller, G. J., Oberoi, R. S., and Callaway, J. (1971). Phys. Rev. 3,2006. Sinapius, G., Fromme, D., and Raith, W. (1986). In “Positron (Electron)-Gas Scattering’’ (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Stein, J., and Sternlicht, R. (1972). Phys. Rev. A 6,2165. Stein, T . S., Kauppila, W. E., Pol, V., Smart, J. H., and Jesion, G. (1978). Phys. Rev. A 17,1600. Stein, T . S., Gomez, R. D., Hsieh, Y.-F., Kauppila, W. E., Kwan, C. K., and Wan, Y. J. (1985). Phys. Rev. Lett. 55,488. Tisenko, Yu. A.(1981). Sov. Phys. J. 24,99. Ward, S. J., Humberston, J. W., and McDowell, M. R. C. (1985). J. Phys. B 18, L525. Ward, S. J., Humberston, J. W., and McDowell, M. R. C. (1986a). J. Phys. B, to be published. Ward, S. J., McDowell, M. R. C., and Humberston, J. W. (1986b). Europhysics Lett. 1, 167. Wheeler, J. A. (1946). Ann. N.Y. Acad. Sci. 48,219. Winick, J. R., and Reinhardt, W. P. (1978). Phys. Rev. A 18, 925.

ADVANCES IN ATOMlC AND MOLECULAR PHYSICS. VOL 22

EXPERIMENTAL ASPECTS OF POSITRON AND POSITRONIUM PHYSICS T. C. GRIFFITH Department of Physics and Astronomy University College London London WClE 6BT, England

I. Introduction Positron physics has, during the past few years, experienced a period of rapid expansion resulting from the application of positron beams of well-defined energies to an increasing range of experiments. The development of positron beam technology has been crucially dependent upon the discovery and design of efficient moderators to produce the low-energy (about 1 eV) positrons. The re-emission of a substantial number of such positrons on thermalizingfast positrons in certain materials was first reported by Groce et al. ( 1968)and this was followed by the discovery by Canter et al. ( 1972)and Coleman et al. (1973) that MgO powder could be employed to provide positron beams of such intensities that accurate cross-section measurements were possible. Further major improvements in the yield of low-energy positrons came from the use of heat-treated tungsten moderators by Dale et al. (1 980) and then by the use of clean single crystal metal targets, such as Cu(III), under ultra-high-vacuum conditions as reported by Mills (1980). There are currently at least two laboratories where intense beams of nearly monoenergetic positrons are being generated either from strong radioactive positron emitters produced in situ near a nuclear reactor or from an electron linear accelerator via the bremsstrahlung process. It is intended that these beams should have intensitiesof up to 1O8 or 1O9 positrons per second with a narrow energy spread and capable of being focussed and collimated into a narrow pencil suitable for the performance of precise experiments. There is little doubt that the motivation for the development of these beams had stemmed, initially, from the challenge of performing atomic physics cross-section measurements with positrons of a quality that would compare with those already in existence for electrons. Today, however, their 37 Copyright 0 1986 by Academic Rws,Inc. All rights of reproduction in any form -4.

38

T. C.Grijith

use has been extended to embrace several other fields of research such as: measurements of low-energy positron diffraction, LEPD (Canter, 1982, 1984); a study, using positron beams in combination with 2 y-annihilation angular correlation apparatus, of the momentum density of electrons in surfaces and of defects in material samples as a function of spatial position (Lynn and Frieze, 1984; Lynn et al., 1985; Howell et al., 1985); the use of polarized positron beams for studies of surface magnetism and of optically active molecules (Rich, 1980, 1985);a detailed study of positronium formation in very clean surfaces and of positronium spectroscopy and fundamental experiments testing certain aspects of quantum electrodynamics (Berko and Pendleton, 1980; Gidley, 1980; Gidley and Coleman, 1984). The content of this article is mainly concerned with recent investigations of and results on the interactions of positrons with gaseous atoms and molecules. Experimental determinations of total cross sections for the scattering of positrons ofenergiesin the range 1 - 1000eV in different gases have played a dominant role for several years, but the emphasis has recently shifted to the measurement of partial cross sections for the various inelastic channels in positron interactions. Among these channels positronium formation is a unique capture channel which has no analog in electron scatteringand is, for this reason, particularly important. Positronium itself can now be produced in abundance and the stage is already poised for qualitative studies of positronium scattering in gases. The impact of the availabilityof intense positron beams on atomic physics experimentsas well as on other groups of experiments is only just beginning. It is now, for example, possible to think in terms of planning accurate measurements of positron differential scattering cross sections for some of the inelastic channels; positron scattering by atomic hydrogen as well as other similar experiments have already reached an advanced stage of planning. Alongside the above, perhaps rather dramatic developments, the older technique of studying positron lifetime spectra is, in a modified form, continuing to produce significant results. As a measure of its importance, and of the varied insight gained by studying the annihilation of positrons in gases at various pressures and temperature and under the influence of electric fields, a substantial discussion of the recent results obtained by this method will be included. Several reviews have been written on the topics covered by the present article; most of the relevant background to its content has been reviewed by Griffith and Heyland ( 1978), Griffith ( 1979), Rich (1980), Stein and Kauppila (1982),Heyland ef al. (1982),and Charlton (1985a).Progress in the field has, however, been so rapid that there is, one hopes, little overlap with earlier works.

POSITRON AND POSITRONIUM PHYSICS

39

11. Annihilation Spectra for Positrons in Gases A. LIFETIME PARAMETERS

In contrast to the modem investigations using positron beams, the basic principles behind the methods of measuring lifetime spectra for positrons in gases have remained unaltered for more than 30 years. The electronic equipment employed for the present work has, however, become so sophisticated that data can be accumulated at a rate which is a few orders of magnitude faster and, with the aid of elaborate computer programs, can be subjected to much more precise analysis than was possible I5 years ago. The cross sections for annihilation are only comparableto those for atomic interactionsat very low velocities and it has to be emphasized that lifetime spectra, therefore, provide information at positron energies which are, as yet, not attainable with positron beams. This information is usually displayed in the form ofa number of parameterswhich describeboth the free positron annihilation rates and the annihilation ofthe positronium, Ps, formed during the positron slowing-down process in the gases under study. In terms, respectively, of the gas density p, of the classical radius of the electron r,, of c, the velocity of light, and n, the number density of the gas in the immediate vicinity of the positron (with u p = nricn), the five main parameters can be defined by where Ifand 1,are the density-dependent equilibrium decay constants for thermalized free positrons and for orthopositronium, 0-Ps, respectively, and ,,Ip is the vacuum decay rate of orthopositronium. The theoretical value of ,,Ip = 7.039 0.006 ps-l predicted by Caswell and Lepage (1 979) is close to the various measured values quoted by Gidley and Coleman ( 1984). 2, is the effective number of atomic or molecular electrons available to the positron for annihilation at agiven gas density and temperature,while z,,is the effective number of electrons available in similar circumstances for annihilation of the positron in the 0-Ps. The bars over these quantities signify that they represent the weighted mean for annihilation over the thermalized positron or positronium energy distribution. Another important lifetime parameter is the positronium formation fraction F, which is defined as the ratio of the total number of positronium events (ortho plus para) to the total number of gas events in a given lifetime spectrum. Positronium is formed by positrons with incident energy E,,, greater than the Ps formation threshold energy Eh. In the inert gases positrons surviving with energy below Eh can only lose energy by elastic collisions; the energy loss then occurs at such a slow rate that the positrons

+

40

T. C.Griflth

frequently annihilate before thermalization, thus giving rise to the characteristic shoulders observed in the spectra of these gases. In the molecular gases,with the exception of nitrogen, the thermalizationtime is so short that shoulders are not observed. Being dependent on the combined effect of elastic scattering and annihilation as a function of velocity, structural details of the shoulders for the inert gases are obviously critically dependent on the momentum transfer and annihilation cross sections.Ps atoms can be formed at positron energies of several hundred electron volts, but those with energy greater than 6.8 eV are believed to be rapidly dissociated in subsequent atomic or molecular collisions in the gas. A large contribution to F is, therefore, expected to be from Ps formed at energies in the range Em < E,, < E,, known as the Ore gap, with E, defined as the first excitation energy. As discussed by Schrader and Svetic (1982), there will be some contribution to Ffrom Ps formed in the range E, < E,, < Ei, where Eiis the ionization energy. Jacobsen et al. (1982) have argued that there is also a contribution to Ps formation in dense gases from spurs formed near the end of the positron trajectory and there is now strong experimental evidencethat the simple Ore model for Ps formation has only a limited range of validity. In most gases the measured values of F are found to be density dependent and this is a feature that cannot be explained by the Ore model because its predictions rely solely on threshold energies and cross-section values. It is of interest to note that important advances have lately occurred through the application of dc electric fields across the annihilation volume. Under these conditions the lifetime parameters are modified and the information contained in the data is analogousto that in the work performed with electron swarms. Methods have now been developed for the determination of drift velocities and, as reported by Davies et al. (1985) and by Charlton (1985b),the preliminary results, discussedin Section II,E, are very encouraging. All the above parametershave been carefully discussedby Heyland et al. (1982) and Charlton (1985a) and these articles review most of the experimental background to the determination and analysis of lifetime spectra and contain a comprehensive survey of the significance of the vast amount of data accumulated for all the inert gases for a wide range of molecular gases and for various mixtures of gases.

B. RESULTS FOR THE INERT GASES Several factors of interest have been revealed by a study of the shouldersin the inert gases and of the parameters Z,and F. It was shown by Campeanu and Humberston ( 1977)and by Humberston and Campeanu (1980) that an investigation of the shape and width of the shoulder in the spectrum for

41

POSITRON AND POSITRONIUM PHYSICS

helium together with the measurement of pe5provide a sensitive and independent test of the accuracy of the wave functions used for the evaluation of the total cross sections for e+-He scattering using the Kohn variational principle. The agreement between theory and experiment is very good. Similar investigations, with varying degree of success, have been carried out for the other inert gases. Campeanu ( 1981, 1982) has performed calculations relating to the shoulders in Ne, Ar, Kr, and Xe, while McEachran er af. (1978, 1979, 1980),using the polarized orbital method, have evaluated for the same gases. In Table I a summary of these results for is given, together with the measured values and estimated maximum and minimum values of F as predicted by the Ore model. The results for pure Kr and Xe gases at low densities are particularly important because of the small values of Fobserved in these gases. Another feature ofinterest for both these gases is illustrated for the case ofXe in Fig. 1, where there is definite evidence for the existence of two fast components close to r = 0 in the annihilation spectrum. The fast components have been linked to the observed dearth of 0-Ps in these gases. The rapid rise in the values of the total crosssection, a, for Kr and Xe at the Ps threshold together with the direct measurement of the Ps formation cross section, a,, for these gases presented in Section II1,C show that, under single collision conditions, 0-Ps is in fact copiously formed. The addition of small amounts of Kr and Xe to He as a base gas as reported by Charlton et af.(1979)also confirms that a, for these gases, as determined in a lifetime experiment, is large. It has been suggested by Griffith er af.(1982)and Wright et al. (1985)that

z,

TABLE I

EXPERIMENTAL AND THEORETICAL VALUESOF Gas ~

Zca(expt)

Zcr(theor)

Fmin F,,

3.85d 7.0f0.3e 27.6d 57.6f2.9e 2 1 7 f 11'

0.14 0.09 0.17 0.20 0.26

zcr

AND

Fa

F(expt)

~

He Ne Ar Kr Xe

3.94 f0.02b 5.99f0.08* 27b 65.7f0.3C 320fgc

0.28 0.32 0.43 0.49 0.56

0.23b 0.26b 0.33b O.llc 0.03'

a At room temperature for the inert gases (Fmin and F- are Ore model lower and upper limits). Griffith and Heyland (1978). Wright et of. (1985). Grover et al. ( 1 980); based on work of Mchchran ef al. (1977, 1979). 9 Schrader and Svetic (1982), based on work of Mchchran et af. (1978, 1980).

ze5

T. C.Grifith

42

I.

Xenon

. . . SHOULDER

lNFR I 150

10’

.

I

200

Channel Number

0-Ps

I I 80

I

I

I

1..

300

520

7LO

960

,

Channel Number

FIG.1. Lifetime spectrum for annihilation of positrons in Xe at 9.64 amagat and 297 K. Details of the “fast” (FF)and “less fast” (SF) components are shown in the inset.

POSITRON AND POSITRONIUM PHYSICS

43

both the small values of F and the existence of fast components in the annihilation spectra of pure Kr and Xe can be explained by postulating that the 0-Ps atoms, moderating in the gas, pass through at least two energy regions where resonant states of the form Kr-Ps and Xe-Ps are formed. These states are short lived and, if the cross sections for these processes are large, then two fast components would be detected in the spectra. These conjecturesare given further support by the observation that the addition of small amounts of He (or Ne) to either Kr or Xe as a base gas lead to an increase in the value of F. This is precisely what would be expected if the slowing-down process for 0-Ps is enhanced by the light He or Ne atoms in such a way that it makes fewer collisions in the energy regions where the resonant states are formed. For Kr-He mixtures the value of F increases from 0.1 1 at zero concentration of He to 0.39 at a He concentration Of 5090. The latter value of F lies within the limits of the Ore model. The addition of molecular gases, such as H2, D,, and N,, give similar results but are effective at much lower concentrations. In all cases the addition of the contaminant gases leads to a rapid decrease in the intensity of the fast components and to a drastic reduction in their lifetimes. In pure Kr the values of the decay constants for the fast components are 5 1.2 k 2.2 and 177 f 9 p s - l amagats-I and the corresponding values in pure Xe are 91.6 f 2.6 and 396 -t 11 ps-' amagats-I.

c. ANNIHILATIONFROM CLUSTERS In their study ofthe values of Fdand Ifasa function ofdensity for pure Xe and for mixtures of Xe with the molecular gases H,, D,, and N,, Wright et al. (1985) have proposed that positrons in Xe are annihilating after selftrapping in clusters of gas molecules in the gas at room temperature. This observation confirms that the phenomenon of clustering is now reasonably well understood both experimentally and theoretically.Cluster formation in different gases at various temperatures and densities has been investigated in several laboratories. It was first reported for helium at low temperature by Roellig and Kelly ( 1965)and later by Hautojarvi et al. ( 1977).The phenomenon has also been studied in detail for N, by Rytsola et al. ( 1984), and some of their observations are reproduced in Fig. 2. The clustering effect has been treated theoretically for the inert gases He, Ne, and Ar by Manninen and Hautojarvi (1978) using a density-functional formalism in which the static properties of the cluster were calculated with the aid of an assumed (simple) positron -atom potential in order to evaluate If and FeBat each density and temperature. Rytsola et al. (1984) have interpreted their results for N, in a similar fashion using a Van der Waals

I

I

i 120 K

0

50

100 Density (Amagotl

150

FIG. 2. Annihilation rates illustrating the effect of positrons trapped in clusters of gas molecules. The top diagrams are the results for “He(right)and ’He (left) observed by Hautojarvi et al. (1 977). The lower diagram is for nitrogen, as observed by Rytsola et al. (1984). Numbers on the graphs are temperature in degrees K.

45

POSITRON AND POSITRONIUM PHYSICS

equation of state and an optical pseudopotential for the positron - molecule interaction. In a full description of the N, data the zero-point energy of the positron on localization had to be evaluated and an allowance for the correThe sponding velocity incorporated in the final calculation of Ifand effect of clustering manifests itself as an increase of the measured values of 1, above that expected for a linear extrapolation from low densities. The divergence becomes prominent at certain densities for temperatures T 5 2T,, where T, is the critical temperature of the gas. As illustrated in Fig. 2, the rise for He is almost a step function but in N2the divergence is more gradual. A general review of theoretical aspects of the clustering phenomenon has been given by Iabukov and Khrapak (1982). In Xe the critical temperature is 289.9 K, so that clustering might be expected at room temperature. It was therefore rather surprising to find that instead of diverging upwards from the linear extrapolation from low density the values of Ifin pure Xe diverged in the opposite direction. It was argued that the observed discrepancy between the measured and calculated values of 2,given in Table I could be explained if in pure Xe at room temperature the positrons are not fully thermalized and that the full effects of clustering would also be masked if this was the case. The addition of small amounts of a molecular gas to pure Xe should ensure full thermalization of the positrons should then be close to the true values and the observed values of Ifand for thermalized positronsin pure Xe. Typical results for a mixture formed by the addition of 8.4% H2 to Xe at various densities at T = 297 K are shown in Fig. 3. The values of Af are now consistentlyhigher than those expected for a linear extrapolation from low densities and are, therefore, exhibiting a behavior that would be expected for pgitrons annihilating after self-trapping in clusters in the gas. The values of 2,rise sharply from the value at low density and are everywhere higher than for pure Xe. Other examples of the characteristicrise of 2,have been reported for different gases by Canter and Roellig (1975) for Ar, McNutt et al. (1975) for CH,, Wright et al. (1983) for C02,Heyland et al. ( 1985)for CO, and SF,, as well as for N, by Rytsola et al. (1984), and in each case the effect has been attributed to cluster formation.

zd.

z,

D. SPURMODEL FOR POSITRONIUM FORMATION It has already been demonstrated that a study ofthe positronium fractions F i n Kr an Xe has led to unexpected results. The proposed explanation of these observations implies a strong interaction between the 0-Ps and Kr and Xe atoms, but there is hardly any other experimental or theoretical information on the elastic or inelastic interactions of Ps atoms. As discussed by Schrader and Svetic (1982), it is observed that, as a function of gas density,

T. C. Grijith

46

1 4

lb)

I.;

0 0

0

0

350

ZCff

300

2 50

200

140 Density of Xe (amagats)

FIG.3. Values of (a) Ifand (b) Z,,in Xe as a function of de_nsityat297 K: (0)pure Xe; (0) Xe after the addition of 8.4%of H,; (0)theoretical value of Z,, (see Table I).

the values of F are only constant as required by the Ore model for He, Ne, and Ar; pronounced density dependence is observed for most other gases, In the molecular gases there are factors other than the Ps formation cross sections, ops,and Ps interactions which influence both the magnitude and density dependence of F. One such factor which has assumed increasing importance of late is the formation of Ps in spurs. In this model, proposed for the liquid phase by Mogensen ( 1974)and extended to dense molecular gases by Jacobsen (1982) and Mogensen (1982),the spur, formed by a positron as it loses energy and approaches the end of its range in any medium, consistsof a localized concentration of positive ions, electrons, excited molecules, etc., as well as the positron itself. If the e+- e- separation is small enough for there to be significant Coulomb attraction between them, the pair can unite to form Ps. The process is described by the reaction [e+ e-] + [PSI in contrast to the Ore model reaction e+ M + M + Ps. Spur reactions are expected to occur when the free electrons and the positron become thermalized in a medium of permittivity E , at temperature T, and their separation is of the same order of magnitude as the Onsanger distance r,, which is given by

+

r,

= e2/4mkT

+

+

(2)

47

POSITRON A N D POSITRONIUM PHYSICS

This is the separation for which the Coulomb potential energy between the (e+,e-) pair is equal to the thermal energy and Ps formation may occur if the spur radius R is also of the same order of magnitude as r,. Other processes such as e--ion recombination, negative ion formation, etc., may occur and both the positron and electrons may diffuse out of the spur. An expression for F which incorporatesboth the Ore and spur models has been written in the following form by Jacobsen ( 1 984):

- F,) exp(-&)[

F = Fo 4- (1

1 - exp(- r,-/R)]

(3) where Fo is the Ore model contribution and zh is the Ps formation time in the spur which is defined in terms of p, the sum of the electron and positron mobilities, by the expression

zR = 4eR3/3e/

(4)

Equation (3) takes account of both the density and temperature dependence of F and, as shown in Fig. 4, has been applied reasonably successfully by Jacobsen (1 985) to describe the data of Heyland et al. (1 985) for C 0 2 .As

07

06

F

05

O L

0

I

1

I

1

5

10

15

M

~~

-

1 25

I

I-

-1

5

10

15

. I 20

p Lamagalsi

FIG.4. Density dependence of the positronium fractions F in CO, at different temperatures. (a) ExpeFimental results of Heyland et a/. (1984): (-0-) 273 K, (--V--) 297 K ( ...A ...) 314 K, (-V-) 350 K, (-0-) 423 K. (b) Correspondingvalues calculated by 273 K, (---) 297 K, (---) 350 K (lower curve). Jacobsen (1985) using Q. (3): (-)

T. C.Gr.@th

48

expected, Fis dominated at low densitiesby Fo,the Ore model contribution. In most gases F increases with density up to a certain point and then decreases. Qualitatively, this can be explained by the argument that the spur contribution increases with gas density because more Ps is formed for the smaller R expected at high densities; at a certain density, however, the positronsin the spur may have a higher probability for free annihilation than for forming Ps so that F tends to decrease beyond this point. Further evidence in support of the spur model has recently been reported by Curry and Charlton (1985) from an investigation of the effect of adding small concentrations of electron-scavenging molecules CC1, and CC12F2to C02.As illustrated in Fig. 5 , the values of Fwere observed to decrease as the concentration of CCl, and CC12F2was increased. The results are interpreted as being due to the inhibition of Ps formation in the spur by the presence of the alien molecules. On repeating these experiments with Ar instead of C02, the values of Fwere, however, observed to increase with concentration. This behavior conforms to the view that Ps in Ar is formed according to the Ore model, i.e., F = F,, and that the increase in Fisdue to the contribution to Ps formation from the alien molecules.

X X

0

c

POSITRON AND POSITRONIUM PHYSICS

49

E. POSITRON MOBILITIES IN GASES Many lifetime experiments have been performed with a uniform e.xtric field applied over the annihilation volume. Most of the measurements, such as those of Lee and Jones ( 1 974), were taken before the present-day techniques of analysis had been developed and are, therefore, of limited usefulness. The calculationsof Campeanu and Humberston (1977) include some predictions of the variation of Z,with electric field in the case of He and measurements currently underway at University College London should soon close this gap between theory and experiment. Measurements of this nature in different gases should, indirectly, provide information on the momentum transfer cross sections below 1 eV. Another important advance in this direction has been fostered by Charlton ( 1985b),who has developed a method of measuring the drift velocity, w, of positrons in low-density ( < 1 amagat) gases under the influence of a uniform electric field. The positron mobility, p+, determined in such an experiment, provides information on a , the momentum transfer cross sections. Other methods that have been used for similar measurements were discussed by Davies et al. (1985). Mills and Pfeiffer (1976, 1977)have used a Doppler shift method for positrons annihilating in condensed media, while in low-density gases Paul and Tsai ( 1979), Bose et al. (198 I), and Paul and Bose (1982) have taken some measurements by monitoring the fraction of positrons reaching a target, after drifting in an electric field, using the known annihilation rate as a clock. The success of the method used by Charlton ( 1985b)and by Charlton and Laricchia (1985) is largely due to the refined techniques of analysis that can these days extract relevant information from very weak signals. As illustrated in Fig. 6, the simple vessel used for this work consists of two brass electrodes with a potential difference Vbetween them, which also constitute the walls of the chamber containinggas at subatmosphericpressures. With the use ofthe technique developed by Coleman et al. (1972), positrons from a 10 pCi 22Nasource were detected as they traversed a thin plastic scintillator of thickness 0.25 mm before entering the volume between the electrodes through a thin Melinex window. The annihilation y rays were detected in the 125 X 100 mm plastic scintillator on the other side. The timing system successfully identifies the 0.1% of the positrons that annihilate in the gas from the 99.9% that annihilate in the vessel walls to appear as a massive prompt peak. The gas signal correspondsto the annihilation of free positrons that have been thermalized in the gas and also the annihilation of Ps atoms formed in the cell. Application of the electric field means that some of the

50

T. C,Grifith

FIG.6. Schematicdiagramofthe apparatus used by Charlton(1985b)to measure positron drift velocities.

thermalized free positrons will annihilate prematurely following field-induced drift to the electrode. This happens if the maximum drift time r,d is and the typical lifetime spectrum,illustrated in Fig, 7, comparable with (If)-' is characterized by an abrupt terminus at rmd, which is preceded by the 0-Ps and free positron decay components. z,d is observed to be dependent on 4, the applied potential. Assuming that the thermalized free positrons are uniformly distributed between the electrodes, then z,d is specified by those positrons which have drifted the entire drift length, d, between the electrodes. The drift speed is then o = d/rmdand the positron mobility is In the approximation that, in the low-energy region, the positron momentum transfer collision rate averaged over the positron energy distribution is independent of the velocity u (viz., a , a l/u), it follows that where n,u+ is the density-normalized mobility.

105

%

..)r

-

*

0-Ps + free1 Y

95

-

Y Y

Y

I

t

YI d .

C

s

I

$

85

free only

-1 I I I

1 Y

I

75

65

FIG.7. Typical lifetime spectrum for SF, gas at 98 torr and 827 V cm-' taken by Charlton and Laricchia (1985). Maximum drift time is T , ~ . The prompt peak lies at channel 155 ( I channel = I .97 ns). x 106

a

b

!

10

-

5

6

7

8

06

10

15

E I P ( Vcm-' tor,.')

FIG.8. Positron drift velocitiesas a function of ( E / p )for (a)SF, ( 100torr, 297 K)and (b) N, (400 torr, 297 K), as measured by Charlton and Laricchia (1985).

T. C. GriJith

52

TABLE I1 VALUESOF p+n, p-n,"

AND a, FOR VARIOUSMOLECULAR GASESAT T = 297 Kb

H2

D2

0,

N,

CO

C02

CH,

SF,

1940 1200 29

2200 1080 26

1440 4460

1560 930 36

1430 1800 40

480 660 119

370

300

154

190

~~

p+n p-n. a,(A) a

40

-

-

In units of cmz V-* s-l amagat.

* Average errors in p+n are typically 10- 1 5%. Examples illustrating the dependence of the measured values of the drift velocities on E/p, where E is in V cm-' and p is the gas pressure in torr, is given for SF, and N, in Fig. 8. Preliminary values of the positron mobilities and of a , calculated using Eqs. ( 5 ) and (6) are given in Table 11. Some of the electron mobilities reported by Peisert and Sauli ( 1984) and by Crompton and Elford (1973) are also tabulated.

111. Cross-Section Measurements with Positron Beams A. TOTALCROSS SECTIONS Only a brief account of some of the more recent results of total cross-section measurements for positrons and electrons at various energies in different gases need to be discussed in this article because a number of comprehensive review articles have already been written on the subject. Gnffith and Heyland (1978), Kauppila and Stein (1982), Raith (1983), and Charlton ( 1985a) have all described the different experimental techniques, and their limitations, used for these investigations. They have also summarizedall the main results that were available up to the end of 1984. As a general observation it is fair to comment that, considering the diversity of experimental methods that have been used, the agreement between measurements from different laboratories is exceedingly good. All the measurements are based on an observation of the transmission or attenuation of an incident positron (or electron)beam of energy Eand intensity Z,(E) in a gas column oflength 1, at temperature T and pressure p , to its transmitted intensity Z(E).The total

53

POSITRON AND POSITRONIUM PHYSICS

cross section (TT(E)at the given energy is then written as

%w)= l n [ ~ o ( ~ ) / W ) I / n ~

(7) lo2’

where the number of target molecules per unit volume, n = 9.67 < (PIT), with p in pm Hg and Tin degrees kelvin. One area of continuing interest is that at intermediate energies- from roughly 20 eV upwards-where it is likely that, as the various theoretical approachesbecome stabilized,there will be a need for more precise measurements of U, and more detailed comparison of positron and electron data taken under the same experimentalconditions as has been done by Hoffman et af. (1982), Dababneh et al. (1982), and Kwan et af. (1983). Another important measurement would be that of e+- atomic hydrogen scattering, which is of considerable theoretical interest and also of interest for the interpretation of certain astrophysical observations (Drachman, 1982; Leventhal, 1985)-at present it still remains as a challenge to the experimentalists. Some recently published measurements of the e+-He total cross sections at energies between 0.6 and 22 eV by Mizogawa et al. (1985) are illustrated in Fig. 9a. As shown in Fig. 9b, these results, after applying corrections for forward scattering, are very close to the theoretical values of Campeanu and Humberston (1 977).

1

I

I

1

I

0 .15

I

(a) 0 .10

0.15 I

N

-

05 - 00.15

NO

E

0

b ’

0.10

b ’

0.10

O.OE

1

2

3

L

Positron Energy lev)

5

6

0.05

0

1

2

3

L

5

6

Positron Energy (eV)

FIG.9. Total cross sections for positrons in He. (a) The results of Mizogawa et al. (1985) with axial magnetic field values of 8 gauss (W) and 13 gauss (O),respectively,compared with the data of Stein et al. (1978) (A); Sinapius ef al. (1980) (V); and Canter ef al. (1973) (0).The theoretical curve (solid line) has been deduced from the work of Campeanu and Humberston (1977), Humberston and Campeanu (1980), and OMalley ef al. (1962). (b) Corrections for forwardscatteringappliedtothedataofMizogawaef al.( 1985). Solidlinesarethesameasin(a).

T. C.Grifith

54

Other recent work of interest has been the measurement by Floeder et al. (1985) of oT for the scattering of both electrons and positrons by a large family of hydrocarbon gases. CH,, C?H,, CzH4,C&, C,H, (propene and cyclopropane), C,H (n-butane and isobutane), and C,H, have been studied, and at positron (and electron) energies between 100 and 400 eV all the cross sections conform to an empirical fit represented by

o$(E)= uN,E-I/~[1 f b exp(- cE)]

(8)

where a, 6, and c are constants and N, the number of electrons in the molecule. The proportionality between o$ and N, at 100 and 200 eV is demonstrated in Fig. 10. These authors have also noted that for a given molecule a;(,!?) is always less than a&!?) and that both cross sections increase with the size of the molecule. They also note that atoms with the same number m of carbon atoms have cross sections of very similar magnitude and that these values were well spaced from those for the molecules with m 1 carbon atoms. The measurements of o, for alkali metal vapors by Stein et al. (1985) are an important technological achievement. Apart from atomic hydrogen the alkali metal atoms are the most important “single-electron’’ atoms to be studied. They also have the additional feature that their ionization potential lies below 6.8 eV so that they have no Ps formation threshold. Technically this experiment has almost the same degree of difficulty as that envisaged for atomic hydrogen experiments. The problem of confining the vapors to the

+

0

10

20

Ne

30

10

20

30

Ne

FIG.10. Total cross-sectionsfor (a) 100-and (b)200-eV positrons(+) and electrons (0)as a function of the number of electrons in the hydrocarbon target gas molecules, as measured by Floeder et al. (1984). The straight lines have been fitted using Eq. (8).

POSITRON A N D POSITRONIUM PHYSICS

55

interaction region has been solved using the high-temperatureoven or target illustrated in Fig. 1 1. Two-channel electron multipliers, out of direct line of the hot oven apertures and protected behind the cold end caps, are used to measure the incident and transmitted positron beam intensities after deflection of the beams in the repeller cages on either side of the oven. Results for 0;and a , determined for potassium with this apparatus are given in Fig. 12, and the notable feature of these results, not seen for any other target atoms, is the close similarity between the values of the cross sections for electrons and positrons. It is suggested that this is due to the fact that the high polarizability of the K atoms essentially swamps the static part of the interaction at these energies so that little difference between of and a, should be expected. Some new results by Kwan et al. (1985)for the scattering of electrons and positrons of energies in the range 1 to 500 eV in 0, and similar data by Kauppila et al. (1985a) for N,O, CH,, and SF, are very interesting because they show structural variations in the values of oTwhich are in remarkably close agreement with similar measurements reported by Charlton et al. ( 1980a) and Charlton et al. ( 1983a) for positrons in H,, N,, O,, CO,, and CH,. A series of measurementsfor electrons ofenergiesup to 50 eV taken in the same gases under the same conditions as for positrons was reported by Griffith et al. (1 982). Magnetic Reld Coils,

Input Aperture Plate,

output CEM

Piotes

Removable Cylinder’ (wlth Heater)

-

0

I

2 3 4

5cm

Cold Cap with Aperture

0

Thermocouple

FIG.1 1. Schematic diagram of the apparatus used by Stein et al. (1985) for measurements of alkali metal scattering cross sections.

T. C. Grifith

56 n

0

0

P

P

P 0

0

P

Energy (ev)

FIG. 12. Total cross sections for (0)positrons and (0) electrons scattered by potassium atoms as measured by Stein er al. (1985). Theoretical estimates for the elastic cross sections QB (--,Guha and Mandal, 1980;-,Bordonaro et al., 1976)andpositronium formation cross sections QPs(Guha and Mandal, 1980) obtained usingthe distorted wave approximation (---) and the first Born approximation (V) are shown.

Sueoka and Mori ( 1984) have also reported measurements of gT for positrons in N2, CO, and CO, . The abrupt rise in 0, at the threshold for the formation of excited-statepositronium, Ps*, in C02noted by Charlton et al. ( 1983a)and by Kwan et al. ( 1984) is quite noticeable in Fig. 13 and will be discussed again in Section II1,D. It is of interest to note that the same feature is also present for the N20 data of Kwan et al. (1984). B. DIFFERENTIAL ELASTIC SCATTERING CROSS SECTIONS

Differential cross sections for the elastic scattering of positrons were first reported by Coleman and McNutt (1979) for positrons in argon. They made

POSITRON AND POSITRONIUM PHYSICS

51

16

15 14

f

13 I

T

12 a,

I

1’

inad)

10 9

8

I 6

5 L

0

I

I

I

I

I

4

8

12

16

20

I 24

I

1

28

32

Energy lev)

FIG. 13. Total cross sections for positrons scattered in CO,: (B)Hoffman ef al. (1982); (0) Charlton ef al. (1 980a); (0)Charlton er al. ( 1 982a). The theoreticalcurves are by Horbatschand Darewych (1983). Curve (---) uses a fixed cutoff, and curve (-) an energy-dependent cutoff parameter in the polarization potential.

use of the fact that in the time-of-flight method discussed in Sections III,E and III,F the time-of-flight spectrum of the scattered positrons in the inert gases is solely dependent on the angular distribution for the scattering provided the incident positron energies are less than ER (in which case there is no inelastic scattering).As the angle of scatteringincreases so does the time of flight of the scattered positron and the time-of-flight spectrum can be partitioned according to the scattering angles. The spectra for positrons of energies between 2.2 and 8.7 eV scattered in argon were analyzed to yield the values of a(@, the differential scattering

T. C.Griflth

58

cross sections, in the range 20" < 8 < 60".They were found to be in reasonable agreement with the theoretical calculations of McEachran et af. ( 1979) and Schrader (1979). Coleman et al. (1980) have given a detailed analysis of the limitations of their method. Ideally, measurements of this nature would be better performed with narrow positron beams of well-defined energies in a crossed-beam arrangement. Such an experiment, for 100-eV positrons in Ar, has recently been reported by Kauppila et af. (1985b). Their system is illustrated in Fig. 14, where a low-energy positron beam (-2 X lo4 e+ s-l) is generated from a 50-mCi 22Nasource with an electrostatic beam transport system and an atom -beam source consisting of a multichannel capillary array in a differentially pumped scattering chamber. The incident positron beam is deflected on emerging from the chamber by the repeller into a channeltron detector placed off the direct line of fast positrons from the source. The scattered beam is detected at various angles by another channeltron detector whose entrance window is defined by a set of collimators viewing the scattering region. Retarding potential elements are used to reject inelastically scattered positrons. The system can be used with either positrons or electrons. The preliminary results for a range of angles between 30" and 120" are shown in Fig. 15, where the experimental data have been normalized at 45" to the theoretical curve computed by A. D. Stauffer and R. P. McEachran (private communication) using the polarized orbital approximation.

\*Y

Chonneltron Detector rY2 ,-Collimators

Lens Elements

U

L G o s Beom

[ e* Differential

Scattering Setup

FIG. 14. Apparatus used by Kauppila et al. (1985) for positron and electron differential elastic scattering cross-sectionmeasurements.

POSITRON AND POSITRONIUM PHYSICS 4

59

i

FIG. 15. Comparison of the differential elastic scattering cross sections for positrons (O), measured by Kauppila ef al. (1985), with the theory (-) of Stauffer and McEachran (private communication).

C. TOTALCROSSSECTIONS FOR THE FORMATION OF POSITRONIUM Experimental determinations of o , the positronium formation cross sections, have been discussed at length in recent review articles by Griffith ( I 984), Coleman ( 1 985a,b),and Charlton (1 985a). Positronium formation is undoubtedly the most interesting of the inelastic channels for positron atom/molecule interactions and its study continues to present surprising o were first obtained from the plots of and unexpected results. Estimates of , the total cross sections 0, for the inert gases as a function of positron energy. A linear extrapolation of the elastic cross sections gel,from below the Ps formation threshold Epsup to Eex,the first electronic excitation energy, gave a value of , o at Eexusing the simple difference expression, , o = 0, - o,,. It was also possible, on adopting a number of plausible assumptions,to deduce a value for both oh and the excitation cross section oa, at the ionization threshold Eifor all the inert gases by using the values of the Ps fractions F determined in the lifetime experiments reported by Coleman et al. ( 1 975) and from the gas mixture results given by Charlton et al. (1 979). All the above work is now superseded by three differentand independent

60

T. C,Grifith

methods that have been used for a direct determinationof a, as a function of positron energy using positron beams. In the first of these methods the University College London (UCL) group (Charlton et al., 1980b, 1983b; Griffith, 1984) have measured the 37 coincidence rate for 0-Ps formed in a differentially pumped gas cell through which the variable-energy positron beam was steered in the same manner as for the total cross-section measurements. Fornari et al. ( 1983)at the University ofTexas at Arlington (UTA),in a sharply contrasting method, have observed the transmission of positrons in a strong axial magnetic field through an extended gas cell; the number of positrons “lost” during transmission through the gas was attributed to the formation of either paraps or 0-Ps. The third measurement, also distinctly different, has been that by Fromme et al. (1985), where a, in helium was deduced by distinguishing between the ions formed by impact ionization and those due to formation of para-Ps and 0-Ps (see Section 111,E).All three measurementsshow a rapid rise in a, from the threshold E, for all the gases investigated,and they also indicate that a substantialamount of Ps is formed at positron energies above 100 eV. There has, however been a serious disagreement between the values for , a measured by the first two groups, with UTA reporting cross sections of magnitude about three times greater than the UCL results at some energies. The preliminary results for helium by Fromme et al. (1985) appear to favor the UTA data. The magnitude of the values of , o are large, but recent calculations for He by Khan and Ghosh (1983) give values which are only slightly lower than the UTA results. The results for He and H2 are reproduced in Figs. 16a and 16b, respectively. The UCL results cover a range of energies from 5 to 150 eV in all the inert gases and the molecular gases H,, N, ,0, ,C02,and CH, .The earlier UTA measurements were restricted to He, Ar, and H and for these gases there is good agreement with the UCL data at energies up to about 2 eV above the E, thresholds and then the measured values diverge sharply as the energy increases. The UTA workers have recently extended their Ar data to 441.3 eV (Diana et al., 1985a) and have also reported some preliminary results for Kr and Xe at energies up to 350 eV. Both experiments may be subject to systematic errors, but, as discussed by Charlton (1985a),the UCL data were subjected to various tests to check the effect of premature quenching of the 0-Ps on the cell walls and also to investigate the escape of fast positronium through the exit aperture. The UCL data were normalized using the values of , a obtained from a, on the assumption that the elastic cross sections could be extrapolatedlinearly from below E,. Recent calculations by Brown and Humberston (1985)for atomic hydrogen indicatethat a, is in fact decreasingat energiesgreaterthan &and that the normalization procedure used by the UCL group might involve an underestimate of the cross sections by as much as 30%. Brown and Humberston (1985) have also

61

POSITRON AND POSITRONIUM PHYSICS

o'67 !N I

I

2

01

I

hsitron Energy (eV)

0

10

20

30

40

50

60

70

80

Positron Energy (eVI

FIG. 16. Positronium formation cross sections as a function of energy. (a) Helium: (H) Charlton et al. (1983); (V)Fornari et al. (1983); (A) Diana ef al. (1985); (0)Fromme et al. (1985); (-*-) theoretical curve by Khan and Ghosh (1983). (b) Molecular hydrogen: (0) Fornarief al.(1983);(0)Charltonetal.(1983);(-X-)LifetimedataofCharltonetal.( 1979); ()curve for u,, = (ar- ue,);(- .. -)theoretical curve by Bussard et al. (1979) for chargeexchange cross sections.

noted that, for atomic hydrogen, the emission of positronium formed in the collisions becomes increasingly confined to the forward direction as the positron energy increases. If this is true then the tests that were performed to check for the escape of fast Ps were possibly limited to aperturesthat were too large to observe significant variations. These two factors alone should bring the measurements closer together, but there are plans at UCL for remeasur, a ing . It is of interest to re-emphasize that the UCL method is based on direct observation of 0-Ps, while the other methods observe both 0-Ps and para-Ps without distinguishing between them. It has been suggested that (e+,e-) correlations (see Section II1,F) might account for some of the difference between the measurements at the lower energies but, as noted by Coleman (1 985b),the difference increases with energy and is much too large to be due to correlation effects. Coleman (1985a)has suggested that the discrepancy is due to 0-Ps quenching on the walls of the gas cell. It is observed that curves diverge sharply at roughly 2 eV above the positronium thresholds and, if this effect is due to wall quenching, it might be of some significancefor the study of Ps- surface interactions. There are good reasons for investigating the source of the discrepancy and the ultimate check will be that, at any energy, the sum of the elastic, impact ionization, Ps formation, and excitation cross sections, all measured independently, should add up to the measured value of a,.

T. C. Grijith

62

The possibility that the Ps formed in e+-atom collisions is peaked in the forward direction, as noted by Brown and Humberston (1985), has important consequences for the production of Ps beams and hence for studying Ps scattering and Ps- surface interactions. It is a way of producing Ps beams of well-defined energy. The alternative proposal for such beams, suggested by Mills ( 198l), would involve a much more complicated system; his method is based on the production of a beam of Ps negative ions, Ps-, by passing positrons through a thin carbon foil and then photoionizing the Ps-to get the 0-Ps beam. Mills and Crane (1985) have, however, recently reported that energetic 0-Ps is also produced by positrons traversing thin carbon foils. Preliminary measurements on the angular dependence of Ps emission in e+-argon collisions have been reported by Laricchia et al. (1985a) and look very promising. The positron beam was passed axially through an 8-mm input aperture into a gas cell of length 20 mm. The 0-Ps formed in the gas emerged through the upper half of an 8-mm-wide slot in the output plane to be detected, 120 mm downstream from the center of the gas cell, in a channel electron multiplier (CEM) positioned initially on the axis of the positron beam. Suitably biased grids in front of the CEM excluded any background due to electrons and slow positrons so that only the 0-Ps atoms impinging on the CEM were counted. The measurements displayed in Fig. 17 show the CEM counting rates, obtained from alternate gas and vacuum runs, as a

8r---------,

0

’

”

”

5

”

10 Positron Energy

(eV)

”

”

’

7

XlO’

l5

FIG. 17. Channeltron detector counts due to DPS formed in e+-Ar collisions with the detector placed (0)on the beam axis; (A) 1 cm off the axis; and ( x ) 2 cm off the axis. Measurements by Laricchia et d.(1985a).

POSITRON AND POSITRONIUM PHYSICS

63

function of positron energy. The counts are seen to increase as expected at the positronium formation threshold for argon. On displacing the CEM off-axis through transverse distances of 10 and 20 mm, respectively, the counting rates at the same positron beam energy fell rapidly, and it is concluded that 40% of the 0-Ps formed in argon by positrons of energy greater than 20 eV are confined within a forward cone of angle less than 5 '. There are grounds, therefore, for reasonable optimism that substantial beams of positronium of well-defined energies, given by E+, = E,, - E,, will soon be available for experiments.

D. THEFORMATION OF EXCITED-STATE POSITRONIUM IN GASES Conclusive demonstration that positronium in its first excited state, Ps*, could be detected was given by Canter et al. ( 1975). This was accomplished by detecting the uv radiation of energy 5.1 eV from the 2P- IS transition of Ps* using a photon counter and an NaI(T1) scintillator, which detected one of the annihilation y rays from the decay of the ground state Ps, in delayed coincidence. The reaction involving the Ps* transition is as follows Ps;-2+

.1 3.2 ns

hv

+

Ps

-37

3. 142 ns

Ps* does not survive to be detected in lifetime experiments,but in positron beam experiments, using gas at low density, the chance ofa Ps* collision with a gas atom/molecule is so small that it has a high probability of experiencing a 2P- 1Stransition to its ground state. Laricchia et al. (1985b) have shown that it is possible to detect Ps* under these conditions and have investigated its production yield as a function of positron energy in a number of gases. The binding energy of the first excited state is only 1.7 eV, so that the formation threshold is very close to the ionization threshold and the results Ar, and Ne show the characteristic rise from the Ps* given in Fig. 18 for H2, threshold that was familiar for the work on ground-state Ps discussed in Section III,C. The scattering chamber used by Laricchia et al. (1985b) is illustrated schematically in Fig. 19. A positron beam of variable energy traverses the differentially pumped gas cell placed between the Ps Lyman-a counter and the NaI(T1) counter which detects the annihilation y ray. The photon counter was a water-cooled EM1 9829A photomultiplier with a sensitivity range for photons of energy between 2.1 and 6.2 eV and a dark count of about 50 s-*. The walls of the chamber were lined with glass which was

T. C. Grifith

64

70

12

14

16

18

20

22

24

26

28

30

32

34

36

Positron Energy 1 eV)

FIG.18. Measured yield of Ps*as a function of positron energy for (0)HZ; (0)Ar; and (A) Ne, as measured by Laricchia el al. (1985b).

coated with A1 and M C 2 to enhance the uv reflectivity. Provisions were made for the insertion, when required, of a 3-mm-thick borosilicate glass shutter between the gas cell and the photon counter in order to stop photons of energy 24.3 eV from reaching the counter. The time resolution of the system was 8 ns FWHM and a typical spectrum for argon gas, taken without the glass shutter, is shown in Fig. 20; it shows the long-livedcomponent with a decay rate of 7 - 10 pus-' to the left of the prompt peak at t > 0 and also a significant signal to the right at t < 0. Insertion of the borosilicate shutter usually removed the long-lived component, thereby confirming that this signal was due to the photons of energy 2 4.3 eV attributed to the Ps* transitions. The signal at t < 0 was always unaffected by the shutter and has been attributed to reactions of the type e+ A + e+ A**, where the prompt annihilation of the positron into 27, on striking an aperture in the scattering cell, givesthe start pulse and a stop pulse results from a delayed photon due to the highly excited (n 3 3) atomic (or molecular) state, A**. Laricchia et al. (1985b) have estimated the maximum yield of Ps* in most Ps* per incident e+, and it is noted that this yield gases to be around 5 X is a factor of 14greater than that quoted by Schoepf et al. (1 982) for positrons incident upon a cleaned polycrystalline tungsten surface in high vacuum. It was mentioned in Section II1,A that the results of both Hoffman et al. (1982) and Charlton et al. (1983a) show a sharp rise in the values of oT for CO, at the formation threshold for Ps*. CO, was, therefore, an obvious

-

+

+

POSITRON AND POSITRONIUM PHYSICS

65

uv reflecting surface of glass tubing with Al + Mg F, overcoats

/cornposed

1

Nal ( T I 1 counter

FIG. 19. Schematic illustration of the scattering cell used for the Ps* measurements by Laricchia ef al. (1985b).

candidate for investigation to see whether a large amount of Ps* was indeed formed in this gas. The spectrum given in Fig. 2 1 shows a large signal both at t > 0 and t < 0. Insertion of the borosilicate shutter, however, only removed a small amount ofthe (supposedly)Ps* signal at 1 > 0. It has to be concluded, therefore, that the sharp rise in 0, and the large signal observed in this experiment is due to some process that may involve ground-state Ps but not the formation of Ps*.

T. C.Grijlith

66

'1 -*

30

25

In

;20

a

u z c

0

z

10 35 10

5

.*

0

0

5

10

15

20

25

Channel

30

35

40

45

50

55

I

XlOl

60

Number

FIG.20. Photon-annihilation p r a y spectra (4.0 ns per channel) obtained for 17-eV positrons in Ar gas: (0)glass shutter retracted (run time = 53 000 s); (*) glass shutter inserted.

E. POSITRON IMPACTIONIZATION CROSSSECTIONS The first attempt to unravel the inelastic processes for positron-atom/ molecule collisions was reported by Griffith et al. (1979), where incident positrons of well-defined energy were used in a time-of-flight method and the energy distribution ofthose positrons inelastically scattered by helium atoms was recorded and analyzed. It was concluded that, at incident positron energies between 100 and 500 eV, impact ionization appeared to be the dominant process and, assuming known values for the excitation cross sections 0, for electrons and the positron elastic cross sections,they were able to estimate some values for aznat these energies. Using a similar time-of-flight technique Coleman and Hutton ( 1980)and Coleman et al. ( 1982) have investigated and obtained values for the excitation (over a limited range of energies) and for the sum of excitation plus ionization cross sections at positron energies up to 50 eV in He, Ne, and Ar. A similar method, modified in such a way (see Section II1,F) that excitation and ionizing collisions can be separately assessed, has been used by Sueoka ( 1982)to extend the measurements in He up to 120 eV. At energies 22 30 eV the values reported for ofnare close to those observed for electrons and this observation may, cautiously, be interpreted as implying that the exchange contribution in electron-atom scattering is not very large at the higher energies.

67

POSITRON AND POSITRONIUM PHYSICS

lo3 0

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FIG.2 1. Photon-annihilation pray spectrum for 14-eV positrons in CO, gas, as measured by Lancchia e? al. ( I985b). The run time was 64 000 s ( 1 channel = 4.0 ns).

More recent investigationsof positron ionizing collisionshave used different methods from those discussed above. Diana et al. (1985b)have investigated positron impact ionization of helium atoms at energies up to 200 eV by detecting the ejected electrons using a retarding field analyzer. Their results are included among others in Fig. 22b. Another novel method of measuring both impact ionization and Ps formation (see Section III,C) has

T. C.Grifith

68

I Positron Energy (eV)

Positron Energy (eV)

FIG.22. (a) Sum of impact ionization and positronium formation cross sections for psitrons in He as a function ofenergy:(0)Fromme ef al. (1985);(-X-) theoretical calculations by Willis and McDowell (1982); (-) qo, for , electrons by Montague et al. (1984). (b) Positron impact ionization as a hnction of energy: (0)Fromme ef al. (1985); (*) Sueoka (1982); (A) Diana ef al. (1985); (- * -) theoretical curve by Basu et al. (1985); (-) electron results of Montague ef al. ( I 984).

recently been developed at Bielefeld, and results from this work have been reported by Fromme et al. (1985). In their method the ions produced by positron collisions with gas atoms in a long thin tube are extracted by being accelerated down the tube in a weak electrostatic field. As shown in Fig. 23, beam transport is accomplished using a longitudinal magnetic guiding field and electrostatic lenses (the ions are also confined by the magnetic field). An E X B field analyzer at the exit of the scattering chamber is used to separate the ions and the positrons. The positrons are only slightly affected by the analyzer so that they continue on their path to the microchannel plate detector at the end of the system. The ions are, however, deflected sideways through 90" and accelerated up to 4 kV and into another microchannelplate (MCP) detector at the side. Delayed coincidences between the two detectorsare recorded, and a peak in the time distribution corresponds to detection of the positron and ion involved in the impact ionization represented by He e+ ---* He+ ee+. When Ps is formed according to the reaction He e+ 4He+ Ps only the ions alone are recorded. A count of all the ions on one hand and of those ions in coincidence with the positron on the other gives the ratio of

+

+ +

+ +

69

POSITRON AND POSITRONIUM PHYSICS

Stainless steel tube Glass tube Tunpsten spiral

Detector 2

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Moderator

n l I To pump

lions]

ector 1 ’ (positrons)

Scattering tube

I

0

PUMP

E Y 6 mass spectrometer

FIG.23. Apparatus used by Fromme ef al. (1985) to measure ionization cross sections for positron- He collisions.

total ionization (Ps plus impact) to impact ionization. The authors report that fewer than 10%of the expected number of ions were detected, probably because the axial magnetic field was not large enough. This factor, together with the uncertainty about the efficiency of the MCP for ion detection, meant that some normalization procedure had to be adopted to evaluate cross sections. This was accomplished by repeating their observations with electrons instead of positrons and matching the relative values of a , obtained with electrons to the electron data of Montague et al. (1984). At energies above 200 eV the convergence of the values of a;f and:a reported by Kauppila et al. ( 1981) was used for the normalization of (a, of) to a ; , assuming that ,a + 0 at these energies. The normalized results for (oh a;,) as a function of energy are compared with the theoretical estimates of Willis and McDowell(l982) in Fig. 22a. In Fig. 22b the normalized values of a&, obtained from the delayed coincidence counts are shown to be in good agreement with the theoretical values of Basu et al. (1985). They are, however, significantly larger than the correspondingvalues measured by Sueoka (1982) and also the values of a ; , for electrons measured by Montague et al. ( 1984). It is difficult to avoid the conclusion that there may be a systematicerror in one of the measurements. The values of a, obtained from the difference between (a, of) and a ; , (coincidence) are given in Fig. 16b and were discussed in Section II1,C.

+

+

+

70

T. C.Grijith F. CROSS SECTIONS FOR ATOMICEXCITATION IN POSITRON COLLISIONS

In their estimates ofa, for the inert gases from lifetime spectra (seeSection III,C), Coleman et al. (1975) were also able to extract approximate values of the excitation cross sections a& for these gases at the ionization energy Ei. Direct measurements of at; were later performed by Coleman and Hutton ( 1980) using the time-of-flight technique mentioned in Section III,E. These authors analyzed the energy distribution of positrons with incident energies E,+ of 5 30 eV scattered by He atoms and concluded that at energies up to 10 eV above the first excitation threshold E,, the secondary peak in the spectrum of the scattered positrons was due to positrons that were exclusively associated with atomic excitation. At higher energies it was not possible to distinguish between positrons that were involved in excitation from those associated with ionizing collisions; only the sum (ot; a;,,) could, therefore, be deduced at the higher energies. Coleman et al. (1982) have used the same apparatus to extend these measurements to incident positron energies of 50 eV in He, Ne, and Ar. It was concluded that at positron energies up to 30 eV in helium the dominant channel appears to be that due to 1'S-2'S excitation. Careful examination of the secondary peaks at low energy have also led to the conclusion that, after the collisions, the positrons involved with atomic excitations in He were confined to angles 5 50" of the forward direction and to smaller angles in Ne and Ar. These observations are in agreement with the recent calculations for He by Purcell et af. (1983). The recent work by Sueoka (1 982), mentioned in Section III,E, was performed using a similar time-of-flighttechnique, and values of af at energies up to 120 eV have been determined. This was achieved by using a retarding field on a grid in front of the CEM detector at the end of the flight path to exclude positrons involved in ionizing collisions, viz., scattered positrons experiencing an energy loss ?(lie+- Ei).The results for G;,, discussed in Section III,E were obtained by subtracting the spectra obtained at a given positron energy with the retarding field applied from those with the retarding field switched off. The author reports that no ionizing collisions were detected in He at incident positron energies less than 30 eV. This is the energy region where the (e+,e-)correlations (see Section II1,C) might be expected to be strong and where the positron may either form positronium or some other complex which results in its annihilation. Klar (1981), Temkin (1982), and Geltman ( 1983)have all estimated the values of a;,, and ,a at energies near the threshold Ei, but their predictions are not in good agreement with one another. It might not, therefore, be unreasonable to expect that the measured values of o;,, for electrons are larger than those for positrons in the energy

+

POSITRON AND POSITRONIUM PHYSICS

71

0.1-

0.08

-

-

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range Ei < E,, 5 30 eV. This appears to be the case for the values of &c determined by Sueoka (1982)but, as seen in Fig. 22b, the results of Fromme et a/. (1985) for positrons, discussed in Section III,E, lie above the corresponding electron values at all energies. The excitation cross sections for He are given in Fig. 24, where it is seen that for positrons of energy greater than 30 eV the values are appreciably greater than those for (,a l1S-2IS) alone for electrons. This observation implies that the contribution from higher excited states is appreciable in positron collisions at the energies under consideration. Most of the crosssection measurements discussed in this section and in Section III,E may be subject to systematic errors which have not yet been thoroughly assessed. Much further work is required before arriving at any firm conclusions regarding the true magnitude of the various cross sections that have been measured.

ACKNOWLEDGMENTS The author is much indebted to Dr. Michael Charlton for many valuable comments and discussion. Thanks are also due to the other members of the positron physics group at University College, namely Dr. G. R. Heyland, Dr. C. J. Beling, Dr. P. J. Curry, Dr. F. M. Jacobsen, Mrs. G. Laricchia Drinkwater, and Miss S. A. Davies. Much of the University College work

72

T. C. Grrfith

discussed in this article has evolved from the inspired encouragement given by the late Sir Harrie Massey and as a tribute to his memory it is immensely pleasurable to acknowledge our gratitude to him. Grateful acknowledgement is given to Miss Una Campbell for preparing many of the diagrams and to the Photographic Unit for their invaluable assistance with the prints. Most of the University College work discussed in this article has been funded by the Science and Engineering Research Council, to whom we are duly grateful.

REFERENCES Basu, M., Mazumdar, P. S., and Ghosh, A. S. (1985). J. Phys. B 18, 369-377. Berko, S., and Pendleton, H. N. (1980). Annu. Rev. Nucl. Part. Sci. 30,543-581. Bordonaro, G., Ferrante, G., Zarcone, M., and Cavaliere, P. (1976). Nuovo Cimento SOC. Ital. Fis. 35B, 349- 362. Bose, N., Paul, D. A. L., and Tsai, J-S. (1981). J. Phys. B 14, L227-232. Brown, C. J., and Humberston, J. W. (1985). J. Phys. B 18, L401-406. Bussard, R. W., Ramaty, R., and Drachman, R. J. (1979). Astrophys. J. 228,928-934. Campeanu, R. I. (1981). J. Phys. B 14,157-160. Campeanu, R. I. (1982). Can.J. Phys. 60,615-617. Campeanu, R. I., and Humberston, J. W. (1977). J. Phys. B 10, L153-158. Canter, K. F. (1982). In “Positron Annihilation” (P. G. Coleman, S. C. Sharma, and L. M. Diana, eds.), pp. 138- 141. North Holland Publ., Amsterdam. Canter, K. F. (1984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), pp. 219-225. Plenum, New York. Canter, K. F., and Roellig, L. 0. (1975). Phys. Rev. A 12,386-395. Canter, K. F., Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1972). J. Phys. B 5, L167- 169. Canter, K. F., Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1973). J. Phys. B 6, L201-203. Canter, K. F., Mills, A. P., Jr., and Berko, S. (1975). Phys. Rev. Lett. 34, 177- 180. Casewell, W. E., and Lepage, G. P. (1979). Phys. Rev. A 20, 36-43. Charlton, M. (1985a). Rep. Prog. Phys. 48,737-793. Charlton, M. (1985b). J. Phys. B. 18, L667-671. Charlton, M., and Laricchia, G. (1985). In “Positron(e1ectron)-Gas Scattering”(W. E. Kaug pila and T. S. Stein, eds.). World Scientific, Singapore, in press. Charlton, M., Griffith, T. C., Heyland, G. R., and Lines, K. S.(1979). J.Phys. B 12,633-639. Charlton, M., Griffith, T. C., Heyland, G. R., and Wright, G. L. (1980a). J. Phys. B 13, L353-356. Charlton, M., Griffith, T. C., Heyland, G. R., Lines, K. S., and Wright, G. L. (1980b).J.Phys. B 13, L757-760. Charlton, M., Griffith, T. C., Heyland, G. R., and Wright, G. L. (1983a). J. Phys. B 16, 323-341, Charlton, M., Clark, G., Griffith, T. C., and Heyland, G. R. (1983b).J.Phys. B 16, L465-470. Coleman, P. G. (1985a). Proc. Int. Con/: Positron Annihilation, 7th (P. C. Jain., R. M. Singru, and K. P. Gopinathan, eds.), pp. 3 17- 327. World Scientific, Singapore. Coleman, P. G. (l985b). In “Positron(e1ectron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,VOL. 22

DOUBLY EXCITED STATES, INCLUDING NEW CLASSIFICATION SCHEMES Department of Physics Kansas State University Manhattan, Kansas 66506

I. Introduction Since the birth of nonrelativistic quantum mechanics, the independent particle approximation has served as the backbone of almost all areas of microscopic physics. In atomic physics, the independent electron approximation assumes that, to first order, an atom is made of a collection of independent electrons, and the motion of each electron is determined by an averaged potential due to the nucleus and the other electrons. This approximation, whether it is in the form of the Hartree-Fock model or its equivalents, has been used to explain qualitatively as well as semiquantitatively a wealth of experimentalobservations.Over the last half-century,a major part of the effort in theoretical atomic physics has been devoted to finding different ways of accounting for the deviations of experimental results from the predictions of the independent electron approximation. Different methods, such as many-body perturbation theory, the configuration-interaction (CI) method, and many other perturbative approaches, have been shown to be capable of accounting for these deviations accurately. When the deviation from the prediction of the independent electron approximation is large, as happens in several isolated spectral lines, the situation can often be attributed to localized “interactions” between a few states. Such situations are amenable to the treatment of the configuration interaction method. Since the early observation of the absorption spectra of doubly excited states of He by Madden and Codling (1963, 1965) using synchrotron radiation, it was recognized immediately by Fano and coworkers that a complete understandingof these new states requires a fundamental departure from the conventional independent particle approach. Not only should the spectral observation be explained, but a desirable new approach should also provide 77 Copynght 0 1986 by Academic Press. Inc. All rights of reproductionin any form reserved.

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C.D. Lin

the framework whereby all doubly excited states of atoms and molecules could be studied. In other words, a new approach should supply the proper language such as new quantum numbers, new systematicsof spectral behavior, approximate selection rules, etc., which are also applicable to doubly excited states of other atoms. Thus one of the goals in the interpretation of doubly excited states of He is to provide this language, analogousto the study of hydrogen atoms to provide a proper language for the independent particle approximation. The early photoabsorptionspectra of doubly excited states of He indicated that among the three possible IPoRydberg series that converge to the N = 2 limit of He+, only one series is prominently observed,while a second seriesis weakly visible and a third series is completely absent (Madden and Codling, 1963, 1965).In a later experiment, Woodruff and Samson (1982)measured the photoelectron spectra at higher photon energies. Their results for doubly excited states of He below the N = 3,4, and 5 limits of He+ are reproduced here in Fig. 1. According to the conventional selection rules for photoabsorption, there are 5,7, and 9 possible Rydberg series, respectively,ofdoubly excited states converging to each of the limits. There was, however, only one prominent series observed in each case. Similarly, in the photodetachment of H- for IPodoubly excited states below the H(N= 6) limit, all the resonances observed belong to the same series (H. C. Bryant, 1981; private communication). A desirable theoretical approach should provide not only a method of calculating the position and width of each doubly excited state but also the approximate selection rules for different excitation processes. There are many theoretical approaches which are capable of predicting an accurate position and width of each doubly excited state. These methods, such as the configuration interaction method, the Feshbach projection technique, the close-coupling method, and the complex coordinate rotation technique and others, provided a wealth of “numerical” data which are essential to sorting out the systematics of doubly excited states. The contribution from these calculations cannot be underestimated. This is particularly true for doubly excited states since experimental data are so scarce. Even if these data do exist, the resolution is not good enough to extract their systematics. Furthermore, it seems clear now that some doubly excited states are not easily populated in some experiments. The main limitation of the above-mentioned approaches is that each doubly excited state is calculated separatelywhile experimental data indicate that the selection rule is a property of a series (or a channel). Furthermore, the results from these types of calculations are sometimes unexpected or difficult to explain. As an example, one can predict the approximate positions of doubly excited states by performing a limited CI calculation. In Table I, the results of such a calculation for the IPo doubly excited states

79

DOUBLY EXCITED STATES

I

I

1

N:3 THRES-D

n.3

69

5

4

70

,“ao5-

8

I

1

I

72

71

I I I

K=Z[

5

n=4

6

b

678

7

J

I

73

N=STHRESHOLt

T

N=4 THRESHOLD

I

I

I

,

FIG.1. The cross section for the autoionizing region of He doubly excited states below the N = 3,4, and 5 thresholds of He+. The quantum number Kand the principal quantum number n of the series are indicated (Woodruff and Samson, 1982). TABLE I CI COEFFICIENTS OF THE FIRSTTHREE LOWEST DOUBLY EXCITED STATES OF He ‘P BELOW THE He+(N = 3) THRESHOLD State

Energy (Ry)

3s3p

3p3d

3s4p

3p4s

3p4d

3d4p

3d4f

I 2 3

-0.667 -0.563 -0.554

0.683 -0.003 0.503

0.616 -0.005 -0.551

-0.127 0.630 -0.226

-0.172 -0.630 -0.317

-0.239 0.330 -0.054

-0.203 -0.304 0.231

-0.104 0.068 0.476

80

C.D. Lin

below He+(N= 3) are shown. According to conventional wisdom, one would expect that the wave function of the two lowest states are the linear combination of 3s3p and 3p3d. The calculation shows that this is indeed the case for the lowest state. The second-loweststate, however, is actually mostly a linear combination of 3s4p, 3p4s, 3p4d, 3d4p, . . . , etc. It is the thirdlowest state which is again predominantly a mixture of 3s3p and 3p3d. This example serves to illustrate the limitation of the conventional approaches. When the admixture of many configurations is substantial for a given state, the meaning of configuration for that state is lost. Information about electron correlationsin these approaches is embedded awkwardly in the mixing coefficients.These coefficientsprovide no direct clues as to how the electrons are correlated. One of the goals of studying doubly excited states is to find a new way of characterizing electron correlations. More precisely, we want to find a new set of quantum numbers which characterize the correlations between two excited electrons. We also want to know the physical or geometrical interpretation of these quantum numbers and possible new spectroscopic regularities. In this article, our objective is to present the progress toward this goal up to this time. The study of doubly excited states described in this article is based mostly upon the geometrical interpretation of the motion of two excited electrons. Our major task is to unravel how electrons are correlated by examining the wave functions in hyperspherical coordinates. This coordinate system is particularly suitable for analyzing electron correlations. By assuming that the mass of the nucleus is infinite, the configuration of the two electrons is described by six coordinates. Three of these coordinates are used to describe the rotation of the whole atom. In hyperspherical coordinates, among the three remaining we use one coordinate to describe the size of the atom and the two others to describe the relative orientations of the two electrons. The correlation quantum numbers are related to the nodal structure in these two angles. The rest of this article is organized as follows. In Section 11, we discuss the qualitative aspects of radial and angular correlations. The correlation quantum numbers and the classification scheme are presented in Section 111.This section also contains the illustration of isomorphic correlations of states which have identical correlation quantum numbers and the existence of a supermultipletstructure. After a short digression on computational methods in Section IV, the correlation quantum numbers are re-examined by analyzing the wave functions in the body frame of the atom in Section V. The existence of approximate moleculelike normal modes of doubly excited states and its limited interpretation are also discussed in Section V. In Sec-

DOUBLY EXCITED STATES

81

tion VI, the effects of strong electric fields on the resonances of H- are discussed. Doubly excited states of multielectron atoms are briefly discussed in Section VJI. Several final remarks and future perspectives are given in Section VIII. There are other studies aimed at the understanding of the systematics of doubly excited states. These include the group-theoretical approach (Wulfman, 1973; Crane and Armstrong, 1982; Hemck, 1983, and references therein), the algebraicapproach (Iachello and Rau, 198l), and the analysisof the electron correlation of model two-electron systems (Ezra and Berry, 1982, 1983). The group-theoretical approach also aims at the classification of doubly excited states. All of these approaches treat the correlations of individual states. In the hyperspherical approach the correlation is studied for each channel and thus any state belonging to that channel has similar correlation properties. These other approaches, particularly the group-theoretical approach, complemented the analysis of correlationsin hyperspherical coordinates presented here. A review of the group-theoretical approach has been given by Herrick (1 983). The applicationsof the complex-coordinate rotation method to doubly excited states have been reviewed recently by Ho (1983). The analysis of electron correlations from the hyperspherical coordinates viewpoint has also been reviewed by Fano (1983). References to earlier works can be found in that article. In this review, we concentrate on the progress made since then.

11. Analysis of Radial and Angular Correlations In this section we describe the correlations of doubly excited states as revealed through the examination of wave functions in hyperspherical coordinates. After a brief outline of the basic equations and a discussion of the quasiseparable approximation where the concept of channels is defined, we examine the meaning and the nature of radial and angular correlations for some typical channels. The discussion in this section is limited mostly to L = 0 states. In describing correlations, we always concentrate on the correlation of a given channel rather than that of each individual state. This is possible because the correlationsfor states belonging to the same channel are similar. Graphical display of correlations for each individual state has been explored by Berry and coworkers (Ezra and Berry, 1982, and references therein) using a density function p(r, ,BI2 ,rz)which measures the probability of finding electron 1 at a distance r, from the nucleus and with interelectronic angle BI2 given that electron 2 is at a distance r, from the nucleus.

C.D.Lin

82

A. THEHYPERSPHERICAL COORDINATES To describe the motion of two electrons in the field of a nucleus, six coordinates are needed. One can choose three coordinates, such as the three Euler angles, to describe the overall rotation of the system and the other three to describe the internal degrees of freedom. Let us start with atomic states which have L = 0; their wave functionsdo not depend on external rotational coordinates. The internal coordinatescan be chosen as the distances rIand r2 of the two electrons and the angle el,.It is also possible to replace rl and r2by R and a,where

R = (r: 4- ri)l’z;

a = tan-1(r2/r,) (1) (see Fig. 2). This latter set has the advantage that R specifiesthe “size” of the atom and does not enter into the description of electron correlationsdirectly. Electron correlationsare then described by the two anglesa and OI2 only. We refer to the correlationdepicted by the angle a as radial correlation and to the correlation described by the angle 012as angular correlation. The correlation quantum numbers for characterizingdoubly excited states provide information about radial and angular correlations of the two electrons. For L # 0 states, the overall rotation of the atom has to be considered. Instead of using the Euler anpJes,computationally it is more convenient to use a,i,,and i2,where ii= (ei&) denotes the spherical angles of electron i, as the five hyperspherical angles. To describe the internal correlations for L # 0 states, the rotation of the atom will be averaged (see Section 111,C). At this point it is convenient to introduce the Schrodinger equation for two-electron atoms in hyperspherical coordinates. Denoting the five angles a,i,,i2collectively by Q, the Schrodinger equation for two-electron atoms, written using atomic units, is (Macek, 1968; Lin, 1974b) R2

R

‘I

FIG.2. (a) Diagram ofthe two-electronconfiguration. (b)Diagram to illustratethe relation between Cartesian and hypersphericalradial coordinates (Fano and Lin, 1975).

DOUBLY EXCITED STATES

83

where A2 = -

d

1

sin2 a cos2 a iii

(

is the square of the grand angular momentum operator and

c=----2

cos a

z +

sin a

1

(1 - sin 2a cos 812)1/2

(4)

is the effective charge. This effective charge C includes both the electronnucleus and electron -electron interactions. In Eq. (4), 2 is the charge of the nucleus. Equation (2) shows that the eigenvalue of A2/R2acts like a centrifugal potential barrier for the simultaneous penetration of the two electrons into the small-R region. It depends not only on the orbital angular momentum of each electron, but also on the degree of radial correlation as represented by the a-dependent operator in Eq. (3). The effective charge Cdepends only on the relative coordinates a and 4,. In Fig. 3 we display the relief map of C on plane for 2 = 1. The ordinates represent the potentials at R = 1. the (a,OI2) In the limit of a 0 (or a n/2), the potential surface has a sharp drop caused by the electron - nucleus attraction. This potential valley corresponds

- -

-C

a (deg)

FIG.3. Potential function C(a,B,,)in Rydberg units for a pair of electrons in the field of a proton.

C.D. Lin

84

to the case in which one electron is near the nucleus and the other is far out. In the region where rl = r,, which corresponds to a = 45”, the potential energy depends critically on whether 1 9 ,is~ approximately 0 or II.When 012= 0 and a = 45 the two electrons are nearly on top of each other where the electron -electron repulsion causesthe sharp spike seen in Fig. 3. We also note that a = 45” and O,, = 180” is a saddle point; the potential is unstable away from a = 4 5”, while it is stable at 012= 180”along the coordinate &. The Schrodinger Eq. (2) can be solved by expanding the total wave function as O,

Y$R,fi) =

2 F3R)@p(R;fi)/(R5/2 sin a cos a)

(5)

/r

where p identifies the channel and n denotes the nth state within that channel. The channel function @,,(R;fi)satisfies the differential equation @JR;fi)= U,(R)@,(R;R) (6) and the hyperradial function F(R) satisfies the coupled equations,

where the coupling terms W are defined as

By dropping all the coupling terms and keeping only the diagonal terms, Eq. (7) becomes

-+-dRz

4R2

UJR)

+ W,(R) + 2E,,

(9)

Notice that the second-order diagonal W , ( R )term is included in Eq. (9) as part of the effective potential. This term IS usually excluded in the BornOppenheimer expansion in diatomic molecules, but it is included in the “adiabatic approximation” of Eq. (9). Under this approximation, the wave function for the nth state within channel p is given approximately by

Y;(R,n)= Flf(R)@p(R;R)/(R5/2 sin a cos a)

(10)

The adiabatic approximation was first introduced by Macek (1968) to study doubly excited states of helium. The energy levels calculated from this approach were found to be in good agreement with experimental results and with other calculations. Later work was directed at understandingthe correlation properties hidden in the “channel functions” @JR;Sl).The major task

DOUBLY EXCITED STATES

85

of understanding and classifyingelectron correlationsis then to untangle this multivariable function in appropriate display and to sort out the order and regularities. To this end, sectional views of the channel function @,(R;Q) on the relative angles a and 012are appropriate. We will proceed with simple examples and then to the complete spectra of doubly excited states. For simplicity, we will first consider L = 0 states only.

B. ANGULARCORRELATIONS Angular correlation is quite familiar. The wave function for an L = 0 two-electron state has the general form v(R,a,42) =

c

W Y 1 ,~2)Y/mV1 f2)

( 1 1)

/

where

Therefore, if the two-electron state can be designated as s2 or any linear combination ofss’, there is no OI2dependencein the wave function and there is no angular correlation between the two electrons. If it is designated asp2or pp’, then the wave function is multiplied by an overall cos OI2 factor. Accordingto the traditional picture, correlation is defined as the deviation from the prediction of the independent particle approximation. Therefore, the angular correlation for a state designated by pp’, for example, is defined to be the deviation of its wave function from the cos 0,2dependence. We will not adopt this definition. Instead, we describe how electrons are correlated. Thus if the 012distribution of agiven state is well described by P,(cos 42), then that state can be designated by the independent particle notation l2 or 11’. We will search for new designations for all doubly excited states where the independent particle approximation fails. C. RADIAL CORRELATIONS Similarto angular correlations,radial correlationsare characterized by the distribution of the wave function in the hyperangle a.In the foregoing discussion, no distinction has been made between singlet and triplet states for angular correlations;their difference comes mostly in radial correlations. Radial correlation is less familiar. For the purpose of illustrating radial correlations, we examine the solution of the Schrodinger equation by neglecting the 4, dependence in the potential. Under this approximation, I,

C.D.Lin

86

and 1, are good quantum numbers and states can be labeled as 1s2,ls2s,and 2s2,etc. Ifthe wave functions are approximated as in Eq. (lo),then these two variable functions can be displayed graphically. In Fig. 4 we show the absolute value of the wave functions for 1s2s lSe,2s2 lSe,and 2s3s ISe(we use the independent-particle designation here) of He on the (r, 7,) plane. We notice that the ls2s ISehas a circular node, corresponding to R, = constant. The wave function for this state is concentrated in the region where rl << r, and in the region where r, << r, (by symmetry). In the rl << r, region, the wave function along r, for a given r1 behaves like a hydrogenic 2s wave function. The wave function has noticeable amplitudes in the r, = r, region only when R is inside R,. For 2s2 'Se, there are no circular nodes, but there are two radial nodal lines running almost parallel to the rl and the r, axes, each one corresponding to a = constant. For this state, the wave function has large amplitudes mostly in the region where r, = r2.In this example, the ls2s ISehas a node in the hyperradial coordinate R and no node in the hyperangle a. For 2s' lSe,there is no node in R but one node at a,, where a, depends on R and lies between 0 and 45". (By symmetry the other nodal line is given by 90" - a,.) We can differentiate each state by the nature of its nodal lines. Let nR and n, denote the number of nodes in the wave function for the R (0 < R < m) and a (0 < a < 45") coordinates, respectively; then ls2s ISe has (n,,n,) = (1,O) while 2s2 ISe has (n,,n,) = (0,l). The ground state, usually designated as ls2 ISe,has (nR,n,) = (0,O). Using this notation, the 2s3s ISestate has (nR,n,) = (1,l); so that this state has one node in R and one node in a. This is indeed the case, as shown in Fig. 4c. So far we have discussed ISestates only. Since the wave function for a lSe state is symmetric under the interchange of the two electrons, the wave function is Symmetric with respect to a = 45". For 3Se states, the wave function has a node at a = 45 This node is fixed at a = 45 and does not change with R.To account for the fact that the wave function is symmetric or antisymmetric with respect to a = 45 it is convenient to introduce a superscript A (= 1 or - 1). The superscript A is not an independent quantumnumber, since A = (- l)s for L = 0 states; nevertheless it helps to bring out the symmetry property in the a coordinate with respect to a = 45 '. Thus all the IS states have the new designations of (n, ,n,)+ and all 3Sestates have ( n R ,TI,)- designations. Since ls2s 3Seis the lowest 3Sestates, it is given by (0,O)-, indicating no node in R nor in a except for the fixed node at a = 45 '. In terms of the "total" number of nodal lines, both ls2s ISe and ls2s 3Se states have one nodal line; the nodal line for the former is R = constant and for the latter is a = 45 '. From this, it is clear that 2s3s 3Sehas the designation of (0,1)-. In Fig. 5 , we see that the corresponding density plot shows that the O.

+

O

O ,

DOUBLY EXCITED STATES

87

(0.0)

FIG. 4. Square root of the volume chargedensity distribution of helium plotted in the (rl ,r2)plane. The dependence of wave functions on OI2 has been neglected. (a) for 1.92s ISe;(b) 2s2 I S'; (c)2s3s I S ' .

C.D. Lin

88

0

5

10

15.

20

25

30

r2

FIG.5. Same as Fig. 4 except for the 2s3s 3Scof He.

number and nature of the nodal lines are consistent with the (0,l)- designation. By neglecting the 8,2dependence in the potential, an L # 0 state can be expressed as Yl,lzLM= ~(~)1g(a)”Y,,,zLA4v1 YF2)

+ (- l)rl+rz-L+s g(W2 - a)”y,*,,Ld~lY~2)1 (13) in the quasiseparable approximation. In Eq. ( 13), the symmetry requirement with respect to the interchange of the two electrons does not impose any condition on the function g(a),since the symmetry is accounted for by the

0

FIG.6. Channel functions g(a;R)for the two IJPo channels of helium converging to the N = 2 limit of He+. Shown are the [I, ,I,] = [0,1] components of each channel. The dipole componentofthe electron-electron interactionsis neglected.(a) ShowsA = I type behavior; (b) shows A = - 1 type behavior in radial correlations.

+

DOUBLY EXCITED STATES

89

second term on the right. States where the function g(a)itselfdoes not have a well-defined or approximate nodal or antinodal structure at a = 45” are assigned A = 0. All singly excited L # 0 states have A = 0. For L # 0 doubly excited states, in addition to A = 0 channels, there are channels where g(a) exhibits near-antinodal or nodal structure at a = 45”.These channels are classified with A = 1 and A = - 1, respectively (Lin, I974b). For example, the two ISP0 channels of helium converging to the N = 2 limit of He+ have these behaviors. By neglecting the OI2 dependence in the potential, we show in Fig. 6 the [I,,12] = [0,1] component of the channel functions. The upper figure shows approximately antinodal structure at a = 45“,similar to the channels. The lower figure shows an approximate node at an angle close to a = 45 ”, similar to the - channels. This approximate +/- symmetry is one of the most striking features of doubly excited states.

+

+

D. RADIALAND ANGULARCORRELATIONS In discussing radial correlations we purposely neglected angular correlations for simplicity. However, angular and radial correlations are not separable. Consider L = 0 states in the quasiseparable approximation: All the information about electron correlationsis contained in the channel function @(R;a,O,,).To show the correlation pattern of two excited electrons, we exhibit the surface densities on the hypersphericalsurface, R(R) = constant, on the (a,O,,) plane. by displaying plots of I@(R;a,OI2)I2 A few general remarks will be helpful in understandingthe structure of the charge-densityplots to be given below. All the channel functions solved from Eq. (6) at a given value of R are orthogonal, corresponding to the surface harmonics on the R(R) = constant surface. The higher harmonics are orthogonal to the lower ones with an increasing number of nodal lines on the (a,OIz) plane. In Eq. (6), the channel function @(R;a,OI2) and the eigenvalue U(R) depend not only on the kinetic energy opefators, but also on the Coulomb interactions between the three charges. To avoid large kinetic energies, the channel functionsmust be smooth with respect to a and OI2and possess few nodal lines. To achieve lower potential energies, the electron nucleus interaction favors the small-a (or a = ~ / 2 )region, while the electron-electron repulsion term favors the region where a = n/4 and O,, = n. Thus the excitation energies U(R)and the pattern of electron correlations are “decided” by these competing factors. The lowest channel is “allowed” to have all the favorable factors at a given R, while the higher channels approach these favorable factors under the constraint of orthogonality to the lower ones. These constraints and the nature of Coulomb potentials set up the pattern ofelectron correlationsfor doubly excited states.

C.D. Lin

90

We first illustrate how the correlation pattern for a given channel evolves as the hyperradius R changes. In Fig. 7 we show the potential curve U(R)for for four the ground channel of H- and the surface plots of I@(R;a,012)12 values of R. At R = 1 and 2, the kinetic energy term, which is proportionalto l/R2,is large and the charge cloud spreads over the whole (a,012)plane. Along the ridge, a = 45', the two electrons tend to stay closer to OI2 = 180". At larger R, R = 4 and 8, the potential energy term dominates so that the two electrons tend to stay near small a (or a = 90"),where the electronelectron repulsion is small. Therefore the channel function becomes nearly independent of OI2. This lack of angular correlation is quite evident in the density plot for R = 8. To get an estimate of how important the angular or radial correlations are for a given state for this channel, it is necessary to consider the hyperradial wave function of that state. For example, if the state has large amplitudes in F(R) for small R , then the angular correlation (or the deviation from the

1

2

3

4

5

8

7

R (OIL)

FIG. 7 . (a) Hyperspherical potential curve for the ground ISechannel of H-.(b) Surface charge distribution for the ISeground channel of H- plotted on the (a$,,)plane at selected values of R.

DOUBLY EXCITED STATES

91

independent particle approximation) is large. If the amplitude F(R) for the state is mostly in the large-R region, then there is little angular correlation, since in the large-R region the channel function is similar to that shown for the R = 8 plot, which shows little angular correlation. We next discuss the correlations for the two 'S'doubly excited channels of H- that converge to the N = 2 limit of H. The two potential curves are shown in Fig. 8a; they are labeled as ( 1,O)+ and (- 1,O)+ channels. The labeling will be explained in the next section. For the moment we note that the (1,0)+ channel has an attractive potential well while the (- 1,O)+ channel is completely repulsive. The surface charge-density plots for the two channels are given in Fig. 8b at R = 8, 12,20. It is obvious that the correlation patterns for the two channels are quite different. They are also quite different from the

a

Rb.u.1

FIG.8. (a) Hyperspherical potential curves for the two ISechannels which converge to the N = 2 of H. (b)Surface charge-densityplots for the two channelsat the values ofR shown. Note the difference in the orientation of the figures along the two columns.

C. D. Lin

92

ground channel shown in Fig. 7. The ( 1,O)+channel has large charge densities in the la1-ge-8,~region; it also has a nodal line near small a (and, by symmetry, another one near a = 90'). For a given value of R, say R = 8, when the ground channel occupies the small-a region (and the a = 90' region), its amplitudes are vanishingly small in the a 45 ' region. At this same value of R, we notice that the ( 1,O)+ channel occupies most of the large-&, region of the (a,&,)plane not occupied by the (O,O)+channel. By concentrating the charge distribution in the a 45' and large-O,, region, the (1 ,O)+ channel minimizes the kinetic energy and the electron-electron repulsion. The repulsive (- 1,0)+ channel exhibits charge distribution mostly in the 0 < 4, < 90" region. The two electrons tend to stay on the same side of the nucleus and thus experience a large electron-electron repulsion. This region, however, is still preferable under the circumstances. Forcing the two electrons to the large-&, region would require additional nodal lines, which would increase the expectation value of the kinetic energy and the excitation energy U(R). As R increases, we notice that the major change in the channel density plots is that the density in the middle a = 45" region drops while the 4, dependence remains nearly constant. The drop in the a = 45 ' region occurs when the two electrons in that channel become confined in the two potential valleys. With this type of R dependence in mind, we can now look at the correlations of higher channels for a given value of R only. In Fig. 9 we show the charge-density plots at R = 20 for the three ISedoubly excited channels of H-that converge to the N = 3 limit of H. The three channels are labeled (2,0)+, (O,O)+,and (- 2,0)+. We note that the charge-density distribution for the (2,0)+ channel is quite similar to that for the (1 ,O)+channel shown in Fig. 8b except that the (2,0)+ channel has a sharper structure around the Wannier point (6= 45' and OI2 = 180'). The (O,O)+channel has a pronounced peak = 90°,in addition to some density in the large-O,, region. The near (- 2,0)+channel is marked by a large charge density in the ~ m a l l - 8region. ,~ One can continue this type of display for doubly excited states that converge to the higher channels. It is obvious, however, that among the channels that converge to a given hydrogenic Nlimit, the charge density for the lowest channel tends to peak at 4, = 180°,while the highest (or the most repulsive) one tends to peak near 4, = 0 and the intermediate channels occupy the intermediate-& region. Physically this means that the most energetically stable state is the one where the two electrons are on opposite sides of the nucleus. Our discussions so far in this subsection have dealt with ISestates only. The different channels presented differ only in their angular correlations. For 3Sestates, the lowest channel is labeled (0,O)-, the two channels that converge to the N = 2 of H are ( 1,O)- and (- 1,O)-,and the three channels

-

-

DOUBLY EXCITED STATES

N

93

FIG.9. Surface-density plots at R = 20 for the three ISechannels of H-converging to the limit of H. Note the difference in the orientation of the last figure.

=3

that converge to the N = 3 of H are (2,0)-, (0,O)-, and (- 2,0)-.The difference between the corresponding ISeand 3Sechannelsis in the radial correlation. For 3Sechannelsthe symmetry condition is such that the charge density has to vanish at (Y = 45'. Thus, for example, the ( 1,O)+ ISeand ( 1,O)- 3Se channels have a similar OI2 dependence; i.e., they have similar angular correlations, but different radial correlations; the wave function at a = 45" is an antinode for ISeand a node for 3Se(Lin, 1982a).

E. THEVALIDITY OF THE QUASISEPARABLE APPROXIMATION At this point we will make a short diversion to discuss the question of the validity of the adiabatic approximation (Lin, 1983a),which was used in the study of doubly excited states in hypersphericalcoordinates. In the conventional Born - Oppenheimer approximation for diatomic molecules, quasiseparability was often attributed to the small ratio of the electron mass to the mass of the nuclei. The corresponding ratio in two-electron problems is unity. Therefore it is not obvious why one can use the quasiseparable approximation. We emphasize that the reason for the validity of the quasiseparable ap-

C. D.Lin

94

proximation is dynamical in origin. It is due to the large difference in the quantization energies along different coordinates. This quasiseparability is independent of the choice of hyperangles and is not limited to two-electron problems. In recent years, it has been established that many atomic and molecular problems can be solvedin the quasiseparableapproximationif the problems are expressed in hyperspherical coordinates (Lin, 1986; Manz, 1985). For two-electron problems, it is possible to check if the wave functions calculated using different approaches resemble those calculated using hyperspherical coordinates in the adiabatic approximation. This has been examined for the configuration-interaction(CI)wave functionsof Lipsky et al. (1977). (See Lin, 1983a.) If we rewrite the lv3Se CI wave function y(r,,r2) in hyperspherical coordinates, then Y(r,

,r2) = Jx~)@;(R;a,~l,)

(14)

where state n belongs to channelp. In Eq. (14), @;(R;a,B,,)is normalized on the surface at R = constant. From the known CI wave functions, both F,(R) and 0;(R;a,012) can be determined. Here we consider the three lowest lSe states of helium belonging to the ( 1,O)+ ISechannel, which lies below the He+ ( N = 2) limit. The hyperradial functionsdeduced from Eq. ( 14)are shown in Fig. 10. These functions behave as expected: The lowest state does not have any node in R, while each higher state acquires one more node in R. In Eq. (14), our notation indicates that we do not assume that @;(R;cu,5,)is independent of n. In the quasiseparableapproximation in hypersphencal coordi0.6

r

I

I

FIG.10. R-weighted hyperradial wave functions of the three lowest CI states of the (1,O)J below the N = 2 limit of He+. The curves are shown in solid lines in regions where the angular part of the CI wave function has a large overlap integral (>95%)with the adiabatic channel function. In regions of R where the overlap is less than 95%, the curves are shown as dashed lines. lSeseries ofhelium

DOUBLY EXCITED STATES

95

nates, as indicated in Eq. (lo), each wave function is given by F,,(R)@:(R;a,012).We can calculate the overlap integral

I = ( (P:(R;~,o,,) i q qRP,o,,) ) (15) as a function of R to determine the region where the two functions differ. We indicate the results in Fig. 10. If the overlap [ Eq. ( 15)] is larger than 95% in that region of R, the curves are shown in solid lines. If the overlap is less than 95%, the curves are shown in dashed lines. From Fig. 10 we notice that the overlap is larger than 95% in the region where the hyperradial function F(R) is larger. This clearly illustrates that wave functions calculated from other approaches, when expressed in hyperspherical coordinates, also exhibit quasiseparability in the region where the charge density is large. We can also display the correlation patterns of wave functions calculated using other approaches using the conversion equation [Eq. (14)]. In Fig. 1 1, we show the surface charge densities of the lowest state of each of the (2,0)+, (O,O)+, and (- 2,0)+ 'S'channels of He which lie below the He+ (N = 3) limit calculated using the CI method (Lipsky et al., 1977). These surface plots are quite similar to those shown in Fig. 9 for H-.

(-2 .of R.16

0

FIG.1 1. Surfacedensity plots for the lowest states of each of the three ISeRydberg seriesof helium calculated from the CI approximation.These plots are similar to those shown in Fig. 9, which were calculated using the adiabatic approximation.

96

C. D. Lin

111. Classification of Doubly Excited States In this section we shall describe the classification of doubly excited states in terms of a set ofcorrelation quantum numbers, K, T, and A. The enumeration of these quantum numbers and their approximate physical meaning will be given. A more precise mathematical definition of these correlation quantum numbers will be postponed until Section V. Surfacecharge-density plots will be used to help visualize the correlation patterns described by these quantum numbers. It will then be shown that states having identical correlation quantum numbers have isomorphic correlations. This isomorphism is the underlying reason for the existence of supermultiplet structure of doubly excited states. The last subsection discusses how the independent electron picture fits into the present classification scheme. A. THECLASSIFICATION SCHEME

In the present scheme, a given state of a two-electron atom is designated by the notation .(K,T)A, =+lLX,where L, S, and n are the usual quantum numbers, Nis the principal quantum number of the inner electron, and n is the principal quantum number of the outer electron. The spin-orbit interaction is not considered throughout our discussion but can be easily included in a perturbative treatment. A given channel or a Rydberg series p is described by the notation / = (K,T)$ =+'L*. Here the principal quantum number Ndenotes the hydrogenic pnncipal quantum number in the dissociation limit. The rules for the assignment of K, T, A, and n for a given L, S, N, and n are discussed below. 1. Assignment ofK and T

Following Hemck and Sinanoglu (1979, the possible values of K and T for a given N, L, and n are determined by

T = O , 1,2,

. . . ,min(L,N-

K=N-l-T,N-3-Ty..

1)

. ,-(N-1-T)

(16)

For states where n = (- l)L+l,T = 0 is not allowed. Notice that the assigned values of K and T do not depend on S. Roughly speaking, T is the projection of the total angular momentum L onto the interelectronic axis and K = - ( r , cos 012)

(17) where r, is the radius of the inner electron. These two quantum numbers K

97

DOUBLY EXCITED STATES

and T were originally used by Herrick and Sinanoglu ( 1975a,b)to characterize approximate doubly excited state wave functions for intrashell states. Based upon a group-theoretical analysis, they showed that the configuration-interaction wave functions for doubly excited states can be approximated by “doubly excited symmetry basis” (DESB) functions. The validity and the limitation of DESB functions for representing doubly excited states were examined by Lin and Macek (1 984).

2. Assignment of A This radial correlation quantum number A was supplementedto emphasize the radial correlation of the two electrons (Lin, 1983d, 1984). Its meaning has been illustrated in Section II,C in connection with the model problem, where angular correlation was neglected. The quantum number A can take values of 1, - 1, and 0 only. Both A = 1 and - 1 states can have large amplitudes on the potential ridge. We stress that the A = 1 channel has an antinodal structure at or near a = a/4 (this statement will be made more precise in Section V), while the A = - 1 channel has a node at or near a = n/4. Electrons in the A = 0 states are confined in the two potential valleys. These states are similar to singly excited states. The radial correlation quantum number A is not independent of Kand T for a given L, S, N, and a. It is given by the following simple relations (Rau, 1984)

+

A

= a(-

+

IF+==a(-

lF+N-K+l

+

if K > L - N

if K S L - N

A=O

(18)

With the relations in Eqs. ( 16) and ( 18), all the correlation quantum numbers K, T, and A for statesconvergingto a hydrogeniclimit Ncan be assigned. For L 2 2N - 1, all the channels have A = 0. In terms of these correlation quantum numbers, K, T, and A, all the doubly excited states of two-electron atoms can be uniquely designated. From the correlations characterized by these quantum numbers, it is possible to understand the photoabsorption data systematically.

3. Selection Rules for Photoabsorption According to the present recommended notation, the three ‘PoRydberg series of He below the N = 2 limit of He+ are (0,1,:) ( 1,O);, and (- 1,O);. They are to replace the 2snp 2pns, 2snp - 2pns, and 2pnd notations used by Cooper el al. ( 1 963). The designations of Cooper et a/. emphasize radial correlations only and cannot be generalized to other doubly excited states.

+

C.D. Lin

98

From the meaning of the correlation quantum numbers, one can deduce from the notation that for states belonging to the (0,I); channel, the interelectronic angle 8,: is nearly 90°, and the two electrons have in-phase radial oscillations, meaning that both electrons approach or leave the small-R region simultaneously. For states belonging to the (1 ,O)y channel, the two electrons are on opposite sides of the nucleus with large probabilities near OI2 = 180°, but they have out-of-phase radial oscillations, meaning that when one electron is approaching the nucleus the other is moving away from the nucleus. For states belonging to the (- 1,O); channel, the two electrons are confined in the potential valleys; there is no radial correlation although the two electrons tend to stay on the same side of the nucleus. The first photoabsorption data for the excitation of helium doubly excited states (Madden and Codling, 1963, 1965) indicated that only the (0,l)t channel is prominently excited, the (1 ,O): channel is barely visible, and the (- 1,O)$ channel is completely absent. From the data of Woodruff and Samson ( 1982),as shown in Fig. 1, the prominent series below each of the N = 3, 4, and 5 series, respectively, are the (1, l):, (2,1)Q,and (3,l)t channels. There are some indications that the (- I ,I): and (0,l): channels are also slightly populated. Since the ground state of He belongs to the (0,O)t channel, these experimental data indicate that the selection rule for photoabsorption is AA = 0 and AT = 1,and the most probable K for a given Nis the maximum Kfortheallowed T = l,i.e.,K=N-2. 4. Assignment of n

To be consistent with the principal quantum numbers used in the independent particle model, the smallest principal quantum number nminof the outer electron is chosen as follows.

+

(a) The lowest n for all A = 1 channels is nmia= N. (b) The lowest n for all A = - I channels is nmin= N 1. (c) The lowest n for the lowest A = 0 channel is n, = N 1, and successive higher A = O channels have nmin increases by one unit for each A K = - 1. Channels having identical K but different T have the same nmin.

+

+

+

Accordingto these rules, all intrashell states have A = 1 with n = N. The lowest doubly excited states for each of the five 'Pochannels below N = 3 are 3(1,1):, 4(2,0);, 3(- l,l)$, 4(0,0);, and 4(-2,0)8. These rules also apply to high-angular-momentum states where all the states belong to A = 0. For have the following lowest states: 4(2,0)!, example, the six channels for 1,3H0 &l,l)!, 6(0,2)8,6(0,0)8,,(- l,l)g, and &-2,O)!. Recall that these states are 3s6h, 3p5g, 3p7i, 3d4f; 3d6h, and 3d8j, according to the independent particle picture. Therefore, the lowest n's are 4, 5 , 6, 6, 7, and 8, as predicted by

DOUBLY EXCITED STATES

99

rule (c) above. The (K,T)" designation is preferable to the independent particle notation even for the A = 0 states because it provides information about angular correlations; there is no such information available in the independent particle description. The number of nodes in the hyperradial function F(R)for a given n of the outer electron is given by n - n- ,where nmiais the minimum n of the given channel.

B. POTENTIAL CURVES In the quasiseparable approximation in hyperspherical coordinates, the wave functions are given by F;(R)@,(R;R).The channel function @,(R;R) contains information about electron correlations, which is reflected in the shape of the channel potential U(R).Now that the channels are identified by p = (K,T)A,2S+1L", channels with identical correlation quantum numbers K, T, and A should have nearly identical correlation patterns and nearly identical potential curves if the correlation determines predominantly the energies of the channel. In this subsection, we discuss the potential curves. 1*3P0, and 1,3Dethat In Fig. 12 we show the potential curves of He converge to the He+(N = 3) limits. Similar curvesfor higher L's are shown in Fig. 13. Only channels that have n = (- l)L are shown. Each potential curve I p 3 S e ,

-0. I 0 -0.12 -0.14 -0. I 6

-0.20

u)

W > K

3 -0.12

F

' t-

-0.14

-0.16

-0.18 -0.20 -0.22

10 I6 22 28 3 4 4 0 4 6

16 22 28 34-46 R (0.u.)

16 22 28 344046

FIG. 12. Potential curves for all the '.'Se, 1.3P0,Iv3Dcchannels for He that converge to He+(N = 3). Curves are labeled in terms of K, T,and A correlation quantumnumbers. Reduced units with Z = 1 are used.

C.D.Lin

100 -0.10 -0.12

-0.14

9 a 2 u

-0.18

-I

a

5

-0.14 -0.16 -0.18

-020 10 16 2 2 2 8 3 4 4 0 4 6 16 2 2 2 8 3 4 4 0 4 6 16 2 2 2 8 3 4 4 0 4 6 R(o.u.1

FIG. 13. Same as in Fig. 12 except for l J F O ,',3Ge,and 1.3H0channels.

is labeled with K, T, and A quantum numbers. We first note that curves which have identical K , T, and A quantum numbers are quite similar and are nearly degenerate. (All the calculations for He are done using reduced units with Z = 1 .) The assignment of correlation quantum numbers for each manifold follows these rules: (1) For a given L, S, n, and N, find the allowed combinations of (K,T)A from Eqs. (1 6) and (18). (2) Order the asymptotic potential curves from the bottom starting with the maximum allowable Kand then in order of decreasingK. If there is more than one values of Tfor a given K, order from below according to decreasing values of T. (3) At small R,the lower or the more attractive curves belong to the A = 1 channels. Among the channels, the large K goes with the lower curve and for a given K , the larger Twith the lower curve. First all the A = 1 curves are assigned, then the A = - 1 channels, and last the A = 0 using the same rule for each A group. (4) Connect curves with identical Kand Tin the two regions. Only the and - curves are allowed to cross.

+

+

+

+

The possible number of channels of doubly excited states for states converging to higher He+(N) limits is quite numerous. As an illustration, we show in Fig. 14 the potential curves of 's3Se,133P0,1,3De,and 1*3F0 states of He below the He+(N = 4) limits. There are 4,7,9, and 10 channels for L = 0, 1, 2, and 3, respectively. The correlation rules discussed above can be used to construct the "diabatic" curves shown. Notice that the curves are dis-

+

DOUBLY EXCITED STATES

21

33 45 57 69

101

27 39 51 63 75

R(o.u.)

FIG.14. Same as in Fig. 12 except for the l s 3 S e , I.,Po, 'JDe,and I.'Fo channels that converge to the N = 4 limit of He+. (-) The channels; (---) the - channels; and (- -) the A = 0 channels. Labels for all the +channels are indicated. The K and Tquantum numbers for - channels for singlets are obtained from the K and Tof -tchannels of triplets, and vice versa. The K and T of A = 0 channels are the same for singlet and triplet.

+

-

tinctly more attractive and are capable of supportinglow-lying bound states. Experimental data on these high-lying doubly excited states are very scarce. The correlation rules presented here can be understood more rigorously in a later analysis (see Section V). Qualitatively, at large R,the electrons for all states are confined to the potential valleys where radial correlation is not important so the ordering of potentials at large R is independent o f A .If the state has a large K, the angle OI2 between the two electrons is large, producing a smaller electron -electron repulsion which in turn results in lower U(R).At

C.D. Lin

102

small R,radial correlation is more important. Channels that have antinodal structure at a = n/4 have lower U(R).For a given A, a large Kagain relates to smaller electron - electron repulsion and thus lower U(R).The crossing between and - channels is due to the change of the relative importance of radial and angular correlations as R changes.

+

C. CORRELATION PATTERNS AND ISOMORPHISM To display the correlated motion of two electrons in a given channel, we of the channel have to exhibit the surface charge distribution l@p(R;i2)(2 functions, similar to those shown in Figs. 7-9. For L # 0 channels, the channel functions depend on five angles, but three of these angles describe the rotation of the whole atom. To exhibit the internal correlation structure, we calculate the averaged surface charge densities (Warner, 1980; Ezra and Berry, 1982; Lin, 1984) CJ,,(R;~,B,~) = ( CDp(R;S2’)l&(~’ - a) ~ ( C OB;2S - cos (?,2)l@p(R;i2’))(1 9) The explicit expression of opwas given in Appendix A of Lin ( 1984). With the definition of averaged surface charge density given by Eq. (19), we can now display the correlated motion ofany two-electronchannel functions for arbitrary L, S, and R. Channels which have identical correlation quantum numbers have isomorphic correlation patterns. To show this, we display in Fig. 15 and (2,O)z channel for ISe,3P0,IDe,and 3F0at R = 20. Refemng to Figs. 12 and 13, at this value of R the potential for each channel is near the minimum. From Fig. 15, it is obvious that the correlation patterns for all

FIG.15. Surfacecharge-density plots for the (2,O)l channels of ISe,3P0,IDe, and 3F0of He at R = 20.

DOUBLY EXCITED STATES

103

FIG.16. Surface charge-density plots for the ( 1,l): 'I)=,( 1,1 ID', ( 1,l )f *Po, and ( 1,1 )T 'Poat the values of R indicated. Notice that all four plots have similar 0,2dependenceas the K and T quantum numbers are identicat. The A = structure at (Y = n/4.

+ and A = - channels differ in the nodal

these four channels are quite similar. They all exhibit a peak at a = 45", 4,= 180", with little charge concentration for 4,< 90". The difference is mostly in the region of small a (and a = a/2). Further remarks on the origin of the difference will be given in Section V,C. We note that for a given N , channels which have K = N - 1 always have maximum densities in 8,, at eI2= 1800. In Fig. 16 we show two more examples of isomorphism. The correlation patterns of ( 1,l): 3De,( 1,l): IPO, ( 1,l)? IDe, and (1, l)? 3P0are displayed at the values ofR given. The angle 4, where the density is maximum occurs at OI2 = 120". The nodal structure near a = 45 for each channel is consistent with the or - values assigned for the quantum number A for that channel. Since the K and Tare identical for all these channels, we note that the 4, distribution is identical for all of them. Surface plots for other values of K, T, and A can also be pictured. As the value of K decreases, the charge distribution shifts toward the smaller 8,, region. For negative values of K, the two electrons are mostly on the same side of the nucleus. The surface plots of A = 1 channels have antinodal structure and the A = - 1 channels have a fixed nodal line at a = 45". In A = 0 channels the two electrons never reach the potential ridge region.

+

O

+

D. SUPERMULTIPLET STRUCTURE According to the quasiseparable approximation in Eq. (10) for the solution of the two-electron wave functions, the approximate energy eigenvalues

C,D. Lin

104

for a given channel p are obtained by solving the one-dimensional equation [Eq. (9)] with the channel potential U(R). Channels that have identical correlation quantum numbers (K,T)Abut different L, S, and a,as we have shown in Section III,B, have nearly degenerate potential curves. This neardegeneracy in U,(R) gives near-degenerate eigenenergies. Thus doubly excited states exhibit new spectroscopic regularities if the energies are ordered according to correlation quantum numbers. This regularity was first discovered by Herrick and coworkers for intrashell doubly excited states from a group-theoretical analysis. It can be interpreted in terms of the moleculelike rovibrational modes. We will come back to this interpretation in Section V. In Fig. 17 we plot the effective principal.quantum number n* of He** below the He+(N= 3) limits versus the correlation quantum numbers, (K,T)A.The and - statesare grouped separately. Two new spectral regularities are obvious: (1) The rotorlike structure of states which have the same (K,T)” but different L, S, and a. The “string” for each rotorlike series is determined from Eqs. ( 16) and ( 18). For a given K, T,and N, the allowed values of L for a rotor series is L = T, T 1, . . . ,K N - 1. Whether the rotor series is a or a - series is determined by Eq. (1 8). There are situations where the number of states in a given string is small. For example, there is only one member for each of the (0,2)$ IDe, (- 1,1): ‘Po,and (- 2,O)f ISeseries. (2) There is a repetition of (K,T)+and (K,T)- rotor structure. The - rotor series for a given (K,7‘)- can be obtained from the (K,T)+rotor series simply by interchanging the spins. The energy levels shown in Fig. 17 were taken from the extensive CI calculation of Lipsky et al. (1977). These authors have classified the levels into different series according to the calculated quantum defects. Some of their assigned classifications were changed in order to preserve the regular rotor structure shown. In Fig. 17 we also notice that states which have identical K, T, A, N, and L but different S and a are nearly degenerate. The small splitting is called T doubling. We notice that the n = (- l)L+l state always has slightly lower energy. The origin of this behavior will be explained in Section V,F. Doubly excited states that converge to the higher thresholds exhibit more pronounced rotor structure. As an example, we display in Fig. 18 the energy levels of H-** that lie below the H(N = 5 ) thresholds. These data were taken from the calculationsof Ho and Callaway (1983). By assigning(K,T)+quantum numbers to these states and ordering the states according to the (K,T)+ quantum numbers, the energies of these intrashell states are seen to exhibit rotorlike structure. The length of each “string” is much longer in this example. In fact, some higher members of the group have not been calculated.

+

+

+

+

DOUBLY EXCITED STATES

105

5.0

4.5 *c

5

s

4.0

2

2

3.5

3 0 1 -

0

3.0

L

5

2.5

FIG.17. Effectivequantum number n* grouped according to the (a) (K,T)+and(b) (K,T)quantum numbers for doubly excited states of helium below the He+(N = 3) limit. The rotorlike structure is evident for each given (K,T)+and (K,T)-. Energy levels are taken from the calculation of Lipsky et al. (1977).

C.D. Lin

106

- 0.040 I

a

K

v)

w0

-0.045

a w z Y

- 0.050

(

K,TIA

FIG. 18. Energy levels of H- resonances lying below the H(N= 5) limit of hydrogen grouped according to the (K,T)+quantum numbers.Data from Ho and Callaway (1983).

+

We must emphasize that the rotorlike structure appliesto A = 1 and - 1 channels only, but the T doubling applies to A = 0 channels as well.

E. SINGLYEXCITEDSTATES AND DOUBLY EXCITEDSTATES WITH A = 0 The classification scheme discussed here applies to all states of two-electron atoms. It incorporates singly excited states as a subset. All singly excited states have (K,T ) = (0,O).For ISe,A = 1, for 3Se,A = - 1, and all of the other L's have A = 0. According to the independent-particle model, the energy for lsnL 3Lis always lower than the energy for 1mL ' Lfor two-electron atoms. This is understood in terms of Pauli exchange correlations: In triplet states, the two electrons have parallel spins and thus they tend to stay away from each other spatially, thus reducing the electron -electron repul-

+

DOUBLY EXCITED STATES 'PO

107

3PO

FIG.19. Surfacecharge-densityplots for the lsnp 'Poand lsnp )POchannelsofHeat R = 2, 4, and 6. Notice that the Pauli exchange correlation is reflected as angular correlation in chargedensity distributions.

sion energy. From the hyperspherical viewpoint, this Pauli exchange correlation is reflected in the difference in angular correlation between the two electrons. To illustrate this point, the surface charge density plots for 'Pand 3Pareshown in Fig. 19 for three values of R . At large R,say R = 6, there is no evidence of angular correlation, and the charge distributions for singlet and triplet are identical. At smaller R, say R = 2, we notice that the triplet state has large charge concentration near O,* = 180" while the singlet has larger concentration in the small-O,, region. Thus the Pauli exchange correlation in the independent-electron picture is reflected in terms of angular correlations if it is visualized from the hyperspherical viewpoint. The A = 0 states were mentioned earlier to be states where the electrons reside in the two potential valleys. In this respect, they are similar to singly excited states. In fact, their spectral behaviors do resemble those of singly excited states, as shown in Fig. 20. By displaying these states according to (K,T)O,we notice that the triplet state in each case does lie lower in energy than the corresponding singlet state except in cases where the irregularity may be due to the numerical inaccuracy. Data were taken from Lipsky et al. (1977).

C.D. Lin

I08

1 (0,210

(-1, I 10

CO,O,O

(K,T)

(-2,OP

A

FIG.20. Effective quantum numbers n* for the A = 0 states of helium doubly excited states converging to the N = 3 limit of He+grouped accordingto (K, T)O.Notice that the triplet state is always below the singlet state for a given K,T, and I[.Notation like ' J Findicates that the two states are nearly degenerate, but IF is slightly above 'F. The two ' J F cases are likely due to numerical inaccuracy. Data from Lipsky ef al, (1977).

The discussion up to now summarizes the classification scheme and the spectroscopic regularities revealed through the introduction of correlation quantum numbers. Channels that have identical designation of correlation quantum numbers exhibit isomorphiccorrelation patterns and near-degenerate potential curves. This isomorphism is the underlying reason for the origin of the rotorlike supermultiplet structure of doubly excited states that have radial correlation quantum number A = 1 or - 1. The discussion so far has been very descriptive for the purpose of presenting the classification scheme itself and for a general qualitativeunderstandingof the correlationof doubly excited states. The rest of the article will provide an in-depth quantitative analysis on correlations in hyperspherical coordinates.

+

DOUBLY EXCITED STATES

109

IV. Solution of the Two-Electron Schrodinger Equation in Hyperspherical Coordinates In the previous two sections we discussed the results of the Schrodinger equation for two-electronatoms in the quasiseparableapproximation for the classification of doubly excited states. In this section, we describe the computational methods used for the solution of the eigenvalue equation [Eq. (6)] and present some typical results.

A. HYPERSPHERICAL HARMONICS AND SOLUTIONS AT SMALL R

We first examine the solution of Eq. (6) in the small-R limit. At small R, the kinetic energy term in Eq. (6) is proportional to l/Rz while the Coulomb potential energy is proportional to 1/R. In the limit of R = 0, Eq. (6) becomes

(20) where v = 1,

+ 1, + 2m and the eigenfunction ullIz,is U/l/2m=

h l ~ 2 m ( ~ ) Y l l ~3 2z) , d ~ ~ (21) In Eq. (21),Y I I 1 2 ~,Fz) F Iis the coupled angular momentum function of the two electrons, (~,~l12m21~~)Y~lml(i,)Y~zm,(~2) (22)

Y , I I z ~ ( f ~= m m

and

h,,Ja) = N(cos a)/1+l(sina)lZ+lF(-m,m + 1, + 1, + 211, + #Isin, a)

(23)

where N is a normalization constant and F is proportional to a Jacobi polynomial (Morse and Feshbach, 1953). A properly (anti)symmetrized hyperspherical harmonic with respect to the interchange of two electrons is given by

C. D.Lin

110

+ +

In Eq. (24), the allowed values of m are such that L S rn = even if I, = 12. Furthermore, the eigenvalue depends only on the sum, v = I,

l2

+ 2m.

-

+

In the R 0 limit, the quantum numbers I, and l2 measure the barrier for the penetration of each individual electron into the inner region, while the quantum number v measures the degree of simultaneous penetration of the two electrons into the small-R region. In this limit, the higher eigenvalues v have a high degree of degeneracy. An analysis of v = I, 1, 2m alone can provide some indications about the nature of angular correlations that are missed in the independent-particle approximation. Strong correlation occurs when two or more eigenfunctions u with the same LS quantum numbers are degenerate. For example, this degeneracy occurs normally at R = 0 for IDe channels with (I, ,I2) = (0,2) and with ( I , ,I2) = (1,l) because they have the same I, I,. This degeneracy occurs for all even values of rn so that the coupling between sd and p p states remains strong. This explains the strong interchannel coupling between the ksnd lDeand kp2 'Destates ( k < n) of alkaline earth atoms along the whole IDe series (OMahaony and Watanabe, 1985; O'Mahony, 1986; Lin, 1974b). Such mixing also explains the strong configuration mixing between 2snp2 2Deand 2s2nd 2Dein aluminum (Lin, 1974a; O'Mahony, 1986; Weiss, 1974). Similar analysis for the degeneracy of N-electron systems has been carried out recently by Cavagnero (1984).

+ +

+

B. THEASYMPTOTIC LIMITAND THE LONG-RANGE DIPOLEAPPROXIMATION In the asymptotic limit when one electron is inside and the other is far outside, corresponding to the limit that R + and a 40, the two-electron wave function is represented by the product of two independent-electron functions. In this limit, Eq. (6) can be easily solved by transforming R and a back to the independent-particle coordinates r2 = R sin a and r, = R cos a = R. Ifwe expand the transformed equation in powers of 1/R, the resulting asymptotic potential (Macek, 1968; Lin, 1974b) is [ UJR) - 1/4RZ- W,,,,(R)]

z2

+

-- n2

2 ( Z - 1) R

1 + -(@,,ll: + 2r2 cos B12(@,,) R2

+

(25)

Ifa,, is the expectation value ofl: 2r2cos 012for channelp, then the channel potentials within a given Nmanifold are distinguished by the different dipole terms a,JR2. Such dipole potentials were first derived by Percival and Seaton

I11

DOUBLY EXCITED STATES

(1957) and by Gzilitis and Damburg ( 1963) from the close-couplingapproximation in e- H* scattering. It is called the dipole representation. In previous works on dipole representation, each channel is labeled by the eigenvalues of Eq. (25). This method does not provide convenient integer quantum numbers. In order to be able to label the channels in the asymptotic limit, a “zero-order dipole basis” was proposed by Hemck (1975). In this approximate representation, only the dipole term r, cos O,, is diagonalized. Each channel’for a given (N,L)is characterized by two quantumnumbers, K and T. The eigenvalue of r, cos 4, in this representation INKTL) is

(NKTLJr,cos Bt2JNKTL)= - 3NK/Z

(26) Notice that this zero-order dipole operator is degenerate with respect to T. This degeneracy is removed if the centrifugal potential l! is included. In a perturbative calculation, Hemck (1975) showed that the dipole potential in the asymptotic limit is given by aJR2, with

+ L(L + 1) + K N 2 - 1 - K 2 - 3T2) - (KZ/12N)[8L(L + 1) + N 2 - 1 - K 2 - 15T2] +

CU, = - 3NK/Z

*

(27)

in the JNKTL)basis space. Equation (27) provides the basis for labeling the potential curves in the asymptotic region, as discussed in Section II1,B. According to Eq. (27), the effective dipole potential is most attractive for large positive K and, for a given K, a large value of T. Each INKTL) channel basis is given as the linear combination of the product of hydrogenic NI states and the spherical harmonics of the outer electron coupled to a total L. This channel function contains information about angular correlations but not radial correlations. The asymptotic dipole potential for a given N , K, and Tdoes not depend on the spin, nor on the parity of the channel. The fact that it does not depend on the parity of the channel is surprising since the 1, and 1, pairs which form the same L but different panty are quite different. This degeneracy is not the result of the perturbation calculation of Eq. (27). It can be shown that this degeneracy is exact from numerical calculations. Nikitin and Ostrovsky (1 976, 1978) have derived the same conclusion from the group-symmetry viewpoint. The fact that the eigenvaluesof Eq. (27) are independent of parity is important for discussing the T doubling in Section V,F.

C. NUMERICAL SOLUTION OF THE CHANNEL EQUATIONS The partial differential equation Eq. (6), can be solved by using an expansion (28) =A g:,,(R;@Y,,I2LM(f*

@/mQ c

[hIzI

7 4 )

C.D. Lin

112

where A is the proper symmetrization or antisymmetrizationoperator (the spin function is not explicitly considered) and Yf,f2LM(F, ,4) is the coupled angular momentum function defined in Eq. (22). We use the convention that Il < 1, in the summation in Eq. (28). With the substitution of Eq. ( 28) into Eq. (6), a set of coupled differential equations in the angle a are obtained. The number of equations is equal to the number of [/,,I2] pairs included in Eq. (28). The resulting eigenvalue equations have been solved by different methods: ( 1) numerical integration of the coupled equations (Macek, 1968); ( 2 ) diagonalization using hyperspherical harmonics (Lin, 1974b; Klar and Klar, 1978, 1980); and (3) the finite difference method (Lin, 1975a, 1975b, 1976). All these methods have some limitations. The numerical integration method often suffers from instability and the finite difference method requires the solution of a large matrix if the number of [I, ,I2] pair is large. The diagonalization method is inaccurate at large R. At large R, the solutions are linear combinations of hydrogenic functions which cannot be expanded in terms of a small set of hyperspherical harmonics. The calculation of the channel functions and the correspondingeigenvalues U(R) is significantly simplified with the introduction of an analytical channel function for a given [I, ,I2] pair (Lin, 1981). The idea behind this is quite simple. It is best illustrated in terms of a few examples. Consider the lowest ISechannel in the [I, ,12] = [O,O] subspace. In the large-R limit, g$(R;cr)

r2 e-'2

=R

sin

mn a

(29)

to within a normalization constant. There are many different ways to generalize Eq. (29) to the small-R region. To do so, we require that the generalized functions reduce to the hyperspherical harmonics in the limit of R = 0. For the channel considered, this is proportional to sin 2 a . A reasonablegeneralized function for this channel is then gg(R;a) = N(R) sin 2ae-R

(30)

This form reduces correctly to the known solutionsin the R = 0 and R -,00 limits. The normalization N(R) satisfies f&(R;a)12 da = 1 For the lowest 3Sechannel in the [O,O] subspace, the generalized function is gg(R;a) = N(R) sin 2a cos 2ae-R 5 n a m a

(32) Notice that both Eqs. (30) and ( 32) satisfy the proper particle exchange symmetry under a -,71/2 - a for lS and for 3S, respectively.

113

DOUBLY EXCITED STATES

This procedure can be extended to obtain analytical channel functions for other L, S, and a states and excited channels. The description for the construction of these functions was discussed by Lin ( I 98 1). With these analytical basis functions the calculation of channel functions @,(R;n)becomes very easy. In a typical calculation we include analytical channel functions and hyperspherical harmonics as basis functions to diagonalize the coupled differential equations. For example, to calculate all the potential curves for ISthat lie below the H(N = 3) or He+(N = 3) limits, a maximum of about fifteen basis functions including [I, ,lzJ= [O,O], [1,1], . . . , up to [3,3] or [4,4] is needed. The simplicity of the computational procedure allows us to study the properties of doubly excited states easily. Other numerical methods have been proposed recently; see Christensen Dalsgaard ( 1984a).

D. REPRESENTATIVE RESULTSFOR HIn this article, we are concerned mostly with the correlations of doubly excited states and the classification scheme. To show that the hyperspherical approach also gives reasonable quantitative results, we present in this section some representative results of H-calculated using hyperspherical coordinates. Consider the *Poresonance states of H- near the H(N = 2) limit. The three potential curves that converge to this limit are shown in Fig. 2 1. Notice that

5

10

15

20

25

R (bohr) FIG.2 1. Potential curves for the three ' P ochannels of H-that converge to the N = 2 limit of hydrogen. The -, and pd notations refer to the (O,l):, (l,O)r, and (- 1,O): channels, respectively.

+,

C.D. Lin

114

the (0,1)+channel has a relatively deep attractive potential well at small R, R 2: 8 a.u., but becomes repulsive at large R with a 2/R2 dependence (Lin, 1975b).The (1,O)- curve is quite repulsive at small R but has a very shallow attractive potential well at large R. The (- 1,O)O curve is completely repulsive. This example highlights the many aspects of correlation behaviors discussed in Sections111. The channel has a more attractive potential at small R because of the in-phase radial correlation of the two electrons. It becomes repulsive at large R because for K = 0 the two electrons tend to stay near 90" from each other. The (1,O)-channel potential is not very attractiveat small R because of its - character. It has a shallow potential well at large R, behaving asymptotically as - 3.7 1/R2(Lin, 1975b),because of the favorable angular correlation that the two electrons maintain an angle close to 180" (K = 1). The (- 1,O)" channel is completely repulsive owing to its unfavorable radial and angular correlations. In this channel, the two electrons are on the same side of the nucleus (K < 0), and they stay primarily in the valley region of the potential surface (Fig. 3). The potential curves shown in Fig. 21 clearly suggest that resonances associatedwith the (1 ,O)- channel and those with the (0, 1)+channel are quite different. The - channel is very repulsive at small R and thus the radial wave function (in R) is quite diffuse. By solving the hyperradial equation [Eq. (9)] using the - potential, the energy of the lowest state was found to be at -0.25 191 Ry. Neglecting the small Lamb shift between the hydrogenic 2s and 2p states, this channel in principle can support an infinite number of states (Lin, 1976). But all the higher states are very close to the H(N = 2) threshold and have not been observed experimentally. For the (0,1)+channel, the potential curve suggests the existence of shape resonances. In actual calculations, the potential was found to support a shape resonance which is 32 meV above the threshold and has a width of 28 meV. Experimentally these resonancescan be observed in electron - hydrogen atom scattering, but better data were obtained from the photodetachment of H-. Because of the lack of suitable intense photon sources in the 10- 15 eV region, such measurements were not done until recently. By taking advantage of the existence of the 800 MeV (v/c = 0.83) relativistic H-beam from the LAMPF facility at Los Almos, Bryant et al. (1977) achieved the desired photon energy range by aiming lasers toward the incoming H- beam at different angles. The lasers were blue-shifted to the desired photon energy range in the H- frame. The results of Bryant et al. (1977) for the photodetachment cross sections near the H(N= 2) limit are shown in Fig. 22. It is clearly seen that the Feshbach resonance associated with the (1 ,O)- channel is quite narrow, and the shape resonance associated with the (0,1)+ channel is much broader. The shape resonance was found to have a width of 23 t- 6 meV, and the separa-

+

+

DOUBLY EXCITED STATES

I15

FIG.22. Photodetachment cross sections of H- near the N = 2 excitation threshold of hydrogen. The solid line is the result of the theoretical calculation of Broad and Reinhardt (1976) (Bryant el a/., 1977).

tion of the Feshbach and shape resonances was found to be 53 meV. From the simple calculation using the quasi-separable approximation in hyperspherical coordinates, the corresponding results were 28 and 58 meV, respectively. More accurate calculations on these resonances have been performed by other methods. The solid line shown in Fig. 22 was due to Broad and Reinhardt ( 1976).

V. Body-Frame Analysis of Correlation Quantum Numbers In Section I11 we presented a classification scheme of doubly excited states using the correlation quantum numbers K, T, and A. The angular correlation quantum numbers Kand T, as discussed in Section IV, were adopted from

C.D. Lin

116

Henick’s work for the approximate description of Stark states in the asymptotic limits. The adoption of quantum numbers in the asymptotic region for the description of doubly excited states seems unsatisfactory since important correlations occur in the region where the two electrons are close to each other. From Fig. 3, we notice that the potential surface is quite smooth along the OI2 coordinate. This smooth dependence allows us to expect that angular correlations do not vary significantly as R changes adiabatically. Similar conclusions have been obtained through actual numerical calculations (Lin, 1982b).We thus expect that the same quantum numbers Kand T used in the asymptotic limit can be used to describe angular correlations in the inner region and also of the whole atom. To incorporate radial correlations, the quantum number A was introduced semiempirically (Lin, 1983d, 1984). In this section we re-examine these quantum numbers by analyzingthe channel functions in the body frame of the atom (Watanabe and Lin, 1986). A. CHOICE OF THE BODY-FRAME AXES

We choose the interelectronic axis = (r1 - r2)/Ir1 - r2l

(33) as the internal axis of rotation. This choice is democraticwith respect to the exchange of the two electrons. The general behavior of this axis is similar to that ofthe vector B = b, - b2(Henick and Sinanoglu, 1975a,b)exploited in the O(4) theory of doubly excited states since for a pure Coulomb field the Lenz vector b is related to r as il*

r (3n/2Z)b (34) where 2 is the charge and n is the principal quantum number. For intrashell states the B in the O(4) theory is proportionalto the interelectronic axis in Eq. (33). The choice of Eq. (33) as the internal axis also has the advantage of not specifying the principal quantum numbers n and N of the two electrons. +

B. DECOMPOSITION INTO ROTATIONAL COMPONENTS

In displaying the correlation patterns shown in Section 111, the charge densities were averaged over the rotational angles of the whole atom. In this section, we decompose the whole wave function or channel function into components along the body-frame axis. Starting with a chosen laboratory frame, the rotation from the laboratory frame to the body frame is effected

DOUBLY EXCITED STATES

117

through a rotation matrix Y / , , * L M ( ~ l , 4 )=

I]YlIl2LQv~ ,WD&(W

(35)

Q

where are defined in the laboratory frame and (pi,?;) in the body frame, and D is a rotation matrix. Suppose that the wave function is known in the laboratory coordinates, W

l ,r2) =

c

q 1 2 ( y I , T 2 ) Y , , I 2 d ~ 2l 2 )

(36)

1111

Substitution of Eq. (35) into Eq. (36) gives

where &Rff,42)

=

2 ~ , , ( cos R

ff,

R sin

~)~,,/,,Qv; ,pi3

(38)

1111

and - L d Q d L. Let us consider the symmetry under particle exchange. A careful analysis in the Appendix of Watanabe and Lin (1986) shows that under a n/2 - a,each rotational component satisfies the property

-

&R, ~ / -2a, $12) = N- 1PQ ~&R,~012) By introducing a phase factor A as A = K(-

(39)

l)S+T

where T = lQl, the index A determines the reflection symmetry of the radial wave function with respect to the a = 7r/4axis. ThusA serves as an index for radial correlations. In the special case L = 0, we have T = 0, A = (- I)S, which is the well-known symmetry requirement for IS and 3S states. For L not identical to zero, there are more than one rotational components. Ifthere is only one dominant rotational component in Eq. (37), then the radial correlation quantum number A is determined from Eq. (40).In fact, Eq. (40) is identical to Eq. ( 1 8) for A = 1 or - 1. We assigned A = 0 for those channels which do not have a major rotational component, even though each rotational component has its own well-defined symmetry in a. Each rotational component also has a well-defined symmetry with respect to the OI2 = K axis. In fact, it can be shown that (Watanabe and Lin, 1986)

+

&(R;a,2n - 612) = (- I)= V/Le(R;a,42) (41) which provides the relation between the motion in OI2 and the rotational quantum number T.

C.D. Lin

118

C. PURITYOF ROTATIONAL STATES The symbol A , as given by Eq. (40), has a close connection with the value of T. According to the decomposition of Eq. (37), if there is only one rotational component, then the radial correlation quantum number A will be either A = 1 orA = - 1. Thus the purity of+/- radial correlation is related to the purity of rotational states. To enrich our picture of the purity of rotational states, we show in Fig. 23 the decomposition of the ( 1 , l ) ~3P0 and (1,l); IPo channel functions at the values of R where their respective potentials bottom out. The percentage represents the contribution to the normalization from each T component. For 3P0,the T = 1 component has 91% of the integrated density. According to Eqs. (40) and (4l), for this component A = - 1 and the function vanishes along 4, = R. The density plot for the T = 1 component clearly exhibits these properties. Figure 23 also shows that there is a 99/0contribution of the T = 0 component for 3P0at R = 23. This component has A = 1 and an = R; the density plot for T = 0 clearly shows this antinodal structure at behavior. Similarly, for ]Po,the T = 1 component represents a 90% contribution and the T = 0 a 10%contribution at R = 16. In this case, the T = 1 component has A = 1 and a nodal structure at OI2 = n, while the T = 0 component has A = - 1 and an antinodal structure at el, = R. The surface plot for each component exhibits these relations. The purity of rotational states maximizes roughly in the range where the potential is near the minimum. To illustrate the dependence of the purity of

+

+

+

T=O

T= I

91 %

10 %

90 %

TOTAL

R.23

R = 16

FIG,23. Decomposition of the density plots into rotational components of the (1 ,l)r 3P0 and (],I): ' P o channels of He at the values of R indicated. The percentages represent the contribution to the normalization from each T component.

119

DOUBLY EXCITED STATES

He ( N = 3 , 'Po)

K

osymptotic limits

i

\

N,

\

-',\ L(I.1). 50.-

W,Ik' /I-

(-2.oP _c---

0

10

20

+--(-2.OP ,or

>'

(0.01-

I

(0.0)-

/

(2

( 0 .or

30

40

R ( i n reduced a.u.1 FIG.24. Normalization coefficientsfor the T = 1 component for all five ' P ochannels of helium below the He+(N= 3) limit. The asymptoticlimits of the coefficientscorresponding to R m for each channel are indicated on the right. Dashed lines are used to indicate the region where diabatic crossing has been imposed.

-

rotational states on R for each channel, we show in Fig. 24 the normalization percentage of the T = 1 component for all of the five ,Po channels that converge to the He+(N= 3) thresholds. The dashed lines represent the interpolated region where the potential curves exhibit crossings. We note that the low-lying channels show greater purity of rotational states, while the higher channels violate the purity of T more severely. We further note that the rotational quantum number T is ill respected in the asymptotic region. The reason is that the angular motion of the outer electron is represented by the term 1:/R2, which is not diagonal in the pure ( K , T )basis.

D. VIBRATIONAL QUANTUM NUMBERS Zero-order vibrational states do not emerge automatically from the bodyframe analysis. To understand the quantum number K , we first assume every channel to be in a pure T state. Suppose that the outer electron approaches the inner one from a large distance. Take the axis of the approaching electron rl to be the z axis of the laboratory frame. In this frame, the O,,), two-electron angular momentum function is proportional to P:(COS where Q = L * i,.The number of nodes in OI2 (0 < OI2 < R ) is I, -IQI, which vanes between 0 and N - 1 - T. The transformation from the laboratory frame to the body frame is identity at large rl ; the transformation evolves smoothly to the small-R limit. Thus, for agiven T, we can use the number of

C. D. Lin

120

nodes n in 012as a label for the vibrational motion in 0,2. In molecular physics, the vibrational quantum number u is related to n by

(42)

u=2n+T

The quantum number K used for labeling hyperspherical channels is related to n and u by

K = N - 2 n - T - 1= N -

U-

1

(43)

When T is fixed, both u and K change in steps of 2. The quantum numbers K and u have thus far been used as labels. According to the definition of Kand Tfrom the asymptotic solution, if a channel is a pure (K,T)state, the expectation value of the dipole moment r2 cos OI2 is -(3NJ2Z)K.We can define a similar leading term in the dipole approximation which contributes to the vibrational energy 1 R sina v,(~;a,e,,) = --COS e12, o s R2 cos2a

-

1 R cosa cos eI2, RZ sinZa

sn~4

7114 s a s n ~ 2

(44)

Here R2V, determines the polarizability of the system. To examine the purity of an effective g, we define

The results for K(R) for He(N = 3, ]Po)channels are shown in Fig. 25. We note that K(R)indeed varies slowly with R and is very close to the integers K used to label the channels, particularly for the low-lying channels. 1

I

.--

2 -

(2

,'

____

I - -

R(R)-0

-2

-___- _ _ _ _ _ _ _ _----

-

'PO (N = 31 1

i

,or

(l,l)* (0,o)( - 1 , I I* (- 2 ,o

1"

121

DOUBLY EXCITED STATES

E. MOLECULELIKE VIEWPOINT OF TwoELECTRON CORRELATIONS The body-frame analysis so far indicatesthat the quantum numbers Kand T can be related to the vibrational and rotational quantum numbers used in molecular physics. This rovibrational viewpoint has been explored extensively for intrashell states (Herrick et al., 1980; Kellman and Herrick, 1978, 1980; Herrick, 1983) and for model two-electron atoms (Ezra and Berry, 1982, 1983). By generalizingto intershell states, one can identify (Watanabe and Lin, 1986) the +/- radial quantum numbers as the symmetric stretch (for A = +) and antisymmetric stretch modes used by quantum chemists (see the review by Ma-, 1985). The potential curves illustrated in Figs. 12- 14 and the energy levels shown in Figs. 17 and 18 indicate that the magnitude of the correlation energies follow the hierarchical order

+

where UA, UK,U,are the separation of the A = and - doublet curves and the local vibrational and rotational energies, respectively. The higher excitations, particularly the A = 0 channels, lead to the less clear-cut order and to a noticeable admixture of other modes. These higher excitations do not exhibit moleculelike modes. After the moleculelike modes have been divided into A = and A = groups, angular correlations can be classified by their degree of excitations. There are two well-defined schemes (Herrick, 1983) which can be easily understood from Eq. (46). One is the d-supermultiplet scheme which utilizes the number of nodes in &, namely the n discussed in the previous subsection, to regroup angular correlation patterns. An example is shown in Fig. 26 for the lowest n = 0 states for the A = and for the A = - subgroups for doubly excited states of He(N = 3). The vertical axis corresponds to L and the horizontal axis is labeled by & T, where we have used - T to designate states which have rotational quantum number T but with parity given by n = (in order to distinguish it from states with identical T but with panty n = (- 1)L. Note that there is a clear correspondence between the "+"-type and "-"-type supermultiplets,namely the interchange of the spin label 1 * 3. Another scheme is known as the I supermultiplets. Defining I = L - T, loosely speaking, I corresponds to the rotational degree of freedom orthogonalto that represented by T (Watanabe and Lin, 1986). With K as the vertical axis and k Tas the horizontal axis, a diamond similar to Fig. 26 can be constructed for each I (Herrick et al., 1980). The moleculelike normal modes motivated Kellman and Herrick ( 1980)

+

+

C. D.Lin

122

nF.3FOIFO (3107)

-2

(3325) (.3039)

0

-I

+I

+2

T

'F' 'F" 3F0 --(2684) (2779) (2673)

3 t

(2789) (2859)

0

(

-2

-I

(2787)

2872)

0

+I

+2

T FIG.26. d-Supermultipletstructure of helium doubly excited states below the N = 3 limit of He+. (a) Intrashell states;(b) the lowest states of all the A = - 1 channels. Energy levels from Lipsky er al. (1977).

to fit intrashell energy levels to the molecular term formula E=E,+o(V+

l)+X(V+

1)2

+ G T 2+ [ B - a(V+ l)][L(L+ 1) - T2]- D[L(L+ 1) - T 2 ] (47) This formula attempts to attribute all the higher-order corrections to the anharmonicity of the bending vibrational potential, centrifugal distortion, etc. In atoms, the impurity of the ( K , T )states owing to angular excitation is an equally important contributor to the departure from the lowest-order formula

E = EN

+

O( V

+ 1) + B[L(L + 1) - T2]+ GT2

(48)

DOUBLY EXCITED STATES

123

A more detailed discussion on the limitation of the moleculelike interpretation of doubly excited states is given in Watanabe and Lin ( 1986).

F. THET DOUBLING In Fig. 17 we note that each pair of T # 0 states which have identical n,N, A , L, and K have near-degenerate energies. The splitting of each pair is called T doubling. T doubling occurs for A = 1 as well as for A = - 1 states. In fact, it also occurs for A = 0 states, as shown in Fig. 20 for states belonging to the (- 1,l )O channel. The effective principal quantum numbers shown in Figs. 17 and 20 clearly indicate that between each pair of states the energy of the state with parity a = (- 1)”+’ is slightly lower than that of the state with parity a = (- l)L. This differencecan be attributed to the wave functions near BIZ = 0. It can be shown (Rehmus et a/., 1978; Ezra and Berry, 1982)that the wave functions vanish identically at BIZ = 0. There is no for states with parity a = (such constraint for states with parity a = (- 1)”. In general, the wave functions for these latter states are small at OI2= 0 and CI = n/4,but they do not exactly vanish. A nonzero amplitude near t9,2 = 0 and a = a/4 tends to increase the electron -electron repulsion energy. If all the other quantum numbers, n,N , A , L, K, and T, are the same for the pair of states, such a stronger electron electron repulsion would result in a higher energy for the state with parity a = (- 1)=. The energy levels in Fig. 17 are in agreement with this prediction [see the (1, l)+, (0,2)+, and (- 1,1)+series in the upper frame and the (l,l)-, (0,2)-, and (- 1,l)- series in the lower frame]. This prediction, however, is not completely confirmed by the results shown in Fig. 18 for the resonances of H- below the H(N = 5 ) threshold (Ho and Callaway, 1983). The calculated energy ordering for the (2,2)+and (1, l)+channelsis opposite to what we have expected. Whether this irregularity in the T doubling is due to some other unaccounted effects or due to the inaccuracy in numerical calculations remains to be resolved. Similar irregularities in this respect can also be found in the calculated energies for the resonances of H- below the H(N= 4) threshold [see Table I1 of Ho and Callaway (1983)J.

+

G. SYSTEMATICS OF AUTOIONIZATION WIDTHS One of the most striking features of the earlier photoabsorption data of doubly excited states of helium is that the autoionization widths of different Rydberg series are dramatically different. To characterize the width of a

C,D. Lin

124

Rydberg series, we define a reduced width, i= = n*3 r,, where r, is the autoionization width of state n with effective principal quantum number n*. (It is well known that the reduced width defined this way is nearly constant along the series.) From the data of Madden and Codling (1963,1965),as well as the results of early close-coupling calculations(Burkeand McVicar, 1965; Burke and Taylor, 1966;see also Fano, 1969),it was shown that the ratios for the reduced widths of the three ]Poseries, (0,,;)I (l,O)?, and (- l,O)q, are 3000:30: 1.Such drastic differences in widths are typical when we compare A = 1, - 1, and 0 channels. The systematics of autoionization widths with respect to other quantum numbers are less clear, although fragmentary evidences and “rules of thumb” have been discussed (Herrick, 1983; Rehmus and Berry, 1981; Watanabe and Lin, 1986). These rules are “understood” in terms of the correlation properties or in terms of the moleculelike normal modes of doubly excited states. (1)The partial width is largest when the continuum channel corresponds to AN = - 1, A K = - 1 (Le., Av = 0), AT = 0, with A unchanged. This rule is easily understood because the overlap between this continuum channel and the quasi-bound resonance is largest owing to their similar correlation patterns. The overlap occurs mostly near the locus of the ridge (Fano, 1981) where the pair correlation in the continuum channel is just breaking up and the quasi-bound resonant wave function is gaining amplitude. (2)The widths for the pair of Tdoublets are nearly identical. This is well supported by existing calculations (Ho and Callaway, 1983,1984).Since the lower state of the Tdoublet has less amplitude near a = n/4and OI2 = 0, it is expected that this state has smaller width. The results given by Ho and Callaway ( 1983)for the resonances of H- converging to N = 4 and 5 of H thresholds do not support this prediction. It is not clear that this discrepancy is due to the neglect of other effects or because of numerical inaccuracies. (3) Along a rotor series, states with higher L have larger widths. This is because the higher rotor states have larger amplitudes near OI2 = 0. This effect can be overtaken by the fact that the higher rotor states have less amplitude near a = n/4.For example, the (4,O)f rotor series of H-, ISe,3P0, IDe, 3F0,‘Ge, 3H0,]Ie,and 3K0,as shown in Fig. 18,have widths, in units of lOV3Ry,of1.1,1.2,1.3,1.2,1.5, 1.8,0.54,and0.2(HoandCallaway,1983). (4)When n, N, L, S, n,and Kare the same, the states with the larger Thave the larger widths. This is due to the increase in the amplitude near OI2 = 0 for higher T states. For example, the (0,2)flDe state of He is a factor of 3.5 broader than the state (0,O)fIDe (Hemck and Sinanoglu, 1975a).There are very few other calculations to check the validity of this rule. These rules were drawn from the few data currently available with “plau-

+

DOUBLY EXCITED STATES

125

sible” explanations provided from the correlation properties. Since the autoionization width depends sensitively on the details of the wave functions of the quasi-bound state and the continuum state, it is interesting to find out what systematics of autoionization widths can be drawn from the correlation properties of the quasi-bound-state wave function alone. Extensive compilation of widths in future calculations will help to check the generality of these rules.

VI. Effects of Strong Electric Fields on Resonance Structures in H- Photodetachment A. EXPERIMENTAL RESULTS

In this section the effect of electric fields on the resonances of H- is discussed. We will not address the large area of the Stark effect of Rydberg electrons; rather, we will concentrate on the effect of electric fields on the doubly excited states of H-, where experimental data have been obtained by Bryant and coworkers at the LAMPF facility at Los Almos. In their experiments, an external magnetic field up to several kG is applied to the relativistic H-beam (see Section IV,E) which corresponds to an electric field of up to a few MV/cm in the H- frame. Their results are summarized as follows. (1) The ‘PoFeshbach resonance below the H(N = 2) discussed in Section IV,E was found by Gram et al. (1978) to split into three components; the two outer components exhibit linear Stark shifts and the middle one exhibits quadratic Stark shift. Later experiments by the same group (Bryant et al., 1983) with the use of polarized laser light confirmed that the two outer components belong to states which have a magnetic quantum number M = 0 and the middle one has IMl= 1. Their results are displayed here in Fig. 27. We notice that the lowest component was observed to quench at E - 130 kV/cm, the middle branch was found to vanish at around 270 kV/cm, while the upper branch appears to burrow into the shape resonance for fields higher than 400 kV/cm. (2) The shape resonance was observed to be quite stable against the electric field. The results from Bryant et al. ( 1983) for the width of the shape resonance in the electric field are shown in Fig. 28. Initially the width decreasesas the field is increased to about 0.2 MV/cm, then it increases rapidly with the field until 0.7 MV/cm, where its rate of increase begins to decline. There is no

C. D. Lin

I26

E(eV)r I

I

l0.9lOC

0

I

0

0.1

I

0.2

1

0.3

ELECTRIC FIELD (MV/cm) FIG.27. Energies of the centroids of Feshbach multipletsas a function of applied electric field. Solid lines are from the theoretical results of Callaway and Rau (1978) (Bryant et al., 1983); (0)“u polarization,” curved prism; (.f “71 polarization,” flat prism; (a)nitrogen laser, unpolarized.

experimental information about the field at which the shape resonance is quenched. (3) The lDeresonance which has acorrespondingphoton energy of 10.874 eV can be excited by single-photon transitions in an electric field. Indeed, a structure at approximately the expected energy of this resonance was observed for fields in excess of 400 kV/cm. It appears that this resonance splits into two at the higher field. (4) The stripping of the ground state of H-in a strong electric field has also been measured by Bryant’s group (private communication, 1983). Their results, together with the earlier weak-field data from other groups, are displayed in Fig. 29. These strong-field results are quite interesting in several respects. Since the resonances studied belong to different channels, the energy shift as well as the change of the width for each resonance is characterized by the correlated motions of the two electrons and their relation with respect to the direction of the electric field.

DOUBLY EXCITED STATES

127

ELECTRIC FIELD (MV/cm) FIG.28. Width of shape resonance versus applied electric field. (-) From the theory of Wendoloski and Reinhardt ( 1 978). (- . -) and (---) are two different fits to the experimental data (Bryant ef al., 1983).

ELECTRIC FIELD (MV/cm) FIG.29. Quenching lifetimes for the ground state of H-in an electric field. The thick solid line is from the data of Bryant (1983, private communication). The crosses are the results calculated from the field-modified hyperspherical potential curves.

C.D. Lin

128

B. THEORETICAL INTERPRETATIONS A preliminary study on the effect of strong electric fields on the resonances of H- has been initiated using the quasiseparable approximation in hyperspherical coordinates (Lin, 1983c). The basic method is similar to that which was described in Section I1 except that the potential due to the Stark field is included nonperturbatively in the new Hamiltonian. Using the analytical basis functions of Section IV, the effective potential curves for each given electric field can be obtained. These potential curves are used to interpret semiquantitatively the observations of Bryant et al. (1983). ( 1) We first show that the lifetime of the ground state of H- in an electric field can be understood using this simple picture. In Fig. 30 we show the adiabatic potential curves of the lowest channel of H- in different electric fields in units of MV/cm. We notice that the effect of the electric field is to introduce a linearly decreasing potential in the large4 region, while the small-R region is hardly affected. In terms of this picture, the lifetime of the ground state of H- can be estimated using the tunneling model similar to that used to describe the a-decay of nuclei. With potentials as shown in Fig. 30, the lifetime of the ground state can be estimated using a WKB approximation. The results of such an estimate are shown by the crosses in Fig. 29. They are in good agreement with the measured results from the low-field to the high-field region. The discrepancy at the higher fields is probably due to

-

I

1

I

1

I

-

-

-z-l

K -1.05

I

'

I

H'(n=l)

--

.-

a

CI

3

-

I

t

-

-

I

l

l

I

l

l

40 60 80 R(a.u.1 FIG.30. The hyperspherical potential curves for the ground channel of H-in an electric field. 20

129

DOUBLY EXCITED STATES

the breakdown of the WKB approximation when the barrier penetration is large. (2) The linear Stark shifts of the zero-field Feshbach resonances can be understood in terms of mixing with a nearly degenerate state with the same spin, but opposite parity: the second recursion of a ISeFeshbach sequence converging to the N = 2 series limit. Such shifts can be calculated using the diagonalization of a large set of basis functions.The results of such a calculation by Callaway and Rau (1978) are shown as solid lines in Fig. 27. Such calculations, however, give only the shifts and provide no information about the quenching. Using the quasiseparable approximation in hyperspherical coordinates,the potential curves for the two M = 0 Stark states shown in Fig. 3 1 can be calculated. In Fig. 3 1a we see that the dependence of the potential curves with the electric field is similar to those shown in Fig. 30 for the ground channel. As the field increases, the potentials in the outer regions decrease linearly with R with little change in the inner region. From the dependence of the potentials with the E field, a simple estimate based upon first-order perturbation theory indicates that the energy shift depends lin-

- 0.250

J ' " " " " 1

(a 1

- -0.255 (L v)

w -a260

>

a

3 V

d

I-

-0.250

z W

+ 2

-0.255 -0.260 0

20

40

60

R (0.u.1

80

100

FIG.3 1. (a) The variation of the zero-field ( 1 ,O)i ISepotential curve of H-in an electric field. The electric fields are given in units of kV/cm. (b) Same as (a) but for the zero-field ( 1 ,O)? IPOchannel of H-. The two horizontal arrows indicate the position of field-free resonances.

C. I). Lin

130

early on the strength of the electric field. The classical field ionization occurs at E 1.00 kV/cm, which is consistent with the experimental value of 140 kV/cm. The upper linear Stark component exhibits a somewhat unfamiliar dependence on the electric field. As shown in Fig. 3 1b, when the electric field is applied the potential curve in the inner portion is shifted upward, while the outer portion of the curve is shifted downward with increasing electric field. In fact, in Fig. 3 1b we notice that the barrier height is above the field-free threshold at -0.25 Ry. The behavior of the potential curves in Fig. 3 1b clearly indicates an upward Stark shift of its eigenvalue; a first-order perturbation calculation indicates that the shift is linear with E a t small electric fields. The decay width of the resonance, because of the increase in the height of the barrier with the electric field, is expected to become narrower at lower fields before it broadens again at higher fields. A simple WKB estimate based on the calculated potential curves indicates that the inner potential is no longer attractive enough to support a bound state at E 350 kV/cm. Experimental data do not give the field where this state is quenched since it lies in the shoulder of the broadened shape resonance. (3) The effective potential curves in electric fields for the shape resonance behave similarly to those shown in Fig. 31b; in a weak field the potential barrier becomes higher while the potential at large R decreases linearly with R [see Fig. 3 of Lin ( 1983c)l. Such dependence implies that the width of the shape resonance becomes narrower in a weak electric field before it becomes broader as the field increases. The narrowing of the shape resonance was observed by Bryant et al. (1 983), as shown in Fig. 28, but the data also indicate that the width increases rapidly for E > 400 kV/cm. This broadening cannot be explained by the calculated effective potentials. The blue shifts of the spectral lines and the narrowing of the resonances in an electric field are not difficult to understand. In a given electric field, electrons in the lower channels tend to line up opposite to the direction of the electric field. In the higher channels, the orthogonality condition ofthe wave functions with respect to lower channels requires that the electrons occupy regions perpendicular to the field or toward the direction of the field. Such rearrangement of the charge cloud tends to increase the energy of the state as well as to render the state more stable against field quenching. The simple interpretation presented here for the strong field effects on the resonances is not complete. In an electric field, a resonance state can be quenched. Its energy is shifted by the electric field in addition to the autoionization. A complete quantitative evaluation of all these effects requires a full treatment of the multichannel scattering aspect of the problem.

-

-

DOUBLY EXCITED STATES

131

VII. Doubly Excited States of Multielectron Atoms So far our discussions have been centered on the doubly excited states of two-electron atoms, He and H-. In this section, we briefly describe the progress made in the understanding of doubly excited states of multielectron atoms. A. ALKALI NEGATIVE IONS AND ALKALINE EARTHATOMS

For these two-valence-electron systems, the electron pair of interest is attracted to an ionic core which is spherically symmetric when both electrons are outside the core. Under this restriction, the electron pair experiences primarily an attractive Coulomb potential plus a weaker polarization potential. On the other hand, penetration of either electron within the core exposes that electron to a stronger field and to substantial exchange of energy and angular momentum with the core electrons. These effects are minimal for two-valence-electron systems where the core can be regarded as “frozen.” Therefore, these systems are similar to two-electron systems. For the two-valence-electron systems, the electron -core interaction is no longer Coulombic, and the single particle states in the asymptotic limits within the same N manifold are no longer degenerate. Thus K and Tquantum numbers, as defined according to the analysis of Stark states, are no longer valid when such degeneracy is removed. On the other hand, our body-frame analysis of the correlation quantum numbers does not rely upon such degeneracy. The interesting question to be answered is whether the classification scheme and the properties of doubly excited states unraveled for the pure two-electron systems remain valid for doubly excited states of multielectron systems. By approximating the electron - core interaction by a suitably chosen model potential, these two-valence-electron atoms can be solved in hyperspherical coordinates (Greene, 1981; Lin, 1983b). The two ISe potential curves of Be which converge to the 2s and 2p states of Be+ are shown in Fig. 32. They are labeled in terms of the independent-particle designations, 2 s ~ s and 2 p p . These notations are by no means adequate. In Fig. 33, the surface charge density plots for the two channels at different values of R are displayed. For R = 2 and 6, we notice that these plots are quite similar to the plots for the (1,0)+ channel of H- shown in Fig. 8. At large R, especially R = 14, the surface plot becomes similar to what one would have expected for the 2 s a ISe,where there is little angular correlation and the channel function shows little OI2 dependence. Similarly, for the “2pp” channel, the

132

C,D. Lin

----k2

L

R (a.u.1

FIG.32. Adiabatic potential curves of (a) Be IScand (b) Be ’Se channels converging to 2s and 2p states of Be+. The channels are labeled using quantum numbers according to the independent-particle approximation.

charge distributions shown in Fig. 33b for small values of R resemble the (- 1,O)+ channel shown in Fig. 8. At large R, these plots are consistent with the designation 2 p p ISe,as the densities show a cos2OI2 dependence. These s ” “ 2 p p ” are suitable for plots clearly indicate that the designations“ 2 ~ ~and the large-R region and the ( 1,O)+ and (- 1,O)+ notations are more suitable for the small-R region. In terms of the description of individual states, the single-particle designations 2sns and 2pnp are more appropriate for excited states ( n >> 2), and n(K,T)+designationsare more appropriate for intrashell states. The adiabatic approximation was found to be valid for the two ISechannels of Be shown, as the coupling between the two channels was found to be small. Despite the fact that the angular correlation does not remain constant for each channel as R changes (as in the pure two-electron case), the angular correlation does evolve smoothly with R. Energy levels calculated from each adiabatic potential were found to be in good agreement with experimental data and with other calculations (see Lin, 1983b). It is interesting to ask if radial correlations are preserved for the two-valence-electron systems along the adiabatic channel. The adiabatic potential curves for the three IPo and three 3P0channels of Be below the 2s and 2p states of Be+are shown in Fig. 34. They were labeled as 2snp, 2pns, and 2pnd (Lin, 1983b). By examining the [1,,12] = [O,l] component of the channel of Eq. function, Greene ( I98 1) has shown that the a-dependent part [g(a) (1 3)] of the 2snp channel exhibits “+”-type behavior at small R, but it

DOUBLY EXCITED STATES a

133 b

R.2

FIG.33. Surfacechargedensity plots for (a) Be 2sa ISeand (b) Be 2-p ISechannels on the (a,&)plane for different values ofR. Notice that the graphs are oriented differently along the channels. The surface plots for the 2ses channel at small R are similar to the plots for the ( 1 ,O); channel, and those for the 2 p p ISeat small R are similar to the plots for the (- 1,O)g channel of two-electron atoms. (See Fig. 8.)

evolves into a function similar to the hydrogenic 2 s at large R. The 2pns channel exhibits "-"-type behavior at small R and evolves into a function similar to a hydrogenic 2 p function at large R. These results are shown in Fig. 35. These plots are to be compared with the two graphs shown in Fig. 6 for He. In Fig. 34 we also note that the 2snp and 2pns curves have a pronounced avoided crossing at R = 5 a.u. It was found that the coupling term between the two adiabatic channels is relatively large. Unlike the +/- crossing for the IP0 (see Fig. 2 1) in H-, this avoided crossing cannot be treated diabatically.It turns out that the physical states are better represented by a linear combination of the adiabatic channels. By solving the coupled radial equations,

C. D.Lin

134

-9 2

-

-

+ c 01 +

-2.5-

-3.0

N I l

i

l

l

l

l

l

l

l

~

a .

0.00

.

.

.

I

'

0.25

a/ 'TT

"

'

0.50

0.00

0.25

0.50

a/ IT

FIG.35. Variations of the a-dependent part of the channel functions [see g(a)of Eq. (13)] at various values ofR for the (a) 2st.p and (b) 2 p ~'Po s channels of Be. For small values of R,g(a) shows behavior similar to the radial correlations for the 2sep channel and- radial correlations for the 2 ~channel s (cf. Fig. 7). At large R, g(a)for the 2st.pchannel reduces to a 2s-type radial wave function in the small-a region with vanishing amplitudes for (Y n/2. For the 2 p s channel, g(a) is very small in the small-a region but it behaves like a 2p radial function for (Y n/2.

+

-

-

135

DOUBLY EXCITED STATES

Greene (1 98 1 ) has shown that the quantum defects for the 2snp and 2pns states are much improved over those obtained from a single adiabatic channel calculation. Further work along this line has been shown recently by O'Mahony and Watanabe (1985) on the IDe spectrum of Be. Their work departs from the pure hyperspherical procedure in that they use the R-matrix method to obtain reliable data, but hyperspherical wave functions were used to present a more transparent picture of electron correlationsas well as to delineate the regions of space at which channel coupling occurs. OMahony (1986) also studied the Mg I IDe spectrum and extended the method to analyze the channel interactions in the zDespectrum of A1 I, thus casting the qualitative analysis of Section IV,A on a quantitative basis. As we proceed to doubly excited states converging to the higher N manifold, the correlations and channel behaviors of the states become closer to those exhibited in the corresponding channels in H- and in He. In Fig. 36 we show the potential curves of the three lowest IPo and 3P0states of Li- that lie below the 3s, 3p, and 3d levels of Li. Note that the curves show diabatic crossing similar to those shown in Fig. 12 for He. No systematicstudies ofthe correlations of these systems have been accomplished yet. It would be interesting to examine how the supermultipletstructure ofsection IV is modified for systems like beryllium. Doubly excited states of other alkalineearths and alkali negative ions have

-

-

-020. 1

10

1

I

20

I

1

30

1

40

I

10

I

I

20

I

FIG.36. Potential curves for (a) ' P oand (b) 'Po of Li- convergingto the N

I

30

I

40

= 3 limits of Li.

C. D. Lin

136

also been studied by Greene (198 1) and by Watanabe and Greene (1980). These subjects have been reviewed by Fano (1983). B. DOUBLY EXCITEDSTATESOF HeDoubly excited states of multielectron atoms in general consist of an electron pair outside a compact open-shell core. The core can be viewed as a perturber that scatters individually one of the outer electronswith a possible exchange of spin. For He-, these doubly excited states appear as resonances in e- He scattering. An ever-increasingvolume of experimental results for resonances associated with the N = 2 and 3 limits of He have been accumulated (Brunt et al., 1977; Buckman et al., 1983; Schultz, 1973). The treatment of doubly excited states of complex atoms as an effective two-electron hyperspherical problem relies upon the division of the twoelectron configuration space into three physically distinct regions (Watanabe, 1982, see Fig. 37). Region I corresponds to the close simultaneous approach of the two outer electrons to the core which is practically forbidden by the centrifugal effects in the energy range of less than 25 eV. Regions I1 and 112 correspond to the penetration of one of the outer electrons into the core, independently of the other electron: The problem here reduces to the

I lro

r

0

r2 FIG.37. Divisionof the (r,,r2)plane into three regions. In Region I, both electronsare in the inner region occupied by the core electrons. I n Regions 11, and 11, ,one of the electronsentersthe core region while the other stays outside. In Region 111, the electron pairs stay outside the core; they are under the influence of the attractive potential due to the nucleus and of the core electrons.

DOUBLY EXCITED STATES

137

scattering of a singleelectron by the core. In Region 111, the core is seen by the electron pair as a mere positive charge. It is in this region where the problem can be reduced to the pure two-electron problems which are to be treated using hyperspherical coordinates. An efficient method for treating these problems has been initiated by Watanabe (1982).By limiting the solution ofthe two-electron problems only to Region 111, he incorporated the core penetration by one of the electrons by means of a boundary condition at the interfaces between Regions I1 and 111. This is conveniently achieved by adding to the electron-pair Hamiltonian a singular surface operator which enforces the correct boundary condition at the core limits (Bloch, 1957).The Hamiltonian in the restricted Region I11 reads then:

H( 1,2)= Hdresidual)

+ L,,(E)

(49)

where cq,= tan-*(r,/R) and n/2 - cq,define the two boundaries between Regions I1 and 111, and Lc(E)defines the logarithmic derivative matrix for the correct emergency of the scattered electron from the core. Watanabe (1982)used this method to study doubly excited states of Henear the He( ls2s, ls2p) limits and compared with the results of the experiment of Brunt et al. (1 977). In their more recent work, Le Dourneuf and Watanabe (1986)extended the method to the doubly excited states of Henear the He( ls31) limits. The two-electron normal modes of the doubly excited stateswere found not to be broken by the core’s perturbation and the states can be classified similarly to the doubly excited states of He and H-. A more detailed introduction to these voluminous works is not possible here. We only mention that their work provides a good example that doubly excited states of multielectron atoms can be interpreted based upon our understanding of the doubly excited states of two-electron atoms. Detailed discussions of their work can be found in Watanabe ec al. (1983)and in Le Dourneuf and Watanabe ( 1986). There have only been a few preliminary studies of doubly excited states of other atoms. Clark (1984)has examined the CI wave functions of the negative ions of rare gas atoms Ne-, Ar-, Kr-, and Xe-. By expressing the two-electron part of the wave functions for some of the resonance states in hyperspherical coordinates, it was found that their basic correlationpatterns are identical to the corresponding doubly excited states in He and H-. A more detailed and systematic study of these negative ion resonances is needed to sort out the spectral regularity.

C.D. Lin

138

VIII. Concluding Remarks and Perspectives The study of doubly excited statesin this article has centered on the nature and characteristics of correlations of two excited electrons. By representing wave functions in hyperspherical coordinates and examining the symmetry and regularities of the surface charge densities on the (a,8,,) plane, angular and radial correlation are conveniently visualized. The adoption of approximate correlation quantum numbers K , T, and A allows us to systematize correlationsand exploit new spectroscopicorder of doubly excited states. By analyzing the wave function in the body frame of the atom, these correlation quantum numbers can further be interpreted as being analogous to the moleculelike rovibrational normal modes. It is recommended that each doubly excited state be designated as ,,(K,T)A, 2s+ ‘L”.This notation contains information about how the two electronsare correlated. This new classification scheme incorporates many important features which are easily revealed by changing one or several quantum numbers: (1) States with fixed K, T, A , N , L, S, and 71 but different n belong to the same series. States within the same series have similar correlation patterns. For neutral atoms, the quantum defect along the series is nearly constant. The selection rule for excitation is characteristic of the whole series. (2) A different “series” can be formed by changing the quantum numbers n and N simultaneously. For example, the series N ( N- 1,O); ISechanging with the value of N forms a “double Rydberg series” (Read, 1977; Rau, 1984a). All the states in this series are characterized by having rl = r2 and OI2 = 180”.For large N, each state behaves like a long linear molecule (Lin, 1982c). The correlations of these states are similar to the Wannier state of two continuum electrons near the double ionization threshold. (3) Stateswith identical n, N, K , T,and A but differentL, S, and nexhibit a rotorlike structure if A = 1 or - 1 (Section 111). Different supermultiplet structure can be obtained by ordering the states according to the number of vibrational nodes in the angle 8,*(Section 111). (4) Singly excited states as well as the independent-electron picture are included as a subset of this more general classification scheme. Most of the works using hyperspherical coordinates have been directed at understanding the structure, particularly resonance states. This success so far has not been extended to scattering problems (Lin, 1975a; Miller and Starace, 1980),nor to excited stateswhere the principal quantum numbers of the two electrons are very different (Park et al., 1985;Fink and Zoller, 1985). The origin of this failure is obvious.Although the hypersphericalcoordinates are very close to the independent-particle coordinates in the asymptotic

+

DOUBLY EXCITED STATES

139

region when one electron is inside and the other is far outside, physically when the two electrons are well separated it is more appropriate to represent the system using the independent-particlecoordinates. The small difference between hyperspherical coordinates and independent-particle coordinates at large R introduces a small but ever-present radial coupling between the adiabatic channels. Macek ( 1985) has shown that it takes the coupling of an infinite number of adiabatic channels at large R to reproduce the independent-particle states in the asymptotic region. Experience (Lin, I975a) has shown that the elastic scattering phase shifts in e - H scattering at higher energies are not well reproduced by one-channel or a few-channel calculations. This difficulty in obtaining accurate continuum states is responsible for our inability to obtain more accurate decay widths using hyperspherical coordinates. Attempts to improve the numerical results using the “post-adiabatic” method (Mar and Fano, 1976; Mar, 1977) have not been very successful, Some of the more recent works on the low-lying alkaline earths (O’Mahony, 1985; Watanabe and O’Mahony, 1985;O’Mahony and Greene, 1985)have adopted the R-matrix method to calculate eigenstates and the use of hyperspherical coordinates to analyze the region of configuration space where the coupling occurs. Recognizing the difficulties in applying hyperspherical coordinates to the large-R region, Christensen - Dalsgaard (1 984b) proposed a new procedure by matching the inner (small-R) hyperspherical coordinate wave functions onto the outer (large-R) close-couplingwave functions at a hyperradius R = R,. The value of R, was chosen where the effects of electron exchange and correlations are small. Preliminary results for the elastic phase shifts in e-H scattering indicate that this procedure eliminates the need of coupling many hyperspherical or many close-coupling channels in each region. Further investigationsare needed to test the general usefulness of this procedure. Extensive analysis of doubly excited states so far has been limited to two-electron atoms only. For doubly excited states with a core structure, the designation presented in this article is adequate for describing the states or channels in the region where correlationsare prevalent. In the outer region, the channels are labeled more appropriatelyin terms of independent-particle quantum numbers. General rules for connecting the two regions have not been established yet, The circumstanceswhere the adiabatic approximation is violated also need to be examined. Systematic experimental data on doubly excited states are scarce. The approximate selection rules for photoabsorption from the ground state of helium is well established. It is not clear, in view of the lack of experimental data as well as extensivecalculations, whether the same selectionrules can be applied to photoabsorption from metastable states of helium. There are little

C.D. Lin

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systematics on the cross sections for forming doubly excited states via electron impact or ion impact. Preliminary data from van der Burgt and Heiderman (1985) for e-He collisions seem to indicate that only the 4- states are excited. Doubly excited states can be populated using multistep laser excitationsin which each of the two valence electrons of alkaline earth atoms are excited separately (Gallagher, 1986). So far, almost all the data pertain to doubly excited states of barium, and the principal quantum numbers of the two electrons are quite different. Because of the lack of calculations, it is not possible to know whether the doubly excited states populated in these experiments belong to the A = 1 channels only or whether the A = - 1 and 0 channels are also populated. Doubly excited states can also be selectively populated via double charge-transfer processes by suitably choosing the projectile-target combinations. Although there are many doubly excited states produced this way, the limited resolutions available so far do not permit the identification of individual states. The theoretical methods and procedures discussed in this article can be further extended to three-electron systems to study triply excited states. Although there are a few calculations for triply excited states using hyperspherical coordinates (Clark and Greene, 1980; Watanabe et al., 1982; Greene and Clark, 1984), there is very limited information about the correlations of these systems. Preliminary study of excitations beyond triply excited states so far is limited to the properties of hyperspherical harmonics (Cavagnero, 1984, 1985).

+

ACKNOWLEDGMENTS

I am greatly indebted to Professor U. Fano, who introduced me to the subject of doubly excited states many years ago. I also want to thank many colleagues, whose works are inadequately represented here, who have contributed significantlyto the present-day understanding of doubly excited states. The preparation of this article and the underlying research have been supported by the U.S.Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences.

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Bryant, H. C., Clark, D. A., Butterfield, K. B., Frost, C. A., Donahue, J. B., Gram, P. A. M., Hamm, M. E., Hamm, R. W., Pratt, J. C., Yates, M. A., and Smith, W. W. (1983). Phys. Rev. A 27,2889. Buckman, S . J., Hammond, P., King, G. C., and Read, F. H. (1983). J. Phys. B 16,4219. Burke, P. G. (1968). Adv. A f . Mol. Phys. 4, 173. Burke, P. G., and McVicar, D. D. (1965). Proc. Phys. Soc. (London)86,989. Burke, P. G., and Taylor, A. J. ( I 966). Proc. Phys. SOC.(London)88,665. Callaway, J., and Rau, A. R. P. (1978). J. Phys. B 11, L289. Cavagnero, M. J. (1984). Phys. Rev. A 30, 1169. Christensen-Dalsgaard,B. L. (1984a). Phys. Rev. A 29, 470. Christensen-Dalsgaard,B. L. (1984b). Phys. Rev. A 29,2242. Clark, C. W. ( 1 984). In “Invited Papers to Electron Correlation Effects and Negative Ions” (T. Anderson ef al., eds.) p. 49. Aarhus, Denmark. Clark, C. W., and Greene, C. H. (1980). Phys. Rev. A 21, 1786. Cooper, J. W., Fano, U., and Prats, F. (1963). Phys. Rev. Left. 10, 518. Crane, M., and Armstrong, L., Jr. (1982). Phys. Rev. A 26,694. Ezra, G. S., and Berry, R. S. (1982). Phys. Rev. A 25, 1513. Ezra, G. S., and Berry, R. S. (1983). Phys. Rev. A 28, 1973, 1989. Fano, U. (1969). In “Atomic Physics” (B. Bederson, V. W. Cohen, and F. M. Pichanick, eds.), Vol. 1, p. 209. Plenum, New York. Fano, U. (1981). Phys. Rev. A 24,2402. Fano, U. (1983). Rep. Prog. Phys. 46,97. Fano, U., and Lin, C. D. (1975). In “Atomic Physics”(G. zu Putlitz, Weber, and A. Winnacher, eds.), Vol. 4, p. 47. Plenum, New York. Fink, M., and Zoller, P. (1985). J. Phys. B18, L373. Gailitis, M., and Damburg, R.(1963). Proc. Phys. SOC.London 82, 192. Gallagher, T. F. (1985). Proc. Int. Con$ Phys. Electron. At. Collisions,14th, Palo Alto. Gram, P. A. M., Pratt, J. C., Yates-Williams,M. A., Bryant, H. C., Donahue, J. B., Sharifian,H., and Tootoonchi, H. (1978). Phys. Rev. Left. 38,228. Greene, C. H. ( I98 I). Phys. Rev. A 23,66 1. Greene, C. H., and Clark, C. W. (1984). Phys. Rev. A 30,2161. Hem’ck, D. R. (1 975). Phys. Rev. A 12,4 13. Hemck, D. R. (1983). Adv. Chem. Phys. 52, 1. Hemck, D. R., and Kellman, M. E. (1980). Phys. Rev. A 21,418. Hemck, D. R., and Sinanoglu, (1975a). Phys. Rev. A 11,97. Hemck, D. R., and Sinanoglu, (1975b). J. Chem. Phys. 62,886. Hemck, D. R., Kellman, M. E., and Poliak, R. D. ( 1980). Phys. Rev. A 22, 1517. Ho, Y. K. (1 983). Phys. Rep. 99, I . Ho, Y. K., and Callaway J. (1983). Phys. Rev. A 27, 1887. Ho, Y. K., and Callaway J. (1984). J. Phys. B 17, L559. Iachello, F., and Rau, A. R. P. (1981). Phys. Rev. Letf.47, 501. Kellman, M.E., and Hemck, D. R. (1978). J. Phys. B 11, L755. Kellman, M. E., and Hemck, D. R. (1980). Phys. Rev. A 22, 1536. Klar, H. (1977). Phys. Rev. A 15, 1452. Klar, H., and Fano, U. (1976). Phys. Rev. Lelf. 37, 1132. Klar, H., and Kalr, M. (1978). Phys. Rev. A 17, 1007. Klar, H., and Kalr, M. (1980). J. Phys. B 13, 1057. Le Dourneuf, M., and Watanabe, S. (1986). J. Phys. B, preprint. Lin, C. D. (1974a). Asfrophys. J. 187, 385. Lin, C. D. (1974b). P h p . Rev. A. 10, 1986.

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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL 22

MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION IN COLLISIONS INVOLVING HYDROGEN A TOMS H. B. GILBODY Department of Pure and Applied Physics The Queen's University Belfassl BT7 INN, Northern Ireland

I. Introduction A detailed understanding of collision processes involving hydrogen atoms, while of considerable fundamental interest, is relevant to the physics of controlled thermonuclear fusion devices and to problems in astrophysics. In this short review, the scope is limited to consideration of the charge-transfer process X4+ + H + X(q-I)+ + H+ (1)

involving one-electron capture by singly or multiply charged positive ions and the corresponding ionization process Xq+

+H

+ Xq+

+ H+ + e

(2)

In fusion devices based on the use of magnetically confined high-temperature plasmas, the processes in Eqs. (1) and (2) involving collisions with H', He2+,and with partially or fully ionized plasma and wall-effect impurity ions are of special importance (see, for example, the review by Gilbody, 1979). Such collisions are relevant to energy loss, particle transport, neutral beam heating, and plasma diagnostics. Data are required over a wide energy range. In astrophysics the process in Eq. (1) involving multiply charged ions (which can lead to selective electron capture into excited states) has been invoked as a possible source of X rays in the interaction of low-energy cosmic rays with interstellar hydrogen (Rule and Omidvar, 1979). In addition, a possible explanation of the observed underabundance of certain multiply 143 Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form WSeNed.

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ionized speciesin the interstellar medium may be due to the influence of Eq. (1) at electron volt collision energies (Steigman, 1975). Theoretical studies of Eqs. (1) and (2) are difficult, and experimentaldata are required to determine the range of validity of the approximations used. Cross sections for collisionswith H atoms are not easily inferred from corresponding measurements in molecular hydrogen. It has therefore been necessary to develop suitable targets of atomic hydrogen and the appropriate experimental techniques to permit accurate measurements over a wide energy range. Although experimental studies of Eqs. (1) and (2) commenced about 25 years ago, it is only since 1977 that data on Eq. (1) for multiply charged ions have become available. Corresponding measurements of Eq. (2) commenced in 1981, and it is only during the past two years that some experimental data on state-selective capture in Eq. (1) for multiply charged ions have been obtained. Measurements with highly stripped ions at low impact energies have required the development of special ion sources to provide beams of the required quality. This article is confined to a short description of the different experimental approaches and consideration of some of the experimentaldata in relation to current theoretical predictions. Most of the recent work has been concerned with multiply charged ions, and a discussion ofthe now-extensivetheoretical treatments of charge transfer may be found in the reviews of Greenland (1982), Janev and Bransden (1982), and Janev and Winter (1985).

11. Outline of Experimental Methods A. CROSSED-BEAM METHODS EMPLOYING THERMALENERGY HYDROGEN ATOMS 1. The Modulated Crossed-Beam Technique

The modulated crossed-beam technique pioneered by Fite et al. (1 958) provided the first experimental data on charge transfer and ionization in collisions with hydrogen atoms. It is designedto utilize low-density(typically lo9atoms ~ m - beams ~ ) of highly dissociated hydrogen effusing from a small aperture in either a simple gas discharge tube or a tungsten tube furnace. Most measurements have been carried out using tungsten tube furnace sources which conveniently provide ground-state H-atom beams with a dissociation fraction of at least 90%.

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145

The primary ion beam is arranged to intersect (at right angles) the H-atom beam in a high-vacuum region. Measurements of the yield of slow ions or electrons from the crossed-beam intersection region using parallel-plate collectors and/or mass analysis of the slow ions enable cross sections for charge transfer or ionization to be determined. The signals from the process of interest are generally much smaller than those arising from interactions of the primary beam with the background gas. By mechanically chopping the H-atom beam at a fixed frequency, the required signal can be selectively distinguished by its specific frequency and phase. Calibration is achieved by comparing signals from highly dissociated hydrogen with those from an undissociated hydrogen beam with the same total mass flow. A ratio of the cross section in H to that in H2 is obtained so that, with a knowledge of the latter from static gas target measurements, the cross sections in H can be determined. Measurements of charge transfer using this technique have been limited to H+, N+, 0+,and He2+ impact (Fite et al., 1958, 1960; Stebbings et al., 1960; Gilbody and Ryding, 1966). The ionization of H atoms by H+ impact has been studied by Fite et al. ( 1960) and by Gilbody and Ireland (1963). Cross sections for electron capture into specific excited states may, in principle, be determined by recording the yield of photons emitted in the spontaneous decay of the excited collision products formed in the crossedbeam region. However, since photon detectors with high sensitivity are required, measurements of this type have been confined to studies of Lyman-a emission in H+ impact (Stebbings et al., 1965; Young et al., 1968; Morgan et al., 1973, 1980; Chong and Fite, 1977; Kondow et al., 1974; Hill et al., 1979). 2. The Crossed-Beam Coincidence Method The crossed-beam coincidence method developed in Belfast by Shah and Gilbody (198 la) has made it possible to use low-density highly dissociated hydrogen beams derived from a simple tungsten tube furnace to carry out high-precision measurementswith low-intensityprimary ion beams of both singly and multiply charged ions. The method was intended primarily for studies of ionization but has also been used for measurements of electron capture. In this technique, the two beams intersect at right angles in a high-vacuum (-