Positron scattering by atomic hydrogen at intermediate energies: 1s 3 Is, 1s + 2s and 1s + 2p transitions

Transcription

1 J. Phys. B: At. Mol. Opt. Phys. 21 (1988) Printed in the UK Positron scattering by atomic hydrogen at intermediate energies: 1s 3 Is, 1s + 2s and 1s + 2p transitions H R J Walters Department of Applied Mathematics and Theoretical Physics, The Queen s University of Belfast, Belfast BT7 INN, UK Received 7 January 1988 Abstract. Calculations are reported for Is-, 1s elastic scattering and for Is+2s and Is+ 2p excitation of atomic hydrogen by positron impact in the energy range eV. A close-coupled pseudostate approximation is used to represent transitions between s, p and d states of the target while couplings to target states of higher angular momentum are treated perturbatively through the second Born approximation. Comparison is made with similar calculations of van Wyngaarden and Walters for electron scattering. The cross section for elastic scattering is smaller than its electron counterpart, the difference being as much as a factor of three at 54.4 ev. The cross sections for Is+ 2s and Is+ 2p excitation are larger than for electron collisions. By contrast the total cross sections for the two projectiles remain almost equal down to the lowest energy. Comparison is also made with other sophisticated theoretical calculations. At 100 ev and above good agreement is found with the previous theoretical work on elastic scattering. For the ls+2s and ls+2p transitions agreement with the other theories is fairly good in the forward angular region, but at large angles significant discrepancies arise, although these reduce with increasing impact energy. Values for the angular correlation parameters A, R and I are also presented for 1s+2p excitation at 54.4eV and compared with the corresponding electron impact results. 1. Introduction The development of good positron sources has resulted in a considerable amount of experimental activity in the area of positron-atom and positron-molecule collisions (Armour and Humberston 1988). The measurements produced to date have nearly all been confined to total or integrated cross sections but now differential data are becoming available (Coleman and McNutt 1979, Hyder et a1 1986, Smith et a2 1987). It is expected that in the not very distant future results will be obtained for positron scattering by the simplest of all targets, namely atomic hydrogen (see Lube11 and Sinapius in Armour and Humberston 1988). It is therefore an appropriate time to look in some detail at positron-hydrogen collisions. That is the purpose of the present work. Recently van Wyngaarden and Walters (1986) have published extensive calculations for electron-hydrogen scattering in the energy range ev using a multipseudostate close-coupling approximation, supplemented by the second Born approximation to make allowance for target states of angular symmetries not included in the pseudostate set. Here we shall use more or less the same approximation to study positron-hydrogen scattering. Even if there were no experimental impetus for this work, there would still be a theoretical interest in seeing how positron and electron cross sections compare when calculated in a similar approximation /88/ IOP Publishing Ltd 1893

2 1894 H R J Walters In D 2 the theory is briefly reviewed. Results for elastic scattering and excitation of the 1s + 2s and Is+ 2p transitions are given in 3 for the intermediate-energy range ev. Atomic units (au) in which h = m, = e = 1 are used throughout. The symbol a, denotes the Bohr radius. 2. Theory The theory is essentially the same as for the electron case (van Wyngaarden and Walters 1986). The scattering amplitude ffo( kf, k,) for a transition in which the hydrogen atom is excited from state +, to state Gf while the incident positron is scattered from momentum ko to kf is approximated by Here fro( PS) is the amplitude for the transition calculated in a coupled-pseudostate approximation using a pseudostate set containing only s-, p- and d-type states. To allow for the contribution of target states with angular momentum (AM) greater than or equal to 3 a second Born termffo(sba, AM> 3) is added on. The approximation (1) differs from that adopted in the electron scattering calculations of van Wyngaarden and Walters in that ffo(s~~, AM 3) is a plane-wave second Born amplitude rather than a distorted-wave term as used by van Wyngaarden and Walters. While a distortedwave second Born approximation would be better, the extra effort involved to evaluate it is not merited at the present time, especially in view of the fact that the contribution from target states with AM 3 3 is not large. The pseudostate amplitude ffo( PS) is calculated by solving the usual coupledpseudostate equations where r, labels the positron coordinates, r, the electron coordinates, and the elements of the pseudostate sett are denoted by $,,. The total energy of the system is (k:/2+ E,) where so is the energy of the initial state +,. The momentum k;, is related to the total energy by E: = kz,+ 2(Eo- E,) (3) where it is assumed that the pseudostates are orthonormal and diagonalise the atomic Hamiltonian with energies E,,, i.e. ((Ln I H al qm> = ZnSnm. (4) Following -- van Wyngaarden ---- and Walters (1986) we have used the nine-member (Is, 2s, 3s, 4s, 2p, 3p, 4p, 3d, 4d) pseudostate set of Fon et a1 (1981) for the calculation at an incident energy of 54.4 ev (k,= 2 au). At 100, 200 and 300 ev we have used the 21-member (s, p and d states from n = 1-8) improved pseudostate set of van Wyngaarden and Walters but, like them, keeping only open channels. This means that at 100, 200 and 300 ev we only retain states up to n = 6, n = 6 and n = 7 respectively in the coupled equations (2). f In general the pseudostate set contains real eigenstates as well as true pseudostates. In particular it must include the initial and final atomic eigenfunctions Go and Qt.

3 Positron scattering by atomic hydrogen 1895 The second Born term ~~O(SBA, AM 3 3) has been extracted from the calculations of Kingston and Walters (1980)i. This amplitude is evaluated in the closure approximation using the average energies given by Kingston and Walters. Since Kingston and Walters do not give an elastic amplitude for 54.4eV we have separately calculated foo(s~~, ~ ~ 2 for 3 this ) case using the average energies ER= au and E, = au for the real and imaginary parts respectively (see table 1 of Kingston and Walters (1980)). We study elastic scattering off the 1s state and excitation of the ls+2s and Is+ 2p transitions. The quantities of interest are the differential cross section U( f) for exciting the state &, the corresponding integrated cross section Qf, the total cross section QT and the angular correlation parameters A, R and Z for the 2p state. These are given by 4T QT=-Imfoo(t9=0) k0 (7) In (7) foo(8 = 0) is the forward elastic scattering amplitude. In (9) and (10) fo and fl stand for the 1s + 2p0 and 1s + 2p+, amplitudes respectively, where the quantisation axis is taken to lie along the direction of ko; it is assumed that a factor e-" has been removed from f,, where 4 is the azimuthal angle of kf measured about ko as z axis. The asterisk denotes complex conjugation. The parameters A, R and I are not independent in this case; it is easily verified that (Fargher and Roberts 1983) R'+ I' = :A (1 -A). (11) 3. Results 3.1. Elastic scattering Our results for the elastic differential cross section in the energy range ev are illustrated in figure 1 and given in table 1. In figure 1 comparison is made with the corresponding electron collision cross sections of van Wyngaarden and Walters (1986) and with other theoretical values for positron scattering. Van Wyngaarden and Walters do not quote any values for the elastic electron cross section at 54.4 ev; that shown in figure l(a) has been calculated, in the manner of van Wyngaarden and Walters, using the nine-member s, p, d pseudostate set of Fon et a1 (1981) supplemented by a distorted-wave second Born term to allow for couplings to target states with AM 3 3. t Note that the plane-wave second Born term is the same for both positrons and electrons.

4 1896 H R J Walters i i 1 L l l l l l " l " l l i I " l Scattering angie ldegl b I I l l, I t / I I I I I, Scattering angle ldegl 1 " I " I " I " l " 10 IC1 1 " l " I " I ' ~ l ' ' 10 Id I ~ l l l l l l l l l l l l l l l Scattering angle ldegl., Figure 1. Differential cross sections for elastic e+-h(ls) scattering at (a) 54.4eV, (b) 100eV, (c) 200eV, (d) 300eV: -, present results; - - -, cross sections of van Wyngaarden and Walters (1986) for electron scattering; A, third-order optical model of Byron and Joachain (1981); 0, UEBSZ of Byron er a/ (1985); SOPM of Makowski er a1 (1986); x, CCOM of Bransden e? ai (1985).

5 Positron scattering by atomic hydrogen 1897 Table 1. Present differential cross sections for the elastic scattering of positrons by H(ls), in units of a: sr-i. Powers of ten are denoted by a superscript. Energy (ev) Angle (deg) I 4.73-I I 1.61-' Y ' 6.23-' 3.71-I 2.52-' 1.75-' 1.23-' ' Y ' 6.77-I 4.63-I 2.64-' 1.65-' 1.04-I Y ' 3.71-I 1.88-' 1.0s-l 6.37T * Figure 1 shows the now familiar behaviour (Walters 1984) of the elastic electron cross section exceeding its positron counterpart. At 54.4 ev the electron numbers are about three times larger than the positron values; this is clearly reflected in the integrated cross section of table 2. At angles away from the forward region, the difference between electron and positron cross sections narrows with increasing energy, becoming about 30% at 300eV. At the forward direction, however, there is little change in the electron/positron ratio, this ratio remaining at the approximate value of 3.1 over the energy range ev. To the extent that second-order perturbation theory is valid, and it usually is for forward scattering by light atoms at not too low energies (Walters 1984), the explanation of the large difference between the electron and positron cross sections at forward angles in figure 1 lies with the relative signs of the first Born (fb,) and second Born (fb2) terms. In elastic scattering fbl is real, while RefB2 contains the important dipole polarisation interaction which is prominent for forward scattering; ImfB2 represents flux loss (Walters 1984). For electron scatteringf,, and RefBZ have the same sign with the result that the two amplitudes add to give a large forward peak. Changing to positron Scattering reverses the sign of fb1 but leaves fb2 unaltered; consequently there is now cancellation between fb1 and RefB2 and as a result a smaller cross section than in the electron case. The other theoretical calculations shown in figure 1 are?: (i) the third-order optical model OPT^) of Byron and Joachain (1981); (ii) the unitarised eikonal-born series t The reader is directed to Walters (1984) and van Wyngaarden and Walters (1986) for a discussion of these approximations and comparison with the present approach.

6 ~ ~~ 1898 H R J Walters Table 2. Integrated cross sections for the elastic scattering of positrons by H(ls), in units of nu:. Energy (ev) Electrons" Present results OPT3b UEBS2' SOPMd CCOMe Morgan' First Born a van Wyngaarden and Walters (1986). At 54.4 ev, as calculated in this paper, see text. Third-order optical model, Byron and Joachain (1981). The result at 54.4 ev has been extrapolated from the value given at 50 ev. Unitarised eikonal-born series (version 2), Byron et a1 (1985). Second-order potential method, Makowski et al (1986). e Coupled-channel optical model, Bransden et a1 (1985). ' Twelve-state close coupling, Morgan (1982). (version 2) (UEBSZ) of Byron et a1 (1985); (E) the coupled-channel optical model (CCOM) of Bransden et al (1985); (iv) a one-channel second-order potential method (SOPM) (Bransden and Coleman 1972, Winters et al1974) as implemented by Makowski et a1 (1986) for positron scattering. From figures l(b)-(d) it is clear that the OPT^ results agree quite well with our calculations. The UEBSZ values of Byron et a1 (1985) at 100 and 200 ev are indistinguishable from the OPT^ values on the scale of figures 1( b) and (c), except for angles beyond 40" at 100eV where the UEBSZ cross section lies below both OPT^ and our present calculations. At 300 ev the unitarised eikonal-born series approximation (version 1) (UEBSI) of Byron et al (1982) (not shown in figure 1) also agrees well with the OPT^ and present cross sections. The SOPM results of Makowski et a1 (1986) are close to the values predicted by the present, OPT^, UEBSZ and uebsi approximations at 100, 200 and 300 ev, so close that in the interests of graphical clarity we do not show them in figures l(b)-(d). At 100 ev the large-angle SOPM cross section lies between the OPT^ and UEBS~ points. The SOPM calculation at 54.4eV is shown in figure l(a). Here agreement with the present work is not as good as at the higher energies; at forward angles there is fair accord, but at large angles the SOPM cross section is significantly smaller than the present values. Finally, we compare with the CCOM calculation of Bransden et a2 (1985). At 54.4 ev (figure l(a)) the CCOM gives a noticeably smaller cross section at forward angles than the present or SOPM approximations; at large angles the CCOM values lie between our present predictions and the SOPM points, tending to be closer to the latter. At 100 ev (figure l(b)), the CCOM cross section is seen to be generally lower than the other theoretical predictions; however, by 200 ev (figure l(c)) it has essentially converged to the other theoretical values. It is clear from the comparisons of the previous paragraph that there is a fair measure of agreement between the various theoretical approximations, particularly above 100 ev. This situation is analogous to that of the electron case (van Wyngaarden and Walters 1986) and, as there, the consistency and pattern of the theoretical predictions prompts us to suggest that our present cross sections are probably accurate to better than 5% at any angle at 200 and 300 ev.

7 Positron scattering by atomic hydrogen 1899 Integrated cross sections for elastic scattering are shown in table 2. These numbers reflect the trends of figure 1. Thus the electron cross section is significantly larger than its positron counterpart, even at 300 ev. The OPT^ and UEBSZ cross sections are a little bigger than the present values while the CCOM results are noticeably smaller. The present calculations are well supported by the SOPM numbers and especially by the calculation of Morgan (1982) at 54.4 ev who also used a coupled pseudostate approximation involving six eigenstates and six pseudostates, the pseudostates being different from those adopted here (Morgan et a1 1977). Total cross sections, QT, are given in table 3. Here again we see another well known pattern (Stein and Kauppila 1982, Walters 1984), namely the close agreement between electron and positron total cross sections at lower energies than might be expected. For example, at 54.4 ev the elastic electron and positron cross sections of table 2 differ by a factor of three, yet the total cross sections of table 3 deviate by a mere 3%. This agreement is 'explained' by a very important result proved by Dewangan (1980). Dewangan has shown that in a closure approximation, and neglecting exchange, all odd Born terms, from third order upwards, are zero for forward elastic scattering. From the optical theorem (7) it then follows that electron and positron total cross sections should be identically equal at all energies (see Walters 1984). While this result is limited by the closure approximation and neglect of exchange, it does at least show that there is a strong tendency for electron and positron total cross sections to be equal at non-asymptotic energies. The values of table 3 and the experimentally observed (approximate) equality in other cases (Stein and Kauppila 1982) are a manifestation of this effect. Table 3. Total cross sections, QT, for the scat:ering of positrons by H( Is), in units of xu:. Energy (ev) Electrons" Present results PT UEBSZ' SOPMd CCOMe Van Wyngaarden and Walters (1986). At 54.4 ev, as calculated in this paper, see text. bthird-order optical mode, Byron and Joachain (1981). The result at 54.4eV has been extrapolated from the value given at 50 ev. Unitarised eikonal-born series (version 2) Byron et a/ (1985). Second-order potential method, Makowski ef a/ (1986). e Coupled-channel optical model, Bransden et al (1985). The electron and positron numbers in tables 2 and 3 imply that the total inelastic cross section (0)- Qls) for positron scattering must exceed that for electron scattering (Walters 1984). On the basis of the present calculations the positron total inelastic cross section is 40% larger than the corresponding electron cross section at 54.4 ev, this decreasing to 5% at 300 ev. All the theoretical calculations of QT shown in table 3 are generally in agreement, except for the CCOM number at 54.4 ev which seems to be somewhat low.

8 1900 H R J Walters 3.2. The 1s + 2s transition Differential cross sections for the 1s + 2s excitation are illustrated in figure 2 and given in table 4. Figure 2 shows that the 2s cross section is generally larger for positron collisions than for electron collisions, particularly at the important forward direction. The CCOM results of Bransden et a1 (1985), the UEBsi numbers of Byron et a1 (1981) at 100 ev and the UEBSZ calculations of Byron er a1 (1985) at 200 ev (as reported by Joachain in Armour and Humberston (1988)) agree quite well with the present work at forward angles, figure 2. However, at large angles the UEBS cross sections are smaller 101 ' ' I " I I " I " I ' r X x I I I I I I, I I I I I I I I I, Scattering nngle ldegl 10-5 I I I, I I, I I, I I,, 1, b Scattering angle ldegl Scattering angle ldegl (1988). Figure 2. Di..-rential cross sections for e++ H( Is)+ ef+h(2s) at (a) 54.4eV, (b) 100eV, (c) 200eV:, present results; - - -, cross sections of van Wyngaarden and Walters (1986) for electron scattering; x, CCOM of Bransden er al (1985); 0, UEBSI ofbyron etal(1981);a,uees2ofbyron eral(1985) by Joachain in Armour and Humberston

9 ~ Positron scattering by atomic hydrogen 1901 Table 4. Present differential cross sections for the ls-2~ excitation of atomic hydrogen by positron impact, in units of a&-'. Powers of ten are denoted by a superscript. Energy (ev) Angle (deg) I 7.70-' ' 4.92-' ' 3.01-' I 4.59-' 2.68-' 1.75-' ' 3.18-' 1.70-' ' 2.22-' 1.066' ' 1.56-' 6.W ' ' 1.11-' ' W' * K ; Y Y T T * P Y ; ' W' ' ' K ' Y ' 9.W Y ~ 2.72-' Y j than the present results. This is also true of the CCOM cross sections at 54.4 and 100 ev, but not at 200 ev. It is interesting to observe the convergence of the large-angle CCOM cross section to the present values with increasing energy. Integrated 1s + 2s cross sections are given in table 5. At 54.4 ev the positron cross section exceeds the electron result by a factor of two; by 300 ev the gap has narrowed Table 5. Integrated cross sections for e*+h(ls)+e++h(2s), in units of mi. Energy (ev) Electrons" Present UEBS~~ CCOMC Morgand First Born a Van Wyngaarden and Walters (1986). Unitarised eikonal-born series (version I), Byron et a/ (1981). Coupled-channel optical model, Bransden et al (1985). Twelve-state close coupling, Morgan (1982).

10 H R J Walters to 10%. The present numbers are very well supported by the umsi values at 100 and 200 ev and by Morgan's (1982) result at 54.4 ev, which has also been calculated using pseudostate close-coupling (six eigenstates + six pseudostates). The CCOM cross sections at 100 and 200 ev are somewhat larger. Note how quickly the present and umsl cross sections converge to the first Born value, while the electron results still remain noticeably different even at 300 ev The 1 s + 2p transition Differential cross sections for the Is+ 2p transition are given in table 6 and shown in figure 3. Because this is an optically allowed transition the cross section at small angles is dominated by the first Born amplitude. As a result, forward electron and positron cross sections are similar and different theoretical approximations tend to converge at forward angles. Away from the forward direction the positron cross section exceeds the electron cross section. The large-angle cross sections of the CCOM (Bransden et ai 1985) at 54.4 and 100 ev are significantly smaller than the present results, but it is clear that with increasing energy they approach the present values, figure 3( c). By contrast the WEBS~ calculation at 200 ev (Byron et a1 (1985), as reported by Joachain in Armour and Humberston (1988)) gives a bigger cross section at large angles. Table 6. Present differential cross sections for the Is-, 2p excitation of atomic hydrogen by positron impact, in units of ai sr-i. Powers of ten are denoted by a superscript. Energy (ev) Angle (ded ' ' 6.55' 7.22' 5.96' ' 3.19' 2.14' 1.42' ' 1.58' I ' 2.80-l ' 1.20-' ' 1.82-' ' ' 1.19-' ' ' 2.43-' * 1.30-* * * * j ' K5 1.5-' ' 1.3-'

11 11 1 rl' 10 L NO - a c 5 10 U VI z U Scattering angle ldegl Positron scattering by atomic hydrogen ,, I, I I, I I I I I I I I Scattering angle idegl Figure 3. Differential cross sections for e++ H( 1s) + e*+h(2p) at (a) 54.4eV, (b) 100eV, (c) 200 ev: -, present results; - - -, cross sections of van Wyngaarden and Waiters (1986) for electron scattering; x, CCOM of Bransden er a/ (1985); A, UEBSZ of Byron et a/ (1985) as reported by Joachain in Armour and Humberston (1988).

12 1904 H R J Walters Table 7. Integrated cross sections for e*+h(ls) +e++. H(2p), in units of naz. Energy (ev) Electrons" Present CCOMb Morgan' First Born Van Wyngaarden and Walters (1986). Coupled-channel optical model, Bransden et a1 (1985). Twelve-state close coupling, Morgan (1982). Integrated 1s + 2p cross sections are reported in table 7. As is already clear from figure 3, the positron cross section is larger than its electron counterpart. The present calculations are in good agreement with the twelve-state calculation of Morgan (1982) at 54.4 ev and the CCOM values at 100 and 200 ev. At 54.4 ev the CCOM cross section is 8% lower. Both the electron and positron cross sections approach the first Born limit from below. The angular correlation parameters A, R and I for electron and positron impact at 54.4eV are compared in figure 4. Unlike the case of electron scattering, the A parameter for positrons shows a single minimum at 22". This minimum is lower and significantly wider than the corresponding first minimum in the A parameter for electrons. The first Born values for A are also shown in figure 4(a) for reference. The R parameter for positrons is somewhat different from the electron result in the angular range "; here ihe positron parameter comes to a maximum at about 62" while the electron parameter plunges through zero. There is a pror,ounced difference between the I parameters for positron and electron scattering, figure 4(c). 4. Conclusions Cross sections for 1s -, Is, 1s +. 2s and Is + 2p scattering of positrons by atomic hydrogen in the energy range ev have been calculated using a coupled pseudostate approximation augmented by a second Born term. The pseudostates are employed to represent target states of s, p and d symmetries while the second Born term results from a perturbative treatment of couplings to target states of higher angular momentum. Comparison has been made with the similar calculations of van Wyngaarden and Walters (1986) for electron scattering. Consistent with now familiar patterns which have been observed elsewhere (see, for example, Stein and Kauppila 1982, Walters 1984), it is found that the Is+ 1s elastic cross section is smaller than its electron counterpart, substantially so at the lower energies, while the total cross sections for electron and positron scattering remain almost equal down to the lowest energy. These two results taken together imply that the total inelastic cross section for positron scattering must exceed that for electron scattering. For the individual inelastic transitions ls+2s and ls+.2p it is found that positron scattering is indeed larger than electron scattering. Comparison of the present work with the best of previous calculations shows a good measure of agreement for Is-, 1s elastic scattering at energies of 100 ev and above. The agreement at the higher energies is so good that we estimate that our elastic

13 ~ Positron scattering by a tom ic hydrogen 1905 Scattering angle (degl I I I 1 I I I I I I I I I I I I I ' Scattering angle ldegl 1 ' ' 1 ' ' 1 ' 1 ' ' ' 1 ' 1 -o.+ 'J I 1, 1, I I,, I,, I I L Scattering angle (degl Figure 4. Angular correlation parameters A, R and I at 54.4 ev: -, present results; - - -, results of van Wyngaarden and Walters (1986) for electron scattering; -. -, first Born approximation. differential cross sections at 200 and 300 ev are probably accurate to better than 5% at any angle. For the 1s + 2s and 1s -j 2p transitions there is a fair degree of harmony between the various theories for scattering in the forward region, but at large angles significant discrepancies arise, although they reduce with increasing impact energy. Comparison of the angular correlation parameters A, R and I for electron and positron excitation of the 2p state shows interesting differences.

14 1906 H R J Walters Acknowledgments Most of the calculations were carried out on the Cray XMP/48 computer at the Rutherford-Appleton Laboratory under a grant provided by the UK Science and Engineering Research Council. References Armour E A G and Humberston J (ed) 1988 Proc. NATO Advanced Research Workshop on Atomic Physics with Positrons, London, 1987 (New York: Plenum) Bransden B H and Coleman J P 1972 J. Phys. B: At. Mol. Phys Bransden B H, McCarthy I E and Stelbovics A T Phys. B: At. Mol. Phys Byron F W Jr and Joachain C J 1981 J. Phys. B: At. Mol. Phys Byron F W Jr, Joachain C J and Potvliege R M 1981 J. Phys. B: At. Mol. Phys. 14 L J. Phys. B: At. Mol. Phys J. Phys. B: At. Mol. Phys Coleman P G and McNutt J D 1979 Phys. Rev. Lett Dewangan D P 1980 J. Phys. B: At. Mol. Phys. 13 L595-8 Fargher H E and Roberts M J 1983 J. Phys. B: At. Mol. Phys Fon W C, Berrington K A, Burke P G and Kingston A E 1981 J. Phys. B: At. Mol. Phys Hyder G M A, Dababneh M S, Hsieh Y F, Kauppila W E, Kwan C K, Mahdavi-Hezaveh M and Stein T S 1986 Phys. Rev. Lett Kingston A E and Walters H R J 1980 J. Phys. B: At. Mol. Phys Makowski A J, Raczynski A and Staszewska G 1986 J. Phys. B: At. Mol. Phys Morgan L A 1982 J. Phys. B: At. Mol. Phys. 15 L25-9 Morgan L A, McDowell M R C and Callaway J 1977 J. Phys. B: At. Mol. Phys Smith S J, Hyder G M A, Kauppila W E, Kwan C K, Mahdavi-Hezaveh M and Stein T S 1987 Proc. 15th Int. Conf. on Physics of Electronic and Atomic Collisions, Brighton ed Geddes et a1 (Amsterdam: North-Holland) Abstracts p 404 Stein T S and Kauppila W E 1982 Ado. At. Mol. Phys van Wyngaarden W L and Walters H R J 1986 J. Phys. B: At. Mol. Phys Waltets H R J 1984 Phys. Rep Winters K H, Clark C D, Bransden B H and Coleman J P 1974 J. Phys. B: At. Mol. Phys