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
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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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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 formation are q l and o12, respectively. 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
0.8!
0
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 .
m
E
I I
1
1
t
I
-0
-
L I
0
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-
+ +
k
0.8
-
0.75
0.7
-
0 80
0.71
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
I
I
I
1
I
I
I
0
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
I
/B
r/ 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 positronium. Ho (1 984) has studied the ISe and 3 P r e s ~ n a n cin e ~considerable detail, 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 n
ry =
C cie-qrpr$r7$(1 + Plz)(e-("I+Yr+pp) 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.
36
J. W. Humberston
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 where Eiis the contribution to Ffrom Ps formed in the range E, < E,, < Ei, 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 corresponding velocity incorporated in the final calculation of Ifand The 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 and the observed values of Ifand should then be close to the true values 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
I
I-
-1
25
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 comparable with (If)-' and the typical lifetime spectrum,illustrated in Fig, 7, 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 YI
I
t
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 , calculated using Eqs. ( 5 ) and (6) are given in Table 11. Some of the and of a 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
0.05 1
2
3
L
Positron Energy lev)
5
6
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
I a,
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 , the positronium formation cross Experimental determinations of o 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 and unexpected results. Estimates of , o were first obtained from the plots 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 disa measured by the first two groups, with agreement between the values for , 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 o are large, but recent calculations for He by Khan and Ghosh values of , (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 a obtained from a, on the assumption that the elastic using the values of , 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 remeasuring . , a 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
hv
Ps;-2+
.1 3.2 ns
+
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 given in Fig. 18 for H2, Ar, and Ne show the characteristic rise from the Ps* 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 gases to be around 5 X Ps* per incident e+, and it is noted that this yield 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
10 35
0
z
10
5
.*
0
I 0
5
10
15
20
25
30
35
40
45
50
60
55 XlOl
Channel
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
8
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,
160
l
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1
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1
1
LOO
Time in ns re!ative to t = 0
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
Coils
Positron source
n l I To pump
lions]
ector 1 ’ (positrons)
Scattering tube
Moderator
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 , obelectrons instead of positrons and matching the relative values of a tained 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
-
-
0.06 -
NO
n
ti
1's - 2's
6 0.01-
-
b
0.020.0-
-
A
I
A I
I
I
I
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. Ital. Bordonaro, G., Ferrante, G., Zarcone, M., and Cavaliere, P. (1976). Nuovo Cimento SOC. 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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Coleman, P. G., and Hutton, J. T. (1980). Phys. Rev. Lett. 45,2017-2020. Coleman, P. G., and McNutt, J. D. (1979).Phys. Rev. Lett. 42, 1 130- 1133. Coleman, P. G., Griffith, T. C., and Heyland, G. R. (1972). J. Phys. E 5,376-378. Coleman, P. G., Gritfth, T. C., and Heyland, G.R. (1973).Proc. R. SOC.London Ser. A 331, 56 1 - 569. Coleman, P. G., Griffith, T. C., Heyland, G . R., and Killeen, T. L. (1975). J. Phys. B 8, L185-189. Coleman, P. G., McNutt, J. D., Hutton, J. T., Diana, L. M., and Fry, J. L. (1980).Rev. Sci. Instrum. 51,935-943. Coleman, P. G., Hutton, J. T., Cook, D. R., and Chandler, C. A. (1982). Can. J. Phys. 60, 584- 590. Crompton, R. W., and Elford, M. T. (1973).Ausf.J. Phys. 26,771 -782. Curry, P. J., and Charlton, M. (1 985). Chem. Phys. 95, 3 13- 320. Dababneh, M. S., Hsieh, Y.-F., Kauppila, W. E., Pol, V., and Stein, T. S. (1982).Phys. Rev. A 26, 1252- 1259. Dale, J. M., Hulett, L. D., and Pendyala, S . (1980). Su$ Interjke Anal. 2, 199-203. Davies, S. A., Curry, P. J., Charlton,M., and Jacobsen, F. M. (1985).In “Positron(e1ectron)Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. de Heer, F. J., and Jansen, R. H. J. (1977).J. Phys. B 10,3741-3758. Diana, L. M., Coleman,P. G., Brooks, D. L., Pendleton,P. K., Seay, B. E., Norman, D. M., and Sharma, S. C. (1985a). In “Positron(e1ectron)-Gas Scattering’’ (W. S. Kauppila and T. S. Stein eds.). World Scientific, Singapore, in press. Diana, L. M., Fornari, L. S.,Sharma, S. C., Pendleton,P. K., andColeman,P. G. (1985b).Proc. Int. Conf: Positron Annihilation. 7th (P. C . Jain, R. M. Sinm, and K. P. Gopinathan, eds.), pp. 342 - 344. World Scientific, Singapore. Drachman, R. J. (1982). Can. J. Phys. 60,494-502. Fliieder, K., Fromme, D., Rakh, W., Schwab, A., and Sinapius, G. (1985). J. Phys. B 18, 3347-3359. Fornari, L. S. Diana, L. M., and Coleman, P. G. (1983).Phys. Rev. A 51,2276-2279. Fromme, D., Raith, W., and Sinapius,G. (1985).Proc. Int. Conf:Phys. Electron.At. Collisions, 14th (M. J. Coggiola., D. L. Huestis, and R. P. Saxon, eds.),Abstracts, p. 325. Palo Alto, California. Geltman, S . (1983). J. Phys. B 16, L525-528. Gidley, D. W. (1982). Can. J. Phys. 60, 543 - 550. Gidley, D. W., and Coleman, P. G. (1984).In “Positron Scatteringin Gases” (J. W. Humberston and M.R. C. McDowell, eds.), pp. 65-84. Plenum, New York. Griffith, T. C. (1979). Adv. At. Mol. Phys. 15, 135-166. Griffith, T. C. (1984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), pp. 53-64. Plenum, New York. Griffith, T. C., and Heyland, G. R. (1978). Phys. Rep. 39C. 169-277. Griffith, T. C., Heyland, G. R., Lines, K. S., and Twomey, T. R. (1979). J. Phys. B 12, L747-753. Griffith, T. C., Charlton, M., Clark, G., Heyland,G. R., and Wright, G. L. (1982).In “Positron Annihilation” (P. G. Coleman, S . C. Sharma, and L. M. Diana, eds.), pp. 61 -70.North Holland Publ., Amsterdam. Groce, D. E., CosteUo, D. G., McGowan, J. Wm., and Hemng, D. F. (1969). Proc. Int. Conf. Phys. Electron. At. Collisions, 6th p. 757. Grover, P. S. (1980). 2. Nutur/brsch. 354 350-354. Guha, S., and Mandal, P. (1980). J. Phys. B 13, 1919-1935.
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Hautojarvi, P., Rytsola, K., Tuovinen, P., Vahanen, A., and Jauho, P. (1977). Phys. Rev. Lett. 38,842-844. Heyland, G. R., Charlton, M., Griffith, T. C., and Wright, G. L. (1982). Can. J. Phys. 60, 503-5 16. Heyland, G. R., Charlton, M., Griffith, T. C., and Clark, G. (1985). Chem. Phys. 95,157- 163. Hoffman, K. R., Dababneh, M. S., Hsieh, Y.-F., Kauppila, W. E., Pol, V.,Smart,J. H., and Stein, T. S. (1982). Phys. Rev. A 25, 1393-1403. Horbatsch, M., and Darewych, J. W. (1983). J. Phys. B 16,4059-4064. Howell, R. H., Meyer, P., Rosenberg, I. J., and Fluss, M. J. (1985). Phys. Rev. Lett. 54, 1698- 1701. Humberston, J. W., and Campeanu, R. I. (1980). J. Phys. B 13,4907 -4917. Iakubov, I. T., and Khrapak, A. J. (1982). Rep. Prog. Phys. 45,697-751. Jacobsen, F. M. (1984). In “Positron Scattering in Gases”(J. W. Humberston and M. R. C. McDowell, eds.), pp. 85-97. Plenum, New York. Jacobsen, F. M. (1985). Chem. Phys., in press. Jacobsen, F. M., Gee, N., and Freeman, G. R. (1982). In “Positron Annihilation” (P. G . Coleman, S. C. Sharma, and L. M. Diana, eds.), pp. 92-95, North Holland Publ., Amsterdam. Kauppila, W. E., and Stein, T. S. (1982). Can. J. Phys. 60,471-493. Kauppila, W. E., Stein, T. S., Smart, J. H., Dababneh, M. S., Ho, Y. K., Downing,J. P., and Pol, V. (1981). Phys. Rev. A 24,725-742. Kauppila, W. E., Dababneh, M. S., Hsieh, Y.-F., Kwan, Ch.K., Smith, S. J., Stein, T. S., and Uddin, M. N. (1985a). Proc. Int. Conk Phys. Electron. At. Collisions, 14th, Abstr. p. 303. Kauppila, W. E., Hyder, G. M. A., Dababneh, M. S., Hsieh, Y.-F., Kwan, C. K., and Stein, T. S. (1985b). Proc. Int. ConJ Phys. Electron. At. Collisions, 14th. Abstr. p. 328. Khan, P., andGhosh, A. S. (1983). Phys. Rev. A 28,2181-2189. Klar, H. (1981). J. Phys. B 14,4165-4170. Kwan, Ch.K., Hsieh, Y.-F., Kauppila, W. E., Smith, S. J., Stein, T. S., Uddin, M. N., and Dababneh, M. S. (1983). Phys. Rev. A 27, 1328-1336. Kwan, Ch.K., Hsieh, Y.-F., Kauppila, W. E., Smith, S. J., Stein, T. S., anduddin, M. N. (1984). Phys. Rev. Lett. 52, 1417-1420. Kwan, Ch.K., Hsieh, Y.-F., Kauppila, W. E., Smith, S. J., and Stein, T. S. (1985). Proc. Int. ConJ Phys. Electron. At. Collisions, 14th, Abstr. p. 332. Laricchia, G., Charlton, M., Griffith, T. C., and Jacobsen, F. M. (1985a). In “Positron(e1ectron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Laricchia, G.,Charlton, M., Griffith, T. C., and Clark, G. (1985b). Phys. Lett. 109A, 97- 100. Lee, G. F., and Jones, G. (1974). Can. J. Phys. 52, 17-28. Leventhal, M. (1985). In “Positron(e1ectron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Lynn, K. G., and Frieze, W. E. (1984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), pp. 165- 177. Plenum, New York. Lynn, K. G., Mills, A. P., West, R. N., Berko, S., Canter, K. F., and Roellig, L. 0.(1985). Phys. Rev. Lett. 54, I702 - 1705. McEachran, R. P., Ryman, A. G., and Stauffer, A. D, (1978). J. Phys. B 11,55 1-561. McEachran, R. P., Ryman, A. G.,and Stauffer, A. D. (1979). J. Phys. B12, 1031-1041. McEachran, R. P., Stauffer, A. D., and Campbell, L. E. M. (1980). J. Phys. B 13, 128I - 1292. McNutt, J. D., Summerour, V. B., Ray, A. D., and Huang, P. H. (1975). J. Chem. Phys. 62, 1777- 1789. Manninen, M., and Hautojarvi, P. (1978). Phys. Rev. B 17,2129-2136.
POSITRON AND POSITRONIUM PHYSICS
75
Mills, A. P., Jr. (1980). Appl. Phys. Lett. 37,667-668. Mills, A. P., Jr. (1981). Phys. Rev. Lett. 46,717-720. Mills, A. P., Jr., and Crane, W. S. (1985). Phys. Rev. A 31,593-597. Mills, A. P., Jr., and Pfeiffer, L. (1976). Phys. Rev. Lett. 36, 1389- 1393. Mills, A. P., Jr., and Pfeiffer, L. ( 1977). Phys. Lett. 63A, 1 18- 120. Mizogawa, T., Nakayama, Y., Kawaratani, T., and Tosaki, M. (1985). Phys. Rev. A 31,2171 2179. Mogensen, 0. E. (1974). J. Chem. Phys. 60,998- 1004. Mogensen, 0.E. (1982). In “Positron Annihilation” (P. G. Coleman, S. C. Sharma, and L. M. Diana, eds.), pp. 763-772. North Holland Publ., Amsterdam. Montague, R. G., Harrison,M. F. A.,andSmith,A.C. H.(1984).J. Phys. B17,3295-3310. OMalley, T. F., Rosenberg, L., and Spruch, L. (1962). Phys. Rev. 125, 1300- 1310. Paul, D. A. L., and Bog, N. (1982). Proc. Insi. Swarm. Semin.. 2nd. pp. 65-82. Paul, D. A. L., and Tsai, J.-S. (1979). Can. J. Phys. 57, 1667- 1671. Peisert, A., and Sauli, F. (1984). C.E.R.N. Report, No. 84-08, pp. 84-108. Purcell, L. A., McEachran, R. P., and Stauffer, A. D. (1983). J. Phys. B 16,4249-4257. Raith, W. (1984). In “Positron Scattering in Gases” (J. W. Humberston and M. R. C. McDowell, eds.), pp. 1 - 13. Plenum, New York. Rich, A. (1981). Rev. Mod. Phys. 53, 127-166. Rich, A. (1985). In “Positron(e1ectron)-Gas Scattering” (W. E. Kauppila and T. S. Stein, eds.). World Scientific, Singapore, in press. Roellig, L. O., and Kelly, T. M. (1965). Phys. Rev. Lett. 15, 746-748. Rytsola, K., Rantaspuska, K., and Hautojarvi, P. (1984). J. Phys. B 17,299-317. Schoepf, D. C., Berko, S., Canter, K. F., and Weiss, A. (1982). In “Positron Annihilation” (P. G. Coleman, S. C. Sharma, and L. S. Diana, eds.), pp. 165- 167. North Holland Publ., Amsterdam. Schrader, D. M. (1 979). Phys. Rev. A 20,9 18- 932. Schrader, D. M., and Svetic, R. E. ( 1982). Can. J. Phys. 60,5 17 - 542. Sinapius, G., Raith, W., and Wilson, W. G. (1980). J . Phys. B 13,4079-4090. Stein, T. S., and Kauppila, W. E. (1982). Adv. At. Mol. Phys. 18.53-96. Stein, T. S., Kauppila, W. E., Pol, V., Smart, J. H., and Jesion, G. (1978). Phys. Rev. A 17, 1600- 1608. Stein, T. S., Gomez, R. D., Hsieh, Y.-F., Kauppila, W. E., Kwan, Ch. K., and Wan, Y. J. (1985). Phys. Rev. Lett. 55,488-491. Sueoka, 0. (1982). J. Phys. Soc. Jpn. 51,3757-3758. Sueoka, O., and Mori, S. (1984). J. Phys. Soc. Jpn. 53,2491 -2500. Temkin, A. (1982). J. Phys. B 15, L301-304. Willis, S. L., and McDowell, M. R. C. (1982). J. Phys. B 15, L31-35. Wright, G. L., Charlton, M., Clark, G., Griffith, T. C., and Heyland, G. R. (1983).J. Phys. B 16, 406 5 4088. Wright, G. L., Charlton, M., Griffith, T. C., and Heyland, G. R. (1985). J. Phys. B 18,43274347.
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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.
78
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 the (a,OI2) plane for 2 = 1. The ordinates represent the potentials at R = 1. 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, by displaying plots of I@(R;a,OI2)I2 on the (a,O,,) plane. 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 the ground channel of H- and the surface plots of I@(R;a,012)12 for four 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 near = 90°,in addition to some density in the large-O,, region. The (- 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.
C. D. Lin
96
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 example, the six channels for 1,3H0 have the following lowest states: 4(2,0)!, &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. In Fig. 12 we show the potential curves of He 1*3P0, and 1,3Dethat 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
'
-0.14
t-
-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
-0.18
a 2 u
-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 The channels; (---) the - channels; and (- -) the to the N = 4 limit of He+. (-) 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 have to exhibit the surface charge distribution l@p(R;i2)(2 of the channel 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 T)O.Notice that the triplet state is converging to the N = 3 limit of He+grouped accordingto (K, 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
are defined in the laboratory frame and (pi,?;) in the body where 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 antinodal structure at = R; the density plot for T = 0 clearly shows this 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
\
\
-',\ L(I.1).
N,
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 two-electron angular momentum function is proportional to P:(COS O,,), 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
-___- _ _ _ _ _ _ _ _----
(l,l)* (0,o)( - 1 , I I* (- 2 ,o
-
'PO (N = 31 1
i
,or
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)
-I
0
+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 for states with parity a = (vanish identically at BIZ = 0. There is no 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
20
-
-
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.
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
80
100
R (0.u.1
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 plots clearly indicate that the designations“ 2 ~ ~and s ” “ 2 p p ” are suitable for 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 function, Greene ( I98 1) has shown that the a-dependent part [g(a) of Eq. (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-
i
-3.0
N I l
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
I
20
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
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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.
REFERENCES Bloch, C. (1957). Nucl. Phys. 4, 503. Broad, J., and Reinhardt, W. P. (1976). Phys. Rev. A 14,2 159. Brunt, J. N. M., King, G. C., and Read, F. H. (1977). J. Phys. B 10,433. Bryant, H. C., Dieterle, B.D., Donahue, H., Sharifian,H., Tootoonchi, H., Wolfe, P. A. M., and Yates-Williams, M. A. (1977). Phys. Rev.Left.38,228.
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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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Lin, C. D. (1975a). Phys. Rev. A 12,493. Lin, C. D. (1975b). Phys. Rev. Lett. 35, 1150. Lin, C. D. (1976). Phys. Rev. A 14, 30. Lin, C. D. (1981). Phys. Rev. A 23, 1585. Lin, C. D. (1982a). Phys. Rev. A 25,76. Lin, C. D. (1982b). Phys. Rev. A 25, 1535. Lin, C. D. (1982~).Phys. Rev. A 26,2305. Lin, C. D. (1983a). Phys. Rev. A 27,22. Lin, C. D. (1983b). J. Phys. B 16,723. Lin, C. D. (1983~).Phys. Rev. A 28,99. Lin, C. D. (1983d). Phys. Rev. Left. 51, 1348. Lin, C. D. (1984). Phys. Rev. A 29, 1019. Lin, C. D. (1986). Proc. Int. Symp. Few Body Methods, Nanning, PRC. Lin, C . D., and Macek, J. H. (1984). Phys. Rev. A 29,2317. Lipsky, L., Anania, R., and Conneely, M. J. (1977). At. Data Nucl. Data Tables 20, 127. Macek, J. H. (1968). J. Phys. B 1,831. Macek, J. H. (1985). Phys. Rev. A 3 1 , 2162. Madden, R. P., and Codling, K. (1963). Phys. Rev. Lett. 10, 5 16. Madden, R. P., and Codling, K. (1965). Astrophys. J. 141,364. Manz, J. (1985). Comments At. Mol. Phys. (to be published). Miller, D., and Starace, A. F. (1980). J. Phys. B 13, L525. Morse, P. M., and Feshbach, H. (1953). “Methods of Theoretical Physics,” p. 1730. McGrawHill, New York. Nikitin, S. I., and Ostrovsky, V. N. (1976). J. Phys. B 9, 3 134. Nikitin, S. I., and Ostrovsky, V. N. (1978). J. Phys. B 11, 1681. OMahony, P. F. (1985). Phys. Rev. A (submitted). OMahony, P. F., and Greene, C. H. (1985). Phys. Rev. A. 31,250. OMahony, P. F., and Watanabe, S. (1985). J. Phys. B (submitted). Park, C. H., Starace, A. F., Tan, J., and Lin, C. D. (1985). Phys. Rev. A (submitted). Percival, I. C., and Seaton, M. J. ( 1 957). Proc. Cambridge Philos. Soc. 53,654. Rau, A. R. P. ( 1 984a). In “Atomic Physics” (R. S. Van Dyck, Jr. and E. N. Fortson, eds.), Vol. 9. World Scientific, Singapore. Rau, A. R. P. (1984b). Private communication. Read, F. H. (1977). J. Phys. B 10,449. Rehmus, P. and Berry, R. S . (1979). Chem. Phys. 38,257. Rehmus, P. and Berry, R. S. (1981). Phys. Rev. A 23,416. Rehmus, P., Kellman, M. E., and Berry, R. S. (1978). Chem. Phys. 31,239. Schulz, G. J. (1973). Rev. Mod. Phys. 45, 378. van der Burgt, P. J. M., and Heideman, H. G. M. (1985). J. Phys. B (submitted). Warner, J. W., Bartell, L. S., and Blinder, S. M. (1980). Int. J. Quantum Chem. 18,921. Watanabe, S. (1982). Phys. Rev. A 25,2074. Watanabe S., and Greene, C. H. (1980). Phys. Rev. A 22, 158. Watanabe S., and Lin, C. D. (1986). Phys. Rev. A to be published. Watanabe, S., Le Dourneuf, M., and Pelamourgues,L. (1982). Colloq. Int. 334, J. Phys. Suppl. 43, 223. Weiss, A. W. (1974). Phys. Rev. A 9 , 1524. Wendoloski, J. J., and Reinhardt, W. P. (1978). Phys. Rev. A 17, 195. Woodruff, P. R., and Samson, J. A. (1982). Phys. Rev. A 25,848. Wulfman, C. E. (1973). Chem. Phys. Lett. 23, 370.
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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H. B. Gilbody
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 THERMALATOMS ENERGY HYDROGEN 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%.
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
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 (- lo-’ ton-) chamber, and the slow ions and electrons formed as collision products in the beam-intersection region are collected by a transverse electric field between two high-transparency grids and separately counted by particle multipliers. The H+ product ions from Eqs. ( 1 ) and (2) are distin-
H . B. Gilbody
146
guished from H2+ and the various background gas product ions by their different times of flight to the ion counter in accordance with the charge-tomass ratios. In addition, H+ ions from the ionization process of Eq. (2) are separated from those from Eq. ( 1 ) by counting them in coincidence with electrons from the same events. In order to obtain a measurement of the yield from the charge-transfer process in Eq. ( l), the fast X(q-l)+product ions are separated from the X g + primary ions by electrostatic deflection beyond the crossed-beam region and counted in coincidence with the time-of-flight analyzed H+ ions. Figure 1 shows a typical time-of-flight electron -ion coincidence spectrum correspondingto ionization by proton impact. An analysis of the e-H+ and e-H2+ signals both with the furnace source hot (mainly H) and with the furnace cold (entirely H,) permits determinations of ionization cross sections to an accuracy of about 7%. All the ionization cross sections for different primary ions have been measured relative to those for H+ impact, which in turn were normalized to theoretical values predicted by the Born approximation in the range 1000- 1500 keV. Measurements of cross sections for Eq. (2) have been carried out for H+, He2+ Li(1-3)+ C(2-6)+ N(2-5)+ 0(2-6)+, and Ar(3-9)+ impact at energies within the range 38-9756 keV(ShahandGi1body 1981a,b, 1982a, 1983a,b).
18
I I
H*x I\
28
100
16
I
0
-!-
n, 250
I
750
500 CHANNEL
1000
NUMBER
FIG. 1. Time-of-flight electron-ion coincidence spectrum observed for 400 keV H+ in hydrogen from (hot and cold) furnacesource.Mass numbers shown correspond to products of background gas interactions (Shah and Gilbody, 198 la).
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
I47
The method, in its present form, is not suitable for extending measurements to low impact energies.
3. Photon Emission Spectroscopy and Electron-Capture Studies Using High-Intensity H-Atom Beams Huber et al. (1983) have utilized a Wood's tube (dc gas discharge) to provide a beam of 7090dissociated hydrogen of density - 1O I 3 atoms ~ m - ~ large compared with that of the background gas for studies ofcharge transfer by a number of multiply charged ions. The Amsterdam/Groningen group (CiriC et al., 1985a,b;Dijkkamp et al., 1985; Dijkkamp, 1985) have obtained hydrogen beams of density lOI3 atoms and about 7690dissociated from an rfdischarge source similar to that described by Slevin and Stirling ( 1981). They have used such beams crossed with intense beams of multiply charged ions from an electron cyclotron source to obtain total electron-capture cross sections and data on selective capture into particular excited states of the X(q-I)+ product ion in Eq. (I). Measurements have been camed out for HG+, C3+,C4+,C5+,N5+,N7+, 06+, and 08+ ions within the range 0.8-7.5 keV amu-l. In these experiments, the vuv photons emitted from the spontaneous decay of the excited collision products were detected and analyzed by means of a grazing-incidence spectrometer fitted with a position-sensitive microchannel plate detector. The absolute accuracy of photon detection was estimated to be within 20%. The determination of cross sections for electron capture into particular states from the measured line emission cross sections requires detailed information on energy levels and accurate transition probabilities for all the relevant states. The rather involved optical calibration procedure and the need to take proper account of cascading contributions and polarization anisotropy of the radiation can severely limit the overall accuracy of measurements based on photon emission spectroscopy (PES).
-
-
B. INTERSECTING-BEAM METHODS EMPLOYING FAST HYDROGEN ATOMS There have been a few measurements based on the use of fast intersecting beams. A target beam of H atoms for such measurements may be conveniently produced by electron-capture neutralization of a proton beam in passage through a gas target. While the use of keV energy beams in such experiments facilitates ion collision product detection by means of particle counters, low beam densities and the interaction of both beams with the background gas (even at ultra-high vacuum) generally results in very low
148
H. B. Gilbody
signal-to-backgroundratios. It is also necessary to take proper account ofthe excited population of the H-atom beams. Metastable H 2s atoms and longlived excited atoms in high-n states may be removed or controlled by electric field ionization. Kim and Meyer ( 1982)have carried out the only measurements in which the two beams are arranged to intersect at right angles. They used electric field ionization to define and control the n-state population of highly excited H atoms and obtained cross sections for electron removal from H atoms in states n = 9 - 24 in collisionswith N, 0,and Ar ions in charge statesq = 1- 5. Other measurements have been carried out using the merged-beam configuration. The two beams with slightly different laboratory energies are arranged to merge while moving in the same direction so that the particles collide with very low center-of-massenergies. In this way Koch and Bayfield (1975) obtained cross sections for electron removal from highly excited H atoms by proton impact for c.m. energies in the range 0.4-61 eV. A field ionization technique was used to define the H-atom excited-statepopulation to a band of principal quantum numbers n ranging from 44 to 50. A merged-beam configuration has also been used by Belyaev et al. (1967) and Newman et al. (1982) to obtain total cross sections for electron capture in H+-H and H+-D collisions. In the measurements of Newman et al. (1982), which extend down to c.m. energies of 0.1 eV, cross sections for the resonant charge-transfer process were determined. It was necessary to discriminate against H+products arising from collisionswith H atoms in moderately high-n states between 9 and 19. Atoms in these states could not easily be removed by electric field ionization and had lifetimes long enough to survive in transit to the collision region. The H+ products arising from collisions with such species had laboratory energies rather less than those arising from the resonant charge-transferprocess and could be successfully rejected by the use of a retarding potential analyzer.
C. FURNACE TARGETMETHODS 1. Total and Differential Cross-Section Measurements
In this method, the primary ion beam is passed through a directly or indirectly heated tungsten tube furnacecontaining highly dissociated hydrogen. In most experiments, the beam is passed along the axis of the furnace, and hydrogen gas, which is fed in at a constant rate, is confined mainly to the hot central region by tungsten buttons with suitable beam entrance and exit apertures; the latter must be large enough to accommodatethe fast scattered
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
149
collision products. The method was first employed by Lockwood and Everhart (1962) to study resonant electron capture in small-angle scattering by H+ in H. It has since been used extensively in a number of laboratories mainly for measurements of total cross sections for electron capture over a very wide energy range. Electrostatic analysis of the charge-statepopulation of the emergent beam allowsthe fractional yield ofthe X(@-I)+ions formed in Eq. ( 1 ) as products of single collisions and hence the total electron-capture cross section can be determined provided the effective H-atom target thickness is known. The fraction of H, molecules remaining undissociated in the furnace (typically - 10%at 2500 K) is easily determined by the use of a fast probe beam of H+ or He2+ionsto study the formation of fast H- or He by two-electron capture in single collisionswith H,. A precise determination ofthe total density of H nuclei present is very difficult. In the method used by McClure ( 1966), H+ions were passed through the furnaceand the angulardistribution of fast H and H- collision products was determined.The scattering from hydrogen nuclei was then described assuming a simple Coulomb potential interaction. In a method used by Phaneuf et al. ( 1978), a H S + probe beam is used to record the fraction of He atoms formed in the beam when He instead of H, is fed into the cell at different temperatures. The relative change in target thickness for He and the H H, mixture is assumed to be the same after allowance is made for the fact that each H, molecule dissociates into two H atoms which effuse from the oven& times faster than H2molecules. Bayfield ( 1969) has used a technique in which argon is substituted for hydrogen and the measurements normalized to known H+-Ar and H+-H, cross sections. Others (cf. Shah and Gilbody, 1978)have normalized their measurementsto the H+-H and H+- H, charge-transfercross sections measured by McClure (1966). For measurements with multiply charged ions at low velocity u < 1 a.u., the furnace target technique has been in conjunction with a number of specially developed sources of ions. At Oak Ridge a Penning Ion Gauge (PIG) type source has been used in a series of measurements on multiply charged ions with q up to 12 (cf. Crandall et al., 1980) for velocities down to about lo7cm s-l. Phaneuf (198 1) has developed a technique in which pulses ofslow highly charged ions are produced by focusinga CO, laser beam onto a solid target. The resulting ions are energy selected, passed through the furnace, and the emergent beam analyzed by time-of-flight methods. This technique has been used in a series of measurements extending, in some cases, down to velocities of about 5 X lo6 cm s-l. At Kansas State University, Can et al. ( 1985) have used a recoil-ion source in which fast heavy ions
+
H . B. Gilbody
150
(- 1 MeV amu-l) are passed through a gas and form slow highly charged
ions. This has been used to study electron capture by multiply charged rare gas ions of q up to 10with velocities down to about lo6cm s-l. Very recently at Oak Ridge, Meyer et al. ( 1985a, b) have successfully utilized an electron cyclotron resonance (ECR) source to obtain electron-capture cross sections for fully stripped and H-like ions ofC, N, 0,F, and Ne in the range 0.18 - 8.5 keV amu-l.
2. Translational Energy Spectroscopy If the primary ion energy is well defined, a careful study of the translational energy spectrum of the forward-scattered ions emerging from the furnace can provide detailed information on the collision processes. In this way, Park et al. ( 1977)have obtained cross sections for ionization ofH by 25 - 200-keV protons from an analysis of the differential energy-loss spectra of the forward-scattered protons. In Belfast we have successfully used the translational energy spectroscopy (TES) approach to carry out the first studies of state-selective capture for processes of the type
+
+
XQ+ H 1s 3 X(Q-l)+(n,l) H+ k A E
(3) in which the main excited product collision channels can be clearly identified. Measurements have so far been camed out for N2+,N3+,C2+,and C3+ ions of energies within the range 0.6- 18 keV (McCullough et al., 1983, 1984; Wilkie et al., 1986). In this method, primary ions X*+ of well-defined energy TI are passed through the furnace containing highly dissociated hydrogen. If the kinetic energy of the forward-scattered X(Q+')+ions formed as products of single one-electron-capture collisionsis measured as T, ,then the change in kinetic energy is given by A T = T2 - TI = A E - A K, where A K is the small recoil energy of the target. Provided that A E/Tl << 1 and the scattering is confined to small angles, the observed change in translational energy AT = AE, thereby permitting (within the available energy resolution) measurements of the product yields from specified collision channels. Cross sections for capture into particular states n, 1 of the X(Q-I)+product ions can then be determined from the measured yields by reference to known total one-electroncapture cross sections. The method also permits an unambiguous identification of collision channels associated with any metastable ions present in the XQ+primary ion
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
15 1
beam. However, a detailed analysis is precluded unless the metastable population is known. Care must be taken to ensure that the furnace geometry and angular acceptance of the product ion detector are sufficient to ensure a negligible loss of scattered particles, especially for higher charge states q (see Olson and Kimura, 1982). In our apparatus two hemispherical analyzers were used to select ions with an energy spread down to 0.17q eV FWHM. After acceleration to the required energy the beam was passed through two diametrically opposed apertures in the middle of a directly heated tungsten tube furnace. The potential drop across the entrance and exit apertures did not exceed 0.15 eV. The forward-scattered X ( Q - l ) +product ions were energy analyzed by a third hemispherical electrostatic analyzer operated with a FWHM energy resolution down to 0.17(q - 1) eV. The geometry of the system was such that X(Q-')+product ions scattered through a mean half-angle of 0.5" were recorded. In our latest measurements a position-sensitive detector was used to count the product ions and provide a very efficient means of recording the energy-change spectra. For the cases investigated (see Section III,D) an energy resolution of up to 1.5 eV FWHM has been sufficient to identify most of the important collision channels. A dissociation fraction of at least 92% ensured a negligible contribution from channels involving undissociated H, .
3. Measurements of Electron Capture into Metastable States In the furnace target approach, while any fast excited atoms formed by electron capture remain in the beam, unless the states of interest are sufficiently long lived they undergo spontaneous decay before emerging from the furnace. In a number of studies of electron capture into the 2s metastable state of H by proton impact (Bayfield, 1969; Ryding et al., 1966; Hill et al., 1979; Morgan et al., 1980), measurements were carried out by recording the Lyman-a radiation emitted during electric field quenching of the fast H 2s atoms formed as products of single collisions. Care must be taken to minimize quenching caused by stray electric fields or magnetic fields associated with the furnace operation. The main uncertainties arise from the difficulty of absolute calibration of the Lyman-a detector (see Shah et al., 1980). Similar studies have been carried out by Shah and Gilbody ( 1974, 1978) and by Khayrallah and Bayfield (1975) for capture into the 2s metastable state of He+ by He2+ impact. In the measurements of Shah and Gilbody ( 1978) a rare-gas-filled photoionization chamber was used to provide an accurate absolute calibration of the 304 A radiation detector.
H . B. Gilbody
152
111. Charge Transfer A. ELECTRON CAPTURE BY SINGLY CHARGED IONS The simple one-electron-capture process
H+
+ H 1s
+
HXn,l + H+
(4)
(where Xn, 1 indicates all bound states) is of considerablefundamental interest. While there have been many theoretical descriptions of this process, experimental measurements are still relatively sparse. At low impact energies cross sections for Eq. (4) are dominated by resonant charge transfer. The measurements of Newman et al. ( 1982), using a merged-beam technique (see Section 11), have provided data down to collision energies (i.e., center-of-mass energies) of 0.1 eV. Their results (Fig. 2) are somewhat smallerin the energy range of overlap than the previous results of the merged-beam experiment of Belyaev et al. (1 967) and the modulated crossed-beam experiment of Fite et al. (1962). However, the results of Newman et al. (1982) are in good agreement with the predictions of the semiclasCOLLISION ENERGY
N
5
v
s! =
0 w I- 60:
m
50-
v) v)
-
...-I -...
A
0
g
20-
I
I
,
I
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
153
sical theory of Dalgarno and Yadav (1953) and the fully quantum-mechanical partial-wave treatments of Hunter and Kuriyan ( 1977a,b). The latter calculations predict a cross section for the slightly endothermic H+-D 1 s process a little smaller than that for the corresponding H+- H 1 s process at energies below 4 eV but a negligible difference at higher collision energies. There are a number of measurements of cross sections for Eq. (4) at higher impact energies. The modulated crossed-beam measurements of Fite et al. (1 960, 1962) provide cross sections up to 40 keV. The subsequent measurements by Gilbody and Ryding (1966), using a modulated crossed-beam technique, and by McClure (1963) and Wittkower et al. (1966), using furnace target methods, span the range 2 - l 17 keV. Significant differences between the values of these measured cross sections preclude a detailed assessment of theorectical predictions in this energy range (see Massey and Gilbody, 1974). More recently Hvelplund and Andersen (1982) have used the furnace target method to obtain cross sections in the range 250- 500 keV. These measurements are normalized indirectly to the earlier results of McClure ( 1966) and are in good agreement with cross sections predicted by Belkie and Gayet ( 1977) using the continuum distorted-wave approximation. Cross sections for capture into individual H(n,l) states up to 4s were calculated and the l/n3 scaling rule was then applied to estimate the total capture cross section. Rille et al. (1 982) have observed an interesting isotope effect in the electron-capture differential cross sections for 40 keV amu-I protons and deuterons scattered from either H or D atoms in a furnace target arrangement. Differential cross sections measured at laboratory angles below 0.5 mrad exhibit a significant dependence on projectile species in accord with theoretical estimates based on the classical-trajectory Monte Carlo method. No dependence on target species could be detected. Rille et al. ( 1982) also show that integration of their measured differential cross sections for different projectiles provides essentially identical total electron-capture cross sections for equivelocity protons and deuterons. Total electron-capture cross sections for O+ and N+ impact have been measured using the modulated crossed-beam technique (Stebbings et al., 1960; Fite et al., 1962), while the furnace target method has been applied to measurements for impact of He+ (Hvelplund and Andersen, 1982), Li+ (Shah et al., 1978), B+ and C+ (Goffe et al., 1979), and C+, N+, and O+ (Phaneuf et al., 1978; Nutt et al., 1979). Theoretical studies of these processes have been very limited. In the case of C+, N+, and O+ impact, the possible influence on the measured cross sections of an unknown fraction of metastable ions in the primary beam must be considered. Such an admixture can seriously affect the measured cross sections. Nutt ez al.,( 1979) used a simple beam-attenua-
H . B. Gilbody
154
tion techniqueto show that, in their measurements, while no metastable ions could be detected in C+ and N+ beams from their ion source, the O+beams contained about 20% metastable ions; this prevented accurate measurements. However, earlier measurementsby Fite et al. ( 1962)for 0+-H collisions provided cross sections al0in the range 20 eV-40 keV. These exhibit an energy dependence in accord with the relation a#,*= a - b In E (where a and b are constants) characteristic of a near-resonant reaction. The energy defect for the reaction is in fact A E = 0.019 eV for ground-state products and collidants.
B. ELECTRONCAPTURE BY SLOWMULTIPLY CHARGED IONS 1. Introduction
At velocities u < 1 a.u. (correspondingto 2.2 X lo8cm s-l) electron capture in Eq. (1) may occur very effectively through pseudo-crossings of the adiabatic potential curves of the molecular system formed during the collision. The pseudo-crossings arise in exothermic collisions with a positive energy defect AE at internuclear separations R, = (q - l)/AEa.u. (neglecting polarization) through the influence of Coulomb repulsion between the collision products. For endothermic reactions and for the special case of accidental resonance when A E = 0,crossings of this type cannot occur and the curves diverge as the internuclear separation increases. Experimental measurements in this velocity range are still not extensive, especially for highly stripped ions Xq+ where special ion sourcesmust be used to provide ion beams of the required quality. As noted in Section 11, the furnace target technique has been used in conjunction with PIG, pulsed laser, and ECR ion sources at Oak Ridge and with a recoil ion source at Kansas State University to obtain total cross sections aq,q-lfor ions with q in some cases up to 14 at energies which extend down to about 10 eV amu-'. Measurements of aq,q-!in other laboratories have been limited to ions of lower q and at rather higher energies. Experimentsbased on translationalenergy spectroscopy (TES)carried out in Belfast and measurements using photon emission spectroscopy (PES) by the AmsterdamlGroningen group (to be discussed in Section II1,D) are designed to identify the main collisionchannels and determinecross sections for the production of X(q-I)+ product ions in specified excited states. The TES measurements can also identify the presence of metastable ions in the primary beam which, in some cases, can have a dramatic influence on measured cross sections aq,q-l.While data on state-selective capture are essential for a detailed and unambiguous interpretation of measured values
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
155
of total electron-capture cross sections, some interesting comparisonscan be made between measured values of oq,q- and current theoretical predictions.
2. Measurements with Partially Stripped Heavy Ions with q > 3 For partially stripped (many-electron)heavy ions with qgreater than 3, the number of curve crossings is large, and several generalized theoretical models may be used to express cross sections oq,q-I in terms of q, the target ionization energy and the collision energy. The absorbing sphere model used by Olson and Salop ( 1976)based on the Landau - Zener method requires a high density of curve crossings around an easily defined critical internuclear separation. Olson and Salop (1 976) have calculated cross sections oq,q-!for ions with q ranging from 4to 54 and velocities u S lo8cm s-l. Cross sections only weakly dependent on u and increasingapproximately linearly with q are predicted. A generalized Landau -Zener method used by Presnyakov and Ulantsev (1976)for multiple curve crossingspredicts an approximately q2dependence for total cross sections when q > 10. In the electron tunneling method used by Chibisov (1976) and Grozdanov and Janev (1978) charge exchange is described in terms of a tunneling of the electron from the electron potential well into the quasicontinuous energy spectrum of the ion. For q > 5 an approximately linear dependence of the cross sections on q is predicted for velocities u - lo8 cm s-l. The classical over-bamer model of charge transfer due to Ryufuku et al. ( 1980)predicts a strong oscillatory q dependence of oq,q- . In this model, the electron to be captured must have sufficient energy to overcome the Coulomb potential barrier between nuclei of the colliding systems. An effective charge qeffderived from spectroscopic information is used for partially stripped ions. Ryufuku et al. (1980)also show that the unitarized distortedwave approximation (UDWA)based on the use of traveling atomic orbitals (Ryufuku and Watanabe, 1978) also predicts strong oscillations in the dependence of cross section on q at energies below 10 keV amu-I. Measurements of oq,q-lby Crandall et al. (1980) for Xe(2-12)+, Ar(2-9)+, and Fe5q6+in the range 0.1 - 5 keV amu-' and other low-energy measurements by Huber ( 198l), Seim et al. (1 98 I), and Can et al. (1985) for multiply charged neon and argon ions confirm some of the predictionsof the generalized theoretical treatments. Low-energy values of oq,q-I due to Crandall et al. ( 1980)for Ar(2-6)+includedin Fig. 23 can be seen for q > 3 to increase with q and be almost independent of velocity. Crandall et al. (1980) have also observed that, for values of q in the range 5 -9, cross sections for Arq+ and Xeq+ions with the same charge are not greatly different. Crandall et al. (1980) observed that equivelocity cross sections uq,q-l
H . B. Gilbody
156
measured for 0, Ar, Fe, and Xe ions with the same q (for q = 4, 5, or 6) exhibit a steady increase through the sequence as the number of electrons remaining on the primary ion increases. This feature has subsequently been considered by Larsen and Taulbjerg ( 1 984). They have used a simple model with quantum-defect wave functions for the final states of the captured electron to calculate cross sections for a sequence of low-energy ions with q = 6. These theoretical values (Fig. 3) at a velocity of 0.1 a.u. can be seen to be in good accord with experimental values. In this case cross sections increase by a factor of three in going from C6+to ions of high atomic number where the cross section attains a “saturation” value. Larsen and Taulberg ( 1984) have noted that the influence of the electrons on the ion core is even stronger at lower velocities. For u = 0.05 a.u. the corresponding cross sections exhibit an increase of about a factor of ten as the atomic number increases over the same range. Figure 4 shows the dependence of cross sections o ~ , ~on- q for FeO- 14)+ in H for various velocities u in the range 0.6 - 1.3 X lo7 cm s-l and for Xe(3-12)+ in H for u = 4 X lo-’ cm s-I measured by Phaneuf (1983) and Crandall et al. ( 1980), respectively. In view of the very weak dependence on u, the fact that the cross sections were measured at different velocities can be ignored. Although there are large experimental uncertainties, the measured cross sections increase with an approximately linear dependence on q and are cm2. There are significant roughly in accord with the relation o = q X deviations for Feq+ at q = 4 and 12 and for Xeq+ at q = 12. Cross sections 80
I
I
Xs*+ H -X
01
I
6
1
1
I
I
5*+H+
1
+r
I
T
I
I
I
10 11 18 22 26 ATOMIC N U M B E R OF P R I M A R Y ION
“
’’
I
S4
FIG.3. Total electron-capturecross sections 0 6 5 for X6+ions (of specified atomic number) in H at 0 = 0.1 a.u.(fromLarsen and Taulbjerg, 1984).(0),theory (Larsenand Taulberg, 1984); (O),experiment (Crandall et al., 1980 Phaneuf el al., 1982; Phaneuf, 1983).
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
157
FIG.4. Total electron-capturecross sections 09.q-I for Fe'+-H and Xe9+-H collisions at velocity B = lo7cm s-I (from Phaneuf, 1983). (O),Fe9+ H (Phaneuf, 1983); (0),Xeq+ H (Crandall ef a/., 1980); (A), theory-absorbing-sphere model (Olson and Salop, 1976); (T), theory-electron-tunneling model (Grozdanov and Janev, 1978); (C), theory-classical overbarrier model (Ryufuku et al., 1980);(E), empirical relation u = q X lopt5cm2.
+
+
predicted on the basis ofthe absorbing sphere model (Olson and Salop, 1976) and the electron tunneling model (Grozdanov and Janev, 1978), also included in Fig. 4,can be seen to be between 50% and 80% larger on average than the experimental values. Values based on the classical over-barrier model of Ryufuku et al. (1980) exhibit an oscillatory dependence on q and are mainly smaller than the experimental values. Dips in the theoretical values at q = 4 and 12 do correlate with the experimentally observed dips, but there are anticorrelations for other values of q. Recent measurements of for A1(*-lO)+ions in H by Phaneuf et al. (1 985) show a similar dependence on q. Observed departures from simple linear scaling do not correlate closely with the strong oscillations predicted by both the classical calculationsof Ryufuku et al. ( I 980) or the multichannel Landau-Zener calculations ofJanev et al. ( 1983)for bare projectile ions. However, Phaneuf el al. ( 1 985) suggest that the absence of strong oscillations may be ascribed to the presence of ionic core electrons which for partially stripped AIQ+ ions, remove the 1 degeneracy characteristic of hydrogenic product ion states.
H. B. Gilbody
158
3. Measurements with Partially Stripped Heavy Ions with q < 3 Cross sections for (many-electron) heavy ions of q < 3 do not exhibit behavior in accord with the predictions of generalized theoretical models since only a limited number of pseudocrossings are generally available. In Belfast we have measured cross sections a21for electron capture by Ba2+, Ti2+,Mg2+,Cd2+,Zn+, Kr2+,and B2+in H within the energy range 0.8 -40 keV (McCullough et al., 1979). While a detailed analysis is precluded without a knowledge of the relative importance of possible excited-stateproduct channels, a few general comments can be made. The cross sections 021for the moderately exothermic processes Cd2+ H and Zn2+ H, which are likely to involve pseudocrossingsat the moderate internuclear separations8.3 and 6.2 a.u., respectively, are very large at low velocities. Cross sections for B2+, Kr2+,and Mg2+ H which have either small or large exothermicities, for the endothermic process Ba2+ H, and for the near-resonant Ti2+ H process are found to decrease rapidly with decreasing velocity. For Cd2+in H, for which R, = 8.3 a.u. for ground-state species, Crothers and Todd (1980a) have used a two-state impact-parameter treatment to predict cross sections cr21in good accord with experiment. In the case of Mg2+ H, for which R, = 18.9 a.u. for ground-state species (which are believed to be dominant), two-state impact-parametercalculations by both Bates et al. ( 1984)and by Crothers and Todd (1980a)are in good accord with the measured cross sections. Cross sectionsa,, and a32 for A12+and A13+inH measured by Phaneuf et al. (1985) at 5 1.8 and 35.9 eV amu-I, respectively, are in reasonable accord with their theoretical estimates based on perturbed stationary-state close-coupling calculations.
+
+
+
+
+
+
4. Measurements with Bare Nuclei and Highly Stripped Ions
Measurements of cross sections crq,4.4-1 for electron capture by bare nuclei or highly stripped (few-electron)ions at low velocities are still not extensive since suitable sources of ions are only just becoming available. Crandall et al. (1979) have used PIG source to obtain crosssections for Heand Li-like ions of ByC, N, and 0 in H in the velocity range 0.4 - 1.1 X lo8 cm s-'. A pulsed laser ion source has been utilized by Phaneuf et al. (1982)to measure cross sections for C(3-a)+and 0(2-a)+ in H within the range 0.0 1 - 10 keV amu-I. Panov et al. ( 1983)have reported some measurements for fully stripped and H- and He-like ions of C, N, 0,and Ne as well as for fully stripped Ar ions in the range 0.33 - 8.8 keV amu-'. Similar measurements have also been carried out by Bendahman et al. (1985) for fully stripped N, 0,and Ne ions in Hat energieswithin the range 1 - 5 keV amu-I. Recently an ECR source has been used by the Amsterdam/Groningengroup
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
159
(Cirit et al., 1985a,b; Dijkkamp et al., 1985) to obtain cross sections aq,4-1 for highly and fully stripped C, N, and 0, ions as well as line emission cross sections (see Section II1,D) within the range 0.8-7.5 keV amu-I. At Oak Ridge an ECR source has also been used successfully(Meyer et al., 1985a,b) in very recent measurementsof aq,q-I for fully stripped and H-like ions of C, N, 0, F, and Ne ions in H within the range 0.18 - 8.5 keV amu-I. The process of electron capture by fully or highly stripped (few-electron) ions in collisions with atomic hydrogen at low velocities involves one or a limited number of curve crossings leading to selective population of excited states of the product ion. There have been many theoretical studies of such processes, most of which have been based on a molecular orbital (MO) approach. Early few-state MO calculations (cf. Hare1 and Salin, 1977; Vaaben and Briggs, 1977)provide a satisfactory general description of the electron-capture mechanisms but lead to cross sections of limited accuracy. For example, in the case of C6+-H collisions, the calculations correctly predict (see Section III,D) that transitions occur primarily through radial coupling through the Sgc~-4fc~pseudocrossing at an internuclear separation of 8 a.u., thereby leading to C5+product ions predominantly in the n = 4 state. The most sophisticated of the more recent calculationshave been camed out by Green et al. (1982) for C6+-H collisionsand by Shipsey et al. (1 983) for 08+H collisions. These are based on expansions of the time-dependent electronic wave function into a large set of traveling molecular orbitals and the solution of the semiclassicalcoupled equations. Similar results have also been obtained for a number of bare nuclei by Fritsch and Lin ( 1984b)using a modified atomic orbital expansion in which molecular binding effects in close collisions are taken into account by including united-atom orbitals at the two collision centers. for C6+ Figure 5 shows both measured and calculated cross sections aq,4-1 in H within the range 0.1 - 10 keV amu-I. The measurementsof Meyer et al. (1985a),which are the most extensive and involve the smallest uncertainties, can be seen to be in reasonable accord with the low-energy measurements of Phaneuf et al. (1982) and the higher-energy measurements of Dijkkamp et al. (1985). The cross sections measured by Panov et al. (1983) exhibit a different energy dependence and a large scatter which might reflect errors arising from the use of a hydrogen target with a dissociation fraction of only 35%. In Fig. 5 cross sections based on the 33-state MO calculations of Green et al. ( 1982)can be seen to be larger than the measured values, but above 2 keV amu-1 agreement is within the experimental uncertainty limits. The values calculated by Fritsch and Lin (1984b) using a 35-state A 0 expansion are in better agreement with experiment. Cross sections predicted by Janev et al. ( 1983)based on a multichannel Landau -Zener calculation with rotational
-
H . B. Gilbody
160
W W
2
W
0.1
1
10
Energy (keV amu-1) FIG. 5. Electron-capture cross sections for C6+ ions in H (from Meyer ef al., 1985b). Experimental data: (m), Meyer et al. ( I 985b);(+), Phaneuf el al. (1982);(V), Panov ef al. (1983); (O), Dijkkamp ef a/. (1985). Theoretical data: (-), 35-state A 0 expansion (Fritsch and multichannel Lin, 1984); (---), 33-state MO expansion (Green et al., 1982); (---), Landau-Zener calculation with rotational coupling (Janev et al., 1983).
coupling, also shown in Fig. 5, can be seen to be in reasonable accord with measured values above about 1 keV amu-l. At lower energies the calculated values become increasingly larger than experiment. A similar degree of accord with theory has also been observed by Meyer et al. ( 1985b)in compar- , W +and 08+ in H. isons with measured values of C T ~ , ~for Meyer et al. (1985a,b) have noted that, in their measurements ofaq,,-, for fully stripped and H-like C, N, 0, F, and Ne ions below 3 keV amu-I, the magnitude ofthe cross section at the same velocity is primarily dependent on the projectile charge q. The insensitivity to the presence of the 1s core electron in the H-like projectiles is believed to reflect the fact that electron capture, in all the cases considered, is expected to occur mainly into excited states n > 3 ofthe product ion, the binding energiesof which are not strongly influenced by the 1 s core. Meyer et al. (1 985a,b) have also observed that their measured cross sec~ -both ~ H and H, targets exhibit an oscillatory dependence on q tions C T ~ , in at low velocities. This is illustrated in Fig. 6, which shows the charge depen- , fully stripped projectiles on H and H, at velocities ranging dence of C T ~ , ~for from 0.3 to 8.0 keV amu-I. In both cases the oscillations become more pronounced at lower velocities. For H, cross sections for primary ions of even charge are up to a factor of 3 smaller than those of odd charge, while for H2 the opposite is observed. Above about 3 keV amu-I, the oscillations are absent and cross sections can be seen to exhibit a nearly monotonic dependence on q.
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
16 1
I I
I
I
I
I
6 7 8 9 4 0 PROJECTILE CHARGE
6 7 8 9 4 0 PROJECTILE CHARGE
FIG. 6. Dependence of measured electron-capture cross sections a,,-, on q for fully stripped primary ions of C, N, 0, F, and Ne in H and H, (from Meyer et al., 1985b). (U), 0.3 keV amu-I; (a),0.6 keV amu-'; (A), 1.0 keV amu-l; (A), 3.0 keV amu-l; (e),8.0 keV amu-I.
The well-defined oscillatory dependence of c ~ on~ q in ~ these ~ -cases, ~ which is not clearly observed for partially stripped many-electron ions (see Section III,B,2), reflects the strongly selective nature of the capture process which, at low velocities, is dominated by a single excited state. The observed oscillatory behavior of og,q-lis correctly predicted by Ryufuku et al. ( 1980) in terms of both the UDWA treatment and the classical over-barrier transition model as well as the multichannel Landau - Zener calculations of Janev ef al. (1983). Meyer et al. (1985a) have also used a simple Landau -Zener curve-crossing description to explain why relative maxima in cg,q-, occur for odd values of q for H and for even values of q for H, . The higher binding energy of H, shifts the region over which curve crossings occur to smaller internuclear separations while crossings corresponding to a particular product n state occur at larger internuclear separations. This is believed to result in a change from a large transition probability for electron capture by projectiles of odd q in H to a large transition probability for projectiles of even q in H, . 5. Collisions Involving Excited H Atoms
-
Burniaux et al. (1977) have studied the process
+
He2+ H*
He+
+ H+
H. B. Gilbody
162
using the merged-beam technique to provide center-of-massenergies in the range 0.25 -478 eV. Field ionization was used to specify the populations of highly excited atoms in the H beam which was produced by charge-transfer neutralization of protons in passage through a gas. Cross sections for H atoms with principal quantum numbers n ranging from 8 to 24 were found to have very large values between about and cm2 almost independent of energy below 50 eV.
c. ELECTRONCAPTURE BY FASTMULTIPLYCHARGED IONS At high velocities, v > 1 a.u., the now quite extensive measurements of total cross sections crq,q-l for the process in Eq. ( 1) carried out in Oak Ridge, Belfast, and a few other laboratories have been based on the furnace target technique. Results are available for multiply charged ions of He (Shah and Gilbody, 1978; Bayfield and Khahrallah, 1975; Hvelplund and Andersen, 1982), Li, B, and C (Shah et al., 1978; Goffe et al., 1979), C, N, and 0 (Phaneufet al., 1978),Si (Kim et al., 1978a),Fe (Gardner et al., 1977),and 0, Fe, Mo, Ta, and Au (Meyer et al., 1979). At velocities v > 1 a.u. cross sections for all the cases investigated decrease with increasing velocity and, for a given velocity, increase with q. However, for q > 3 cross sections at a particular velocity are only weakly dependent on the projectile species. For example, Meyer et al. (1979) show that the variation of measured cross sections aq,q-Ifor 0,Si, Fe, Mo, Ta, and W ions with q = 7 and 14 over the range 2.5 -6.5 cm s-' can be described to within about 30% by single curves corresponding to each charge state. A typical set of data for C(I-@+ions which illustrates the strong dependence of oq,q-lon q is shown in Fig. 7. Values of cross sections aq,q-lfor the electron-lossprocess Cq+
+ H 1s
-+
+
C(q+I)+ H(X:c)
+e
(6)
(where H(Z:c) indicates all final bound and continuum states of H ) are also shown for C(1-3)+ions. These can be seen to begin to exceed the corresponding values of oq,.q-lat high velocities. Included in Fig. 7 are cross sections a65 for fully stripped ions calculatedby Ryufuku and Watanabe (1979a) by the use of the unitarized distorted-wave approximation UDWA. In this velocity range, the theoretical values can be seen to be considerably larger than the experimental values of 065. Better agreement is obtained with the cross sections 0 6 5 calculated by Olson and Salop (1977) using the classical Monte Carlo trajectory (CTMC) method first developed by Abrines and Percival(l966). The method, which Olson and Salop ( 1977)have applied to velocities in the range 40 - 200 keV amu-I,
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
163
1
v)
v)
0
a 0
1
1
ENERGY ( keV ornu-'
FIG.7. Cross sections u ~ , ~for - , ane-electron capture and a,,,, for one-electron loss for Cq+ionsin H. (0),(O),uq~q-,,a,,q+l (Goffeef al..1979);(+),u,,,-,(Phaneufetal., 1978);(0S), a65CTMC theory (Olson and Salop, 1977);(RW), a6sUDWA theory (Ryufuku and Watanabe, 1979).
is based on the solution of the twelve coupled Hamilton's equations of motion for a three-body system with a random distribution of initial conditions. Between 2000 and 10,000trajectories are consideredin order to obtain cross sections with statistical uncertainties less than about 10%. By making allowance for electron screening by the use of effective charges q,,derived from spectroscopicdata, Olson and Salop ( 1977) have estimated cross sections ( T ~ , ~ - for partially stripped ions which are in generally good agreement with experimental data. This is illustrated by the comparison shown in Fig. 8 for 0 ions ranging from q = 3 to 8. Electron capture by fast partially stripped ions has also been considered by Eichler et al. (1 98 1 ) in calculations for Li, C, N, and 0 ions within the Oppenheimer- Brinkman -
H. B. Gilbody
164
PROJECTILE ENERGY (keV/omu)
2
3
200
to0
50
4
5
6
7
RELATIVE VELOCITY (!08 cm/sl
FIG.8. Cross sections u ~ , ~for - ,electroncapture by 0 4 + ions in H (from Meyer et al., 1979).
(m), Meyer et a/. ( 1 979), Phaneuf et al. (1979); (O), CTMC theory, Olson and Salop (1977).
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
165
Kramers (OBK) approximation, taking into account both the departure from the Coulomb potential and the effect of occupied projectile states. These calculations indicate that for q > 3, except at high energies, there is no appreciable screening effect so that q =
[email protected], Eichler et al. (1 98 1) also note that the unjustified use ofa smaller value of qeEintheir calculations would improve the agreement with experiment. At velocities beyond the peak value of the cross section, experimental data ~ q - > ~ 3 are found to scale according to simple relations of the for ~ 7 ~ ,for form (7) 0g.q-I - ooq" where the scaling parameters ooand n depend on the velocity. Values of n are found to vary from about 1.5 to 3 over the velocity range 3-6.5 X lo8cm s-I. For example, in the case of Li, B, and C ions, Goffe et al. (1979) find that at about 6 X lo8 cm s-l, n = 3,2.4, and 2.5, respectively. Crothersand Todd (1980b) have shown that on the basis of the OBK approximation, the continuum-distorted-wave, and the continuum-intermediate-state description, a q3 cross-section scaling might be expected at the highest velocities considered experimentally. , Fe and Mo ions in H obtained by Meyer et al. Cross sections ~ 7 ~ , ~ - for ( 1979), respectively, show localized departures from the simple qn scaling than expected relation. For 60 keV amu-' Mo ions, smaller values of og,q--l are observed when q = 6 and 14. These values of q correspond to 4p6 and 3d1° closed-shell ground-state configurations. A similar anomaly is observed for Fe8+ 3p6 ions. In measurements with fast multiply charged ions ofTa, W, and Au, Meyer et al. (1979) find cross sections ~ 7 ~ ,which ~ - ~ exhibit oscillatory deviations about the smooth qn power-law function. This behavior is not correlated with ion binding energies and electron configurations. Kim et al. (1978b) have suggested that the effect is due to interference between the scattering amplitudes arising from the long-range Coulomb and short-range screened Coulomb forces acting on the transferred electron during the collision. However, the more recent calculations of Eichler el al. (198 l) within the OBIS approximation, in which it is shown that for q > 3 electron screening effects are negligible, fail to account for the observed oscillations and suggest that another mechanism is involved. For the fully stripped ions HeZ+,Li3+, B5+,and c6+ (for which q = 2 ) Goffe et al. (1979) show that measured ratios ag,q-l / Z 3when plotted against velocity approach a common curve at sufficiently high velocities (which increase with 2 )consistent with q3 scaling. In this region it is found that gq,g-l= (28.28)
[email protected]
a.u.
166
H . B. Gilbody
Crothers and Todd ( 1980) have subsequently used the Bohr-Lindhard model of electron capture from light target atoms to predict that, at high velocities, = (25.14)q3a:v-’
a.u.
an expression which is in good agreement with that obtained experimentally. measured in H and H, is of The relationship between cross sections some interest since, in the absence of expenmental data on H, it has often been assumed that at sufficiently high velocities H, may be considered to be equivalent to two free hydrogen atoms. At velocities u > 1 a.u., cross sections oq,q-,for H, are found to be well described by the simple relation of Eq. (7) but the scaling parameters n and a,,differ from the corresponding values in H. The ratio R = a( H)/a(H,) ofthe electron-capture cross sectionsin H and H, for all the available experimental measurements shows a definite trend (see Greenland, 1982; Knudsen et al., 1981) to values less than f at high velocities.At v = 3 a.u. the average value ofthe observed ratio R is about 0.4, a value which compares favorably with the theoretical estimates of R by Tuan and Gerjuoy (1960) for high-velocity proton impact. D. ELECTRONCAPTURE INTO SPECIFIED EXCITEDSTATES
I. Collisions Involving H+ and Hez+ Ions As noted in Section I1 (with references therein) the modulated crossedbeam technique has been used at impact energiesin the range 0.5 - 30 keV to obtain cross sections for H 2 p and H 2s formation in H+-H collisionsfrom observations of the spontaneous and electric-field-induced Lyman-a radiation. The furnace target has also been used to extend the measurements of cross sections for H 2s formation to 100 keV. Results of the most recent measurementsof H 2s formation carried out by Hill et al. ( 1 979) and by Morgan et al. (1980) using the furnace target technique are shown in Fig. 9 together with a number of theoretical predictions. The experimental data, while based on the use of different normalization procedures, are in good agreement at energies about about 3 keV. They are also in good agreement with the earlier results of Morgan et al. (1 973) and Chong and Fite (1977) (not shown) obtained using the modulated crossedbeam technique in the respective energies ranges 5 - 26 and 6 - 25 keV. At energies below 3 keV the cross sections measured by Hill et al. ( 1979) are seen to be substantially larger than those of Morgan et al. (1980) and in better agreement with the results of the seven-state (4 hydrogenic and 3 pseudostates) close-coupling calculations of Cheshire et al. ( 1970). On the
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
ENERGY
(keV 1
+
-
167
+
FIG.9. Electron-capturecross sections for the process H+ H 1s H 2s H+. Experiill er al. (1979). Theory:(A), Born approximation (Bates ment: (O),Morgan et al. (1 980);(A), H and Griffing, 1953); (B), 4-state close coupling (Cheshire, 1968); (C) 7-state close coupling (Cheshire et al., 1970); (D), 10-state molecular treatment (Schinke and Kruger, 1975); (E), 5-state molecular treatment (Chidichimo-Frank and Piacentini, 1974).
other hand, the low-energy results of Morgan et al. (1 980) are in better agreement with the calculations of Chidichimo -Frank and Piacentini ( 1 974) and of Schinke and Kruger (1 976) based on a multistate molecular approaches. Results (not shown) due to Fritsch and Lin (1982) based on a two-center atomic-expansion method and by Kimura and Thorson (198 1) based on multistate molecular orbital calculationslie between the two sets of experimental data below 3 keV, but are in good accord with both experiments at higher energies up to about I5 keV. It can be seen that the two-state impact-parameter calculations of Cheshire ( 1968) with exchange provide results which are in poor accord with experiment. At high energies the experimental cross sections are approaching the Born values (Bates and Griffing, 1953) but are considerably smaller than the close-coupling estimates of Cheshire et al. (1970). The electron-capture process
+
He2+ H 1s
-
He+(n,l)
+ H+
(8) takes place predominantly through the accidentally resonant He+ 2s and He+ 2p product channels. The furnace target technique has been used to measure cross sections uzsfor He+ 2s formation for 4 - 343 keV 3He2+ions
H. B. Gilbody
168
(Shah and Gilbody, 1974, 1978) and for 7-70 keV 4He2+ions (Khayrallah and Bayfield, 1975) through observations of the 304 .k Lyman-a radiation arising from the electric-field-quenched metastable collision products. The measurements of Shah and Gilbody (1978), which are believed to be the most accurate, utilized a rare-gas-filled photoionization chamber to provide an absolute calibration of the 304 .k radiation. There have been a number of multistate calculations of tsZS(see, for example, discussions by Bransden et al., 1980; and Bransden and Noble, 1981) and the results exhibit large differences which in some cases are difficult to understand. An eight-state close-coupling calculation by Rapp (1974) gives cross sections which are in good accord with experiment for 3He2+energies of about 80 keV. An impact-parameter approximation by Piacentini and Salin ( 1977) based on molecular states provides results in reasonable accord with experiment below 10 keV, while the results of a translationally invariant 10-statemolecular orbital expansion (Hatton et al., 1979)is in satisfactory agreement with experiment for energies up to about 50 keV.
1
ENERGY (krV arnu") 2 4 6 8 10 12 1416
- 0 2
He2'+H S
P
d
I
f
+
-
+
FIG.10. Cross sections uZpfor the process (shown 0) He2+ H He+ 2 p H+determined by CirZ et al. (1985a) from observations of the He 11transitions indicated (wavelengths are given in nm). Also shown are theoretical estimates of q,,: (-), molecular orbital calculations(Winter and Hatton, 1980);(- * -), molecular orbital calculations (Kimura and Thorson, 1981); (---), atomic orbital calculations (Bransden and Noble, 198I); ( ), atomic Hylleraas expansion calculations (Ludde orbital calculations (Bransden et al., 1983); (--), and Dreizler, 1982).
.-.
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
169
The recent measurementsofCirit et al. ( I985a)based on photon emission spectroscopy (PES) provide the first experimental data on the He+ 2p formation channel in Eq. (8). In measurements within the range 1 - 10 keV amu-I they obtain emission cross sections gem for np- 1s transitions with n = 2, 3, 4 (see Fig. 10). These are related to the previously measured 2s capture and total electron-capture cross sections a(2s) and 02,(Shah and Gilbody, 1978; Nutt el al., 1978) based on the furnace target technique through the expression 4
a2,- ~ ( 2 s=)
C C,.,,(np
- Is)
n-2
Results were shown to be in good accord with this expression, thereby demonstrating satisfactoryaccord between the calibration procedures of the two experiments. ) He+2p formation obtained by Figure 10 showsthe cross sections ~ ( 2 pfor CiriC el al. (1985a) after correction for cascade from the n = 3 level and a correction (of not more than 15%) for polarization anisotrophy of the Lyman-a radiation. Also shown in Fig. 10 are theoretical predictions based on the molecular orbital calculation of Winter and Hatton (1980) and Kimura and Thorson ( I 98 I b), the atomic orbital calculations of Bransden and Noble ( 198I ) and Bransden et al. (1983),and the Hylleraasexpansion calculations of Ludde and Dreizler (1982). All these calculations are in good general agreement with experiment. 2. Collisions Involving Slow Multiply Charged Ions
As noted in Section II1,B at low impact velocities (v < 1 a.u.), where a molecular description of the collision is appropriate, electron capture in exothermic processes of the type
+ H 1s
-
+ + AE
X(g-*)+(n,l) H+
(9) may take place very effectivelythrough pseudo crossings of the adiabatic potential energy curves describing the initial and final molecular states. The crossings take place at internuclear separationsR , (q - 1)/AE neglecting polarization and may lead to selectivecapture into a number of excited states (n,l) of the product ion. Transitions involving fully stripped or highly stripped few-electron ions generally involve one or a small number of such crossings. For many-electron heavy ions with q 6 3 only a small number of suitable crossings are generally available, but for q > 3, where the total cross sections are practically independent of energy, a large number of crossings are involved. Xg+
2
170
H. B. Gilbody
As noted in Section 11, the first detailed studies of such processes have been camed out in Belfast using the technique of translational energy spectroscopy (TES) in highly dissociated hydrogen within a tungsten tube furnace. Measurements A T of the forward-scattered X(9--)+ product ions allows, within the available energy resolution, determination of the product ion yields correspondingto particular collision channelscharacterizedby energy defects A E = A T. Measurements have so far been camed out for N2+,N3+, C2+, and C3+ ions of energies within the range 0.6 - 18 keV (McCulloughet al., 1983, 1984; Wilkie et al., 1985, 1986). In the recent measurements based on photon emission spectroscopy (PES) the Amsterdam/Groningen group have derived cross sections for capture into specified states from measured line-emissioncross sectionswith a knowledge of the energy levels and transition probabilities of the relevant and 08+ excited states. Data are available for C3+,C4+,C6+,NS+,N7+,06+, for a number of energies within the range 0.8 -7.5 keV amu-' (Cine et al., 1985a,b;Dijkkamp et al., 1985; Dijkkamp, 1985). While the rather involved calibration procedure necessarily means that the cross sections are subject to greater uncertainties than those obtained using the PES approach, the TES technique does provide a much higher energy resolution. However, it cannot detect electron capture into ground or metastable states or into states which decay by autoionization.Clearly the TES and PES methods are complementary. TES studies of state-selectivecapture by N2+ionsinH (McCullough et al., 1983; Wilkie et al., 1985) were camed out at impact energies in the range 0.6-8 keV. Figure 1 1 shows the energy-change spectra obtained at 2 and 6 keV. The resolution is sufficient to reveal the presence of four product N+ 1 s22s2p3 channels correspondingto the formation of N+ 1s22s2p3300, 3P0,N+ ls22s2p3IDo, and N+ ls22s22p3p3Pions with energy defects A E of 4.55,2.45, - 1.89, and - 5.17 eV, respectively. The 2Doand 3P0exothermic product channels correspond to curve crossings at internuclear separations R, of about 6.0 and 1 1 . 1 a.u., respectively. The individual cross sections for capture into the four product channels were determined from the observed relative yields by reference to separately measured total cross sections aZlfor capture into all final states of N+. N+ 3P0,N+ Figure 12 shows the individual cross sections for the N+ 300, IDo,and N+'Pproduct channelsobtained in this way. Also shown are recent theoretical estimates of the cross section for capture into the N+ 3D0channel obtained by A. Dalgarno et al. (personal communication, 1984) using a molecular approach (Heil et al., 198 1). They predict that the N+ 3D0moderately exothermic channel should provide the main contribution to a2,at low impact energies, and it is evident that this prediction is well supportedby the experimental data. At the higher velocities, the increasing divergence be-
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
171
- 2 -1 0 1 2 N' ' P o
.
I
I
-8
. I
. ?
-6
I
-4
-2
0
2
Energy change A T
4
6
8
3
IeVI
FIG.1 1 . Energy-change spectra for one-electron capture by (a) 6 keV and (b) 2 keV N2+ ions in atomic hydrogen. Insert (c) shows the energy profile of the primary N2+ions. The vertical lines indicate values of A T = A E for the indicated N+ product-ionstates. (From Wilkie er al., 1985.)
tween experimental values of 021and the theoretical estimates of3D0capture and is in accord with the observed increasingcontribution from the 3P0,*Do, 3Pproduct channels. It should be noted that the N+ product channels in Fig. 12 are all correlated with N2+2s22p 2Poground-state primary ions. Although there are, in addition, possible channels involving an unknown fraction of N2+2s2p2 4Pmetastable primary ions which could not be resolved, Wilkie et al. (1985) show that these are unlikely to have a significant effect on
172
H . B. Gilbody
Energy ( k e V )
FIG.12. Cross sections for electron capture by N2+in atomic hydrogen (from Wilkie et al., 1985). (O), uZI,total electron-capture cross section (Wilkie et a/., 1985); (V),(M),(O), (A), cross sectionsfor electron capture into the 3D0,,Po, IDo, and 3PstatesofN+measuredby Wilkie ef al. (1 985) using TES;(-), theoretically predicted cross sections for electron capture into ’Do state of N+ (A. Dalgarno et al., personal communication, 1984).
the analysis of the energy-changespectra. The cross sectionsshown in Fig. 12 are therefore believed to pertain to ground-state N2+2Poprimary ions. Our measurements of electron capture by C2+ions in H (McCullough et al., 1984) illustrate how the TES technique, when the energy resolution is adequate, can detect the presence of product channels arising from both ground-state and metastable primary ions. Figure 13 shows the energychange spectrum obtained for 4 keV C2+ions with the positions of product channelscorrespondingto ground-stateC2+2s2 ‘Sand metastable C2+2s2p 3P0primary ions. An analysis of energy-changespectra for C2+ions over the range 2 - 8 keV showed that between 60 and 65% of the total C+ product-ion yield arises from exothermic channels correlated with C2+3P metastable primary ions. Only between 20 and 25% of the yield arises from exothermic channels involving ground-state C2+ions with an additional 10% from unresolved endothermic channels with A E > -2 eV. The dominant reaction channel observed for ground-state ions corresponds to C+ 2s2p2 2Dformation with A E = 1.48 eV. This is in accord with the theoretical predictions of Heil er al. (1983), who have used ab inirio configuration-interaction methods to show that this channel takes place through the three 2X+states of CHr .Avoided crossings between the incoming molecular state and the outgoing state leading to C+2Dformation are predicted to occur at internuclearseparationsat both 3 and 24 a.u. At 8.1 eV,
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
173
-C‘ ‘0
3 Y
+. U
c
r
c‘
w
“25
-a
-c
*D
-
7
-2
TC* ’p
0
I
Energy change
10
12
A T lev)
FIG. 13. Energy-change spectrum for one-electron capture by 4 keV C2+ions in atomic hydrogen. An asterisk indicates channels involving metastable C2+’Poas distinct from C2+IS ground-state primary ions. (From McCullough eta!., 1984.)
the highest energy they considered, C+’ D formation through the innermost crossing is believed to provide the main contribution to the C+yield. TES measurements of electron capture by N3+ions in H (Wilkie et al., 1986) in the energy range 2.1 - 15 keV exhibit energy-change spectra (Fig. 14) which contain many unresolved collision channels possibly involving both ground-state N3+ls22s2IS and metastable N3+ ls22s2p 3P primary ions. The major peak in Fig. 14 corresponding to an energy change of 6.4 eV accounts for 90%of the N2+yield at 2.1 keV and about 62%at 15 keV. This peak can be correlated with the two unresolved channels N3+IS
+ H 1s
N3+3P
+ H Is
+
N2+1 ~ ~ 2 . 2S ~ ~+ 3H+ 5 + 6.38 eV
(10)
+ + 6.48 eV
(1 1)
and +
N2+2s2p3d 4P0 H+
Although the 3P metastable population of the beam is not known, it seems unlikely to be large so that, unless cross sections for Eqs. (10) and ( 1 1 ) (for which A Ediffers by only 0. I eV) differ greatly, the N2+yield associated with the major peak should be approximately the same as if the N3+ions were entirely in the ground state. Quanta1calculations for electron capture by ground-state ions camed out
H . B. Gilbody
174
G
I1111I11111 II I I I l l I 1 .I .I
I I
0 -!
I‘
I
w
.I I* .I I. - 1 .I.
I
>
I
N
z
I
I
W
1 l-
1
a w J
I . I . I . I .
..I.: ..-,.< I .:’
K
1
1
..-K.--.*-
1
II I I I 1 I
I
1 I
....--......
I
,-.... 1
1
1
..
....-.....-....
1
1
1
1
--.:-.--.*-*.--* 1 1
. 1
1
1
FIG. 14. Energy-change spectrum for one-electron capture by 15 keV N’+ ions in atomic hydrogen.The positionsof collision channelscorrespondingto ground-state(G)and metastable (M) primary ions are also shown. (From Wilkie el al., 1986.)
by Bienstock et al. (1984) and by Gargaud and McCarroll(l985) predict that Eq. (10) accounts for at least 94% of the total N2+yield over the energy range of the measurements. While the measurements do confirm the dominance of Eq. (10) predicted by theory, there is also evidence of other significant collision channels. The peak at about 3.5 eV (Fig. 14), which accounts for about 16% of the N2+yield at 15 keV, probably has a major contribution from the channel
+
+ + 3.35 eV
N3+ IS H 1s + N2+2s23p 2P0 H+
(12) but may also contain contributions from several channels involving metastable 3Pprimary ions. Unresolved channels with energy defects between 8 and 22 eV can also be seen to make a significant contribution to the N2+ yield. Our TES studies of electron capture by C3+inH (McCulloughet al., 1984) for impact energies in the range 0.6- 18 keV provided the first detailed experimental data on such a process for comparison with theoretical predictions. Full quantal calculations for both total and state-selectivecapture have been camed out by Bienstock et al. (1982). The use of a position-sensitive detector instead of a simple particle multiplier in our apparatus (see Section
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
175
I1,C) has resulted in improved energy resolution and increased product-ion detection efficiency. The new measurements (Wilkie et al., 1986) may also be compared with recent PES measurementsby the Amsterdam/Groningen group (Cirie et al., 1985b)in the range 8 - 55 keV. In this reaction, analysis is simplified by the fact that no metastable states of C3+exist. Figure 1 5 showsthe observed energy change spectrum for electron capture by 15 keV C3+ions in H (Wilkie et al., 1986) obtained with an energy resolution of 1.7 eV FWHM compared with our previous observations (McCullough et al., 1984)obtained with an energy resolution of only 3.0 eV FWHM. Additional structure is apparent in the higher-resolution measure-
-! c
.
>
N
c
U
2
-t (b) U
F-
m
1' I
..I
-G
rH
ENERGY CHANGE A T lev]
FIG.15. Energy-change spectra for one-electron capture by 15 keV C3+in H using (a) a position-sensitivedetector (Wilkie er 01.. 1986)and (b) a particle multiplier (McCulloughet al., 1984).
H . B. Gilbody
176
ments, and the possible collision channels are
C 3 + ls22s2S+H--, C2+ 1 s22s3d D - 0.02 eV
(A)
+ C2+ 1 s22s3p 3P0 + 2.05 eV, C2+ 1 s22s3p IPo + 2.15 eV, C2+ ls22s3s IS + 3.61 eV, C2+ 1 s22s3s 3S + 4.72 eV, C2 ls22p2IS + 1 1.63 eV, C2+ ls22s3d 3D 0.78 eV,
t
C2 ls22p2ID
+ 16.17 eV,
+
+
with R,= 69.7 a.u.
( B)
with R, = 26.5 a.u.
(C)
with R, = 25.1 a.u.
( D)
with R, = 15.1 a.u.
(El
with R, = 11.5 a.u.
(F)
with R, == 4.7
(GI
a.u.
with R,==3.4 a.u. (H) Relative cross sections were derived from the energy-changespectra for the channels (G) and (H) and the combined contributions from (E) (F), (C) (D), and (A) (B). An analysis of peak shapes indicate that the contribution from (E) is negligible. Cross sections for the formation of these channels shown in Fig. 16 were obtained by deconvolution of the energychange spectra and normalization to total one-electron capture cross sections 4 3 2 calculated by Bienstock et al. (1 982), which can be seen to agree well with experimental values of 4 3 2 over a wide range. Our cross sections measured previously with a lower energy resolution (McCullough et al., 1984) and the recent PES data of CiriC et al. (1 985b) are also included in Fig. 16. While there are differences between the results obtained with the TES and PES techniques, the overall degree of accord is not unreasonable in terms of the maximum combined uncertainties. CiriC et al. (1985b) estimate systematic errors of approx 30%and random errors of approx 10%(not shown) for the major collision channels but larger unspecified uncertainties for the minor channels. The calculations of Bienstock et al. (1982) predict strong adiabatic coupling for the collision channels (F), (G), (C), and (H) with cross sections shown in Fig. 16. The TES cross sections for (F) and (G) are seen to be in good accord with theory while the PES cross sections at the higher energies fall increasingly below the theoretical values. The TES cross sections for (C) (D) are in good agreement with the theoretical estimates for (C) alone. The high-resolutian PES measurements in fact show that the contribution from (D) is small below 12 keV, but becomes comparable with (C) at higher energies. For channel (H), the TES measurements agree with the theoretically predicted values only at low im-
+
+
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
177
I-
2s3d 3D
+ IB + A 1
2s3d 'D
4
u 1.0
10
100
ENERC. I k e V l
FIG. 16. Cross sections for one-electron capture in C3+-H 1 s collisions leading to C2+in specified states [collision channels (A)-(H) in text]. Total cross sections aj2for capture into all final states are also shown (from Wilkie et al., 1986). Experimental state-selectivecapture cross sections: (V),(M), (A),(O),(+), TES (Wilkie et al., 1986); (X), TES (McCullougb et al., 1984); (V), (O), (A), (0),(0),PES (CiriC ef a/., 1985b). Experimental total cross sectionsa,,: (@),jO), Phaneufetal. (1978,1982);@),Crandalletal. (1979);(0),YousifandGeddes(1983);(0),CiriC a,,; (---), ef al. (1985b);(a),Gardneref af. (1980). Theory(Bienstock ef al., 1982): (-), cross sections for capture into states indicated.
pact energies. At higher energies neither the TES nor the PES values are in accord with theory. The PES measurements show that below 30 keV the contribution from (A) to (A) (B) is small. The theory does not account for the significant contribution from (B) observed experimentally, but otherwise the general degree of accord between the TES measurements and theory is very encouraging. Figure 17 shows the CIV transitions observed and cross sections measured by Cirit et a!. (1 985a) in their PES studies of electron capture in C4+- H 1s
+
H. B. Gilbody
178
ENERGY ( k e V l a m u l
(a1
N
P
S
d
f
5
c
V
W In v)
In
0 a 0
-
1 0.1
ENERGY (keVlamul 4 6 8 10 12
0.2
1
a3 0.4 0.5 VELOCITY (a.u.1
a6
0.7
ENERGY I k e V l a m u I 2
~
-03s 2 0.1
\i
In
; u)
0.2
7
0.1
0.2
0.5 0.6 VELOCITY (a.u.1
0.3
0.4
0.7
0.1
0.2
0.3 0.4 0.5 0.6 VELOCITY (a.u.1
0.7
FIG.17. Cross sections for electron capture into specified states of C3+in C4+-H collisions obtained by Cirik el al. (1985a) using PES. (a) Decay scheme and branching ratios for CIV indicate observed transitions and (---) the unobserved transitions upon emissions. (-) which the measurementsare based. (b) Full symbols-total cross section uqjand cross sections for capture into n = 3 and 4 states of C3+.Open symbols-total cross sections u,, measured by Crandall ef al. ( 1979). Smooth curves- theoretical predictions by Fritsch and Lin ( 1984a).(c) Full symbols connected by lines-cross sections for electron capture into 3s, 3p, and 3d states of C3+.Smooth curves- theoretical predictions by Fritsch and Lin (1984a). (d) Full symbols connected by lines-cross sections for electron capture into 4s, 4p, 4d, and 4fstates of C3+. Smooth curves-theoretical predictions by Fritsch and Lin (1984a).
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
179
collisions in the range 0.8 -6.6 keV amu-I. Measured emission cross sections were used with known branching ratios and corrections for cascade for electron capture into all contributionsto determine cross sections o(n,/) subshells with n = 3 and 4, cross sections for n-shell capture a,, = Z, a(n,l) and total cross sections = Enan.The total cross sections a,, deduced from these spectroscopic measurements are seen to be in good agreement with the total cross sections measured by Crandall et al. (1979) using the furnace target technique. Values of a, and u4 are seen to be in good agreement with the theoretical estimates of Fritsch and Lin (1984a) based on a modified two-center atomic orbital expansion method. The results also confirm earlier theoretical predictions (see, for example, Hare1 and Salin, 1977) that n = 3 capture should be predominant. Reasonable agreement is ob-
FIG. 18. Total capture cross sections u6,and line-emission cross sections a,& - n’) for C6+in H measured by Dijkkamp ef d.(1985) using PES. (O),(W), (A), (e),Dijkkamp ef al. (1 985); (v),Phaneuf el ul. (1 982); long curves (G)-theory, Green et ul. ( 1 982); short curves @)-theory, Salin (1984).
180
H . B. Gilbody
tained in most cases between the experimental subshell capture cross sections o(n,l)and the theoretical predictions of Fritsch and Lin (1984a). Line-emission cross sections measured recently by Dijkkamp et af.( 1985) in H in the range 3 - 7.5 keV amu-* provide the first for C6, N7+,and 08+ data on state-selective capture by fully stripped ions and permit a more sensitive test of current theory than total electron-capture cross sections alone. Emission cross sections oem(n- n’)for (n - n’) = 1 and 2 were measured for principal quantum number n = 5 4 , and 3. Total electron-capture cross sections were also determined. In order to compare the experimental results with theory, emission cross sections gemwere derived from the theoretically predicted values of on/using known branching ratios for H. Data obtained for C6+-H collisions by Dijkkamp et af.(1985) for 465 and for o,(4 - 2), 3(,4 - 2), acm(4- 3) at 13.5, 18.2, and 52 nm are shown in Fig. 18. The low-energy values of oG5measured by Phaneuf et af.( 1982) are also shown. The experimental data may be compared with values derived from calculations Green et al. (1982) based on a 33-state molecular orbital model. These agree closely with results of a 46-state atomic orbital calculation by Fritsch and Lin (1984b) (not shown). Results based on the complete l-mixing model of Salin (1984) are also included in Fig. 18. While there are discrepancies and there is a need for more experimental data, the general degree of accord between experiment and the predictions of Green et af. (1 982) is quite encouraging, especially in view of the 25% estimated calibration uncertainty of the vuv spectrometer used in the measurements.
E. GENERAL SCALING RELATIONS FOR ELECTRON CAPTURE BY MULTIPLY CHARGED IONS As noted in Sections III,B and III,C, the collision mechanisms governing electron capture by multiply charged ions in the velocity ranges v < 1 a.u. and v > 1 a.u. are different. Measured cross sections in these two velocity ranges exhibit a different dependence on the initial charge state q, and simple scaling relations, where they apply, take on different forms. There have been a number of attempts to obtain an approximate description of the available experimental data in terms of general scaling relationships based on various theoretical models. In the high-velocity region, Chan and Eichler (1 979) have used an eikonal treatment which shows that at velocities v > 2 a.u. experimental cross sections og,g-lfor bare nuclei scale to values og,g-l(OBK) based on the
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
18 1
Oppenheimer- Brinkman - Kramers approximation. They obtain the expression (13) bq,q-I - aoq.q-1 (OBK) where (Y is a scaling factor. Theoretical values of (Y are found to be only weakly dependent on both q = Z and v and in rough accord with experimental values of (Y obtained from the ratio aq,q-,(expt)/~q,q-,(OBK) for a range of fully and partially stripped ions for energies up to 1000 keV amu-I. For the partially stripped ions an effective nuclear charge was used to determine the OBK cross sections. A general scaling relation based on the unitarized distorted-wave approximation (UDWA) treatment, which is applicable over an unusually wide velocity range, has been used by Ryufuku and Watanabe (1979a,b) to develop a general scaling relation for completely stripped ions. UDWA cross sections were calculated for ions with Z ranging from I to 20. The ssaled cross sections 5 = oz,z-l/ Z , when plotted against a scaled energy E = E(keV a m ~ - ~ ) / Z ~are . ~shown " to fall on a single universal curve for Z 5 3 at velocities v > 10 keV. This curve provides a reasonable description of the energy dependence of experimental cross sections of a range of fully and highly stripped ions for values of E up to about 1 MeV amu-I but overestimates the magnitude of individual cross sections by up to a factor of three. Knudsen et al. ( 1981) have considered scaling of cross sections uq,q:I in terms of the semiclassical Bohr and Lindhard (1954) model. Calculations based on this model show that the scaled cross section o = oq,q-l/ q depends only on the scaling parameter k = E(keV a m ~ - ~ ) / qAn ~/~ analytical . expression for aq,qI for ions ofq 2 5 is obtained in terms of the atomic number and ionization potential of the target atom and an adjustable parameter a which determines the smallest effective target velocity for electron capture. The value of a is adjusted to obtain the best fit to experimental data in plots of d against k. In this way, a curve is obtained which provides a reasonable fit to experimental data for ions with q 2 5 for values of k ranging from about 5X to 3 X keV amu-I q 4 1 7 . Janev and Hvelplund (1 98 1) have shown that in the case of experimental values of L T ~ , ~for - ~q 5 5, a plot of d = oq,q-l / q against k = E(keV arnu-') q112can be fitted by a single curve of the form (14) For velocities v between 0.1 and 1 a.u. a = 1 and is only weakly dependent on 5 = v/q ,I4. For u" > 1, a gradually increasesand approaches an asymptotic value of 5 when v" >> 1. d = go( v")q*(#)
182
H. B. Gilbody
IV. Ionization A. IONIZATION BY BARENUCLEI The first experimental studies of ion-impact ionization of atomic hydrogen were confined to the simple process (15) H + + H ls-,H++H++e Measurements based on the modulated crossed-beam technique by Fite et al. (1 960) in the range 0.04-40 keV and by Gilbody and Ireland (1963) in the range 50-400 keV provided cross sections which were determined by reference to known cross sections for the ionization of H, by proton impact. Ionization cross sections determined by Park et al. (1977) in the range 25 200 keV were based on an analysis of the differential energy-loss spectra in the passage of protons through highly dissociated hydrogen in a furnace target. The results of these experiments are only in rough accord (see Shah and Gilbody, 198la), and a detailed comparison is precluded by large experimental uncertainties and the use of different normalization procedures. The development of the crossed-beam coincidencetechnique by Shah and Gilbody ( 1981a) (see Section 11) has made it possible to measure ionization cross sections with high precision for a wide range of projectile species and thereby assess the range of validity of current theoretical descriptions. Figure 19 shows cross sections measured by Shah and Gilbody ( 1981a,b, 1982a, 1983a,b) for the ionization of H 1s atoms by the bare nuclei H+, HeZ+,Li3+,and C6+compared with a number of theoretical estimates. All the experimental cross sectionsare measured relative to those for H+impact, which in turn were normalized to theoretical values based on the first Born approximation (Bates and Griffing, 1953) in the range 1000- 1500 keV amu-l. The estimated 6.7%uncertainty associated with this normalization procedure is, in this particular case, believed to be much less than would be incurred by normalization to experimental measurements. It is well known (Batesand Griffing, 1953)that the Born cross sections for ionization of H by equivelocity bare nuclei of atomic number Z scale according to Z2.In Fig. 19 it will be seen that the experimental cross sections for He2+and Li3+impactapproach the Born and Bethe values (seeGillespie, 1981) at progressively higher velocities as Z increases; the single experimental point for C6+impact is also consistent with this trend. Observed peak values in the cross sectionsare also shifted to higher velocities with increasing 2. Theoretical values obtained by McGuire ( 1982)using the Glauber approximation are in much better agreement with the experimental data, particularly at intermediate velocities. For C6+impact, the
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
183
FIG.19. Cross sectionsfor ionizationof H 1satoms by bare nuclei (from Shah and Gilbody, 1983a). Experimental data: (O),cross sections for H+, He2+, Li3+,and C6+impact (Shah and Gilbody, 1981a,b, 1982a, 1983a); (0),1/2 cross section for electron removal from H by C6+ impact (Schlachteret al., 198I). Theoretical data: the numbers 1,2,3, and 6 refer to H+, He2+, Li3+,and C6+impact; (B), Born approximation (Bates and Griffing, 1953); (BT), Bethe a p proximation (Gillespie, 198I); (G), Glauber approximation (McGuire, 1982); (U), UDWA method (Ryufuku, 1982); (CDW), CDW treatment with eikonal initial state (Crothers and McCann, 1983); (0),(0),(0),(A), CTMC method for H+, HeZ+,Li3+,and C6+(Hardie and Olson, 1983).
value calculated by Ryufuku (1 982) at 400 keV amu-1 using the unitarized distorted-waveapproximation is in good agreement with the experimental cross section. It is interesting to compare the experimental cross sections with those calculated using the classical-trajectory Monte Carlo (CTMC) approach. This was first applied by Abrinesand Percival ( 1966)to H+- H collisionsand subsequently by Olson and Salop (1977) to other projectiles. While the method is useful at intermediate velocities,the classical approach must fail at high velocities since it leads to a 1/Erather than the experimentally observed 1/EIn E energy dependence in accord with the first Born approximation. In Fig. 19 the cross sections calculated by Hardie and Olson (1983), which use an improved representation of the H 1 s electronic radial distribution in the
184
H . B. Gilbody
CTMC approach, significantly overestimates the maximum value of the cross section. At high velocities the CTMC cross sections for proton impact converge to values (not shown in Fig. 19) calculated by Bates and Kingston ( 1970) using the classical impulse (binary encounter) approximation which fall increasingly below the Born values. Crothers and McCann (1983) have used a theoretical model of ionization of H by bare nuclei in which distortion is accounted for in the entrance channel by the eikonal approximation and in the exit channel by the continuum distorted-wave (CDW) approximation. Cross sections calculated using this model (Fig. 19) may be seen to provide the best overall agreement with experiment at intermediate velocities. Calculations by Belkil: (1 978, 1980) based on the CDW approximation alone (not shown) provide a gross overestimate of the cross sections at velocities near the observed peak values. This failure has been ascribed by Crothers and McCann (1983) to incorrect normalization of the CDW initial state. Apart from the single cross section measured at 400 keV amu-I by Shah and Gilbody (1983a), the only other experimental estimates of the cross section for ionization by C6+ impact are based on one-half of the cross section for electron removal from H, measured by Schlachter et al. ( 1981) using the condenser plate method at 0.3 1,l. 14,and 4.75 MeV amu-I. These data included in Fig. 19 involve comparatively large uncertainties. While the charge-transfer contribution to the measured cross sections is small at the energies considered, the assumption that H, is equivalent to two free hydrogen atoms, even at high velocities, is not borne out by detailed crossed-beam coincidence studies (Shah and Gilbody, 1982b) of the ionization of H, by a range of multiply charged ions. B. IONIZATIONBY PARTIALLY STRIPPED IONS Cross sections for ionization of H by C(2-6)+,N(2-5)+, 0(2-6)+, Li(1-3)+,and ions have been measured by Shah and Gilbody (1981b, 1982a, 1983b) using the crossed-beam coincidence method for impact energies within the range 12-243 keV amu-l. Results for C(2-4)+,N(,4+, and 0(2-5)+ impact are shown in Fig. 20. Cross sections for different ions with the same charge q may be seen to agree closely in both magnitude and energy dependence, while the velocity at which cross sections attain ,peak values increase as q increases. At velocities beyond the peak values cross sections increase with q. This simple dependence on q is not observed at low velocities where there is competition from the corresponding electron-capture processes. Ionization cross sections calculated by Olson and Salop (1 977) using the
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
185
15’ 10
-
4 -
2 -
ln ln 0
0
a
CI+
c 3+ C 2‘
I
10
40 100 ENERGY ( k e V omu“
LOO
1
FIG.20. Cross sections for ionization of atomic hydrogen by Cq+,N@+,and 0 4 + ions (from Shah and Gilbody, 198 1b). Experiment: (O),q = 2; (0),q = 3; (O),q = 4; (a)q = 5 (Shah and Gilbody, I98 1b). Theory: (O), q = 3; (O),q = 4; (O),q = 5 (Olson and Salop, 1977).
CTMC method for ions q 3 3 are also shown in Fig. 20. These calculations make use of effective charges Z,, which allow for electron screening. It will be seen that while these theoretical cross sections are only about one-third of the experimental values at 50 keV amu-*, they tend to approach the experimental values at the highest velocities. The agreement also improves as q increases. At high velocities, where the ionization cross sections have decreased
186
H. B. Gilbody
beyond their peak values, cross sections for ionization scale (like the corresponding cross sections for electron capture)accordingto simple expressions of the form u = aoqn,where uo and n are constants for a given velocity. Berkner et al. (1978) first showed that cross sections for ionization of H, by very fast Fe(9-22)+ ions measured using the condenser plate technique could be described by this expression where n = 1.43 and 1.40 at 1100 and 227 keV amu-l, respectively. For an atomic hydrogen target Shah and Gilbody ( 198l b) have shown that for carbon, nitrogen, and oxygen ion impact measured cross sections are well described by the expression when n = 1.46 & 0.05 at an energy of 145 keV amu-l common to these ions. In measurements with Li(l-3)+ ions, measured ionization cross sections (Shah and Gilbody, 1982a) are found to scale according to q”, with n increasing from 1.24 to 1.42 as the energy is increased from 57 to 387 keV amu-I.
RELATIONS FOR IONIZATION C. GENERAL SCALING While a simple q* scaling relation is useful for the prediction of ionization cross sections at high velocities beyond the peak value, there have been attempts to obtain scaling relations of wider applicability. Knudsen (1982) has used an approach based on a model due to Bohr in which the Rutherford cross section for an energy transfer to a free stationary electron is considered. Cross sections for ionization are calculated by integrating the Rutherford cross section from a lower limit correspondingto the ionization energy to an upper limit correspondingto the maximum transferable energy 2m Vz,where rn is the electron mass and V is the ion velocity. Since the target electrons are not free and stationary, two types of collision are identified. There are “free collisions” characterized by a collision time less than the revolution time and “resonance collisions” in which the field from the ion is approximately uniform across the electron orbit. Collisions are described in terms of the parameters x = 2quo/Vand q = 2 V/v,where v is the velocity of the target electron and uo is the Bohr velocity. When x > q > 1 it is shown that the ionization cross section may be expressed as
+6
0 = 4na~q2(vo/V)2(lo/r)[(x/ll)-’ln(q2/x2)- q-2]
(16) where 6 is the fraction of resonant collisions leading to ionization in a particular target atom; for atomic hydrogen S is estimated to be 0.28. In Eq. ( 16) Knudsen ( 1982)has shown that, for ions of high charge q, the ratio u/q depends only on the scaling parameter E/q (where E is expressed in keV amu-1) for a wide range of values of this parameter. In Fig. 21 are shown plots of a/q against E/q derived from Eq. (16) for values of q ranging from 3 to 9, compared with the experimental data of Shah
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FIG.2 I . Reduced cross sections u/q for ionization of H by multiply charged ions of argon, for carbon, and lithium compared with theoretical predictions of Knudsen (1982) [(-) different initial charge states q] based on the Bohr model. Experimental data obtained by Shah and Gilbody (1983b). (D),Ar9+;(+), ArB+;(X), Ar7+;(A),Ar6+;(0),Ar5+;(O), Ar4+;(A), Ar3+; (A),C6+; C5+;0, C4+;(A),C3+;(o),C2+;(A),Li3+;(a), Li2+.
(a),
and Gilbody for Ar(3-9)+,U2-@+, and Li(2-3)+impact. The general scaling relation can be seen to fail at the lower velocities where the curves corresponding to different values of q diverge and are in poor accord with experiment. For high values ofE/q the experimental points do tend to approach the common curve predicted by the Bohr scaling relation. Gillespie (1 982) has developed a scaling relation within the Bethe- Born approximation which provides a good description of the experimental data. In this approach the ionization cross section is described by an expression of the form where cMeis the Bethe cross section for the ionization of H by fast protons, /3 the relativistic speed v/c,and a is the fine-structure constant. The term q2a-( p) predicts the correct q2 Born scaling at asymptotically high velocities, while the term Jis designed to correct the overestimation of 6- at
H . B. Gilbody
188
1.0
0.1
FIG.22. Plot ofthe scaling function u/q2aesthe for ionization plotted against reduced energy (from Shah and Gilbody, 1983b). Theory: (-), Gillespie (1982). Experiment: (A),Ar9+; (h),A?+; (V),Ar7+;(0), Ar6+;(U), Ar5+; (0),Ar4+;(A), Ar3+;(@), C6+;( O ) ,Cs+;(a),C4+; (A),C3+;(A), Li3+;(O), Li2+(Shah and Gilbody, 1983b).
moderate velocities. Gillespie has shown that the experimental data are well described by a single curve fitted by the simple function.
fW2Q/P)= exP[-4q1/2Q/P)21
(18)
when 1 = 0.76. This good agreement is illustrated in Fig. 22 in which f = u/q2u,, is plotted against E/q. The representative set of experimental data shown for AI-(~-~)+, U3-'3+,and Li(2-3)+ions measured by Shah and Gilbody do not deviate by more than 13Yo from the theoretical curve. In the development of this scaling relation, Gillespie ( 1982) has suggested that only ions with q > Z/2 should be considered since ions with more electrons cannot adequately be described as point particles. The inclusion of experimental data points for q G Z/2 in Fig. 22 illustrates that the scaling relation has a wider range of validity than Gillespie suggests.
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V. Cross Sections for Electron Removal from Hydrogen Atoms in Collisions with Positive Ions A. GENERAL SCALING RELATIONS FOR MULTIPLY
CHARGED IONS For fusion research applications, a knowledge of the total cross section 6, for electron removal from H by the combined processes of charge transfer and ionization is of particular importance. The experimental data in Fig. 23 for multiply charged ions of Ar in H shows that for initial charge states q 3 3, the energy dependence of the separate cross sections for charge transfer and ionization is such that the cross section ue for the combined processes is not strongly dependent on energy over a very wide energy range.
FIG.23. Cross sections for electron capture and ionization in collisions between Arq+ ions Ionization cross and H atoms (from Shah and Gilbody, 1983b). (O),(O),(m), (A),(o),(X), (a), sections (Shah and Gilbody, 1983b); (0),electron-capture cross sections (Shah and Gilbody, 1983b);(a),electron-capture cross sections (Crandall et a/., 1980).
H. B. Gilbody
190
In an attempt to derive a general scaling rule for u,,*Olson et al. (1978) obtained an analytical fit to cross sections calculated using the classical-trajectory Monte Carlo (CTMC) method (Olson and Salop, 1977)for q between 1 and 50 and energies between 50 and 5000 keV amu-l. In this approach, values of a,/q plotted against E/q in keV amu-I should lie on a universal curve given by the expression u,/q = (4.6)(32q/E)[1 - exp(-E/32q)] X
cm2 (19) Olson et al. (1978) and Schlachter et al. (198 1) showed that this expression provided reasonable agreement with the experimental data available at that time. Estimates of the ionization contribution to a, were then based mainly on values of one-half the cross section for the ionization of H,,a procedure which is now known (from the measurements of Shah and Gilbody, 1982b) to be of limited accuracy. The availability of accurate experimental data for the ionization of atomic hydrogen (Shah and Gilbody, 1981a,b, 1982a, 1983a,b) has now made it possible to obtain a better assessment of the
I
I
I
1
I
10 E l q IkeV arnd'I
I
I
I
1
100
FIG.24. Reduced total cross sections a,/q for electron removal from H atoms by the combined processes of charge transfer and ionization in collisions with multiply charged ions (from Shah and Gilbody, 1983b). Classical scaling predictions: (A), Olson et al. (1978) with limits of validity indicated by lower bars; (B), Hardie and Olson (1983b). Experimental data (Shah and Gilbody, 1983b):(A), (0),Ar6+;(O), Ar5+;(0),Ar4+;(A), Ar3+; C6+;(a),Cs+; (a),
(v),
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
19 1
validity of this scaling relation. A scaling relation based on recent classical calculations by Janev (1 983) coincides almost exactly with Eq. (1 9). In the modified CTMC approach used by Hardie and Olson ( 1983),cross sections up to 40% larger are predicted and the scaling relation becomes n,/q = (6.4)(23q/E)[1 - exp(-E/23q)] X
cm2 (20) In Fig. 24 representative experimental data for and C(2-6)+in H are compared with the curves corresponding to the scaling relations of Eqs. (19) and (20). The modified expression in Eq. (20) may be seen to be in rather better accord with the experimental data, especially at the lower impact energies where some of the experimental points lie strictly outside the predicted limits of validity of the CTMC theory. At high velocities, where the experimental data are expected to be in accord with Born rather than with classical scaling, it is perhaps not surprising that the curves corresponding to both Eqs. (19) and (20) begin to fall increasingly below the experimental points.
B. ELECTRON REMOVAL FROM HIGHLY EXCITED H ATOMS Koch and Bayfield ( 1975) have used the merged-beam approach to measure cross sections n, for electron removal from highly excited H atoms by proton impact at c.m. energies in the range 0.4-6 1 eV. This is equivalent to 0.8- 122 eV protons colliding with stationary H atoms. A field ionization technique was used to define the excited states of the H atoms to a band of principal quantum numbers n ranging from 44 to 50. Measured cross sectionsdecrease from 8 X lov9cm2at 0.8 eV to 1 X cm2at 122 eV. Banks et al. (1976)have used the CTMC method to calculate cross sections n, which take into account direct ionization and electron capture into both bound and continuum states. When the cross sections measured by Koch and Bayfield ( 1975)are normalized to the CTMC prediction for 122 eV proton impact on H(n = 47) atoms, the experimental cross sections are in good agreement with theory down to 22 eV. At c.m. energies below about 2 eV, where the experimental cross sections n, measured by Koch and Bayfield can be mainly attributed to electron capture, there is rough agreement (within the factor of 2 uncertainty in absolute values) with theoretical estimates of Janev et al. ( 1984)of resonant charge-transfer cross sections; these take into account both tunneling (under bamer) and over barrier (classically allowed) transitions. There is also reasonable agreement with classical calculationsof Smirnov (197 I), but cross sections based on the perturbed stationary-statecalculations of Toshima ( 1979)are about a factor of six too small.
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In the crossed-beam experiment carried out by Kim and Meyer (1982), electron removal from highly excited H(n = 9 -24) atoms in collisions with N('-5)+,0(1-5)+, and ions was studied for impact velocities u in the range 1 -2 a.u. The population of excited atoms in the beam had a measured n-3 dependence for n 3 9. An electric-field ionization technique was used to change the n-state population of the beam and thereby determine a sum of electron-removal cross sections weighted according to the population of the excited states concerned. An analysis of these measured cross sections for different n populations was used to show that electron-removal cross sections 0, are independent of ion species for a given value of q and u. In addition, within the accuracy of the measurements, 0,increases with q2 for a given value of u and decreases as u-2 for a given value of q. This behavior is in accord with classical predictions (Bohr and Lindhard, 1954) of ionization which provides the dominant contribution to 0,at the velocities considered. A detailed study ofthe dependence ofthe electron-removal cross section on n was precluded in this experiment by the presence of an electric field in the collision region which significantly reduced the ionization energy of excited H atoms.
REFERENCES Abrines, R., and Percival, I. C. (1966). Proc. Phys. SOC.88, 873. Banks, D., Barnes, K. S., and Wilson, J. McB. (1976). J. Phys. B 9, L141. Bates, D. R., and Griffing, G. (1953). Proc. Phys. SOC.A 66,961. Bates, D. R., and Kingston, A. E. ( 1 970). Adv. At. Mol. Phys. 6,269. Bates, D. R., Johnston, H. C., and Stewart, I. (1964). Proc. Phys. SOC.84, 5 17. Bayfield, J. E. (1969). Phys. Rev. 185, 105. Bayfield, J. E., and Khayrallah, G. A. (1975). Phys. Rev. A 12,869. BelkiC, Di. (1978). J. Phys. B 11, 3529. BelkiC, Di. (1980). J. Phys. B 13, L589. BelkiC, Di., and Gayet, R. (1977). J. Phys. B 10, 191 1 . Belyaev, V. A,, Brezhnev, B. G., and Erastov, E. M. (1967). Sov. Phys. JETP 25,77. Bendahman, M., Bliman, S., Dousson, S., Hitz, D., Gayet, R., Hanssen,J., Harel, C., and Salin, A. (1985). J. Phys. 46,561. Berkner, K. H., Graham, W. G., Pyle, R. V., Schlachter, A. S., Steams, J. W., and Olson, R. E. (1978). J. Phys. B 11, 875. Bienstock, S., Heil, T. G., Bottcher, C., and Dalgarno, A. (1982). Phys. Rev. A 25,2850. Bienstock, S., Dalgamo, A., and Heil, T. G. (1984). Phys. Rev. A 29,2239. Bohr, N., and Lindhard, K. ( 1 954). Dan. Vidersk. Silsk. Mat. Fys. Medd. 28, 1. Bransden, B. H., and Noble, C. J. (1981). J. Phys. B 14, 1849. Bransden, B. H., Newby, C. W., and Noble, C. J. (1980). J. Phys. B. 13,4245. Bransden, B. H., Noble C. J., and Chandler, J. (1983). J. Phys. B. 16,4191, Burniaux, M., Brouillard, F., Jogneaux, A., Govers, T. R., and Szucs, S. (1977). J. Phys. B 10, 242 1 .
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
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Can, C., Gray, T. J., Varghese, S . L., Hall, J. M., and Tunnell, L. N. (1985). Phys. Rev. A 31,72. Chan, F. T., and Eichler, J. (1979). Phys. Rev. Lett. 42, 58. Cheshire, I. M. (1968). J. Phys. B 1, 428. Cheshire, I. M., Gallaher, D. F., and Taylor, A. J. (1 970). J. Phys. B 3, 8 13. Chibisov, M. I. (1976). Sov. Phys. JETP Lett. 24, 46. Chidichimo-Frank, V. C., and Piacentini, R. D. (1974). J. Phys. B 7, 548. Chong, Y. P., and Fite, W. L. (1977). Phys. Rev. A 16,933. Cirik, D., Dijkkamp, D., Vlieg, E., andde H e r , F. J. (1985a). J. Phys. B18, L17. Cirii., D., Brazuk, A., Dijkkamp, D., deHeer, F. J., and Winter, H. (1985b). JPhys. B 18,3629. Crandall, D. H., Phaneuf, R. A., and Meyer F. W. ( I 979). Phys. Rev. A 19, 504. Crandall, D. H., Phaneuf, R. A., and Meyer F. W. (1 980). Phys Rev. A 22,379. Crothers, D. S. F., and Todd, N. R. (1980a). J. Phvs. B 13,547. Crothers, D. S. F., and Todd, N. R. (1980b). J. Phys. B 13, 2277. Crothers, D. S. F., and McCann, J. F. (1983). J. Phys. B 16, 3229. Dalgarno, A., and Yadav, H. N. (1953). Proc. R Soc. London, Ser. A 66, 173. Dijkkamp, D. (1985). Ph.D. Thesis, University of Utrecht. Dijkkamp, D., Cirii., D., and de Heer, F. J. (1985). Phys. Rev. L e f t 54, 1004. Eichler, J. K. M., Tsuji, A., and Ishihara, T. (1981). Phys. Rev. A 23,2833. Fite, W. L., Brackmann, T. R., and Snow, W. R. (1958). Phys. Rev. 112, I 16 1. Fite, W. L., Stebbings, R. F., Hummer, D. F., and Brackmann, R. T. (1960). Phys. Rev. 119, 663. Fite, W. L., Smith, A. C. H., and Stebbings, R. F. (1962). Proe. R. Soe. Ser. A 268,527. Fritsch, W., and Lin, C. D. (1982a). J. Phys. B 15, L281. Fritsch, W., and Lin, C. D. (1982b). Phys. Rev. A 26,762. Fritsch, W., and Lin, C. D. ( I 984a). J. Phys. B 17, 327 I . Fritsch. W., and Lin, C. D. (1984b). Phys. Rev. A 29,3039. Gardner, L. D., Bayfield, J. E., Koch, P. M.. Kim, H. J., and Stelson, P. H. (1977). Phys. RPV.A 16, 1415. Gargaud, M., and McCaroll, R. (1985). J. Phys. 3 18,463. Gilbody, H. B. (1979). Adv. At. Mol. Phys. 15,293. Gilbody, H. B., and Ireland, J. V. (1963). Proc. R. SOC.Ser. A 277, 137. Gilbody, H. B., and Ryding, G. (1966). Proc. R. SOC.Ser. A 291,438. Gillespie, G. H. (1981). Phys. Rev. A 24,608. Gillespie, G. H. (1982). J. Phys. E 15, L729. Goffe, T. V., Shah, M. B., and Gilbody, H. B. (1979). J. Phys. B 11, L233. Green, T. A., Shipsey, E. J., and Browne, J. C. (1982). Phys. Rev. A 25, 1364. Greenland, P. T. (1982). Phys. Rep. 81, 132. Grozdanov, T. P., and Janev, R. K. (1980). J. Phys. B 13, L69. Hardie, D. J. W., and Olson, R. E. (1983). J. Phys. B 16, 1983. Hare], C., and Salin, A. ( 1 977). J. Phys. B 10, 35 1. Hatton, G. J., Lane. N. F., and Winter, T. G. (1979). J. Phys.B 12, L571. Heil, T. G., Butler, S. E. and Dalgarno, A. (1981). Phys. Rev. A 25, 1100. Heil, T. G., Butler, S. E., and Dalgarno, A. (1983). Harvard Smithsonian Center for Astrophysics Preprint Series No. 1732. Hill, J., Geddes, J., and Gilbody, H. B. (1979). J. Phys. B 12, L341. Huber, B. A. (1981). Z. Phys. A 299, 307. Huber, B. A., Bumbel, A., and Wiessmann, K. (1983), J. Phys. E 145, 145. Hunter, G., and Kuriyan, M. (1977a). Proc. R. Soc. London Ser A 358, 321. Hunter, G., and Kuriyan, M. (l977b). Proc. R. Soc. London Ser. A 353, 575. Hvelplund, P., and Andersen, A. (1982). Phys. Scripta 26, 370.
194
H . B. Gilbody
Janev, R. K. (1983), Cited by Shah and Gilbody (1983b). Janev, R. K., and Bransden, B. H. (1982). IAEA Report INDC (NDS)- 135/GA. Janev, R. K., and Hvelplund, P. (1981). Comments At. Mol. Phys. 11,75. Janev, R. K., and Winter, H. (1985). Phys. Rep. 117,266,2463. Janev, R. K., Belic, D. S., and Bransden, B. H. (1983). Phys. Rev. A 28, 1293. Janev, R. K., Joachain, C. J., and Nedeljkovic, (1984). Phys. Rev. A 29. Khahrallah, G . A., and Bayfield, J. E. (1975). Phys. Rev. A 11,930. Kim, H. J., and Meyer, F. W. (1982). Phys. Rev. A 26, 1310. Kim, H. J., Phaneuf, R. A,, Meyer, F. W., and Stelson, P. H. (1978a).Phys. Rev. A 17,854. Kim, H. J., Hvelplund, P., Meyer, F. W., Phaneuf, R. A., Stelson, P. H., and Bottcher, C. (1978b). Phys. Rev. Lett. 40, 1635. Kimura, M., and Thorson, W. R. (1981). Phys. Rev. A 24, 1780. Knudsen, H. (1982). Proc. Int. Conf Phys. Electron. At. Collisions, 12th. Gatlinburgpp. 65769. Knudsen, H., Haugen, H. K., and Hvelplund, P. (1981). Phys. Rev. A 24,2287. Koch, P. M., and Bayfield, J. E. (1975). Phys. Rev. Lett. 34,448. Kordow, T., Girnius, R. J., Chang, Y. P., and Fite, W. L. (1974). Phys. Rev. A 10, 1167. Larsen, 0.G., and Taulbjerg (1984). J. Phys. B 17,4523. Lockwood, G. J., and Everhart, E. (1962). Phys. Rev. 125,567. Ludde, H. J., and Dreizler, R. M. (1982). J. Phys. B 15,2713. McClure, G. W. (1966). Phys. Rev. A 148.47. McCullough, R. W., Nun, W. L., and Gilbody, H. B. (1979). J. Phys. B 12,4159. McCullough, R. W., Lennon, M., Wilkie, F. G., and Gilbody, H. B. (1983).J.Phys. B 16, L173. McCullough, R. W., Wilkie, F. G., and Gilbody, H. B. (1984). J. Phys. B 17, 1573. McGuire, J. H. (1982). Phys. Rev. A 26, 143. Massey, H. S. W., and Gilbody, H. B. (1974). “Electronicand Ionic Impact Phenomena,” Vol4. Clarendon, Oxford. Meyer, F. W., Phaneuf, R. A., Kim, H. J., Hvelplund, P., and Stelson, P. H. ( 1979).Phys. Rev. A 19,5 15. Meyer, F. W.,Howald, A. M., Havener, C. C., and Phaneuf, R. A. (1985a).Phys. Rev. Lett. 54, 2663. Meyer, F. W., Howald, A., Havener, C. C., and Phaneuf, R. A. (1985b).Phys. Rev. A 32,33 10. Morgan, T. J., Geddes, J., and Gilbody, H. B. (1973). J. Phys. B 6,2 1 18. Morgan, T. J., Stone, J., and Mayo, R. (1980). Phys. Rev. A 22, 1460. Newman, J. H., Cogan, J. D., Zeigler, D. L., Nitz, D. E., Rundel, R. D., Smith, K. A., and Stebbings, R. F. (1982). Phys. Rev. A 25,2976. Nutt, W. L., McCullough, R. W., Brady, K., Shah, M. B., andGilbody, H. B. (1978).J. Phys. B 11. 1457. Nutt, W: L., McCullough, R. W., and Gilbody, H. B. (1979). J. Phys. B 12, L157. Olson, R. E., and Kimura, M. (1982). J. Phys. B 15,423 1. Olson, R. E., and Salop, A. (1976). Phys. Rev. A 14,579. Olson, R. E., and Salop, A. (1977). Phys. Rev. A 16,53 1. Olson, R. E., Berkner, K. H., Graham, W. G., Pyle, R. V., Schlachter, A. S., and Steams, J. W. (1978). Phys. Rev. Lett. 41, 163. Panov, M. N., Basalaev, A. A., and Lozhkin, K. 0. (1983). Phys. Scripta T3, 124. Park, J. T., Aldag, J. E., George, J. M., and Peacher, J. L. (1977). Phys. Rev. A 15, 508. Phaneuf, R. A. (1983). Phys. Rev. A 28, 1310. Phaneuf, R. A., Meyer, F. W., and McKnight, R. H. (1978). Phys. Rev. A 17,534. Phaneuf, R. A., Alvarez, I., Meyer, F. W., and Crandall, D. H. (1982). Phys. Rev. A 26, 1892. Phaneuf, R. A., Kimura, M., Sato, H., and Olson, R. E. (1985). Phys. Rev. A 31,2914.
MEASUREMENTS OF CHARGE TRANSFER AND IONIZATION
195
Piacentini, R. D., and Salin, A. (1 977). J. Phys. E 10, 1515. Presnyakov, L. P., and Ulantsev, A. D. (1976). Sov. J. QuanfurnElectron. 4, 1320. Rapp, D. J. (1974). J. Chem. Phys. 61, 3777. Rille, E., Olson, R. E., Peacher, J. L., Blankenship, D. M., Kvale, T. J., Redd, E., and Park, J. T. (1982). Phys. Rev. Left. 49, 1819. Rule, D. W., and Omidvar, K. (1979). Asfrophys. J. 229, 1 198. Rpding, G., Wittkower, A. B., and Gilbody, H. B. (1966). Proc. Phys. Soc. 89, 547. Ryufuku, H. (1982). Phys. Rev. A 25,720. Ryufuku, H., and Watanabe, T. (1978). Phys. Rev. A 18, 1005. Ryufuku, H., and Watanabe, T. (1979a). Phys. Rev. A 19, 1538. Ryufuku, H., and Watanabe, T. (1979b). Phys. Rev. A 20, 1828. Ryufuku, H., Sasaki, K., and Watanabe, T. (1980). Phys. Rev. A 21,745. Salin, A. ( 1984). J. Phys. 45,67 1. Schinke, R., and Kruger, H. (1976). J. Physique E 9,2459. Schlachter, A. S., Berkner, K. H., Graham, W. G., Pyle, R. V., Steams, J. W., and Tanis, J. A. (1981). Phys. Rev. A 24, 1 1 10. Seim, W., Muller, A., Wirkner-Bott, I., and Salzborn, E. (1981). J. Phys. B 14,3475. Shah, M. B., and Gilbody, H. B. ( I 974). J. Phys. E 7,630. Shah, M. B., and Gilbody, H. B. (1978). J. Phys. E 11, 121. Shah, M. B., and Gilbody, H. B. (1981a). J. Phys. B 14,2361. Shah, M. B., and Gilbody, H. B. (1981b). J. Phys. B 14, 2831. Shah, M. B., and Gilbody, H. B. (1982a). J. Phys. E 15,413. Shah, M. B., and Gilbody, H. B. (1982b). J. Phys. B 15,3441. Shah, M. B., and Gilbody, H. B. (19838). J. Phys. E 16, L449. Shah, M. B., and Gilbody, H. B. (1983b). J. Phys. E 16,4395. Shah, M. B., Goffe, T. V., and Gilbody, H. B. (1978). J. Phys. B 11, L233. Shah, M. B., Geddes, J., and Gilbody, H. B. (1980). J. Phys. E 13,4049. Shipsey, E. J., Green, T. A., and Browne, J. C. (1983). Phys. Rev. A 27,821. Slevin, J., and Stirling, W. (1981). Rev. Sci. Insfnrrn. 52, 1780. Smirnov,B. M. (1971).Sov. Phys. JETP32,670. Stebbings, R. F., Fite, W. L., and Hummer, D. G. (1960). J. Chem. Phys. 33, 1226. Stebbings, R. F., Young, R. A., Oxley, C. L., and Erhardt, H. (1965). Phys. Rev. A 138, 1312. Steigman, G. (1975). Asfrophys. J. Leu. 195, L39. Toshima, N. (1979). J. Phys. Soc. Jpn. 47, 257. Tuan, T. F., and Gejuoy, E. (1960). Phys. Rev. A 117, 756. Vaaben, J., and B n w , J. S. (1977). J. Phys. B 14, L521. Wilkie, F. G., Yousif, F. B., McCullough, R. W., Geddes, J., and Gilbody, H. B. (1985). J. Phys. B 18,479. Wilkie, F. G., McCullough, R. W., and Gilbody, H. B. (1986). J. Phys. B 19,239. Wittkower, A. B., Ryding, G., and Gilbody, H. B. (1966). Proc. Phys. Soc. 89, 541. Winter, T. G., and Hatton, G. J. (1980). P h p . Rev. A 21,793. Young, R. A., Stebbings, R. F., and McGowan, J. W. (1968). Phys. Rev. 171,85. Yousif, F. B., and Geddes, J. (1983), cited in McCullough, R. W., Wilkie, F. G., and Gilbody, H. B. (1984). J. Phys. E 17, 1573.
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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 22
ELECTRON-ION AND ION- ION COLLISIONS WITH INTERSECTING BEAMS K. DOLDER AND B. PEART Department of Atomic Physics The University Newcastle upon Tyne NEI 7RU, England
I. Introduction In this article we will be concerned primarily with two-body, electronion, and ion-ion collisions that have been studied with intersecting beams, and perhaps one should start by asking why these collisionsare so interesting. In the “thin” plasmas encountered, for example, in thermonuclear research or stellar atmospheres, the ionization equilibrium (and hence the principal properties) is largely determined by the balance between electronimpact ionization and dielectronic recombination; more recently ion - ion collisions have also been found to be important. Some of the results we will describe are directly applicable to planetary atmospheres, flames, and “heavy-ion fusion,” and we will mention applications as they arise. Excitation of ions is also significant because the ensuing line radiation is often a major source of energy loss from plasmas. But a particular fascination arises from opportunities to study the simplest types of structure and reactions. For example, we will mention collisions involving He+ (H-like), Hf (simplest molecular structure), and Hf (simplest polyatomic) or one-electron heavyparticle reactions between, for example, H+- He+ or He2+- He+. In this way the experimentalist is able to test theory at a very fundamental level. Moreover, techniques have been developed which enable reactions to be studied at interaction energies from less than 0.1 eV to more than lo6eV so that a wide range of theoretical approximations can be examined. The first successful experiment with charged beams explored the ionization of He+ ions by electron impact; i.e., He++ e+ He2++ 2e
(1)
197 Copyright 0 1986 by Academic Press, Inc. AU rights of reproduction in any form reserved.
K. Dolder and B. Peart
198
but the field soon broadened to include a wide range of electron-ion and ion-ion collisions. The types of reaction which have so far been studied are: (1) Single and multiple ionization of atomic ions by electron impact; i.e.,
e
+ An +Am++
(rn - n + 1)e
(2) (2) Single and double detachment from atomic negative ions by electron impact; i.e., A 2e (3) e \A+ 3e (4) (3) Excitation of atomic and molecular ions by electron impact, e.g.,
+ +
+
e + Ca+(42S,Iz) Ca+(4 ’P,,) +
+e
(5)
(4) Dissociation ionization, excitation, recombination, and “pair production” from molecular ions by electron impact, e.g.,
&::+:5”
(6)
e+Hf,
\H
(7)
+H H+ H-
(8)
+
(9)
(5) Dielectronic recombination of atomic ions, e.g.,
+
+
e C+ **C -,C hv (6) Inelastic scattering of electrons by atomic positive ions, e.g.,
e + Cd+(5 2S112) -,Cd+(5 zPo)+ e
(10) (1 1)
(7) Mutual neutralization of atomic (or molecular) ions; i.e., A+
+ B- -*A + B
(12)
(8) Detachment from atomic negative ions by positive ion impact; i.e., A+
+ B--,A+ + B + e
(13)
(9) Association between H+ and H- ions; i.e., H + + H - + *H2 Hf (10) Ionization by positive ion impact; i.e., A+
+e
+ B+ + A + + B2++ e
(14)
(15)
ELECTRON -ION AND ION - ION COLLISIONS
199
(1 1) Charge transfer between positive ions; i.e.,
+
A++ B++A Bz+ (16) (12) Double charge transfer between positive and negative atomic ions; i.e., A+
+ B-
A-
+ B+
(17) In this article we will discuss some of the more interesting experimental results for each type of reaction and compare the results with theory. We will also endeavor to list the experiments that have been performed and briefly indicate applications of the results. But before proceeding we will comment very briefly on experimental techniques. +
11. Notes on Experimental Techniques This section also includes a few points of interest to theoreticians. These have been discussed many times, and the reader can refer to reviews by Harrison (1968, 1978), Dunn (1969, 1980, 1985), Dolder, (1969, 1983), Dolder and Peart (1976, 1985), Crandall, (1983), Salzborn (1983) and Brouillard and Claeys (1 983). A newcomer to this field might start by consulting the articles by Dolder and Peart ( 1976, 1985), Salzborn, Brouillard and Claeys, and Dunn (1985). All the experiments require two well-collimated beams of charged particles which are made to intersect so that their collision products can be detected and counted. The first point to stress is that the beam intensities are limited by space charge (e.g., Harrison, 1968; Dolder and Peart, 1976),and sometimes by other considerations, so that they are extremely tenuous. ~ and this is comparaParticle densities are typically of order lo6~ m or- less, ble to the density of residual gas in the ultrahigh vacuums through which the beams pass. This point is sometimes overlooked by experimentalists contemplating new types of measurements, especially if they are more familiar with atomic-beam or static-gas experimentswhere particle densities usually exceed loi2~ m - ~ . It follows that the “signals” from electron -ion or ion - ion reactions may be masked by “backgrounds” due to much more frequent collisionsbetween the beams and residual gas, but sophisticated beam-modulation techniques (e.g., Hamson, 1968; Dolder, 1969) have been devised to extract signals from backgrounds even when they are three or four orders of magnitude larger.
200
K. Dolder and B. Peart
The experimentalist in this field does, however, enjoy certain advantages. Charged beams can be quite energetic (1 - 100 keV) so that the collision products are easy to detect and, moreover, if they are charged, they can be very effectively isolated by various arrangements of static electric and magnetic fields. More remarkable advantages may ensue (e.g., Brouillard and Claeys, 1983) because the beams can be made to collide perpendicularly (6’ = 90°) or obliquely (6’ - loo),or they may be “merged”(6’ = Oo)to move in confluence over a path of several centimeters. Merged or inclined beams provide access to thermal interaction energies (-0.1 eV) even when working with conveniently fast beams ( E > 1 keV). Moreover, the energy resolution in the center-of-mass frame is also enhanced so that fine details of a reaction can be explored. A word of caution (especially to theoreticians!). It has sometimes been claimed that merged-beam experiments attained interaction energies far below 0.1 eV with comparably good resolution. Dunn (1980) has, however, pointed out that a beam is not composed of strictly parallel rays and that there are transverse components of velocity due to thermal energies and other effects (see also Dolder, 1983). When these are taken into account it becomes clear that energies and resolutions of, say, eV can only be attained if the beams are extremely well collimated, and it is doubtful whether this precaution was included in the relevant experiments. One might bear in mind a remark by Bardsley (made very much in jest) that “experimentalists can’t measure anything below lo-‘ eV.” This is of course untrue, but it is not so very wide of the mark for colliding-beam experiments. Another point to remember when presented with a measured cross section is the initial state of excitation of the reactants. Experimentalists sometimes gloss over this point, but it was illustrated quite dramatically by Latypov et al. ( 1964), who demonstrated that the apparent cross section for Xe++e-+Xe2++2e (18) can be made to increase fourfold simply by increasing the anode voltage in the ion source until metastable Xe+ionswere formed. Even greaterambiguities may exist when working with molecular ions since these can possess considerable vibrational and rotational internal energy. The most difficult experimental problem so far encountered in this field has been the absolute radiometric calibration of photomultipliers used in measurements of optical-excitation cross sections. The huge difference in intensity between light emitted from a crossed-beam experiment and a standard lamp was just one of the difficulties, but calibrations can now be achieved to within about 10%for visible and uv wavelengths, (e.g., Taylor, 1972). One should be very wary indeed of absolute excitation cross sections
ELECTRON-ION AND ION- ION COLLISIONS
20 1
that do not rely upon the most careful radiometric calibration. Some spectacularly incorrect results have been published.
111. Ionization of Positive Atomic Ions by Electron Impact In hot, thin plasmas the ionization equilibrium is primarily determined by the balance between electron-impact ionization and dielectronic recombination. Both processes have been studied with colliding beams. An extremely valuable and detailed collection of experimental and theoretical results for electron-impact ionization of atoms and positive ions has been assembled by Tawara et al. (1985), and this contains far more detail than we can accommodate here. In addition there is a recent and valuable review by Dunn (1985). Crossed-beam measurements for the ionization of positive ions have been listed in Table I, where the x's denote results for single ionization. For some ions measurements have also been reported for multiple ionization, e.g.,
+ Ar2++ e
Ar2+ e
-+
+ 3e Ar5++ 4e Ar4+
(19) (20)
and this is indicated by the addition of a circle. Nearly all references were listed by Tawara et al. (1985). A glance at the table shows that relatively little attention has been paid to heavier or more highly charged ions, and further inspection reveals a tendency for positions to have been filled in diagonal sequences (e.g., B+, C2+, N3+,04+ and C3+,N4+,05+). This was done to study isoelectronicseries and thereby reveal the effect of increasing the nuclear charge. There has naturally been a tendency to study ions that are most common in astrophysical or fusion plasmas but there are still some notable omissions. For example, calcium and silicon are commonly found in stellar atmospheres but they have, as yet, received little attention. Table I is not intended to inspire a young experimentalist to forsake stamp collecting and embark on filling empty boxes, but the blanks immediately reveal significant gaps in our knowledge. There are three attractive directions for further work. First, one might look at more isoelectronic series (e.g., He+, Li2+,Be3+,etc.) to examine the relative effectsof central and noncentral forces on bound electrons, or one might proceed horizontally across the periodic table (e.g., C+, N+, 0+,F+)to see the
TABLE I OF ELECTRON-IMPACT IONIZATION OF POSITIVE IONS" CROSSED-BEAM MEASUREMENTS Initialionization Atom + I
He Li
+2 + 3 +4 +5
+ x x x x
N
X
X
0 F
x x x x x
X
X
X N e X X X Na Mg X X
+
x x X
X + + + + X
+
cax sc
+2 + 3 4-4 +5
Ti
X
X
X
Cr Mn Fe X X
c
A K
Atom + I
v
X
a
+6 + I
X
B e x B X X
Al Si P S
Initial ionization
x
+6 +7 Atom + I
+2 +3 +4 +5
X
cu Zn
x
+7 Atom + 1 4-1 + 3 +4
Gd Tb
Pd
DY
a X
Sn
x x
Sb Te I
G a x Ge
+6
Ru
x
In
Ni
Initial ionization
Tc Ag
co
X
Initial ionintion
+
X
X
Ho Er Tm Yb Lu
Hf 0
X e + + + + + +
+
Ta W
cs
Br Kr Rb Sr
La
ce
Ir Pt
Pr
Au
Nd
Hg
zr
Nb Mo
B a x
+ + + o
+ X
X
X
Re 0s
As
se
X X
Pm
n
x x
Sm Eu
Pb Bi
+ + 0
" (X), Single ionization; (0),multiple ionization; (+), both. For references, see Tawara et nl. (1985) and Dunn (1985).
+5
+6
+7
ELECTRON-ION AND ION-ION COLLISIONS
203
effect of adding electrons, one by one. This type of measurement, when combined with theory, might reveal something subtle about ionic structure and behavior. It also provides immediate tests for the various empirical formulas used by plasma physicists to predict cross sections when theory or measurements do not exist. These formulasare particularly useful for highly charged ions which are beyond the range of most experiments. A second line of investigation is a more detailed study of autoionization (i.e., ejection of outer-shell electrons by an impact-excited, inner-electron core) which sometimes enhances ionization very dramatically. Thirdly, experimentaltechniques should be extended towards the study of very highly charged ions. We will deal briefly with these topics and illustrate them with a few examples. A. AUTOIONIZATION
Figure 1 illustrates measured (Peart and Dolder, 1975) cross sections for the single ionization of Mg+, Ca+, Sr+, and Ba+ and it shows that, except for Mg+, the cross sections increase abruptly at energies greater than the outershell ionization threshold. This is attributed to the onset of excitation autoionization. A number of attempts have been made to calculate the magnitude of these effects. Burke et al. (1983) considered the direct ionization of Ca+
+
Ca+(3p64s) e
-
+
Ca2+(3p6) e
(21)
I
Ekctron mrrpy (eV) FIG. 1 . Measured cross sections for electron impact ionization of Mg+, Ca+,Sr+,and Ba+ ions. With the exception of Mg+ the ionization functions show a steep rise due to the onset of autoionization.
K. Dolder and B. Peart
204
-
and its enhancement by,
+
Ca+(3p64s) e
+
*Ca+(3p54snd) e
-+
+
Ca2+(3p6) 2e
(22)
Although the process is conceptually simple, it is difficult to calculate. There are many contributing autoionizing states of *Ca+and they can be strongly coupled by interaction with the scattered electron. Moreover, close to the autoionization thresholds (when the scattered electron is moving slowly) there is strong “postcollision interaction” between the excited and scattered electrons, and there may also be considerable interference between direct ionization and autoionization. In spite of these difficulties, it can be seen from Fig. 2 that Burke et al. obtained good agreement with experiment for Ca+ at electron energies greater than 50 eV. Autoionization might be expected to be particularly strong for ions in which a small number of valence electrons surround a closed shell which has the same principal quantum number. This situation exists for Ti3+,Zr3+, and Hf3+,which have been studied experimentally by Falk et al. (1 983) and theoretically in the distorted-wave approximation with exchange, by Bottcher et al. (1983). Transitions of the type np6nd
+
npsnd2
(23)
E
0
+ Y al VI
e
Incident elettron energy ( e V )
FIG.2. Electron-impactionization of ground-stateCa+. Experimental results (-) are McGuire, 1977) and with the compared with a Born calculation to direct ionization (---, addition of an R-matrix calculation for autoionization (---, Burke ef al., 1983).
ELECTRON-ION AND ION-ION COLLISIONS
205
were calculated with n = 3, 4, and 5 for Ti3+,Zr3+, and Hf3+,respectively. Figure 3 illustrates results for Ti3+.The dashed curve is an estimate of direct ionization obtained from the Lotz empirical formula, and it is seen to be an order of magnitude smaller than the measurements. The solid, steplike curve shows the calculated autoionization added to the Lotz estimate. The effect of convoluting this with a 2 eV Gaussian (to model the experimental resolution) is represented by the dot-dashed curve. It must be added that the theoretical result for autoionization has arbitrarily been divided by 2.5 to fit the measurement, and this indicates how difficult these calculations are.
r:l
-10 40
45
*
50 ENERGY ( e v )
55
) FIG.3. The curve (---) is a Lotz estimate of direct ionization ofTi'+ ions. The (curve shows the effect of adding a DWA estimate of autoionization, which has been arbitrarily divided by 2.5. The (- . -) curve, obtained by convoluting this result with the experimental energy spread, can be directly compared with the experimental points.
206
K. Dolder and B. Peart
It has also been shown (Crandall et al., 1979) that for certain isoelectronic sequences (e.g., C3+, N4+, 05+) autoionization becomes relatively more important as the charge increases, although recent measurements for Ne7+ (Chantrenne et al., 1985) suggest that this trend does not persist for very highly charged members of the series. Figures 4 and 5 show measurements for electron-impact ionization of N4+and 05+, where the onset of autoionization at electron energiesin the range 400- 500 eV can clearly be seen. Very recent measurements using an electron cyclotron resonance (ECR) source, in place of a Philips Ion Gauge (PIG) source by D. H. Crandall et al. (private communication) show the autoionization structure with greater precision and detail and permits useful comparisons with the theories of Younger (198 1) and Jakubowicz and Moores (198 1). Inner-shell effects occur for a wide variety of ions and sometimes their threshold is close to that for outer-shell ionization. This is important in plasmas since it is usually the overlap between the high-energy “tail” of the plasma electron distribution and the near-threshold ionization cross section that primarily determines the ionization rate. Figures 6 and 7 illustrate measured cross sections for Ga+ (Rogerset at., 1982) and Xe3+(Gregory et
FIG.4. Ionization of N4+showing a secondary peak due to autoionization. (0)and (0) denote results with crossed beams (Crandall er al., 1979) and an ion trap (Donets and Pikin, 1976).()and (---) curves representCB calculationsby Moores (1978)and Golden and Sampson ( 1980).The (- . -) curve is a Lotz estimate of direct ionization and the (---) curve illustrates Henry’s (1979) estimate of autoionization.
207
ELECTRON-ION AND ION-ION COLLISIONS
FIG.5. Measurements (Crandall ef al., 1979) of ionization of Os+,showing a secondary peak due to autoionization compared with a Lotz estimate of direct ionization (---), CB (---, Golden and Sampson, 1980), and CC (--, Henry, 1979).
-
N
I00
5 2I0
80-
z
60-
-c
I
I
I
//-: 1 . *...**
0
8
40-
:
’ :.:..*.,‘ 0
20-
.0..
0
V
0
I
L’
.Y
/
-
0
*:
v)
: .....-./.../ * -
...*... .). ..--(’ *
v)
g K
I
I
0
0
0
-
0
I
I
1 with
K. Dolder and B. Peart
208
I
70
30
I
1
110
1
I
150
ENERGY i e V 1
FIG. 7. Measured cross sections for Xe3+ compared with DWA calculations of direct ionization and excitation autoionization (Griffin et a!., 1984).
al., 1983)and in both cases the substantial autoionization was found near the outer-shell threshold. LaGattuta and Hahn (1 98 1) showed theoretically that cross sections may be further enhanced by another mechanism known as capture autoionization. In the reaction Fel5+ + e -+ **Fe14+-b Fel6+ + 2e (24) the incident electron is captured to form a doubly excited ion which decays with the ejection of two electrons. The process is similar to dielectronic recombination (Section IV) except that the doubly excited ion now decays with the emission of electrons rather than radiation. Their result for FeI5+is illustrated by Fig. 8. An extremely powerful new technique for studying autoionizing transitions in positive ions has just been developed from the work of Lyon et al. ( 1984), who described a merged-beam experiment on the photoionization of Ba+ by vuv synchrotron radiation; i.e., Ba+
+ hv -,Ba2++ e
(25)
Further experiments (as yet unpublished) revealed great detail. More than 75 resonant structures were found in the photon energy range 16.3- 28 eV, and for the largest structures the cross section exceeded cmz. The cross sections were measured absolutely with a resolution of 4 meV, which greatly exceeds the best so far obtained (- 100 meV) for electron-impact ionization of heavy ions (Peart et al., 1973). Similar results have been obtained for the photoionization of Ca+ and Sr+. Figure 9 shows part of the photoionization
ELECTRON- ION AND ION-ION COLLISIONS
209
E, f k e v )
FIG.8. Calculated cross sections for the ionization of Fe15+showingthe effects ofexcitation autoionization and capture autoionization. ( . . . ), Lotz estimate of direct ionization; (---), enhancement by autoionization; and (-), effect of adding capture autoionization.
cross section of Ba+, but these ordinate scales will be revised when calibrations of the ion and photon detectors are complete.
FORMULAS FOR IONIZATION CROSSSECTIONS B. EMPIRICAL Empirical formulas are widely used when measurements or theory are not available. The expressions most widely used were proposed by Lotz (1968, 1969),Moores et al. (1 980), Golden and Sampson (1 977,1980), and Burgess and Percival(1968), and the most detailed tests of their validity were made by Itikawa and Kato ( 198 1). For ions with an initial charge greater than 3 Lotz suggested
+
for the cross section at incident electron energy E. Here rj represents the number of electrons in the sublevel of ionization energy 4. For less highly charged ions he proposed the more complicated expression
K. Dolder and B. Peart
t
Wavelength I nm I
7
1
4
Wavelength ( n m 1
FIG.9. Part of the photoionization cross section of Ba+. The inset shows the largest resonances. Note: final calibration of the ordinate scale has not yet been performed.
wherexi = E/4and aj,bj,and cjrepresentadjustable parameters. Numerical values of these parameters were given for (He -Ga)+, (Li - Zn)*+, and (Be Ga)’+. Equation (27)is therefore restricted to ions for which parameters were evaluated and for the limited range of energies over which they have been fitted. Golden and Sampson suggested an expression which contains four adjustable parameters which depend only on the subshell of the ejected electron.
ELECTRON-ION AND ION-ION COLLISIONS
21 1
Numerical values have been determined forj = 1sto 4fand a choice is made to ensure the correct Bethe asymptote for a hydrogenic ion. Their expression can be written
+(;+a)
.
.
(1 -u;1)]
where n is the principal quantum number of levelj, IH= 13.6 eV, uj = E/Ij and Z($is the effective charge binding an electron in levelj of the initial ion. Aj, Dj, c j , and dj represent the adjustable parameters. Burgess and Percival derived the "ECIP" (exchangeclassical result added to a long-range impact parameter contribution) formula widely used for highly charged ions in astrophysics,but it is more complicated to apply and will not be discussed here. Itikawa and Kato compared the Lotz and the Golden and Sampson formulas with experimental data for no fewer than 27 ions. The agreement was frequently within 25%. In many cases agreement was much better, but large deviations were noted for Na+, O+,and C3+. Crandall also discussed the validity of the Lotz formula for some highly charged ions. He found (with the exception of C3+) that it agreed with experiment within a few percent but that the agreement was partly fortuitous. The simple Lotz formula [Eq. (3.8)] usually overestimates direct ionization and omits autoionization, but for many ions these errors roughly balance and so there is often remarkably good agreement above the autoionization thresholds. Unfortunately, this is not true when autoionization is particularly strong. C. IONIZATION OF VERY HIGHLY CHARGED IONS The study of highly charged ions reveals systematictrends in isoelectronic sequences and provides direct information about ions encountered in very hot laboratory plasmas and stellar atmospheres. Ionization reaction rates can be deduced from observations of vuv emission from hot plasmas, and a recent example is the work of Jones and Kallne (1983), who studied Krn+ (n = 8,9,10,11). More accurate and detailed information can be obtained from crossedbeam experiments, although these encounter three major difficulties. First, it is convenient to have resolved ion beams with currents of at least lo-* A, but to achieve this for very highly charged ions (A"+with n 2 10) inevitably
212
K. Dolder and B. Peart
requires a large, expensive source. Second, if the ions possess metastable or long-lived Rydberg states they will almost certainly be populated in the source and it may not be possible to define their initial excitation. Third, the charge ratio of parent and product ions (e.g., A"+ and A("+*)+)approaches unity as n increases and so it becomes more difficult to separate them. An excellent introduction to this topic is provided by Volume T3 of Physica Scripta (1983) and Volume B9 of Nuclear Instruments andMethods ( 1 985). It seems that large PIG sources operating with low gas density and high anode voltages are suitable for the production of ions with four- or fivefold charge. Beyond that, one must consider sources employing electron-beam ionization (EBIS), electron cyclotron resonance (ECR), or lasers. The most notable work with PIG sources is the measurement of ionization and excitation of multiply charged ions (e.g., C3+, N4+, and Os+)at Oak Ridge ( e g , Crandall et al., 1979), but more refined measurements have recently been obtained with an ECR source (Meyer, 1985). Measurements for Ne7+have just been reported by Chantrenne et al. (1985), who observed structure due to autoionization similar to that seen for N4+and Os+. Donets (1983) described EBIS sources and their application to electronion collisions. In his source ions were trapped by electric fields for about one second while they were bombarded by an extremely energetic (10 keV), intense (600 A cm-2) electron beam which produced progressive ionization by multiple collisions. Kr34+andXe52+were produced in this way, although such highly charged states were not sufficiently prolific for electron -ion experiments. The electronic excitation of the ions is generally unknown, Geller and Jacquot (1 983) described a source employing electron cyclotron resonance heating to produce C6+,N7+,08+, and NeIo+with total ion yields of order A. These sophisticated types of source point to one direction in which the study of electron - ion collisions is likely to develop, and Fig. 10 illustrates measurements with a PIG source by Muller et al. (1980) for the ionization of ATR+( n = 1 - 5 ) which fit quite well to the solid curves given by o = 1.4 X In x / I i x cm2 (29) where x is the electron energy in units of the ionization energy I,, . Measurements for Ar"' ( n = 8 - 17)by Donets and Ovsyannikov ( 1977)with an EBIS source have also been reported. Since the current of highly charged parent ions and their cross sections tend to be small, Muller et al. (1985a) have developed a carefully designed high-current electron gun in which electron space-charge forces are balanced by potentials applied to external electrodes. It has recently been used to measure (Muller et al., 1985b)cross sections for single and double ionization of Ar"+ ( n = 1-4).
ELECTRON - ION AND ION - ION COLLISIONS 1
I
I
I
I
I l l 1
1 0 %-
,
I
I
1
1
0
1
213
-
1
, - - -
-6
N
-
n.1
z
0
n.2
+
w un; m
-
-
n=3
I
I
n.6
-
v)
E
-
n.5
XI-* 1
K)
I
I
I I I I I I
100
I
I
I
1
, , , , I
-
1000
A In EIX~E, A = 1.4X
IV. Measurements of Dielectronic Recombination In hot thin plasmas, ionization is primarily balanced by dielectronic recombination (DR), in which an incident electron collides with a positive ion and loses energy by exciting a discrete transition in the bound electrons. As a result, the projectile becomes trapped in a discrete state, so that a doubly excited atom or ion is formed which subsequently decays radiatively. The process is very important in plasma physics, but measurements with electron - ion beams are extremely challenging, for two reasons. First, the cross sections tend to be small but, more seriously,the projectile must excite a discrete transition in the ionic electrons. Consequently the process is sharply resonant and it depends sensitively upon the environment in which the ions are found. Until recently almost all information about DR was obtained theoretically and early work has been reviewed by Seaton and Storey ( 1977), Dubdu and Volante ( 1980), and Roszman ( 1982).A detailed comparison of calculations with approximate formulas for the DR of hydrogenlike ions has been prepared by Fujimoto er al. (1982). The first published beam measurement appears to be that of Belit et al.
K. Dolder and B. Peart
214
(1983), which studied Mg+(3s)
+e
Mg(3p, nl) + Mg(3s, n l )
+ hv
(30) They measured the rate of production of neutral Mg atoms formed when beams of electrons and Mg+(3s)ions intersected. By observing the production of 3s-3p photons and Mg atoms in delayed coincidence, they were able to discriminateagainst atoms produced by extraneousprocesses and identify those produced solely by the reaction in Eq. (30). Their next problem was to define the exact energy and energy spread of their electron beam. This was done by using the same apparatus to make absolute measurements of electron-impact excitation cross sections for Mg+(3s+ 3p). Now excitation cross sections for atomic ions are known to rise vertically from a well-defined threshold, but measured cross sections do not rise abruptly, due mainly to the energy spread (=0.3 eV) ofthe electron beam. By studyingthe threshold behavior of an experimentalcross section, it is possible to define the electron beam energy precisely and also estimate its spread. This spread can then be convolutedwith the sharply resonant theoretical cross section for DR so that direct comparisons between theory and experiment are possible. Another problem encounteredby Belie et al. concerned the efficiencywith which Mg(3s, nl) atoms were detected. A small (36 V/cm) electric field was used to separatethese atoms from parent Mg+ions,and it was calculatedthat the field was sufficient to field ionize atoms in states with n 2 64. When this and other uncertainties were taken into account, it was estimated that DR cross sections were measured to within f 58%. Figure 11 compares the experimental points with theoretical results for n d 64 which have been convoluted with the experimental energy distribution. Curve B is Burgess’s result (private communication to G. Dunn, 1982). Curves LHI and LH2 (LaGattuta and Hahn, 1982, 1983) correspond, respectively, to field-freeconditions and to the Lorentz (V X B) field encountered by the ions as they passed through the magnetic field used by Belie et al. to constrain their electron beam. Although this field was only 24 V cm-’, it appears to enhance the cross section by an order of magnitude. Hickman (1985) recently demonstrated that the cross section in the field-free limit is larger if the fine structure of the ionic core is taken into account since each state contributes separately to recombination. He also used more accurate autoionization rates for states with large angular momentum. Curves H2 and H 1, respectively, show Hickman’s results with and without allowance for fine structure. Measurements for Ca+ have been briefly reported by Williams ( 1983), who used a coincidence technique similar to that just described. A different approach which used merged beams of electrons and 450 keV C+ ions was described by Mitchell et al. (1983). This enjoyed the good resolution +
ELECTRON-ION AND ION-ION COLLISIONS
z
215
-
0
c
0.8-
U
w
-
0.6v1
w
2
-
0.4-
-
U
0.2 0
5.0
ENERGY ( e V )
FIG.1 1. Measurements of the dielectronic recombination of Mg+ compared with calculations by Burgess (---), Hickman (HI and H2), and LaGaauta and Hahn (LHI and LH2). Experimental points are represented by (x).
(-45 meV) typical of merged-beam experiments and the energetic neutral products were easy to detect with almost 100%efficiency. However, the ratio of signal to background was poorer than in coincidence experiments and the C+ beam contained a proportion of metastable ions, estimated to be much less than 30%. It was therefore claimed that the results represent a lower limit. Merged beams have also been used by Dittner ef al. (1983) to study B2+ and C3+and their results differ only slightly from theory. The effects of fields on their apparatus have not yet been examined but they are expected to be less significant than in the apparatus of BeliC et al. Dunn ef al. (1984) concluded “the important process of DR is at once complex and beautiful, yet fragile and sensitive. It remains a challenge to both experimentalists and theoreticians to understand DR and how it is affected by the environment in which it occurs.”
V. Measurements of Electron-Impact Excitation of Positive Ions More details of this topic from the experimental standpoint can be found in articles by Crandall ( 1981, 1983),Crandall ef al. ( 1979),Dunn ( 1980),and
K. Dolder and B. Peart
216
TABLE I1 OF ELECTRON-IMPACT EXCITATION OF CROSSED-BEAM MEASUREMENTS POSITIVE IONS
Transition
Reference
Dance et al. (1966); Dolder and Peart (1973) He+(IS- 2 P) Daschenko et al. (1975) Li+(2 IS-2 )P) Rogers et al. ( 1978) Zhmenyak et al. (1983) Na+ Na2+ Zhmenyak et al. (1983) Taylor et al. (1980); Be'(2S- 2P) Gregory et al. (1 979) Kohl and Lafyatis (1983) C'(2p 'pO-2~' '0) C3+(2S-2P) Taylor et al. (1 979) Gregory et al. (1979) N4+(2S-2P) Kel'man et al. (1975); Mg'(3 'S- 3 'P) Zapesochnyi et al. (1984) Ca+(4 'S-4 2P) Kel'man and Imre (1975) Ca+(4S, 4P; 4P, 4 0 ) Zapesochnyi et al. (1976) Taylor and Dunn (1973) Ca+(4S-4P,/2.3/,) Zapesochnyi et al. (1976) Sr+(5S-5P) Pace and Hooper ( 1 973); B a V 2 S- 62P,12,3,2) Crandall et al. ( 1974) Ba+(22P312-72S,6 '0) Zapesochnyi et al. (1973) Cd+(5p =P,6s 2S,12,5s2 '0) Hane et al. (1983) Fe+ Kolosov et al. (1983) Zapesochnyi et al. (1973) Ar+(4S- 4P) Kr+(SS- 5 P) Zapesochnyi et al. (1973) Crandall et al. (1975) Hg+(6 'S-6 'P3/2) Hg'(6S-7S) Phaneuf et al. (1976) Zn+(4p zP, 5s ' S ) Rogers ef al. (1982) Ga+(4 IP) Stefani et al. (1982)
Comment" Nb N* N* A' A
A Ad Ad Ad Ad Ad A
A A A' Ae A A
A U A A U U A
A A A
The symbols A, N, and U indicate whether the measurements were absolute, normalized to theory, or expressedin arbitrary units, respectively. H-like. AS # 0, He-like. Li-like. Also measured polarization.
ELECTRON-ION AND ION-ION COLLISIONS
217
Dolder and Peart ( 1976). Crandall(l983) has also given a lucid outline of theoretical methods which will appeal to experimentalists, but for more profound reviews oftheory one should consult Seaton ( 1979, Henry ( 198I), and Robb (1980). Theoretical results for more than 100 transitions were listed by Mertz et al. (1980), and the application of excitation cross sections to fusion and astrophysics have been discussed by many authors including Gabriel and Jordan (1972), Lorenz (1978), and Post ( 1982). Table I1 lists most of the reactions which have been studied by measuring radiation emitted from intersecting beams. An alternative crossed-beam approach has recently been developed in which inelastically scattered electrons are collected and counted. This will be discussed in Section VI. A. BRIEFCOMMENT ON THEORY
There is a striking qualitative difference between electron excitation of positive ions and neutral atoms. In the case of atoms the cross section is zero at threshold, whereas for ions it is finite and often has almost its maximum value. This is due to the attractive ionic field, which causes a number of partial waves to contribute even at threshold. At high energies (i.e., one or two orders above threshold) the Bethe-type energy dependence ln(BE) (31) is followed for A1 = 1, As = 0 transitions, but when A1 # 1 and As = 0, 4 = (A/E)
G=
C/E
(32)
and when As # 0 Q = DIE3
(33) where A, B, C,and D are constants for a given reaction. Equation (31) can also be written
wherefis the oscillator strength, while A E and E represent energies of the transition and the incident electron (Rydbergs). A valuable check on theory, which does not rely on radiometric calibration, is provided by measuring the polarization of emitted radiation. McFarlane ( 1974)showed that B can be evaluated in the Bethe approximationfrom the energy at which the polarization passes through zero, and since A is defined by the oscillator strength (which is known for many transitions)the
218
K. Dolder and B. Peart
high-energy cross section is defined absolutely by the polarization. A simpler check is provided by comparingthe measured and calculated energy dependence of polarization. Plasma physicists often speak of a “collision strength” (a)defined by
i2=ociE (35) where o represents the statistical weight of the lower (initial) state of the target ion. This scales approximately as where Zcffisthe effective charge on the commuting electron. For ionization we discussed empirical formulas. Astrophysicists sometimes use the “g-bar formula” for excitation
c:,
wheref is the oscillator strength for the particular transition, E is the incident electron energy (Rydbergs),AEis the energy of the transition (Rydbergs),gis the “effective Gaunt factor,” and a, is the radius of the first Bohr orbit. Reliable oscillator strengths for very many transitions can be obtained from tables and so the prime uncertainty is usually associated with i.Crandall (1983) critically reviewed the choice of values for For resonant transitions of singly charged ions one may assume S = 0.2, but larger values are needed for highly charged ions. Younger and Wiese (1 979) suggested g = 1 .O for An = 0 transitions of more highly charged ions, whereas when An # 0, the best values ranged from 0.05 to 0.7. Generally, it is assumed that Eq. (36) is valid within a factor of 2 or 3 , but clearly there can be exceptions. Just as autoionization could play havoc with empirical formulas for ionization, one might expect that resonances and other correlation effects will upset simple predictions for excitation.
B. MEASUREMENTS OF THE EXCITATION OF He+ IONS Daschenko et al. (1975) made nonabsolute measurements of the He+ (IS-2P) transition which, when normalized with theory at higher energies, agreed quite well with the predicted shape of the cross-section curve. The situation is, however, less satisfactory for He+ ( IS-2s). Measurements of nonabsolute cross sections by Dance et al. (1966) and Dolder and Peart (1973) indicated a much slower variation of cross section with electron energy than predicted by theory. The anomaly has been discussed by Seaton ( 1973, Henry (1 98 1), Crandall ( 1983),and others, but no convincingexplanation has yet been found. However, it has been suggested that subtle effects can arise from slow ions trapped in the space charge of electron beams when
219
ELECTRON-ION AND ION-ION COLLISIONS
these ions interact with the fast ion beam in a crossed-beam experiment. Evidence was obtained by Defrance et al. ( 1982) in their study of H-+e+H++3e
(37)
and it is not inconceivable that some interaction between slow, trapped ions and He+(2S) might have influenced the measurements of He+(IS-2s). Recent unpublished measurements on reaction (37) in this laboratory (Wilkins et d . )support the evidence of Defrance et al. Other unexplained spurious effects, which depended upon electron current, were noted by Gregory et af.( I 979) in course of their measurements ofthe excitation ofC3+andN4+ and so further measurements of He+( IS- 2 s ) are clearly needed.
c. RESULTSFOR HELIUMLIKE IONS Rogers et al. ( 1978) made absolute measurements for the Li+(1 IS-2 3P) transition by observing 1 = 5485 A radiation from Li+. This was notable because it was the first measurement of a spin-changing transition and the cross sections (- lo-'* cm2) are consequently small. The results are illus-
0.11
50
I I
60
I
I
80
I I I 100 120 ELECTRON ENERGY ( e V ) I
1
I I 110
I I 1M)
I
!a0
I
200
FIG.12. Electron-impact excitation of Li+(1 IS-23 P )compared with two distorted-wave calculations (---), a Coulomb-Born-Oppenheimer calculation (-), and three closecoupling calculations (open symbols).
220
K, Dolder and B. Peart
trated by Fig. 12, where evidence of near-threshold resonances can be seen. At high energies the results follow the predicted E-3 energy dependence and, although close-coupling calculations (Rogers et al., 1978; Henry, 1981) reproduce the structure, they do not agree with the magnitude of the measurements between 60 and 160 eV to better than & 50%. More experimental and theoretical work is needed on spin-changingcollisions of two-electron ions because they are believed to persist over a broad range of temperatures in fusion plasmas, and it has been predicted by Pradhan et al. ( 1981) that in highly charged ions there are extremely large resonances in intercombination transitions which could enhance energy-averaged rate coefficients by factors of about six.
D. RESULTSFOR LITHIUMLIKE IONS Measurements for Be+, C3+,and N4+provide the best available set ofdata for an isoelectronic sequence, and the very careful work on Be+(2S-2P) has been cited as “the benchmark experiment.” It is therefore interesting to compare the measurements by Taylor et al. (1980) for Be+ with theoretical predictions. Figure 13 illustrates the gratifying result that, in this case, as theory became more sophisticated the agreement with experiment improved.
Electron Energy (eV)
FIG. 13. Measurements of excitation of Be+(2S-2P) compared with Coulomb-Born (CBI and CBI I), Coulomb-Born exchange (CBXIand CBXII), close-coupling(2CCand 5CC), and the g empirical formula.
22 1
ELECTRON-ION AND ION-ION COLLISIONS
M
E l e c t r o n Energy ( e V )
FIG. 14. Measurements of Ca+ K line excitation compared with 3-state close-coupling (CC), Coulomb distorted-wave (CDW), and classical binary encounter (CL) calculations.
Measurements for resonant excitation of C3+and N4+were discussed by Gregory et al. (1 979), and they were found to agree very well with two-state close-couplingcalculations which included exchange (van Wyngaarden and Henry, 1976). Theory might tend to become more reliable as ionic charge increases because the central field becomes relatively more important and fine-structure levels are more widely spaced.
r
It
1.I
6
.r
‘9fi I
U
0.
c,
N L
.r
2 -0.10
n
-0.2
1-
t
d.
1
t f t
,
FIG. 15. (0)and (0),respectively, show measured polarizations of Ca+ H and K line radiation. (), theoretical predictions.
222
K. Dolder and B. Pearl E. RESULTS FOR MORECOMPLEX IONS
There is insufficient space to discuss many of the various other transitions that have been studied, but measurements of the H and K transition in Ca+(4S-4P3,,, are worth mentioningin view of their astrophysicalsignificance. The polarization of radiation from the two transitions has also been measured, and it was noted in Section V,A that the constant Bin Eq. (34)can be determined from this result. Figures 14 and 15, taken from Taylor and Dunn (1973), illustrate these results and compare them with theoretical predictions.
VI. Scattering of Electrons by Ions Hundreds of measurementshave been made on scatteringof electrons by atomic or molecular beams and the prospect of analogous experiments with ion beams springs readily to mind. But the experimental difficultiesare very daunting indeed. For space-charge reasons it is impractical to produce welldefined ion beams with particle densities much greater than lo6 ~ m - and ~, this is 6 or 7 orders of magnitude less than in most molecularbeams. Clearly, any attempt to scatter electrons from such tenuous beams will encounter severe problems of signal to background. But the prospect is enticing. By collectingelectrons that have lost specified energies one can measure excitation cross sections without recourse to radiometric calibration, and if one were able to work with more highly charged ions (where electrons exciting resonant transitions lose greater energies) the experiments might, in one sense, become easier than corresponding radiometric measurements, because in the far vuv (where efficient filters or lenses are unavailable) these could well be impossible. Scattering experiments provide diyerential cross sections which might provide more searchingtests for theory and they would reveal any resonance states with lifetimes 5 1O-I2 s. The results are also free from cascading, and “forbidden” transitions can be excited. On the other hand, they are not well suited to the study of excitation thresholds (where cross sections are often largest and most interesting) and the integration of results to produce accurate, absolute total cross sections would, at best, be extremely laborious. Scattering measurements therefore tend to complement radiometric experiments and they call for the brand of heroic optimism displayed by Chutjian and his colleagues who have pioneered the field. Their apparatus (Chutjian, 1984)illustrated by Fig. 16 shows an electron beam crossing an
ELECTRON-ION AND ION-ION COLLISIONS
ION SOURCE
EXTRACTION LENSES
/I
223
DUMP
ON SOURCI
FIG.16. Apparatus developed by Chutjian's (1984) group to study inelastic scattering of electrons by positive ions.
ion beam which is selected by momentum and velocity filters. A hemispherical analyzer selects electrons scattered through an angle which, in experiments with Zn+ ions, was set typically at 14". The ion beam energy and current were typically 6 keV and 4 X 10-6 A, which corresponds to particle densities of order 5 X 1O7 ~ m - Electron ~. beam currents were 2 - 4 X 10-6 A and the gun and analyzer had an overall resolution of 0.45 eV (FWHM). Typically only 1 in loL2of the primary electrons was detected! The pressure in the interaction region (1 - 5 X lo-* torr) was surprisingly high and could certainly be reduced by better pumping. It is therefore not surprising that the peaks of scattered electrons rested upon a steeply sloping background, due primarily to scattering from molecular nitrogen. Relative differential cross sections can be deduced from the peak heights, and in Fig. 17 the results of Chutjian and Newel1 (1982) are compared with 5-state close-coupling theory by Msezane and Henry (1983). The inset also compares normalized measurementswith the theoretical curve, but here the theory is illustrated over a wider angular range. The results are encouraging and the technique has appeal and potential, but one might ask how useful this type of measurementwill be if it cannot be made absolute over the whole angular range. Chutjian (1984) has obtained similar results for Cd+(5 2S- 5 2po).
K. Dolder and B. Peart
224
-
I \
b
u)
\
0.1
ftq.01
cy
E U
2I
10
20
10.
loo
60
140 1;
+
0
n
cc c
a 'z1
,-42pc
\
b
' I I
1.0.
I
.
.
.
'
'
0 4 8 12 16 2 0 SCATTERING ANGLE (deg) FIG.17. Measured differential cross sections for the scattering ofelectrons by Zn+normalized to and compared with 5-state close-coupling theory. (+), Results for 75 eV electrons at a scattering angle of 14". The inset also normalized to the theoretical curve, (-), compares normalized measurements (0)with the theoretical curve, (-).
VII. Collisions between Electrons and Negative Ions Attention was first drawn to detachment from H-ions by electron impact,
+
2e (38) H-+ e - H because it seemed that this might be significant in the destruction of H- in stellar photospheres and thereby control their opacity to visible radiation. It transpired, however, that
+
H-+ H + H2 e
(39) is a much more rapid process (Schmeltekopf et al., 1967). Early work on electron - negative ion collisions was summarized by Dolder and Peart (1 976a) and, as these processes are rarely of much practical importance, we will deal only briefly with them. The early work did, however, illustrate how
ELECTRON-ION AND ION-ION COLLISIONS
225
sensitive the theory of negative ion collisionscan be to the choice of accurate wave functions and, apart from this, there are only three aspects on which we will comment. First, in the course of measurements of the reaction in Eq. (38) with inclined beams, well-defined structures were observed in the detachment function. The larger and smaller of these structures were attributed (Taylor and Thomas, 1972; Thomas, 1974) to the formation of short-lived excited states of H2- with wave functions that were, respectively, 2s22p 2P0and a mixture of 2p3and 2s22p.Theorists have contested the possible existence of H2- but the matter does not appear to have been resolved. We only wish to point out that all the experimental evidence comes from this laboratory (although it was obtained with two pieces of apparatus) and we would like others to investigate these structures, perhaps with merged beams. Evidence of two smaller structures has been observed in the detachment function of 0-(Peart et al., 1979). The second point concerns the search for a simple scaling law analogous, perhaps, to Thomson scaling for positive ions. Measurements of electronimpact detachment are available for H-, C-, 0-, and F-, and this led Esaulov ( 1980) to apply the well-known split-shell model in which, for example, H-is considered to have one strongly bound and one weakly bound electron. Thereafter, using classical methods, he derived a law which fitted the four measured cross sections to within about 20%. Finally, a more disturbing result unearthed by Defrance et al. (1982) in their measurements of H-
+e
+
H+
+ 3e
(40)
They obtained cross sections much smaller than those published by Peart et al. ( 197 1 ) and they argued persuasively that slow ions, trapped in electronbeam space charge, may greatly influence some measured cross sections in a surprisingly large and subtle way. Moreover, crossed-beam measurements usually include several powerful internal checkswhich give one considerable confidencein the results, but for reasons that Defrance et al. explain, these slow-ion effects may elude most of the checks. Similar results have recently been obtained in this laboratory (Wilkins et al., unpublished). The nagging doubt therefore arises that subtle flaws may exist in some other measurements. Elsewhere in this article we have noted difficulties associated with H+/H- neutralization, e/He+ ( 1 s-2s) excitation and e/N4+ excitation. Having made this point, it must be emphasized that the very great majority of electron - ion and ion -ion experiments are very accurate and that agreement between measurements in various laboratories and with theory is usually very good indeed.
226
K. Dolder and B. Peart
VIII. Collisions between Electrons and Molecular Ions Merged and inclined beams have been used to study a variety of collisions between electrons and molecular ions. One attraction of these methods is that they are applicable to the simplest molecular structure ( H f ) for which Schrodinger’s equation can be solved exactly. Moreover, its chemical activity in plasmas makes H f unsuitable for study by afterglow techniques. Cross sections have been measured for each of the processes and the early work was
reviewed by Dolder and Peart ( 1976b). Inclined beams have also been used to study e - D t collisions. The most useful measurements are probably those of dissociative recombination [reaction of Eq. (43)], which have been excellently described and discussed by Mitchell and McGowan (1983). The inclined-beam measurement of e- H$ recombination by Peart and Dolder ( 1974a)was amplified by Vogler and Dunn ( 1975)and Phaneuf et al. (1979, who, respectively,combined optical and crossed-beam techniques to measure partial cross sections for
Hf+e+H(2p)+H
(45)
and
H f + e--,H(n =4)
+H
(46) They demonstrated that neither channel contributes a large fraction to the cross section and this agrees with theoretical evidence that n = 3 dominates. Other attractions of beam methods (as opposed to afterglow techniques) are that they provide cross sections (rather than reaction rates) which may exhibit structure which can be correlated with theory; they also give results over two or three orders of electron energy with constant molecular internal energy. This helped to resolve disputes which existed about the temperature dependence of dissociative recombination. There have recently been a number of merged-beam measurements of the dissociative recombination of molecular ions but a fundamental difficulty is the production of molecular ions with known initial internal excitation. This is important because calculations (e.g., Derkits et al., 1979) predict that the
227
ELECTRON-ION AND ION-ION COLLISIONS
recombination of Hf (and presumably of other molecular ions) is very sensitive to vibrational excitation. For Hf it is possible to produce ions with a known distribution of vibrational states (von Busch and Dunn, 1972) and this makes measurements interpretable. Efforts are underway (Mitchell and McGowan, 1983) to build sources in which ions are vibrationally cooled before extraction. This is insufficient space to describe in detail measurements on molecular ions and the reader is referred to the review by Mitchell and McGowan. Ions that have been studied with merged beams include, Hf, HD+, Df, H:, Of, Nf, NO+, CH+, H30+, D30+, NHf, OH+, and H20+. The most direct competitor to the merged-beam technique is the trapped-ion method (e.g., Heppner et al., 1976),in which molecular ions can be held in a trap for many
+
I
--
I
I l l l r r
--
-
-
10-1'
-
---
---
-
-E,
-
w
-
6"
\
-
Y) Y)
F
u ,0-'5
-
---
10-4 0.01
I
I I I IIIII
I
0.1 Center-of-moss energy I eV I
I 11lIld 1.0
FJG.18. (0)with error bars illustrate merged-beam measurements of the recombination of H,O+. These are compared with trapped-ion measurements by Heppner ef al. (1976), represented by (0)and (A).
228
K. Dolder and B. Peart
hours until they are vibrationally de-excited. An energy-resolved electron beam is then injected into the trap and the measured rate of decay of ion density enables the recombination to be deduced. This technique enjoys the great advantage that the initial ionic excitation is defined, although (compared with beam techniques) it is harder to obtain absolute results. Figure 18 compares measurements for H 3 0 +by trapped-ion (Heppner et al., 1976) and merged-beam techniques (Mu1 et al., 1983) and the results are in fair agreement, although recombination rate coefficients derived from the results differ from those obtained by plasma techniques. Perhaps this is due to the different initial excitation of the ions. A final word: When looking at merged-beam measurements of recombination, remember Dunn’s arguments (Section 11) and be cautious about any measurement below 0.1 eV and downright sceptical about anything below 0.0 1 eV. Also ask whether observed “structure” is real and reproducible.
IX. Collisions between Positive and Negative Ions We will deal relatively briefly with ion -ion collisions because these have recently been extensively reviewed by Dolder and Peart (1985). The most important interaction between positive and negative ions is mutual neutralization since this controls the properties of the upper atmosphere and other cool ionized gases by destroying their free charges. The simplest example is H+
+ H-
H
+H
(47) which was studied with merged beams by Moseley et al. (1970) and with inclined beams by Rundel et al. ( 1969), Gaily and Harrison (1970), and Peart et al. (1976a). The results of all these experiments agreed very closely and, in particular, they revealed large, well-defined structures in the crosssection curve which were not reproduced by any theory. As theory became more sophisticated this disagreement seemed more remarkable, and it was particularly worrying because, until one could account for the uniquely simple example of mutual neutralization, there seemed little prospect of understanding the more complicated processes that occur in nature. The situation was discussed by Sidiset al. ( 1981,1983) in a paper entitled “Developments in the H+- H- Problem” and the mystery deepened when Sziics et al. (198 1) obtained new experimental results that were very different from previous measurements and were in much better accord with theory. A detailed account of these events was given by Dolder and Peart (1 985). Very briefly, they described an exhaustive series of new experiments (Peart et al., +
229
ELECTRON- ION AND ION - ION COLLISIONS
1985)which failed to reproduce the earlier results (which had been obtained by three independent, experienced groups) but their new measurements were consistently and unshakeably in quite good accord with the latest theory and with the measurements of Szucs et al. Recent experimental and theoretical results are illustrated by Fig. 19, where the experimental points show cross sections measured by Peart et al. with H+ and H- beams which intersected either at 84' or 15 the energy of the H+ ions is also noted in the inset. The continuous curve shows the results obtained by Szucs et al. with merged beams, while the broken curves (S and B), respectively, illustrate recent theoretical results by Sidis et al. and Borondo et al. (198 1). In spite of a most detailed series of tests, no explanation has been found for the large discrepancies between recent experimental results and those obtained before 1977, but the anomaly seems peculiar to H+-H-. For example, new measurements (Peart et al., 1985) for He+ H- -,He H (48) O;
+
+
agree will with much earlier results by Olson et al. (1970), Gaily and Harrison ( 1970b), and Peart et al. ( 1976~). Another relatively simple example of one-electron transfer between a positive and negative ion is
+
He2+ H-
01
20
+
He+
+H
1
50
loo
(49)
1
200
500
lo00
2000
Interaction energy lev)
FIG.19. The experimental pointsillustratemeasurements(Peart et a/., 1985)of the mutual neutralization of H+ and H- ions obtained with the angles of beam intersection and proton energies shown by the inset. These are compared with measurements by Szucs et a/. (Sz) and theoretical predictions by Sidis er al. (S) and Borondo ef a/. (B).
K. Dolder and B. Peart
230
0'
I
1
I
I
2 RELATIVE
3 SPEED
I
4 I l0'm
I
I
5
6
I
1
8
1
1
10
s-'I
FIG.20. Recent measurements for mutual neutralization of H+-H- (0),He+-H- (A), H,+-H-(X), and one-electrontransfer between 3He2+(0)plotted on a common velocity scale.
for which resultswere presented by Dolder and Peart ( 1985).Figure 20 shows our results for H+-H- (circles), He+-H- (triangles),3He2+-H+(squares), and Hf -H- (crosses)plotted on a common velocity scale and it can be seen that at the higher velocities results for the singly charged positive ions are very similar. An interesting experiment which points to one direction in which experiments might develop were measurements for Na+
+ 0-
4
Na
+0
(50)
by Weiner et al. (1971), in which an attempt was made to measure the intensities of spectral line radiation emitted by the sodium atoms to determine the electronic states in which the atoms were formed. State selection of collision products is an important goal for experimentalists and very little work has yet been done. Unfortunately, it seems that Weiner et al. did not master the very difficult task of accurate radiometric calibration and their partial cross section for the formation of Na(3 2 D ) was some three times larger than cross sections measured by Moseley et al. (1972) for the production of Na in all states. But pioneering experiments often encounter the toughest problems!
ELECTRON- ION AND ION - ION COLLISIONS
23 1
Most examples of mutual neutralization that are important in nature involve molecular ions and the situation can be extremely complicated because cross sections may depend quite simply on the initial vibrational and electronic excitation. Moreover, one can rarely be sure that the initial excitation of ions formed in the laboratorywill be the same as those found in nature. The difficulty is further compounded by the tendency of atmospheric ions to form “clusters” such as NO+ - H 2 0or ( H30)+.Some relevant comments on these problems can be found in the excellent review of mutual neutralization by Moseley et al. ( 1 975). Attempts have been made to estimate the influence ofclustering on reaction rates (a)for mutual neutralization and, according to Hickman (1979), the effect might not be too large for small clusters because the additional relative mass of the motion will tend to decrease the rate while it tends to be counterbalanced by the additional internal excitation. Hickman also developed Olson’s ( 1 972) absorbingsphere model for recombination to obtain,
a = (2.28 X low5)(T/300)-0.5p-0.5x-0.4cm3 s-’ (51) wherep is the reduced mass(a.u.)oftheionpairandx(eV)isthedetachment energy of the negative ion. Measured values of a-l (at T = 300 K) plotted against p0.5,y0.4for a wide variety of ion pairs tend to be linear and in rough accord with Eq. (5 1). Before leaving the discussion of mutual neutralization we must draw attention to a disturbing experimental result. Recently measurements were made in this laboratory (Peart et al., 1986) for the mutual neutralization of HZ and H- ions. A merged beam experiment on Hf-D- had previously been performed by Aberth et al. ( 197 1) and there is also a very brief account of a merged-beam experiment on HZ-H- by Szucs et al. (1983). We found that our apparent cross sections increased with HZ beam energy and this was attributed to our failure to collect all the neutral collision products,even with beam energies of 20 keV, Now, the experiments of Aberth et al. and Szucs et al. both used beam energies of about 3 keV and their results agree quite well with ours at a similar energy. We therefore believe that the earlier results are too low. This raises a serious doubt. Nearly all our experimental data about the recombination of molecular ions (other than those obtained by plasma techniques) come from the work of the Stanford (SRI) research group, and their data were all obtained with beam energies much less than 20 keV. We are thereforetempted to speculate that at least some of their other results are also too low and that this area needs reinvestigation. These remarks are of course tentative and they are not intended to devalue the fine pioneering work at the SRI.
232
K. Dolder and B. Peart
Two other reactions that have been studied are, H + + H-+ H+
+H +e
(52)
and H++ H - 4 *H2+Hf+ e (53) Proton-impact detachment from H- was measured by Peart et al. (1976d) and at higher energies the measurements converged to results of a first Born calculation by Bell et al. ( 1978),which employed 20- and 33-parameterwave functions for the ground state of H-. At these energies the results also coincided with electron-impact detachment cross sections, scaled to the same projectile speed. The association of H+ and H- (reaction 53)was studied by Poulaert et al. (1978) with merged beams, and the results imply that measurements were possible at interaction energies as low as eV. In view of Dunn’s arguments outlined in Section 11, this seems improbable.
X. Collisions between Positive Ions A. INTRODUCTION
This subject was recently reviewed by the authors (Dolder and Peart, 1985) and so we shall do no more than summarize the more interesting topics and mention very recent results. One application occurs in fusion plasmas where charge transfer between D+ and impurity ions may neutralize the deuterons and simultaneously enhance the charge of the impurity. This leads to escape of fuel and an increase in energy lost by bremsstrahlung. Other applicationsarise in “heavy ion fusion” (HIF), where intense, pulsed ion beams might be used to bombard a pellet and cause fusion and, in astrophysics, Baliunas and Butler (1980) showed that charge transfer between Si+-H+ and Si2+-He+ significantly influences the distribution of silicon ions in coronas. Clearly, these ion -ion reactions will be influential in a variety of hot plasmas. B. EXPERIMENTAL RESULTS Consider first the one-electron reactions H + + He++
H++ He2++ e
(54)
ELECTRON-ION AND ION-ION COLLISIONS
233
and H + + He++ H
+ He2+
(55) which are usually called ionization and charge transfer, respectively. The cross sections will be represented by a,and a,. Mitchell et al. (1 977), Peart et al. (1977b), and Angel et al. (1978a) measured cross sections (a) for the formation of He2+by H+-He+ collisions and so, to good approximation, a = oi a,. Coincidence techniques were subsequently developed by Angel et al. (1978b) and Peart et al. (1983a)to measure cross sections (a,) specifically for charge transfer and this one-electron process has, in the words of Fritsch and Lin ( 1982) “evolved into a testing ground for different models, since calculated cross sections are rather model sensitive.” Certainly several very sophisticated calculations have been performed to determine a, and some of the results are illustrated in Fig. 2 1. This shows results of molecular state calculationsby Winter e?al. (1980) and Kimura and Thorsen (198I), a Sturmian calculation by Winter (1982), and a pseudostate expansion by Fritsch and Lin (1 982). A more elaborate Sturmian calculation has just been
+
2j 50
21
1
I
I
10
20
30
Interaction Energy I k e V 1
FIG. 21. (0)and (0)by Peart et al. (1977b, 1983a) represent measurements of charge transferbetweem H+ and He+.These are compared with theoretical predictionsof Winter et al. (1980, O), Kimura and Thorsen (198 I , X), Winter (1982, W), and Fritxh and Lin (1982, A).
234
K. Dolder and B. Peart
completed by Winter (personal communication) which confirms the earlier results. Measurements of a by Peart et al. (1977b, 1983a)are also indicated. At these energiesionization is negligible and so a = a,. Excellent agreement can be seen between experiment and the various theories even though the results span the “intermediate energies” where the relative speed of the ions and bound electrons are comparable and where theory is most difficult. The only disappointing feature of H+-He+ to collisions was a relatively small disparity between measurementsof by Angel et al. and Peart et al. (see Dolder and Peart, 1985). However, it transpires that this arose from an arithmetic error by Angel et al. and when this is corrected the agreement is excellent; the revised results will be published by Dunn et al. (1986). Further confirmationof the experimentalresults has recently been obtained by Rinn et al. ( 1985), who have obtained values for a and a, in the range 8 -90 keV. The excellent accord between a wide variety of experimental and theoretical results for these one-electron collisions provides a firm base from which theory can advance. One might, for example, experiment with H+/Li2+,for which Winter (personal communication) has recently performed coupled state Sturmian calculations. Measurements of “resonant” or “near-resonant” collisions are particularly interesting. Here the commuting electron does not gain or lose significant energy so that the “energy defect” (AE) is small. The reaction He+
+ He2+
+
+
He2+ He+
(56)
is an example of resonance, but since both collision products are positively charged they repel and so it is difficult for experimentaliststo collect them all. The experiments by Jognaux el al. (1978) and Peart and Dolder ( 1979)both encountered this problem and neither is entirely satisfactory.New measurements are needed. Two examples of accidental near resonances are, H+
+ Ti+
+
H
+ Ti2+
H
+ Mg2+
(57)
and H+
+ Mg+
-+
where the energy defects are, respectively, 0.02 and 1.4 eV. In these cases at intermediate energies one would expect ionization to be negligible (a a,) and that charge transfer would proceed almost exclusively through unexcited states. If so, cross sections would be the same for the reverse processes
+
Ti2+ H + Ti+
+ H+
(59)
and Mg2+
+H
-+
Mg+
+ H+
(60)
235
ELECTRON-ION AND ION-ION COLLISIONS
t
1
1
0
2
-
1 5
10
6
20
6
Interaction Energy ( k e V I
FIG. 22. Measurements of near-resonant charge transfer between H+ and Ti+ (0)and between H and Ti 2+ (0).
Measurements for the reactions in Eqs. (58) and (60) by Peart et al. (1977a) and McCullough ( 1979),respectively, are in excellent agreement,Similarly, Fig. 22 shows that results for Eqs. (57) and (59) by M. F. A. Harrison et al. (private communication) and McCullough et al. agree within experimental error. Rather than repeat details of our earlier review (Dolder and Peart, 1985), we conclude by mentioning some new measurementsby Watts et al. ( 1986) and Dunn et al. (1985) for
H++X++XZ++
*
(61)
* *
N
. y L - - I c
VI
I-
!!
6
1
I '
10-1~ I
100
I
1
1
I
200
300
LOO
500
Interaction Energy I k e V
1
a0
I
FIG. 23. Measurements of cross sections for the formation of A12+, Ga2+, and TI2+by collisions between protons and Al+,a+, and TI+ are indicated, respectively, by (0),(A), and (0).Ionization cross sections for equivelocity electrons are, respectively, illustrated by (---), (- * -), and (--).
K. Dolder and B. Peart
236
where X = Al, Ga, In, or T1, and experiments by Hopkins et al. (1986) for charge transfer between H+/C+ and H+/N+. The results for Al+,Ga+, and Tl+are illustrated in Fig. 23, which includes experimental electron-impact-ionization cross sections for the three ions scaled to the same relative speed. We notice that for ion energiesgreater than about 200 keV the ion-ion and electron-ion results do not differ very greatly and this suggests (as one would expect) that ionization dominates at these energies. At lower energies the ion-ion curves rise above those for electron-impact ionization, indicating the presence of ion -ion charge transfer. Very recently these studies have been extended (K. F. Dunn, private communication) to include measurements of charge transfer (~JJso that ionization cross sections can be deduced. These are shown to be roughly compatiblewith a modified version of a simple scalinglaw suggestedby Neill et al. (1983), which has the form
where R is the Rydberg constant, 2, the projectile charge, and ui represents the ionization energy of the n, electrons in the ith subshell. Figure 24 illustrates measurements of a,for several ions that have been scaled in this way. Dolder and Peart (1 985) noted that measurements by Neill et al. for C2+
H t t Het Ht+Lit
0
H t t Gat H + + Int H++TI+ H+t H Hez'+ H
A
d
'4
14 Ep / hU
16
A
2
4
6
e 10 ScPled energy
e
12
v
H++ 'C
o
H++ N+ H t + Alt
A
v
4
18
FIG.24. Scaled measurements of ionization cross sections for several ion pairs.
20
ELECTRON - ION AND ION- ION COLLISIONS
237
and N2+production by H+/C+ and H+/N+collisions tended to converge at lower energies with results for charge transfer between C2+/Hand N2+/H; but the comparisonswere not unambiguous, because the results ofNeil1et al. included unknown contributions from ionization. Recently, however, Hopkins et al. (1986) studied charge transfer between H+/C+ and H+/N+ at energies between 40 and 150 keV, and extrapolationsof their results to lower energies agree very satisfactorily with measurements(see Dolder and Peart, 1985, Figs. 27 and 28) for the reverse charge transfer processes. Two points are worth noting. First, the agreements are good in spite of the fact that the energy defects are 2 1 and 16 eV, respectively. Second, the cross sections deduced by Hopkins et al. for the ionization of N+ by H+ agree quite well with calculations by McGuire (1984), who showed that, between 100 and 400 keV, autoionization contributes about 10%of the ionization.
REFERENCES Aberth, W., Moseley, J. T., and Peterson J. R. ( I 97 1). Air Force CambridgeRes. Rept. AFCRL71-048 I . Angel, G. C., Dunn, D. F., Sewell, E. C., and Gilbody, H. B. ( 1978a).J.Phys. B 11, L49;(1978b). J . Phys. B 11, L297. Baliunas, S., and Butler, S. K. (1980).Asfrophys. J. Left 235, US. Belit, D. C., Dunn, G. H., Morgan, T. J., Mueller, D. W., and Timmer, C. (1983). Phys. Rev. Lett. 50, 339. Bell, K. L., Kingston, A. E., and Madden, P. J. (1978). J. Phys. B 11,3977. Borondo, F., Macias, A., and Riera, A. (198 I). Phys. Rev. Left.46,420. Bottcher, C., Griffin, D. C., and Pindzola, M. S. (1983). J. Phys. B 16, L65. Brouillard, F., and Claeys, W. (1983). In “Physics of Electron-ion and Ion-ion Collisions” (F. Brouillard and J. W. McGowan, eds.). Plenum, New York. Burgess, A., and Percival, 1. C. (1968). Ads. At. Mol. Phys. 4, 109. Burke, P. G., Kingston, A. E., and Thompson, A. (1983). J. Phys. B 16, L385. Chantrene, S., Defrance, P., Rachafi, S., Belic, D., and Brouillard, F. (1985). Proc. ICPEAC, 15th, Palo Alfo. Chutjian, A. (1984). High Temp. Sci.17, 135. Chutjian, A., and Newell, W. R. (1982). Phys. Rev. A 26,227 I. Crandall, D. H. (1981). ICPEAC, 12th. Amsferdam. Crandall, D. H. (1983). In Physics of Electron-ion and Ion-ion Collisions” (F. Brouillard and J. W. McGowan, eds.). Plenum, New York. Crandall, D. H., Taylor, P. O., and Dunn, G. H. (1974). Phys. Rev. A 10, 141. Crandall, D. H., Phaneuf, R. A., and Dunn, G. H. (1975). Phys. Rev. A 11, 1223. Crandall, D. H., Phaneuf, R. A., Hasselquist, B. E., and Gregory, D. C. (1979).J. Phys. B 12, L249. Dance, D. F., Harrison, M. F. A., and Smith, A. C. H . (1966). Proc. R. SOC.Ser. A 290,73. Daschenko, A. I., Zapesochnyi,I. P., Imre, A. I., and Bukstich, V. S. (1975).Sov.Phys. JETP40, 249.
238
K. Dolder and B. Peart
Defrance, P., Claeys, W., and Brouillard, F. (1982). J. Phys. B 15, 3509. Derkits, C., Bardsley, J. N., and Wadehra, J. M. (1979). J. Phys. B 12, L529. Dittner, P. F., Dab, S., Miller, P. D., Heath, C. D., and Bottcher, C. (1983). Phys. Rev. Lett. 51, 31. Dolder, K. (1969). I n “Case Studies in Atomic Collision Physics” (E. W. McDaniel and M. R. C. McDowell, eds.), Vol. I , p. 249, North Holland, Amsterdam. Dolder, K. (1983). In “Physics of Electron-ion and Ion-ion Collisions” (F. Brouillard and J. W. McGowan, eds.). Plenum, New York. Dolder, K., and Peat, B. (1973). J. Phys. B 6,2415. Dolder, K., and Peart, B. (1976a). Rep. Prog. Phys. 39,693. Dolder, K., and Peat, B. (1976b). Comments At. Mol. Phys. 5,97. Dolder, K., and Peart, B. (1985). Rep. Prog. Phys. 48, 1283. Donets, E. D. (1 983). Phys. Scripta T3, 1 I . Donets, E. D., and Ovsyannikov, V. P. (1977). Joint Inst. Nucl. Res. Reprint P7-10780 (Dubna). Donets, E. D., and Pikin, A. I. (1976). Sov. Phys. JETP43, 1057. Dubau, J., and Volante, S. (1980). Rep. Prog. Phys. 43, 199. Dunn, G. H. (1969). At. Phys. 1,417. Dunn, G. H. (1980). Invited lecture SPIG-80 Boris KidriC Inst. Nucl. Sci., Belgrade. Dunn, G. H. (1985). I n “Electron Impact Ionization” (T. D. Mark and G. H.Dunn, eds.). Springer-Verlag,New York. Dunn, G. H., BeliC, D. S.,Morgan, T. J., Mueller, D. W., and Timmer, C. ( 1984).In “Electronic and Atomic Collisions” (J. Eichler, I. V. Hertel, and N. Stolterfoht, eds.). Elsevier, New York. Dunn, K.F., Watts, M. F., Ange1,G. C., andGilbody, H. B. (1985a). Proc. ICPEAC, 14th, San Diego. Dunn, K. F., Watts, M. F., and Gilbody, H. B. (1986). J. Phys. B (in press). Esaulov, V. A. (1980). J. Phys. B 13, 1625. Falk, R. A., Dunn, G. H., Gregory, D. C., and Crandall, D. H. (1983). Phys. Rev. A 27,762. Fritsch, W., and Linn, C. D. (1982). J. Phys. B 15, 1255. Fujimoto, T., Kato, T., and Nakamura, Y. (1982). Nagoya Inst. Plasma Physics, Report IPPJAM-23. Gabriel, A. H., and Jordan, C. (1972). In “Case Studies in Atomic Collision Physics” (M. R. C. McDowell and E. W. McDaniel, eds.), Vol. 2, p. 221. North Holland, Amsterdam. Gaily, T. D., and Harrison, M. F. A. (1970a). J. Phys. B 3, L25. Gaily, T. D., and Harrison, M. F. A. (1970b). J. Phys. B 3, 1098. Geller, R., and Jacquot, B. (1983). Phys. Scripfa T3, 19. Golden, L. B., and Sampson, D. H. (1977). J. Phys. B 10,2229. Golden, L. B., and Sampson, D. H. (1980). J. Phys. B 13,2645. Gregory, D. C., Dunn, G. H., Phaneuf, R. A., and Crandall, D. H. ( 1979). Phys. Rev. A 20,4 10. Gregory, D. C., Dittner, P. F., and Crandall, D. H. (1983). Phys. Rev. A 27,2338. Griffin, D. C., Bottcher, C., Pindzola, M. S., Younger, S. M., Gregory, D. C., and Crandall, D. H. (1984). Phys. Rev. A 29, 1729. Hane, K., Goto, T., and Hattori, S. (1983). J. Phys. B 16,629. Hamson, M. F. A. (1968). In “Methods in ExperimentalPhysics” (W. L. Fite and B. Bederson, eds.), Vol. 7B. Academic Press, New York. Harrison, M. F. A. (1978). Inst. Phys. ConJ Ser. No. 38. Henry, R. J. W. (1979). J. Phys. B 12, L309. Henry, R. J. W. (1981). Phys. Rep. 68, 1 . Heppner, R. A., Walls, F. L., Armstrong, W. T., and Dunn, G. H. (1976). Phys. Rev. A 13,1000.
ELECTRON-ION AND ION-ION COLLISIONS
239
Hickmann, A. P. (1979). J. Chem. Phys. 70,4872. Hickmann, A. P. (1985). J. Phys. B 18, 3219. Itikawa, Y., and Kato, T. (1981). Rept. IPPJ-AM-17, Inst. Plasma Phys., Nagoya Univ. Jakubowicz, H., and Moores, D. J. (1981). J. Phys. B 14,3733. Jognaux, A., Brouillard, F.,and Szilcs, S. (1978). J. Phys. B 11, L669. Kel’man, V. A., and Imre, A. I. (1975). Opt. Spectrosc. 38,709. Kel’man, V. A., Daschenko, A. I., Zapesochnyi,andImre, A. I. (1975). Sov. Phys. DOH. 20.38. Kimura, M., and Thorson, W. R. (1981). Phys. Rev.A 24,3019. Kohl, J. L., and Lafyatis, G. P. (1983). Proc. ICPEAC, 13th Berlin p. 195. Kolosov, P. A., and Smirnov, Y. M. (1983). Sov. Astron. AJ26, 364. LaGattuta, K. J., and Hahn, Y. (1981). Proc. ICPEAC, 12th. LaGattuta, K. J., and Hahn, Y. (1982). J. Phys. B 15,2101. LaGattuta, K. J., and Hahn, Y. (1983). Phys. Rev. Lett. 51,558. Latypov, Z . Z., Kuprianov, S. E., and Tunistkii, N. N. (1964). Sov. Phys. JETP 11, L669. Lorenz, A. ( I 978). Phys. Rep. 37C, 56. Lotz, W. (1968). Z. Phys. 216, 241. Lotz, W. ( 1 969). Z. Phys. 220, 466. Lyon, 1. C., Peart, B., West, J. B., Kingston, A. E., and Dolder, K. (1984). J. Phys. B 17, L345. McCullough, R. W., Nutt, W. L., and Gilbody, H. B. ( I 979). J. Phys B 12,4 159. McFarlane, S. C. (1974). J. Phys. B 7, 1756. McGuire, E. J. (1977). Phys. Rev. A 16, 73. McGuire, E. J. ( 1 984). Phys. Rev. A 29,3429. Mertz, A. L., Mann, J. B., Robb, W. D., and Magee, N. H. (1980). Los Almos Rept. LA-82 67MS. Meyer, F. W. (1985). Nucl. Instrum. Methods B9, 532. Mitchell, J. B. A., and McGowan, J. W. (1983). In “Physics of Electron-ion and Ion-ion Collisions” (F. Brouillard and J. W. Maowan, eds.). Plenum, New York. Mitchell, J. B. A., Dunn, K. F.,Angel, G. C., Browning, R., andGilbody, H. B. (1977).J.Phys. B 10, 1897. Mitchell, J. B. A., Ng, C. T., Forand, J. L., Levac, D. P., Mitchell,R. E., Sen, A., Miko, D. B., and McGowan, J. W. ( 1 983). Phys. Rev. Lett. 50, 335. Moores, D. L. ( 1978).J. Phys. B 11,403. Moores, D. L., Golden, L. B., and Sampson, D. H. (1980). J. Phys. B 13,385. Moseley, J. T.,Aberth, W., and Peterson, J. R. (1970). Phys. Rev. Leti. 24,435. Moseley, J. T., Aberth, W., and Peterson, J. R. (1972). J. Geophys. Res. 77,255. Moseley, J. T., Olson, R. E., and Peterson, J. R. (1975). In “Case Studies in Atomic Collision Physics” (E. W. McDaniel and M. R. C. McDowell, eds.), Vol 5, p. 1. North Holland, Amsterdam. Msezane, A. Z., and Henry, R, J. W. (1983). BUN.Am. Phys. SOC.28,798. Mul, P. M., McGowan, J. W., Defrance. P., and Mitchell, J. B. A. (1983). J. Phys. B 16,3099. Muller, A., Salzborn, E., Frodl, R., Becker, R., Klein, H., and Winter, H. (1980).J. Phys. B 13, 1877. Muller, A., H u h , K., Tinschert, K., Becker, R., and Salzborn, E. (1985a). J. Phys. B 18,2993. Muller, A,, Tinschert, K., Achenback, C., Becker, R., and Salzborn, E. (1985b). J. Phys. B 18, 3011. Neill, P. A., Angel, G. C., Dunn, K. F.,and Gilbody, H. B. (1983). J. Phys. B 16, 2185. Olson, R. E. ( 1972). J. Chem. Phys. 56,2979. Olson, R. E., Peterson, J. R., and Moseley, J. T. (1970). J. Chem. Phys. 53, 3391. Pace, M. O., and Hooper, J. W. ( 1 973). Phys. Rev. A 7,2033. Peart, B., and Dolder, K. (1975). J. Phys. B 8, 56.
240
K. Dolder and B. Peart
Peart, B., and Dolder, K. ( 1979). J. Phys. B 12,4 155. Peart, B., and Dolder, K. (1974). J. Phys. B 7,236. Peart, B., Walton, D. S., and Dolder, K. (197 1). J. Phys. B 4,88. Peart, B., Stevenson, J. G., and Dolder, K. (1973). J. Phys. B 6, 146. Peart, B., Gray, R., and Dolder, K. (1 976a). J. Phys. B 9, L369. Peart, B., Gray, R., and Dolder, K. (1976~).J. Phys. E 9, L373. Peart, B., Gray, R., and Dolder, K. (1976d). J. Phys. B 9,3040. Peart, B., Gee, D. M., and Dolder, K. (1977a). J. Phys. B 10,2683. Peart, B., Grey, R., and Dolder, K. (1977b). J. Phys. B 10,2675. Peart, B., Forrest, R. A., and Dolder, K. (1979). J. Phys. B 16,2735. Peart, B., Rinn, K., and Dolder K. (1983). J. Phys. B 16, 1461. Peart, B., Bennett, M. A., and Dolder, K. (1985a). J. P h p . B 18, L439. Peart, B., Bennett, M. A., and Dolder, K. (1986). J. Phys. B (in press). Phaneuf, R. A., Crandall, D. H., and Dunn, G. H. (1975). Phys. Rev. A 11,528. Phaneuf, R. A., Taylor, P. O., and Dunn, G. H. (1976). Phys. Rev. A 14,2021. Post, D. E. (1983). In “Atomic Collision Processesin Thennonuclear Plasmas” (F. Brouillard and J. W. McGowan, eds.). Plenum, New York. Poulaert, G., Brouillard, F., and Claeys, W.(1978). J. Phys. B 11, L671. Pradhan, A. K., Norcross, D. W., and Hummer, D. G. (1981). Phys. Rev. A 23,619. Rinn, K., Melchert, F., and Salzborn (1985). J. Phys. B 18, 3783. Robb, W. D. (1980). In “Atomic and Molecular Processes in Controlled Thermonuclear Fusion” (M. R. C. McDowell and A. M. Ferendeci, eds.). Plenum, New York. Rogers, W. T., Olsen, J., and Dunn, G. H. (1978). Phys. Rev. A 18, 1353. Camilloni, R., Dunn, G. H., Msezane, A. Z., and Henry, R. J. W. Rogers, W. T., Stefani, G., (1982). Phys. Rev. A 25,737. Rogers, W. T., Dunn, G. H., Ostgaard Olsen, J., Reading, M., and Stefani,G. (1982). Phys. Rev. A 25, 681. Roszman, L. J. (1 98 1). Invited Review ICPEAC, 12th. Rundel, R. D., Aitken, K. L., and Harrison, M. F. A. (1969). J. Phys. B 2,954. Salzborn E. (1983). In “Physics of Electron-ion and Ion-ion Collisions’’ (F. Brouillard and J. W. McGowan, eds.). Plenum, New York. Schmeltekopf,A. L., Fehsenfeld, F. C., and Ferguson E. E. (1967). Astrophys. J. 148, L155. Seaton, M. J. (1975). Adv. At. Mol. Phys. Seaton, M. J., and Storey P. J. (1 977). In “Atomic Processesand Applications” (P. G. Burke and B. L. Moisewitsch, eds.). North Holland, Amsterdam. Shimon, I. I., Golovchak, N. V., Garga, I. I., and Goldovskii, V. I. (1983). Opt. Spectrosc. 55, 137. Sidis, V., Kubach, C., and Fussen, D. (198 1). Phys. Rev. Left.47,1280; (1983).Phys. Rev. A 27, 243 I . Sziics, S., Karema, M., Terao, M., and Brouillard F. (1984). J. Phys. B 17, 1613. Tawara, H., Kato, T., and Ohniski M. (1985). Report IPPJ-AM-37, Inst. Plasma Phys. Nagoya, Japan. Taylor, P. 0. (1972). Ph.D. thesis, University of Michigan. Taylor, P. O., and Dunn, G. H. (1973). Phys. Rev. A 8,2304. Taylor, P. O., Phaneuf, R. A., and Dunn, G. H. (1980). Phys. Rev.A 22,435. Taylor, H. S., and Thomas, L. D. (1972). Phys. Rev. Lett. 28, 1091. Thomas, L. D. (1974). J. Phys. B 7, L97. van Wyngaarden, W. L., and Henry, R. J. W. (1976). J. Phys. B 9, 146. Vogler, M. K., and Dunn, G. H. (1975). Phys. Rev. A 11, 183. von Busch, F., and Dunn, G. H. (1972). Phys. Rev. A 5, 1726.
ELECTRON- ION AND ION-ION COLLISIONS
24 1
Watts, M. F., Dunn, K. F., Angel, G. C., and Gilbody, H. B.(1986). J. Phys. B (in press). Weiner, J., Peatman, M. B., and Berry, R. S. (1971). Phys. Rev. A 4, 1824. Williams, J . F. (1983). Proc ICPEilC, 13th p. 209. Winter, T. G . (1982). Phys. Rev. A 25,697. Winter, T. G., Hatton, G. J., and Lane, N. F. (1980). Phys. Rev. A 22,930. Younger, S. M., and Wiese, W. (1979). J. Quantum Spectrosc. Radiat. Transfer 22, 161. Zapesochnyi, I. P., and Daschenko, A. I. (1983). Proc. ZCPEAC, 13th, Berlin. Zapesochnyi, I. P., Imre, A. I., Daschenko, A. I., Vokstich, V. S., Danch, F. F., and Kel'man, V. A. (1973). Sov. Phys. JETP36,1056. Zapesochnyi, I. P., Kel'man, V. A., Imre, A. I., and Danch, F. F. (1976). Sov. Phys. JETP42, 989.
Zapesochnyi, I. P., Imre, A. I., and Daschenko, A. I. (1983). Proc. ICPEAC, 13th. Berlin.
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ADVANCES IN ATOMIC A N D MOLECULAR PHYSICS. VOL. 22
ELECTRON CAPTURE BY SIMPLE IONS EDWARD POLLACK AND YUKAP HAHN Department of Physics University of Connecticut Stnrrs, Connecticut 06268
I. Introduction A significant amount of experimental and theoretical effort is currently being directed to studies of electron-capture processes in ion -atom and ion - molecule collisions. These processes are of intrinsic interest as manybody rearrangement collisions. They are also of interest because of many applications, for example, in neutral beam injection to tokamak devices for temperature enhancement, the production of fast neutral beams by charge exchange being a necessary step. Electron-capture collisions are important in plasmas where capture by singly as well as multiply charged ions generally occurs to excited levels, which subsequentlydecay by radiation emission resulting in energy losses by the system. The capture channels must be included in modeling astrophysical problems and also in addressing a large number of chemical processes since these channels strongly compete with the reactive scattering. They present challenging theoretical problems that test the adiabatic potentials and approximations. There have been a number of excellent reviews of the field; we cite those of Park (1983), Macek (1983), McCarrol(1982),Hasted (1979), Belkic et al. (1979), and Bayfield (1975). The present article will complementthe earlier reviews and is directed at topics which have not been fully covered in these earlier reviews. The discussion will be limited to a small number of ion - atom and ion - molecule systems that are selected to illustrate the dominant underlying processes. The He+ H, case is of particular interest and will be presented in some detail. It is simple, isoelectronic with He+ He, and yet sufficiently complex to provide a test of collision models. Electron capture is generally studied experimentally by: (1) the energy dependence of the total cross section, (2) the probability of electron capture
+
+
243 Copynght 0 1986 by Acadermc Press,Inc.
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rights of reproduction in any form reserved.
Edward Pollack and Yukap Hahn
244
(or charge exchange) as a function of scattering angle at fixed energy or as a function of energy at a fixed scattering angle, (3) the doubly differentialcross section for excitation of a particular outgoing channel, (4)the optical-emission cross section as a function of energy, ( 5 ) photon-scattered atom coincidence measurements, and (6) analysis of recoiling target ions or dissociation fragments. Experimental results from the above types of measurements are mutually complementary. The direct scattering channels (A+ B ---*A+ B/B*) are often strongly coupled to the exchange channels, and experimental results from both direct and exchange scattering are important. In this article we will emphasizecollisions at energies from about 100eV to the low-keV range, where a quasimolecular model is applicable, although processes at very low and very high energies relevant to our discussion will also be commented on. Particular attention will be given to studies in which the scattered neutral beam is detected; this type of experiment generally provides information on the modes of excitation (Section 11). Studies of optical spectra from collisionally excited projectiles or targets allow identification of the emitting states. However, the major shortcomings of such experiments are that: (1) no information on capture to ground-state channels is obtained, (2) the emission spectra include contributions from cascade processes which must be accounted for, (3) the emissions may follow excitation of the direct and exchange channels (asan example, Lyman-a is emitted both in He+ H2 --* He+ H H* and in He+ H, ---* Heo H* H+), and (4)emissions following simultaneous excitation of both collision partners can contribute to the spectra. On the other hand, an important advantage of optical experiments is that the energy distribution of the incident ion beam is not a critical factor, unlike the case in energy measurements on the scattered beams, where the energy spread in the ion source and the energy resolution of the detectors may result in uncertaintiesin the identification ofthe collision channels. More detailed studies includingthe polarization of the emission spectra can also provide valuable information on the underlying processes such as those resulting in sublevel distributions. The idealized experiments, employing coincidence techniques that allow the angular distributions of neutral particles in particular states to be studied, benefit from the advantages of both the angular distribution and emission studies but prove to be difficult to perform except in selected cases.
+
+
+ +
+
+
+ +
11. Theory Much of our understanding of ion -atom and ion- molecule processes comes from qualitative analyses of experimental data using various simple
ELECTRON CAPTURE BY SIMPLE IONS
245
transition models. In particular, quasimolecular models have successfully been used to analyze low- and medium-energy direct and exchange-collision processes. The collisional excitation results from electron transitions which are most probable at or near crossings of the potential energy curves for the molecular states involved. These crossings(or near crossings)result from the promotion of molecular orbitals of the transient molecule. The calculations involved in finding the states and the interactions between them are complex, but a simple model based on the molecular orbitals (not the states) of the quasimolecule provides some insight into the type of processes involved and lays the groundwork for more rigorous theoretical treatments. It will be shown later, with particular emphasis on the He+ H2 Heocase, that the molecular-orbital (MO) model gives a good qualitative account of many of the experimental findings to date. Two basic types of processes are introduced in the “MO collision model” and are termed diabatic-Iand diabatic-I1(Brenot et al., 1975b,c).The difference between them is that in the case of the diabatic-I process an initial MO crossing is required for a state crossing and in the diabatic-I1processes there are no MO crossings. Consider an initially filled MO (A), with its two electrons described by a state A2, which becomes promoted as the collision progresses. At a particular internuclear separation if this MO crosses an initially empty MO (B), one or both of the electrons in the orbital can make transitions which result in final molecular states corresponding toA2,B2and AB MO electrons in a single configurationpicture, as shown in Fig. 1. Ifthe u, g symmetries of the two orbitals are different, only A2 and B2 are allowed. As shown below the same MO crossings may further result in an infinite number of state crossings A2 BC, where C corresponds to an empty MO lying above B in energy and not necessarily crossing A. The continuum resulting in A 2 B e- is included. The diabatic-I process occurs as indicated schematically (at 0)in Fig. 1. A diabatic-I1process requires an initial vacancy in an MO (labeled B in Fig. 1) which is only weakly promoted when compared to filled higher-lying MOs correspondingto outer electrons in the quasi-molecule.A state crossing occurs at an internuclear separation where 2eA - (ee ec) = 0 (e is the MO energy) as indicated by the arrows on the MO diagram. A similar situation is possible in the diabatic-I case after a vacancy has been created at the first crossing. Under the assumption that the total electronic energy is the sum of the MO energies in both the diabatic-I and -11 cases an infinite series of quasi-diabatic state crossings is generated. In a more rigorous treatment that includes the Coulomb and exchange terms the energies and crossing positions are different but the basic process is the same. The diabatic-I1 case generally results in “core-excited” configurations which become highly repulsive with decreasing internuclear separation and correlate to autoioniz-
+
-
-+
-
+
246
Edward Pollack and Yukap Hahn diabaticJI
diabatic I
FIG.I. Schematic representation showing the diabatic-I and diabatic-I1 processes.
ing states in the united-atom limit. As shown in Fig. 1, the state @A2) must therefore cross infinite series of singly excited levels that are associated with lower core excitation. These crossings occur since the higher-lyingC orbitals generally are weakly bound when compared to B. The diabatic-I1process is especially important in rare-gascollisionssuch as in He+ He and results in important contributions from inelastic channels at low energy. Since the crossings for the diabatic-I1 processes occur at larger internuclearseparation than for the diabatic-I case, they result in inelastic processes at small scattering angles. The following theoretical discussion is limited to providing a coherent picture of the D-I and D-I1(D, diabatic)transition models as they are applied to a qualitativeinterpretation of the existingexperimental data on ion -atom and ion - molecule charge-exchangecollisions. No attempts will be made to present a complete theoretical treatment.
+
A. PRELIMINARY DISCUSSION
The theory of charge-exchange (CEX)collision has been the subject of vigorous development for the past 25 years, not only because of its intrinsic
247
ELECTRON CAPTURE BY SIMPLE IONS
interest as a typical three-body scattering problem with particle rearrangements and an associated nonorthogonality (Hahn, 1969, 1970, 197 1 ) property, but also for its varied applications. Thus, for example, the asymptotic energy dependence of the CEX cross section is by now well documented (Shakeshaft and Spruch, 1979) in terms of the classical Thomas (1927) picture, and the importance of the asymptotic boundary conditions in the perturbed stationary state (PSS)representation is fully recognized (Bates and McCarroll, 1958) in terms of the electron translation factor (ETF). Along with the detailed calculations, several simple models for CEX have been developed to provide a qualitative, and often semiquantitative, understanding of an otherwise extremely complicated collision process. We attempt to present in this section a coherent summary of these models, which will then be used to interpret some of the recent experimental data in ion-atom (IA) and ion - molecule (IM) collisions at intermediate energies. We limit our discussions to energy regions where the relative collision velocity uI is much smaller than the typical electron orbital velocity u,, but where E/a.m.u. >> E, = +meuf . For E, = 1 Ry in light ions, we have 10 E, = 140 eV 5 E/a.m.u.
5 (O.l)(MH/me)Ee==
3 keV
(1)
This is the energy range most difficult to treat theoretically; for higher-energy collisions, simple Coulomb - Born, Coulomb - secondBorn and, eikonal approximations, etc., are available, while at energies much lower than that given by Eq. ( I), many of the standard low-energy approximation methods, such as coupled-equations and variational methods, should work well. Nevertheless, we can often make the following simplifying assumptions in the region of Eq. (1):
(1) Ion trajectories are taken to be straight line in the evaluation of the amplitude. The deviation of the ion beam due to the collision can be taken into account through the momentum-transfer dependence Q = K i - Kj of the amplitude. (2) The Born-Oppenheimer separation of the scattering amplitude in terms of the adiabatic states is generally valid except in the region of nonadiabatic level crossing. (3) The region of internuclear coordinates where some levels are close is localized and only a few adiabatic states are involved. A two-level model is often locally valid, except in some special situations, to be discussed later. Various theoretical as well as experimental progress has been periodically reviewed by many authors, and there are several detailed summaries available (Nikitin, 1970; Nikitin and Smirnov, 1979; Janev, 1976; McCarroll, 1982; Crothers, 1981; Bransden and Janev, 1983; Sidis and Dowek, 1984). In particular, we mention the article ofJanev ( 1976), which covered the basic
Edward Pollack and Yukap Hahn
248
nonadiabatic coupling in IA collisions in a coherent fashion, and a more recent review by McCarroll ( 1982),with emphasis on high-energy collisions. The IM-CEX was reviewed by Sidis and Dowek (1984). The theoretical basis for different models may be presented in several different ways, depending critically on the choice of the basis functions which are used to expand the scatteringfunction. As will be shown, a unified picture can be developed by leaving the choice of the basis functions open to the end (Janev, 1976).In the following we present the formalism for CEX in the time-independent framework; much of the available literature is based on a time-dependent picture, which is valid only in the limit of a straightline-trajectory approximation. An explicit demonstration of their equivalence is given in the impact-parameter representation (McCarroll, 1982). B. ADIABATICAND DIABATIC REPRESENTATIONS
The IA-CEX reaction is schematically described by
+ ( e + b)
+
+ e) + b;
a e=A e + b = B; (2) where a and b may have internal structures of their own, with N, and Nb electrons, respectively. These “core” electrons are treated collectively, and produce distortion fields for the single active electron. Thus, Eq. (2) represents essentially a three-body problem. The Hamiltonian for Eq. (2) is given, in obvious notations, in many different partitions a
+
(a
H = T,+ h B + V,= H = TbA
+ hA + v~ =
H = Tab He H e = Te+ V,
H,+
V , = T,+ HP
(3)
HbA
+ v~ = TbA + Hg’
(4) (5)
V = V d + V,+
+
Vbe
+
+
where, for example, h, = Teb Veby h, = T, V,, and V, = Vd V,,, etc. Over the years, all possible combinations of the Ts and Vs for many different coordinate representations have been tried. For example,
hA and h,
+
simple diabatic sets,
{y;) and {d)
He + adiabatic set {4,,)
Hg and H:
(6)
-,distorted cluster states {4fl)and {&)
In particular, He generates (q&,) as (see Fig. 2 for the coordinates) He(r,RMn(r,R) = En(R)+n(r,R)
(7)
ELECTRON CAPTURE BY SIMPLE IONS
249
n
+ + +
FIG.2. Coordinates used for a (e b) charge-exchangecollisions. The locations of the centers of mass of A = (e a) and B * (e b) are exaggerated.
+
and the total scattering function is expanded as Y=
C unWMn(r,R) n
HY=EY where we have neglected the inner-shell electron excitations. As is well known, although the expansion in Eq. (8) is exact, the series is often truncated in practice, which immediately introduces complicationsin satisfying the asymptotic boundary conditions. This problem was resolved by Bates and McCarroll(l958, 1962) by introducing the ETF, which also eliminates the spurious long-range coupling. Of course, such problem can be easily avoided altogether by using the sets (y;) and ( y f ) ,which are suited especially for high-energy collisions, or more appropriately (Hahn, 1967; Bates, 1958)the sets (4f}and {$:) generated by Hg’ and H:. We then have, instead of Eq. (81,
u;4;
Y= n
+ c u#);
(10)
rn
Returning to the form of Eq. (8) and as we truncate the set, ETF Fn may be introduced as N
y
C un(R)Fn(r,R)$n(r,R)
( 1 1)
n-l
such that
The Fn are in general difficult if not impossible to construct except in the
Edward Pollack and Yukap Hahn
250
simple case of a straight-linetrajectory; even then the form is unique only in the asymptotic region F, = eik“. r& and eik!: * re (13) where k; = (me/Mab)K;, a = a, b Except for the original discussion on the need of F, in Eq. ( 1 1) by Bates and McCarroll(1958),the ETF problem has received perhaps more attention in the literature than it deserved, mainly in terms of the problem of switching functions. The question is totally an arbitrary one, in so far as the short-range behavior of F, is concerned; any “optimum” choice for a fixed number of terms in Eq. (1 1) would be equivalent to incorporating configuration mixing. We note that the simple choice Eq. ( 13) for all r, and raeis in a perfectly acceptable form everywhere, and are so used in the first Born, second Born, and eikonal approximationsfor high-energy CEX collisions, where the effect of F, is important. Of course, a slightly modified form of the adiabatic representation should be used; 4, + 4; = (1/&)[4, k 4,J & for the degenerate case, and 4, + yf or @ for R -,CQ directly for the nondegenerate case. Since many ofthe complicationsassociated with the expansion in Eq. ( 1 1) and F,are well known by now, we simply omit F, in the following qualitative discussion, without loss of generality. Thus from Eqs. (8) and (9), we have, with Tab= T R ,
-
[TR+ EAR)+ W A R ) - EIuAR) = - 2
wnm
um
(14)
mtn
where
w,,= w,;+ w;;+ w;:
(15)
and
Equation ( 14)is the scattering equation in the adiabatic representation. The basis set (4,) is diagonal in He, and W,, are presumably small, 11 Wnml1-c I ~ E , - E,J, except in some (pseudocrossing)limited regions in R. Evidently, transitions E , E,, are realized by either WyAor W;:. In practice, it is rare
-
ELECTRON CAPTURE BY SIMPLE IONS
25 1
that both these operatorsbecome important simultaneouslyat some point in R; W E are generally effective near the R of closest approach, where the rotation of R with respect to the fixed 2 axis is the largest, while W;%are more effective at large impact parameters. Alternative to the adiabatic representation, any representation in which the effect of TR is "minimized" is termed diabatic. Thus, for example, the sets ( y i ) and (YE} in Eq. (6) are independent of R and diabatic; the corresponding W's are all zero. On the other hand, they are not diagonal in He, and are not very useful in describing the short-range behavior of a slowly colliding system. In order to retain the distortions (i.e., the R dependence) and at the same time incorp$rate the large effect of one of the W's in Eq. ( 16), we may define a new set (4,Jin a finite subspace of (c$J as
such that, for any one of the Ws, C-' WC = Wd = diagonal
(18)
Obviously, He is no longer diagonal in this diabatic representation,
(&lHe1&), = En,,,f diagonal
(19)
but a great deal of distortionsare still retained. (Near the pseudocrossing, the n and rn states are nearly degenegte and Enmare nearly diagonal.) In this new representation, with Y = XUon4,, and W = Wd Wd where W d are the residual W's which are not included in Eq. (1 8), i.e., Wd = W d or Wra, we have
+ N
[ TR
+ Enn(R)+ W:$(R) - E]ii,(R) = - C ( W $ + Enm)iim
(20)
m#n
The two different physical pictures described by Eqs. (14) and (20) are important in describing the different modes of CEX, to be discussed in the following subsections. More generally, we define an unspecified set {&J, and write
{4,Jcan assume any of the above (&I,
(&}, (y:}, and (yfp},etc. Then, the
structure of the scattering equations becomes
[TR + Enn + Wnn - EIun = -
N
m#n
[En,
+ WnmIum
(22)
252
Edward Pollack and Yukap Hahn
where all the operators in Eq. (22) are defined now in terms of 4n
w,,= wy;+ W E
(24)
For later discussions, it is convenient to further separate Em into two parts En, = ECnm O“ + EcXnCmh, n+m (25) where
Of course, depending on the particular choice of the {&), some of these quantities could be zero or become very large in certain regions of R. The complications due to the ETF and nonorthogonality overlaps are omitted again for simplicity. The structure of Eq. (22) will in general depend on the choice of the basis set which presumably reflects best the particular physical situation. However, the final CEX amplitude obtained should be independent of this choice so long as enough terms are included in Eq. (2 1). The theoretical problem is then to choose the optimum basis set so that the number of termsNis small. Connection between the time-dependent and time-independent theory is sketched below. The CEX theory is usually given either in the time-dependent (TD) or time-independent (TI) formalism, in most cases without justification. We critically compare these two approaches and point out some of the advantages peculiar to each of them. The task is easier in the case of the medium-energy collisions of interest here, where straight-line trajectories may be assumed. As a general comment, TD and TI scattering formalisms are completely equivalent if the entire interacting system is included in the formalism so that the total energy is conserved. Naturally, in that case, the TI approach is simpler for theoretical calculation, while physical pictures can be made more realistic in the TD language. On the other hand, when some energy sink or source is included in the theory as a time-dependent external field, then obviously only TD approach is applicable. In the present case of CEX, the heavy particle motion is often taken to be “classical,” acting as a source of energy for the electronic part of the system. In the impact-parameter representation and in the straight-line trajectory
ELECTRON CAPTURE BY SIMPLE IONS
253
approximation, we may set
R= b
+ z = b + vt
(26) where vll.2 and b v = 0. For simplicity, we again neglect the complication due to the presence of the ETF, F,,.Then, for Eq. (22), we let
u,,(R) = o,(R)e"n and assume that T, W,, = 0. Here we used
(27)
*
-
ti 2 K 3 2 M d = E - En where En are the asymptotic values of En, at R
+
(28) m;
i.e.,
En,,= V,, En
(29)
Equation ( 2 2 ) becomes
-
2 [Em + W,,]~,(R)e~~m-~n).
m+n
At this stage, we may set v,, = v and t i 2 ~ , ,vR/M* = u,,a/az= a/at 9
exp[i(K, - Kn) * R] = I Obviously, V,,, W,,,, and Em are dependent on t through their R dependence. If one starts with a time-dependent equation
(H, - i a/at)+(r,t,b) = o
(31)
instead of Eq. (9),then, with 'k = Z,,&,,(t,bM,(r,t,b), we obtain (McCarroll, 1982) a set of coupled equations for &,, similar to Eq. (30),except for the En, term on the left-hand side instead of V,,. (This will result in the factor exp[iJr(E,, - E,)dt'] in the coupling term, which damps the large t contribution.) If the slat representation for v a/aZ is introduced at a much later stage, as was done above in Eq. (3 I), the (Em- En)contribution to o,, does not appear. That is, while the collision dynamics for CEX is entirely contained in V,, and Em Wmy the collision energies needed for CEX are supplied by the ionic cores to the electronic system, in the time-dependent picture, as described by Em - En,,.Here, conservation of energy for the electronic part alone is lost during the collision, mainly because of the approximation v, v.
+
-
Edward Pollack and Yukap Hahn
254
C. TRANSITION PROBABILITIES We return to Eq. (14) or Eq. (22) and examine the structure of the equations and their solutions, neglecting again for simplicity the complications which arise due to the presence of ETF in a more realistic theory. We first note that, for reactions in which Eq. ( 1 ) and the assumptions (1)-(3) are valid, E , - ,E for different n and m are usually very large compared with W,, for most R, except near the level crossings, say R = RJn,m). Therefore, forlR - R,I large, Eq. (2) are essentially uncoupled and {4,) is a "good" representation. On the other hand, for R = R, for a particular pair (n,m), W,, can have a very large effect on the solution u, ,in which case {#Jis not a very realistic set. It is then critical to determine which of the three operators in Eq. (22) is the most dominant, (En, and W,,) in the region of R,. As summarized in Table I, detailed studies of this problem for a number of cases over the years have shown that each of the four couplings ( W g ,Wz,,Eyl, and E g ) have different R dependences. From the localization assumption (3), a two-state model near a particular R, should be a good first approximation to a realistic physical picture. (Many level crossings are possible, especially when ionic channels are present near the ground-state channel.) Thus, near R = R, and K K , we let
+ W,)w,( R) ( V, -I- W, - iv d/dZ)w,(R) = -(Eba+ Wba)o,(R) ( V, -I- W, - iv d/dZ)o,( R) = -(Ed
(32)
where again the basis set 4, and &,used in the evaluation of E,,, V,, W,, TABLE I DIFFERENT TRANSITION MODELS FOR ION-ATOM CEX COLLISIONS Basis
Hamiltonian
4"
He
6*
He
~9
W:
cpi, 4:
Coupling
Range (4)
W%
1 -2
WZ!
2-4
W$, Emrr nm
3 -6
E;zh
4-8
h,and h~
EY;h or E Z
4-8
H i and H :
Both W,, and En,
1-8
+ Wr* or Wmt
1-4
Remarks
Adiabatic basis, AA = k 1 D-IL D-IR, M= 0 D-I, diabatic basis D-IIC, correlation (two-electron) D-IIX, exchange (one-electron) Diabatic asymptotic method Distorted cluster states adiabatic basis
ELECTRON CAPTURE BY SIMPLE IONS
255
w a b , etc. is left arbitrary. The effect of all the other channels neglected in Eq. (32) can be incorporated (Hahn, 1978) in Eq. (32) in terms of additional effective interactions. The diagonal parts ofthe interaction in Eq. (32) can be simply disposed of by defining the new functions (a= a and b)
C, = omexp( -
f
/
Z
(V,
+ W,) dZ’)
(33)
which gives for Eq. (32)
a
iv - c b
az
=
v, c,
(34)
where
v& = (A!?& + W&)eXp(-- (vb+ Wb - V, - W,) dZ’ = V%W+ V Z h + v:y+ v3
;jz
(35)
The boundary conditions are C,(Z = - C Q ) = 1 and cb(z= -a) = 0. Here we neglected the contribution from W s . For the following discussion, we simply set
The set of coupled equations (34) with (36) is still difficult to solve even numerically. It is more instructive if some analytic solutions are available. Different expressions for the transition probabilities P& are possible for different approximations on YOb and qab: (1) Landau -Zener- Stuckelberg model. It is assumed that, in the vicinity of R Rx i=:
y = yo = constant
(37) qo = constant q = qoZ, Then it is possible to evaluate the solution of Eq. (34) in terms of Weber functions (Zener, 1932), and obtain pd = ICb(Z= a)12
e-*@
(38)
where
P =YiYvIo),
Ox
= UR(R = Rx)
(39)
256
Edward Pollack and Yukap Hahn
This form was critically examined by Bates (1960) and by CoulsonZalewski (1962), and later by Bates et al. (1964). It was found that the LZS form generally underestimates the CEX cross section by as much as a factor of 2, as compared with the more exact solution of Eq. (34). On the other hand, we know that the multichannel effect, which was neglected in Eq. (34), will reduce the cross section in the more exact treatment, so that the apparent effectivnessof the LZS formula in many cases may be fortuitous, and may be due to cancellation of errors. (2) Rozen-Zener-Demkov model. An early study of Eq. (34) by Rosen and Zener (1932) gave an analytic solution in terms of hypergeometric function for a special choice of U, y = yo sec(aZ/uz);
q = qo = constant (40) Demkov ( 1964)examined the same problem assumingan exponential decay of y = yo exp(- aZ/vz). The result is
(-1 -1
Pd = sech2
avo dR
-l
rm
) sin2(
y dZ)
dzRx
Forms similar to Eq. (41) were also studied by several groups, Gurnee and Magee (1957), Vainstein et al. (1 962), and Ellison and Borowitz (1964). (3) In case when the interaction at large R is important and U z h is dominant, we have from Eq. (41)
(1:
Pd = sin2
U z h dZ)
with the sech2term being a constant. The resonant and quasiresonantcharge exchange is often treated quite successfullyusing Eq. (42), or its variations, in some limited energy region. For a more quantitative study, however, the t P x should be calculated by solving a set of coupled equations of the type of Eq. (34), with additional channels which are strongly coupled to channels a and b, plus some additional pseudochannels if necessary. A more effective and practical method to treat CEX collisions at medium energies is lacking, and, as the scattering systems become more complicated, the simple transition models such as Eqs. (38), (4 l), and (42) are particularly useful in providing a qualitative and sometimes even semiquantitative understanding of the process. D. ION- MOLECULE COLLISIONS Mainly because of their importance in many applications, ion - molecule collisions are receiving increasing attention in recent years, both at thermal
ELECTRON CAPTURE BY SIMPLE IONS
257
and at medium energies. Some high energy processes are also being studied, often as substitutes for electron-ion collisions, such as in the resonanttransfer excitation (RTE), to be described in Section VII. From a purely scattering theory point of view, even the simplest of all ion - molecule systems, H+ H,, is too complicated to analyze rigorously and to provide detailed scattering information. On the other hand, it is a fertile ground for developing and testing simple models to explain complex reaction phenomena both in IA and IM systems. Eventually, detailed calculations should be carried out to determine the applicability of these models; in fact, several attempts have already been reported (Sidis, 1985; Kimura 1985; Pascale, 1985). The presently available data on the IM-CEX are predominantly experimental, and their theoretical understanding is mainly based on the various transition models developed previously for I A collisions, such as those summarized in Table I. For a CEX reaction
+
A+
+ (B+ C )
+
A
+ ( B + C)+
B+C=M,
(B+C)+=M+
(43)
in the energy region of Eq. (l), the assumptions (1)-(3) of Section II,A are still valid, and the adiabatic molecular states (4,} for the complex (A B C)+are needed. Here, beside the A - M coordinate R,we have an additional internuclear coordinate s (Fig. 3) which is to be fixed in 4,. With
+
+
H=He+TR+T,=He+T,, He= H$
+H f c +
HFt
(44)
"Y FIG.3. Coordinates used for ion- molecule collisions,A -I-BC. The rcoordinateis reserved for electrons as in the ion-atom system.
Edward Pollack and Yukap Hahn
258
we have
He(r$,s)&(r;RYs)= M 0 M k ; R y s ) (45) For each fixed set of variables (O,,& - +,), E,(R,s)give the adiabatic potential energy surfaces in the variables R and s. Their intersections are the nonadiabatic (pseudo)crossinglines, analogous to the crossing points R, in the case of IA collision. Therefore, the interplay between the manifold of adiabatic hypersurfaces provides the essential collision dynamics. The total scattering function is expanded approximately in a finite set as
where we have again neglected for simplicity the ETF associated with the variable R. Obviously, unthemselves are complicated functions to evaluate. For the reaction in Eq. (43), it is convenient to further expand u, as
where (xP)(s)} are the nuclear vibration - rotational wave functions for the system (B C), and are generated by
+
+
[ T8 En(s,R= 03) - E k(n)]Xk(n)(s) = 0
(48)
Substitution of Eqs. (46) and (47) into the Schrodinger equation (HE)" = 0 gives a set of coupled equations for the expansion coefficients yd(R), analogous to Eq. (1 4) but now dependent on the indices n and k. Instead of proceeding further with Eqs. (46) and (47), we make the following simplification on u,; in place of Eq. (47), let
which is equivalent to a closure approximation. Other forms are also possible, such as ~ y ' ( s ) y , , &"(s)y& and P)(s,R)y,(R),etc., but Eq. (49) is the simplest. For the CEX collisions in the Eq. (1) energy range, p ) ( s )for the initial state may be taken as the ground-state nuclear configuration (B C)o.The approximation of Eq. (49) may be justifiable in light of the experimental finding that the average vib- rot excitation energies are independent of the electronic excitation (Section IV). With Eq. (49), the R dependence in Y will be contained in 4, and y , .For a more general basis set (&(r;R,s)),as with Eq. (23), we then have
+
+
[TR+ Enn + w n n - EIyn = - 2 [En, + w n m I ~ m m2n
(50)
ELECTRON CAPTURE BY SIMPLE IONS
259
The Woperators corresponding to dpsand Ly, are omitted here on physical ground, but may be added if necessary. As in Eq. (25), we further separate Enm as Em = EEm Ecmh nm (53)
+
In the straightline trajectory and two-state approximations,Eq. (50)reduces to a form similar to Eqs. (32) and (34); with
( f lZ( + W,) dZ'
y,(R) = elk,' exp --
V,
(54)
where a = a and b channels, we have
iv
a
(55) c b = ubaca
and the boundary conditions C,(Z=-m)=
1,
C,(Z=-m)=O
(56)
As in Eq. (35)
u*=u~fl+up+u$+u~
(57) corresponding to different transitions, as summarized in Table I. Thus, the discussion given for IA-CEX in Section II,C can be taken over for the IMCEX. However, we recall that here we made one additional (perhaps drastic) approximation in Eq. (49), in order to reduce the scattering equations to the form of Eq. (50). Very recently, some theoretical calculations of the adiabatic potential energy surface r,(s,R) have been reported for the He+ H2system (Kubach et af.,1985) and also the H+ H, system (Kimura, 1989, using the model potential approach (Valiron et af.,1979) and diatoms-in-molecule (DIM) procedure, respectively. A simpler and potentially powerful procedure based
+
+
Edward Pollack and Yukap Hahn
260
on a pseudopotential construction is also being explored (Rossi and Pascale, 1985). Table I1 summarizesthese methods, all ofwhich attempt to bypass the complications due to the presence of many electrons embedded in manynuclear-core configurations. Wave functions for the active valence electrons are calculated such that the known energy levels for the subsystems are reproduced. While some differences are noted in the resulting energy surfaces, it is still quite remarkable that seemingly different, and somewhat ad hoc, approaches give results which are consistent with each other. On the other hand, sensitivity of the CEX cross sections on the details of the potential energy surfaces and on the wave function as they affect the operators U,, is not yet very well understood. Much more theoretical and experimental efforts are needed both on IM and IA systems to obtain a semiquantitative understanding.
TABLE I1 APPROXIMATEPROCEDURES FOR THE EVALUATION OF ADIABATIC ENERGY SURFACES
Method Model potential (MP)
Pseudopotentials
(PP)
Diatom in molecule (DIM)
References
Characteristic features Screened Coulomb potentials for A, B, C, separately; short-range attractive potentials added to produce virtual orbitals; correct number of nodes produced for valence electrons; fit low-energy phase shifts Only valence electrons involved; orthogonality constraints suppressed by adding short-range repulsive potentials (for low I); fewer orbitals, fewer basis sets Wave functions constructed from individual atomic orbitals; Hamiltonian matrix diagonalized separately for atomic and diatomic fragments; parameters introduced for these fragments; effective at large R,but less accurate at small R
Hanssen ef al. ( 1979); Masnou-Seeuws (1982); Sidis (1985); Peach (1982)
Baylis (1969); Pascale (1985); Rossi and Pascale (1986); Peach (1982)
Ellison ( 1963); Tutly and Preston (197 1); Kimura (1985)
ELECTRON CAPTURE BY SIMPLE IONS
26 1
111. Experimental Background A. A TYPICAL EXPERIMENTAL ARRANGEMENT The experimental techniques used for studying charge-exchange collisions via direct measurements on the scattered neutrals are well known. Except for some specificdetails (such as the relative detection efficiencies of neutrals to ions and energy references) we only outline the procedures. The experimentsrequire the usual beam production, scattering, and detection of the scattered neutral atoms. They must be done with checks on the singlecollision conditions,and the analysis of the results requires an understanding ofthe geometry of the scatteringregion (the geometry problem may in fact be largely responsible for disagreements in cross sections reported by different laboratories). Since these points are well understood, they are not discussed here [a review of the basic techniques is given by Kessel et al. (1978)l. The experimental arrangement used by Heckman et al. (1984) to study H+ H, ---* Ho is typical and will be outlined. The apparatus, shown in Fig. 4, was also used in a study of Heo D2and is described by Jakacky et al. (1985). H+ is generated in an ion source (a), extracted, and focused by an einzel lens (b).The H+beam passes through a set ofshim fields and through a collimating hole into a beam-choppingregion consistingof two plates (about one cm long and separated by 0.5 cm). One of the plates is maintained at a fixed bias while the other has a voltage pulse (= 18 V, 0.1 psec wide at a frequency of 300 kHz) impressed upon it. After chopping the H+ beam passes through additional shim fields into a Wien filter (j)for velocity analy-
+
+
FIG.4. A schematic of the experimental arrangement. Ion source (a), extractor and einzel lens system (b, c, d), deflector plates (e-i), Wien filter (j),chargeexchange cell (k), deflector plates (I), scattering cell (m), valve (n), electrostatic energy analyzer (o), cryopump (p), and time-of-flight (TOF) detector (9).The distance from the scattering cell to the TOF detector is 4.2 m.
Edward Pollack and Yukap Hahn
262
sis, and through a charge-exchangecell (k) where ion beams may be partially neutralized to generate a beam of fast neutrals for scattering experiments (such as for Heo D2as an example). When experiments with fast neutral beams are performed the charge-exchange cell is filled with gas and the emergingion and neutral beams pass through sweep plates (1) where remaining ions can be deflected out of the beam. In the H+ H2+ Hoexperiments the charge-exchangecell is under vacuum and the sweepplates are grounded. The H+ beam enters the scattering cell containing H2target gas and scatters through an angle 8 into the detector chamber. The scattered beam enters (0) an electrostatic energy analyzer used for energy analysis of scattered ions. The neutral scattered beam passes through the analyzer into a drift tube to a detector (9).The detectorwill be described below. The pulse generated by the detected particle is amplified,passed through a constant fraction discriminator, and serves as a start pulse for a time-to-amplitudeconverter. The stop pulse to the converter is provided by the chopping voltage pulse which is suitably delayed. The data are recorded by a multichannel analyzer. For cross-section measurements an automated system is used to control a stepping motor for changing angles. The incident beam is collimated by two circular apertures. The first of these also serves as the exit aperture on the charge-exchange cell, and the second as the entrance to the scattering cell (m). The scattered beam passes through a third and fourth collimating hole positioned on both sides of (n) before amving at the entrance slit (effective width 2 mils) of the electrostatic energy analyzer. The Ho passing straight through the analyzer is finally detected by a metal surface (0.5 in. in diameter) on the TOF detector (9).In the H+ H2+ Hostudiesthe angular resolution is primarily determinedby the third collimating hole and the diameter of the detecting surface. As is generally the case (because of unavoidable misalignments), the measured angular resolution is better than that given by the geometricaldefinition. The FWHM of the detected beam is typically less than 0.1 '.
+
+
+
B. DETECTION EFFICIENCY The starting point in many studies involves a determination of Po(8),the probability of charge exchange (or electron capture) at scattering angle 8. Po is defined as N(O)/[N(8) Z(8)], where N and Z represent the number of neutrals and ions arriving at the detector (or detectors) in a given time interval. The relative detection efficiency must be known in comparing the scattered ion to the scattered neutral signals for determiningPo. In resonant charge exchange such as in He+ He the intensity of the scattered neutral He beam exhibits a marked oscillatory structure when plotted as a function
+
+
263
ELECTRON CAPTURE BY SIMPLE IONS
of angle. A similar structure differing in phase is obtained for the scattered He+ signal. At small angles where the contribution from inelastic channels is small, the combined neutral and ion signals should decrease monotonically with increasing scattering angle. If the detection efficiencies of the neutral and ion beams are different an oscillatory structure will be seen in T, the scattered total (ion plus neutral) signal. At angles where the neutral signal predominates the lower relative detection efficiency for He would result in a signal which falls below the expected smooth curve. This can be seen in the I .O keV data of Nagy et d.( 197 1) shown in Fig. 5 where minima in Tare seen close to scattering angles where the N peak has maxima. Analysis of the data showed that for the detector used the detection efficiency at 1.O keV for Heo is about 0.85 that of He+. A similar plot of Tat 3.0 keV does not display the 10
10'
w
!az
100
I-
2 3
E w
;
10
J W
a
I .o
0.I
I
0.5
I
I
1.0
1
I 1.5
I
I
2
.o
I
I
b I
25
SCATTERING ANGLE ( Degrees 1
+
FIG.5. The He+ He collision at an incident energy of 1.O keV. The curve labeled N shows the relative count rate ofthe detected Heoversus angular position ofthe detector, while curve T represents the relative count rate of all the scattered He particles regardlessofcharge state. An interesting feature of curve T is the presence of a subtle oscillatory structure. This structure is due to the different detection efficiences of the detector for He+ and Heo.
Edward Pollack and Yukap Hahn
264
oscillatory structure, showing that at the higher energy the detection efficiency for Heo and He+ is essentially the same. It is clear that care must be taken in establishing the relative importance of the direct to exchange processes (especially at the lower energies) in these experiments. C. A TIME-OF-FLIGHT DETECTOR
At beam energies of the order of several hundred to several thousand eV, TOF (time-of-flight)techniques are generally used to measure the energy of the scattered neutral particles. These techniquesare well understood (Brenot etal., 1975; Hodgeetal., 1977; Kesseletal., 1978)andaredescribedindetail by Raith ( 1976). If both ions and neutrals enter a TOF,detectorthe resulting spectra from the two charge states may overlap and the data cannot be cleanly interpreted. A detector used by Hodge et al. (1977) which permits simultaneous acquisition of the neutral and ion signals is shown in Fig. 6. C = 0.1 pF
R -100K
CHANNELTRON
SHIELD
SCATTERED BEAM
~
-
SURFACE
-FIG.6. A time-of-flight detector. The scattered neutrals and ions (shielded from stray charges by a drift tube) enter the detector, strike the circular metal detecting surface ejecting are adjusted to maximize secondary electrons. Voltages on the surface and housing (“BOX”) the electron signal reaching the channeltron cone. Ions entering the detector are accelerated by the internal fields, arrive at the surface earlier in time than the neutrals, thereby separating the spectra corresponding to the two charge states.
ELECTRON CAPTURE BY SIMPLE IONS
265
The scattered beam strikes a flat metallic detecting surface which is maintained at a voltage of about - 500 V while the surrounding housing is biased at a slightly greater negative voltage. The secondary electrons ejected from the surface are detected by a channeltron (a channelplatecan also be used but the detection efficiency may be lower). The scattered positive ions are accelerated when they enter the detector and arrive at the surface earlier in time than the neutrals. This allows a complete separation of the spectra from the direct and exchange channels permitting simultaneous measurements on them. In these measurements the cone of the channeltron is grounded and the ejected electron signal is maximized by adjusting the bias voltage on the housing. The detector is described in detail by Goldberger (1984).
D. ENERGYREFERENCES In electron-capturecollisionsthe projectile changes charge state and there is no reference channel in the spectrum from which absolute energy losses can be determined. In some cases a reference can be obtained from known results in another collision system. In the H + + H 2 + H 0 studies the H+ Ar collision provided the necessary reference since it was known that at small 0 the dominant process is H+ Ar H( 1s) Ar+(2P).Figure 7
+
+
-+
+
c
+
+
FIG. 7. Spectra showing electron capture in 1.0 keV H+ H, collisions. (a) The H+ Ar H(1s) Ar+(2P)(Q = 2 eV) reference peak is used to identify the final channels as H( 1s) HZ(ZZ:) (Q = 2 eV). (b) A typical spectrum.Separation between the maxima of the two peaks is 10 eV.
+
+
Edward Pollack and Yukap Hahn
266
+
+
shows energy spectra of the Ho at 1.0 keV from H+ H2 and H+ Ar collisions (0 = 0.2").Using the peak from the Ar target as a reference (Benoit et al., 1977)and the known energy calibration ofthe detector, the main peak in H+ H2 is attributed to the H( 1s) HZ(?ZZ) channel.
+
+
IV. Typical Studies The next section on ion - molecule collisions will show that electron capture by He+ from diatomic molecules is very similar to the capture processes resulting from collisions of He+ with rare-gas targets. The small-angle direct inelastic and exchange collisions, in both the rare-gas and diatomic molecular target cases result from curve and surface crossings, respectively, which are attributed primarily to the vacancy in the He+ MO. This vacancy in the incident channel is most important at small scattering angles and generates processes involving the transition of two electrons (diabatic-11). For harder collisions promotion of a He MO gives rise to both one- and two-electron excitations (diabatid), primarily generating outgoing p states in the direct and exchange channels. The He+ He and He+ Ar cases are discussed to illustrate the basic ion -atom exchange collisions. The early experimental results on Ar+ N2by Fernandez et al. ( 1975)are also reviewed here. This work is particularly significant and shows that at a given scattering angle the average vibro-rotational excitation is the same in collisions involving the electronic ground and electronicallyexcited molecular states. In addition the work confirms the applicability of the FranckCondon principle to collisions in the medium-energy range. These findings are very important since they allow the outgoing electronic states to be identified.
+
+
+
A. He+
+
+ He
The He+ He system has been studied extensively but only a limited summary is presented here. The most distinguishing characteristic of this collision is the marked oscillatory structure found in the cross sections for both the direct elastic and resonant exchange channels (Everhart, 1963; Lockwood et a/., 1963; Lorents and Aberth, 1965; Marchi and Smith, 1965; Olson and Mueller, 1967; Nagy et al., 1971; Eriksen et al., 1971; Brenot et al., 1975a). The cross sections for these are shown in Fig. 8 (Brenot et al., 1975a) and indicate a well-defined phase difference between the direct and exchange channels in accordance with Eq. (35), where a and b correspond to
ELECTRON CAPTURE BY SIMPLE IONS
267
Ground stotc processes
5 100: + .E
5
-P
<
-
'
:
.
Y
.0, 10: I1 P
-
. .
b n = 3 exeitotion
2
4 T
6
=EO ( krV drg 1
FIG.8. The reduced differentialcross sections for direct ( -) and exchange (- - -) scattering in 500 eV He+ He.
+
the 2Zgand 2Zuchannels. This phase difference in the cross sections is well understood and due to interference resulting from collisions involving the 2Zgand 2Zustates in the He$ quasi-molecule. Excitation of the direct inelastic and exchange channels are qualitatively well described by the He; correlation diagram which is shown in Fig. 9 (Brenot et al., 1975a). As the collision progresses, the A 2Z~1s0g)(2p~u)2 incident channel crosses the B 'ZP
800 eV the energy-loss spectra were found to have maxima for He+ He He(2p) He+, which are consistent with earlier results on the direct target
+
+
+
+
-
268
.*
*..-*
.. .
..-*.... .-*
.."--'(1s n = 2
2; ID 1s2p
*
/.L ,.
Is nf
lr2p
ls2s 2
Is
He United atom
lntrrnuclcor distance
Separated atoms
FIG. 9. A schematic correlation djagram for He:. Rotational couplings (0)and radial couplings (0) are shown for selected P (-) and lT (---) states.
excitation (Barat eta].,1972). Thep excitation was attributed to the 2pu --* 2pn rotational coupling at small internuclear separation. Interference between theXZZ, CZn,andA '2, + C211gexcitationmodes for the n = 2 states (the resolution was not always adequate to resolve 2s and 2p) is seen in Fig. 8. For small T the direct and exchange cross section oscillate in phase as expected from the Landau - Zener - Stuecketberg interference associated with the A zZg-,B ' 2, crossings. For the large t value collisions, however, C 'H, modes are effective and, as a both the X 'Xu + C 'nuand A 'Z, result, the direct and exchange cross sections are out of phase. Additional features to note are: ( 1) the cross section for n = 3 excitationsshown in Fig. 8 is relatively weak since the 2pu 2pn rotational coupling leads to n = 2 channels and ( 2 )the cross sections for two-electron excitation processes were found to be smooth and as expected showed similar behavior in the direct and exchange channels because both were attributed to excitation by the same D 'Ag outgoing channel.
-
-
-
269
ELECTRON CAPTURE BY SIMPLE IONS
B. He+ + Ar
+
Large contributions from inelastic processes in He+ Ar were clearly expected from the observed rapid decrease with increasingscatteringangle in the cross section for elastic scattering (Aberth and Lorents, 1966).However, the strong loss from the elastic channels could not be accounted for from the magnitude of the measured cross sections for the direct inelastic scattering (Baudon et al., 1970), indicating that the capture channels are strongly populated. These channels were studied by Smith et al. ( 1 970) for energies in the range from 65 to 300 eV. The cross sections obtained show several families of oscillations. The direct and exchange channels were investigated in detail by Sidis et al. ( 1 977) for 100 < E < 2000 eV using TOF techniques for identifying the exchange channels. The dominant exchange processes in this system result in production of Ar+*excited states with only weak contributions from the ground-statechannel (which is exothermic). Energy spectra, at small angles, show the presence of He( ls2) Ar+*(3s3p6),S (Q= 4.65 eV) and two important peaks resulting from He( Is2) Ar+*(3p4ID 4s),D (Q = 9.6 eV) and He*(l~zs)~SAr+ (3ps) (Q = 11 eV) with additional features corresponding to channels at higher-energy losses (which were difficult to identify because the possible levels are too close in energy). Capture to the ground state is weak and found at higher energies. At small 7 the He( Is2) Ar+*(3p4 ID 4s) and the He( Is2) Ar+*(3s3p6)channels dominate and p [the reduced cross section = 8, a(@ at small angles] shows an oscillatory structure with a first peak at 7 values of 420 and 500 eV deg, respectively. The oscillations in the cross sections for these two channels have the same period but they are out of phase. The dominant exchange processes exhibit peaks with maxima for 7 = 2.0 keV deg. The He( ls2) Ar+(3s3p6)channel was found to decrease strongly for 7 > 3 keV deg and a population-sharingmechanism was proposed (Sidis et al., 1977)for its excitation. It is also of interest that the weakness of the He+ Ar + He( ls2) Ar+(3ps) ground-state exchange channel confirms that the diabatic MO correlation rules for outer-shell processes must be modified (Brenot et al., 197%) in asymmetric systems.
+
+
+
+
+
+
+
C. Ar+
+
+
+ N,
In a study of Ar+ N, at low keV energies, Fernandez et al. (1 975) found that the small-angle direct scattering was characterized by spectra showing two peaks and a shoulder (labeled A, B, and C, respectively). Peak A corresponds to an electronically elastic collision, peak B was associated with a Q
Edward Pollack and Yukap Hahn
210
(the excitation energy)ofabout 8 eV at threshold and was attributed primarily to Ar+ N2 + Ar+ N2(B311E),and C (with a Q = 16 eV) was attributed to Ar+ Nt(,X;). Experiments using either TOF (for the scattered neutral component) or electrostatic energy analysis (generally used on ions because of superior energy resolution) techniques provide only information on A E, the change in projectile kinetic energy, not on the physically significant excitation energy (Q). Q must be determined from the measured A E and the computed A E,, , the energy loss for elastic scattering. When an incident projectile of energy E and mass mp is elastically scattered through a small angle 8 by a target of mass m,,conservation of energy and momentum in the lab system require a projectile kinetic energy loss, A E,, = (mJm,)E02(which is equal to the recoil energy of the target). For small-angle inelastic scattering the energy loss is given by AE = AE,, Q (Bray et al., 1977). In collisions involving atomic targets the electronic states excited are identified by Q, since it is the electronic excitation energy. In collisions with diatomic molecular targets even the electronically elastic (quasielastic)scattering results in an energy-loss distribution reflecting vibro - rotational excitation of the target molecule(Andersenet al., 1980;Jakackyetal., 1985;SnyderandRussek, 1982; Sigmund, 1978). The energy-lossdistribution depends on the range of impact parameters, molecular orientations,and internuclearseparation that contributes to the scattering at 8. In the limiting case of a small-angleelectronically elastic binary Ar+ N, collision, the energy loss A Ebinof the scattered projectile is given by A Ebin= (mP/m)EB2, where m is the mass of an N atom in the molecule. This follows from the definition of a binary collision and does not necessarily reflect an actual physical situation. In the case of an elastic collision A E,, = (mp/2m)EB2.When A E is plotted versus EB2,the binary and elastic limits yield straight lines of slopes ( m p / m and ) (mP/2m),respectively. For an inelastic collision with a homonuclear diatomic target A E may be written as
+ +
+
+
+
A E = (mP/2m)EO2 + &(EB2)
+ Q,,
where &(E02) is the average vibro-rotational excitation energy, and Q, is the electronic excitation energy. The vibro - rotational excitation energy in the ground electronic state (Q, = 0) is seen to be the difference between the measured A E and the computed (mP/2m)EB2at each EB2 value. Figure 10 shows plots of A E vs E02 for peaks A, B, and C at beam energies of 1.O, 2.0, and 3.0 keV. Since the curves are essentially straight and parallel to each other the experimental results show that (1) Q, is roughly linear in EB2 and ( 2 )for a given EB2the are the same for the different electronicstates. This suggeststhat on the average the vibro-rotational excitation occurs independently of the electronic excitation. The measured energies in Ar+ N,
ow
+
ELECTRON CAPTURE BY SIMPLE IONS
I
2
3
4
5
6 7 8 91011 EBe ( k e V - d e g 2 )
27 1
121314
FIG. 10. The most probable energy losses of Ar+ for peaks A, B, and C versus EB2.
showed that the electronic excitation of the target molecule occurs in accordance with the predictions of the Franck-Condon principle. Moore and Doering ( 1969)also reported agreement with the principle for projectile-ion velocities greater than lo8 cm/sec for the relative band intensities of the Av = - 1 sequence of the first negative system ofN:(B2C;f X2C:) excited by collisions with several atomic and molecular ion projectiles. For lower velocities some disagreement with the Franck -Condon principle was found (Moore and Doering, 1969), contrary to what is expected from the BornOppenheimer picture. The work of Fernandez et al. also included a study of the electron capture in Ar+ N, and found Po (the probability of electron capture) to increase monotonically with increasing scattering angle and then maintain a rela-
-
+
Edward Pollack and Yukap Hahn
212
00
2 kaV
I -
Ar*
+
1
I
1
I
2 .o
1.0
N2
4.0
3.0
SCATTERING ANGLE ( d r q I
2 krV N'+ .I
Np
0 I
0.2
I
I
I
I
I
I
0.4 0.6 0.8 1.0 1.2 1.4 SCATTERING ANGLE ( d r q I
FIG. 1 1 . The probability of charge exchange in (a) 2.0 keV Ar+ N, collisions.
N+
+
+ N, and (b) 2.0 keV
tively constant value close to 0.5 as shown in Fig. 1 1. It is noteworthy that over most of the angular region investigated charge-exchange processes occur in approximately one-halfthe collisions.Also shown in the Fig. 1 1 is Po for N+ N, .In this system charge exchangeis again seen to occur in approximately one-half the collisions in the angular range studied. The Po behavior in both collision systems exhibits a similar angular dependence which is in fact characteristic of many ion - molecule systems. Charge exchange is important in Ar+ N2 since at small scattering angles it is a quasiresonant process. This was directly confirmed by Hodge et al. (1977) using TOF techniquesfor energy analysis. A typical spectrum from these studies showed that
+
+
Ar+
+ N,
-P
ArO
+ Nt(X 'ZZ),
Q =0
213
ELECTRON CAPTURE BY SIMPLE IONS
is the dominant small-angle exchange process. The studies of Hodge ez al. (1977) on a number of collision systems suggested that since the direct and exchange channels are nearly degenerate at asymptotic distances, Demkovtype couplings must be involved in the charge exchange. [See Eqs. (41) and (421.1
V. Ion - Molecule Charge-Exchange Collisions: He+ H,
+
It is well known that electron capture is an important (and in some cases the dominant) process in collisions between atomic ions and molecules, but until recently only little theoretical work was done on these more complex systems. This is due in large part to the presence of many electrons around several centers and many available channels in the case of molecular targets; the recoiling molecular ions can be electronically and vibro - rotationally excited or dissociated,and the electron may be captured to an excited state of the atom. Experimental studies of electron capture in ion-molecule collisions in many cases, however, suggest processes that have an unexpected simplicity. This is illustrated, for example, by the early work of Lockwood and Everhart ( 1962)on the H+ H and H+ H, systems.Figure 12is a plot of the probability ofelectroncapture at 0 = 3 as a function ofenergy. Except
+
+ O
"-1
0.78
1.11
1.57
20.1 hcu
3.9'2
I
7.69 krv
2.39 1
I
I
I Ill
5
I0
PROTON ENERGY
20
L
1
50
keV
+
FIG.12. The probability of electron capture asa function ofenergy for 8 = 3 H+ H, and H+ H collisions.
+
Edward Pollack and Yukap Hahn
274
for damping, the H, target exhibits the same behavior as the resonant electron capture from H targets. This striking result (discussed in more detail below) strongly suggests tractable underlying processes. Recent work on a number of systems has shown that the ion - molecule problem is beginning to be successfully addressed. The discussion of electron capture in ion- molecule collisions will focus on the He+ H, system since it is in many ways typical of the other diatomic target cases which were studied in detail by direct measurements on the neutralized He projectile. The He+ N,, O,, NO, and CO collisions are reviewed in the next section. These are followed by discussionsof Ar+ CO and H+ H,. Electron-capture processes in He+ H2 have been and continue to be studied over a wide energy range. In the energy range from several hundred to several thousand eV the neutral scattered He can be directly detected, its angular distribution measured, and sufficient energy resolution (allowing reasonable state identification) is obtainable by TOF techniques. Experimental studies of the scattered Heo, which are differential in energy and angle, allow for rigorous tests of collision models. The discussion of the He+ H,, D, case includes a short review of selected early work on the total cross sections for electron capture and cross sections for selected processes such as for H+ production and for capture to specific He* states. The early work was very successful in leading the way to our present understanding. For completeness a limited discussion of the collision at low and thermal energies is also presented. Many of the conclusions obtained from the low-energy work are useful in interpretingthe results at the higher energies. Stedeford and Hasted (1 954) measured the total cross section for electron capture in He+ H,. The dominant capture process was tentatively attributed to He( Is2) Ht(2pc7")final states, and the total cross section results are shown in Fig. 13. In a later experiment the Lyman-a radiation from these collisionswas studied by Dunn et al. ( 1962)in an energy range from 100 eV to 2 keV. Lyman-a emission can result from either the direct or capture collisions (with additional cascade contributions from collisionally excited states). However, the shape and magnitude of the cross section was found to be similar to that reported by Stedford and Hasted for charge exchange (into all states) and it was concluded that He+ H, + He H+ H* is the dominant source of the Lyman-a emission. The magnitude of the cross section showed that about one-half the collisions in the energy range studied yield H(2 2P) products and also suggested that the dominant capture processes excite dissociating channels of Hfleading to H+ H*. Van Zyl et al. (1 967) extended the study of the Lyman-a emission from He+ H,, D, up to energies of 25 keV. Young et al. (1968) also reported on
+
+
+
+
+
+
+ +
+
+ +
+
+
275
ELECTRON CAPTURE BY SIMPLE IONS
Lyman-a production and found basic agreement with the two previous results above 1 keV but at lower energies the cross sections were reported to be larger. In addition these later studies found a very small polarization of the Lyman-a. The combined results of the three measurements cited above show a cross-section maximum near 300 eV with a second broader maximum at higher energies, as shown in Fig. 13. Isler and Nathan (1972) investigated the Lyman$, Balmer-cY and -p, and He(3 3D-2 3P, 3 '0-2 'P, and 4 3D-2 3P)emissions at energies below 700 eV. The total cross sections for these processes are presented in Fig. 13 [which also shows the results of Gusev et al. (1969) on Balmer-a and -/?I. Polarization measurements on the Balmer-a and -/? showed that only the a
01
1.0
10.0
100.0
ENERGY ( k e V )
+
FIG. 13. The total cross sections for selected processes in He+ H,: He, total charge transfer (Stedefordand Hasted, 1954);H+, H+ production (Browningef al., 1969);L,, LymanLY emission (Dunn et al., 1962; Van Zyl et al., 1967; Young ef al., 1968);H, and Hp H emission (Isler and Nathan, 1972; Gusev er al., 1969); and He*, He* emission from 3 'D 2 'P (Isler and Nathan, 1972).
-
Edward Pollack and Yukap Hahn
276
line has any significant polarization. The polarization was found to be strongly energy dependent with a maximum value of about 12% at 50 eV. Following Van Brunt and Zare ( 1968)this was interpreted as resulting from unequal populations of magnetic sublevels of the upper state of the spectral line and a nonisotropicdistribution of the molecular dissociation axes. Visible spectra from 200 and 700 eV He+ Hz collisions show that optical emissions from H dominate at both energies, and at 700 eV collisionsresulting in capture to He* are seen. Capture to excited states of He therefore require a harder collision. The studies of Browning el al. (1969) on H+ production at energies between 5 and 45 keV again confirm that charge exchange primarily results in dissociative processes for energies less than about 6 keV. The cross section for H+ production is also shown in Fig. 13. Electron capture in He+ H, was studied as part of several broader investigationsof the He+-H, interactions in the range from thermal energies to about 60 eV (c.m.) (Joneset al., 1980;Wu and Hopper, 1981). The related theoretical work (Hopper, 1978, 1980a,b), using a multiconfiguration self-consistent-field method, included electronic-structure calculations on several states of (HeH,)+. The emphasis of one aspect of the experimental work (Jones et al., 1980) was to obtain energy thresholds for selected final channels. Of particular significance is the finding that the threshold for H+ and D+production in He+ H, ,D,is at 7.1 0.2 eV (c.m.). Thresholds for production of H(2p), H(3s, 3p, 3 4 , and H(4s, 4p, 4 4 were reported to be 9.0 0.5, 12.7 1.0, and 14.4 1.5 eV, respectively. Excitation of these channels involves sevqral potential surfaces of (HeH,)+ and the threshold values show that the energies at which the surface crossings occur are relatively low. An important conclusion of the studies is that
+
+
+
+
+
+
He+
+ H2(X%,:
+
+
+
u = 0) -,He( Isz) H( 1s) H+
collision must occur in the C,symmetry “just off of Czvsymmetry.”The 7.1 eV threshold is the energy at which the incident channel reaches the C,, surface intersection. Excitation of this channel was attributed to a two-electron transition and is in marked contrast to the mechanism effective in electron capture at thermal energies. Absolute total cross sections for Hf production were reported by Wu and Hopper (1981) in an energy range from about zero to 60 eV (c.m.). The cross section showed an interesting behavior “dropping precipitously almost to zero” as the energy increased to 1.8 eV. A threshold was seen at 2 eV and the cross section increased with energy to 4.5 eV and was then flat out to 13 eV. Capture processes resulting in target Rydberg states such as Hf*(2s08, 2p08, 2paU, etc.) involve a two-electron transition while the He*( lsnl) Hi(X ’St) final channels were attributed to one-electron transitions at the relevant avoided crossings. It\addition, Hopper (1 98Oa)proposed a radiative
+
277
ELECTRON CAPTURE BY SIMPLE IONS
mechanism for the production of Hfat thermal energies. Electron capture to He( ls2) via dissociative charge transfer is found at the very lowest energies. Johnsen et al. ( 1980) investigated He+ H,, D, He +H+, D+ H, D over the range from 78 to 330 K using a drift tube mass spectrometer. Although this channel is associated with a relatively large exothermic energy it is indeed seen. From measurements of the amval-time spectra it was concluded that at least 80%of the capture collisions result in H+ production with outgoing Hf and/or HeH+ products contributing the balance. The channel has a small rate coefficient and shows a strong isotope effect with the two-body charge transfer from D, a factor of 10 slower than from H, . This finding is in good agreement with the theory ofPreston et al. (1978)although the absolute values of the rate coefficients predicted by theory are smaller by a factor of 4 to 5 when compared to experiment. At fractional eV energiesthe vibrational state of the H, plays a very significant role and the cross section increases by almost two orders of magnitude between collisions involving the initial u = 0 and 1 levels (Preston et al., 1978).
-
+
+
A. STUDIES BASED ON COLLISION SPECTROSCOPY TECHNIQUES
The work of Bray et al. (1973, 1975, 1977; Bray, 1975) on the direct and exchange scattering in Het H2 extended collision spectroscopy techniques (see, for example, Kessel et al., 1977; Smith, 1969) from the ionatom to the ion-molecule case. It was found that the techniques can successfully be used even in these more complex collisions. The collisions were studied in some detail for energies in the range of 1 to 3 keV. The “total” (independent of charge state) differential cross section was measured at five energies in this range for scattering angles out to about 7” and the results showed only a rapid falloff with angle and no marked structure. The probability of electron capture was determined for energies between 0.5 and 3.0 keV. The Po measurements were made simultaneously on the scattered ions and neutrals which were detected by two different channeltron multipliers (the detection efficiencies were verified to be the same for both detectors). Figure 14 shows some of the results of these measurements in Po vs T plots. The charge-exchange processes are seen to become dominant with increasing T. Although it was not possible to identify the specific processes from these early studiesthe general structure ofthe curves suggests that at least two different processes are involved in agreement with the total cross sections shown in Fig. 13. The energy-loss spectra in the direct He+ H, 4He+ collision showed the presence of four dominant peaks labeled A, B, C, and D. The lowest-lying peak (A) in the direct He+ -tH2collision corresponds to a quasielastic scat-
+
+
Edward Pollack and Yukup Hahn
27 8 10-
05-
.d” 0 10
20
30
EB ( keV d e g )
FIG.14. The probability of charge exchange as a function of reduced scattering angle in He+ H,.
+
tering process. Figure 15 shows A E, at the position of the maximum of peak A, plotted as a function ofEB2at beam energies of 1 .O, 2.0, and 3.0 keV. The data for EB2 < 8 keV degZare well fitted by a common curve corresponding to elastic scattering from a target of molecular mass (scattering from an H target would result in energy losses that lie on the upper AE,,, curve). The data show that the collision is basically elastic at small EB2.For EB2 > 8 keV deg2 the measured energy losses “break away” from the elastic curve and indicate the onset of vibro - rotational excitation of the ground electronic state of the H, target (as mentioned earlier, the vibro-rotational excitation energy is the difference between the measured A E and the value obtained from the A Em,, curve at the same EB2 position). The other peaks labeled B, C, and D were associated with Q values of 13.3, 17, arid 3 1 eV at threshold, respectively. The energy-loss curves ( A E vs EOz) for these channels were found to be basically parallel to the one for peak A showing also the same breakaway point. This finding is of particular significanceand confirms that the average vibro - rotational excitation is the same in the several participating electronicstates(as was also found in the Ar+ N, case discussed earlier) at these energies. The measured 13.3 eV energy loss for peak B would allow excitation of several electronically excited H, states into vibrational levels which are in agreement with the Franck-Condon principle. Peak C was attributed to excitation of the HZX22: state into a vibrational level, again in agreement with a Franck - Condon transition. Peak D represents excitation of the repulsive 2puu state of Hf. Experimental cross-section results are conveniently presented in p [the reduced differential cross section =02u(0) at small O] vs t plots. In the ion-atom and atom-atom cases (Kessel et al., 1977; Brenot et al., 1975;
+
ELECTRON CAPTURE BY SIMPLE IONS
279
.-
- 1.0 kaV 2.0 keV
D- 3.0 k e V
0 2
6
10
Eo
14
18
e2 [ keV dog2)
FIG. 15. Measured kinetic energy loss at the position of the maximum of peak A. The curves labeled AEare the kinematically required kinetic energy losses for elastic scatteringfrom H(m = I ) and H,(m = 2). The results show the collision to be basically elastic (since they lie along the m = 2 curve) at small EP.
Smith, 1969) results presented in this form are generally used to infer features of the collision dynamics such as the type ofprocess involved (radial or rotational coupling), or the interparticle separation at which a particular inelastic channel opens. Thep vs 7 presentation was also found to be useful in the He+ H, collision. Peak A (the quasielastic channel) showed a monotonic decrease in p with increasing z. The reduced cross section for the direct channel corresponding to peak B is shown in Fig. 16. The experimental results (un-normalized) for this peak are seen to lie on a common curve, at the energies studied, displaying a well-defined maximum for z sz 3.0 keV deg. A similar behavior in the p vs z curves is seen for inelastic channels excited via a “curve crossing” in the quasimolecule for an ion-atom collision (Kessel et al., 1977). Common p vs z curves were also found for processes C and D. The reduced cross section for the exchange channel is shown in Fig. 17. The maximum in the cross section for electron capture occurs for
+
280
Edward Pollack and Yukap Hahn
*-1.0 keV A - 2 . 0 keV m - 3.0 k e V
2
0
4 6 EB (keVdeg)
8
FIG.16. The reduced differentialcrosssection as a function of reduced scattering angle for peak B. The experimental points for the different energies are unnorrnalized.
He*t H2
P,
0
2
4
20kW
vs E,@
6
8
EoB ( k e V d o g )
FIG. 17. The reduced differential cross section as a function ofreduced scatteringangle for small-angle exchange scattering.
28 1
ELECTRON CAPTURE BY SIMPLE IONS
7 = 3 keV deg which is at the same location as found for peak B in the direct inelastic channel. Time-of-flight techniques were used by Hodge et al. ( 1977)to identify the dominant process in small-angle He+ H,, Ar+ N,, Ar+ H,, and He+ N, charge-exchange collisions. In the absence of reference channels in the neutral energy spectra the energy-loss scales were referred to those found from resonant charge exchange in He+ He and Ar+ Ar exchange collisions. He+ H, (and D,) collisions were studied at selected angles from 0.1 to 3.0" and at beam energies of 0.5, 0.75, and 1.O keV. Although the energy resolution was limited by the short flight path (64 cm) in the apparatus, the dominant exchange channels were found to correspond to 13 < Q < 18 eV. The spectra showed some structure but suggested a strong capture process with a maximum near Q = 13 eV. Capture to the ground-state channel (Q = - 10 eV) was found to be weak. Figure 18 is a schematic representation of the participating direct scattering states reported by Bray ei al. (A, B, C, and D) as well as some selected exchange states. As is evident, quasiresonant exchange scattering is not possible since there is no charge-exchange state at the energy corresponding to peak A. Within the energy
+
+
+
+
+
D-
-> El
30
-
25
-
20
-
- H e ( 152s I t H o tH'
-He(ls2s)*H'*H(2sl - He'( Isl+H'+ H ( Is 1
c-
I
>
+
EXCHANGE
DIRECT
-H e ( l s 2 p ) + H * + H ( I s ) -He(ls2s)+H'+ - H e ( l s 2 p ) + H ' (2 I S )
8-
-He(Is2~)+H:~(f~~
z W
-He(ls2 l + H ' t H ' 5-
- He ( I s 2 ) + H ** H ( 2 s 1
O-
AT He'* H ~ ( X ' 1 i l
5-
10
-
+
-He(1s2)
*Ht+H(ls)
- He( 1s')
+Hi
(2s;1
Edward Pollack and Yukap Hahn
282
resolution available to Hodge et al. the dominant small-angle charge-exchange process has its maximum near Q = 13 eV and lies close in energy to the He( Isz) HZ* and He* HZ(X) channels. As shown above, the maxima in p for the neutral and for the B channels were found at the same 7 value, suggesting a common primary mechanism (Bray, 1975; Hodge et al., 1977) for their excitation. A surface crossing between the incident and an inelastic channel (such as B) followed by a second Demkov-type coupling between the inelastic and exchange channels (or vice versa) is responsible for determining the common position of the maxima in p, provided the direct inelastic and exchange channels are asymptotically near degenerate. After excitation of an exchange state additional interactions with the direct and with other exchange states occur (Hodge et al., 1977). If the direct scattering state is close in energy to a large number of exchange states, these should become important in the collision process. A similar Demkov coupling was used in explaining ion-atom charge exchange in He+ Ne (Nagy et al., 1971) and in the nonsymmetrical alkali ion - alkali atom (Perel and Daley, 1971;Olsen, 1972) cases. In these systems the charge exchange proceeds via couplings between initial and final states that lie close in energy at large internuclear separation. Recent work by Dowek et al. (1981, 1982, 1983) on electron capture by He+ from H,, N,, O,, CO, and NO has provided significant additional insight into ion - molecule charge-exchange collisions. The analysis of the results again followed an extrapolation from an understanding gained in earlier work on ion-atom collisions. The interpretation of the electroncapture results in He+ H, (D, was actually used as the target gas since it provides higher-resolution data than are obtainable from H,) was in fact guided by earlier work on the isoelectronic He+ He system (Barat et al., 1972; Brenot et al., 1975a).As an example in the He+ He, and also in the general He+ plus rare gas, collisions the 1svacancy in the incident projectile was found to play a significant role in the collision, and it was expected to play an equally important role with H, targets. The experimental findings were interpreted in terms of a simple “MO model” which was followed by a more detailed triatomic molecular approach extending the MO promotion model. The initial analysis emphasized the separate roles of the core and Rydberg electrons in the collision and, as suggested earlier by Bray et al. (1975, 1977), assumed a primary excitation mechanism. The proposed mechanism (Dowek et al., 1982) involves either a two-electron rearrangement in a diabatic-I1type process or an MO promotion (occurring at small interparticle separation), resulting in a diabatic-I process. As the collision partners separate the active electrons are shared by the projectile and or the target, exciting either the direct inelastic or exchange channels.
+
+
+
+
+
+
ELECTRON CAPTURE BY SIMPLE IONS
283
The experimental techniques used by Dowek et al. (1981, 1982, 1983) to study the ion- moleculecollisionsemployed a 127' electrostatic analyzer for energy analysis of the scattered ions and time-of-flight methods for energy analysis of the neutrals. A flight-pathlength of from 1.25 to 7.5 m resulted in good energy resolution. The relative importance of the direct and exchange processes in He+ H2was determined at 1 keV and compared to the results of Bray et al. (1977). Good agreement was found and the Bray results were then used for calibration at other energies. An important point establishedin these studies is that an understanding of the electron-capture processes in He+ H, requires also an understanding of the direct scattering in both the He+ H, and Heo H2collisions. The Heo results are needed to establish the separate roles of the diabatic-I and -11 processes; the correlation diagrams are the same for the He+ H, and Heo H, cases but no type-I1 processes are possible with Heo. Energy spectra from the exchange collisionsare shown in Fig. 19. At small angles the dominant capture processes are seen to be
+
+ +
+
+
+
(A) He+
+ H,
O(aW
+
He( Is),
30
20
+ Ht*(nU)
10
0
-lo
FIG.19. Energy-loss spectra of Heo in 2 keV He+ + H, collisions.
284
Edward Pollack and Yukap Hahn
and
(B) He+
+ H,
He*( lsnl)
4
+ Hf(X)
Processes (A) and (B) are in two Rydberg series that overlap in energy. Contributions from the Hf continuum are also seen in the spectra. The energy resolution was sufficient to confirm the presence of the He( ls2) Hf*(2pgU)outgoing channel. The location of the maximum of the peak corresponding to this channel was seen to be shifted by 1 eV below the value expected from a Franck-Condon transition. This is due to the strongly repulsive shape of the potential energy curve so even a small departure from the expected transition would result in a significantly different energy loss. Figure 19 shows that capture to n = 2 and 3 states in He* are strong processes at small angles. At the larger scattering angles He*( ls2p) Hf(X) is selectively populated. In addition two-electron excitation processes attributed to
+
+
+ He*( ls2p)+ Hf*(2pnU) He*( ls2p) Hf*(2pau)
and
were identified by extrapolatingfrom the curves of Sharp (1971). Since these Hf states are repulsive (several bound excited states of Hf were recently reported by Kirchner et al., 1984), their excitation contributes to the H+ production cited earlier. Figure 19 directly confirms that capture to the ground state (the only exothermicchannel) is always weak as was previously found (Hodge et al., 1977). The relative weakness of the exothermic to endothermic exchange processes is a striking characteristic in He+ H,. This relative weakness is in fact found in all the diatomic target systems studied and also in the He+ plus rare gas collision systems (Sidiset al., 1977). The results of both the direct and exchange scattering can be summarized as follows: At small 7 (7 = 1 keV deg) and for E < 1 keV (where this 7 range could be studied) all the singly excited channels, He+ H2+ He+ HT(n 3 2) direct and the He( Is2) Hf* and He*( lsnl) Hf(X) exchange, exhibit maxima in their probability of excitation. The He Hf*(2pou) is seen at the lowest E and 7 values. With increasing E and T, processes populating
+
+
He+(1s) He( Is2)
+
+
+
+
+ Ha(n = 2),
+ Hf*(n > 2),
and He*( 1 s2p)
+ Hf(X)
successively open for the intermediate, 1 < 7 < 2 keV deg range, followed by
ELECTRON CAPTURE BY SIMPLE IONS
285
direct excitation and charge exchange involving higher singly excited levels. The results are consistent with previously reported total cross-section measurements on H+ and Hf production as well as with the optical studiescited earlier. For larger 7 values (>3 keV deg) the dominant one-electron processes result in excitation of the direct He+ Hf(n = 2) and exchange He*( ls2p) H&Y)channels. In this large 7 range the two electron-excitation processes in both the direct He++ Hf*(n =2) and capture He*( ls2p) Hf*(n = 2) channels exhibit sharp thresholds and then become more important than the single excitation processes. It is very significant that at large 7 (3- 4 keV deg), the relative probabilities for excitation of the dominant capture channels involving double- and single-electron processes is found to be the same as that for the dominant double- to single-electron processes in the direct channels. This situation is similar for the Heo H, case. Also of importance is the finding that at these large 7 values the total single-excitation probability (in the summed direct and exchange channels) and the total double excitation probability are, respectively, equal to the corresponding probabilities in the Heo H, system. These results imply that there is a common excitation mechanism operating in both the ionic and neutral systems for the hard collisions. Collisionsleading to final ground-state products were seen at very small 7, and found to be weak, but this channel showed a strong energy dependence. The underlying process was attributed to a direct transition of the DemkovNikitin type. The probability of charge exchange was estimated from this model (Dowek et al., 1982)and a comparison with the experimentalresults showed the correct exponential dependence on the velocity but predicted exchange probabilities a factor of 5 to 10 larger than found experimentally. The similarity of the inelastic processes in He+ H, and in He+ He collisionssuggested an extension of the electron promotion model similar to one used for asymmetric systems (Fano and Lichten, 1965;Barat and Lichten, 1972). In those He+- atom cases where at least two of the atomic electrons are in orbitals that have energies higher than the 1s electron in He+,the collisionally formed quasimolecule is in a “core excited state” since the He+ “hole” is associated with an inner vacancy. This results in the two electron diabatic-I1process discussed earlier. In this process, as shown in Fig. 20, one electron fills the He Is, vacancy in the la’ MO and the other makes a transition to an empty higher-lying MO (na’).This gives rise to an infinite series of crossings which populate He( Isz) Hf*(nfiZ) channels. After the electron rearrangements the primary excitation is shared and populates Rydberg series: (a) He( Is2) Ht*(nU), (b) He*( ls,nf) Hf(X), and (c) He+ Hf( 1sqnfA). This mechanism was justified by the results found in He H,, where diabatic-I1 processes are not possible and no states due to
+
+ +
+
+
+
+
+
+ +
+
+
Edward Pollack and Yukap Hahn
286
FIG.20. A schematic quasidiatomic MO correlation diagam for He+ -tH,. The diabaticI1 mechanism (---) and the 2a‘- la” rotational coupling (O), which occurs at small R, are
shown.
Rydberg series are found in the collision processes. This mechanism is effective for 100 < E < 1000 eV and for 0.5 < o < 2 keV deg. The first of the diabatic-I1crossings (Joneset al., 1980)is positioned at 7 eV (c.m.) above the incident channel and at an interparticle distance < 3a,. The experimental results in the large z range are consistent with a diabatic-I-typeprocess. This process, also shown schematically in Fig. 20, involves both one- and twoelectron transitions between the promoted 2a’ MO and the initially vacant 3a’ and 1a” MOs and accountsfor the triple-peak behavior seen by Dowek et al. ( 1982)at larger o. Transitionsto the 3a’ and 1a” MOs selectivelypopulate the “n = 2” levels. This would give rise to He*(ls2p) H;(X) and He*( ls2p) Hf**(2pa or 2pa) final states and He+ H,*(n= 2) and He+ H2(2paZor 2pa,2pn). These interpretations (Dowek et al., 1982)are based on the generalizations of the He+ He rotational coupling excitation model. A more realistic analysis of the He+ H, collision problem requires energy surfaces which involve the variables s (H-H separation), R (He+-H, center-of-mass distance), and y ( s - R angle). The promotion model was extended to the triatomic case by the introduction of a “correlation cube” (Dowek et al., 1982;Dowek and Sidis, 1983).The cubes were constructed by fixing y at a/2 and 0”,correspondingto the C,, (isoscelese triangle) and C,, (linear conformation) cases, respectively. As the collision progresses y varies. A correlation diagram is constructed for each face of the cube and the energy surface is then drawn from continuity and symmetry-conservationconsiderations. Three faces of the cube correspond to limiting diatomic cases. The
+
+
+
+
+
+
ELECTRON CAPTURE BY SIMPLE IONS
287
case of C,, symmetry is shown in Fig. 21. Face 1 is the s = 0 limit which resultsin the He-Hecorrelation diagram (the lsaand 2pacurvesareextrapolated to the Be united-atom limit as R 0). Face 2, for R --., corresponds to the H - H correlation diagram together with unchanged He atomic orbitals. Face 3, s m represents three isolated atoms He, Ha, and H,.The remaining face (4) for R 0 is effectively the only triatomic case. At large s three linear combinations of atomic orbitals, of specific symmetries and nodal structure, are formed and then correlated with the Be united-atom limit at s = 0. The MO surfaces are constructed by traversing the faces and following molecular orbitals of the same symmetry by continuity. As an example the lowest surface, 1a, corresponds to the limiting curves 1 ag,1sHe, IsHe,and 1sagon faces 4,3,2, and 1, respectively.The resulting surface( 1 a is seen in Fig. 2 1. The surfaces for the Cmu symmetry are constructed in a similar way. Both the C,, and Cmucases show promotion of the 2aI(2a) surface correlated with the 1st~~ H2orbital which implies that the diabatic-I1 process occurs in both configurations.Here one of the 2a:(2a2) electrons fills the initial vacancy in the bonding 1a 1a) MO surface while the other electron is excited into a Rydberg MO. Forbidden transitions in one symmetry can occur in the general case and the occurrance of MO surface crossings dependson the conformation. Figure 2 1 suggeststhe presence of two types of processes: the first involving one- and two-electron transitions along the 2a, - lb, intersection and the second process involving one- and two-electron transitions by rotational coupling near the contact of the 2a,- lb, surfaces at R -0. The 2a,- lb, crossings result in H~(1sa~paU), H?*(2pa,2) and are only seen in the tnatomic picture. The rotational coupling was previously seen in the diatomic picture and populates H;(2p7rU),
-
- -
101
FIG.2 1. Triatomic correlation diagram in the C,,conformation ( y = 71/2): la, (lowermost surface), 2a, (dotted surface), 1b2 (hatched surface), and I b , (transparentsurface).
Edward Pollack and Yukap Hahn
288
He*( ls2p), etc. Although the discussion above is valid only for C,,,this conformation is attained in a typical collision which involves a wide range of y values. In the energy range investigated s = s, (the equilibrium separation of H,) a simplification is possible by drawing cuts at this s for the two configurationsresulting in a “correlation cylinder.” Typical paths associated with elastic scattering and excitation due to a rotational coupling from a rectilinear trajectory can then be inferred from the resulting diagram (Dowek et al., 1982).
B. OPTICAL EMISSION STUDIES Detailed studies ofcapture to He(3 3P)by Eriksen and Jaecks ( 1976,1978) and Goldberger et al. (1984) lead to problems not yet addressed in our discussion. These include coherent interferenceamong the excited channels and differences between the H, and D, target cases. Coincidence measurements (on the Heo and its decay photons) with and without linear polarization analysis were made on the 3889 A line from He(3 3 P - 2 3S)produced by He+ H2 collisions at 1.5 and 3.0 keV. The experimental techniques employ a momentum-analyzed He+beam which crosses a thermal H, beam. An optical system placed perpendicular to the collision plane detects the emitted radiation which is analyzed according to wavelength and linear polarization. The axis of maximum transmission of the polarization analyzer makes an anglep with respect to the incident He+direction. Projectiles scattered through an angle 8 enter a parallel-plate electrostatic energy analyzer. With voltage on the back plate, only the neutralized Heo beam passes straight through the analyzer and is detected by an electron multiplier placed behind a slit. Pulses from the photomultiplier tube (PMT) are detected in delayed coincidence with those from the Heo detector. The coincidence techniques employed discriminate against cascade contributions. Measurements of this type which provide information on the angular distributions and polarization ofthe emitted light are very difficult (and time consuming)to make, but they reduce the number of undetermined collision variables and allow for decisive tests of the collisional model. The optical distributionsdepend on the relative populations of the radiating substates. If 3P0and 3P, states are coherently excited the light emitted reflects interference between the excitation amplitudes. There has been a considerable amount of work along these lines and the papers by Eriksen and Jaecks contain referencesto the literature. A large alignment (due to the nonstatistical populations of magnetic sublevels where the rn = f 1 population is different from that for rn = 0) of the He(3 3P) was originally reported by Eriksen and Jaecks ( 1976).At particular beam energies and scatteringangles
+
289
ELECTRON CAPTURE BY SIMPLE IONS
+
the alignment was found to be even larger than for capture in He+ He (Vassilev et al., 1975;Jaecks et al., 1975; Eriksen et al., 1976). Strongcoherence effects are present in He+ 4- H, collisions. The coincidence rate at 1.5 keV was determined as a function of 8 (without linear polarization analysis). This rate was taken to be proportional to the differential probability for excitation of He(3 3P) and indicates a threshold 7 < 1.1 keV deg (in He+ He the threshold is 1.5 keV deg). Figure 22 shows the coincidence rate as a function of /3, the polarization angle, at four selected combinations of energy and scattering angle. The z axis is along the direction of the incident He+ beam. In each case data were takenat polarizeranglesofO", 45", 90°, and 135O.Theactualdatapointsare shown with error bars in the plots; the points without error bars are best fits to the data as described by Eriksen and Jaecks (1976). The data at 2.33"correspond to E8, = 8 keV deg2 which still lies along the elastic curve in the data of Bray shown in Fig. 15. The patterns in Fig. 22 were interpreted by using the results of Fano and Macek (1973). The polar Z vs /3 plot is generally described by an hourglass-shaped pattern with major axes lying along the solid lines in Fig. 22. The ratio R = JZ,-, (minimum to maximum inten-
+
He++ H2 1.5 kOV I.oo'
I
1.5 k e V 1.50.
1.5 LOV
3.0 k.V
2.33.
1.25.
FIG.22. Polar plots of the average number of real coincidences per lo9scattered neutral particlesvsp for several energy-angle combinations. The incident beam direction is along the z axis and the neutrals are detected above this axis.
290
Edward Pollack and Yukap Hahn TABLE I11
MEASURED AND CORRECTED VALUESOF THE RATIO R4
He*-H, 1.5 I .5 1.5 3.0
1.00 1S O 2.33 1.25
0.90 f 0.09 0.62 f 0.04 0.82 f 0.07 0.72 f 0.10
0.89 f 0.09 0.60 f 0.04 0.79 f0.07 0.70 f 0.10
He* - He I .5 1.5 3.0 3.0 3.0 3.0 3.0
1.25 1.50 1.00 1.25 1.50 1.75 2.00
0.75 f 0.1 1 0.72f0.14 0.72 f 0.09 0.72 f 0.10 0.84 f 0.09 0.79 f 0.09 0.82 f 0.1 1
0.73 f 0.1 1 0.70f0.14 0.70 f 0.09 0.70 f 0.10 0.83 f 0.09 0.78 & 0.09 0.81 f0.11
Toand 0 are the laboratory incident kinetic energy in keV and laboratory scattering angle in degrees, respectively.
sity) is of particular significance and was determined by computer fits for each of the plots shown in the figure. The results are shown in Table I11 along with those for He+ He. The table suggests several interesting features. At 1.5 keV- 1S oand 3.0 keV- 1.25" the He+ H, collisions resulting in capture to He(3 'P)have at least as much polarization as is found for any of the reported data in He+ He. A pure state with m = 0 relative to the z' axis (the direction of the maximum in the polarization pattern) would have a minimum value ofR = 0.46. The R = 0.60 0.05 at 1.5 keV- 1.5" isinterpretated as being sufficiently close to this minimum to suggest the excitation of a nearly pure He(3 ' P )state. Table IV lists the angle ofthe major or minor axis of the polarization pattern (Om) and the angle of the momentum transfer axis (Omom) calculated for selected energy losses (Q). The Omom were calculated using energy and momentum conservation, assuming no dissociation during the collision, and were made for Q values expected from several possible excitations. The results show that the direction of the observed symmetry axis basically agrees with the direction of the momentum transfer axis. Photon-correlation measurements were made by Goldberger et al. (1984)
+
+
+
+
29 1
ELECTRON CAPTURE BY SIMPLE IONS TABLE IV COMPARISON OF ANGLES 6,,,
3.05
1.25
109
+ I I (maj)
1.52
1.00
112
+ lO(maj)
1.52
1.50
107 + 9 (maj)
1.52
a
2.33
6,-
13.9 14.9 16.3 29.5 13.9 14.9 16.3 29.5 13.9 14.9 16.3 29.5 13.9 14.9 16.3 29.5
97 98 98 104 105 106 107 118
102 102
103 112 100 100
101 107
Inelastic energy loss.
for 1.5 keV He+
125f23(min)
AND
+ H,, D2
- He(3 3P)
+
He(2 3 S ) y
No differences in the electron-capture processes from these targets were previously expected since the electronic structure of H, and D, is the same. There are differences in mass, vibrational, and rotational structure but these should not substantially affect properly presented experimental results. In these most recent studies, linear and circular polarization measurements were made (no initial or final target states were identified). The data were analyzed to best yield information on the relative populations of the magnetic sublevels in the radiating 3P states. In He+ H,, D, collisions the availabilityof a number of H, and D, initial and Hzand Dtfinal target states would result in a collision system characterized by a weighted incoherent superpositionof pure states (Fano, 1957) with the resulting 3 3P 3Sradiation characterized by alignment and orientation parameten (Fano and Macek, 1973; Gau and Macek, 1975). The results were, however, presented using Stokes parameters, which are expressible in terms of the scattering cross sections for the magnetic sublevels. All measurements on H, and D, were made at 1.5 keV laboratoryenergy so that the relative projectile- target
+
-
Edward Pollack and Yukap Hahn
292
velocity would be the same. In order to compare scattering that probes the same region of potential (in the case of a central potential and small scatter( p is the reduced mass, ing angle) data were presented in plots using pOc.m. /3c.m.is the c.m. scattering angle). The behavior of
PI = k[o(O)- 2 4 1)]/[0(0) + 2a(I)] with k = 15/41 [calculatedfrom results given in Fano and Macek ( 1973)]for He(3 3P 2 3 S )is shown in Fig. 23 and seen to be different for the H, and D2cases, indicating that the shape of the electron distribution in the He 3 3P is different with the two target molecules because of differences in the populations of substates. At p&,. = 6: a(0) = 0.66a(l) for H, and a(0) = 4 . 1 5 41) for D, . The degree of circular polarization P3 was found to change from left to right handed with increasing ~ 6 , . ~The . . probabilities for He(3 ' P ) production is also different for the two molecules, as shown in Fig. 24. The D2case shows some evidence of oscillatory structure. If the chargeexchange processes result in dissociating H t and D t states, then the two molecular targets should give different results because excitation of these channels is stronglyp dependent and the value of the pUe,.,. plot in Fig. 24 is questionable.The excitation mechanisms for the two isotopes are the same, as is suggested by the overall similarity of the Stokesparameters Pz and P3,as a function ofpLJ,.,, (Goldbergeret al., 1984).However the degree to which aI and q, are excited is different for the two isotopes. --j
40
20
d
0 0 X
\
ci -20
-40 -60 0
2
4
tJ8<
6
6
1
0
( m u deg)
FIG.23. Stokes parameter P,/ k for He(3 3P)excitation from He+ selected PUB values.
+ H,, D2collisions at
ELECTRON CAPTURE BY SIMPLE IONS
293
+
FIG.24. Probability for He( 3 3P)excitation in He+ H, ,D, at selected&,, values. Open circles and open squares for D, represent the probability computed from linear and circular polarization measurements, respectively.
VI. Ion - Molecule Collisions: Other Systems A. He+
+ N,, 0,
The availability of a quasiresonant channel makes charge exchange an important collisional process in He+ N, (here quasiresonant refers to channels which are near degenerate in energy asymptotically). The total cross section is large, decreases slowly with energy, and has a value close to cmz at 1 keV (Stebbings et al., 1963; Moran and Conrads, 1973; Koopman, 1968; Grovers, 1975).The major contributions to the total CEX cross section come from channels resulting in dissociating Nf states (Gustafsson and Lindholm, 1960; Stebbings et al., 1965).The small-angle charge exchange was studied by Fernandez et al. ( 1971) at energiesin the range from 0.5 to 3.0 keV. The dominance of the charge-exchange processes is clearly seen in Fig. 25, which is a plot of Po as a function of 8. Additional measurements made at 3" and 4" at 3 keV indicated that the curves could simply be extrapolated to these larger angles. In comparing the results at different energies, Po was found to increase with increasing collision time at fixed E8 values. A detailed study of both the direct and electron-capture processes in He+ N, in an energy range from 0.2 to 4 keV and for angles out to 3" was made by Dowek et al. (198 1). Over the z range studied the direct quasielastic channel showed only weak vibrational excitation which decreased with increasing energy. The energy spectra of the Heo resulting from exchange
+
+
Edward Pollack and Yukap Hahn
294
-
10-
:F 1.0 keV
H & t N, 0.1
l ' ' ' l l ' L 1.0
PO
-
2.0keV
-
1 05
10
20
I5
0 5
10
I5
20
SCATTERING ANGLE ( Deg 1
SCATTERING ANGLE (Deg )
FIG.25. Charge-exchange probability in He+
+ N,.
collisions showed quasiresonant, exothermic (Q < 0), and endothermic (Q > 0) capture processes. At low energy (E = 200 V) and for 8 < 1 ", a single peak due to the quasiresonant He+(1s)
+ N2
+
He( Is2) N f ( C zZz, v = 5,6)
4
Q = 0.3 eV channel was seen. The measured FWHM of the peak was 1.3 eV, compared with the corresponding value of about 0.6 eV for He+ He (which was used as a reference), showing the excitation of several vibrational states. A significant feature of the experimental results was the strong contributions from endothermic channels when compared with the exothermic processes. The electron capture into the exothermic channels yields He( Is2) with resulting Nfproducts in B 2Zu,v = O,l, A 211u, v = 2,3, and X 2Zi, v = 0,l states. No structure corresponding to Ng(4Z3was seen, confirming the Wigner spin-conservation rule. Endothermic channels were found for z 2 0.3 keV deg and they increased strongly with increasing z and E. The spectra showed a broad peak for 10.5 < Q < 15.6 eV corresponding to He*( 1s21) Nf(X2Zi)and He( 1 snl) Nf(XZZ;). A second broad peak for 18 < Q < 24 eV was attributed to He* ( l s n l ) Nf*. This peak was superimposed on a high-Q tail resulting from transfer ionization processes with Q > 18.1 eV. At higher collision energy a third broad peak appeared for Q = 30 eV which could correspond to autoionizing Nf** product states (involving two-electron excitation). The relative magnitude of these excitation peaks shows that the quasielasticchannel is dominant at small z with the endothermic channels becoming dominant with increasing z. The exothermic processes remain weak in the entire z range investigated.
+
+
+
+
295
ELECTRON CAPTURE BY SIMPLE IONS
+
In He+ 0, the total cross section for electron capture is again large (lo-'' cm2 at 1 keV) and decreases with increasing collision energy (Stebbings et al., 1963; Moran and Conrads, 1973). This behavior was attributed by Stebbings et al. to the importance of the near-resonant He O;(c4Z;), Q = 0.16 eV channel in the scattering. The O;(c4Z;) state is strongly predissociated (98 k 2% Guyon et al., 1978) and yields a large O+/O;ratio (Stebbings et al., 1963; Mauclaire et al., 1979; Anicich et al., 1977). Emission spectra from He+ 0, were studied for energies between 25 and 4300 eV (Harris et al., 1974). The emission primarily resulted from capture collisions and show that the He 1 lines increase in intensity with increasing kinetic energy. The dominant Ofemissions found were those of the first negative system ( b4Z:g+ a 411,).Emission from Of(c4ZJ was found to be weak since the state is strongly predissociated. A detailed direct study of electron capture in He+ 0, was made by Gillen and Kleyn ( 1980). Time-of-flight techniqueswere used to identify the final channels for laboratory energies of 104 and 3 16 eV at angles out to 25 '. The measured angular distribution of the Heo at 356 eV shown in Fig. 26 is strongly peaked in the forward direction and then decreases monotonically by five orders or magnitude in the laboratory angle range from 0 to 24". Energy spectra of the Heodirectly confirm the importance of the quasiresonant exchange process in the forward scattering. The near-resonant He 0; peak includes contributions from the c4Z;(v = 0) and other close-lying states. At larger scattering angles the He spectra reflect increasing contributions from both endothermic and exothermic exchange channels. A shift in Q value with increasing scattering angle was seen and associated with increasing average vibrational excitation energies. The near-resonant process was found to contribute 60% of the total charge-transfer cross section. The endothermic processes contribute 35% and the exothermic processes about 596. Dissociative charge transfer in He+ 0, was studied by Bischaf et al. (1983) for energies in the range from 1 to 100 eV. It was found that the 0; ion yield was less than 1 % of the O+ yield. The results show four dominant channels leading to O+. The angular distribution of the O+ was found to be strongly nonisotropic for a11 four channels and the branching ratios (integrated over angle) were found to be strongly dependent on collision energy. At the lowest energies O(3P) 0+(,Po) was the dominant product state, while at 100 eV the O( ' D ) O+(4S0)dominates. Electron capture by He+ from N, and 0, shows the same basic features. It is the dominant collision process, and the availability of a quasiresonant exchange channel results in a large total cross section. A large number of states contribute to the endothermic processes which dominate the scattering at the larger angles. The molecular-ion states initially excited in the
+
+
+
+
+
+
+
Edward Pollack and Yukap Hahn
296 100
10-1
1
o-~
10-5
0
4
8
12
16
0 (degrees)
20
24
+
FIG.26. The angular distribution of Heo from 356 eV He+ O2charge-exchangecollisions.
quasiresonant processes are strongly predissociating and give rise to large "+I/[ N a and [0+]/[03ratios. Although the exothermic processes remain weak at all angles their excitation probabilities have a significant energy dependence. They result from single-electron transitions from the molecule to the He 1s orbital. The Demkov model was used (Dowek et al., 1981) to calculate the probability of a direct transition to the least exothermic channels in N, and 0,. Agreement was found in the velocity dependence although a direct application of the model overestimates the experimental results by a factor of 5 to 10. Application of this model to the more exothermic channels yield probabilities orders of magnitude smaller than found experimentally. This was also found in He+ Ar (Sidis et al., 1977). Excita-
+
297
ELECTRON CAPTURE BY SIMPLE IONS
tion of the endothermic processes is primarily due to the initial vacancy in the He 1sorbital (Dowek ef al., 1981 ). The vacancy, as in the He+ H, case discussed earlier, gives rise to core-excited repulsive states of the triatomic quasimolecule and results in the two-electron transitions and the diabatic-I1 curve crossings between states having the same spatial and spin symmetries. They primarily populate He( Is2) Ng* and Ot* channels with the excited electron either in a valence or Rydberg state. Some evidence suggesting DI processes for the harder collisions is also found (Dowek et al., I98 1 ).
+
+
B. He+
+ NO, CO +
Electron capture is an important collisional process in He+ NO with a total cross section that decreases from about 1 1 to 6 X cm2 with increasing energy from 1 to 3 keV (Moran and Conrads, 1973). This is again primarily due to the availability ofquasiresonant exchangechannels. Dowek et al. (1 983) investigated the direct and exchange collisions in He+ NO. In addition the direct scattering in He+ NO was compared with results from Heo NO and at small 7 clear differences in the spectra were seen. They were primarily associated with significant excitation of the A ,X+ state with He+ projectiles and its weak contribution to the collision when He is used. On the other hand at larger 7 the direct He and He+ collision channels showed similarities in the energy spectra. These findings are again consistent with a picture that the diabatic-I1 mechanism is effective in the He+ case at small scattering angles while a diabatic-I mechanism in both He and He+ becomes important at larger scattering angles. In the small-7 range the capture channels were primarily due to processes corresponding to - 3 < Q < 3 eV. The endothermic channels increased in importance with increasing 7 and the state identifications were consistent with collisions resulting in excited He with and without simultaneous NO+* production. Several of the processes could not be identified with known states of NO'. The exothermic channels were seen to be weak at all r values. They were indeed found at the smallest experimentally studied 7 and their relative contributions to the exchange channels were seen to decrease with increasing energy defect. The total cross section for electron capture in He+ CO is = 8 X cmZ(Moran and Conrads, 1973) at energies in the range of 1 to 3 keV. For energies below 400 eV Coplan and Ogilvie ( 1974)found the cross section to rise with decreasing energy. In the energy range from 200 to 1500eV the total cross section was also found to be much larger than the sum of the measured emission cross sections from the A-Xand B-Xtransitions in CO+. Coplan and Ogilvie found contributions from He* in the emission spectra, and inferred the importance of endothermic processes (Lipeles, 1969).
+
+
+
+
Edward Pollack and Yukap Hahn
298
+
+
The He+ CO (and Heo CO) collision system was studied in some detail by Dowek et al. (1983), who found that for z > 1 keV deg the electron capture probability is larger than that for the direct scattering.For low energy and small scattering angle ( E < 500 eV, 8 < 0.5”) quasiresonant charge exchangewas again dominant. The CO+ states that could contribute (Okuda and ,A but the and Jonathan, 1974; Locht, 1977) are the C ,C+, D relatively narrow FWHM of the Heo energy-loss peak suggested that the C state is most important [a similar argument was used by Fernandez et al. (1975)to assign states in Ar+ N,]. Of particular significance(in relation to the discussion of Ar+ CO in the next section) is that the
+
He+
+
+ CO
+
+
He( ls2) CO+(B,X+)
channel was clearly seen in all the spectra of Dowek et al., being the most important exothermic process. A number of endothermic channels whose importance increases with E and z are found but identification of the participating CO+ states was not always possible. The spectra suggested contributions from He*( lsnl) CO+(X) which are consistent with the findings of Coplan and Ogilvie (1974).
+
C. DISCUSSION There are basic similarities in the electron capture from H,, N,, O,, NO, and CO by He+. In these systems at energies in the range from several hundred to several thousand eV, collisionsresulting in electronicallyexcited states of the molecule (or molecular ion) occur via a “vertical transition,” in agreement with predictions of the Franck-Condon principle, and in addition the Wigner spin rule is found to be obeyed when it can be tested. The electron-capture processes primarily yield dissociating (or predissociating) states of the molecular ion leading to the production of recoil atoms and atomic ions. In collision systems where a quasiresonant channel is available the cross section for electron capture is large and the quasiresonant channel is dominant in the small-angle exchange scattering. Exothermic processes are always found to be weak but their excitation is strongly energy dependent. Excitation of these channels is attributed to a “Demkov coupling” and once excited they can populate other channels with larger Q values (Dowek et al., 1983). The endothermic processes become important at the larger scattering angles. These processes result from state crossings in the triatomic molecular ion. Scattering at small angles is attributed to a correlated two-electron transition due to an initial vacancy in the He MO. For harder collisions, the state crossings result from MO crossings and give
299
ELECTRON CAPTURE BY SIMPLE IONS
rise to both one- and two-electron excitation processes. The molecular states generated are not always seen in conventional studies of the emission spectra. It is of particular significance that the processes in these systems are very similar to those previously found in the He+ plus rare gas cases.
D. Ar"
+ CO
+ CO collision was studied by Goldberger et dence of Po was found to be very similar to that previously seen in He+ + H, (Fig. 14), He+ + N, (Fig. 25), and in Ar+/N+ + N, (Fig. 1 1). The results The relatively complex Ar+
a/. ( 1977) and Goldberger ( 1984) at low keV energies. The angular depen-
again indicate that for a given 7 value, Po increases with increasing collision time (decreasing energy) and that the exchange processes dominate the scattering in harder collisions. Energy spectra for the direct scattering show several peaks resulting from the excitation of CO to the a 311and A ' l I / D 'A, and to CO+ (X ,Z+) states. With increasing scattering angle inelastic collisions exciting CO(A n/D A) are found to become more important than those resulting in CO(a 311).The excitation of the singlet and triplet states cited is comparable in intensity at 7 = 1.4 keV deg at both 2 and 3 keV. As found in earlier studies ofHe+ H, and Ar+ N, the electronic excitation in Ar+ CO is seen to occur independently of the vibro-rotational excitation. For E02 < 2 keV deg2the collision is "vibro-rotationally elastic." The capture channels were studied in detail at E = 1 .O keV using TOF techniques. Figure 27 shows a typical spectrum from Ar+ CO 4Aro CO+ at 8 = 0.25 '. Capture to the ground-state channel, ArO CO+(X ,Z+), an exothermic process with Q = - 1.7 eV, is weak. The dominant exchange channel at small angles is found to be Aro CO+(A ,lI), with Q = 0.8 eV. The small peak in Fig. 27 is consistent with excitation of ArO CO+(D 211)/C+(2P) O(3P)and broadens toward higher-energy loss with increasing scattering angle. Electron-capture processes leading to dissociating channels are only weakly excited in the 7 range studied, in sharp contrast to capture collisions involving He+ projectiles where the molecular ions are predominantly formed in dissociating or predissociating states. The reduced cross sections for both the elastic scattering and for the
+
+
+
+ +
+
+
Ar+
+ CO
4
ArO
+
+
+ CO+(A,lI)
channels are peaked in the forward direction and found to decrease with increasing 7.The reduced cross sections for Ar+
+ CO
-
Ar+
+ CO(a 311)
Edward Pollack and Yukap Hahn
300
' 10
0
AE ( e V )
+
FIG.27. Energy spectra ofArofrom 1.O keV, 0 = 0.25", Ar+ CO charge-exchange collisions. X + Aro + CO+(X22), A Aro CO+(A TI),D + Aro + CO+(D211)/C+(2P) +
00~).
+
+
and for Ar+
+ CO
+
ArO
+ CO+(DZll)/C+(ZP)+ O('P)
have maxima at z = 0.75 keV deg, which is consistent with excitation of these channels via a common primary mechanism. The reduced cross section for Ar+
+ CO
-+
Ar+
+ CO(A I l l / D 'A)
has a maximum at z = 2.5 keV deg. An energy level diagram for selected direct and exchange states is shown in channel does not Fig. 28. As may be seen in the figure the ArO CO+(BT ) lie close to a direct scattering state. It is particularly significant that excitation of CO+(B %+) is weak if at all present (this state would be located at the position of the minimum between the peaks in Fig. 27). This is in sharp contrast to the He+ CO collision, where it was found that excitation of
+
+
30 1
ELECTRON CAPTURE BY SIMPLE IONS DIRECT A r + + CO'
Ar'
+
+
EXCHANGE
e-
c ( ~ P )o ( ~ P ) +
A r ' + C+('P) Ar'
+
CO(A'T)
Ar+
+
CO(a3T)
Ar'
Ar*
+
+
+
0 ('0)
CO+(Dz?r)
CO( X'c')
+
FIG.28. Selectedstates, at the locations of the ground vibrational levels, in Ar+ CO and Aro CO+. The peaks found in the direct scattering are at the positions of the arrows.
+
CO+(B%+) is the most important exothermic process. The weak excitation of this state in the Ar+ case is consistent with electron capture that primarily occurs via couplings between close-lying direct and exchange states at asymptotic distances. E. H+
+
+ H,
H+ H2 is the most fundamental ion-molecule collision system. The small-angle elastic scattering here is determined primarily by the H,+ ground-state potential energy surface. The ground- and lowest-excited-state
302
Edward Pollack and Yukap Hahn
surfaces are involved in excitation of the
H+
+ H2
+
H( 1s)
+ H,+(X ’ZQ)
capture channel which can easily be studied by TOF techniques. Detailed studies of this system can provide an ideal testing ground for the simplest potential energy surfaces and couplings between interacting states. In the early experimental study (previously cited) by Lockwood and Everhart ( 1962)the peaks in the electron-capture probability were seen to occur at the same energies (Fig. 12) for both the H and H2 targets. The probability was calculated by Piancentini and Salin (1978) at large scattering angles for 1 < E < 20 keV. In the calculations it was assumed that the initial H, and final H,+ nuclear separations remain fixed at the H2ground-state equilibrium separation and that the molecular axes are randomly distributed during the collision. The two lowest potential energy surfaces were used and the transition probabilities were found assuming an impact-parameter approximation. The theory reproduced the essential features of the experimental results. A more recent two-state calculation by Yenen et al. (1984) used a formulation of Pfeifer and Garcia (198 1) that incorporates electron translation factors to calculate the velocity dependence of the electron-capture probability. It was assumed that the projectile is scattered by one H atom of the molecule [the binary limit is attained at small 6 for E > 3 keV as shown by Vedder et al. (1 980)], which allowed a determination of the scattering angle from the impact parameter. The electronic transitions were treated using the “quasi-diatomic” approximation (Dowek et al., 1982; Jaecks et al., 1983) with potential energy curves of Bauschlecher et al. (1973) for a single geometrical configuration (fixed s and 7). The calculations reproduced the frequency of the oscillations to within 4%. The phase, however, was not well reproduced. Balmer a emission was studied in H+ H2 collisions for 1.2 < E < 100 keV by Williams et al. (1982). Interference filters separated the Doppler-shifted radiation resulting from electron capture to H(n = 3) from the a line generated by dissociated H2+.Plots and tables of measured cross sections are reported. The maximum cross section for H, emission resulting from the capture collisions is found to be 15 X cm2 and occurs at 45 keV. Cross sections for capture to 3s and the summed 3p and 3dlevels are reported. At energies below 5 keV D+beams of equivalent energy were used. For 5 < E < 10 keV, results using D+and H+ beams of equal velocities were shown to yield the same cross sections. A comparison to the results of Morgan et al. (1973) and Birely and McNeal(l972) showed that although the total cross section for capture to H(n = 3) differs in magnitude from that for capture to H(n = 2) there is a general similarity in the energy depen-
+
ELECTRON CAPTURE BY SIMPLE IONS
303
+
dence of 4 3 s ) and 4 2 s ) and also in 4 3 4 0.12a(3p) and a(2p). The absolute cross section for H+capture to H(2s) for several target gases including H, were reported by Shah et al. (1980). The cross section for Lyman-a emission was shown to have a maximum for energies near 14 keV by Van Zyl et al. (1967). Total cross sections for electron capture for 5 < E < 150 keV in H+ H, (and a number ofother gases)were obtained by Rudd et a/. ( 1 983). A detailed study at E = 49 keV for H+ H, + H(n = 3) was made by Knize et al. ( 1984). The cross sectionswere found to monotonically decrease with increasing L and IMLI.The total cross section for capture to n = 3 was found to be about 3% of the total electron-capturecross section and seemed to be in agreement with the ratio predicted from the simple n-3 scaling law. Heckman et al. ( 1984) used TOF techniques (describedin Section III,A)to investigate electron capture in H+ H, at energies of 1.0,2.0,and 3.0 keV for 7 < 3.0 keV deg. These studies were a continuation ofearlier work on the direct (Vedder et al., 198 1 ) and exchange (Peterson el al., 1979) scattering in D+ H, . The H+ Ar collision provided the necessary energy reference (see Fig. 7). Figure 29 shows the probability of electron capture to
+
+
+
+
+
0.9
H++H*-- H ( 1 s ) + H$Z;
-
)
1.0 keV 2.0 keV
t
o' 0.1 0
0
2.0
1.0
3.0
T ( keV deg )
FIG.29. Probability of electron capture to H( Is) tering angle in an exchange collision.
+ H:(*Z:)
as a function of reduced scat-
Edward Pollack and Yukap Hahn
304
+
H( 1s) H;('Z,'), as a function of z, following an exchange collision at energies of 1.O and 2.0 keV. The results at 1.O keV show some structure since the resolution in z is better at the low energy. Capture to final ground states is seen to be the dominant small-angle exchange process at these energies. Measurements at 3.0 keV again show that at the largest z (3 keV deg) investigated approximately one-half the exchange collisions result in ground-state products. At the larger angles capture to H(n 2 2) Ht(2Z,') accounts for one-half the collisions. Plots of the reduced cross section as a function of reduced scattering angle for the summed processes (electron capture into all final channels) are peaked in the forward direction, decrease monotonically, and show no structure. A p vs z plot for the
+
+ H, H( 1s)+ HZ(X *Z8) channel at 3.0 keV showsbasically the same angulardependenceas found for elastic scattering in D+ + H, (Vedder el al., 1981). This was expected H+
+
(Hodge el al., 1977) since the incident channel lies close in energy (Q = 2 eV) to the exchange channel. The better energy and angular resolutions at 1 .O keV allow identification of the channels corresponding to H(n z= 2)
+
-
6.0 c v)
c .C
a
40
4.0
v
Q 2 .o
u 1.0 2.0
0
-
T ( keV deg 1
-
+
FIG.30. The reduced cross section at I .OkeV for H+ -tH, H* H;(%:). Solid curve is for E = 3 keV D+ H, D+ H: (Q = I 3 eV). The peaks in the cross sections, each plotted
+
+
in arbitrary units, occur near z = 1 keV deg. The basic similarityin these cross sections suggests that a common primary interaction is responsiblefor the direct and these exchange channelsin the hydrogen collision system.
ELECTRON CAPTURE BY SIMPLE IONS
305
H,+(,Z;). The reduced cross section for capture to these excited states is shown in Fig. 30. Also shown in the figure is the reduced cross section for the direct inelastic channel in D+ H, (Q = 13 eV) at 3 keV (Vedder et al., 198 1 ). The two cross sections (each plotted in arbitrary units and determined from data taken on different apparatuses) are seen to exhibit a similar behavior. The maximum in the cross section for capture into excited states of H (with the Hf in its ground state) occurs at approximately the same z value at which the cross section for the direct inelastic channel exhibits a maximum. This figure also confirms that t,which has approximately the same value in laboratory and center-of-mass coordinates (Jones et al., 1963: Smith et al., 1966), is a useful parameter in describing ion - molecule collisions. Electron capture to H(2s) from H2 and D, was investigated to test for a possible isotope effect (Lee et al., 1985) due to differences in the vibrational levels of the targets. Differential cross sections for
+
H+
+ H,
H+
+ D,
and
H(2s)
+ H:
H(2s)
+ Dt
4
-
were reported at a laboratory scattering angle of 1 ' for energies from 2 to 20 keV. The differential cross section showed a peak at about 5 keV and a subsidiary peak near 10 keV for both the H, and D, targets. Within experimental error no isotope effect was seen.
VII. Electron-Transfer-ExcitationCollisions Recently, several exotic processes in high-energy ion - atom and ion molecule collisions, some of which lead to charge-exchanged final states, have been studied. As an example, the dominance in the high-energy limit of the second Born contribution to the CEX cross section was studied in great detail. In addition, resonant transfer and simultaneous excitation (RTE) as well as its nonresonant counterpart (NTE) has been investigated experimentally by a number of groups (Tanis et al., 1981, 1982, 1984; Clark et al., 1985). In RTE, we have a high-energy (- 100 MeV), highly charged ion beam interacting with a gas target, for atomic ( A )and molecular ( M )targets, B Z + + A + (B(z-l)+)**+ (A+)* (58) Bz+ + M + (B(z-l)+)**+ (M+)*
Edward Pollack and Yukap Hahn
306
and subsequently
The intermediate state formed, in Eq. (58), is an autoionization (resonance) state which decays either by photon emission (RTE) or by an (Auger) electron emission [Eq. (60)].The process in Eq. (60) leads in effect to ionization of the target A or M. The experimental signature of RTE is the K x ray in Eq. (59),in coincidence with the “neutralized” beam ions The measurements reported to date involve mostly lithiumlike ion beams B = s, Si, V, Ca, Ar and A = He, Ne, Ar, M = H2.The formation of (B(z-l)+)** may proceed via two different modes: ( 1 ) The electron being transferred interacts with one of the inner-shell electrons in Bz+ and forms a resonant intermediate state, This is the RTE, and the effective interaction involved here is an electron - electron correlation. ( 2 )Alternatively, an electron is transferred from A to B via an electron Bz+ interaction, while the target core A+ interacts with one of the core electrons in Bz+ and excites it. These combined effects lead to the same intermediate states as in (l), and this is the NTE. These two modes are experimentally distinguished as NTE is dominant at low energy, while RTE cross sections peak in the region where the translational velocity of the active electron in the initial state with respect to Bz+matches the resonance condition (Figs. 3 1 - 33). The main interest in RTE is related to the corresponding electron - ion collision.
Bz+ + e +( B ( Z - l ) + ) * *
...
A(B(Z-l)+)*
+
+y
-(BZ+)* e’
(61)
(62)
which is a dielectronic recombination (DR) process; the projectile electron e is captured to form an intermediate state. Obviously A and M in Eq. (58) provide the “electron beam” required for the process in Eq. (6 1 ) . Data by Tanis et al. ( 198 1,1982,1984)were analyzed (Brandt, 1983a)in terms of the corresponding DR process ( McLaughlin and Hahn, 198 1 ) by foldingthe DR cross section into the target Compton profile. Agreement between the theory and experiments for RTE is found to be excellent. Usually two peaks are observed, the first peak corresponding to the 1s 2s, 2p excitations of Bz+ with an electron capture to 2s or 2p, and the second peak for the capture to all the higher states. They are not resolved in the S and Si cases. In addition to direct cross-section comparisons, the change in the peak heights was examined in an isonuclear sequence; presumably peak ratios are less sensitive to
-
ELECTRON CAPTURE BY SIMPLE IONS
307
E (MeV)
FIG.3 I . Comparison between the measured RTE cross sections for Car'+ions on H,and He. The theoretical predictions are given by the dashed and solid lines.
systematic errors. Table V shows the comparison to be in good agreement between theory and experiment. In particular, we note that for CaI2+the first peak in the experimental data (Tanis et al., 1986)essentially disappeared, as confirmed by the theory (McLaughlin and Hahn, 1986). The NTE cross sections have also been measured (Tanis et al., 1984;Clark et al., 1985) for Si and S beams (Figs. 32 and 33). As noted above, the NTE
I
N
5 9
..
3--
-
2.-
.
TUNL
BNL
N
L
- a
b"
t;
I -
0
20
40
ENERGY ( M e V )
+
FIG.32. RTE and NTE cross sections for Sir'+ He are compared with the preliminary theoretical calculation.
308
Edward Pollack and Yukap Hahn
I He
+ I)
E (MeV) FIG.33. RTE and NTE cross sections for S3+ -I-He are compared with the preliminary theoretical calculation.
309
ELECTRON CAPTURE BY SIMPLE IONS TABLE V THEVARIOUS RESONANCE PEAKRATIOS" Ion
Theory (DR)b
23v'9+
Experiment (RTE)
I .o
1.o
1 1 .O
1.1 0.9
.o
23V20+
23V*'+ ,&ai2+
6
2 5
, Ratio = Peak(II)/Peak(I). The theoretical values are directly from the DR calculation,
cross section peaks just above the threshold and decreases rapidly before the energy reaches the RTE zone. A theoretical analysis of NTE and RTE is being carried out by several groups (Feagin et al., 1984; Reeves et al., 1984; Hahn, 1986; Brandt, 1983a,b). Schematically,
+ b) + (e2+ a) -(el + e2+ b)** + a el + b = Bz+; e, + a = A
(el
(el + e, + b)** = (B(z-l)+)**4( B ( Z - I ) + ) *
(63)
+y
with an amplitude given by
where d = intermediate resonance states,
vj = v12 +
+ + v& vb2
(65)
and D is the electric dipole, electron - photon coupling. It was shown (Hahn, 1986) that VITand Val mediate the RTE and NTE processes, respectively, while the contnbution from the last two terms in Viis small due to the orthogonality between the iand dstates. It should be pointed out that, contrary to the NTE case here, the potential Vb2plays an important role in the direct charge exchange (without simultaneous excitation) when Zb >> Z,, i.e., Bz+ is highly stripped. The distortion effect of Bz+ on e2 is still very important, however. The cross sections for RTE and NTE are given by
2 sUW,(&)Ae JK,
aRTE =
d
Edward Pollack and Yukap Hahn
310
where W, =
eR
I
dqulcA(q2)12= the Compton profile ofA (or M)
and is the resonant DR cross section, averaged over a small energy bin size. Similarly,
,.
where is the Bz+ excitation probability mediated by Val and Pamthe electron (e2)capture probability, as given by the overlap ofthe wave function in A+ and B+. Gerber el al. ( 1972)observed the formationof the doubly excited autoionization state 2p2 ' D in a He+ He collision
+
+
+
He+(1s) He -+ He**(2p2) He+
' \
+ e' 2p) + y
He+(Is)
\
\---+He*(ls
where the ejected electron spectra have been detected. Since A, >> A,, the decay of this intermediate state is predominantly by electron emission, rather than by radiation. Nevertheless, this is very closely related to the RTE (and NTE) procedure. High-precision Auger spectroscopy associated with (4) is potentially a powerful method for carrying out a detailed study of (1) (Itoh et al., 1985).
ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under Grants No. PHY-83-03618 and PHY-85-07736 and the Department of Energy under Grant No. DE-FG02-85 ER53205. We would like to thank many colleaguesat the University ofconnecticut, Orsay, and Saclay, and in particular, Dr. Victor Sidis, Dr. Michel Barat, and Dr. J. Pascale for several helpful discussions. We would also like to thank Dr. R. K. Janev for his useful comments on the manuscript.
REFERENCES Aberth, W., and Lorents, D. C. (1966). Phys. Rev.144, 109. Andersen, N., Vedder, M., Russek, A., and Pollack, E. (1980). Phys. Rev. A 21,782 (1980).
ELECTRON CAPTURE BY SIMPLE IONS
31 1
Anicich, V. G., Laudenslager, J. B., Huntress, Jr., W. T., and Futrell, J. H. (1977). J. Chem. Phys. 67,4340. Barat, M., and Lichten, W. (1972). Phys. Rev. A 6,211. Barat, M., Dhuicq, D., Francois, R., McCarroll, R., Piacentini, R. D., and Salin, A. ( 1 972). J. Phys. B 5, 1343. Bates, D. R. (1960). Proc. R. SOC.Ser. A 257, 22. Bates, D. R., and McCarroll, R. (1958). Proc. R. Soc. Ser. A 245, 175. Bates, D. R., and McCarroll, R.(1962). Adv. Phys. 11,39. Bates, D. R., Johnston, H. C., and Stewart, 1. (1964). Proc. Phys. SOC.84, 5 17. Bayfield, J. E. (1957). In “Atomic Physics” (G. Zu Putlitz, E. W. Weber, and A. Winnacker, eds.), Vol. 4, p. 391. Baylis, W. E. (1969). J. Chem. Phys. 51,2665. Baudon, J., Barat, M., and Abignoli, M. (1970). J. Phys. B3,207. Bauschlicher, Jr., C. W., ONeil, S. V., Preston, R. K., Schaefer, 111, H. F., and Bender, C. R. (1973). J. Chem. Phys. 59, 1286. Belkic, Dz., Gayet, R., and Salin, A. (1979). Phys. Rep. 56,279. Benoit, C., Kubach, C., Sidis, V., Pommier, J., and Barat, M. (1977). J. Phys. B 10, 1661. Birely, J. H., and McNeal, R. J. (1972). Phys. Rev. A 5, 692. Bischof, G., Reinig, P., and Linder, F. (1983). Int. Conj Phys. Electron. At. Collisions. 13th, Berlin p. 628. Brandt, D. (1983a).Phys. Rpv.A 27, 1314. Brandt, D. (1983b). Nucl. Instrum. Methods 214,93. Bransden, B. H., and Janev, R. K. ( 1 983). Adv. At. Mol. Phys. 19, 1. Bray, A. V. ( 1 975). Ph.D. thesis, University of Connecticut. Bray, A. V., Newman, D. S., Drozd, D. F., and Pollack, E. (1975). Int. Conf Phys. Electron. At. Collisions, 9th, p. 627. Bray, A. V., Newman, D. S., and Pollack, E. (1977). Phys. Rev. A 15,2261. Brenot, J. C., Pommier, J., Dhuicq, D., and Barat, M. (1975a). J. Phys. B8,448. Brenot, J. C., Dhuicq, D., Gauyacq, J. P., Pommier, J., Sidis, V., Barat, M., and Pollack, E. (1975b). Phys. Rev. A 11, 1245. Brenot, J. C., Dhuicq, D., Gauyacq, J. P., Pommier, J., Sidis, V., Barat, M., and Pollack, E. (197%). Phys. Rev. A 11, 1933. Browning, R., Latimer, C. J., and Gilbody, H. 8. (1 969). J. Phys. B 2, 534. Clark, M., Brandt, D., Swenson, J. K., and Shafroth, S. M. (1985). Phys. Rev. Lett. 54, 544. Coplan, M. A., and Ogilvie, K. W. (1974). J. Chem. Phys. 61,2010. Coulson, C. A., and Zalewski, K. (1962). Proc. R. SOC.Ser. A 268,437. Crothers, D. S. F. (1981). Adv. At. Mol. Phys. 17,55. Demkov, Yu. N. (1964). Sov. Phys. JETP18, 138. Dowek, D., Dhuicq, D., Pommier, J., Tuan, Ngoc Vu, Sidis, V., and Barat, M. (1981). Phys. Rev. A 24,2445. Dowek, D., Dhuicq, D., Sidis, V., and Barat, M. (1982). Phys. Rev. A 26, 746. Dowek, D., Dhuicq, D., and Barat, M. (1983). Phys. Rev. A 28,2838. Dunn, G. H., Geballe, R., and Pretzer, D. (1962). Phys. Rev. 128, 2200. Ellison, W. 0. (1963a). J. Am. Chem. SOC.85,3540. Ellison, W. 0. (1963b). J. Am. Chem. Soc. 85,3544. Ellison, W. O., and Borowitz, S. (1964). At. Collision Proc. (M. R. C. McDowell, ed.).North Holland, Amsterdam. Eriksen, F. J., Jaecks, D. H., deRijk, W., and Macek, J. (1976a). Phys. Rev. A 14, 1 19. Eriksen, F. J., and Jaecks, D. H. (1976b). Phys. Rev. Lett. 36, 1491. Eriksen, F. J., and Jaecks, D. H. (1978). Phys. Rev. A 17, 1296.
312
Edward Pollack and Yukap Hahn
Everhart, E. (1963). Phys. Rev. 132,2083. Fano, U. (1957). Rev. Mod. Phys. 29,74. Fano, U., and Lichten, W. (1965). Phys. Rev. Lett. 14, 621. Fano, U., and Macek, J. (1973). Rev. Mod. Phys. 45,53. Feagin, J. M. et al. (1984). J. Phys. B 17, 1057. Fernandez, S. M., Eriksen, F. J., and Pollack, E. (1971). Phys. Rev. Lett. 27, 230. Fernandez, S . M., Eriksen, F. J., Bray, A. V., and Pollack, E. (1975). Phys. Rev. A 12, 1252. Gau, J. N., and Macek, J. (1975). Phys. Rev. A 12, 1760. Gerber, G., Morgenstern, R., and Niehaus, A. (1972). J. Phys. B 5,963. Gillen, K. T., and Kleyn, A. W. ( 1 980). Chem. Phys. Lett. 72, 509. Goldberger, A. L. (1984). Ph.D. thesis, University of Connecticut. Goldberger, A. L., Newman, D. S., Vedder, M., and Pollack, E. (1977). Int. Conf: Phys. Electron. At. Collisions, 10th. Paris p. 862. Goldberger, A. L., Jaecks, D. H., Natarajian, M., and Fornari, L. (1984). Phys. Rev. A 29,77. Covers, T. R. (1975). Chem. Phys. Lett. 9,285. Gurnee, E. R., and Magee, J. L. (1957). J. Chem. Phys. 26, 1237. Gusev, V. A., Polyakova, G. N., Erko, V. F., Fogel, Ya. M., and Zats, A. V. In Sixth International Conference on the Physics ofElectronic and Atomic Collisions, Abstracts of Papers, Boston, 1969 (MIT, Cambridge Mass., 1969), p. 809. Gustafsson, E., and Lindholm, E. (1960). Ark. Fys. 18,219. Guyon, P. M., Baer, T., Ferreira, L. F. A., Nenner, I., Tabche-FouhailC, A., Botter, R., and Govers, T. R. (1978). J. Phys. B 11, L141. Hahn, Y. (1968). Phys. Lett. B 30,595. Hahn, Y. (1970). Phys. Rev. C 1, 12. Hahn, Y. (1971). Ann. Phys. 67,389. Hahn, Y. (1978). J. Phys. B 11, 3221. Hahn, Y. ( 1 986). Phys. Rev. A (to be published). Hanssen, J., McCarroU, R., and Valiron, P. (1979). J. Phys. B 12,899. Hams, H. H., Crowley, M. G., and Leventhal, J. J. (1974). Chem. Phys. Lett. 29,540. Hasted, J. B. (1979). Adv. AI. Mol. Phys. 15,205. Hodge, Jr., W. L., Goldberger,A. L., Vedder, M., and Pollack, E. (1977). Phys. Rev. A 16,2360. Hopper, D. G. (1978). In!.J. Quantum Chem. Quantum Chem. Symp. 12,305. Hopper, D. G. (1980a). J. Chem. Phys. 73,3289. Hopper, D. G. (1980b). J. Chem. Phys. 73,4528. Heckman, V., Martin, S. J., Jakacky, Jr., J., and Pollack, E. (1984). Phys. Rev. A 30,2261. Ioup, J., and Russek, A. (1 973). Phys. Rev. A 8, 2898. Isler, R. C., and Nathan, R. D. (1972). Phys. Rev. A 6, 1036. Itoh, A., Zouros, T. J. M., Schneider, D., Stettner, U., Zeitz, W., and Stolterfoht, N. (1985). J. Phys. B 18,4581. Jaecks, D. H., Yenen, O., Natarajan, M., and Mueller, D. (1983). Phys. Rev. Lett. 50, 825. Jakacky, Jr., J., Pollack, E., Snyder, R., and Russek, A. (1985). Phys. Rev. ,4 31,2149. Janev, R. K. (1976). Adv. At. Mol. Phys. 12, 1. Johnsen, R., Chen, A., and Biondi, M. A. (1980). J. Chem. Phys. 72,3085. Jones, E. G., Wu, R. L. C., Hughes, B. M., Tiernan, T. O., and Hopper, D. G. (1980). J. Chem. Phys. 73,563 1. Jones, P. R., Costigan, P., and Van Dyk, G. (1963). Phys. Rev. 129,211. Kessel, Q.C., Pollack, E., and Smith, W. W. (1977). In “Collisional Spectroscopy(R. G. Cooks, ed.), Ch. 2. Plenum, New York. Kimura, M. (1985). Phys. Rev. A 802.
ELECTRON CAPTURE BY SIMPLE IONS
313
Kirchner, N. J., O’Keefe, A., Gilbert, J. R., and Bowers, M. T. (1984).Phys. Rev. Lett. 52,26. Knize, R. J., Lundeen, S. R., and Pipkin, F. M. ( 1984).Phys. Rev. A 29, 1 1 14. Koopman, D.W.(1968).Phys. Rev. 166,57. Koopman, D.W.(1969).Phys. Rev. 178, 161. Krupenie, P. H. (1972).J. Chem. ReJ Data 1,423. Kubach, C., Courbin-Gaussorgues,C., and Sidis, V. (1985).Chem. Phys. Left. 119,523. Lee, A. R., Williams, D. G., and Butcher, E. C. (1985).Phys. Lett. A 107, 218. Lipeles, M. (1969).J. Chem. Phys. 51, 1252. Locht, R.(1977).Chem.Phys. 22, 13. Lockwood, G. J., and Everhart, E. (1962).Phys. Rev. 125,567. Lockwood, G. J., Helbig, H. F., and Everhart, E. (1963). Phys. Rev. 2078. Lorents, D. C., and Aberth, W. (1965).Phys. Rev. A 139, 1017. Lorents, D.C., Aberth, W., and Hesterman, W. (1966).Phys. Rev. Letf. 17,849. McCarroll, R. (1982).In “Atomic and Molecular Collision Theory” (F. A. Gianturco, ed.), p. 165. Plenum, New York. McLaughlin, D., and Hahn, Y. (1981).Phys. Lett. A 88, 394. McLaughlin, D., and Hahn, Y. (1985).Phys. Left.A 112,389. Macek, J. (1984).In “Electronic and Atomic Collisions” (J. Eichler, I. V. Hertel, and N. Stolterfoht, eds.), p. 317. Elsivier, Amsterdam. Macek, J., and Jaecks, D. H. (1971).Phys. Rev. A 17,2288. Marchi, R. P., and Smith, F. T. (1965).Phys. Rev. A 139, 1025. Masnou-Seeuws, F. (1982).J. Phys. B 15,883. Mauclaire, G., Derai, R., Fenistein, S., Marx, R., and Johnsen, R. ( 1 979). J. Chem. Phys. 70, 4023. Moore, Jr., J. H., and Doering, J. P. (1969).Phys. Rev. 177,218. Moran, T. F., and Conrads, R. J. (1973).J. Chem. Phys. 58, 3793. Morgan, T. J., Geddes, J., and Gilbody, H. B. (1 973). J. Phys. B 6,2I 18. Nagy, S. W.,Fernandez, S. M., and Pollack, E. (1971). Phys. Rev.A 3,280. Nikitin, E. E.(1970).Adv. Quanfum Chem. 5, 135. Nikitin, E. E., and Smirnov, B. M. (1979).USPEKHI 21,95. Okuda, M., and Jonathan, N. (1974).J. Electron Spectrosc. Relat. Phenom. 3, 19. Olson, R. E., and Mueller, C. R. (1967).J. Chem. Phys. 46,3810. Park, J. T. (1983).Adv. A f . Mol. Phys. 19,67. Pascale, J. (1985).In “Spec. Line Shapes” (F. Rostas, ed.),vol. 3,p. 564. Peach, G. (1982).Comment. Af. Mol. Phys. 11, 101. Peterson, R. S., Vedder, M., and Pollack, E. (1979).Abstr. Int. Con$ Electron. At. Collisions, 11th. Kyoto p. 826. Pfeifer, S. J., and Garcia, J. D. ( I 98 I). Phys. Rev. A 23, 2267. Piacentini, R. D.,and Salin, A. (1978).J. Phys. B 11,L323. Preston, R. K.,Thompson, D. L., and McLaughlin, D. R. (1978).J. Chem. Phys. 68, 13. Raith, W.(1976).Adv. At. Mol. Phys. 12,281. Rapp, D., and Francis, W. E. ( I 962).J. Chem. Phys. 137,2631. Reeves, T.,Merzbacher, E., and Feagin, J. (1984).Bull. Am. Phys. SOC.29,792. Rosen, N.,and Zener, C. (1982).Phys. Rev. 40, 502. Rossi, F., and Pascale, J. (1985). To be published. Rudd, M. E., DuBois, R. D., Toburen, L. H., Ratcliffe,C. A., andGoffe, T. V. (1983).Phys. Rev. A 28, 3244. Shah, M. B., Geddes, J., and Gilbody, H. B. (1980).J. Phys. B 13,4049. Shakeshaft, R., and Spruch, L. (1979).Rev. Mod. Phys. 51,369.
314
Edward Pollack and Yukap Hahn
Sharp, T. E. (1971). At. Data 2, 119. Sidis, V., and Dowek, D. ( 1 984). I n “Electronic and Atomic Collisions” (J. Eichler, 1. V. Hertel, and N. Stolterfoht, eds.), p. 403. Elsevier, Amsterdam. Sidis, V., Brenot, J. C., Pommier, J., Barat, M., Bernardini, O., Lorentz, D. C., and Smith, F. T. (1977). J. Phys. B 10,2431. Sigmund, P. (1981). J. Phys. B. 14, L321. Smith, F. T. (1969). In “Atomic Physics” (B. Bederson, V. W. Cohen, and F. M. J. Pichanik, eds.), p. 353. Plenum, New York. Smith, F. T., Marchi, R. P., and Dedrick, K. D. (1966). Phys. Rev. 150,79. Smith, F. T., Reischmann, H. H., and Young, R. A. (1970). Phys. Rev. A 2, 379. Snyder, R., and Russek, A. (1982). Phys. Rev. A 26, 1931. Stebbings, R. F., Smith, A. C. H., and Ehrhardt, H. (1963). J. Chem. Phys. 39,968. Stedford, J. B. H., and Hasted, J. B. (1954). Proc. R. SOC.London Ser. A 227,466. Tank, J. A., Shafroth, S. M., Willis, J. E., Clark, M., Swenson,J., Strait, E. N., and Mowat, J. R. (1981). Phys. Rev. Lett. 47, 828. Tank, J. A., Bernstein, E. M., Graham, W. G., Clark, M., Shafroth, S. M., Johnson, B. M., Jones, K. W., and Meron M. (1982). Phys. Rev. Lett. 49, 1325. Tanis, J. A,, Bernstein, E. M., Graham, W. G., Stockli, M. P., Clark, M., McFarland, R. H., Morgan, T. J., Berkner, K. H., Schlachter,A. S., and Steam, J. M. (1 984). Phys. Rev.Left. 53, 255 1. Thomas, L. H. (1 927). Proc. R. SOC.114,56 1 . Tully, J. C., and Preston, R. K. (1971). J. Chem. Phys. 55, 562. Vainshtein, L., Presnyakov, L., and Sobel’man, I. (1982). Zh. Eksp. Thor. Fiz. 4 3 , s 18. Van Brunt, R. J., and Zare, R. N. (1968). J. Chem. Phys. 48,4304. Van Zyl, B., Jaecks, D., Pretzer, D., and Geballe, R. (1967). Phys. Rev. 158, 29. Valiron, P., Gayet, R., McCarroll, R., Masnow-Seeuws, F., and Phillippe, M. (1979). J. Phys. B 12,53. Vassilev, G., Rahmat, R., Slevin, J., and Baudon, J. (1975). Phys. Rev. Lett. 34,447. Vedder, M., Hayden, H., and Pollack, E. (1981). Phys. Rev. A 23,2933. Williams, I. D., Geddes, J., and Gilbody, H. B. (1982). J. Phys. B 15, 1377. Wu, R. L. C., and Hopper, D. G. (1981). Chem. Phys. 57, 385. Yenen, O., Jaecks, D. H., and Macek, J. (1984). Phys. Rev. A 30,597. Young, R. A., Stebbings, R. F., and McGowan, J. W. (1968). Phys. Rev. 171,85. Zener, C. ( 1 932). Proc. R. SOC.Ser A 137,696.
ADVANCES IN ATOMIC AND MOLECULAR PHYSICS,VOL. 22
RELATIVISTIC HEA VY-IONATOM COLLISIONS R. ANHOLT Department of Physics Stanford University Stanford, California 94305
HARVEY GOULD Materials and Molecular Research Division Lawrence Berkeley Laboratory university of California Berkeley, California 94 720
I. Introduction The coupling of Lawrence Berkeley Laboratory’sheavy-ion linear accelerator (Super-HILAC)with its relativistic synchrotron (Bevatron) to produce the Bevalac has opened up new frontiers in atomic-collision and atomicstructure physics at relativistic velocities. This upgrading of 1950s and 1960s accelerator technology has produced an accelerator capable of accelerating uranium ions to energies of 1 GeV/amu (total energy 238 GeV), which makes possible, for the first time, the study of collisions and interactions of the heaviest few-electron ions. From the point of view of atomic-collisionphysics, the high relative velocities obtainable with relativistic projectiles is of foremost interest. Many fundamentalprocesses in atomic-collision physics scale with the ratio of the ion velocity to the velocity at the active electron. With relativistic projectiles one can obtain velocities that are high with respect to the fastest electron in the heaviest atom, the uranium Kelectron. The ions can be stripped, so that one can study one- or two-electron, high-2 ions where both the atomicstructure problem and the treatment of scattering processes are greatly simplified. Most theories of scattering processes are greatly simplified also. While scatteringtheories at low relative velocities are becoming increasingly refined only by employing numerically intensive coupled-channel calcula-
-
315 Copyright Q 1986 by Academic Press,Inc. AN nghts of reproduction in any form resewed.
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R. Anholt and Harvey Gould
tions including large numbers of basis states, at relativisticvelocities one can still make use of high-velocitytheories such as the first Born approximation (Bethe, 1930, 1932) for ionization and excitation processes and the secondBorn (Drisko, 1955), eikonal (Eichler, 1985), strong potential Born (Macek and Alston, 1982), or impulse approximations (Briggs, 1977; Amundsen and Jakubassa, 1980) for capture processes. Several effects have been closely studied with relativistic heavy ions. For example, the binding and polarization effectson ionization and excitation processes at intermediate velocities (Basbas et al., 1978), target atomic screeningand antiscreeningeffects on ionization and excitation ( McGuire et al., 198l), and the importance of electron transfer from excited target states into excited projectile states (Meyerhof et al., 1985) have been examined. What is most attractive, however, is that the atomic-structure and collision physics at high relative velocities are sufficiently simple that one can begin to address important problems long thought to be intractable. One example of this is the complex interactions occurring inside solid targets (Betz, 1972). The influence of excited states of the projectile on producing higher charge states of ions in solid-targetstripping foils over gas targets can be studied experimentally in a regime that also can be approached theoretically. Discouraged by the difficultiesposed by solid-targetresearch, in many atomic-collision-physicsstudies, experiments are done exclusively with gas targets, thus limiting experimentalpossibilities due to the paucity of monoatomic gases in nature. However, in relativistic heavy-ion atomic physics one can confront these problems head on, solve them, and derive rules and formulas that are possibly applicable to both relativistic and nonrelativistic projectiles in gas and solid targets. From the point of view of atomic-structurephysics, the study of one- and two-electron uranium and other high-2 ions allows the examination of higher-order corrections in Za (ais the fine-structure constant) to the quantum-electrodynamic (QED) self-energy contribution to the binding energy and to the anomalous magnetic moment of the electron bound in a Coulomb field. Some of these cannot be measured using low-2 ions. For example, in few-electron uranium, the self-energy contribution to the binding energy comes predominantlyfrom terms which are of high order in Zcr. These terms make almost no contribution in low-Z Lamb-shift measurements. Consequently, even a modest-precision Lamb-shift measurement in high-Z ions like uranium can provide a powerful test of QED in the high-Z, strong-field limit. In this article, we will depart from the structure normally used in scientific reviews, and will merge the theoretical discussion with the discussion of the experiments and results. We retain a discussion of the experimental methods in Section 11,and discuss ionization processes (Section 111),electron-capture
RELATIVISTIC HEAVY-ION -ATOM COLLISIONS
317
processes (Section IV), collisions in solid targets (Section V), x-ray continuum production in Section VI, ultrarelativistic collisions in Section VII, atomic-structure experiments in Section VIII, and the conclusions are in Section IX. We define at the outset the projectileatomic number Z,, the target atomic number Z,, the ion velocity v, and p = v/c, where c is the speed of light. The ion kinetic energy is equal to ( y - 1)Mc2,where M is the nuclear mass and y = (1 - p2)-1/2. The K-shell electron binding energy is EK,a = e2/hc= 1/137 is the fine-structure constant, and, in general, in theories of ionization and capture where Z , and Z, are not specified, we denote by Z, the perturbing charge and by 2 the charge to which the active electron is initially attached. v,is the initial velocity ofthe bound Kelectron and is equal to Zin atomic units.
11. Experiments At the end of 1985 only first-generation experiments using relativistic heavy ions have been completed (Crawford, 1979; Salamon et al., 1981; Ahlen and Tarle, 1983; Waddington et al., 1983; GouId et al., 1984, 1985; Anholt et al., l977,1984a,b, 1985,1986;Heckman et al., 1984;Meyerhofet al., 1985; Thieberger et a/., 1985). The methods employed to study atomic collisionsof very heavy ions at relativisticvelocities have been limited by the 1O5 to l O7 ions/pulse (one pulse every 4 to 6 seconds)which can be delivered by the only currently available relativistic heavy-ion accelerator, the Bevalac. The combination of a low intensity and a fairly large emittance of 30n mm-mrad has so far discouraged attempts to measure the impact-parameter or scattering-angledependence of inner-shell ionization or capture. Channeling experiments may be possible. X-ray measurements with very thin solid targets or gas targets is possible, but requires techniques for suppressing background-related target electron bremsstrahlung, secondary electrons (delta rays), and nuclear disintegration. On the other hand, the use of relativisticheavy ions permits several experimental simplifications. For example, the main source of error in x-ray-producing cross-section measurements at low velocities is the beam intensity normalization, determined either by beam current integration or Rutherford scattering normalization(Anholt, 1985b).With relativistic heavy ions, the cross sections are often very large, so one can count every incident ion with = 100%efficiency, using surface barrier or scintillator detectors. The vacuum requirements for relativistic (heavy) ions are not stringent, and if
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one is not interested in the projectile charge state (or has already analyzed it) one can pass the beam through a thin window and count the particles in air. In this section we discuss two kinds of first-generationexperiments: x-ray cross-section measurements and projectile charge-changing cross-section measurements. A. X-RAYCROSS-SECTION MEASUREMENTS In measuring x-ray cross sections using relativistic heavy ions, which can pass through several grams/cm2of matter, an active collimation system has been used (Anholt et al., 1984b). The relativistic beam is arranged to pass through a = 30 cm2thin metal foil target placed between a pair of about 1 cm2 active-area surface-barrier or particle scintillator detectors. The number of projectile ions passing through both particle detectors is counted in coincidence with detected x rays coming from the target. The target is much larger than the particle detectorsto ensure that an ion passing through both particle detectors must also pass through the target. This method has been used by Anholt et al. ( 1984b)for relativistic heavy ions and by Bak et al. (1 983, 1985) for a mixed beam of protons, pions, and electrons (where the particle detectors also identified the particle). At high particle intensities surface barrier detectors and scintillatorphotomultipliers begin to saturate, so ionization chambers have been used. For absolute measurements they are calibrated against single-particle detectors at lower particle fluxes or against known cross sections such as that for the 12C(p,pn)11C nuclear reaction (Anholt et al., 1976).The conversion of x-ray counts to x-ray-production cross sections is discussed in Hoffmann et al. (1979), Anholt et al. (1976, 1984b),and Bak et al. (1985). Figure 1 illustrates the kinds of x-ray emission processes observed at relativistic velocities. Prominent features in the x-ray spectrum produced by uranium colliding with a uranium target are the K x rays coming from the target uranium atom and K x rays coming from the projectile uranium ion. The target x rays produce a sharp peak in the spectra since the target atom is at rest and does not recoil significantlywhen the projectile ion passes by and excites a Kvacancy. The projectile x rays, viewed perpendicular to the beam, are shifted due to the transverse Doppler shift, and are Doppler broadened due to the range of angles subtended by the x-ray detector. The projectile Ka x rays are clearly seen, but the less intense Kp x rays are not seen above the continuum x-ray background. Also seen in the U Be spectrum (Fig. 1) is a broad peak near 275 keV, due to the radiative capture oftarget electronsinto the Kshell ofthe uranium projectile. Radiative electron capture is the inverse of the photoelectric ef-
+
RELATIVISTIC HEAVY-ION - ATOM COLLISIONS I o5
I
I
319 I
I
422 MeV/amu
u+u
Pro] Ka
85O
I
-
lo4
t
4
2 I
\ Lo
5 n
-
103
z 0
+
w 0 v)
Ln 0
g
102
0
10' 100
200
300
E,
(keV)
400
+
FIG. 1. Double-differentialx-ray production cross sections seen in 422 MeV/amu U Be and U U collisions at ff = 85".The curves show calculated contributionsdue to secondaryand primary bremsstrahelectron bremsstrahlung (---), radiative electron capture (---), curves show the total intensity. (From Anholt ef al., 1986.) lung (- -). The (-)
+
fect. An electron, at rest in the laboratory frame, has a kinetic energy of 230 keV in the rest frame of the 422 MeV/amu projectile. When the electron is captured into the projectile Kshell, a photon is emitted whose energy is equal to the electron kinetic energy plus the K-shell binding energy. In the laboratory frame the photon is Doppler shifted. Viewed perpendicular to the beam direction, the photon energy in the laboratory is E i = EJy, where Ex is the photon energy in the rest frame of the projectile and y = (1 - p2)-1/2 is the relativistic energy factor with /3 = v/c. A smaller peak due to the radiative capture into the Land higher projectile shells is also present, but is difficult to separate from the continuum x-ray background. Primary- and secondary-electron bremsstrahlung account for most of the background. Primary bremsstrahlung is produced by the target electron scattering from the projectile nucleus (Yamadera et al., 1981; Anholt et al., 1986). It is a fundamental process that cannot be removed unless one removes all of the target electrons. For a 422 MeV/amu projectile, the highest-
R. Anholt and Harvey Gould
320
energy photon which one can produce is one in which the electron is left at rest after bremsstrahlung, so that E,, = ( y - l)rnc2 = 230 keV. In the projectile frame, this photon energy is Doppler shifted and extends from 590 keV in the forward direction down to 92 keV in the backward direction and is 164 keV at 90". Primary bremsstrahlung is readily seen in U Be collisions (Fig. 1). For high-Z targets, bremsstrahlung from secondary electrons produced by Rutherford scattering is the dominant process for producing radiation (Folkmann et al., 1973; Ishii et al., 1977). In this two-step process target electrons are first ejected by Rutherford scattering on the projectile, then emit bremsstrahlung in collisions with other target nuclei, or if they leave the target, with other nearby matter. Because the bremsstrahlung cross sections scale as 2: , secondary bremsstrahlung is the dominant source of continuum x rays in high-Z targets. A target electron, scattered by a 422 MeV/amu ion will have an energy of 1.1 MeV if scattered forward and zero if scattered perpendicular to the projectile. Bremsstrahlung of secondary electrons thus produces a photon continuum to a maximum energy in the laboratory frame of 1.1 MeV. To first order, relativistic ion-induced target K-shell ionization cross sections are the Kx-ray-production cross sectionsdivided by the K-shell fluorescence yield. This neglects two background effects, secondary excitation by ionized target electrons (delta rays) or photoelectric excitation by primary and secondary electron bremsstrahlung radiation, and target K-electron capture. The apparent cross sections for ion-induced K-shell ionization in thick targets are as much as 30Yo larger than for thin targets due to background effects. Their subtraction has been discussed by Jarvis et al. ( 1972) and Anholt et al. (1 976,1984b).These effectscan be largely avoided by using very thin targets. The other background effect is the capture of target K electrons by the nearly bare projectiles (Gray et al., 1976).The K-capture contribution to the total target K-vacancy yield is given by
+
YKC = I,'dT' [ ( a , +
aKH)yo(n
+ a,,)Y,(W + 0 K H Y 2 ( T ' ) +
-
1
(1) where Y,,(T ) is the fractional yield of projectiles carrying n electrons, a, is the cross section for the capture of a target Kelectron into the Kshell ofa bare projectile, a, is the cross section for capture of a target K electron into any higher shell of the projectile, and Tis the target thickness (in units of atoms per unit area). For targets of equilibrium thickness, the contribution to the total K-vacancy production cross section is given by (Anholt et al., 1985a) +(+OM
a K C = (oKK
+
aKH)Fo
*
*
+ (+am+ 0m)FI + amF2 + . * *
(2)
RELATIVISTIC HEAW-ION- ATOM COLLISIONS
32 1
where F, is the equilibrium fraction of projectiles carrying n electrons. The evaluation of the capture contribution requires knowledge of the capture cross sections 0, and om and the projectile charge-state fractions. The capture contributions to the total K-vacancy production cross sections can generally be neglected for highly asymmetric collisions. They are important for some measurements with relativistic heavy ions, e.g., 82 to 200 MeV/ amu Xe ions incident on Ag(2 = 47) and heavier targets (Anholt et al., 1984b, 1985). Projectile K x-ray production cross sections, while less straightforward to interpret, provide information about the states of ions in matter. They are discussed in Section V.B. B. CHARGE-CHANGING CROSS-SECTION MEASUREMENTS
Charge-changingcross sectionscan be studied by passing a beam of known charge-statecomposition through a target and measuringthe fraction of ions that have either captured or lost an electron. Figure 2 showsthe experimental arrangement used at the Lawrence Berkeley Laboratory’s Bevalac (Greiner et al., 1974; Gould et d.,1984, 1985). A collimated beam of a single charge state passes through a chamber containing targets that can be remotely inserted into the beam. The resulting charge states are analyzed by a pair of dipole magnets and are detected by a 50-cm-long position-sensitiveproportional counter (Borkowski and Kopp, 1975). Between the targets and the dipole magnets the beam passes through a pair of focusing quadrupole magnets. After the analyzing magnets, the beam passes through a thin window at the end of the vacuum chamber. The detector sits in air. A chargeions through 7.1 state distribution produced by passing 962 MeV/amu mg/cm2 Mylar is shown in Fig. 3. Charge-changing cross sections may be measured by the “thick-target” method (Datz et al., 1970; Betz, 1972) or the “thin-target” method (Nikolaev, 1965; Datz et d , 1970). In both cases one passes a beam of known charge states through targets of known thickness and measures the resulting charge-state fractions. In the thin-target method only the linear part of the charge-statefraction versus target thickness curve is used. In the thick-target method the target thickness is increased as far as possible often until the equilibrium thickness charge-state distribution is approached. The thick-target method is useful in measuring cross sections involving charge states which are not present in the incident beam. Since many charge states may be studied with a single incident charge state, the technique is useful in survey experiments, where a large number of cross sections can be obtained without retuning the accelerator. It is also used in experimentswith
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
323
80
I FIG.3. Charge states of incident 960 MeV/amu U6*+ after passing through 7.1 mglcm2 Mylar. The spatial separation between adjacent peaks is approximately 1 cm. The horizontal focus of the beam and position resolution of the position-sensitivedetector are about 0.2 cm. (From Crawford et al., 1983.)
limited beam intensity (Alonso and Gould, 1982; Gould ec al., 1984). To extract cross sections, one uses a least-squares fit of the calculated cross sections to the thickness-dependent charge fractions (Datz et al., 1970; Betz, 1972). Since many unknown cross sections are to be determined, one must choose the target thicknesses such that each charge state ofinterest is present for as many different target thicknesses as possible. If multiple capture and ionization needs to be included, finding the “best fit’ can be tedious. In the “thin-target’’method (Nikolaev, 1965; Datz et al., 1970) the targets are sufficientlythin so that in principle the yield increases linearly with target thickness. The ionization or capture cross section is just the yield Y, divided by the target thickness. In practice finite-thicknesstargets are used, so one fits YJT to a straight line to obtain the T = 0 intercept. This method is simple and can give unambiguous cross sections (Meyerhof el al., 1985). A disad-
324
R. Anholt and Harvey Gould
vantage of the “thin-target” method is that it requires collecting data using target thicknesses for which only a small percentage of the ions undergo a charge-changing collision. Consequently the counting statistics may be poor. Errors can occur both because of the low absolute count rates, limited by beam intensity, and because the relatively large count rate for the incident charge state can cause pileup and dead time in the detector, leading to measurement error. The thin-target charge-state measurements generally provide a much cleaner way of studying ionization processes than x-ray measurements of target ionization. In target ionization experiments the projectileboth ionizes the target and captures target electrons. In charge-state measurements of projectile ionization the target is neutral and there is no possibility for it to capture projectileelectrons. One also avoidsthe secondaryprocesses(Anholt et al., 1986) and fluorescence-yielduncertainties. However, in the theory of projectile ionization, one must account for the screening of the perturbing target nucleus by the target electrons.
111. Ionization Processes A. OVERVIEW Inner-shell ionization with relativisticheavy ions can be classified according to three types: target K-vacancy production, projectile ionization in low-Zions, and projectile ionization in high-Zions (Anholt et al., 1985). It is important to distinguish the physical effects present for each type, which are summarized in Table I. Since relativistic heavy ions travel at velocitiesthat are high with respect to the electrons, a logical starting point in the analysis of electron ionization is the first (plane-wave)Born approximation (PWBA), which has been formulated by Bethe (1 930, 1932) and Marller (1932), and has been widely applied to analyze inner-shell ionization at nonrelativistic velocities (Merzbacher and Lewis, 1958; Basbas et al., 1973, 1978). These theories generally use hydrogenic electron wave functions. For target K-vacancy production, one uses effective charges (Z, is replaced by Z* = 2, - 0.3 for the K-shell electron), and one replaces the hydrogenic K-electron binding energy i(Z*)2by the experimental value EK = OK; (Z*)2for neutral atoms. Nonrelativistic hydrogenic wave functions and charges Z* are valid for low-Z projectile ionization, but for high-Z projectile ionization, Dirac electronic wave functions and binding energies EK [=( 1 - s)mc2,where s2 = 1 - a2Z2and a = 1 / 1371 should be used.
325
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS TABLE I EFFECTSON K-SHELL IONIZATION On projectile ionization On target ionization
Effect First-order theory
PWBA
Electronic wave function
Low z
High Z
PWBA
PWBA
yo e-2.r. use z* = z, - 0.3
vo- e-a; z = z,
Dirac; z = z, duo - r'-'e-"
Binding energy Perturbing nucleus screening
E~ = eKjz*2
EK = iZ2
EK = (1 - s)mc2, s2 = 1 - a 2 2 2
Bare projectiles, no screening
Cross-section reductions
Z:
Relativistic velocity j3 a interaction
Calculate with Z*, OK
Negligible
Calculate with Dirac wave functions
Wave-function distortion
Polarization plus binding effect
V/VK>
Experimental effects
Target Kelectron capture
-
-
10, no correction
-B
Z:
+ Z,
-
1 to 2, corrections needed
V/VK
An effect not present for target K-shell ionization by nearly bare projectiles is the screening of the perturbing nuclear charge by attached projectile electrons. The cross sections for ionizing l o w 2 projectiles by neutral target atoms are significantly reduced by the screening of the perturbing target nucleus by the target electrons (McGuireet al., 1981). The screening has less effect on the cross sections for ionizing high-Z relativisticprojectiles,but due to contributions to projectile ionization by the perturbations of the 2,separate target electrons, the ionization cross sections vary as 2: 2,. In the nonrelativistic PWBA, the interaction Hamiltonian consists ofjust the Coulomb interaction between the perturbing nucleus and electron. For relativistic ions, the PWBA also includes the current - current interaction between the perturbing nucleus and active electron (Mraller, 1932; Davidovic et al., 1978; Anholt, 1979). Including this interaction gives additional incoherent contributions to inner-shell ionization and excitation, called the transverseexcitation cross sectionsto distinguishthem from the longitudinal contributions coming mainly from the static Coulomb interaction. The transverse ionization cross sections have been calculated for target K-vacancy production using hydrogenic wave functions following similar
+
326
R. Anholt and Harvey Gould
methods used for nonrelativistic projectiles (Merzbacher and Lewis, 1958). At 5 GeV/amu, the transverse and longitudinal ionization cross sections are about equal, but the transverse cross sections continue to increase as In y2 at high energies where the longitudinal ones approach a constant. The transverse contributions are also present for projectile ionization. For low-2 projectiles, target screeningreduces their magnitude considerably (Gillespie et al., 1978; Anholt, 1985a). The PWBA is based on first-order perturbation theory, and tends to break down when the perturbing potential becomes very large. The nonlinear response of the amplitude for ionization to an increase of the perturbing charge is viewed as the effect of the distortion of the electronic wave functions (Basbas et al., 1978). This distortion reduces the ionization cross sections at low ion velocities, but increases them at high velocities. At low velocities, the electrons form diatomic molecular orbitals around the projectile and target nuclei, thereby becoming more tightly bound and more difficult to excite; so smaller ionization cross sections are seen (Basbas et al., 1973;Anholt and Meyerhof, 1977).At high ion velocities,the polarization of the electron clouds decreases the electron- perturbing-nucleus distance, thereby increasing the strength of the perturbing potential, leading to higher ionization cross sections (Basbas et al., 197la,b, 1978). These distortion effects affect high-2 projectile ionization and target K-vacancy production by relativistic projectiles. However, low-2 projectile ionization is less affected because the relative velocity v/vKis very high. Finally, K-electron capture contributions [Eqs. (1) and (2)] to total K-vacancy production cross sections are only present when target K-vacancy production is studied. Fully occupied target atoms cannot capture projectile electrons efficiently: so projectile ionization is relatively unaffected by capture considerations. Section II1,B presents the formulation of the PWBA for ionization by relativistic projectiles, including transverse excitation. Sections C and D discuss the evaluation of the PWBA cross sections numerically and in the dipole approximation. The target-nucleus screening effect on projectile ionization is discussed in Section E, and wave-function distortion effects are discussed in Section F.
B. THEPLANE-WAVE BORNAPPROXIMATION The theory of target-electronionization by incident ions is the basis of our knowledge of electronic stopping powers (Fano, 1963; Bethe and Ashkin, 1953), which are crucial to the design of apparatus and the interpretation of
RELATIVISTIC HEAW-ION - ATOM COLLISIONS
327
measurements in nuclear and high-energy physics. The theory was formulated by Mraller ( 1932),Bethe ( 1930,1932),and Bethe and Fermi ( 1932).The dipole approximation was used to calculate stopping powers (see, e.g., Fano, 1963). In the Lorentz gauge, the interaction between the perturbing nucleus and electron is given by (Moiseiwitsch, 1980, 1985; Amundsen and Aashamar, 1981):
H'
= -eV(r,t)
+ IXY
A(r,t)
(3)
where the Coulomb interaction is given by
The A field is given by
P(R) = Zxe4LpWKpv j(R) = Z x e @ ~ ~ ~ & K , W
(6)
r denotes the electron coordinate, R the nuclear coordinate, Z , is the charge of the perturbing nucleus, Q! is the Dirac matrix, [p ] and [j] denote the charge and current densities at the retarded time, and Ki and h a r e initial and final projectile momenta. In nonrelativistic formulations, the current - current interaction ea! A(r,t) in Eq. (3) is not included. The cross section for the excitation of an electron from state i to f while the projectile goes from momentum states &,(R) to & ( R ) is given by ( Fano, 1963)
where &r) denotes electronic wave functions and q = IKf - Kil is the momentum transferred to the electron. In the PWBA, one sets
- exp(iKi
c$~,
R - air)
(8)
and by Fourier transforming the retarded potentials into momentum space
the integral over nuclear coordinatesgives delta functions; so the cross sec-
328
R. Anholt and Harvey Gould
tion becomes
Ij+w)(
1 - B a,)exp(iq
r)+i(r) d3r
I2
(10)
where qo = A E,/v is the minimum momentum transferred to the electron, A E, is the difference in electronic energies of states i and f, and j9 is the ion velocity. The vector product /3 a can be decomposed as 9
+ P sin lal,
/3 a = /3 cos h,
(1 1)
where cos A = qo/q and ai is the ith component of the Dirac matrix, The cross section is then given by
where F,
=
I@'@) exp(iq * r)+i(r) d3r
(13)
G,= J $ f ( r ) a , exp(iq * r)$i(r) d 3 r (14) and similiarly for Gzg.The first term in Eq. (12) is due to the longitudinal interaction between the perturbing nucleus and electron, and the second is the transverse interaction. One can write the cross section in terms of the incoherent sum because the longitudinal and transverse interactions excite the electron to states with different azimuthal quantum numbers. Commutation relations give (Anholt, 1979) G,, = A Eif F-,/qC so that the term containing F, - P cos AG, can be written as
(15)
which cancels the retardation factor (q2- q@2)-2 appearing in Eq. (12) for the longitudinal interaction. For electron ionization [c#+ = +&)I one obtains, after summing over final states (integrating over ionized electron kinetic energies E )
RELATIVISTIC HEAVY-ION - ATOM COLLISIONS
329
The first term in this expression is identical to that used in the nonrelativistic PWBA (Menbacher and Lewis, 1958). In nonrelativistic derivations of the PWBA, both the current-current interaction and retardation effects are neglected, but the only effect on the final formula is the absence of the transverse ionization term in Eq. ( 17). This allows us to use nonrelativistic theories (Merzbacher and Lewis, 1958) to calculate the most important contribution to ionization by relativistic ions.
C. THEDIPOLEAPPROXIMATION The dipole or Bethe (1930) approximation plays an important role in the calculation of electron stopping powers (Fano, 1963). One sets
Fit = ($,*(r)lexp(iq * r)I$i(r)) z is . ($z(r)IrI$i(r))
(18)
and Gxi,
W (&Yr)IrI4i(r))
(19)
thereby obtaining
For heavy projectiles the maximum momentum that can be transferred to the electron q, = 2McyP, where M is the projectile mass, is essentially infinite; hence in numerical evaluations of Eq. (1 7) one often puts q, = 00, but this would give In 00 in the first term of Eq. (20). The application of the dipole approximation is only valid if q-’ is smaller than the spatial range of the electron wave functions; the expectation value (q r) should be much less than unity. This limits the maximum value of q to be less than q- where Ei is the electron binding energy in atomic units. (For K-shell ionization, q, is equal to Z in atomic units, since r is of the order of Z-I.) For the transverse contribution, most of the integral over q comes from the region near q = qo (and no logarithmic divergences are present), so we can put qmm= m there. Using qo = Ei/u (which neglects the electron kinetic energy E), we get
a,
330
R. Anholt and Harvey Gould
Evaluations of electronic stopping powers (AEo), where A E is the energy transferred to the electron, make use of the fact that ( E Ei)l(45,lrlc#+)l2 is the oscillator strength, and the integral over final states is the sum over oscillator strengths, giving unity for each target electron. For inner-shell ionization, 12 and one could use a sum rule for the matrix elements I(4,1r14i)(Bethe Salpeter, 1957). One can also relate the matrix elements I( 4,1rl$i) I2 to photoelectric ionization cross sections (Pratt et al., 1973) in the dipole approximation.
+
gPE(o = Ei
+ E ) = ( 2 ~ ) ~ a(&lrl$+) ol l2
(22)
Hence the ionization cross sections can be written as
For a rapid estimation of the ionization cross section, one can assume that the photoelectric cross sections vary as
where n(- 2 to 3) is a fitting constant. Then the integral over E is easily done, and one obtains (Anholt et al., 1984b)
The cross sections evaluated so far have been defined per electron, but if one uses atomic photoelectric ionization cross sections in Eq. (25) (Scofield, 1973), the sum over initial states is automatically included. (In numerically evaluatingthis expression,all quantities are dimensionlessexcept the photoelectric cross section; hence the ionization cross section takes the dimension of the photoelectric one.) Equation (25) indicates the general characteristics of ionization and excitation cross sections at relativistic velocities, which are similar to those seen for K-shell ionization of uranium by protons in Fig. 4. The first logarithmic term in Eq. (25) comes from the longitudinalinteraction. At high relativistic projectile energies, the ion velocity approaches c and no longer varies with energy (or 7) so that the longitudinal cross section approaches a constant. The transverse contribution increases as the In y2, however. The additional spin-flip contribution shown in Fig. 4, but not present in Eq. (25), is discussed in the following section.
RELATIVISTIC HEAVY-ION-ATOM COLLISIONS
I
I00
10
33 1
I000
Y
FIG.4. Very-high-energy behavior of proton-induced uranium K-shell ionization cross sections calculated using the plane-wave Born approximation.(From Anholt, 1979.)
D. EVALUATION OF INNER-SHELL IONIZATION CROSS SECTIONS
Although Scofield (1978)and Amundsen and Aashamar (198 1) evaluated the longitudinal and transverse form factors using relativisticHartree - Fock wave functions, inner-shell electronic wave functions are traditionally approximated by nonrelativistic hydrogenic wave functions (Merzbacher and Lewis, 1958).To preserve the spin structure of the electronic wave functions (needed if one does not want to approximate the Dirac matrix a, with its nonrelativistic ansatz), one uses Pauli wave functions (Davidovic et al., 1978; Anholt, 1979). For electrons with spin quantum numbers m, = and - +,the 4-component Pauli wave functions are given by
++
m,=-f where Ni is a normalization factor, K2
- 2c, and 4, is the nonrelativistic
R. Anholt and Harvey Gould
332
hydrogenic wave function. For the K shell, one could use /~ & = (22)3/2 exp(- z r ) Y o o ( ~ , ~ )N~ , = [ I + ( ~ 0 1 / 2 ) ~ 1 - ~(27) where Yoois the spherical harmonic wave function. Following Bethe (1930), the ionization matrix elements are usually calculated in parabolic coordi~ q2/ZZ, , Q, - q,,/Z2= w2/4tlr(y and nates. Defining qK = ( / ~ / Z LQY= W = 1 k2 = 1 2e/Z2, one obtains for the K-shell ionization cross section per electron (Merzbacher and Lewis, 1958; Khandelwal et al., 1969; Anholt, 1979)
+
+
where
A,, =
2' exp(-2/k tan-I[2k/(Q 3[ 1 - exp(-2z/k)][(Q - k2
+ 1 - k2)]) + l)z + 4kZ]"
and
d Z= [ 1
+ (ZC.I/~)~]-~[1 + ( W - l)(Z0(/2)~]-'
(29)
The first two terms in Eq. (28) containing FK and GK come from the longitudinal and transverse contributions to the total K-shell ionization cross sections. In evaluating Eq. (28) for target vacancy production one uses an effective charge 2 = Z* = 2, - 0.3 for K-shell ionization, dictated by Slater ( 1930)screeningrules. The value of Wminis determined by the experimental K-shell binding energy E K yand is equal to O K , 2EK/Z*2.The twofold numerical integration is carried out as described in Merzbacher and Lewis ( I 958). Figure 5 compares calculations of target K-vacancy production by 4.88 GeV protons with measurements of Anholt et al. (1976). The presence of the transverse excitation term is clearly seen. The theoretical longitudinal cross sections alone are about one-half of the measured cross sections. Adding the transverse cross sections nearly brings the measured and calculated total K-shell ionization cross sections into agreement. Equation (28) also contains an electron spin-flipterm (the last term in FK) which is unique to relativistic ion-induced ionization.This term is important only for high-Z atoms, e.g., p U collisions. Since both the spin-flip and transverse cross sections increase as In y2 (Fig. 4), the spin-flipterm is never greater than about 0.25 times the transverse cross sections for K-shell ioniza-
+
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
30
40
50
70
333
90
22
FIG.5. Measured and calculatedK-vacancy production by 4.88 GeV protons versus target Longitudinal contribution;(---), longitudinal and transverseconatomic number. (-), tributions;(- * -), total crosssection, including spin-flipcontributions.(From Anholt, 1979.)
tion. The ratios of the spin-flip cross sections to the transverse cross sections scale as the square of the electron binding energy; hence, for L- and outershell ionization, the spin-flip contributions should be negligibly small. Leung and Rustgi (1983) have explored the possibility of measuring the spin-flip contribution to K-shell ionization using polarized proton beams. The 4.88 GeV proton measurements are the highest-velocity measurements of inner-shell ionization that have been done with heavy particles. Relativistic effects have been explored at greater velocities (or greater values of y ) using incident electrons. In applying this theory to calculate electroninduced K-shell ionization cross sections, we note that the maximum momentum transfer cannot exceed the incident electron momentum, and the maximum energy transfer cannot exceed the electron kinetic energy. Since the contributions to the ionization cross sections peak where the energy transfer is equal to the K-shell binding energy (which never exceeds 133 keV for 2 = 92), electron-induced ionization cross sections for electron energies exceeding - 1 MeV can be calculated with Eq. (28) without modifications. Figure 6 compares electron-induced ionization cross sections for 2 MeV ( y = 5) to 900 MeV ( y c- 2000) electrons with Eq. (28). The measured cross sections clearly are in agreement with theory, except perhaps for 2 MeV electrons incident on high-Z targets, where the electron binding energies are largest compared to the incident electron kinetic energy. One need not evaluate the longitudinal part of the ionization cross sec-
334
R. Anholt and Harvey Gould I000 1
-
w ELECTRONS
20
-
30
40
50
70
90
22
FIG.6. K-vacancy production by 2 MeV (Li-Scholz et al., 1973), 50 MeV (Hoffman et al., 1978, 1980), 300 and 900 MeV (Middlemann ef al., 1970) electrons. (-), Plane-wave Born-approximationcalculations using Eq. (28). (From Anholt, 1979.)
tions numerically since tables of the reduced cross section
or (30) are available (Khandelwal et al., 1969; Choi et al., 1973; Rice et al., 1977; Benke and Kropf, 1978;Johnson et al., 1979)for the K, L, and Minner-shell electrons. One must calculate qK.using qK = (p/Z*a)2instead of the formulas based on ion energy given in the tables, however. For the transverse contribution, the dipole approximation tends to be an accurate approximation down to the smallest qK values where the transverse term is still significant. (This is less true for the longitudinal part, however, where the dipole approximation is only accurate to about a factor of two for qK = 1.) The transverse contribution to the K-shell ionization cross section (per electron) in the dipole approximation is given by (Anholt, 1979) FK(qK/ei9eK) = eKf(qK,eK)/qK
RELATIVISTIC HEAW-ION - ATOM COLLISIONS
335
Although the transverse contribution has not been worked out for any shell but the K-shell, one can estimate its magnitude using Eq. ( 2 5 )
Anholt (1979) gives a formula for calculating the spin-flip contribution to the K-shell ionization cross section. The spin-flip contributions should be negligible for L-, M-, and outer-shell ionization. Scofield ( 1978) has calculated cross sections for electron-induced K- and L-shell ionization, and Amundsen and Aashamer (1 98 1) have calculated cross sections for proton-induced ionization using fully relativistic, Hartree- Fock or Hartree- Fock- Slater electronic wave functions. An important feature missing when one uses nonrelativistic hydrogenic wave functions is the small-r divergence of the wave functions (Amundsen et al., 1976). For example, Dirac Is wave functions vary as
4,s- rS-I exp(-
Zv)
(33) where s2 = 1 - d Z 2 is less than unity. The increased electron density at small r increases the density of high-momentum components in the Is momentum wave function. At low velocities u where the minimum momentum transfer q,, - E,/v is very large, the larger density of high-momentum components in the initial and final electronic wave functions gives increased ionization cross sections (Amundsen et al., 1976; Anholt, 1978). For the present relativistic heavy ions where the momentum transfer qo is small, and where one can put FAq) = 4 * ( 4 e l r l 4 K )
(34)
the divergenceleads to a contraction of the radial wave functions which gives smaller values of the expectation value of r; hence one obtains smaller cross sections. Anholt et al. ( 1985) evaluated the longitudinal contributions to projectile K-shell ionization in Xe, Au, and U projectiles using the relativistic PWBA formulation of Jamnik and Zupancic (1957). This calculation uses Dirac wave functions and makes a multipole expansion of the final-state wave functions. The calculated U K-shell ionization cross sections for p = 0 . 2 to 1 are up to 209’0 smaller when Dirac wave functions and binding energies are used. E. TARGET-ATOM SCREENING
In calculating projectile ionization by neutral atoms, one must take into account the electronic screening of the perturbing target nucleus (Bates and
R. Anholt and Harvey Gould
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Griffing, 1953, 1954, 1955; McGuire et al., 1981; Briggs and Taulbjerg, 1978). McGuire et al. (198 1) formulated a theory of the screening of He atoms, which can be generalized to many-electron target atoms (Anholt, 1985a). For projectile ionization, the perturbing charge 2: = 2: appearing in Eq. (17) is replaced with 1
\2
+ 2,-
CI(v/iIexP(iq r)Iv/i>12
(35)
i
where cyi is the target atomic orbital for the P electron, and the sum includes all target electrons. In this expression, q is the momentum transferred to the electron, hence S(q)must be included inside the integral over q [ Eq. (1 7)] or Q [=q2/Z2in Eq. (28)]. The first term in Eq. (35) is the effective screened target charge; if q approaches zero, the charge vanishes. Then, ionization, which would normally occur at large impact parameters (of the order of q-l), does not occur, because the target nucleus is completely screened; the projectile electron sees a neutral perturbing atom (Anholt, 1985a).However, in cases where q is reasonably large, the matrix elements ( yilexp(iq r)lv/,) are small, so the effective charge is close to 2,.The antiscreening term, given by the middle term in 2,in Eq. (33, is the contribution to projectile ionization by the target electrons. If q is large, the ionization cross section is proportional to 2: Z , , where 2: comes from the Coulomb potential between the target nucleus and active electron, and the factor of 2, comes from 2, separate electron - electron Coulomb interactions. Since at q = 0, ionization by the neutral target atom cannot occur, the final term is the antiscreening correction (ASC) approaching 2, at small q, thus cancelling the Z , factor. Target matrix elements, ( yilexp(iq r)1yi), can be calculated using Hartree - Fock or other suitable many-electron wave functions. In the first term of Eq. (33, Zi(yi(exp(iq r)(yi)isjust the atomic form factor F,(q,Z,) used in Compton scattering calculations, so these values can be taken from tabulations (e.g., Hubbel et al., 1975). Tables of the ASC factor, Zil (v/Jexp(iq r)lyK)12, are not available, however. The ASC factor lies between Ft(q,Z,) and(Ft(q,Z,)(2/Z,(Anholt, 1985a), but one obtains nearly identical cross sections independent of whether Ft or F:/Z, is used for the ASC. Figure 7 shows calculations of projectile 1sand 2s ionization in 400 MeV/amu Ne and 962 MeV/amu U atoms. Target-atom screening significantly reduces Ne 2s and Ne 1s ionization cross sections (where the momentum transfer is smallest), but the U 2s and Is ionization cross sections are only slightly different from the unscreened Born results. For U, the momentum transfer is large, so F,(q,Z,) and the ASC are nearly
+
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RELATIVISTIC HEAVY-ION-ATOM COLLISIONS
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I
---__
"lS
I00
337
5
10
20 30
50
100
Zt
FIG.7. Calculated cross sections for ionizing 2s and 1s projectile electrons for 400 MeV/ amu Ne and 962 MeV/amu U ionsversus 2,.If screeningis neglected, o/Z: is independent of 2, (). Screening without antiscreening gives (- * -). The curves (---) and (---) were calculated using F and 1f12/Z,,respectively, for the antiscreening correction. The difference between the two calculationsis negligible. For U, screening effectsare almost negligible, and the 2, due to the antiscreening term. (From Anholt, 1985a.) cross sections vary as 2: i-
+
zero, and the cross sections vary as Z : 2,. The calculated uranium cross sections are therefore larger than unscreened Born results at low 2,. Small screening effects reduce the uranium cross sections below the unscreened values at large Z,. Equation (35)neglects kinetic energy constraints on the target-electroninduced excitation contributions. These are negligible for relativistic low-2 ions, but can be important for high-2 ions. For a target electron to ionize a projectile electron, the target-electron kinetic energy in the projectile frame must be greater than the binding energy of the projectile electron. Thus at 200 MeV/amu, the target electron (energy 100 keV) cannot ionize U K electrons (binding energy 1 15to 132 keV, depending on the degree of ionization). For high-2 projectiles where, according to Eq. (35), the cross sections vary as (2: ZJa, (aBis the PWBA cross section for protons), Anholt et al. ( 1985)replaced the term Z,aBwith Zp,, where a, is the calculated electroninduced cross section of Rudge and Schwartz ( 1966)for the relevant electron kinetic energy (100 keV in the example given above). The cross sections of Rudge and Schwartz agree with measurements for electron kinetic energies near the ionization threshold. The validity of this approximation for K-shell ionization in atomic H H collisions, where the screening effects can be calculated exactly in the PWBA (Bates and Griffing, 1953, 1954, 1955)has been confirmed by Anholt (1 986).
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+
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The transverse ionization cross sections are more drastically reduced by target-atom screening than are the longitudinal cross sections (Anholt, 1985a;Gillespie et al., 1978).This occurs because most ofthe contributions to the integral over q for transverse excitation comes from the region q qo where S(q) is smallest. The longitudinal cross sections, however, fall off as q-I for high velocities; hence larger contributions to the longitudinal cross sections come from regions of large q where S(q)is larger. In the impact-parameter picture important contributions to transverse ionization occur at larger impact parameters than for longitudinal excitation. Here the projectile electron sees a more effectively screened target atom. For example, in 2.1 GeV/amu Ne Ag collisions, the transverse Ne K-shell ionization cross section is reduced by a factor of 0.06 by Ag target-atom screening, but the longitudinal cross section is only reduced by a factor of 0.5. As a result, transverse ionization is negligible in these collisions.
-
+
F. DISTORTION EFFECTS The PWBA is based on first-order perturbation theory. The ionization amplitudes increase linearly with the perturbing charge 2, so the cross sections (in the absence of target screening)increase as Z:, where Z , comes from the perturbing Hamiltonian, Zxe2/IR- r 1. In high-velocity collisions (characterized for K-shell ionization by v/v. >> 1) excitation can occur at very large impact parameters b where the internuclear distance R,on the average, is much greater than electron distance r. Hence, the potential is always small, and the linear PWBA or semiclassicalapproximationis always valid. However, at small velocities (v < vK) or even intermediate velocities (v = uK), excitation occurs at small impact parameters where the strong perturbing potential can significantlydistort the inner-shell wave functions, giving a nonlinear dependence of the amplitude on Z , (Basbas et al., 1973, 1978). Distortion effects on inner-shell ionization are often analyzed in terms of the binding and polarization effects (Basbas et al., 1978). One can understand these effects by considering the following schematic coupled-channel calculation in an atomic basis. We consider three closely coupled states: the 1s state (amplitude ao)and the 3p (a,) and 3d (a2)states. We thus model ionization by 1s 3p (+3d) excitation. The current-current interaction and the m subshellsare neglected;so we retain only three channels (instead of nine). Also back coupling to the initial state is neglected. The rate of change of the excitation amplitudesis given by (Skinner, 1962;Kingston et al., 1960;
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339
Bates, 1959)
u, = -aoV, u , = a 0 V10eior - V , , a ,
+
+ VI2a2
(36)
u, = aoVzoeior V,,a, - V,,a,
where cr) is the ls-3p or ls-3d binding frequency difference, and
These equations are solved for a , ( m ) with the initial condition ao(-m) = 1. First-order perturbation theories calculate a,(+ 03) by setting ao(t)= 1 for all t ; hence they integrate the equation
u l ( t )= Vlo(t)eiof (38) along an ion trajectory with impact parameter b. It is important to realize though that the perturbing potential appears not only as the off-diagonal term Vlo but as the diagonal part of the perturbation matrix and in the coupling between the p and d states. One usually solves such equations by making a phase transformation (Skinner, 1962)
dj(t>= aj(t) exp
(I,’qj(t’)
dt’)
(39)
obtaining for the desired ls- 3p excitation amplitude
)
~ , ( t ) = d O V , , e x p ( i ~ ( oV+ , - V l l ) d t ’ +d2V12exP(i[(v22-
VIM’)
(40)
As the perturbing charge increases, three effects on the excitation amplitude d,(m) can occur. By comparing d,(t)with u l ( t ) ,we identify these as binding effects [increase in the binding energy difference o due to the addition of V,(l) - V,,(t)], polarization effects (increase in the excitation amplitude d, due to the addition of the term in d,), and unitarity effects [depletion of the initial amplitude do(t)< 11. At high velocities where q - w/v is small and excitation occurs at large impact parameters where R > b and V,(R) and V ,,(R) are small, the addition of these potentials does not affect d, significantly, so that if do(t) = 1, the amplitudes should vary as
dlW
- ~Xdi.l+
Z 3 1 2
(41)
340
R. Anholt and Harvey Gould
where dBis the first-orderamplitude, a from Eq.(38), calculated for Z, = 1, and we call d , , the polarization amplitude. After integrating ldl(w)l2over impact parameters, the ionization or excitation cross sections should vary as 6 -.
+
Z ~ -I-B O(Z3 O(Z$) (42) where O(2;) denotes terms of the order of 2;. This is called the polarization effect. In theories of electron stopping powers, one assumes that the attractive perturbing potential polarizes the electron clouds, decreasing the effective electron-perturbing nucleus distance, thus increasing the ionization cross section. This has been analyzed classically (Ashley et a/., 1973), quantum mechanically (Hill and Merzbacher, 1974) using harmonic-oscillator models of the atom, and quantum mechanically in coupled-channel calculations of K-shell ionization by Ford et al. (1977) and Becker et al. (1 980). The type of harmonic oscillatorwave functionsappearingin the calculations of Hill and Merzbacher ( 1974)suggest intermediate couplingto d states as in our schematic coupled-channel model, but other states are also important. At low ion velocities, the momentum transfer q = w/v is large, and the where n LO (Basbas et al., K-shell ionization probabilities fall off as (v/o)”, 1973).In these collisions, increasingthe energy transfer by adding V, - Vl will further reduce the cross sections. For the Is level, the binding energy at R = 0 in this model increases from &Z2a.u.to jZ2a.u. for symmetric collisions; hence this binding effect significantlyreduces ionization cross sections for large Z, at small velocities (Brandt et al., 1966; Anholt and Meyerhof, 1977). For the intermediate to high velocities relevant to K-shell ionization by high-Z relativistic ions, the theory of Basbas et al. (1 978) which interpolates between theories of the binding effect at low velocity and theories of the polarization effect at high velocities have been applied. The results of this theory agree well with coupled-channel calculations of Becker et al. ( 1980) for p , He, and C-induced Ar K-shell ionization. Figure 8 compares target K-vacancy production cross sections for a variety of high-2 projectiles with PWBA calculations, dipole approximation results, and the theory of Basbas et al. (1978; denoted Borr corr. in Fig. 8). Remarkably, despite the large perturbing chargesand large atomic charges, the measured cross sections are not too different from the first-order PWBA calculations [or the dipole-approximation ones which are here shown to illustrate the accuracy of Eq. (25) compared to the exact PWBA theory]. The Ne- and Kr-induced K-shell ionization cross sections are negligibly affected by distortion effects, which is due to the relatively small perturbing charges there. For low Z, (corresponding to high v/vK)the theory of Basbas et al. gives larger ionization cross sectionsthan the PWBA due to the dominance ofpolarization effects, but for high Z, and low-velocity 82 MeV/amu Xe collisions (corresponding to low
-
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
30 4 0
60 80
30 40
60 80
34 1
60 80 100
zt
FIG.8. Measured and calculated K-shell vacancy production cross sections. The numbers beside each curve denote the ion energy in MeV/amu. The ionization cross sections were calculated with Eq. (28) (-), with the Bethe (dipole) approximation (- -), and with the Basbas et Born approximation with corrections for binding and polarization effects (---; al., 1978). Fully stripped projectiles can also capture target K electrons, giving the estimated contributions shown by (---). (From Anholt ef al., 1984b.)
v/vK),the Basbas theory gives smaller cross sections due to the binding effect. However, the data shown in Fig. 8 are not sufficiently precise to distinguish between the different theories. Also, at large Z , , electron-capturecontributions to K-vacancy production by the nearly bare projectiles are present. (For further discussion of electron capture, see Section IV.) Electron-capture contributions to ionization are absent when one measures projectile ionization. Figure 9 shows reduced Xe ls ionization cross sections versus 2,for 82, 140, and 200 MeV/amu Xe projectiles. For high-Z projectiles, target-atom screeningis expected to be negligible; hence the cross sectionsare expected to vary as(Z: Z,)a, in the PWBA. The reduced cross sections are obtained by subtracting the estimated target-electron-induced ionization cross sections Zp, from the measured reduced cross sections, and dividing by 2:. In the Born approximation, the cross sections should be independent of 2,. These measurements are ideal for studying distortion effects: The velocities are sufficiently small; so that the transverse excitation contributions are only 4% (at the largest energy) of the total cross sections, electronic relativistic effects are small, and the momentum transfer is sufficiently large that the target nuclei are basically unshielded (Anholt ez al., 1985). The theory of Basbas et al. (1978;thin solid lines in Fig. 9) predicts that for these collisions the reduced cross sections increase with Z , , because the
+
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342
40 81.5 MeV/amu
20
---__
C v)
0 0
20
40
60
80
zt
FIG.9. Reduced projectile ionization cross sections u/Z: for 8 I .5,140, and 200 MeV/amu Xe ions plotted against target atomic number. The reduced cross sections uRdwere obtained by subtracting the electron-induced cross sections Z,u, from the measured ones and dividing by Z : . The thick solid lines are PWBA calculations, and the thin solid lines were calculated with the theory of Basbas er al. ( 1978)for binding and polarizationeffects using cx = 1.5 and 3 (---). (- -), Glauber-approximationcalculations. (From Anholt er al., 1985.)
-
polarization effects are more important than binding effects. This theory disagreeswiththe bulk of the high-2, data. At the heart ofthis theory, acutoff impact parameter b, = cKaKis present. The theory assumes that for impact parameters smaller than b,, binding effects are important, and for larger impact parameters, polarization effects are dominant. To obtain better agreement with the data shown in Fig. 9, Anholt et al. (1985) semiempiri1.5 to 3 to reduce the polarization cally increased the c , value from =c, effect and enhance the binding effect. Using c, = 3 brings the theory into better agreement with experiment (dashed lines in Fig. 9). It was found by trial and error that no improvement is obtained with other values of c,; the value c, = 3 gives the best overall compromise fit. The theory of Basbas et al. (1978) is usually applied to calculate target inner-shell vacancy production where 2, -=c2 (2, << 2,).For 2, = 2, several theories which have been developed to calculate ionization in H+ H collisions (Park, 1983) may be applied. To compare these theories with measurements for relativistic heavy ions where 2, = Z,, Anholt et al. ( 1985)obtained reduced cross sections for XeS3+ Xe collisionsby interpolating the reduced cross sectionsin Fig. 9 between measurementsfor 2, = 47
+
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RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
343
and 79. By plotting Z2a(Z= Z , = Z,) versus the proton kinetic energy, they compare Xe Xe with p H ionization cross sections (Park, 1983; Shah and Gilbody, 198 1) at the same value of u/u,. The Xe energy scale in Fig. 10 is related to the p H one using
+
+
+
E,=932(--
1
1).
Ji=p
,
54
D=137.037
J0.04O25Ep
(43)
where Ex,is in MeVJamu and Epis the proton energy in keV. The Xe energy scale ends at E, = 160 keV, where p is equal to unity. This type of scaling is exact for symmetric collisions ( Z = 2,) in the PWBA (Khandelwal et al., 1969) and in molecular perturbed stationary-statecalculations for one-electron systems(Thorson, 1975). Since the target-electron antiscreeningeffects have been removed in derivingthe reduced cross sections, and the electronic relativisticeffects and transverse excitation are negligible, this scaling should be nearly exact in the Xe Xe collisions. The measured Xe Xe points in Fig. 10 are clearly in good agreement with the measurements of Shah and Gilbody (198 1 ) and J. T. Park (1983, and private communication). We also show in Fig. 10 the calculations of Basbas et al. (1978) using c, = 1.5 (thin solid line) and c, = 3 (dashed line). Those using c, = 3 are in reasonable
+
+
Ex, 20
100 200
0
400
I
I
50
I00
E,
+
( MeV/omu)
I50
(keV)
+
FIG.10. Scaled Xe Xe (0)and p H 1s ionization cross sections plotted against proton kinetic energy. The PWBA (thick solid line), Basbas theory using cK= 1.5 (thin solid line) and cK= 3 (---), and Glauber theory (- -) results are shown. The p H data points are from Park ( 1983, and private communication;A) and Shah and Gilbody (198 1 ;0).Some of Park’s points for E,, > 50 keV have been omitted. (From Anholt et al., 1985.)
-
+
R. Anholt and Harvey Gould
344
+
+
agreement with the measured p H and Xe Xe cross sections, but the original theory with cK= 1.5 overestimates the p H cross sections. Theories of ionization and excitation of hydrogen by protons based on the Glauber (1 955) approximation have given some of the best agreement with experiment (see Park, 1983; and McGuire, 1982, for comparisons). An important feature of the Glauber approximation is that it preserves the unitarity of the ionization amplitude for 1s ionization. For symmetric collisions near v - v, the first-order semiclassical approximation (Hansteen et af., 1975) predicts ionization probabilities that are greater than f in smallimpact-parameter collisions. Such large ionization and equally large excitation probabilities deplete the initial 1s occupation amplitudes d&) in Eq. (40), assumed to be unity for all time tin first-order theories like the PWBA. Therefore, smaller ionization probabilities and cross sections can be expected. It is interesting that the Glauber approximation, which accounts for unitarity effects, but neglects binding and polarization effects, agrees well with experiment and with the semiempirical modification of the Basbas theory accounting for (mainly)binding effects,but neglecting unitarity ones. Figure 9 compares Glauber calculations (chain curve) of the reduced ionization cross sections with experiment. Disagreement between the two occurs for 2, < 20, where the data points are higher than the Born calculations, and the Glauber cross sections always lie below the Born ones. The low-2, data points in Fig. 9 do not agree with the original theory of Basbas et al. either. These data points are most affected by target antiscreening. If the full Born electron - electron contribution ZtoBwere subtracted from the measured cross sections instead of Zp,(a,incorporating threshold effects), the reduced cross sections would be in better agreement with the Born and Glauber calculations at low Z, (e.g., the 82 MeV/amu Xe Be reduced cross sections are reduced by 25%). These considerations suggest that the discrepancy at low Z , may be due to our lack of a complete theory of target screening and antiscreening near the electron ionization threshold velocity. Since for ions of a given velocity v/vLthe velocity relative to the projectile L shell, is greater than v / v ~the , binding and polarization effects should have a negligible effect on target L x-ray production. Also, target L electron capture in Xe Pb collisions is likely to be as negligible as target Kelectron capture in Xe Zr collisions(see Fig. 8). In Fig. 1 1 we compare L x-ray cross sections with the first-order PWBA (including transverse excitation, but neglecting distortion and projectile screening effects). The measured cross sections are in good agreement with calculations using theoretical ionization cross sections and measured or semiempirical Lshell fluorescence yields and Koster-Kronig transition probabilities (Salem et al., 197 1; Anholt et af., 1984b).
+
+ +
+
345
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
lo5
-I I - 670 MeV/arnu Ne -
ti ‘1
Io4
<
n
-
104
4I +
Io5
-
-;)
70
80
1 -
-
-
:+-++-+I
-
Kr
- 82 MeV/arnu Xe -
ItJ
-
4
-
b’
I
-
-
197 MeV/arnu Xe
-
-
-
-
-
-
0?
-
-
-
x
I MeV/amu
<
-
-
h
---422 -
lo5?
-
-
I06
<
L j yj “““I++4:
dbb 4 -+
++
I
-1 I
I o4
I0 3
90
100
70
80
90 70
80
90
70
80
90
IV. Electron-Capture Processes A. RADIATIVE ELECTRON CAPTURE
Radiative electron capture is the inverse of the photoelectric process. An electron incident on the projectile ion is captured with the emission of a photon whose energy is the electron kinetic energy ( y - l)rnc2in the projectile frame plus the final electron binding energy Ei.Raisbeck and Yiou ( 1971) noted that REC was needed to explain total electron-capture cross sections for 300 MeV protons incident on low-2 target atoms. The REC photon was observed at nonrelativistic velocities by Schnopper et al. ( 1972), Kienle et al. (1973), and others(e.g., Spindler et al., 1979),and at relativistic velocities by Anholt et al. (1 984a).
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R. Anholt and Harvey Gould
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The cross section for REC into the projectile shell i can be calculated from using the photoelectric cross section oiPE
+
where k = Ei/mc2 y - 1. Under the condition that the binding energy of the target electron can be neglected, there are the 2, electrons per target atom that can be captured with equal probability. One uses atomic photoelectric cross sections (per filled atom) hence the REC cross sections are for bare projectiles. The advantage of writing the REC cross section in terms of the photoelectric cross section is that much is known about photoelectric processes in atomic physics (Pratt et al., 1973). For numerical calculations, we have the choice of many possible photoelectric cross-section formulas. For low-2, relativistic ions, the Sauter formula (Sauter, 193 1 ;Pratt et al., 1973) provides reliable cross sections (Anholt, 1985a).For high-2, relativistic ions at velocities less than 82 MeV/amu for Xe and greater than 400 MeV/amu for U, the Sauter formula with higher-order relativistic corrections (Pratt et al., 1973; Eq. 6.1.8) tends to break down significantly; so photoelectric cross sections calculated with exact Dirac wave functions (Hultberg et al., 1967) are used there. Tabulations of photoelectric cross sections for all shells calculated using many-electron wave functions are available (Scofield, 1 973). Meyerhof et al. ( 1985) used the bound-state normalization theory of Pratt ( 1960; Schmickley, 1966) to approximate Dirac L- and M-shell photoelectric cross sections using many-electron ones. Figure 1 shows a measured photon spectrum for 422 MeV/amu U Be collisions (Anholt et al., 1986). The photon peak at 270 keV is the most prominent feature in this spectrum, and is due to REC into the projectile K shell. A second peak at - 190 keV is due to the REC into L and higher shells of U. The 190 keV peak is nearly inseparable from the continuum primaryelectron bremsstrahlung spectrum ending at - 160 keV. The REC photon peak energies in the laboratory frame are given by
+
-
-
so, although the rest frame energy of the K-shell REC photon is 380 keV, the laboratory frame energy at 8’= 85” in Fig. 1 is -270 keV. The width of the REC lines is determined by Doppler broadening (the spread in angles AO’ subtended by the x-ray detector giving a spread in x-ray energies A E!J, and the Fermi momentum of the target electrons. In Fig. 1 the REC width for U Be is determined entirely by the angular acceptance of the x-ray detector. Equation (45) assumes that the target electrons are at rest. For target electrons with momentum pt, the width of the REC peak is of the order of
+
RELATIVISTIC HEAVY-ION - ATOM COLLISIONS
341
ycb pt (Kleber and Jakubassa, 1975), which, except for the inner-shell electrons of high-2, atoms, is smaller than the Doppler width. The REC line shapes shown in Fig. 1 were calculated by Anholt et af. (1986). One interesting feature of REC is that, although the peak energy is Doppler shiftedwith laboratoryangle, the angular distribution does not have the forward-peakingcharacteristic of a Doppler-shiftedphoton, but varies as sinZ0’ in the laboratory. This is due to the cancellation of electron-retardation and Lorentz-transform effects (Spindler et uf., 1979). In the projectile frame, the incident electron is captured and the emitted photon angular distribution is peaked in the direction of the incident electron, which is backwards in the laboratory. The photon angular distribution varies as
do -dQ
sin2 0 ( I - p cos e)4
where 8 is the angle between the direction of the incident electron and the photon. The Lorentz transform
where 8’ is the laboratoryangle, shifts the angular distribution forward in the laboratory, so that the resulting laboratory angular distribution is nearly exactly sinZ0’. Equation (46) is not exact, however, and small deviations from sinz 8’ are seen in Fig. 12 (Anholt et af.,1984b), and expected when Dirac (Hultberg et af.,1967) or Sauter (193 I) calculationsof the photoelectric angular distribution are used (Fig. 12). Although the measured points are not sufficiently accurate to test photoelectric angular distributions, they do clearly demonstrate the electron-retardation Lorentz-transform cancellation effect. The total REC photon cross sections for thick targets are given by the REC capture cross sections for bare projectiles multiplied by the equilibrium number of projectile K vacancies NKv (48) This equation can be used to measure either the REC cross section or the number of projectile Kvacancies inside the solid targets. If the number of K vacancies inside the solid targets is equal to that measured downstream using a magnetic spectrometer oRElxy
= fNKv%EcK
where Foand Fl are the measured fractions ( S 1) of projectilescarrying0 and 1 electrons, respectively, one can obtain the REC cross section. Figure 13
R. Anholt and Harvey Gould
348
0
1.2
2
1.0
t
0.8 I97 MeWamu Xe +Be
210 MeV/amu %+Be I00
-
75
L
ul
t
B 5 50
25
0 0
90
45
135
180
QLAB
+
FIG. 12. Angular distribution of K REC photons seen in 197 MeV/amu Xe Be collisions and the ratio to sin2 normalized at ebb= 90"(Anholt el a/., 1984a).For the photoelectron angular distributions, from which the REC angular distributions are derived, results from Sauter (1 93 1; ---), Hultberg ef al. (1 967; - -), and neglectingelectron retardation (-) were used. The Hultberg angular distributions were calculated using Dirac wave functions, but are for 150 keV protons incident on Sn, corresponding to similiar 210 MeV/amu Sn Be collisions.
-
+
compares measured and calculated REC cross sections and derived and measured K-vacancy numbers for a variety of collisions. For the cases thus far studied (82 and 200 MeV/amu Xe and 422 MeV/amu U), consistent K-vacancy numbers are obtained using REC and charge-statemeasurements (Anholt et al., 1984a). The equivalence of the in-target and downstream charge-state fractions is consistent with the discussion in Section V.
B. NONRADIATIVE ELECTRONCAPTURE Unlike REC where the difference between the energy of the initial and final states of the captured electron is released by the photon, NRC involves
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
-Y
349
V
102
b’ 2 NKV
I
V
# Id
1
b’
-
I75 MeV/amu -
2
I
l
l
I
I
l
l
cI
a
NKV
I97 MeV/amu Xe
Lo I 0
5
10 2 0
I
l
l
l
l
-
I
-7I
50
5
10 2 0
I
50
zt
FIG. 13. Measured (0)and calculated (-) K REC cross sections and the deduced average number of projectileKvacancies.(x), from post-targetchargefractionmeasurementsof Gould el al. ( 1984), and (-), for NKvare charge-fraction measurementsfrom Meyerhof et al. ( 1 985).
the transfer of energy or momentum between the electron and the projectile motion, mediated by the electromagnetic perturbation potential. For this reason, NRC is a process like ionization. Unlike for ionization, however, electron capture involves a transfer of the electron from the target frame to the projectile frame. It is convenient to use instead of binding energies Eiand E,, total electron energies
E.
= m c 2 - ~&f. which for K-shell electrons in hydrogenic atoms is simply 1.f
(50)
2,= mc2sp
(51) The longitudinal momentum transferto the target where sp= (1 - a22,2)1/2. electron then is given by (Moiseiwitsch and Stockman, 1980; Eichler, 1985) 90
=
1 &- YEi ; Y
the absolute magnitude of which approaches the ratio of the energy transfer to the ion velocity, l/u (Ei- Ef +mu2),at low velocities.
+
350
R. Anholt and Harvey Gould
In the impact-parameter picture, the cross section for the capture of an electron from an initial state ( j p ) to a final state (j’p’) is given by (Eichler, 1985; see also Moiseiwitsch, 1985; Moiseiwitsch and Stockman, 1980; Shakeshaft, 1979)
where the capture amplitude is given by A, = i
dt dr y#(rb,t’)S-ZP fi(rt,t)
(54)
rP
r; is the electron - projectile nucleus distance, r, is the electron -target nucleus distance, S transforms the wave function from the target frame (unprimed coordinates) to the projectile frame (primed coordinates)
and the first-order projectile and target wave functions in their respective frames are written as
yi(rt,t)= +i(rt) exp((56) yf(ri,t’)= +f(ri) exp(- iEft’) Equations (53)-(56) have been evaluated by Oppenheimer (1928) and Brinkman and Kramers ( 1930)for nonrelativistic projectile velocitieswhere y = 1 and S = 1. Shakeshaft ( 1979)and Moiseiwitsch and Stockman (1980) calculated NRC cross sections for relativistic projectile velocities. These first-order OBK theories disagree with experiment at both relativistic and nonrelativistic velocities, exceeding measurements by significant factors (2 to lo), as shown for 1050 MeV/amu Ne collisions in Fig. 14. In recent years, much progress has been made in the theory of nonrelativistic NRC. At relatively high, but nonrelativistic, velocities, the impulse (Briggs, 1977;Amundsen and Jakubassa, 1980),eikonal (Eichlerand Chan, 1975;Chan and Eichler, 1979), second Born (Drisko, 1955;Shakeshaft and Spruch, 1979), and strong-potential Born approximations (Macek and Alston, 1982; Briggs et al., 1982)give better agreement with experiment than do the first-order theories. For relativistic projectiles, calculationshave only been made using the second Born and eikonal approximations. Humphries and Moiseiwitsch(1 984,1985) used the second Born approximation to calculate NRC for low-2 relativistic projectiles. They found that for both relativistic and nonrelativistic p H collisions, the second Born cross sections are about 0.3 times the OBK cross sections. However, this
+
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
t
1050 MeVIarnu Ne
/
t
10
35 1
1
20
30
50
I 00
Zt
FIG.14. Projectile K-electron capture cross sections (per bare ion) for 1050 MeV/amu Ne ions. The OBK results (Moiseiwitschand Stockman, 1980)are shown by (- --) and (O), and the eikonal calculations with shielding for target K, L, and the sum of Kand L capture are shown by (-). A second Born approximation calculation (Humphries and Moiseiwitsch, 1985) is shown by (V). Data (0)from Crawford (1979), as analyzed by Anholt (1985a). (From Anholt and Eichler, 1985.)
reduction is still not sufficient to bring the second Born calculations into agreement with experiment (Figs. 14 and 15). Anholt and Eichler (1 985) found good agreement between relativistic eikonal-approximation calculations of NRC cross sections and experiment. In the prior form of the eikonal approximation, the interaction between the electron and target nucleus is approximately treated to all orders of perturbation theory by replacing the projectile wave functions with
where r, is the electron coordinate relative to the target nucleus, and Zi = 2,. The interaction between the electron and projectile is treated only in first order. This version physically describes a hard collision of the electron with the projectile nucleus followed by multiple soft collisions with the target
352
R. Anholt and Harvey Gould
10 2 0
50 10 20
50 10 2 0
50 100
zt
FIG.15. Projectile K-electron capture cross sections calculated using the eikonal approximation (-) compared with measurements of Crawford (1979) as analyzed by Anholt (1985a) and second Born calculations (Humphries and Moiseiwitsch, 1985; V, always lying immediately above the data points shown by 0).(---), The region where the eikonal approximation, including target K and L capture, may not be valid (see text).
nucleus. Eichler (1985) has shown that the calculation of eikonal cross sections for relativistic projectiles can be simplified considerably using a density-matrix formalism. The numerical evaluation of the cross section consists of integrating over the momentum transfer (which is also done in the OBK approximation) and another variable 1stemmingfrom the integration over d7 in the eikonal phase factor in Eq. (57). Since the eikonal approximation is a high-energy approximation, one expects to find good agreement between theory and experiment if the ion velocity is much larger than the velocity of the active electron. For highly asymmetric collisions, the electron velocity in atomic units is equal to the value of the largest atomic charge. In Fig. 15, this implies that u should be much greater than 2,.Hence the eikonal results are shown as dashed lines in the region where u < 22,. The solid lines are in good agreement with experiment, but where v < 2Z,, the cross sections calculated using the eikonal approximation are below experiment. Impulse-approximation and other second-order Born-approximation formulations exist for relativistic projectiles (Jakubassa- Amundsen and Amundsen, 1980, 1985), but no calculations have been done. Attention has focussed on the asymptotic dependence of NRC (Shakeshaft and Spruch, 1979). The nonrelativistic OBIS cross sections fall offas E-6 or v-** at high
RELATIVISTIC HEAW-ION-ATOM COLLISIONS
353
velocities. At relativistic velocities, u approaches a constant c, hence most of the velocity factorsdrop out, and the relativisticOBK cross sectionsfall off as E-I or y-* (Moiseiwitsch and Stockman, 1980). [One can also see from Eq. (52) that the momentum-transfer approaches a constant at large 7.1 The eikonal (Eichler, 1985) and second Born calculations of Humphries and Moiseiwitsch (1984) also find an asymptotic y-l dependence for capture of 1s target electrons into Is orbitals of the projectile, but Jakubassa/ y depenAmundsen and Amundsen ( 1980, 1985) find a (In ~ ) ~ asymptotic dence in both the impulse and exact second Born approximations. (Humphries and Moiseiwitsch make a peaking approximation which the Amundsens avoid.) Shakeshaft( 1979)has suggested that the question of the asymptotic falloff of the NRC cross sections may be academic, since REC is dominant at high relativistic velocities. However, this is not true for high-2 projectiles and target atoms. For U U collisions, the REC and eikonal NRC cross sections are approximately equal above 15 GeV/amu, where both fall off as y-' (see also Section VI).
+
-
C. ELECTRONCAPTURE IN HIGH-ZIONS Calculations of NRC cross sectionsoften consider only transitions from Is orbitals of the target to Is orbitals of the projectile because these transitions are dominant at velocities that are large compared to the orbital velocities. For very heavy ions, this condition may not be reached even at relativistic velocities, and there may be appreciable capture into outer shells of the projectile or from outer shells of the target. At high velocities, the momentum transferred to the electron for electron capture q,, ( y - l)rnc/yP becomes large, making the small-r (high-momentum) part of the projectile- and target-electron wave functions important. The dominant contribution near the origin is from s orbitals whose electron density scales as (Z/ny. Thus, the total capture cross section from any filled target shell to any unfilled projectile shell should vary as (Meyerhof et al., 1985)
-
For bare projectiles, one can use the full projectile charge Z,, but for manyelectron target wave functions, the electron density at the origin is smaller, and a reduced target charge Z: is used. Calculations of NRC use relativistic or nonrelativistic hydrogenic target wave functions with reduced effective charges Z: = 2, - AZ, where A Z = 0.3 and 4.15 for the K and L shells (Slater, 1930). Anholt and Eichler (1985) examined the validity of using
R. Anholt and Harvey Gould
354
effective charges to simulate many-electron wave functions for electron capture in highly asymmetric relativistic heavy-ion - atom collisions. Figure 16 shows relative state-to-state NRC cross sections for 1050MeV/ amu Ne Ag, 197 MeV/amu Xe Ag, and 82 MeV/amu Xe Au collisions. For asymmetric collisions such as Ne Ag, Eq. (58) is nearly valid. For any given n, the cross sections decrease as np3.For low 2,and constant n p ,the cross sections fall off faster than n;', due to the reduction of the target charge 2: by an n-dependent A 2 value. At high Z,, the relative target L capture cross sections are larger than Eq. ( 5 8 ) predicts (see also Fig. 14), due to the smaller momentum transfer needed for target L-electron capture. For 82 MeV/amu Xe projectiles incident on Au targets, 1s- 1s transitions are no longer dominant (Fig. 16). In these collisions, capture of target L electrons into the projectile Kshell gives the largest cross sections, but other state-to-state cross sections are large also. This occurs because the momentum transfer for transitions such as 2s, 2p 1s is significantly smaller than for 1s- 1s transitions. Since the cross sections fall off very steeply with increasing momentum transfer, the smaller momentum transfer overcompensates the q3or n;' falloffs.
+
+
+
+
-
I00
IU
I050 Ne+Ag
197 Xe+Ag
82 Xe+Au
1 2 3 4
1 2 3 4
1 2 3 4
"P
FIG. 16. Calculated relative cross sections for the capture of electrons from target shells with n,= 1,2,3 ( A is the sum ofall target shells) into bare projectileshellswith np = 1,2,3, and all shells with n, 2 4 for 1050 MeV/amu Ne +Ag, 197 MeV/amu Xe Ag, and 82 MeV/amu Xe Au. The numbers above each bar represent n, quantum numbers, and below the axis represent n, quantum numbers. (From Meyerhof et al., 1985.)
+
+
RELATIVISTIC HEAVY-ION -ATOM COLLISIONS
355
The dominance of capture into excited states of the projectile is directly seen in measurements of electron capture by XeS4+ions (bare nuclei) and Xe52+ions (filled K-shell) incident on high-2, targets (Figs. 17, 18, and 19; Meyerhof et al., 1985). For 82 MeV/amu Xe Au and Ag collisions, approximately equal capture cross sections for XeS2+and Xes4+ionsare found, indicating that capture into the Kshell of Xe is not dominant. The RECcross sections vary approximately as n;'. For Xe Be collisions, REC dominates, and the XeS2+capture cross sections are much smaller than the XeS4+ ones. Figures 17 and 18 also compare measured cross sections for electron capture by XeS2+and XeS4+with eikonal calculations. The eikonal approximation treats the interaction between the electron and one ofthe nuclei to all orders in perturbation theory and the interaction with the other nucleus to only first order. Depending on which is treated perturbatively, one obtains
+
+
I
"
I 06
I o5
lo5 In
E
?5
104
Be
b
I o5
lo3 -
I o4
102 -
I o3 50
100 200 Ep
10' L 5 0 100 200 (MeWarnu)
FIG.17. Electron-capturecross sections for XeS4+ions incident on Be, Mylar (My), Al, Cu, The total REC and NRC cross sections where the NRC cross Ag, and Au targets. (-), sections were calculated with the eikonal approximation according to the higher-potential post-prior prescription. (---), The higher-charge criterion was used.(- -), REC cross sections, which are dominant in the Be target. (From Meyerhof ef al., 1985.)
R. Anholt and Harvey Gould
356
~ e 5 ~ +
I0 6
I o4
I o5
I o3
-2 lo5
I o2
L
0
n
b
lo4
I o2
I o5
10'
Io4
10'
I o3
50
100 200 Ep
10' 50 100 200 (MeVIomu)
FIG. 18. Same as Fig. 17 for Xe'*+ ions, where capture into the Xe Kshell is not allowed; hence only capture into the L, M, and higher orbitals of Xe occurs.
different cross-section results. This is referred to as the post-prior discrepancy. The usual choice for 1s- 1s transitions is to use the prior form if Z , is greater than Z,, and otherwise the post form. Meyerhof et al. (1985) proposed a criterion based on the largest interaction potential ( Z / r ) .Since the expectation value of ( Z / r ) for hydrogenic atoms varies as Z2/n2,independent of the electron subshell, one should use the post form if Z p / n , > ZJn, or the prior form if Zp/nq< ZJn,. The post form is obtained from the prior form, Eqs. (54)-(57), by interchanging Z, and 2, everywhere,putting Z : = Z,, and interchanging initial and final states. The solid lines in Figs. 17 and 18 were calculated with the higher potential criterion, and the dashed lines were calculated using the higher-charge one. In most cases the Z / n criterion (solid lines) for choosing the post or prior form agrees better with the data. Since the Z/n criterion is based on an estimate of the interaction strength using a diagonal matrix element, the effect of the shell size may be overestimated. Therefore, in some cases, the conventional higher-charge criterion (dashed lines) gives better agreement with the data.
RELATIVISTIC HEAVY-ION -ATOM COLLISIONS
357
A
Be
to4 I o 5 M I
o3
50
100 200
E,
lo'
50
100 200
(MeVIamu)
FIG. 19. Measured XeS2+(A) and Xe"+ (0)electron-capture cross sections. For low-energy Xe Au collisions, the Xe"+ and XeS2+cross sections are approximately equal,indicating that capture into excitedstates ofthe projectile is dominant. For Xe Be collisions, REC is dominant, and the XeSZ+cross sections are factors of about f smaller than the Xe%+ones, as expected for REC into the projectile t and higher shells.
+
+
V. Collisions in Solid Targets Although atomic-collision studies are usually done using solid targets, understandingthe manyfold interactions of ions in matter is one of the least tractable problems in atomic physics. At nonrelativistic velocities, one finds higher projectile charge states emerging from solid targets than from gas targets (Betz, 1972). This is a useful effect in accelerator operation where higher-ion charge states produce higher-energy beams. The Bohr - Lindhard ( 1954) interpretation of gas-solid charge-state differences emphasizes the role of electronic excitation. Electrons are more easily ionized from excited states;therefore, if the time between collisionsis shorterthan the decay times of the excited states, higher charge states will be obtained. For low-2 ions, higher charge states are produced in solid targets than in gases. The effect is expected to diminish for few-electron,very high-Zions where the lifetimesof excited states are generally very short. Betz and Grodzins ( 1970) proposed that post-target Auger decay of excited states also gives higher charge states.
358
R. Anholt and Harvey Gould
Relativistic heavy ions allow the detailed study of the role of excited states on the charge states and collisional interactions of ions in matter (Anholt, 1985a; Anholt and Meyerhof, 1986).Low-Z projectiles have small radiative decay rates, so that high excited-statepopulations are present in solid targets. Calculations are simple because only zero- or one-electron projectiles are present. Because the relative velocities are very high, the high-velocity approximations for ionization, excitation, and capture are valid. For high-2, few-electron ions, the allowed electric-dipoledecay rates are very large compared to the rate of charge-changing collisions, so that the relative excitedstate populations are lower. Nevertheless, a few metastable states decay slowly. Their influence on the charge states of high-Zions in solid targets is a matter of investigation. Experimental information about ions in matter is obtained from (1) the equilibrium charge states of ions measured downstream of the target using magnetic spectrometers, ( 2 )the target-thickness dependence of the chargestate yields, (3) projectile K , (and K,) x-ray production solid targets, (4) metastable state populations (observed by delayed photon decay), and ( 5 ) REC photon production, which was discussed in Section IV,A. For relatively thick targets, experimental cross sections are obtained by least-squares fitting the target-thickness dependence of measured chargestate yields. Our models of ions in matter indicate how those cross sections should be interpreted. A. Low-ZIONS For low-2 ions a four-state model consistingof two charge states, zero and one electron, and two excited states, 2s and 2p, was formulated (Anholt, 1985a). This model was used to obtain equilibrium charge-state distributions and target-thicknessdependences for relativistic ions with 2 S 18 incident on solid targets between Be and Au. The model is schematically illustrated in Fig. 20. Electron-capture cross sectionsinto the Is, 2s, and 2p states are designated as a , , a,, and a3.The Is, 2s, or 2p electrons can be ionized with cross sectionssI,s2, or sg.Monopole 1s-2s (x~), dipole 1s-2p (xJ, and 2s- 2p (x3)excitation and de-excitation can occur. Besides collisionalde-excitation, the states can decay by radiative decay, of which the 2p + Is decay rate is the largest. The 2s --* 1s and 2p + 2s radiative decay rates can be neglected compared to the collisionalde-excitationcross sections.In order to compare decay rates with collisional cross sections, we define a decay cross section as
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
s3
359
dz
x2 x3
2p'
N3
N2
Nl
NO
FIG.20. Schematic level diagram showing transitions leading to attachment (a,,a2,a3), ionization (s,,s2,s3) excitation ( x , ,xz,x3),and decay d2. (a) A four-state model including the fully stripped ions No and those with one electron in the Is, 2s, or 2p state. (b) A simplified three-state model.
where 12p,,, is the radiative decay rate (Bethe and Salpeter, 1957), and n2 is the target-atom density. (We could also make calculations on a rate scale by converting the collisional cross sections to collision rates by multiplying by nzPc.1 From diagrams like that in Fig. 20(a), one can immediately write down the rate equations governing the populations Ni of the various states.
+ + ~ 3 ) N +o SIN,+ ~2N2+ ~ 3 N 3 kl= - (sI + + x~)N,+ + ( X J+~ = + X ~ N ,- + + + k3 = ~ 3 N o+ ~ 2 N+ l - + + +~ 3 ) / 3 ] N 3 No = -(a1
a2
U~NO
k 2
~
2
XI
~
0
( ~ 2 XI
~
1
~
2
d2)~3
(60)
~ 3 1 ~ ( 2~ 3 / 3 ) ~ 3
~ 3 N 2 [ ~ 3 d2
( ~ 2
where Ni = dNi/dT, T is the target thickness in atoms/cm2, and ZiNi = 1. These equations are solved with the initial condition Ni(0)= Sipor Sil,depending on the incident ion charge states. The ratio of ions having an electron to fully stripped projectiles was measured, which is given by
The equilibrium ratio R , is obtained by setting all k,= 0, and solving the resulting set of linear equations. With relativistic, low-Z ions, one can calculate accurately most of the required cross sections in Eq. (60), some of which are shown in Fig. 2 I. The equilibrium ratios can then be calculated numerically or using approxima-
R. Anholt and Harvey Gould
360
....... ......... ...... KREC ...... .................... *-*:
102
-
(~103)
5
10
20
50
100
‘t
FIG. 2 1. Bottom: Calculated cross sections for Is + 2s, Is +2p, and 2s -+ 2p excitation (-), Is and 2s ionization (---), 2p + Is radiative decay (- * -), and REC into the projectile Kshell ( * ) for 400 MeV/amu Ne ions. Top: (-), calculatedequilibrium ratios R, of electron-bearing projectiles to bare ions in solid targets, and (---) in gas targets. (Data points from Crawford, 1979.)
tions to the model. In the model of Anholt (1985a), it was assumed that, because of the large magnitudes of the 2s- 2p excitation cross sections, the 2s and 2p populations equilibrate according to the level multiplicity N(2p)/ N(2s) = 3. Combining the equations for the 2s and 2p populations, as depicted in Fig. 20( b), yields three sets of linear equations,which can be solved analytically for the equilibrium ratio:
+
where F2 = 4(3s3 s2). This equation illuminates the origins of gas-solid charge-state differences. In a gas target, the decay cross sectionsd2are dominant because n2 = 0; hence one has
+ +
(63) This equation states that all captured electrons decay to the ground state, so the ratio of projectiles with one electron to zero electrons is the ratio of the total capture cross section to the 1s ionization one. However, for low-Z, Req = (aI
36 1
RELATIVISTIC HEAVY-ION -ATOM COLLISIONS
relativistic projectiles in solid targets, one has S; >> ad, (Fig. 2 1); hence we get R,
=
a, (1 s, +x, +x,
+
-)
+
0 s2
+
In this case, electrons captured into excited states (with cross sections a, a,) do not decay, but are ionized with cross sections S,; so the fraction of projectiles having one electron in the 2s or 2p state is just (a, a,)/?,, An
+
electron captured into the 1s state can be excited into the n = 2 state, where, if the L-shell ionization cross section is very large, it is ionized. Therefore, the fraction of projectileswith 1selectrons is approximately the 1s capture cross section divided by the 1s ionization plus excitation cross sections s, x, x, . However, the L ionization cross sections are not infinitely large; so the term 1 (x, x2)/Szrepresents a correction;a fractional number of excited 1selectrons given by a,(x, x2)/S,(sl x, x,) remain in the n = 2 states, and contribute to the total number of electron-bearing ions. Comparison of Eqs. (63) and (64) predicts differencesbetween relativistic ion equilibrium ratios measured in gas and solid targets. For the case of 400 MeV/amu Ne Cu collisions, Eq. (63) gives R , = 0.3 X for gas tarfor solid targets, a 30%difference. gets and Eq. (64) gives R, = 0.23 X For other Z , values, the differences between gas- and solid-target charge states are shown by the dashed and solid lines for R, in Fig. 21. The solid-targetfractions of one-electron projectilesare smaller, implying higher charge states, in agreement with nonrelativistic-velocity results (Betz, 1972). Approximating R, for solid targets by the first term in Eq. (64), one can write
+ +
+ +
+ +
+
+
R, --
a,
solid
+ a, + a3 s, + x, + x, > 1 a,
SI
The solid-target charge state is higher than the gas-target value [or the oneelectron fraction is lower, as expressed in Eq. (65)], because (1) in a gas, capture into all states leads to attachment, but only 1s capture leads to attachment in solids, and (2) in solids, excitation into excited stateshas a high probability of subsequent ionization; hence the effective ionization cross sections, s, x, x, are larger. This model is similar to the BohrLindhard model since both emphasize the higher likelihood of ionizing excited electrons. In the measurements of Crawford (1979), bare incident projectiles on thick targets were used [Ni(0)= &I, and R(T) was fit to
+ +
R(T) = R,[ 1 - exp(-osT)]
(66)
362
R. Anholt and Harvey Gould
where a, is an effective stripping cross section. Crawford (1979) interpreted a, as the 1s ionization cross section sI. If one interprets the equilibrium charge-state fraction in Eq. (64) as an effective capture cross section divided by an effective ionization cross section, neglecting the terms in u2 a,, the effective ionization cross section is given by
+
a, = (sl
+XI
I(+ "'SX2)
+ x2)
-
1
+
Because excitation to the 2s and 2p levels with cross sections xl x2 enhances ionization,the effective ionization cross section is larger than sI,.but is smaller than s, x I x2by the factor 1/[ 1 ( x I x2)/i2].Effective ionization cross sections may also be obtained by fitting the numerically calculated solutions to Eqs. (60) and (61) for R(T)to Eq. (66). Figure 22 shows results for 250 and 2100 MeV/amu Ne ions. At low Z , , REC dominates, and R, decreases with 2, because the ionization cross sections increase as Z : and the REC ones increase linearly with 2,. Eventually, R, increases with increasing 2, due to NRC contributions to the total electron-capture cross sections. Agreement between the measured and cal-
+ +
Io
-~
+ +
I
I
250MeWN
I
I
I
2100 MeWN
FIG.22. Top: Equilibrium ratios calculated using only the REC cross sections(-) and including the derived NRC cross sections (---) for 250 and 2 I00 MeV/amu Ne ions. Bottom: the effective ionization cross section uscalcuThe radiative 1s capture cross section (-), lated using Eq. (67) (- * -) and by fitting the numerically calculated values of R( T) [ Eq. (66); ---I. The derived 1s nonradiative capture cross sectionsare shown by (---). (The data are from Crawford, 1979.)
RELATIVISTIC HEAVY-ION - ATOM COLLISIONS
363
culated R , values at low Z , indicates that the model is complete, and the necessary cross sections and decay rates can be accurately calculated. Therefore, at higher Z , , one can extract NRC cross sections from the measured equilibrium ratios shown in Fig. 22. This is the origin of the NRC cross sections, which were compared with theory in Section IV (Figs. 14 and 15). In addition, where the target-thickness dependence of the one-electron charge-state ratio R( T ) was measured by Crawford ( 1979), the extracted effective ionization cross sections agree with Eq. (67), and with fits to the numerically calculated target-thicknessdependence of R( T ) .Equation ( 6 7 ) therefore indicates how one should interpret thick-target stripping cross sections. For low-2 ions in solid targets, one measures a sum of the stripping and excitation cross sections. Larger effective solid-target stripping cross sections are consistent with the Bohr- Lindhard ( 1954) model. However, this is only for thick targets where the approach to equilibrium is measured. We have verified numerically that, if one-electron ions are incident on thin targets, the yield of zero-electron ions increases as sIT for sIT << 1. B. HIGH-2IONS
In high-2, few-electronions, the lifetimesof most states that decay directly to the ground state are much shorter than the mean time between chargechanging collisionsor collisionswhere excitationoccurs. For 197 MeV/amu Xe Ag collisions,the cross section for the radiative decay of the 2p state in H-like ions is 4 X lo6barns, while the cross section for 2p ionization is only 3 X 1Os barns. For low-2 ions, the predicted difference between charge states of ions in gas and solid targets occurs because the radiative decay cross sections [ad2in Eq. (62)]are smallerthan the L-shell ionization cross sections S; in solid targets (Fig. 2 1) and are much larger than S; in gas targets. Sincethe radiative decay cross sections are larger than L-shell ionization cross sections in both gas and solid targets in high-Z collisions, this would suggest that differences between gas- and solid-target charge-state distributions should vanish. Furthermore, one should be able to predict the charge-statedistributions with a ground-state model. The ground-state model of equilibrium charge states assumes that independent of whether electrons are captured into the ground state of a bare ion (with cross section a ,) or into excited states (with cross sections into the 2s, a,, and into the 2p, aJ, they always decay to ground state so that the charge-state distribution can be calculated with just the total electron-capture cross sections and the ground-state 1s single-electron ionization st and double-electron Is2ionization cross sections c , . The rate of change of the
+
R. Anholt and Harvey Gould
364
charge-state fractions 5 with j = 0, 1 , or 2 electrons are given by Fo = -(al ~2 ~3)Fo SIF~+ C I F ~
+ + + = (a1 + + ~3)Fo- ~ 1 / +2 a2 + F2 = + a2 + U J F ~ - (cI + 2~1)F2
2~1F2
(s1+
$1
(68)
(~1/2
where Fi= dF,/dT, and T is the target thickness in atoms/cm2. The equilibrium charge-statefractions are found by setting the derivatives equal to zero, and solving the resulting set of linear equations (Allison, 1958) F o = [ l +A(1 +B)]-' Fl = AF,, A=
F2 = BF,
a1+u2+u3.
+
sI Bc, '
......................
(69)
+ + a3
B = 4 1 2 u2 2s, CI
+
2p- 1s
21
---
FIG.23. Calculated cross sections for 2p + Is radiative decay ( ), Is and 2s ionization 2s 2p excitation (- .-), and REC (---) for 197 MeV/amu Xe collisions. The measured Is ionization cross sections are shown by (O), and measured XeS4+and XeS2+ electron capture cross sections are shown by (A and V)with connecting solid lines to guide the eye. (From Anholt and Meyerhof, 1986.) (-),
+
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
365
The measured ionization and electron-capture cross sections for 197 MeV/ amu Xe ions, discussed in Sections 111 and IV, are all that is needed to calculate the equilibrium charge states of 197 MeV/amu Xe ions in matter. The ionization and capture cross sections are compiled in Fig. 23. The cross sections a , and a2 a3are related to the XeS2+and XeS4+electron-capture cross sections using
+
a , = o(XeS4+)- o(XeS*+);
a,
+ a3= o(Xe52+)
(70) Double-electron ionization cross sections were discussed by Anholt et al. (1985). No evidence for simultaneoustwo-electron capture was observed in 197 MeV/amu Xe collisions, so double-electron capture has been omitted in the present calculations. Figure 24 compares measured equilibrium chargestate fractions for zero-, one-, and two-electron ions with the ground-state model calculations (dashed lines). Although the measured and calculated values of F, and Fl agree reasonably well, the ground-state model predicts fractions of two-electron ions that are slightly higher than experiment. The shapes of the charge-state curves in Fig. 24 are due to competition between ionization and two capture processes, similar to that seen in Figs. 2 1 and 22
.'' -
I00
I
-
I
0
10-3
2
5
10
20
50 100
FIG.24. Measured and calculated charge-state fractions for 197 MeV/amu Xe collisions. The experimental uncertainties in the ionization and capture cross sections shown in Fig. 23 give uncertainties in the calculated charge-state fractions extending from the lower to upper solid lines. The ground-state model f---) should be compared with the middle solid lines, based on the most likely electron-captureand ionization cross sections. (From Anholt and Meyerhof, 1986.)
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R. Anholt and Harvey Gould
for low-Zions.At low Z,, REC is dominant. The REC cross section increases linearly with Z,, and the ionization cross sections increase as Zf ;hence the fractions of one- and two-electron projectiles fall off with Z , . Near Z , = 15, NRC becomes significant. The NRC cross sections increase more rapidly with Z , than the ionization cross sections up to Z , 50; hence the one- and two-electron fractions increase. Above Z, = 50, the NRC cross sections do not increase much more rapidly than the ionization ones; hence the chargestate fractions level off. The differences between the measured charge-state fractions and the ground-state model predictions can be traced to the influence of metastable excited states. For instance, in H-like ions, 2p electrons decay quickly to the ground state, but 2s electrons decay more slowly, usually by excitation to the 2p state followed by 2p + 1s radiative transitions. In He-like ions, the ls2p 'PIand 3P1states decay quickly, but the 1s2s 'Soand 'S, and ls2p 3P0and 3P2 states decay by ls2s + ls2p de-excitation and/or excitation to the rapidly decaying 'PIand sf', states. Anholt and Meyerhof ( 1986)describean 1 1-state model of the states of relativistic Xe ions in matter that includes the H-like Is, 2s, and 2p states and the He-like Is2, ls2s, and ls2p states. The same measured ionization and electron-capture cross sections were used as in the ground-state model calculations, but, in addition, 1s- 2s, 1s- 2p, 2s- 2p, and 1s2s - 1s2p excitation and 2s and 2p ionization cross sectionswere calculated absolutely or relative to the 1s ionization cross sections using the PWBA. The results of the 1 1-statemodel calculations are shown by solid lines in Fig. 24. The uncertainties in the measured ionization and capture cross sections give uncertainties in the calculated charge-state fractions. The upper solid lines were calculated using the largest capture and smallest ionization cross sections, and the lower solid lines were calculated using the highest ionization and lowest capture cross sections within the experimentaluncertainties. The charge-state fractions calculated using the most likely capture and ionization cross sections are in good agreement with experiment. Anholt and Meyerhof ( 1986) developed an analytical quasi-ground-state model that illuminates the origin of the difference between the calculated ground-state and 1 1-state charge-state fractions. This model assumes that the ions are usually in their ground 1sor ls2states; hence the small populations of excited states can be calculated from the charge-state fractions and ratios of excitation cross sections to excitation plus decay plus excited-state ionization cross sections. In the quasi-ground-state model, the ratio F2/F1is given by
-
which should be compared with the ground-state-model value. Bin Eq. (69),
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+
where r2, r, ,and r, are all unity. In 197 MeV/amu Xe Ag collisions, one obtains r2 = 0.53, r, = 0.63, and r, -- 1.08, so that FJF, = 0.144, instead of B = 0.197 (the numerical 1 1-statemodel gives F,/F, = 0.140). Formulas for r, ,r, ,and r, are given by Anholt and Meyerhof ( 1986). For other targets, the r values are not significantly different from the Xe Ag ones. For low-Z ions, higher charge states are seen in solid targets than in gas targets principally because the effective ionization cross sections are larger in solid targets. For high-Z ions, higher charge states are seen because the effective cross sections for capture into excited states, r,u2 r3u3,are lower than in gas targets. Equation (67) showed that the effective ionization cross sections for low-Z ions in solid targets are larger than the 1sionization cross sections by the addition of the excitation cross sections x, x,. This increases the effective ionization cross sections in 400 MeV/amu Ne Ag collisions by a factor of 1.4. However, the factor [ 1 (xl xZ)/~J1in Eq. (67) limits the enhancement to only a factor of - 1.2. Electrons excited into the 2s and 2p states of low-Z ions are easily ionized, so that excitation and ionization both lead to electron loss. For high-Z ions, however, dipole excitation populates states that decay rapidly back to the ground state, so that the effective ionization cross sections are not enhanced significantly by dipole excitation. Most of the factor of r, = 1.08 increase in the effective ionization cross sections for He-like ions, rJ2sl cI)in Eq. (7 1), comes from monopole excitation to ls2s states. For high-Z ions, the excited-state capture cross sections u2and u3are reduced by about 50%in thick solid targets. Electrons captured into the metastable 1s2s and ls2p statesdecay slowlyby 1s2s- ls2p excitation,so that there is a high probability that the electrons will be ionized before decaying to the ground state, and hence will contribute less to the fraction of He-like ions. For low-Z ions where electrons captured into even the 2p states do not decay rapidly, excited-state capture contributes negligibly to the one-electron charge-state fractions. However, in low-Z ions, the excited-statecapture cross sectionsare only 20%of the total electron-capture cross sections into bare ions. For high-Z ions, the relative probability of electron capture into excited states of the projectile is much larger, as discussed in Section IV, so the 50%reduction in the excited-statecapture cross sections for high-2 ions in thick targets, gives a significant change in the two-electron charge-state fractions. Information about the states of high-2 ions in matter can be obtained from the projectile K x-ray production cross sections. For example, in the 1 1-state model, the equilibrium Ka! x-ray production cross sectionsare given by
+
+
+
+ +
+
+
OK- = diN(2p) + dZN('P,) + d3N(3P1) (72) where di are decay cross sections [Eq. (59)] for the 2p, P, ,and 'PIstates, and
R. Anholt and Harvey Gould
368 I o5
-
104
I02
10'
5
10
20
50
100
z1
FIG.25. Measured and calculated Xe KO x-ray production cross sections in 82 and 197 MeV/amu Xe collisions. The uncertaintiesin the fraction ofthe XeS2+electron capture into the 2p states give theoretical uncertainties extending from the lower to upper (---). (From Anholt and Meyerhof, 1986.)
Ni are the calculated equilibrium populations of the 2p, ' P I ,and 3P, states. Figure 25 compares calculations of K a x-ray production in 197 MeV/amu Xe collisions with experiment. The major theoretical uncertainty in the model is in the fraction of electrons captured by XeSZ+ions that go into the 2p state. Ifwe assume that a3is equal to the total XeS2+capture cross section, we obtain the upper dashed curves in Fig. 25; ifwe assume that a3 = 0, we get the lower curves. All other cross sections in these calculations are identical to those giving the charge-state fractions agreeing well with experiment in Fig. 24. The fraction of the Xe52+electron capture cross section to the 2p state, calculatedusing the eikonal approximationfor NRC and photoelectric cross sections for REC, gives good agreement with the measured Xe K , x-ray production cross sections. The quasi-ground-state model may be applied to analyze the origin of projectileK , x-ray production in thick targets (Anholt and Meyerhof, 1986). Three mechanisms are present in 197 MeV/amu Xe collisions: 1s- 2p excitation followed by radiative decay, 2p capture followed by radiative decay, and 1s- 2s excitation or 2s capture followed by 2s- 2p transfer and radiative decay. In 197 MeV/amu Xe collisions, 1s-2p excitation is dominant, and 2p capture and the 2s contributions each give about 10% of the total K , x-ray cross section. The quasi-ground-statemodel formulas for projectile K , x-ray production were extended to three- and four-electron ions, so that cross
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369
sections for 82 MeV/amu Xe collisions could be calculated from measured charge-statefractions where significant three- and four-electron fractionsare present (Fig. 25). The uncertainties in the 2p capture cross sections are also present for 82 MeV/amu Xe ions, and give uncertaintiesin the calculated K , x-ray cross sections. The calculations using the relative eikonal and REC capture cross sections are in good agreement with experiment.
VI. X-Ray Continuum Processes A. PRIMARY BREMSSTRAHLUNG
Radiative electron capture is the capture of target electrons into bound states of the projectile. An analogous process is the capture into continuum states of the projectile. Viewed differently: In the projectile frame, the projectile nucleus is bombarded by target electrons with mean kinetic energy ( y - 1)mc2.These may emit bremsstrahlung photons with energy E,as large as the electron kinetic energy in the projectile frame, 109 keV for 200 MeV/ amu projectiles. In the laboratory frame, the endpoint x-ray energy is given by E; = (7 - i)mc2y-yi - p cOS e y (73) which varies in 200 MeV/amu collisions from approximately 56 keV at backward angles to 200 keV at forward ones. As a first approximation, the primary-bremsstrahlung(PB) cross section is just the bremsstrahlung cross section for 2, electrons with kinetic energy T = ( y - l)mc2 bombarding the projectile nucleus Lorentz transformed into the laboratory (Schnopperet a)., 1974; Spindler et al., 1979; Yamadera et al., 1981; Anholt et al., 1986)
where the relation between 6 and 0’ and dS2 and dQ’ are given by Eq. (47), and
Here the unprimed quantities are projectile-frame (center-of-mass for projectile-electron collisions)quantities, and the primed ones are laboratory
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quantities. For the bremsstrahlung cross section, the Bethe - Heitler (1 934) formula [Eq. (3BN) of Koch and Motz, 19591 including the Elwert (1939) correction factor was used. Equations (73)-(75) assume the target electrons are free and have no intrinsic momentum; therefore, the theory gives a sharp cutoff in the continuum spectrum where the x-ray center-of-massenergy is equal to the electron kinetic energy. When one adds the Fermi momentum of the target electrons ps to the translational momentum p = ypmc, larger electron kinetic energies and therefore higher bremsstrahlung endpoint energies are obtained. The continuum shape can be calculated in the impulse approximation using (Jakubassa and Kleber, 1975; Anholt et al., 1986)
where the PB cross section on the right-hand side isgiven by Eq. (74), J(pz)is the Compton profile (Biggs et al., 1975), and the electron kinetic energy is given by
+
+
T = ( [ ( ~ ppZ/mc)’ 11’’’ - l)mc2
(77)
Finally, one averages over the laboratory angles subtended by the x-ray detector. These folding procedures have no effect on the PB cross section for x-ray energies well below the endpoint (Fig. 1). The electron momentum folding causes the spectrum to drop off more slowly above the endpoint; the Doppler folding shifts the position of the endpoint to slightly higher energies, due to the inclusion of smaller laboratory angles, and rounds off the continuum shape near the endpoint. The calculated PB cross sections are generally lower than the measurements of Anholt et al. (1 986), as shown in Fig. 1 and discussed in more detail below. However, the shape of the continua agree reasonably well with experiment, at least where PB is not obscured by REC lines and bremsstrahlung from secondary electrons (see below), as in Be-target collisions. The electron bremsstrahlung cross section has an approximately angular distribution in the center-of-mass frame of the form (Tseng et al., 1979)
where 8 here is the angle between the electron direction and photon direction, so the angular distribution in the projectile frame is peaked in the direction opposite to the beam direction. The first term in this equation is identical in form to the photoelectron angular distribution determining
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37 1
REC,hence gives an angular distribution of PB in the laboratoryofthe form:
Unlike for REC, the PB cross section is differential in laboratory x-ray energy, so instead of obtaining an angular distribution proportional to sin2 B’, an additional term 1-/? cos 0‘ is present. The angular distribution is determined keeping the center-of-mass x-ray energy E. constant. For Ex = 45 keV in 197 MeV/amu Xe Be collisions, the continuum cross sections are taken at E: = 62 keV at 45“, 37 keV at 90°, 27 keV at 135“, etc. Figure 26 shows that the measured angular distributions are indeed peaked at backward angles, as suggested by Eq. (79). The solid lines were calculated with the Bethe- Heitler ( 1934) formula for the bremsstrahlung angular distribution, but were normalized to the data near 8’= 90” (Anholt et al., 1986). The agreement between the measured and calculated PB angular distribution shapes is very good.
+
t
5,
ti” 60
75
90 elab
1b5
IiO
I:
+
FIG.26. The angular distribution of radiation in 197 MeV/amu Xe Be collisions compared with the Bethe-Heitler calculationsof primary bremsstrahiung, normalized at 8’ = 90”. The numbers give the center-of-mass (projectile frame) photon energy in keV.
312
R. Anholt and Harvey Gould B. SECONDARY-ELECTRON BREMSSTRAHLUNG
In a binary collision between a 197 MeV/amu heavy ion and a nearly free electron, electrons with kinetic energies up to 450 keV can be produced. These electrons can collide with other nearby target nuclei in solid targets, emitting secondary-electronbremsstrahlung (SEB)photons with energiesup to -450 keV. To calculate the cross section for SEB, we assume (1) that Z , target electrons are free and scatter elastically from the projectile nucleus, and (2) the ionized electrons follow a tortuous path inside the solid target so electrons never leave the target material, and the angle between the bremsstrahlungphoton direction and the projectiledirection is random. The latter assumption follows from a calculation (Jackson, 1962) of the mean multiple scatteringangle for 50 -200 keV electrons in the 5 - 50 mg/cm2targets used. The mean multiple scattering angle is of the order of several radians; so one cannot assume the electrons travel in a straight line after being emitted. With these assumptions, the cross section is given by (Anholt et al., 1986; Folkmann et al., 1973; Ishii et al., 1977)
where the elastic electron scattering cross section is given by the McKinley Feshbach (1948) equation, E- = 2y2p2mc2,S(EJ is the electron stopping power in the target material (Ahlen, 1980), ye = 1 iE:/mc2, = 1 - ye--2, and da,,/dE, is the angle-integrated bremsstrahlung cross section calculated using the Bethe - Heitler- Elwert formula (Koch and Motz, 1959). SEB is dominant for high-2, targets where the agreement between theory and experiment is very good (Fig. 1 for U U collisions; Anholt et al., 1986).
+
c. THE2,DEPENDENCE OF CONTINUUM X-RAY PRODUCTION For SEB, the bremsstrahlung cross section in Eq. (80) varies as Z:, the stopping power S(E) varies roughly as Z,/A,, where A, is the target atomic number, n, varies as A;.', the elastic cross section varies as 2;, and 2, electrons can participate in SEB, so the SEB cross section for infinite target thickness varies as ZiZ:. For PB, the bremsstrahlung cross section varies as 2; and 2, electrons participate, so the PB cross section varies as ZiZ,. Figure 27 shows the Z , dependence of continuum x-ray production in 82, 197, and 422 MeV/amu Xe and U collisions(Anholt et al., 1986). The x-ray energy chosen falls within the region where both PB and SEB are present. The x-ray energies were chosen to avoid as much as possible interference
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
/ 1 1A I I I t
5
10
I
20
I
I
373
I I I I I
50
100
zt
FIG.27. Cross sections for continuum x-ray emission in 82 MeV/amu Xe collisions at laboratory x-ray energies of 40 keV, 197 MeV/amu Xe collisions at 60 keV, and 422 MeV/amu U collisions at 140 keV versus target atomic number. (---), Secondary-electron bremsstrahlung; (- .-), primary bremsstrahlung; (-), total intensity.
with characteristic x rays and REC (though cross sections for continua falling beneath target x-ray lines could not be obtained at some Z , values). Good agreement between the SEB calculationsand experiment is found at high Z , , but there is a systematic discrepancy at low Z, where PB dominates. Since the shape of the calculated and measured x-ray spectra agrees reasonably well, identical qualitativeresults are expected at other x-ray energies. If one subtracts the calculated SEB contributions from the measured cross sections,the resulting cross sections increaselinearly with Z , , as predicted by the PB theory, but are factors of 1.7 (197 MeV/amu Xe) to 2.9 (422-MeV/ amu U) too high. The reason for the discrepancy between calculations of PB and experiment is not known. The calculated PB cross sectionsare probably accurateto within k 30%.Tseng ez al. ( 1979)and Lee et al. ( 1976)compared double-and single-differentialelectron bremsstrahlung cross sections calculated with the Bethe- Heitler - Elwert formula, with numerical calculations using fully
R. Anholt and Harvey Gould
314
screened atomic electronic wave functions or Dirac wave functions. For the present nearly bare projectiles, Dirac electronic wave functions should be used, but there is generally less than a +30% difference between the three different calculations in the relevant electron energy, Z , and angular range. The reasonably good agreement between measured atomic electron bremsstrahlungcross sections(Quarleset al., 1981) for similiar electron energy and Z ranges and the same calculations (Kissel et al., 1983), suggests that it is unlikely that the origin of the disagreementbetween the PB calculations and experiment lies in the electron bremsstrahlung cross sections. The SEB calculations are less certain than the PB ones, but we have not hypothesized a realistic mechanism that would quantitatively account for the discrepancy between theory and experiment for the Be targets. For example, one could assume that a significant number of electrons escape from the 6.6) Be target and collide with the higher-Z A1 (Zd = 13) or Mylar (Zd chamber walls. The relative contribution from the chamber wall increases with Z,, ,not as Z&,,,,because the electron stopping in the wall cancels one power of Z. A simple model which assumes that the electrons travel in a straight line, then upon leaving the target collide with an A1 wall in the same position as the target and make SEB along the remaining part of their range, increases the U Be SEB contributions by less than a factors of 1.5, which cannot explain the entire discrepancybetween theory and experiment there. In conclusion, the shape of the PB spectrum and the angular distribution of the radiation agree with theory, but the magnitude of the measured cross sectionsdiffer by factors of I .7 to 2.9. The measurementsare not sufficientto ascertain whether the discrepancy between theory and experiment is a Zpdependent or velocity-dependenteffect. The two measurementsfor Xe ions at 82 and 197 MeV/amu where the ratio of experiment to theory varies from 1.7 Ifi 0.2 to 1.8 k 0.2 are not sufficientto tell. Measurements with Xe and U ions up to 962 MeV/amu or U ions down to 100 MeV/amu would be useful.
-
+
VII. Ultrarelativistic Collisions Ultrarelativistic heavy ion accelerators have been proposed for the study of quark matter. The energies of these proposed accelerators range from about 10 GeV/amu for fixed-target machines (Lawrence Berkeley Laboratory, 1982;Brookhaven National Laboratory, 1983)up to 100GeV/amu per beam for colliders (Lawrence Berkeley Laboratory, 1979; Brookhaven National Laboratory, 1984). A 100 GeV/amu collider is equivalent to a fixedtarget energy of 20 TeV/amu. An interesting question is the type and significance of atomic collisions in this energy range.
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Radiative and nonradiative capture cross sections decrease at relativistic energies, but ionization cross sections increase logarithmically, leading, it would appear, to a desert of bare ions. This may not always be the case because electrons from electron- positron pair production may be captured by the ions. The cross section for electron -positron pair production reaches several thousand barns for 15 GeV/amu uranium on fixed-target uranium and an astonishing 1O5 barns for 100GeV/amu colliding uranium beams. If even a small fraction of the electrons are captured by the uranium ions, this would result in a significant charge capture cross section compared to the usual radiative and nonradiative capture cross sections. A detailed theory of electron -positron charge-changing processes at relativistic energies has not yet been worked out, but it is possible to make a crude estimate of the charge-changingcross section from a simple formula for electron-positron pair production and from a classical model for the binding of the electron. The cross section for producing electron -positron pairs from the Coulomb field of two (bare) nuclei was one of the early quantum-mechanics collisions problems (Landau and Lifshitz, 1934; von Weizacker, 1934; Williams, 1935; Bhabha, 1935, 1935a; Nishina et al., 1935;Oppenheimer, 1935;Nordheim, 1935;Racah, 1937).The mechanism for producing these pairs may be thought of as pair-producing absorption of the virtual photons from the motional Coulomb field scattering from a static Coulomb field (Weitsacker, 1934;Williams, 1935).A simple formula for the total cross section for large y is (Bhabha, 1935a, b): = (28/27a)a2Z$Z:ri(ln y)3
(81)
where r, = e2/mc2= 2.8 X is the classical electron radius. For uranium on uranium a- = 98(ln y)3 barns. Cross sections for electron - positron pair production for uranium on uranium at ultrarelativisticenergies, computed from Eq. (8 I), are given in Table 11. In a 100GeV/amu uranium collider with a luminosity of lo2’cm-2 sec-l, TABLE I1 PAIR-PRODUCTION CROSSSECTIONS FOR U92+ ON U92+ Energyham (GeV/amu) 2 4 20 100
Equivalent fixed target energy (GeV/amu)
I5
Cross section (kilobarns) 2
48
6
880
31 98
20,400
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1O8 electron - positron pairs per second would be produced in each interaction region. Such a machine has been proposed (Brookhaven National Laboratory, 1984). Heavier leptons will also be produced in smaller numbers. Upper limits, obtained by neglecting the nuclear size, are, for muons, m,/mp times the electron pair rate, or about 2OOO/sec. Tau pairs and other heavy particles can also be produced, Appreciable fractions of all of these particles would be bound to the uranium nuclei. As the pair production rate is luminosity dependent rather than beam current dependent, pair production may be used as a real-time nondestructive luminometer. Additional discussion of atomic physics effects in relativistic heavy ion collisions may be found in Gould (1984, 1985b) and in Brookhaven National Laboratory (1984). The electrons most likely to be captured after pair production are those having momenta which overlap with the momenta of the final bound K electron. This means that electrons with kinetic energy less than the uranium K-shell binding energy may be captured. The energy distribution ofthe pairs produced by colliding 15 GeV/amu uranium ions with a fixed uranium target extends to about 10 MeV, but is peaked at the lower energies. The fraction of electrons within the K shell with kinetic energies of less than 130 keV would then be roughly 130 keV/ 10 MeV = 0.0 1, hence approximately 1% of the electrons could be captured into the K shell. For muons, which have larger binding energies, the capture fraction could be larger, but the production cross sections are probably much lower.
zt
FIG.28. The calculated equilibrium fraction of electron bearing I5 GeV/amu Au ions for various targets Z , . The equilibrium fraction is the ratio of the total electron-capture cross Calculated including just section into bare Au ions to the Is ionization cross section. (-), REC; (---), calculated including NRC and REC; and (- -), including capture of 1% of the electrons produced by pair creation.
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A synchrotron booster has been proposed to inject heavy ions into the Brookhaven Alternate Gradient Synchrotron (Brookhaven National Laboratory, 1984)allowing acceleration to I 5 GeV/amu. The calculated equilib+ GeV/amu is shown in rium ratio of one-electron Au7*+to bare A U ~at~ 15 Fig. 28. Neglecting pair production, the radiative electron capture (REC), nonradiative electron capture (NRC), and projectile ionization cross sections produce equilibrium ratios as a function of 2, resembling those of lower-energy relativistic ions (Figs. 22 and 24). For low Z,, REC, which scales as Z,, is dominant. Since the ionization cross sections scale as Z: ,the equilibrium ratios decrease with increasing 2, in this region. At high Z, (2,= 80), NRC becomes important so that the equilibrium ratios increase rapidly with Z,. The NRC cross sections shown in Fig. 28 were calculated using the eikonal approximation (Eichler, 1985),and the REC cross sections were calculated using the Sauter formula for photoelectric cross sections (Sauter, 1931; Pratt et al., 1973). If 1% of the electrons made by electronpositron pair production are captured, the equilibrium ratios increase by The increase is independent of 2,because both the pair produc2.5 X tion and ionization cross sections scale as Z : .
VIII. Relativistic Few-Electron Ions in Quantum Electrodynamics Experiments Relativistic, very high-2 atoms present special opportunities for atomicstructure experimentsbecause any atom in the periodic table can be stripped to few-electron or even bare ions. Relativistic and quantum-electrodynamics (QED) effects can then be studied in systems where they are very large, can be experimentally isolated, and where precise and unambiguous calculations can be performed. Tests of QED in the highest-2, few-electron atoms are of particular interest because strong field tests of the theory can not be matched in low-2 experiments of any conceivable precisions. At 2 = 92, the contributions to the Lamb shift are the self-energy, -57 eV(Mohr, 1982),the vacuum polarization, 14 eV(Mohr, 1983), and the finite-nuclear-sizecorrection, -33 eV(Mohr, 1983). The vacuum polarization, but not the self-energy, is well tested in muonic atom experiments. High-2 Lamb-shift experiments are primarily a test of the self-energy in a strong Coulomb field. The self-energy contribution comes almost entirely from terms which are of very high order in zcll (where a is the fine-structure constant). Because these terms are large only at very high Z (strong Coulomb field), they are not
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tested in present lower-Z Lamb-shift and fine-structure experiments. The contribution of the higher-order terms in the self-energy can be seen by comparing the series expansion for the self-energy of the 1 2S,,2state with a numerical evaluation to all orders in Za (Mohr, 1982; Desideno and Johnson, 1971; Erickson, 1971; Cheng and Johnson, 1976; Johnson and Soff, 1985; Mohr, 1974;Sapirstein, 1981). If we write the self-energyZ, in a power series in a(the fine-structure constant) and Za, we have
Z,
+ A4, ln(Za)-2](Za)4 + A50(ZC11)5 -k [Am + h(ZCk!)-2+ h2(&)-2](z~)6+ A ~ O ( Z ~ ) ~ + higher-order terms} (82)
= n-’(a/n)rnoc2{[Alo A61
A62
where n is the principal quantum number and rn, is the electron mass. Values of the coefficients A, to A70 can be found in Mohr (1974) and Sapirstein (198 1). Figure 29 shows the ratio of the higher-order terms in the self-energy [terms of hlgher order than A70(Za)7]to the total self-energy. In neutral hydrogen the higher-order terms in the self-energy contribute about 0.1 parts per million to the Lamb shift, which is nearly 100 times smaller than the uncertainty due to proton structure (see, for example, Lundeen and Pipkin, 1981). At 2 = 18 the contribution is only about 1% of the Lamb
0
state to the total FIG. 29. Ratio of the higher-order terms in the self-energy of the self-energyobtained by comparing the seriesexpansionvalues through terms of the order of& (Z,)’ with numerical calculationsto all orders in Z,. The series expansion changes sign near Z = 60 allowing the ratio of the higher-orderself-energyto the total self-energyto exceed unity for very high Z. (From Gould, 1985a.)
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shift, which is equal to or smaller than the typical experimental uncertainties (Gould and Marrus, 1983; Woods el af.,1982; Strater el af., 1984; Pellegrin et af.,1982). At Z = 92, however, the higher-order terms are essentially the entire self-energy contribution, and make up over half of the total Lamb shift. Previous Lamb-shift experiments are compatible with both the series expansion and numerical calculationsto all orders in &. Again, this is because the higher-order terms in the self-energy are large only at very high 2 hence they are not measured in present QED experiments. A consequence is that a significant deviation from QED at high Z is unobservable in previous experiments, and the difference between the numerical calculation and the series expansion is not well tested. The currently available flux of los uranium ions per second at the Lawrence Berkeley Laboratory's Bevalac and a planned increase of a factor of ten makes several atomic spectroscopy experiments which test high-Z QED possible. These experiments include: (1) measurement of the 2 'Po lifetime in a heliumlike very heavy atom which, with the 2 'P0-2 'S, electric dipole matrix element, and the 2 'Po- 1 'Sotwo-photon decay rate determines the 2 'P0-2 3S1splitting;(2) direct spectroscopic measurement of the uv transitions 2 'P0-2 3S, in heliumlike atoms and the 2 2P1,2-2 2S1, transition in lithiumlike atoms (about 254 and 283 eV in heliumlike uranium and lithiumlike uranium, respectively); (3) measurements of the 2 'P3/2- 1 2S1,2 transition in a hydrogenlike atom (roughly 102 keV in hydrogenlike uranium); (4) measurements of the 2 'P2-2 'S, transition in a heliumlike atom (about 4.5 keV in heliumlike uranium); ( 5 ) measurements of the hyperfine splitting in hydrogenlike thallium (about 3800 A); and (6) measurements of the g factor of the 1 2Sl/2state of a hydrogenlike atom. Measurements ( 5 ) and (6) are tests of the QED contribution to the anomalous magnetic moment of an electron bound in a Coulomb field. A more complete description of very high-Z QED experiments can be found in Gould (1985a,b).
IX. Conclusions Studies of ionization and capture processes using relativistic heavy ions can probe theories of ionization and electron capture in a regime where the ion velocity equals or exceeds the K-electron velocity. For ionization, this has the consequence that we can, for the first time, connect studies of innershell ionization (which have traditionally been made using light ions inci-
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dent on heavy atoms at low to medium velocities where u/uK 5 3 ) with studies ofp H, p He, and He He collisions where u/uk> 1. By studying projectile ionization as a function of the perturbing nuclear charge, one can continuously connect measurements in asymmetric collisions where there are small perturbing charges with studies of symmetric collisionsin the region of v/uK 1. The study of NRC at relativistic velocities has significantly advanced the theory of electron capture. NRC into heavy ions is dominated by capture of outer-shell electrons into outer-shell projectile orbitals. The eikonal theory has evolved to take into account outer-shell capture. The study of relativistic ions allows, for the first time, the formulation of an ab initio theory of ions in matter incorporating excited-state effects. Although minimal models have been used until now, the theories are in good agreement with measurements of projectile charge states emerging from solid targets and projectile x-ray production. The questions we can answer with these models concern the differences between gas- and solid-target charge states, the origin of projectile K, x-ray production in solid targets, and what is measured when one obtains cross sections from fits of the targetthickness dependence of projectile charge states in thin and thick targets. Atomic collisions at ultrarelativistic energies have not yet been explored experimentally.Electron capture from pair production is expected, but reliable estimates of the pair-capture cross sectionshave yet to be calculated. For high-2 ions and targets, REC and NRCcareexpected to be of comparable magnitude; hence it may be possible to test asymptotic theories of NRC. Atomic-structure experiments will grow more sophisticated as facilities improve and experienceis developed. Detailed understanding of relativistic atomic collisions enables the spectroscopiststo populate desired states efficiently, minimizing interfering transitions and background. The measurements that have been made until now have barely begun to explore the myriad of possible studies of relativistic heavy ions. This article has concentrated on studies of cross sections for atomic-collision processes since such measurements are the simplest to do under the present experimental conditions. We look forward to the application of relativistic heavyion storage rings, and new relativistic heavy-ion accelerators to atomic physics.
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ACKNOWLEDGMENTS
We thank our many colleagues who participated in the measurements and analysis of relativistic heavy-ion processes and theorists who have contributed calculations and to the
RELATIVISTIC HEAVY-ION-ATOM COLLISIONS
38 1
interpretation of the data. The measurements presented in this article are the results ofcollaborations with J. Alonso, S. A. Andriamonje, H. Bowman, H. Cradord, D. Greiner, D. H. H. Hoffman, P. Lindstrom, W. E. Meyerhof, J. D. Molitoris, E. Morenzoni, Ch. Munger, J. 0. Rasmussen, D. Spooner, Ch.Stoller,T. J. M. Symons, P. Thieberger, H. Wegner, J. S . Xu, and Z. Z. Xu, We thank J. Eichler, J. H. McGuire, and B. L. Moiseiwitsch for making available unpublished calculations. We especiallythank the operators and the staff and management of the Lawrence Berkeley Laboratory’s Bevalac for making experiments with relativistic heavy ions possible. This work was supported in part by the National Science Foundation under Grant No. PHY-83-13676, and in part by the Director, Office of Energy Research; Office of Basic Energy Sciences, Chemical Sciences Division of the U S . Department of Energy, and by the Office of High Energy and Nuclear Physics, Nuclear Science Division of the U.S. Department of Energy, under Contract No. DE-AC-03-76SF00098.Portions ofthe manuscript were prepared by one of us (H.G.) while visiting the Aspen Center for Physics. REFERENCES Ahlen, S. P. ( 1980). Rev. Mod. Phys. 52, I2 1. Ahlen, S . P., and Tarle, G. (1983). Phys. Rev. Lett. 50, I 1 10. Allison, S . K. (1958). Rev. Mod. Phys. 30, 1157. Alonso, J., and Gould, H. (1982). Phys. Rev. A 26, I 134. Amundsen, P. A., and Aashamer, K. (1981). J. Phys. B 14,4047. Amundsen, P. A., and Jakubassa, D. H. (1980). J. Phys. B 13, L467. Amundsen, P. A., Kocbach, L., and Hansteen, J. M. (1976). J. Phys. B 9, L203. Anholt, R. (1978). Phys. Rev. A 17,976. Anholt, R. (1979). Phys. Rev. A 19, 1009. Anholt, R. (1985a). Phys. Rev. A 31, 3575. Anholt, R. (1985b). Rev. Mod. Phys. 57,995. Anholt, R. (1986). Phys. Lelt. 1 1 4 4 126. Anholt, R., and Eichler, J. (1985). Phys. Rev. A 31, 3505. Anholt, R., and Meyerhof, W. E. (1977). Phys. Rev. A 16, 190. Anholt, R., and Meyerhof, W. E. (1986). Phys. Rev. A., submitted. Anholt, R., Nagamiya, S., Bowman, H., Ioannou, J., Rauscher, E., and Rasmussen, J. 0. (1976). Phys. Rev. A 14,2103. Anholt, R., Ioannou, J., Bowman, H., Rausher, E., Nagamiya,S., Rasmussen,J. O., Shibata, T., and Ejiri, H. (1977). Phys. Lett. 59A, 429. Anholt, R., Meyerhof, W. E., Stoller, Ch., Morenzoni, E., Andriamonje, S. A., Molitoris, J. D., Hoffmann, D. H. H., Bowman, H., Xu, J.-S., Xu, Z.-Z., and Rasmussen, J. 0.(1984a). Phys. Rev. Lett. 53, 234. Anholt, R., Meyerhof, W. E., Stoller,Ch., Morenzoni, E., Andriamonje, S. A., Molitoris, J. D., Hoffmann, D. H. H., Bowman, H., Xu, J.-S., Xu, Z.-Z., Murphy, D., Frankel, R,Crowe, K., and Rasrnussen, J. 0. (1984b). Phys. Rev. A 30,2234. Anholt, R., Meyerhof, W. E., Gould, H., Munger, Ch., Alonso, J., Thieberger, P., and Wegner, H. E. (1985). Phys. Rev. A 32, 3302. Anholt, R., Stoller, Ch., Molitoris, J. D., Spooner, D. W.,Bowman, H., Xu, J . 4 , Xu, Z.-Z., Rasmussen, J. O., and Hoffmann, D. H. H. (1986). Phys. Rev. A 33,2270. Ashley, J. C., Ritchie, R. H., and Brandt, W. (1973). Phys. Rev. A 8,2402. Bak, J. F., Meyer, F. E., Petersen, J. B. B., Uggerhsj, E., Bstergaard, K., Msller, S . P. Sorensen, A. H., and Siffert, P. (1983). Phyt. Rev. Left.51, 1163.
382
R.Anholt and Harvey Gould
Bak, J. F., Petersen, J. B. B., Ugerhsj, E., Ostergaard, K., Msller, S. P., and Sorensen, A. H. (1986). Physica Scripta 33, 147. Basbas, G., Brandt, W., and Laubert, R. (1971a).Phys. Lett. 34A, 277. Basbas, G.,Brandt, W., Laubert, R., Ratkowski, A., and Schwartzchild,A. ( I 97 1 b). Phys. Rev. Lett. 27, 17 I . Basbas, G., Brandt, W., and Laubert, R. (1973). Phys. Rev. A7,983. Basbas, G., Brandt, W., and Laubert, R. (1978). Phys. Rev. A 17, 1655. Bates, D. R. (1959). Proc. Phys. SOC. London 73,227. Bates, D. R., and Griffing, G. W. (1953). Proc. Phys. SOC.London Ser. A 66,961. Bates, D. R., and Griffing, G. W. (1954). Proc. Phys. SOC. London Ser. A 67,663. Bates, D. R., and Griffing, G. W. ( 1 955). Proc. Phys. SOC.London Ser. A 68,90. Becker, R. L., Ford, A. L., and Reading, J. F. (1980). J. Phys. B 13,4059. Benke, O., and Kropf, A. (1978). At. Data Nucl. Data Tables 22,249. Bethe, H. (1930). Ann. Phys. 5,325. Bethe, H. (1932). Z. Phys. 76,293. Bethe, H. A., and Ashkin, J. (1953). In “Experimental Nuclear Physics” (E. Segre, ed.), Vol. I , p. 166. Wiley, New York. Bethe, H. A., and Fermi, E. (1932). Z. Phys. 77, 296. Bethe, H. A., and Heitler, W. (1934). Proc. R. SOC.London Ser. A 146,83. Bethe, H., andSalpeter, E. E. (1957).“Quantum MechanicsofOne- and Two-ElectronAtoms.” Academic Press, New York. Betz, H. D. (1972). Rev. Mod. Phys. 44,465. Betz, H. D., and Grodzins, L. (1970). Phys. Rev. Lett. 25,211. Bhabha H. J. (1935a). Proc. Cambridge Philos. SOC.31,394. Bhabha H. J. (1935b). Proc. R. SOC.Ser. A 152,559. Bigs, F., Mendelsohm, L. B., and Mann, J. B. ( I 975). At. Data. Nucl. Data Tables 16,202. Bohr, N., and Lindhard, J. (1954). K. Dan. Videns. Selsk. Mat. -Fys. Medd. 28, (7). Borkowski, C. J., and Kopp, N. M. (1975). Rev. Sci. Inst. 46,95 1. Brandt, W., Laubert, R., and Sellin, I. (1966). Phys. Lett. 21, 5 18. Brim, J. S. (1977). J. Phys. B 10,3075. Briggs, J. S., and Taulbjerg, K. (1978).In “Topics in Current Physics” (I. A. Sellin, ed.), Vol. 5. Springer, New York. Briggs, J. S., Macek, J., and Taulbjerg, K. (1982). Comments At. Mol. Phys. 12, 1. Brinkmann, H. C., and Kramers, H. A. (1 930). Proc. Acad. Sci. Amsterdam 33,973. Brookhaven National Laboratory (1983). Proposal for a 1 5A-GeV heavy ion facility. Brookhaven National Laboratory Rep. BNL-32250 (unpublished). Brookhaven National Laboratory (1984). RHIC and quark matter, proposal for a relativistic heavy ion collider. Brookhaven National Laboratory Rep. BNL-5 1801 (unpublished). Chan, F. T., and Eichler, J. (1979). Phys. Rev. Lett. 42, 58. Cheng, K. T., and Johnson, W. R. (1976). Phys. Rev. A 14, 1943. Choi, B. H., Merzbacher, E., and Khandelwal, G. S. (1973). At. Data 5,291. Crawford, H. J. (1979). Ph. D. thesis, University of California, Lawrence Berkeley Laboratory, Rep. No. LBL-8807 (unpublished). Crawford, H., Gould, H., Greiner, D., Lindstrom, P., and Symons, T. J. M. (1983). Proc. 6th High-EnergyHeavy Ion Study, 2nd WorkshopAnornalons, Lawrence Berkeley Lab. June 28-July 1. Rep. LBL-16281, p. 583, and LBL-16241. Datz, S., Lutz, H. O., Bridwell, L. B., Moak, C. D., Betz, H. D., and Ellsworth, L. D. (1 970). Phys. Rev. A 2,430. Davidovic, D. M., Moiseiwitsch, B. L., and Nomngton, P. H. (1978). J. Phys. B 11, 847. Desiderio, A. M., and Johnson, W. R. (1971). Phys. Rev. A 3, 1267.
RELATIVISTIC HEAVY-ION- ATOM COLLISIONS
383
Drisko, R. M. (1955). Ph.D. thesis, Carnegie Institute of Technology (unpublished). Eichler, J. (1985). Phys. Rev. A 32, 1 12. Eichler, J., and Chan, F. T. (1975). Phys. Rev. A 20, 104. Elwert, G. (1939). Ann Phys. (Leipzig) 34, 178. Erickson, G. W. (1971). Phys. Rev. Lett. 47,780. Fano, U. (1963). Annu. Rev. Nucl. Sci. 13, 1. Folkmann, F., Gaarde, C., Huus, T., and Kemp, K. (1973). Nucl. Instrum. Methods 116,487. Ford, A. L., Fitchard, E., and Reading, J. F. (1977). Phys. Rev. A 16, 133. Gillespie, G. H., Kim, Y. K., and Cheng, K.-T. (1978). Phys. Rev. A 17, 1284. Glauber, R. J. (1959). In “Lectures in Theoretical Physics” (W. E. Brittin and L. G. Dunham, eds.). (Wiley Interscience), New York. Gould, H. ( I 984). Atomic physics aspects of a relativistic nuclear collider. Lawrence Berkeley Laboratory Rep. No. LBL-18593 (unpublished). Gould H. (1985a). Nucl. Instrum. Methods B9, 658. Gould H. (1985b). In “Atomic Theory Workshop on Relativistic and QED Effects in Heavy Atoms” (H. P. Kelly and Y. K. Kim, eds.), AIP Conference Proceedings No. 136, pp. 66 - 79. Lawrence Berkeley Laboratory Rep. LBL 19884. Gould, H., and Marms, R. (1983). Phys. Rev. A 28,2001. Gould, H., Greiner, D., Lindstrom, P., Symons, T. J. M., and Crawford, H. (1984). Phys. Rev. Lett. 52, 180; ibid., idem 52, 1654 (erratum). Gould, H., Greiner, D., Lindstrom, P., Symons, T. J. M., Crawford, H., Thieberger, P., and Wegner, H. (1985). Nucl. Insfrum. Methods B10,11,32. Gray, T. J., Richard, P., Jamison, K. A., Hall, J. M., andGardner, R. K. (1976). Phys. Rev.A 14, 1333. Greiner, D. E., Lindstrom, P. J., Bieser, F. S., and Heckman, H. H. (1974). Nucl. Instrum. Methods 116,2 1. Hansteen, J. M., Johnsen, 0. M., and Kocbach, L. (1975). At. Data Nucl. Data Tables 15,306. Heckman, H., Bowman, H., Karant, Y., Rasmussen, J. O., Wanvick, A., and Xu, Z. (1985). Lawrence Berkeley Report LBL- I903 1. Hill, K. W., and Menbacher, E. (1974). Phys. Rev. A 9, 156. Hoffman, D. H. H., Gem, H., LOW,W., and Richter, A. (1978). Phys. Lett. A 65,304. Hoffman, D. H. H., Brendel, C., Genz, H., U w , W., Miiller, S., and Richter, A. (1979). Z. Phys. A293, 187. Hubbel, J. H., Veigele, W. J., Brigs, R. A., Brown, R. T., Cromer, D. T., and Howerton, R. J. (1 975). J. Phys. Chem. Ref:Data 4,47 I . Hultberg, S., Nagel, B., and Olsson, P. (1 967). Ark. Fys. 38, I . Humphries, W. J., and Moiseiwitsch, B. L. (1984). J. Phys. B 17,2655. Humphries, W. J., and Moiseiwitsch, B. L. (1985). J. Phys. B 18,2295. Ishii, K., Kamiya, M., Sera, K., Morita, S., and Tawara, H. (1977). Phys. Rev. A 15,2126. Jackson, J. D. (1962). “Classical Electrodynamics” Wiley, New York. Jakubassa, D. H., and Kleber, M. (1975). Z. Phys. A 273,23. Jakubassa-Amundsen, D. H., and Amundsen, P. A. (1980). Z. Phys. A 298, 13. Jakubassa-Amundsen, D. H., and Amundsen, P. A. (1985). Phys. Rev. A 32,3106. Jamnik, D., and Zupancic, C. (1957). K. Dan. Videns. Selsk. Mat.-Fys. Medd. 31, ( 2 ) . Jarvis, 0. N., Whitehead, C., and Shah, M. (1972). Phys. Rev. A 5, 1198. Johnson, D. E., Basbas, G., and McDaniel. F. D. (1979). At. Data Nucl. Data Tables 24, 1. Johnson, W. R., and SoK G. (1985). At. Data Nucl. Data Tables 33,405. Khandelwal, G. S., Choi, B. H., and Merzbacher, E. (1969). At. Data 1, 103. Kienle, P., Kleber, M., Povh, B., Diamond, R. M., Stephens, F. S., Grow, E., Maier, M., and Proetel, D. (1973). Phys. Rev. Lett. 31, 1099.
384
R. Anholt and Harvey Gould
Kingston, A. E., Moiseiwitsch, B. L., and Skinner, B. G . (1960). Proc. R. SOC.Ser. A 258,237. Kissel, L., Quarles, C. A., and Pratt, R. H. (1983). At. Data Nucl. Data Tables 28, 381. Kleber, M., and Jakubassa, D. H. (1975). Nucl. Phys. A 252, 152. Koch H. W., and Motz, J. W. (1959). Rev. Mod. Phys. 31,920. Landau, L., and Lifshitz, L. (1934). Phys Zs. Sov. V. 6, 244. Lawrence Berkeley Laboratory (1979). The VENUS project: Accelerator and storage ring for relativistic heavy ion physics. Lawrence Berkeley Laboratory, PUB-5025. Lawrence Berkeley Laboratory ( 1982).The TEVALAC A national facility for relativistic heavy ion research to 10-GeVper nucleon uranium. Lawrence Berkeley Laboratory, PUB-508 1. Lee, C. M., Kissel, L., Pratt, R. H., and Tseng, H. K. (1976). Phys. Rev. A13, 1714. Leung, P. T., and Rustgi, M. L. (1983). Phys. Rev. A 28,2529. Li-Scholz, A., Coil$, R., Preiss, I. L., and Scholz, W.(1973). Phys. Rev. A 7, 1957. Lundeen, S. R., and Pipkin, F. R. (1981). Phys. Rev. Lett. 46,232. Macek, J. H., and Alston, S. (1982). Phys. Rev. A 26,250. McGuire, J. H., Stolterfoht, N., and Simony, P. R. (1981). Phys. Rev. A 24, 97. McGuire, J. H. (1982). Phys. Rev. A 26, 143. McKinley, W. A., and Feshbach, H. (1948). Phys. Rev. 74, 1759. Merzbacher, E., and Lewis, H. W. (1958). In “Encyclopedia of Physics” (S. Fliigge, ed.). Springer, Berlin. Meyerhof, W. E., Anholt, R., Eichler, J., Gould, H., Munger, Ch., Alonso, J., Thieberger, P., and Wegner, H. E. (1985). Phys. Rev. A 32,3291. Middleman, L. M., Ford, R. L., and Hofstadter, R. (1970). Phys. Rev. A 2, 1429. Moiseiwitsch, B. L. (1980). Ad. At. Mol. Phys. 16, 281. Moiseiwitsch, B. L. (1985). Phys. Rep. 118, 135. Moiseiwitsch, B. L., and Stockman, S. G. (1980). J. Phys. B 13,2975. Msller, C. (1 932). Ann. Phys. (Leipzig) 14, 53 1. Mohr, P. J. (1974). Ann. Phys. (N. Y.)88,26. Mohr, P. J. (1982). Phys. Rev. A 26,2338. Mohr, P. J. (1983). At. Data Nucl. Data Tables 29, 453. Nikolaev, V, S. (1965). Vsp. Fiz.Nauk85,679 (Sov. Phys. Vsp. 8,269). Nishina, Y., Tomonaga, S., and Kobayasi, M. (1935). Sci. Pap. Inst. Phys. Chem. Res. Jpn. 27, 137. Nordheim, L. (1935). J. Phys. Radiurn6, 135. Oppenheimer, J. R. (1928). Phys. Rev. 31, 349. Oppenheimer, J. R. (1935). Phys. Rev. 47, 146. Park, J. T. (1983).Adv. At. Mol. Phys. 19,67. Pellegrin, P., El Masri, Y., Palffy, L., and Priells, R. (1982). Phys. Rev. Lett. 49, 1762. Pratt, R. H. (1960). P h p . Rev. 119, 1619. Pratt, R. H., Ron, A., and Tseng, H. K. (1973). Rev. Mod. Phys. 45,273. Quarles, C. A,, and Heroy, D. B. ( 1981). Phys. Rev. A 24,48. Racah, G. (1937). Nuovo Cim. 14,93. Raisbeck, G., andYiou, F. (1971). Phys. Rev. Lett. 4, 1858; ibid., Phys. Rev. A4, 1858. Rice, R., Basbas, G., and McDaniel, F. (1 977). At. Data Nucl. Data Tables 20, 503. Rudge, M. R. H., and Schwartz, S. B. (1966). Proc. Phys. SOC.London 88,563. Salamon, M. H., Ahlen, S. P., and Tarle, G . (1981). Phys. Rev. A 23, 73. Salem, S. I., and Schultz, C. W. (197 I). At. Data 3,215. Sapirstein, J. (1981). Phys. Rev. Lett. 47, 1723. Sauter, F. (1931). Ann. Phys. (Leipzig) 9, 217; 11,454. Schmickley, R. D. (1966). Ph.D. thesis, Stanford University, unpublished.
RELATIVISTIC HEAVY-ION-ATOM COLLISIONS
385
Schnopper, H. W., Betz, H., Delvaille,J. P., Kalata, K., Sohval,Jones, K. W., and Wegner, H. E. ( 1972). Phvs. Rev. Lett. 29. 898. Schnopper, H. W., Delvaille, J. P., Kalata, K., Sohval, A. R., Abdulwahab, M., Jones, K. W., and Wegner, H. E. (1 974). Phys. Lett. 47A, 6 1. Scofield, J. H. ( 1 973). Theoretical photoionization cross sections. Unpublished report, UCRL-51326. Scofield, J. H. (1978). Phys. Rev. A 18,963. Shah, M. B., and Gilbody, H. B. (1981). J. Phys. B 14,2361. Shakeshaft, R. (1979). Phys. Rev. A 20,779. Shakeshaft, R., and Spruch, L. (1979). Rev. Mod. Phys. 51,440. Skinner, B. G. (1962). Proc. Phys. SOE.(London) 79,7 17. Slater, J. C. (1930). Phys. Rev. 36,57. Spindler, E., Betz, H. D., and Bell, F. (1979). Phys. Rev. Lett. 42,832. Stljdter, H. D., von Gerdtell, L., Georgiadis, A. P., Muller, D., von Brentano, P., Sens, J. C., and Pape, A. (1984). Phys. Rev. A 29, 1596. Thieberger, P., Wegner, H. E., Alonso, J., Gould, H., Anholt, R., and Meyerhof, W. E. (1985). Proc. Particle Accelerator Con$ (IEEE-Nucl. Sci. NS32, 1267). Thorson, W. R.(1975). Phys. Rev. A 12, 1365. Tseng, H. K., Pratt, R. H., and Lee., C. M. (1979). Phys. Rev. A 19, 187. Von Weizsiicker, C. F. ( 1934). Ann. Phys. 88,6 12. Waddington, C. J., Freier, P. S., and Fixsen, D. J. (1983). Phys. Rev. A 28,464. Williams, E. J. (1935). K. Dan. Videns. Selsk. Mat.-Fys. Medd. 13 (4). Woods, 11, 0. R., Patel, C. K. N., Murnick, D. E., Nelson, E. T., Leventhal, M., Kugel, H. W., and Niv, Y. (1982). Phys. Rev. Lett. 48, 398. Yamadera, A., Ishii, K., Sera, K., Sebata, M., and Morita, S. (1981). Phys. Rev. A 23,24.
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ADVANCES IN ATOMIC AND MOLECULAR PHYSICS, VOL. 22
CONTINUED-FRACTION METHODS IN ATOMIC PHYSICS S. SWAIN Department of Applied Mathematics and Theoretical Physics The Queen S University Belfast BT7 INN, Northern Ireland
I. Introduction A. SCOPEAND OUTLINE
The object of this article is to describe the application of a systematic continued-fraction (CF) theory to problems in quantum mechanics with particular reference to atomic and molecular physics. Instead of expressions for the observables being given as perturbation series, they are given in the form of continued fractions. We hope to convince the reader that this approach is powerful and versatile with definite conceptual and computational advantages over conventional perturbation theories. The main advantages can be summarized briefly as follows: First, certain subsets of terms in the conventional perturbation series are summed to all orders; this means, for example, that “self-energy’’ effects are automatically dealt with properly. Second, the rate of convergence is usually much faster than that of power series. Third, the solutions can be written down directly using fairly simple rules without solving the equations of motion. Fourth, the continued-fraction approach is a unifying one, as the various other perturbation methods (Lennard - Jones- Wigner - Brillouin, Rayleigh - Schriidinger, projection operator, etc.) can all be understood within this framework. In this article we limit ourselves to discussing the systematic form of CF perturbation theory first put forward by Feenberg. We are forced to be very selective in our choice of references; to attempt to refer to all papers using continued fractions would impossible. While we restrict ourselves to papers which give applicationsto atomic and molecular physics, we do not attempt to give a complete bibliography of even this limited area. 387 Copyright 0 1986 by Academic Press,Inc. All rights of reproductionin any form reserved.
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The work is organized as follows: In Section I1 we consider the purely mathematical problem of obtaining CF solutions to systems of linear homogeneous and inhomogeneousequations; the remaining sections apply these methods to problems of physical interest. Section II,A gives the results which are most useful when there is no degeneracy; Section II,B gives the generalizations appropriate when there is degeneracy; and Section II,C shows how to reduce the number of linear equations by eliminating a given subset of variables from the original system. The latter is used in comparing the CF and projection-operator approaches to perturbation theory and in deriving the rate equations of Section IV. It also gives another way of dealing with degeneracy. Section I11 derives the main results for quantum-mechanical perturbation theory with a time-independent Hamiltonian and compares the CF resultswith the conventionalperturbation theories. Section IV on the density matrix and rate equations can be considered as an example of the application of CF perturbation theory to the Von Neumann equation of motion for the density matrix, but because of its importance we give it a separate section. Some examplesare given but, except for the section on rate equations, these are chosen to illustrate the theory and not for their own importance. References to explicit use of these methods in realistic problems are given at the appropriate places in the text. Finally, in the Appendix, we derive the basic determinantal expansions on which this work is founded. SURVEY B. LITERATURE
Continued-fraction formalisms for the eigenvalue problem and for scattering theory were first obtained using algebraic techniques by Feenberg ( 1948a,b), who also discussed its relation to the Rayleigh- Schr6dinger and Wigner- Brillouin perturbation methods (Feenberg, 1958). Heuristic methods of obtaining Feenberg’s results were presented by Richards ( 1948) and Feshbach (1948). (See also Morse and Feshbach, 1953.) Methods of improving the convergence of the Wigner - Brillouin perturbation series were derived and related to the CF perturbation theory (Feenberg, 1956; Goldhammer and Feenberg, 1956; Lippmann, 1956; Young et al., 1957). These methods were applied by Amos ( 1970, 1972) to Hartree - Fock calculations of the ground-state energy of helium and comparisons made with other approaches. Bendazzoli et al. ( 1970) made calculations on He-like ions and discussed upper and lower bounds. (See also Brandas and Goscinski, 1970.)
The earlier applicationsto atomic physics that we have cited were mostly concerned with atomic structure. The later applications dealt with the calculation of transition probabilities and the interaction of atomic systems with
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389
the electromagnetic field. Gontier et al. (1975a,b, 1976)developed a continued-fraction approach for the resolvent operator and applied it to a variety of phenomena, including multiphoton ionization, multiphoton resonances, and Raman processes. Mower ( 1980)and Cresser and Dalton ( 1980)developed projection-operatortheories which lead to CF expression for the principal quantities. We have developed a linear-algebraicapproach to obtaining the CF solutions of Feenberg’s (Swain, 1975a, 1976, 1977; Jackson and Swain, 1981) which arose out of earlier adhoc treatments (Swain, 1973a,b, 1974~).This is the approach we adopt here. While we are particularly concerned with quantum-mechanical perturbation theory, we should emphasize that the method is applicable to any set of linear equations. The latter are of frequent occurrence in physics, and we have applied the approach to the calculation of Bloch - Siegert shifts and multiple quantum transitions in spin one-half and spin-one systems (Swain, 1974b; McClean and Swain, 1976; Hermann and Swain, 1977; Jackson and Swain, 1982; Jackson, 1982). Outside the field of atomic and molecular physics, Feenberg’s methods have been applied to the calculation of thermodynamic Green’s functions (Goscinski and Lukman, 1970;Bowen, 1975)and in statistical physics (e.g., Mori, 1965). In this review we are concerned exclusively with analytical approachesto perturbation problems. H a n d et af.(1978) discuss the numerical aspects of CF expansions with particular attention to scattering theory and statistical physics. They also cite a number of wider CF applications. The recursive method is a powerful numerical technique involvingCF methods which has been developed by the Cambridgegroup for calculating the electronic structure of disordered solids (Haydock 1980, 1982). It has been adapted to discussthe theory oflaser-molecule interactionsby Nauts and Wyatt (1983, 1984) using a Floquet theory approach (Chu, 1985).
11. Continued-Fraction Solutions to Linear Equations A. THEBASICSOLUTION
I . Algebraic Expressions Many problems in physics reduce to the solution of systems of linear equations. Two cases which will particularly concern us are the eigenvafue
S. Swain
390
problem, where we have to solve the homogeneous equations
x
(zidij- Hij)uj= 0
i
for the eigenvalues z and eigenvectors u of the Hamiltonian H, and the time-dependent Schrodinger equation, iv‘ = Hv, which, if H is time independent, may be Laplace transformed F(z) = -ilexp(izt)u(t) dt
(2)
and written in the form of inhomogeneous equations
C (zdij- Hij)8j= ~ ~ ( 0 )
(3)
j
where the * denotes the Laplace transform and z is the Laplace variable. For maximum generality we consider the purely mathematical problem of solving linear equations. We will obtain formal solutions involving continued fractions which are, however, rather complicated. We then give simple rules which enable us to write the solutions in a straightforward way. (The situation is similar to the Feynman graph method, where the formal solution is cumbersome but where in practice the solution may be written down using the Feynman rules.) We consider first the set of Ninhomogeneous equations in N unknowns
Caiixj=bi,
i = 1, 2,
. . . ,N
(4)
i
where N is taken to be finite. (We allow N this system is given by Cramer’s rule as
m later.) The formal solution of
where A is the determinant of the matrix A and AJi is the determinant obtained from A by removing thejth row and the ith column. Anticipating the application of these equations to quantum mechanics, we shall often refer to the indices i, j , . . . , etc., as “states.” In order to obtain a continued-fraction expression for xi we make use of the following expansion for the determinant of a matrix which is proved in the appendix
CONTINUED-FRACTIONMETHODS
39 1
where A , , denotes the determinant obtained from A by removing the ith, jth, . . . , and nth rows and columns, and Ai = A”. We have to adopt the convention that A,, . . . = 1, i.e., the determinant obtained by removing all rows and columns has the value unity. For i # j we have the expansion , ,
Henceforth an asterisk on the summation sign will be used to indicate that the indices being summed over must be distinct from one another and from any other indices which appear in the summand. We define the D functions by the relation 0%’ .....7 = Aij . . . J A i j . . mn . r (8) (with D, . ,= A/A, ,). Notice that the superscripted indices refer to rows and columns which have been deleted from both the determinants on the right-hand side of Eq. (8). Thus in the expansion of the D function no superscripted indices appear. The definition also implies that D functions with more than one subscript may be decomposed into products of D functions with fewer subscripts, as the following example illustrates ,
, ,
, ,
, ,
,
(9) For convenience we define GY;;.. to be the reciprocal of the corresponding D function Dij = A/Aij= (A/Ai)X (Ai/Aij)= DiDj = DjD$
(10) GY;;; = 1/DF IJ . ... It will be shown that G plays the role of a Green’s function or propagator. Using the definitions in Eqs. (8) and (10) together with Eq. (7) in the Cramer’s solution [Eq. ( 5 ) ] , we may write
The D functions are obtained by iteration from the expression of Eq. (6) and the definition of Eq. (8)
S. Swain
392
Note that henceforth the asterisk on the summation signs indicates that the variables summed over are not allowed to take on the values of any of the superscripts, in addition to the earlier defined restrictions [given after Eq. (7)] on the values the indices summed over are allowed to take. This constitutes the full definition of Z*, and is consistent with our earlier usage. Next we briefly consider the case of homogeneous equations
Ca,xj=O,
i = 1,
...,N
(13)
1
The condition for consistent solutionsis A = 0: IfA, # 0 this is equivalent to AIA, = 0. That is, we can write the condition for consistent solutions as
Di = 0 (14) where i labels any row for which A, # 0. We can only find the ratio of solutionsto the system of Eq. (1 3): If r is such that A, # 0, we may write the equations as
2 aii(xj/x,)= -a,,,
i, J # r
(15)
j+r
From Eq. (1 l), a nontrivial solution is
+ c*aijaj,/D;i+
xi/x,= - ail/D;
* * *
i
-D(,,)/D[
(16)
where
D : ) . . . " a , - C * a . . aIJ . /JrD fJ" . . . + i
- -
(17)
Note that 0:).. . is closely related to the quantity D$ . . . defined in Eq. ( 12): The only differencesare that the final state in each process is r, not i, and that the states i and I are added to the superscripts on every D function on the right-hand side. We have the relations
D$!).. . = DT . . . D(!$...= O F . . .
. . .) (i # u,v, . . .)
( i # u,v,
(18)
With these definitions the solution of Eq. (1 1) may be written
We should emphasize that our calculations to date are exact and the series terminate for a finite matrix.
CONTINUED-FRACTIONMETHODS
393
2. Rules for the Solutions Let us now obtain the rules for writing down the solutions. We note that the terms in Eq. (1 1) may be represented in the following way
With each term we have associated a process, which we have written above the contribution. All the terms in our solutionsare representedby irreducible processes-that is, processes for which all the intermediate indices are distinct from each other and from the initial and final indices. By inspection, we deduce the following rules for calculating the solution to order M in the coefficients aij: a. x Rules
To calculate xi to Mth order, (1) List all the irreducible processes up to and including those of order M (i.e., all processes involving from 0 to Marrows) which proceed from state i to any accessible state r. (2) With each transition k -+ 1associate a factor aw. (3) The contribution of each process is (- 1y G i . X [the product of all the a, factors from part (2) above] X b,, where i, . . . , rare all the indices listed in the process and r is the final index. (4) Add the contributions of all irreducible processes up to and including the Mth. The Gi . . which appear in these expressions are found by splitting G, . . into factors with only one subscript as in Eq. (9), e.g., ,
,
,
,
Gi.. . r = and using the following:
, ...D ; . . . y
( ~ p , ~ i j
(21)
b. D-Function Rules To calculate Djk . . to Mth order, ( 1) List all the irreducible processes which take one from index i through 0, 1,2, . . . ,(M - 1) intermediate indices (none ofwhich can be equal to any of the indices j , . . . , r which are superscripted) and back to the index i again. (2) With each transition k 1 associate the factor aw. (3) The contribution of each process is (- lyG$k,.,.,.;X [the product of all the aw factors from part (2)] where st . . . u are the intermediate states
-
S. Swain
394
passed through in going from state i to state i. The zeroth-order contribution is simply aii. (4) Add the contributions of all the irreducible processes up to and including the Mth. [The rules for calculating the D&. . . functions are similar except that (a) the processes go from index i to indexj and (b) all the G functions in D (3) have both “i” and “j” included in the superscripts.] In the next section we indicate how Di . . may be calculated directly, i.e., without using the decomposition of Eq. (2 1). However, for nondegenerate systems, the present approach is the simplest. Taking a system of three inhomogeneous equations in three unknowns as an example, we find for xI, say, the following irreducible processes ,
1;
142,
1-3;
1-2-3,
1-3-2
(22)
(Processes of different orders are separated by semicolons.) No further irreducible processes are possible. According to the x rules the solution is
For the D functions implicit in this equation the D rules give the following processes and contributions
D,:
1-2-1,
1-1;
143-1;
1-3-2-1,
1-2+3-1
D1= a11- a12a21G: - a13a31G:
-- - -
f a12a23a32G:3
(24)
+ a13a32a21Gh 0::
Di2:
2 3
2;
2
3,
Di2= a33
3
2,
Dl = a22- a2,a3,Gi2
These give the full solution for xi bl
-
XI = a11
a12b2
a22
-
-
- a23a32/a33 a12a21
a22
- a23a321a33
a13b3
+
a12a23b3
a33 - a32a23Ia22
-
a13a31
a33
- a32a23/a22
a22a33 +
a12a23a31
+ a,3a32b2 - a23a32 + a13a32a21
a22a33
- a23a32 (25)
which is easily confirmed by elementary calculations.
CONTINUED-FRACTIONMETHODS
395
B. EXPANSION OF A DETERMINANT BY Two OR MOREELEMENTS The determinantal expansion of Eq. (6) makes the state i special. In some circumstances the symmetry of the problem may suggest that two statessay thepth and the qth-should be treated on the same footing. An example where this is so is the case where the matrix A has degenerate eigenvalues. By repeatedly using the expansion of Eq. (6), we have shown (Swain, 1977; and Appendix) how an expansion with respect to two (or more) elements analogous to Eq. (6) can be developed. We found
A
= (appaqq- apqawMpq
-
C*
(appaqiaiq
+ aqqapiaip - apqaqiaip
i
From this expression we obtain = (appa,
-
-
x*
apqaqp)
(appaqiaiq + aqqapiaip - apqaqiaip
i
+
- apiaMaw)Dp (27) We note that all the D functions which appear on the right-hand side of this expression have p and q as superscripts, and hence the expansions of these D functions contain no matrix elements belonging to the pth or qth row or column. As we shall see, this is an advantage when solving the degenerate eigenvalue problem by iteration. The contributions in Eq. (27) may be obtained by the following rules: I. D, Rules (In the following, the term “an intermediate state” means any state passed through except p or q.) The rules for calculating Dpqto Mth order are similar to the ordinary D-function rules except that ( 1 ) The lowest-order process that contributes is the second, (2) There are two types of process:
(a) “connected” processes, in which the system passes irreducibly from
S. Swain
396
statep through K(K = 0,1,2 . . . ,M)intermediatestates to state qand then through M - K intermediate states back to state p; (b) "disconnected" processes, in which the processes arise in pairs: one in which the system proceed irreducibly from state p to state p through K intermediate states and the other in which the system proceeds irreversibly from state q to state g through M - K intermediate states; (3) With each transition k + 1 associate the factor a , and take the product of all such factors together with the factor (- 1 . , where a, b, c, . . . are all the intermediate states which appear in the process, with R = 1 for connected processes and R = 2 for disconnected processes. For example, the contributions to D, through to fourth order are
y+R/wk , ,
~
_
_
Order
Disconnected
Second Third
p-p;g+q p+p;q-i-q p+ i-+p; p-p;q-i-j-q p + i p; q +j -+ q p +i j - p ; q - q
Fourth
-
+ +
Connected p-4-4 p-q+i-*p p+i+qdp p+q+i-j-p p + i 4q j + p p - i 4j - q + p
-
(28)
The second- and third-order processes give rise to the terms of Eq. (27). The rules for expansion by, say, three elements are similar except that the disconnected processes come in two's (R = 2) or three's (R= 3). (For more details and examples see Swain, 1977.) Using the relations of Eq. (9) between D functions, we may write
Di = DwD;W/DA; D, = D,DT/Dg; etc. (29) By means of such expressionswe may express the solutionsof Eq. (1 1) to the inhomogeneous equations (4) as
When there are several statesp, g, r, . . . which we wish to be treated on an equal footing, the corresponding equations may be written down by inspection. The special feature of Eqs. (30) and (3 I ) is that all the D functions in the
CONTINUED-FRACTIONMETHODS
397
second factor of each equation involves p and q only as superscripts, so that their expansions contain no explicit reference to either p or q. This is an advantage when the states p and q are degenerate, because then all the complications due to degeneracy are contained in the initial l/D, factor, and these can be handled by means of the expansion in Eq. (27). C. ELIMINATION OF VARIABLES In certain problems, it may be expedient to eliminatesome of the variables from the set of linear equations [Eq. (4)]before making a complete solution. For example, in the case of degeneracy,we may eliminateall the states except the two degenerate ones, leaving us with a 2 X 2 matrix to solve. In the general case, let us suppose that the indices of the variables to be retained are denoted by the set W (for “wanted”) while those of the variables to be eliminated are labeled by U (for “unwanted”) (U W = (1,2, . . . , N ) ) . We write the Eqs. (4) for the latter set in the form
+
C a i r x , = b i - C a . 11. xI . = c i , rE U
~ E U
j€W
Using Eq. (1 l), the solution is
Note that the superscript W on the D functions and the definition of Z* means that the states which are summed over can only belong to the unwanted set. In the second line we have substituted for cj the expression cj = bj - X&yajkxk and simplified. These expressions for the unwanted xr are now substituted into the remaining equations for the wanted xi
2 aijxj+ 2U a,xr = b,,
jCW
iE W
(34)
rE
After a little simplification, we obtain the equations for the wanted variables only
x aiijxj= bi, -
j€W
iE W
(35)
398
S. Swain
where the "renormalized" coefficients uijand bi are given by
Expressions (35) and (36) give the most general formulation for the CF solutions to a set of linear equations; they reduce to the solutions of Eq. (1 1) by taking the wanted set to be just the state i [ W = (2): xi = &/Zii = Eq. ( 1 l)]. The renormalized coefficients Zij may be calculated by rules identical to those for calculating the D(ij,functions. The renormalized vector is calculated similarly by listing all the processes which begin in state i and end up in any other state belonging to the unwanted set.
6.
111. Perturbation Theories In this section we apply the CF solutions of Section I1 to the perturbation problem and consider its relationship with other perturbation theories.
A. THECONTINUED-FRACTION APPROACH: NONDEGENERATEPERTURBATIONTHEORY We first consider the eigenvalue problem defined by Eq. (1) from the CF point of view, assuming that the Hamiltonian H may be decomposed into the sum of an unperturbed Hamiltonian Ho and a perturbation V. We assume that the eigenstates li) and corresponding eigenvalues of Ho are known
H,(i) = eJi) (37) Thus for the matrix A of Eq. (4) we take A , = ziSij- H,, where Hij = (ilHlj}, and b = 0. To determine the eigenvalues zi we need to solve the secular equation A = (z,Z- HI = 0, where I is the unit matrix. Clearly the secular equation is equivalent to Di = A/Ai = 0 (except possibly when A; = 0 -a case that we exclude). Noting that the diagonal elements of A are of the form uj = zi - ej - V, = zi - Cj while the off-diagonal elements in-
CONTINUED-FRACTION METHODS
399
volve only V : aij = - Vij,we may evaluate Dito say fourth order from Eq. ( 12) giving us the eigenvalue equation
In this expression we have evaluated the D functions which appear in the denominators of the terms in the definition of Eq. (12) for Dito the minimum order needed for the Taylor-series expansion of Di to be correct to fourth order in V.This has been done so that we can make comparisonswith conventional perturbation theory but there is no necessity for this in general. For example, when discussing saturation effects, it may be more appropriate to expand the D functions in the denominators of Eq. (38) to higher order in V. It is one of the advantages of the CF method that we have this flexibility. Equation (38) is an implicit equation for the eigenvalue z i .A natural way of solving it is by iteration, working to a consistent order in V. Thus in the final term of Eq. (38) we may replace z i by its zeroth-order approximation, z , = ei, and in the second sum by its first-order approximation, z i = = e, Vii.In the first sum we need to replace z , by its second-order approximation
ei
+
z , = Zi
+ z* IVii12/ei,j i
(39)
where eij= e, - ej. Hence for the CF expression for the fourth-order eigenvalue we obtain
The advantage of this expression is, as we shall show, that it is equivalent to summing certain subsets of terms in the conventional perturbation theory to all orders; its disadvantage is that to extend the calculation to a higher order, one has to recalculate all the lower-order contributions. The method can also be applied to time-dependent problems. We have used it to discuss multiphoton ionization via the calculation ofthe resolvent operator for the system (McClean and Swain, 1978a,b, 1979a,b).
400
S. Swain B. COMPARISON WITH ORDINARY PERTURBATION THEORY
To obtain the WWB (Lennard-Jones, 1930; Brillouin, 1933; Wigner, 1935) perturbation series, we retain the zi terms in the denominators but then expand the denominators in a Taylor series, retaining all terms correct to V4. We then find straightforwardly from Eq. (38) the expression
Here we have rearranged the sums so that 2‘ appears rather than Z*, where Z; means that the term k = i is omitted from the sum. This expression agrees with the one given by Dalgarno (196 l), who derives it by a simple iterative technique (see also Ziman, 1969). We note that (1) Eq. (41) gives zi only implicitly; (2) the higher-order terms are given by an obvious extension of the terms of Eq. (41); (3) we have thrown away infinite series of terms as compared with our original expression in Eq. (38), although these terms have a sum of order V5; and (4) factors such as Vjkappearin the numerators: these are awkward for many-body perturbation theory. As we mentioned in the Introduction, methods of improving the convergence of the LJBW series have been discussed in the literature. To obtain the Rayleigh - Schrodingerperturbation series(Rayleigh, 1894; Schrodinger, 1926),we start from Eq. (40) and expand the denominators so as to obtain a power series for zi correct to order 1 PI4.This yields (Dalgarno, 1961)
In this case the eigenvalue is given explicitly but we have thrown away further infinite sets of terms of order IVI5and the higher-order contributions may not now be inferred by inspection from the lower-order ones. The LJBW series may converge where the RS series diverges. To summarize, the CF method for the eigenvalue zi is obtained by setting D, to zero. The CF series will terminate and give exact results for a finite matrix of arbitrary order: the WBW series will terminate and give exact
40 1
CONTINUED-FRACTION METHODS
results only for a matrix of order 2 or less. The LJBW and RS series to a given order may be obtained from the CF series by neglecting infinite series of terms which sum to a higher order than the one being considered; this may adversely affect the rate of convergence of the series.
c. EXAMPLE: TWO-LEVEL ATOM IN SINGLE-MODE FIELD It may be helpful at this stage to look at a specific example. Let us calculate the eigenvaluesand eigenvectors of a two-level atom interactingwith a single mode of the electromagnetic field. The Hamiltonian is Ho V where
+
~ o = l 1 ) ~ 1 ( ~ I + l 2 ) ~ 2 (aXa,w, 21+
v = (sk*l1)(2I+g,P)(
llXa, - 4)
(43)
El and E2are the free atomic energy levels (we assume E2 > El), a, and al are the annihilation and creation operators, respectively, for a photon of mode k, frequency w,, and g, is a coupling constant whose precise form is unimportant here. We use a system of units for which h = 1, so that energy and frequency have the same dimensions. In this section we henceforth drop the mode index k. (We will need it later on when we discuss spontaneous emission.) Two eigenvalues of Ho are 12,n), where the atom is in its excited state with n photons present and I 1,n 1), where the atom is in its ground state and there are n 1 photons in the field. The corresponding eigenvalues of Ho are e2,n= E2 nw and = El (n 1)w. We wish to calculate that eigenvalue of z which reduces to e2,nwhen ) g (4 0. It is given by the solutions of the equation D&) = 0. According to the D rules we must list all the irreducible processeswhich take us from the statel2,n) through various other states and back to the state 12,n). Because the interaction in Eq. (43) causes the atom to change its state with a photon being either emitted or absorbed by the field, the processescan either take one step “up the ladder” or one step “down the ladder” as shown
+
+ +
2, n-
+ +
1, n
+1
4
2, n;
2, n 4 1, n - 1 + 2, n
and there are no further processes. Thus, exactly D2.n
= Z - Ig12(n
+ 1)/0:5+1-
IgI2n/Dt;-,
(44)
The D functions in the denominators of this expression are determined by the processes
1, n + 1 +2, n + 2 4 l , n + 1; 1, n - 1 4 2 , n - 2 4 I , n- 1 (45)
S. Swain
402
Continuing this procedure we obtain the exact CF expression for D2,Jz)
-
y-EI+w-
y
- E2 +
kI2n IglYn - 1) 2w - lg12(n- 2 ) / y - El
+ 3w -
*
*
(46)
where y = z - n o . Next we find the zeros of Eq. (46) by iteration. The zeroth approximation in Igl is y = E2:provided that we are far from resonance,
lE2J - 01 Igln1’2 (47) the second-order approximation, obtained by setting y = E,, and lg(= 0 in the denominators, is
The procedure may be extended in an obvious way to obtain the eigenvalue to any desired order in 1gI2. The iterative method of solution may fail if the eigenvalues are near-degenerate. Thus if the unperturbed eigenvalues are exactly degenerate, E2 n o = El (n l)o, then the left-hand side of Eq. (47) is zero. When this condition is not satisfied, we can no longer state that the term y - E, - win the denominator of the first continued fraction in Eq. (46) dominates the remaining terms, and we cannot make a consistent sequence of approximations using this approach. [The second continued fraction in Eq. (46) gives no problems.] For these reasons we delay discussion of the degenerate situation. We digress a little to discussthe physical meaning ofEq. (46).The quantity G,(z) = D,(z)-’ may be considered as a “propagator”; i.e., its inverse Laplace transform gives the probability amplitude for the system to remain in the state ti) (e.g., Goldberger and Watson, 1964)
+
+ +
G,(z)exp(- izt)dz
l2
(49)
We may interpret Eq. (46) physically as follows: The system starts in the free state 12,n), and stays in that state until there is an interaction. All that is permitted by the perturbation is for the system to undergo a transition to atomic state I 1 ) and to either emit a photon (indicated by the occurrence of the (gI2(n 1) factor in the first continued fraction) or absorb a photon (the IgI2n factor of the second continued fraction). Let us concentrate on the first
+
403
CONTINUED-FRACTION METHODS
continued fraction: After the first emission the system then propagates in the virtual intermediate state I 1 ,n 1 ) until it undergoes another interaction with the field. All it can do now is to undergo a transition to atomic state 12) and emit another photon. (The only other possibility allowed for by the interaction is for it to absorb a photon and return to the ground state; this would take the system back to the initial state 12,n), which is forbidden by the definition of an irreducible process.) This accounts for the partial denominator beginning with the term y - E2 - 2 0 . From this virtual state the system can only emit another photon, and so on. The second continued fraction in Eq. (46)has a similar interpretation, except that in this case only absorptions are allowed. Clearly the latter continued fraction is finite (for n finite), whereas the first continued fraction is infinite. To obtain the corresponding eigenvector, we use Eq. (16)(or equivalently, the x rules), taking r to be the state 2 , n. The eigenvector is
+
It is easily seen from the form of the interaction that there is no contribution from the first sum unless n - rn is odd and no contribution from the second sum unless n - rn is even. For the coefficients ( X ~ , ~ + ~ /and X ~ (, ~x )~ , ~ + ~ / x ~ , ~ ) , for example, we find from the x rules that the only contributing processes are
l,n
+1
+
2, n;
2, n
+2
-+
1, n
+1
---*
2, n
which give
+ l)’’’/D$+1 ~ 2 , n + 2 / ~ 2 ,= n g2[(n+ l)(n + 2)11’2/ot::+2,1,n+1 x19n+1/x2,n
= -g*(n
(51) (52)
The D functions in these expressions are calculated using the D rules in a similar way to the calculation of One can only go up the ladder (if m > 0) or down the ladder (if rn < 0), but not both. Thus we find the eigenstate to be
S. Swain
404
where the Cs and Bs are continued fractions defined by the relations
- so - Ig12(n + s + l)/C2,fl+,+, G,"+, = Y - Ez - so- Ig12(n + s + 1)/CI,fl+,+l Bl,fl-s = Y - El - so - M2(n - ~)/BZ,fl-s--, Bz,fl-s= y - E2 - sw - Ig12(n - S ) / B ~ , ~ - ~ s- ~>,0 Cl,n+s= Y - El
(54)
[Usingthis notation, Eq. (46) may be written in the form D2,, = C,,,- Ig12n/ Bl,fl-l.] The quantity may be determined by normalization. These solutions were obtained previously by different means (Swain, 1973a,c).
D. THEPROJECTION-OPERATOR APPROACH The projection-operator (PO) method provides a powerful and elegant approach to perturbation theory (see e.g., Ziman 1969;Messiah 1961;Goldberger and Watson, 1964). In the following subsection we show how the CF and PO methods can be combined so as to take advantage of the best features of each; but first we outline the PO approach to perturbation theory. We consider the operator equation equivalent to Eq. (4) Alx) =I& (55) where A is an operator decomposable into an unperturbed part A, ,whose eigenvectorsand eigenvalues are assumed known: A,li) = e,li), and an unperturbed part K A = A, - V(the minus sign is chosen because we have in mind applications where A, = z - If,). Projection operators P and Q = 1 - P are defined such that P commutes with A, and such that P 2 = P.For ', where example, we may take P = P PO= Cli)(il
(56)
iEu
where CJ denotes some subset of the eigenvectors of A,. Writing Eq. ( 5 5 ) as (A, - V)(P Q)lx) = Ib),and operating in turn with P and Q, we obtain the equations
+
P(A, - V ) P J X) P V Q ~ X=) PJb)
(57)
QUO- V Q l x ) - Q W x ) = Qlb)
(58)
Equation ( 5 8 ) has the formal solution Qlx) = (-41- QV-YQlb) + Q W x ) )
(59)
405
CONTINUED-FRACTIONMETHODS
which, when substituted into Eq. (57), gives [Ao - PVP - PV(Ao - QV)-'QVP]Plx) = P[1
+ PV(Ao - QV)-'Q]Ib)
(60)
This may be written in the form (A, - R)Plx) = Slb)
(61)
where R = SVP is the "level-shift operator"
R = PVP
+ PV(A, - QV)-'QVP
The usual procedure is to expand (A, (A, - QV)-' =A,'
- QV)-'
(62)
in a power series
+ AC'QVA,' + A ~ ' Q V A , ' Q V A ~ '+
* * *
(63)
which, on substitution into the expressions for R and S,gives
As a first application we consider the eigenvalue Eq. (1): We take A, =
z - H,, Ib) = 0 and P = Pi = li) (il. Substituting the expansion of Eq. (64) into Eq. (6 1) and taking the matrix element with ( il, we obtain the LJBW perturbation series for that eigenvalue of H = H, V which tends to ei as
+
IVI-,O
As a second example we consider the time-dependent problem of Eq. (2): It is clear that in this case we must take A, = z - H,, ( i ( x )= ui(z) and (iJb= ) ui(0), where z now indicates the Laplace variable. Taking P = P i = li) (il, Eqs. (6 1 ) and (64) then give
It will be noted that in the PO approach we are using the superscripts on the Ps and the Rs to indicate the states included in the definition of the projection operator. In the general case we take Wto denote the set of wanted states, which for convenience we suppose to be the first n states: W = (1,2, . . . ,n),so that
S. Swain
406
= X i , Ji ) ( il. The remaining N - n states will be referred to as the unwanted set. Equation (6 1) then gives the n equations
P
2 (aiiSij- R r ) x j = C Srbj + xU Srb,,
j€ W
j€W
iE W
(67)
rE
In our examples we have taken the projection operators to be sums of operators of the form li) (it; in fact, more general projection operators can be defined. For example, Aganval ( 1974) uses a more general definition in his derivation of the master equation. OF THE PROJECTION-OPERATOR AND E. COMPARISON CONTINUED-FRACTION METHODS
Projection-operator methods have been developed by Cresser and Dalton (1 980) and Mower ( 1980) which give CFexpressions for the level-shift operator. These were based on earlier formalisms due to Mower (1966, 1968). Here we use the approach of Jackson and Swain (1 982) and Jackson (1982). In Section II,C we obtained the CF result
and in Section II1,D we obtained the PO result j€ W
(aiiSij- R r ) x j = 2 Srbj j€W
+
Srb,
(69)
rE U
Comparing the two expressions and making use of Eqs. (36) and (1 9) we obtain the correspondences
aiibij- Rr= iij = D&,
(70)
i,j E W
S r = Sij, i,j E W S r = -air/D," aisas,/Dr+
+ x*
(71)
--
a ,
i E W, r E U
=Ry/D,W (72) Equation (70) is an important relation between the D functions and the level-shift operator. It enables us to use the conventional projection-operator method to obtain the solutions in terms of the level-shift operator (and S), and then to obtain CF expressions for the matrix elements of R, rather than using the power-series expansion of Eq. (64). Further results can be obtained by considering some special cases (Jackson, 1982). Let us take bi = & and P = PIk= I j ) ( j l + Ik)(kl. (This is the choice of projection operator we would make if the states j and k were
CONTINUED-FRACTION METHODS
degenerate.) Then Eq. (69) gives us the pair of equations a .. - Rjk -R$ a&-
(.
.
407
(73)
These have the solution
R$ (aii - R$)()(a, - R&) - RjkR# Jk J
xJ. =
(74)
On the other hand, taking bi = &, Eq. ( 19) yields Xj =
-D(jk)lD,k,
J
#
k
(76)
xk = l/Dk= l/(a& - Rh) (77) [The right-hand side of Eq. (77) follows from setting i =j = kin Eq. (70) and using Eq. (18).] Comparing the denominators of Eqs. (74) and (76), and using Eq. (70), we obtain the identity Dlk. = DkDi J - RjkR$ Jk j z (78) Comparing Eqs. (75) and (77), we obtain a relation between level-shift operators defined for different projection operators J?
R& = R& - RjkRJk/Dk Ik kl J This is a special case of a more general result
(79)
RW.r Jk - R ~ R $ r / D ~ (80) RW= Jk which may be proved fairly simply from the power-series expansion for the level-shift operator, Eq. (64). Equation (80) was first obtained by Cresser and Dalton ( 1980) and Mower (1980).
F. DEGENERATE PERTURBATION THEORY 1. Summary of Methods
In the preceding pages we have obtained the basic results needed to discuss degenerate perturbation theory, and in this section we draw them together and give an example. Let us suppose that the unperturbed eigenvaluesepand eq are equal: ep= eq= e. In the CF approach we have two methods for dealing with degeneracy. The first way is to treat the set Uof all states i # p , q
S. Swain
408
as unwanted states and to eliminate them using the methods of Section I1,C. From Eq. (35) we obtain -
iippx,+ x,Z, = b, ii,,X,+ iiwxp= b,
where the as and the bs are given by Eq. (36).As we have indicated in Section II1,D these are entirely equivalent to the set of Eq. (67) obtained by the PO approach. For the eigenvalue problem, the bs are replaced by zero and the matrix A by z - A; thus the eigenvalues z are given by the solutions of the equation [ z - ZPP(Z)][Z - Z J Z ) ] - q)&Z)Z@(Z) = 0 (82) This equation can be solved iteratively in such a way that, to any order, one has to solve a quadratic equation for z. We illustrate the method by taking A = H, V and the states i to be the (known) eigenstates of H,: H,li)= eili). For convenience we assume Vii= 0. To calculate the eigenvalues to second order, we use Eq. (36) to evaluate the quantities in Eq. (82)consistently to fourth order. We obtain
+
where z ( I ) is the first-order approximation to the eigenvalue z and qi = e - e ; .To proceed we need to specify our model more precisely. First, let us assume I V I, # 0. Then clearly we have
e+lV,I (84) Substituting this expression into Eq. (83)gives us a quadratic equation for the eigenvalues which can be solved immediately. To show that this agrees with conventional perturbation theory we evaluate the second-order approximation to the eigenvaluesby replacing z ( *by ) e and omitting the fourth-order terms in Eq. (82).This gives z ( l )=
CONTINUED-FRACTION METHODS
409
On expanding the square root and using the notation V, = IV,lexp(@) this simplifies to z(2)= e -t
lV,l
+ C*1Vi eW2 &2e.iVi e-iW12
(86)
i
The above expression is identical with the standard expression [Dalgarno, 1961: Eq. ( 175)].The higher-order expressions in the conventionalapproach rapidly become very complicated (Dalgarno, 1961). The eigenvalues for the case V, = 0 can be obtained straightforwardly from Eq. (83). Thus the CF approach leads to the same equations as the PO approach;the difference lies in the way the quantities R , and S,, are evaluated. In the latter they are usually calculated by means of the power-series expansion of Eq. (64), whereas in the former the full CF structure is retained. The second way of dealing with degenerate states depends upon the method described in Section II,B. LRt us concentrate on the eigenvalue problem. SubstitutingA = z - Hin Eq. (27) for D, we find the fourth-order eigenvalue equation D, = 0
=O
(87)
It is not difficult to see that this leads to our previous results of Eqs. (82)(86); it can be shown generally that the two methods leads to identical expressions [cf. Eq. ( 3 1) of Swain, 19771. We give an example ofthis second approach.
2. Example We consider again the system described by th Hamiltonia in Eq. (43), but assume the inequality, Eq. (47), to be violated. Eigenvaluesofthe system are found by solving the equation D, = 0. As before, we look for those eigenvalues which reduce to E2 no when lgl+ 0. In the notation of Section II,B, the degenerate states p = 12,n)and q = 1 I,n 1 ) are to be treated on an equal footing. The advantage of using the rules is that the equations of
+
+
S. Swain
410
motion need not be calculated: The solution may be written down by noting the states connected by the interaction. The interaction in Eq. (43) causes the atom to change its state and a photon to be emitted or absorbed; hence, referring to the Dm rules, or to the table in Eq. (28), the processes which contribute to fourth order in lgl are Second order: 2, n + 2, n; 2, n + 1, n
1, n
+1
+1
--*
1, n
+
2, n
+ 1;
2, n
4
Third order: 2, n + 2, n;
-
1, n
2, n ---* 1, n - 1,
+1
2, n;
Fourthorder: 2 , n + l , n - l + 2 , n ;
+2 1, n + 1
+
+ 1; 1, n + 1
1, n
+
1,n+l+2,n+2+1,n+l (88)
Hence we obtain the eigenvalue equation ( z - E, - nw)[z- E, - (n
+ l)o] - 1g12(n+ 1)
- ( Z - E2 - nw)lgl2/D?$!$?+' - [ z - E , - (n
+ l)o]Igl2n/D::$!l.y+I
+ (gl4n(n+ 2)/D::$!!*y$;+,
=0
(89) The processes which contribute to second order to the D functions in the third and fourth terms are 2, n
+2
---*
1, n
+3
+
2, n
+ 2;
1,n - 1 + 2, n - 2 ---* 1, n - 1
giving D$;$!$?+l=z - E2 - ( n
+ 2)0 - IgJ2(n+ 3 ) / [ ~- El - (n + 3 ) ~ ]
D2*n+',n+1 1.n-1 = z - El - (a - l ) -~IgJ2(n- l)/[z - E2 - ( n - 2)0]
(90) For the final term in Eq. (89) it is sufficient to use the zeroth-order approximations for the D functions of Eq. (90). For simplicity, we assume n >> 1 and set (gI2n= V 2 .Defining y = z - (El E, o)/2 - no and 6 = E2 E, - w , the eigenvalue equation [Eq. (89)] may be written
+ + y 2 - d2/4 - V 2- 2 V 2 [ y 2- S(6 + 4w)/4]/(y2 - (6 + 4 - 2 V 2 [y2 - (6 + 4 ~ ) ( 6 40)/4] / [ y 2- (6 - 4 ~ ) ~ / 4+ ] }2V4[y2- 6(6 - 4 ~ ) / 4 ] i w - (6+ 4 ~ ) * 1-[ (6 ~~ 44211
+ V4/[y2- (6 + 4 ~ )=~0 ]
~
)
~
41 I
CONTINUED-FRACTION METHODS
This agrees with the expression Eq. (12) of McClean and Swain (1976), which was obtained by a semiclassical method. This equation may be solved iteratively for y 2 in the manner previously described; if we develop from this a power-series expansion we obtain
+ 2E2,,V2/(E2,, + 0)
+ +
- 2E2,1V2/(E2,, w ) ~ * * * (92) which is the series obtained by Shirley (1965), Hannaford et af. (1973), and Hioe and Eberly (1975) in their treatments of multiple quantum transitions in spin3 systems. Clearly, no difficulty occurs at resonance with the above series. We should point out that in this particular example there is no necessity to use the two-element expansion -we could have obtained identical expressions by considering the single element D function and multiplying through by the first denominator [cf. Eq. (44)]. However, in some problems the degenerate term may appear not in the first denominator, but deeper in the body of the continued fraction; then there is an advantage in using the two-element expansion. For example, in studying the three-quantum transition, the states 12,n)and I 1,n 3) are degenerate. The degenerate term then occurs in the third partial denominator of the first continued fraction of the D2,,function [cf. Eq. (46)], where y - El - 30 = 0. From the expression for y the positions of the multiple quantum transitions can be calculated (Shirley, 1965). However, our expression [Eq. (9 l)] is not the best starting point for the calculation of these resonance positions. Stenholm (1972a,b) has obtained CF expressions which give the resonance positions directly (see also Tsukada and Ogawa, 1973; Stenholm and Aminoff, 1973). We have shown (Swain, 1974c)that the resonances occur at the energy level spacings which satisfy 2. I v2/1 1 - 2.3.V2/1 E f , = m2 (93) (301~- E? - 2.3V2/2 ,I-... We note that the power-series expression for the position of the three-quanturn resonance does not appear to converge for strong fields (Swain, 1974c) but the CF expression can still be utilized, and gives the positons of the resonances to great accuracy. y 2 = a2/4
+
\-
- I
~
6.L
-
G. CONNECTION WITH DIAGRAMMATIC PERTURBATION THEORY Next we consider the relation of CF perturbation theory to diagrammatic perturbation theory (Swain, I975a). In the latter approach, the contribution
S, Swain
412
of a particular order in perturbation theory is associated with a set of diagrams (called Feynman diagrams) and rules are given for calculating the contribution of each diagram. We do not have the space to list these rules fully or to go into their derivation: An introductory account is given by Ziman (1 969)and more detail by Abrikosov et al. (1963).Important quantities in this approach are the Green's functions, or propagators, which are closely related to our G functions [ Eq. (lo)].As an example we consider the spontaneousemission of EM radiation by a two-level atom in the presence of an EM field. The Hamiltonian is the same as that ofEq. (43)except that there is now a sum over all modes k
We use the notation n = (nl ,n2, . . .) to denote a state of the field in which there are n , photons in mode 1, n2 photons in mode 2, and so on. For spontaneous emission we take the initial state to be li) = (2;n) (atom in excited state with n photons present) and the final state to be If) = I l;n,k') (atom in ground state with one additional photon in mode k'). We first find the eigenvaluesz of the Hamiltonian using the CF method by calculating the D function Di and equating it to zero. The second-order processes which contribute are
1231) + I l;n,k) 4(241); 123) -P 1 l;n,- k )
+
12,n)
(95)
(The "-k" indicates a state in which there is one fewer photon in mode k). Using the D rules we find
where we have calculated the denominators of the final term in Eq. (96)to zeroth order in lgl; i.e., we have made a Rayleigh-Schrodipger type of approximation, and n w E Zknkyk. Proceeding in the usual way we introduce an infinitesimal positive imagnary term into the denominators of Eq. (96)and evaluate the sum over k. We find that the perturbed energy ziis given by zi = E2
+n
w
+ SE2 - i12
(97)
(Swain, 1975a),where SE, is the Lamb shift of level 12) and 1, is its spontaneous decay rate (Power, 1964;Loudon, 1973).
413
CONTINUED-FRACTIONMETHODS
The processes in Eq. (95) may also be depicted diagramatically as k
k
/+-, / \
2‘
\ I
.
\
3-
’I
2’
2‘
’I
x-\
f
1,
\
1
/
The solid lines represent atom states and the dotted lines photon states: an arrow to the left indicates the emission of a photon, whilst an arrow to the right represents absorbtion. For simplicity we do not show those photon states that are unchanged by the interaction. Diagrams of the above type in which the system ends up in the state where it began are called “self-energy” diagrams. If we consider the spontaneousemission case (n = 0) for simplicity, the first of the diagrams in Eq. (98) shows that the system begins in the purely excited atomic state, then spends part of its time in the ground state with one photon present before returning to the excited state. Thus the energy of the atomic state 12) is modified by such a process. The fourth-order processes and diagrams are
2; n 1; n, k 4 2; n, k, k’ -, 1; n, k’ + 2; n 2;n- l ; n , k ~ 2 ; n , k , - k ’ - + l;n,-k’-2;n 2; n 1; n, - k - 2; n, - k, k’ 1; n, k’ 2; n 2; n - 1; n, - k - 2 ; n, - k, - k ’ - 1; n, - k’- 2; n
-
- -
+
/
I
\ l
2‘
\ I \
2‘
,+
\
/“’ \
L \ L \
2‘
I’
‘I
-,
k‘
,+.k
‘I
2‘
‘I
’\
LI\!
~-i 2‘
I \
1 \ 1 \
/
\ 1 \
2’
\ 1
2’
-
‘I
\
’I
-
2‘
I \ \
2‘
‘ \
\ \ 2‘
I’
-
\
\ \
’I
2’
These, and higher-order terms, are important in resonance fluorescence (Swain, 1975b), but we do not evaluate them here. Turning now to diagrammatic perturbation theory we find that the same diagrams arise but the contributions are calculated differently. We assume conservation of energy at each vertex and with each line of the diagram we
S. Swain
414
associate a zeroth-order propagator of the form
+ ie)-I, i = 1, 2 F$(z)= ( z + mk k iIz)-’
Gi(z)= ( z - Ei
for the atoms and photons, respectively. In the latter case the plus sign applies if the arrow on the photon line is to the left and the negative sign ifit is to the right. For example
/
c
9,” \
I
In order to obtain expressions roughly equivalent to our Eq. (40) or to the LJBW series one has to use the Dyson equation to sum an infinite set of diagrams of the form
Apart from the first, these diagrams would be termed reducible according to our definition as they represent processes in which the system repeatedly passes through the same state. Notice that the above reducible diagrams can be considered as a sequence of irreducible diagrams joined by a single Green’s-function line [contrast with the fourth-order irreducible diagrams in Eq. (99)]. Irreducible diagrams are the fundamental building blocks of diagrammatic perturbation theory. An irreducible process in the CF formalism corresponds to an irreducible diagram in diagrammatic perturbation theory. In the CF approach we need consider only irreducible processes because the effects of reducible processes have been implicitly summed to give the continued-fraction structure- the continued fraction represents a repeated or rather nested application of the Dyson equation summation procedure. In summary, the irreducible diagrams that appear in the diagrammatic perturbation theory may also be interpreted in CF terms. However, in the diagrammatic approach the rules associate a Green’s function of a particular type [e.g., atom or photon as in Eq. (loo)] with each line, whereas in the CF approach the Green’s function associated with each line is always of mixed atom -photon character. This is because the CF method is equivalent to the diagrammatic method in which infinite subsets of diagrams have been summed. The diagrams are very useful for a physical classification of the various processes.
415
CONTINUED-FRACTION METHODS
IV. The Density Matrix and Rate Equations A. THEBASICRATEEQUATIONS In many circumstances, the density matrix is a more convenient quantity to work with than the wave function. The main reason is that dissipative effects, such as spontaneous emission and finite laser linewidths, can be easily and accurately incorporated into the Von Neumann equation of motion for the density matrix, which is then usually termed a “master equation” (see, e.g., Agarwal, 1974, 1978; Swain, 1980b). Collisions can also be taken into account in a simple, phenomenological way. We define a “rate equation” to be an equation which involves only the diagonal elements of the density matrix. Such equations are easier to handle and interpret than the master equation, but usually they are only approximately true. Often they are obtained by an adiabatic elimination of the off-diagonal matrix elements, which can be camed out when these elements decay at a much faster rate than the diagonal elements (Wilcox and Lamb, 1960;Ackerhalt and Shore, 1977). In a recent publication (Swain, 1980a)we have used CF techniques to eliminate the off-diagonal elements exactly. In the time domain the resulting rate equations have the form of integro-differential equations, but in Laplace transform space they have a simple algebraic structure. While the rate equations may be derived for any system described by a time-independent Hamiltonian, we have in mind application to the interacting atom-EM field system, and we shall bias our presentation towards this end. For simplicity, we include only the effects of spontaneous emission an3 Lorentzian laser line shapes. We write the Hamiltonian for the system in the form H = H, V,where Ho is the Hamiltonian for the atoms and field alone and Vis their interaction. We use the notation JA) = la)ln,) for the direct product of an atomic state la) and an EM field state In,) in which there are n, photons present in the laser mode. E A is the corresponding eigenenergy: HolA) = EAIA).(For simplicity, we consider only one mode; the extension to several modes is straightforward.) The Laplace transform of the master equation for the density matrix may be written (Agarwal, 1974; Swain, 1980b) as
+
fAB(z)6AB(z)
+
21
VAC6CB(z>
- PAC(.>
C
-JABx C
where
TsCnPnn,c,n,.Xz) = PAB(O)
VCBl
416
S. Swain
and A is the laser phase-diffusion linewidth in the Lorentzian approximation (Glauber, 1965; Haken, 1970; Zoller et al., 1981). The first two terms come from the Laplace transform of the Von Neumann equation, and the third term in each of Eqs. (101) and (102) describes the effects of spontaneous emission. [,,is the usual rate of spontaneoustransitions from atomic state Ic) to atomic state la) (it is zero if E, < E,); yo is the total rate of spontaneous transitions out of level la): y, = Z,C, and yd = (y, yb)/2. Note that the effects due to Lorentzian laser linewidths are taken into account by changing the decay rates which appear in the master equation without laser phase fluctuations and modifying them according to the substitution rule
+
Ynb --* Ynb
+ni.4
(103)
(Osman and Swain, 1980); clearly there is no modification to the diagonal decay rates (as nrsa= 0). To obtain rate equations we need to eliminate from Eq. (10 1) all the off-diagonal elements; this can be accomplished using the methods of Section I1,C. The equation may be written in the form
where
LAB,CD = ~ A B ~ A C ~ B+Di( V A C ~ B -DV D B ~ A-C~)A B ~ C D & , C ~ , ( 105) and for simplicity,we have suppressed the z dependence ofp’and L. We may associate with each ordered pair A, B a single index i so that Eq. (104) is equivalentto Eq. (4).Choosingthe unwanted set to be the set of all off-diagonal elements, we obtain a reduced equation for the diagonal elements alone
zMJB
The reduced elements and/5,(0) are obtained by using the rules given after Eq. (36) to write down the processes which take one from state AA to state BB through 08-diagonal intermediate states only. This is guaranteed by the superscript D on the G functions, where D stands for the set of all diagonal states.
417
CONTINUED-FRACTION METHODS
We illustrate the procedure for EMBB to second order in I/; higher-order contributions are discussed in Swain (1980a), where the elimination of the off-diagonal elementsis carried out in an explicit fashion. Using Eq. (105)to obtain the value of the matrix elements, the processes and contributions are
AA
-aw,
AA
BB;
AB -’y”- BB,
AA-‘y“- BA%BB;.
LMJB = - C d n a , n b - IVAB12(1/D?B + l/DiL) +
.
* *
9
A
..
+B
(109) The D functions are calculated according to the usual D function rules. For example, to second order
Note that the spontaneous processes contribute explicitly to Eonly in first order [the first term in Eq. (109)-they contribute implicitly to all orders through the D functions on account of the fa term]. Similarly, we find
To obtain the familiar rate equation form we introduce the notation -LABB = W AB = Caban,nb + Xa, A + B; L , ~ = z + W, ( 1 12) PAis the Laplace transform of the probability of the system being in state IA ) at time t , and WAz) may be interpreted as the generalized transition rate between stateslil ) and 1B ) . The first term in the latter is the spontaneousrate and the second term is the sum of the stimulated contributions,which, from Eqs. (109) and ( 1 10) is, to second order in V
FM
xAB
= FA;
=
Re(lVAB12/hB)
= 211/,,I2(z
+ Yab + An;$)/[(z + Yab +
+ +
+q B 1
(1 13)
[We note that if (z yd An3@)+ 0, Eq. (1 13)gives the standard “Fermi golden rule” expression for the transition rate: X , = 2 4 Val2 s(EA- EB).] For the interactingatom-fieldproblem, described for example by the Hamil-
S. Swain
418
tonian of Eq. (43),the second-order terms given in Eq. (1 13) describe a single-photon process. The fourth-order terms (Swain, 1980a) describe a two-photon process, and so on. With the definitions of Eq. ( 1 12),Eq. ( 106)assumes the desired gain - loss structure required of a rate equation
where ApAA(0)= pAA(0)- PA(0). If there are no initial coherences [pAB(0)= 0 for A # B], which is often the case, the final term in Eq. (1 14)is zero. Then the right-hand side is equal to the difference between the rate of transitions into and out ofthe state IA), ifwe interpret W'(z) as the total rate of transitions out of the state 1-4).The left-hand side is equal to the Laplace transform of dP,(t)/dt. In general, the inverse Laplace transform of the right-hand side is a convolution so that in the time domain the rate equations are integro-differential equations as remarked earlier; however, if we are interested only in the long-time behavior, it may be a good approximation to replace z by zero in the generalized rates W,&). Then the Laplace transform of Eq. (1 14)is
[assuming the final term of Eq. (1 14)is zero]. The rates W,AO) are identical with those that would have been obtained using the standard adiabatic elimination procedure (Wilcox and Lamb, 1960; Ackerhalt and Shore, 1977). The latter authors have compared the solutions of the approximate rate equation ( 1 1 5) with the numerical solutions of the Bloch equations for a three-level atomic system undergoing photoionization, and found that they cannot reproduce the Rabi oscillations of the system. For short times the approximate rate equation and exact solutions disagree, but for times much greater than the inverse of the characteristic decay rate of the off-diagonal elements, the agreement is good. The approximate rate equations should give exact results for the quantities PA(t)in the limit t + CQ (when these quantities exist). This is a consequence of the Laplace identity limf(t) t--
= lim
zf(z)
2-0
We emphasize that the full rate equations [Eq. ( 1 14)]are exact and equivalent to the master equation [Eq. (lol)]. Equation (1 14)is fully quantum mechanical and thus the effects of quantum-mechanical interference must be built into the rates WAB(z), which
CONTINUED-FRACTION METHODS
419
therefore cannot be interpreted as conventional transition rates. Thus the rates XAB(z),apart from the second-order rate of Eq. (1 13), are not necessarily positive definite. A similar situation was found by Wilcox and Lamb (1960) for the rates obtained by adiabatic elimination. Furthermore, we do not apparently have detailed balance: X A B = X B A . However, the condition that V be symmetric is sufficient to ensure that detailed balance is satisfied (Swain, 1980b).
B. PROJECTION-OPERATOR DERIVATION In this section we give an alternative derivation using the projection-operator approach of Section III,D, and exploiting Liouville space notation (Cohen - Tannoudji, 1975; Dalton, 1982). We define the generalized states IIAB) as the operators IIAB) = IA) (4
(1 17)
so that we may write the density operator p as
m l 1 9 ( z ) )= llV70))
(1 19)
Next we define the projection operator P Dby
P D=
cJIM)(MI1
( 120)
A
This projects p into a sum of its diagonal elements
PDllY)= EPAAIIM)
(121)
A
The rules for manipulating the Liouville space operators such as L(z) are neatly summarized by Dalton ( I 982). The double subscripts (AB)may now be treated just like single subscripts in the ordinary approach. From Eqs. (69)-(72), (102), and (105) we obtain
S. Swain
420
using the notation pM = PA. This equation has the desired rate equation structure [ Eq. ( 1 14)] and gives us the correspondences
A number of sum rules can be derived (Swain, 1980a; Jackson, 1982). From the expression [Eq. (lOS)] for the Liouvillean, we have
From Eqs. (123), ( 1 12), and (107) we have
= Ya
using Eqs. (126) and ( 127). Equation ( 128) may be rewritten as
(128)
42 1
CONTINUED-FRACTIONMETHODS
From Eqs. (125) and (108) we have
IZJ
\
=O ( 130) using Eq. (127). In terms of the generalized transition rates, the sum rules may be written
Relation ( 13 1 ) confirms that WM may be interpreted as the total rate of transitionsout ofstatelA) toall otherstatesJB);withitsaid wemaywriteEq. ( 1 14) in the more familiar rate-equation form zpA(z)
-
=
x’
[ wBA(z>*B(z>
- wAB(z)*A(z)l
+ ApM(o)
( 34)
B
Theform of the rate equations [Eqs. ( 1 14) or ( 134)] is exact; the approximations made in practice in using them consist of (1) truncating the number of states IA) to be considered and/or (2) approximatingthe rates W. (Examples of the use of these rate equations in multiphoton ionization, optical double resonance, and population trapping are to be found in Osman and Swain, 1980, 1982; Jackson et al., 1982; O’Brien and Swain 1983; Swain 1982, 1984a,b.)
C. APPLICATIONS 1. Alomic Rate Equations
It is sometimes desirable to have rate equations which involve the populations of the atomic statesonly. To achieve this, we follow Cohen - Tannoudji
S. Swain
422
and Reynaud (1 977) and make the following approximations: The first is ij,= p,,,,, +pna(OPAZ)
( 1 35)
where p J 0 ) is the initial probability of there being n, photons in the field. This approximation assumes that the field is unchanged by its interaction with the atom. The second is for n’ = n = ii
a,@) = p,(O),
( 136)
where nis the mean number of photons in the field. This requires that the distribution p,(O) is a slowly varying function of n in the neighborhood of %. These approximations are reasonable for a laser operating well above threshold. Recalling that the states A , B, . . . are composite atom-field states, IA) = la)ln,), we may sum Eq. (1 14) over n, to obtain the required rate equations for the atomic populations [ z+ Wa(Z)lpa(z) -
z’
Wbapb(z>
b
where WbdZ)
(Xba(Z))
+ cba =
z
Pn,(o)
nb
= Pa(0) + APaa(0)
( 37)
+ cba
(138)
xXbnb;an.(Z) no
2. The Continuum as a Reservoir In general, the atomic states include the continua as well as the bound states. However, in some problems, the observable of interest may not make explicit reference to the continuum states. For example, the total photoionization probability may be written P,(t) =
cPdf)
=
K
1-
2 PA(t)
(1 39)
A
where we reserve the index K to denote the continuum states, and A , B, C, . . . to indicate bound states. In such cases the continua may be regarded as a reservoir and it is possible to eliminate the “unwanted” continua states from the equations of motion. One way this can be accomplished is to use the procedure of Section IV,A but with the continua states included in the unwanted set -that is, we take the wanted set W to be the set of all diagonal bound states, W = (IIAA)):In the PO approach, this corresponds to using the projection operator
PBD= CIIM)(AAII A
(1 40)
423
CONTINUED-FRACTION METHODS
The resulting rate equation is then identical with Eq. ( 1 14) except that the states A and E refer only to the bound states. In the expressions for the rates WAB,Eq. (107), the sums are now over the diagonal continua states, in addition to the off-diagonal bound and continua states as in Section IV,A. One consequence of the curtailed definition in Eq. ( 140) is that the sum rules of Eqs. (131)-( 133) no longer apply, and thus the rate equations must be used in the form of Eq. (1 14) rather than the form of Eq. (134). An alternative procedure is to take the unwanted set U to be the set of density-matrix elements which involve any continuum state, diagonal or off-diagonal. The equivalent projection operator is
This time we obtain a set of equations which do not have the rate-equation structure
where
-
=LAB,,
L a , ,
-
x*
G B +
LAB,dI,,cD
IJ
...
(143)
IJ
and at least one of the states I, J must be a continuum state. If the continua states are initially unpopulated the term P A B ( 0 ) = pAB(0). Equation ( 142) involves matrix elements of p between bound states only; the continua states appear only in the expressions for L and P(0).The effect of the sums over continua states in L A B , A B is to add energy shifts and decay rates (i.e., ionization losses) to the coefficientsf,, of Eq. (102), while their effect in E A B , c D is to modify the interaction matrix elements, so that Eq. ( 142) may be written in the form I
i
Thus the effects of the continua interactions are represented by the modified coefficientsy, V', and V". This is a consequence of the fact that after the elimination of unwanted states the new set of equations [Eq. (35)] has the same structure as the original set of equations [Eq. (4)] but with modified coefficients [Eq. (36)]. These coefficients can always be regarded as the old coefficients "dressed" by the interaction with the unwanted states. It is clear that this is a general property. In the next section we find that a system interacting with a laser subject to phase fluctuations is described by an equation of the form of Eq. (144). We can now eliminate the off-diagonalelements from the linear equations
S. Swain
424
[Eq. (144)] in exactly the same way as we did in Section IV,A. In the PO approach this is achieved by using the projection operator pDB=
cI I A )(MI1
(145)
A
We obtain the rate equations for the bound-state populations alone (Z
where
-
z + WM
+
WM)FA=
LM.M
+
PA(0) ApM(0)
= Lm,M -
+ C'
( 146)
WBAPB
B
c*-
-
LMjJLIJd
GDB+ iJ
...
( 147)
IJ
and the Zand @(O) are the quantities which appear in Eq. (142). As in the previous case the sum rules, Eqs. ( 13 1)-( 133) will not be valid but the rates X, will be symmetric if the interaction Vis symmetric. The sums in Eqs. ( 147)- ( 149) are over off-diagonal bound states only. Of the two methods, the latter is generallyto be preferred, as the extra labor involved in calculating the modified Liouvillean of Eq. ( 142)is usually more than compensated by the reduction in the number of processes involved in calculating the W's due to the continuum states being excluded.
D. NON-LQRENTZIAN LASERLINEWIDTH EFFECTS The theory of a single-mode laser operating well above threshold indicates that the line shape of an ideal laser (where the source of noise is purely quantum mechanical) should be well approximated by a Lorentzian up to many linewidths from line center. (For a discussion of the line shapes of multimode lasers, see, e.g., Zoller et al., 1981, and references cited.) For certain laser systems, these predictions have been verified (e.g., Hinkley and Freed, 1969; Manes and Siegman, 1971; Gerhardt et al., 1972, and references therein). However, for many of the single-modelasers used in practice, the dominant source of noise is not the quantum-mechanicalphase fluctuations, and the laser line shape is found to fall off more rapidly than a Lmentzian in the wings. This can have important experimental consequencesin, for example, optical double resonance, resonance fluorescence, and multiphoton ionization (e.g., Hogan et al., 1978;Georges and Lambropoulos, 1978,
CONTINUED-FRACTION METHODS
425
1979;Zoller and Lambropoulos, 1979;Osman and Swain, 1980,1982;Yeh and Eberly 1981a,b Dixit et al., 1980; Swain, 1980b, 1984a,b; Zoller et al., 1981; Jackson et al., 1982; OBrien and Swain, 1983; George and Dixit, 1984; Dixit and George, 1984). There is thus considerable interest in the theory of the interaction of atoms with non-Lorentzian line shapes. Recently, there has also been interest in the novel effectswhich arise when the fluctuations of two or more lasers are correlated in some way; this is called cross-correlation (Thomas et al., 1981, 1982; Dalton and Knight, 1982,1983;Kennedyand Swain, 1984a,b;Dalton et al., 1985;Swain, 1985). In this section we indicate how the effect of non-Lorentzian laser fluctuations can be taken into account approximately by means of substitution rules similar to Eq. ( 103) for the Lorentzian case. The details of the calculations can be found in Jackson and Swain (1 982) and Swain (1984b, 1985); here we present just the results. We should point out that these results have been obtained using the CF methods of Section 11. We consider several lasers which we denote by a,p, y, . . . . In the phase-diffusion model (Haken, 1970)the phase of the ath laser is a random variable which satisfies the Langevin equation
+ CUOd = Fu(t)
+fP
(150)
where the random noise sourceF,(t) has the following correlation properties (Fu(t)Ffl(t’)) = 2BMC&&t - t’) For convenience we write the diagonal laser parameters as
The parameter B, is called the “Lorentzian linewidth” of the ath laser and C, its “cutoff.” (It can be shown that in the limit C, -D 03 the line shape is Lorentzian with half-width B,.) For independent lasers, Bd = B. SM. The coefficient B is called the “cross-correlation coefficient.” The inequality lBdl 5 (BUB3l2can be derived (Dalton and Knight, 1982). We have shown that with laser fluctuations the density matrix of the interacting atom-field system satisfies the same equation [Eq. ( 101)] as if there were no fluctuations except that the decay constants, detunings, and interaction matrix elements are modified; the fluctuations form a kind of reservoir which “dresses” or “renormalize” these quantities. We use the same notation as in Section IV,A except that we have several lasers, so that ( A ) = la,n:,r$,n;, . . .), etc. The decay rate yij of the I + J transition is modified to yI,
TI, = yjj +
cB,(ZJ) + 2 c Sd(ZJ)
(153)
S. Swain
426
where
S,(ZJ)
+ Ci)
= B,Ct(n$)2/(d2,
( 154)
dij is the detuning
and the detuning is modified to
aij= Aijdij,where
In these expressionswe have set z = 0 so that they describe the dynamics for times t B C;’, 7;’. We have also assumed that y i j - B, <<
C,
(158)
so that the deviations from Lorentzian line shape are not too great. We do not write out the modified interaction matrix elements because if the system is such that, within the rotating-wave approximation, each laser couples only one step of the ladder of atomic states, the corrections are zero. (That is, the interaction matrix elements are unmodified in this case.) This situation is a commonly occurring one. An example where the corrections are important is in multiphoton ionization with a single laser (Jackson and Swain, 1982). Details of the corrections can be found in Swain (1985). If we take the limit C, + 00 (the Lorentzian limit) we obtain
A, -, 1;
BJZJ)
-
B,[n$]z;
B@(ZJ)-,B@nyJnfJ
(159)
which is consistent with our previous substitution rule [Eq. (103)l. We have applied these methods to multiphoton ionization (Jackson and Swain, 1982; Jackson et al., 1983; Swain, 1984a) and to optical double resonance (Osman and Swain 1980, 1982;OBrien and Swain, 1983;Swain, 1985). From the modified master equation, which includes the effect of non-Lorentzian fluctuations through the expressions in Eqs. (1 53) and (1 57), etc., we can derive rate equations as in Section IV,A. This is done in Swain ( 1984b), where applications to optical double resonance are discussed.
CONTINUED-FRACTIONMETHODS
42 7
V. Conclusions In Section 11, we have used the determinantal expansion derived in the Appendix to obtain continued-fraction expressions for the ratio of the two determinants which appears in the Cramer’s solution of a system of linear equations. Using this solution, we show how the size of the original set of equations can be reduced by eliminating an arbitrary subset of variables from the system. We also obtain an expansion of the determinant by two or more elements which is useful in the case ofdegeneracy. These are our three basic mathematical results, and they are exact. In Section 111they are used to obtain iterative solutions to the time-dependent Schrodinger equation and to the problem of finding the eigenvectors and eigenvalues of a time-independent Hamiltonian. The method is compared with the various conventional perturbation theories and shown to involve the summation of infinite subsets of the conventional perturbation theory terms. (Thus “self-energy” effects are automatically taken into account by the CF perturbation theory.) Several ways of dealing with degeneracy are discussed. Finally, in Section IV, the methods are applied to the perturbation theory of the Von Neumann equation for the density matrix, or to the master equation. Exact rate equations are derived, and examples of their use are cited. It is shown how non-Lorentzian (phase-diffusion)laser linewidths can be incorporated into the formalism. Throughout we have assumed a time-independent Hamiltonian. However, extension to Hamiltonians periodic in time is possible via the Floquet theorem (Swain, 1973a; Chu, 1985).
Appendix: Derivation of the Continued-Fraction Expansion of a Determinant Here we derive the determinantal expansion which is the basis of all the subsequent theory. Using the notation of Section II,A, we have, by the Laplace expansion, A = 2 a ..(- 1)’+iAii U
j
= aiiAii
+ 2 aij(jZi
l)i+jAii
S. Swain
428
where we have separated out the j = i term. Repeating the procedure for the determinant A’’ we obtain aij(-
A = aiJi +
l)i+j
-
l)”+j‘,#J.ki
k+i
j+i
= aiJi
C (-
c aijajiAij+ z caija,(-
ly+j+k’+i’Aij$ki
(161)
k+ij j + i
j+i
where we have separated out the k =j term and put a prime on the final i, j , etc., to show that the labeling of the rows and columns may have been affected by the removal of the kth row and the ith column. A little thought will show that this is the origin of the minus sign before the second term of Eq. (16 1 ) . Continuing in the same way, we obtain the expansion of Eq. ( 6 ) . The expansion of Eq. (7) is obtained by comparing Eq. ( 160) with the expansion of Eq. (6). The expansion ofEq. (26) by two elementsis obtained (with iforpandjfor q) by substituting
Ai = ajjAij-
z*ajkakjAijk+ . . . k
+ -
Aik = aJJ- AI.lk. - z*ajiaijAiju I
*
obtained from Eq. (6), into Eq. ( 6 ) itself.
REFERENCES Abrikosov, A. A., Gorkov, L. P., and Dzyaloshinski, I. E. (1963). “Methods of Quantum Field Theory in Statistical Physics.” Prentice-Hall, New York. Ackerhalt, J. R., and Shore, B. W. (1977). Phys. Rev. A 16,277. Agarwal, G . S. (1974). Springer Tracts Mod. Phys. 70, 1. Aganval, G. S . (1978). Phys. Rev. A 18, 1490. Amos, A. T. (1970). J. Chem. Phys. 62,603. Amos, A. T. (1972). J. Quantum Chem. 6, 125. Bendazolli, G. L., Goscinski, O., and Orlandi, G. ( I 970). Phys. Rev. A 2,2. Bowen, S . P. (1975). J. Math. Phys. 16,620. Brandas, E., and Goscinski, 0. (1970). Phys. Rev.A 1,552. Bnllouin, L. (1933). J. Phys. 4, 1. Chu, Shih-I (1985).Adv. At. Mol. Phys. 21, 197. Cohen-Tannoudji, C. (1975). “Course de Physique Atomique et Moleculaire.” College de France, Pans. Cohen-Tannoudji, C., and Reynaud, S . (1977). J. Phys. B 10, 345.
CONTINUED-FRACTION METHODS
429
Cresser, J. D., and Dalton, B. J. (1980). J. Phys. A 13,795. Dalgarno, A. (1961). In “Quantum Theory. I. Elements” (D. R. Bates, ed.),p. 172. Academic Press, New York. Dalton, B. J. (1982). J. Phys. A 15,2157. Dalton, B. J., and Knight, P. L. (1982). J. Phys. B 15, 3997. Dalton, B. J., and Knight, P. L. (1983). Opt. Commun. 42, 41 I. Dalton, B. J., McDuff, R.,and Knight, P. L. (1985). Opt. Acta 32,61. Dixit, S. N., and Georges, A. T. (1984). Phys. Rev. .4 29,200. Dixit, S. N., Zoller, P., and Lambropoulos, P. (1980). Phys. Rev. A 21, 1289. Feenberg, E. (1948a). Phys. Rev. 74,206. Feenberg, E. (1948b). Phys. Rev. 74,664. Feenberg, E. (1956). Phys. Rev. 113, 1 116. Feenberg, E. (1958). Ann. Phys. 3,292. Feshbach, H. (1948). Phys. Rev. 74, 1548. George, A. T., and W i t , S. N. (1984). Phys. Rev. A 23,2580. George, A. T., and Lambropoulos, P. (1978). Phys. Rev. A 18,587. George, A. T., and Lambropoulos, P. (1979). Phys. Rev. A 20,991. Gerhardt, H., Welling, H., and Guttner, A. (1972). Phys. Lett. 40A, 191. Glauber, R. J., (1965). In “Quantum Optics and Electronics” (C. deWitt et al.. eds.), p. 65. Gordon & Breach, New York. Goldberger, M. L., and Watson, K. M. (1964). “Collision Theory.” Wiley, New York. Goldhammer, P., and Feenberg, E. (1956). Phys. Rev. 101, 1233. Gontier, Y., Rahman, N. K., and Trahin, M. (1975a). Phys. Rev. Lett. 34,779. Gontier, Y., Rahman, N. K., and Trahin, M. (1975b). Phys. Lett. 5 4 4 341. Gontier, Y., Rahman, N. K., and Trahin, M. (1976). Phys. Rev. A 14,2109. Goscinski, O., and Lukman, B. (1970). Chem. Phys. Lett. 7,573. Haken, H. (1970). In “Handbuch der Physik” (S. Flugge, ed.), Vol. 25, Part IIc. Springer-Verlag, Berlin and New York. Hanggi, P., Rosel, F., and Trautman, D. (1978). Z. Naturforsch. 339,402. Hannaford, P., Peg, D. T., and Series, G. W. (1973). J. Phys. B6, L222. Haydock, R. ( 1980). Solid State Phys. 35,2 15. Haydock, R.(1982). In “Excitations in Disordered Systems” (M. F. Thorpe, ed.). Plenum, New York. Hermann, J., and Swain, S.(1977). J. Phys. B 10,2111. HinkIey, E. D., and Freed, C. (1 969). Phys. Rev. Lett. 73,277. Hioe, F. T., and Eberly, J. H. (1975). Phys. Rev. A 11, 1358. Hogan, P., Smith, S. J., George, A. T., and Lambropoulos, P. (1978). Phys. Rev. Lett. 41,229. Jackson, R. I., and Swain, S. (1981). J. Phys. A 14, 3169. Jackson, R. I., and Swain, S.(1982). J. Phys. B 15, 575. Jackson, R. I. (1982). Ph.D. thesis, Queen’s University, Belfast (unpublished). Jackson, R. I., O’Brien, D. P., and Swain, S. (1982). J. Phys. B 15, 3385. Jackson, R. I., O’Brien,D. P., and Swain,S. (1 983). In “Quantum Opticsand Electronics” (P.L. Knight, ed.). Wiley, New York. Kennedy, T. A. B., and Swain, S. (1984a). J. Phys. B 17, L389. Kennedy, T. A. B., and Swain, S. (1984b). J. Phys. B 17, L751. Lennard-Jones, J. L. (1930). Proc. R. SOC.Ser. A 129,604. Lippmann, B. A. (1956). Phys. Rev. 103, 1149. Loudon, R. (1973). “Quantum Theory of Light.” Oxford Univ. Press, London. McClean, W. A., and Swain, S. (1976). J. Phys. B 8, 1673.
430
S. Swain
McClean, W. A,, and Swain, S . ( 1978a). J. Phys. B 11, I7 17. McClean, W. A., and Swain, S. (197813). J. Phys. B 11, L5 15. McClean, W. A., and Swain, S. (1979a). J. Phys. B 12,723. McClean, W. A., and Swain, S. (1979b). J. Phys. B 12,2291. Manes, K. R., and Siegman, A. E. (1971). Phys. Rev. A 4, 373. Messiah, A. (196 1 ). “Quantum Mechanics.” North-Holland Publ., Amsterdam. Mori, H. (1965). Prog. Theor. Phys. 34, 399. Morse, P. M., and Feshbach, H. (1953). “Methods of Mathematical Physics,” Part 11, Ch. 9. McGraw-Hill, New York. Mower, L. (1966). Phys. Rev. 142,799. Mower, L. (1968). Phys. Rev. 165, 145. Mower, L. (1980). Phys. Rev. A 22,882. Nauts, A., and Wyatt, R. E. (1983). Phys. Rev. Lett. 53, 2238. Nauts, A., and Wyatt, R. E. (1984). Phys. Rev. A 30, 872. OBrien, D. P., and Swain, S . (1983). J. Phys. B 16,2499. Osman, K. I., and Swain, S. (1980). J. Phys. B 13,2397. Osman, K. I., and Swain, S . (1982). Phys. Rev. A 25,3187. Power, E. A. ( 1964). “Introductory Quantum Electrodynamics.” Longmans, London. Rayleigh, Lord. ( I 894). “The Theory of Sound,” Vol. I, p. 1 13. Macmillan, London. Richards, P. I. (1948). Phys. Rev. 74,835. Schrodinger, E. ( 1 926). Ann. Phys. 80,437. Shirley, J. H. (1965). Phys. Rev. 138, B979. Stenholm, S. (1972a). J. Phys. B 5, 878. Stenholm, S. (1972b). J. Php. 8 5 , 890. Stenholm, S., and Aminoff, C. (1973). J. Phys. B 6,2390. Swain, S . (1973a). J. Phys. A 6, 192. Swain, S. (1973b). Phys. Lett. 43A, 229. Swain, S. (1973~).J. Phys. A 6, 1919. Swain, S. (1974a). Phys. Lett. 46A, 435. Swain, S . (1974b). J. Phys. A 7, L52. Swain, S. (1974~).J. Phys. B 7,2363. Swain, S . (1975a). J. Phys. A 8, 1277. Swain, S . (1975b). J. Phys. B 8, L437. Swain,S. (1976).J. Phys. A 9 , 1818. Swain, S . (1977). J. Phys. A 10, 155. Swain, S. (1980a). J. Phys. B 13, 2375. Swain, S . (1980b). Adv. At. Mol. Phys. 16, 159. Swain, S . (1981). Opt. Acta 28, 1195. Swain, S. (1982). J. Phys. B 15, 3405. Swain, S. ( 1 984a). Fifth Rochester ConJ Coherence Quantum Opt., 5th p. 40 1. Swain, S . (1984b). 1.Phys. B 17, 3873. Swain, S. (1985). J. Opt. SOC.Am B, to be published. Thomas, J. E., Ezekiel, S., Leiby, C. C., Jr., Picard, R. H., and Willis, C. R. (1981). Opt. Lett. 6, 298. Thomas, J. E., Hemmer, P. R., Ezekiel, S., Leiby, C. C., Jr., Picard, R. H., and Willis, C. R. (1982). Phys. Rev. Lett. 48,867. Tsukada, N., and Ogawa, T. (1973). J. Phys. B6, 1643. Wigner, E. (1935). Math. Naturwiss. Anz. Ungar. Akad. Wiss. 53,477. Wilcox, L. R., and Lamb, W. E., Jr. (1960). Phys. Rev. 119, 1915.
CONTINUED-FRACTION METHODS
43 1
Yeh, J. J., and Eberly, J. H. (198 la). Proc. Int. ConJ Multiphoton Processes, 2nd Budupesf p. 305. Yeh, J. J., and Eberly, J . H. (1981b). Phys. Rev. A 24, 888. Young, R. C., Biedenharn, L. C., and Feenberg, E. (1957). Phys. Rev. 106, 1151. Ziman, J. M. (1969). “Elements of Advanced Quantum Theory.” Cambridge UNv. Press, Cambridge. Zoller, P., and Lambropoulos, P. (1979). J. Phys. B 12, L547. Zoller, P., Alber, G.,and Salvador, R. (1981). Phys. Rev. A 24, 398.
This Page Intentionally Left Blank
207 in ionization of N4+, 207 in ionization of 05+, in ionization of Ti3+,205 in measured cross sections for electron impact ionization, 203 thresholds for, 204 Autoionization widths, 123- 125 of different Rydberg series, 123- 124
A Absolute radiometric calibration of photomultipliers, 200 Absorbing sphere model, 157 Adiabatic and diabatic representations, 248-253 “Adiabatic approximation,” 84, 93 validity of, 93 Adiabatic potential curves of Be, 132- 134 Amsterdam/Groningen group, 158, 170, 175 Angular correlations, 82, 85 definition of, 82, 85 Annihilation spectra for positrons in gases, 39-52 annihilation from clusters, 43-45 annihilation rates, graphs, 44 annihilation volume, 40 and elastic scattering, 40 lifetime parameters, 39-40 positron mobilities in gases, 49-52 results for the inert gases, 40-43 positron - helium scattering, 4 I positron- Krypton, decay constants for, 43 positron-Xenon, lifetime spectrum, 42 spur model for positronium formation, 45-48 fraction F defined, 47 Antiscreening correction, 336- 337,343 Argon, 62, 149,2 12,269,299- 30 1 H+- Ar cross sections, 149 ASC, see Antiscreening correction Auger decay, post-target, 357 Auger electron emission, 306 Auger spectroscopy, 3 10 Autoionization, 203-209,237,245,306,310 definition of, 203
B Basbas theory, 340- 342 and reduced cross sections, 341 -342 Beryllium adiabatic potential curves of, I32 - 134 doubly excited states of, 132- 135 Bethe approximation, 329-33 1, see also Dipole approximation Bethe asymptote, 2 I 1 Bethe-Born approximation, 187 ionization cross section, expression for, 187 Bethe cross section, 187 for ionization of H by fast protons, 187 Bethe- Heitler- Elwert formula, 372 - 373, see also Bethe-Heitler formula Bethe-Heitler formula, 370-371, see also Bethe-Heitler-Elwert formula Bethe-type energy dependence, 2 I7 in electron excitation of positive ions, 2 17 Bevalac, 3 15, 3 17, 32 1, 379 Bevatron, 3 15, see also Relativistic synchrotron Bloch-Siegert shifts, 389 Body-frame analysis of correlation quantum numbers, 115-125 choice of axes, 1 16 decomposition into rotational components. 116-117
433
434
SUBJECT INDEX
moleculelike viewpoint of two-electron correlations, 12 1 - I23 order of correlation energies, 12 1 purity of rotational states, 1 18 - 1 19 decomposition of density plots into rotational components, 118 systematicsof autoionization widths, 123- 125 T doubling, 123 vibrational quantum numbers, 119 - 120 Bohr-Lindhard interpretation of gas-solid charge-state differences, 357, 361, 363 Bohr- Lindhard model of electron capture, 166, 181 Bohr scaling relation, 187 Born approximation, 12, 15 - 16, 19,2 1 Born cross sections, 182 for ionization of H by bare nuclei, 182 Born-Oppenheimer approximation, 93,27 I Born -0ppenheimer separation of the scattering amplitude, 247 Bremsstrahlung cross section, for electron, angular distribution of, 370-371 measured and calculated results for Xe Be, 371 Bremsstrahlungphotons, 369 Bremsstrahlung, primary- and secondaryelectron, 3 19 Bremsstrahlung process, 37
+
C
C,, symmetry, 276,287 C, symmetry, 276 Capture autoionization, 208 -209 CBI and CBII, see Coulomb-Born CBXI and CBXII, see Coulomb-Born exchange CDW, see Continuum distorted-wave CDW, see Coulomb distorted-wave CEM, see Channel electron multiplier CEX, see Charge exchange CEX cross section, 247 CEX theory, 252 time-dependent, 252 time-independent, 252 CF, see Continued-fraction Channel density plots, 9 1- 93 Channel electron multiplier, 62
“Channel functions,” 84- 85, 88-90 Charge-changing cross-section measurements, 32 1- 324 charge states, spectrum, 323 diagram of apparatus, 322 “thick-target’’ method, 32 1 “thin-target” method, 32 1 disadvantage of, 324 Charged beams, 200 Charge-exchange collision, 246 Charge transfer, 152 - 18 I electron capture by singly charged ions, 152-154 electron capture by slow multiply charged ions, 1 54 - 162 Charge transfer and ionization in collisions of hydrogen atoms, 143- 195 CI, see Configuration-interaction CI wave functions of negative ions of rare gas atoms, 137 CL, see Classical binary encounter Classical binary encounter calculations, 22 1 Classical impulse approximation, I84 Classical-trajectory Monte Carlo method, 153, 162, 183 Close-coupling formula, 220 Closed channel projection operator, 9 Cluster formation, 43-45 Collision channels for one-electron capture by C3+in H, eight possibilities, I76 Collisions between electrons and molecular ions, 226 -228 merged-beam measurements of recombination, 226 - 228 Collisions between electrons and negative ions, 224 - 225 Collisions between positive ions, 232- 237 in astophysics, Si+- H+ and Si2+-He+, 232 in fusion plasmas, 232 H+ He+, charge transfer, 233 H+ He+, ionization, 232-233 in HIF, 232 Collisions between positive and negative ions, 228-232 cross section measurements, 228 - 23 1 H+ - H-, 228-230 H: - H-, 229 - 230 He+ - H-, 229-230
+
+
SUBJECT INDEX 3He2+- H+, 230 Na+ 0-, 230 experimental results too low, 231 Collision spectroscopy techniques, 277 - 288 “Collision strength,” 2 I8 defining equation, 2 18 Compton profile, 306, 370 equation for, 3 16 Configuration-interaction method, 77 wave functions of, 94 Continued-fraction expansion of a determinant, derivation of, 427 -428 Continued-fraction methods in atomic physics, 387-431 density matrix and rate equations, 4 15-426 perturbation theories, 398 -4 14 Continued-fraction solutions to linear equations, 389- 398 algebraic expressions, 389 - 392 eigenvalue problem, 389- 390 timedependent SchrGdinger equation, 390 D functions, equation for, 39 1 D-function rules, 393-394 Continuum distorted-wave approximation, 153, 184 “Correlation cube,” 286 “Correlation cylinder,” 288 Correlation patterns and isomorphism, 102- 103 Correlation quantum numbers K, T, and A, 96- 125, passim in doubly excited states, 80- 82 Correlations, radial and angular, 81 -95 Coulomb-Born exchange, 220 Coulomb- Born formula, 220 Coulomb- Born -0ppenheimer calculation, 219 Coulomb distorted-wave calculations, 22 1 Coulomb- second-Born approximations, 247 Coupled-channel model, 340 Coupled static approximation, 8 - 9, 2 1 Cross-beam experiment, 200-202 measurements of electron-impact ionization of positive ions, table of, 202 Cross-correlation,425 Cross-correlationcoefficient, 425
+
435
Crossed-beam coincidence technique, 182, I84 Crossed-beam methods employing thermal-energy hydrogen atoms, 144-151 crossed-beam coincidence method, 145- I47 modulated crossed-beam technique, 144- 145 photon emission spectroscopy and electron-capture studies using highintensity H-atom beams, 147 Cross-section measurements with positron beams, 52-7 1 cross sections for atomic excitation in positron collisions, 70- 7 1 differential elastic scattering cross sections, 56 - 59 apparatus illustrated, 58 formation of excited-state positronium in gases, 63 - 67 Ps* as a function of positron energy, 64 scattering cell illustrated, 65 positron impact ionization cross sections, 66-69 total cross sections, 52-56 experimental and theoretical results compared, 53 scattering by alkali metals, 5 5 - 56 scattering by hydrocarbon gases, 54 total cross sections for formation of positronium, 59-63 as a function of energy, graph, 6 1 Cross sections for electron capture by N2+ in H, 172 for electron capture into specified states of C3+in C4+- H, 178 for one-electron capture in C3+- H, 177 Cross sections for electron removal from hydrogen atoms in collisions with positive ions, 189- 192 electron removal from highly excited H atoms, 191- 192 general scaling relations for multiply charged ions, 189- I9 1 reduced total cross sections, graph, 190 Cross sections, total capture for C6+in H, 179 CTMC, see Classical-trajectoryMonte Carlo
436
SUBJECT INDEX
D
E
D-I and D-11, see Diabatic Decay cross section, 358 equation for, 358 Degenerate perturbation theory, 407 -4 1 1 Demkov coupling, 283,298 Demkov model, 296 Demkov-Nikitin transition, 285 Density matrix and rate equations, 415-426 atomic rate equations, 421 -422 the continuum as a reservoir, 422-424 Laplace transform for, 4 I5 non-lorentzian laser linewidth effects, 424-426 PO derivation, 4 19- 42 1 DESB, see Doubly excited symmetry basis “Diabatic” curves, 100 Diabatic processes, 245-246 Diabatic transition models, 246 Diagrammatic perturbation theory, 41 1 -414 Diatoms-in-moleculeprocedure, 259 Dielectronic recombination, 20 1, 2 13-2 15, 306 for B2+and C3+,2 15 for Ca+, 214-215 definition of DR, 2 13 measurements of, 2 13- 2 15 for Mg+, compared with calculations, 214-215 DIM, see Diatoms-in-molecule Dipole approximation, 329 - 33 1, see also Bethe approximation Dipole representation, 11 1 Dipole transition operator, 26 Dirac electronic wave functions, 324, 346, 374 Dissociative charge transfer, 277 Distorted-waveapproximation, 19,2 1 Distortion effects, 338-345 Doppler shift, transverse, 3 18 Double Rydberg series, 138 Doubly excited states, 77- 142 classification of, 96- 108 Doubly excited symmetry basis, 97 DR, see Dielectronic recombination Dyson equation, 414 to sum diagrams, 4 14
EBIS, see Electron-beam ionization sources ECIP, see Exchange classical result added to a long-range impact parameter contribution ECR, see Electron cyclotron resonance ECR source, 158- 159 Effective charge C,83 potential function C in Rydberg units, 83 “Effective Gaunt factor,” 2 18 Eikonal treatment, 180, 184,247, 352,380 projectile K-electron capture cross sections from, 352 and outer-shell capture, 380 Electron-beam ionization sources, 2 12 Electron capture by fast multiply charged ions, 162- 166 cross sections for electron capture by @+, 164 Electron capture by simple ions, 243 - 3 14 experimental background, 26 1 -266 Electron capture by singly charged ions, 152- 154 cross sections for, 152 Electron capture by slow multiply charged ions, 154- 162 collisions involving excited H atoms, 161-162 measurements with bare nuclei and highly stripped ions, 158- 161 cross sections for C6+ions in H and H,, 160 measurements with partially stripped heavyionswithq>3, 155-157 measurements with partially stripped heavy ions with q < 3, 158 Electron-capture differential cross sections, 153 isotope effect in, 153 Electron capture in high-Z ions, 353-357 Xe projectiles on Au targets, 354-357 Electron capture into specified excited states, 166- 180 collisions involving H+ and HeZ+ions, 166-169 cross sections for H+ H Is, 167 cross sections for HeZ+ H, 168
+
+
437
SUBJECT INDEX collisions involving slow multiply charged ions, I69 - 180 Electron-capture processes, relativistic, 345-357 in high-Zions, 353-357 nonradiative electron capture, 348- 353 radiative electron capture, 345-348 Electron correlations, 80-85 Electron cyclotron resonance source, 150, 206,2 12 Electronic stopping powers, 326, 330 theory of target-electron ionization by incident ions, 326 Electron-impact excitation of positive ions, measurements of, 2 I5 -222 crossed-beam measurements, table of, 2 I6 and polarization of emitted radiation, 217-218 Electron-impact ionization, 20 1 Electron -ion collisions with intersecting beams, 197-241 types of reactions studied, 198 Electron promotion model, 285 Electrons, momentum density of, 38 Electron swarms, 40 Electron-transfer-excitationcollisions, 305 -3 10 dielectronic recombination, 306 doubly excited autoionization state in He+ He, 310 electron -electron correlation, 306 high-energy ion -atom and ion - molecule collisions, 305 in lithiumlike ion beams, 306 NTE, 305-310 RTE, 305-310 autoionization in intermediate state, 306 RTE and NTE cross sections, equation for, 309 RTE and NTE cross sections for S3+ He compared with theory, 307 RTE and NTE cross sections for Sill+ He compared with theory, 308 second Born contribution to CEX cross section, 305 theory and experiment for RTE cross sections for Ca”+ on H, and He, 307 Electron translation factor, 247
+
+ +
Electron tunneling model, 157 Elwert correction factor, 370 Emission cross sections in H, 180 for C6+,N7+,08+ Empirical formulae for ionization cross sections, 209 -2 1 I , see also Ionization cross sections, empirical formulae for Energy-change spectra, 170- 175 for one-electron capture, graphs by C2+, 173 by C3+, 175 by N2+,171 by N3+, 174 “Energy defect,” 234,237 ETF, see Electron translation factor ETF problem, 249 Exchange classical result added to a long-range impact parameter contribution, 21 I Excitation amplitudes, rate of change of, 338 Excitation cross section, 6 1,66, 70 Excitation of He+, measurements of, 218-219
F “Fermi golden rule,” 4 17 Fermi momentum, 370 Feshbach multiplets, energies of, 126 Feshbach projection technique, 78 Feshbach resonances, 9, 114-115, 125, 129 Feynman diagrams, 4 12- 4 13 Feynman graph method, 390 Feynman rules, 390 Field ionization technique, 19 1 to define excited states of H, 19 I First Born approximation, 183 Floquet theory, 389 for laser-molecule interactions, 389 Franck-Condon principle, 266, 27 1, 278, 298 for collisions in medium-energy range, 266 “Free collisions,” 186 Furnace target methods, 148- 151, 162, 166, 179 measurements of electron capture into metastable states, 15 1
438
SUBJECT INDEX
total and differential cross-section measurements, 148- 150 translational energy spectroscopy, 150-151 FWHM energy, 15 1 G
“g-bar formula” for excitation, 2 18 equation for, 2 18 ji empirical formula, 220 Glauber approximation, 182, 342 -344 and Born calculation, 344 for ionization by bare nuclei, 182 and reduced ionization cross sections, 342-344 Grand angular momentum operator, 83 Green’s functions, thermodynamic, 389 or propagator, 39 1 , 412 Ground-state model, 363 - 365
1 1-state model for relativistic Xe ions, 366-367 ground-state model, 363 - 365 for charge-state distributions, 363 higher charge states for, 367 and K x-ray production cross sections, 367 - 369 quasi-ground-state model, 366- 368 relative eikonal and REC capture cross sections compared, 368 - 369 Hultberg angular distributions, 348 Hylleraas expansion calculations, 168 Hylleraas wave function, 23, 24, 28 Hyperangle a,85 Hyperradial function F(R), 84 for CI states, graph of, 94 Hyperspherical coordinates, 82- 85 Schriidinger equation for two-electron atoms, 82 Hyperspherical potential curves for H- and H, 90-91 Hyperspherical surface, 89
H I Hartree-Fock model, 77 Hartree- Fock wave functions, relativistic, 33 1 “Heavy-ion fusion,” 197, 232 Heavy-ion linear accelerator, 3 I5 He+ ions, measurements of excitation of, 218-219 Helium channel functions of, 88 doubly excited states of He-, 77, 136- 137 cross section for autoionizing region of, 79 photoabsorption spectra of, 78 surface-density plots for, 95 volume charge-density distribution, square root of, 87 - 88 Heliumlike ions, excitation of, 2 19-220 Li+, 2 19 HIF, see Heavy ion fusion High-Z ions, on solid targets, 363-369 charge-state fractions, rate of change of, 363-364 equilibrium distribution, equation, 364 measured and calculated values compared, 365-366 electric-dipole decay rates large, 358
IA, see Ion-atom IA-CEX reaction, 248, 259 IM, see Ion-molecule IM-CEX, 257,259 Impact ionization, 66 for positrons, 66-67 Impact-parameter approximation, 302 Impact-parameter treatment, two-state, 158, 167, 168 Independent electron approximation, 77 Independent particle approximation, 77, 78, 91
Inner-shell effects, 206 in plasmas, 206 Inner-shell ionization cross sections, 330-335 distortion effects, 338-345 equation for, 330 K-shell ionization cross section per electron, 332 equation for, 332 K-vacancy production, graph, 333 target-atom screening, 335-338 Inner shell ionization with relativistic heavy ions, 324
439
SUBJECT INDEX projectile ionization in high-Z ions, 324 projectile ionization in low-Z ions, 324 target K-vacancy production, 324 Intersecting-beam methods employing fast hydrogen atoms, 147 - 148 Ion-atom collisions, 247, 266-269 He+ Ar, 269 He+ He, 266-268 Ion-ion collisions with intersecting beams, 197-24 1 types of reactions studied, 198 Ionization, 182- I88 by bare nuclei, 182- I84 general scaling relations for, 186- 188 by partially stripped ions, 184- 186 cross sections for ionization of H by @+, Nq+, @+, 184- 185 plot of scaling function for, 188 Ionization cross sections empirical formulae for, 209-2 I I ECIP, 21 I Lotz formula, 209 for primary ions, 146 of H by 25 -200-Ke V protons, 150 Ionization of positive atomic ions by electron impact, 20 I - 2 I3 Ionization of very highly charged ions, 21 1-213 isoelectronic sequences, trends in, 21 1 Kr”+ studies, 2 1 1 Ionization processes, with relativistic heavy ions, 324- 345 dipole approximation, 329-33 1 inner-shell ionization cross sections evaluated, 33 1 - 335 PWBA, 326-329 Ion kinetic energy, 3 17 Ion -molecule charge-exchange collisions: He++ H,, 273-293 collision spectroscopy techniques, 277-288 correlation diagram for He+ H,, 286 electron capture in, 273-277,282 energy-loss spectra, 277, 283 of Heo, 283 optical emission studies, 288 -293 reduced differential cross section, 280 total cross section for, 275, 277 vibro-rotational excitation energy, 278 definition of, 278
+
+
+
Ion-molecule collisions, 247,256 - 260, 269-273,293-305 adiabatic potential energy surfaces, 259-260 table of, 260 Ar+ CO, 299 - 30 1 capture channels in, 299 electron-capture processes in, 299 energy level diagram for selected direct and exchange states, 300- 30 1 energy spectra of Aro from charge-exchange collisions, 300 Ar+ N,, 269-273 charge-exchange probability, 272 coordinates used for, 257 H+ H2,301-305 Doppler-shifted radiation from, 302 electron-capture probability, 302 - 303 H+ A t as energy reference, 303 and H: ground-state potential energy surface, 301 possible isotope effect, 305 reduced cross section for, 304- 305 He+ N,, 293-297 angular distribution of Heo, graph, 295 - 296 charge-exchange probability in, 294 endothermic versus exothermic processes, 294 - 295 quasiresonant channel for charge exchange, 293,295 total CEX cross section for, 293 Wigner spin-conservation rile confirmed, 294 He+ NO, CO, 297 - 298 electron capture in He+ NO, 297 endothermic channels in, 297 quasiresonant charge exchange in, 298 total cross section for electron capture in He+ CO, 297 He+ 0,, 293 - 297 dissociative charge transfer in, 295 electron capture by He+ from 4, 295 quasiresonant exchange channel, 295 Ion velocity, v, 3 17 Isomorphism, 102- 103
+
+
+
+
+
+
+
+
+
J
Jacobi polynomial, 109
440
SUBJECT INDEX
K K-capture, 320 equation for, 320 Kohn variational method, 5 - 8,25,4 1 matrix form, 5 Koster- Kronig transition probabilities, 344- 345 K-shell electron binding energy EK,3 17 K-shell ionization, effects on, table of, 325 K-vacancy production, measured versus calculated, 333, 341 K x-rays production cross sections, 32 1 from uranium. 3 18
L LAMPF facility, 114, 125 H- beam from, 1 14 Landau- Zener method, 15 5 - 16 1 Landau-Zener-Stuckelberg model, 255, 256 Laplace identity, 418 Lennard- Jones- Wigner-Brillouin perturbation method, 387, 388 LEPD, see Low-energy positron diffraction “Level-shif?operator,” 405 Lithiumlike ions, excitation of, 220-221 Be+, C3+,N4+,220 LIWB, see Lennard-Jones- WignerBrillouin Lorentz gauge, 327 Lorentzian limit, 426 Lorentzian linewidth, 425 Lorentz-transform effects, 347 equation for, 347 Lotz empirical formula, 205 Lotz estimate, 205, 207 Low-energy positron diffraction, 38 Low-Z ions, on solid targets, 358 - 363 four-state model, 358-359 radiative decay rates compared to collisional de-excitation cross sections, 358 rate equations for populations of states, 359 ratio of ions with an electron to fully stripped projectiles, 359 solid- versus gas-target charge states, 361 Luminometer, 376
and pair production, 376 Lyman4 detector, 15 1 LZS, see Landan-Zener- Stuckelberg
M Many-body perturbation theory, 77 McKinley- Feshbach equation, 372 MCP, see Microchannel plate detector Merged-beam experiments, 200 MgO source of positron beams, 37 Microchannel plate detector, 68 MO, see Molecular orbital MO crossing, 245 Model potential approach, 259 Modulated crossed-beam technique, 182 Molecular ions, in collisions with electrons, 226-228 Molecular orbital collision model, 245 Multielectron atoms, doubly excited states Of, 131-137 alkali negative ions and alkaline earth atoms, 13 1 - 136 doubly excited states of He-, 136- 137 Hamiltonian, 137
N Nonradiative electron capture, 348 - 353, 362-363,377 capture amplitude, equation for, 350 eikonal approximation for, 368 OBK results, 350-352 Nonresonant transfer and simultaneous excitation, 305 - 3 10 NRC, see Nonradiative electron capture NTE, see Nonresonant transfer and simultaneous excitation 0
OBK, see Oppenheimer-BrinkmanKramers Onsanger distance r,, 46 Oppenheimer- Brinkman - Kramers approximation, 165, 181, 350-353 cross sections, 350- 353 relativistic and nonrelativistic, 352- 353 and second Born cross sections, 350- 35 1
SUBJECT INDEX
+
Optical emission studies, for He+ H2, 288 - 293 coincidence rates, polar plots of, 289 linear and circular polarization measurements, 29 1 photoncorrelation measurements, 290-291 Oregap, 2-3, 6, 40 Ore model, 40-41 Orthopositronium, 2 Outer-shell ionization, 206 Over-barrier model of charge transfer, 157, 161
P Parapositronium, 2 Pauli wave functions, 33 1 equation for, 33 i PB, see Primary bremsstrahlung Penning ion gauge, 149 Perturbation theories, 398-414 CF and diagrammatic perturbation theory, 41 1-414 CF and ordinary perturbation theory, 400-401 continued-fraction approach nondegenerate perturbation theory, 398- 399 fourth-order eigenvalue, 399 degenerate perturbation theory, 407 -4 1 1 PO and CF methods compared, 406 -407 projection-operator approach, 404 406 two-level atom in single-mode field, 40 I -404 Perturbed stationary state representation, 247 Perturbing charge Z,, 3 17 PES, see Photon emission spectroscopy Phaseaiffiuion model, 425 Photoabsorption, selection rules for, 97 -98 Photodetachment of H-, 78 Photoelectric ionization cross sections, 330, 346 equation for, 330 Sauter formula for, 346 Photoionization, 62,208 -210 of Ba+, Ca+, Sr+, 208,2 10 Photoionization probability, total, 422 equation for, 422 Photon-correlation measurements for He+ H,, D2, 290-291
-
+
44 1
Photon emission spectroscopy, 147, 154, 169, 170, 175 PIG, see Penning ion gauge PIG source, 206,2 12 to measure cross sections for AP+, 212-213 Plane-wave Born approximation, 324-345, passim based on first-order perturbation theory, 326 cross section for protons, 337 for high-Z ions on solid targets, 366 relativistic versus nonrelativistic, 325 PO, see Projection operator Polarization amplitude, 340 Polarization effect, 340 Polarized orbital approximation, 58 Positron-argon collisions angular dependence of Ps emission, 62 scattering, 22 Positron -atom collisions, 1 potential, 43 scattering, 2 - 22 Positron beam technology, 37 Positron differential scattering cross sections, 38 Positron drift velocities, 49 - 52 Positron-electron annihilation radiation, 33 from galactic center, 33 from solar flares, 33 Positron-helium scattering, 19-23,69 apparatus to measure ionization cross sections, 69 experimental and theoretical results plotted, 20 Positron - hydrogen scattering, 3 - 19 angular distribution of positronium, 18-19 cross section for, 5 cross sections, elastic and total, 11, 17 differential cross section for formation of positronium, 18 dwave, 13-15 Hamiltonian for, 3 higher partial waves, 15 fwave, 15 as positronium-proton, 3 pwave, 11-13 energy dependence of, 12
442
SUBJECT INDEX
s wave, 6- 1 1 energy dependence of, 6- 7 total cross sections, 16 - 18 total wave function for, 3-4 Positron lifetime spectra, 38 Positron -lithium scattering, 22 Positron mobility, 49 - 52 Positron-neon scattering, 22 Positron-potassium scattering, 22 Positrons in gases, annihilation spectra for, see Annihilation spectra for positrons in gases Positron-xenon bound state, 23 Positron-xenon scattering, 42 -45 Positronium binds to itself, Ps,, 32 binding energy, 32 dipole polarizability of, 2 excited state, 63 -67 experimental work, 37 - 75 ground-state energy of, 2 interaction with simple systems, 22-33 spectroscopy, 38 theoretical work, 1-36 thresholds of formation, 2 Positronium- electron system, 23 - 28 autoionization, 24 photodetachment cross section of, 26-28 ISeand ’Se resonances, 25 wave function for, 23 Positronium formation cross sections, 1 fraction F, 39-41 threshold, 54, 59 time, 47 Positronium hydride, 28 energy of dissociation of, 28 Positronium- hydrogen system, 28 - 32 interaction potential of van der Waals form, 32 scattering cross sections, 3 1 s-wave phase shifts, 29 s-wave resonance, 29 Post-adiabatic method, 139 “Postcollision interaction,” 204 Post-prior prescription, 355 post-prior discrepancy, 356 Potential curves, 99- 102 for channels for helium, 99- 101 Pratt bound-state normalization theory, 346
Primary-bremsstrahlung cross section, equation for, 369 - 37 1 Bethe-Heitler formula, 370- 373 Elwert correction factor, 370 Product channels, 170- 172 Projectile atomic number Zp, 3 17 Projection operator method, 387, 404-406 Ps, see Positronium Pseudopotential construction, 260 PSS, see Perturbed stationary state PWBA, see Plane-wave Born approximation
Q Quantum electrodynamics, 38 Quantum-electrodynamic self-energy, 3 16 “Quasi-diatomic” approximation, 302 Quasi-ground-statemodel, 366 - 368 Quasiseparableapproximation, 88 - 89, 128-129 validity of, 93-95 QED, see Quantum-electrodynamic Quenching rate of ortho-positronium, 32
R Rabi oscillations,4 18 Radial correlations, 85 - 89 Radiative electron capture, 3 18, 345 -348, 358,362,366, 368,377 angular distribution of K REC photons for Xe Be, 348 cross section for, equation, 346 photoelectric cross sections for, 368 photon peak energies, equation for, 346 photon production, 358 in terms of photoelectric cross section, 346 total photon cross sections, equation for, 347 Rayleigh-Schradinger method, 387, 388, 400,412 REC, see Radiative electron capture Relativistic few-electron ions in QED experiments, 377-379 high-Z Lamb-shift experiments, 377-379 muonic atom experiments, 377 Relativistic heavy-ion-atom collisions, 3 15-385 complex interactions inside solid targets, 316
+
443
SUBJECT INDEX electron-capture processes, 345 - 357 experiments, 3 I7 -324 ionization processes, 324-345 relativistic few-electron ions in QED experiments, 377-379 solid targets, collisions in, 357-369 ultrarelativistic collisions, 374- 377 x-ray continuum processes, 369- 374 Relativistic synchrotron, 3 15 Relativistic-velocity theories, 3 16 eikonal approximation, 3 16 first Born approximation, 3 16 impulse approximation, 3 16 second Born approximation, 316 strong potential Born approximation, 316 “Resonance collisions,” 186 Resonance structures in H- photodetachment, effects of strong electric fields on, 125-130 “Resonant” or “near-resonant” collisions, 234 H+ Mg+, 234 HC Ti+, 234-235 He+ Hi,234 Resonant transfer and simultaneous excitation, 305-310 resonant-transfer excitation, 257 Retarding field analyzer, 67 R-matrix method, I39 Rotorlike structure of states, 104- 105, 108 Rozen-Zener-Demekov model, 256 RS, see Rayleigh- Schriidinger RTE, see Resonant transfer and simultaneous excitation Rutherford cross section, 186 for energy transfer to a free stationary electron, 186
+
+ +
S Sauter formula, 346 Scaling law, simple, 236 equation for, 236 Scaling parameters, 165 Scaling relations, general, for electron capture by multiply charged ions, 180181 Scattering of electrons by ions, 222-224 measured differential cross sections, 223 for Cd+ and Zn+, 223-224
SEB, see Secondary-electronbremsstrahlung Secondary-electronbremsstrahlung, 372 elastic electron scatteringcross section, 372 electron solid target nuclei, 372 Singly excited and doubly excited states with A = 0, 106- 108 Smooth q” power-law function, I65 Solid targets, collisions in, 357 - 369 high-2 ions, 363- 369, see also High-Z ions, on solid targets low-Z ions, 358-363, see also Low-Z ions, on solid targets REC photon production, 358 and relativistic heavy ions, 357-369 target-thickness dependence, 358, 363 Spin-changingtransition, 219 Spin-flip contributions, 330, 332-333 Split-shell model, 225 Spur model for positronium formation, 45 -48 density dependence of fraction F, 47-48 Spur radius, 47 Stark shifts, 125, I30 linear and quadratic, 125 Stark states, 1 16 States-selective capture, I50 Static exchange approximation, 25, 30 Stokes parameters, 29 1-292 graph of, 292 Straight-line trajectory approximation, 250 Stripping cross section, 362 Sturmian calculation, 233-234 Super-HILAC, 3 15, see also Heavy-ion linear accelerator Supermultiplet schemes, 121 Supermultiplet structure, 80, 103- 106 rotorlike structure ofstates, 104- 105, 108 Surface charge density, averaged, 102 Surface magnetism, 38
+
T Target-atomic number Z, , 3 17 Target-atom screening, 335-338 antiscreening correction, 336-337 TD, see Time-dependent Tdoubling, 104, 123 TES, see Translational energy spectroscopy Thermonuclear fusion. 143
444
SUBJECT INDEX
“Thin plasmas,” 197,201 DR in, 2 13 TI, see Time-independent Time-dependent CEX theory, 252 Time-independent CEX theory, 252 Time-of-flight electron-ion coincidence spectrum for H+in H,, 146 Time-of-fight method, 57,66,70,261-265, 272,295,302-303
TOF, see Time-of-flight Tokamak devices, 243 Total scattering function, 249 equation for, 249 Transition probabilities, 254-256 for ion-atom CEX collisions, table, 254 Translational energy spectroscopy,
uranium on fixed-target uranium, 375 Unitarized distorted-wave approximation, 155, 161, 162, 181
UDWA cross sections, 18 1 “Unwanted” variables, 397 Uranium and ASC, 337 cross section for continuum x-ray emission, 373 K-shell ionization in, 335 one- and two-electron ions of, 3 16 proton-induced K-shell ionization cross sections, 33 1 U Be and U U spectra, 318-319,346
+
+
V
150-151, 154, 170-177
TES versus PES for oneelectron capture by C3+in H, 176 Transverse excitation cross sections, 325 Trapped-ion method, 227 Triatomic correlation diagram, 287 Triply excited states, 140 Two-center atomic orbital expansion method, 179 Two electron-excitation processes, 285,299 Two-electron Schrainger equation in hyperspherical coordinates, solution of, 109-115
asymptotic limit and the long-range dipole approximation, 1 10- 1 1 1 hyperspherical harmonics and solutions at small R, 109- 110 numerical solution of the channel equations, I I 1- 113 representative results for H-, 1 13 - I 15 potential curves, 1 13 Tungsten tube furnace, 144, 148
U UDWA, see Unitarized distorted-wave approximation Ultrarelativistic collisions, 374- 377 electron- positron charge-changing processes, 375 pair-production cross sections for U92+on U92+, table, 375 ultrarelativistic h a w ion accelerators to study quark matter, 374
Van der Waals equation of state, 44-45 “Vertical transitions,” 298 Von Neumann equation of motion, 388,415 application of CF perturbation theory to, 388
W
Wannier state of two continuum electrons, 138
“Wanted” variables, 397 Weber functions, 255 Wigner spin-conservation rule, 294,298 WKB approximation, 128
X Xenon angular distribution of K REC photons for Xe Be, 348 REC dominates, 355 cross sections for continuum x-ray emission, 373 electron capture by Xe”+ and Xe52+,355,
+
364 - 36
comparison with eikonal calculations, 355-356
K, x-ray production cross sections, measured and calculated, 368 K-shell ionization in, 335 reduced cross sections for XeS3+ Xe, 342 relative cross sections for electron capture, 354
+
445
SUBJECT INDEX
+ +
Xe Ag, 354 Xe Au, 354 Xe Ag, relative cross sections for electron capture, 354-355 Xe Pb, 344 Xe Zr, 344 Xe projectiles on Au targets, 354-355 Xe Xe ionization cross section, 343 XeS4+and Xe52+incident on Mylar, Be, Al, CU,Ag, Au, 355-356 X-ray continuum processes, 369-374 primary bremsstrahlung, 369- 37 1 secondary-electron bremsstrahlung, 372 2, dependence of, 372- 314
+ +
+ +
in graph of Xe and U collisions, 373 X-ray cross-section measurements, using relativistic heavy ions, 3 18 - 32 1 bremsstrahlung, primary- and secondaryelectron, 3 19 K x-rays, from projectile and target U, 3 18 radiative electron capture, 3 18 transverse Doppler shift, 3 18 U Be spectrum, 318-319 U U spectrum, 3 19
+ +
Z “Zero-order dipole basis,” 1 1 1
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