Charlton M., Humberston J.W. Positron Physics

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4.2 Theory 159

Fig. 4.2. The results of various calculations of the l = 1 partial-wave contribu-

tion to the positronium formation cross section in positron–hydrogen scattering

in the Ore gap: A, Brown and Humberston (1985); B, Chan and McEachran

(1976); C, Winick and Reinhardt (1978b); D, coupled-static approximation; E,

Born approximation.

Born approximation are now only a factor of three too large, and those of

the coupled-static approximation are a little less than half of the accurate

values. Furthermore, because the diﬀerence between σ

and σ

is no

longer very small, the subtraction procedure of Winick and Reinhardt

(1978b) yields quite accurate values for the p-wave contribution to σ

The R-matrix results of Higgins and Burke (1993) and the coupled-state

results of McAlinden, Kernoghan and Walters (1994) and Kernoghan and

coworkers (1995, 1996) are in reasonable agreement with those of Brown

and Humberston, and also with those of Basu, Mukherjee and Ghosh

(1990), although the agreement between the latter results and those of

Brown and Humberston is worse than for the s-wave.

160 4 Positronium formation

Fig. 4.3. The results of various calculations of the l = 2 partial-wave contribu-

tion to the positronium formation cross section in positron–hydrogen scattering

in the Ore gap: A, Brown and Humberston (1985); B, Winick and Reinhardt

(1978b); C, Born approximation; D, coupled-static approximation.

The d-wave contribution to σ

is relatively insensitive to the method

of approximation, and even the results of the Born approximation agree

quite well with the accurate variational calculations. It is therefore to be

expected that reasonably accurate values can be obtained for all higher-

partial-wave contributions to σ

by using the Born approximation, and

this has been conﬁrmed by the results of Gien (1997).

The total positronium formation cross section in the Ore gap, con-

structed from the addition of accurate variational results for the ﬁrst

three partial waves and the values given by the Born approximation for

all partial waves with l>2, is plotted in Figure 4.4. On the scale of the

ordinate, the s-wave contribution is too small to be visible. A very small

s-wave contribution is found to be a feature of the positronium formation

cross section for several other atoms.

4.2 Theory 161

Fig. 4.4. The total positronium formation cross section in positron–hydrogen

scattering in the Ore gap as calculated using the results of Brown and Humber-

ston (1985) for l ≤ 2 and the Born approximation for l>2.

The diﬀerential cross section for positronium formation may be ex-

pressed in terms of the partial-wave K-matrix elements as

dσ

dΩ

(θ)=





(2l +1)



1 − iK



(cos θ)



, (4.27)

and the angular distributions obtained from the accurate data for the

ﬁrst three partial waves are plotted in Figure 4.5 for several energies in

the Ore gap. At energies just above the positronium formation threshold,

rather more than half the positronium is produced at angles greater than

◦

relative to the incident positron beam direction but, as the positron

energy is increased, the angular distribution of the positronium becomes

more peaked in the forward direction. It is this feature of positronium

formation which is exploited in the production of positronium beams (see

subsection 7.6.1).

162 4 Positronium formation

Fig. 4.5. The angular distribution of positronium formation in positron–

hydrogen scattering at various incident positron wavenumbers in the Ore gap.

Most investigations of positronium formation in positron–hydrogen

scattering have been made over a wider energy range than just the Ore

gap. One of the most detailed studies of s-wave scattering over the

energy range up to the n

= 4 excitation threshold of hydrogen is that of

Archer, Parker and Pack (1990), who included all energetically possible

reaction channels in the formulation. Their results reveal interesting

resonance phenomena associated with various excitation thresholds of

hydrogen and positronium. Rather less detailed studies, but extending

over an even wider energy range, up to 60 eV, and including more partial

waves, have been made by Higgins and Burke (1993), Hewitt, Noble and

Bransden (1990), McAlinden, Kernoghan and Walters (1994), Kernoghan

and coworkers (1995, 1996) and Mitroy (1996), all using various forms

of the coupled-state approximation. Higgins and Burke (1991, 1993)

found resonance structure in the s- and p-wave cross sections for both

positronium formation and elastic scattering at incident positron energies

4.2 Theory 163

Fig. 4.6. The two double-binary collision processes resulting in positronium

formation following positron impact at high energies. The positron collides with

an atomic electron: on the left the electron then scatters oﬀ the residual ion into

the same direction as the positron, whilst on the right the process is shown in

which the positron scatters oﬀ the residual ion.

between 2.7 ryd and 3.5 ryd (37–48 eV), well above the break-up energy

of the target atom, and they cannot therefore be Feshbach resonances

associated with degenerate excitation thresholds of the hydrogen or

the positronium atoms. Originally they were believed to be so-called

‘coupled-channel shape resonances’, arising from the coupling between

the hydrogen and positronium channels, but they

have since been shown

by Kernoghan, McAlinden and Walters (1994b) to be artifacts of neglect

of the other open channels.

Shakeshaft and Wadehra (1980) used the distorted-wave Born approxi-

mation to calculate positronium formation cross sections over the positron

energy range 13.6–200 eV. Their results for ground state positronium

formation agree reasonably well with more accurate values at the lower

end of this range, and so they are probably quite accurate at all energies.

Another variant of the distorted-wave Born approximation was used by

Mandal, Guha and Sil (1979), but their results are probably less accurate

than those of Shakeshaft and Wadehra, particularly at the higher energies.

The ﬁrst Born approximation is known to provide a rather inaccu-

rate description of positronium formation, even at high energies, because

the process then becomes essentially two-stage; this can be understood

as follows. In order to form positronium, the positron and an elec-

tron must emerge from the target with very similar velocities, and the

simplest way in which this can be achieved is via one or other of the

processes represented in Figure 4.6. In both cases, ﬁrst the positron

scatters from the electron and then either the electron or the positron is

scattered into the required ﬁnal direction by the nucleus. It is therefore

to be expected that the second Born approximation, with its quadratic

164 4 Positronium formation

Fig. 4.7. The diﬀerential cross sections for positronium formation into the

ground state and the n

= 2 excited states in positron–hydrogen collisions at

an incident positron energy of 5.44 keV: ——, ground state formation; -- -- -- --,

formation into the 2S state; — —, formation into the 2P state (Igarashi and

Toshima, 1992); ·····, ground state formation (Deb, McGuire and Sil, 1987).

Reprinted from Physical Review A46, Igarashi and Toshima, Destructive and

constructive interferences of the second Born amplitudes for positronium forma-

term in the interaction potential, provides a more appropriate represen-

tation of this process than does the ﬁrst Born approximation, although

amplitudes of all orders do of course contribute to some extent. The

distorted-wave Born approximation implicitly includes second and higher

order terms, which explains why the results of Shakeshaft and Wadehra

(1980) and Mandal, Guha and Sil (1979) are rather similar to those

obtained by Deb, McGuire and Sil (1987) and Basu and Ghosh (1988),

who used the second Born approximation directly. These second or-

der results are strongly forward peaked at high energies, as shown in

Figure 4.7, which relates to a positron energy of 5.44 keV, but they

4.2 Theory 165

Fig. 4.8. Cross sections for positronium formation in positron collisions with

atomic hydrogen and helium. (a) Hydrogen: ——, 33-coupled-state results of

Kernoghan, Robinson, McAlinden and Walters (1996) for the total positronium

formation cross section together with the experimental points of Zhou et al.

(1997). (b) Helium: the theoretical results were obtained by Campbell et al.

(1998a) using a 27-coupled-state approximation; — · —, ground state formation;

– – –, sum of the the formation cross sections for n

= 1 and n

= 2; ——,

total positronium formation cross section. Experimental results:

, Fornari,

Diana and Coleman (1983);

, Diana et al. (1986b); , Fromme et al. (1986);

, Moxom et al. (1993); •, Overton et al. (1993). See also Figure 4.17 for more

data on this well-studied system.

166 4 Positronium formation

also reveal a quite prominent secondary peak around 45

◦

; this is the

Thomas peak predicted by the classical kinematics of the two-stage cap-

ture process.

The results discussed above relate to the formation of ground state

positronium in collisions of positrons with hydrogen atoms in their ground

state, but investigations have also been made of positronium formation

into excited states, notably by McAlinden, Kernoghan and Walters (1994)

and Kernoghan and coworkers (1995, 1996) using the coupled-state ap-

proximation mentioned previously. These authors obtained results for

positronium formation into states with n

≤ 4 but, because their positro-

nium states with n

= 3 and 4 were represented by pseudostates, they

considered it to be more accurate to estimate the positronium formation

cross sections for n

> 2 using a scaling rule derived from the Born

approximation. According to this rule, at suﬃciently high energies the

cross section for positronium formation into a state with principal quan-

tum number n

is proportional to 1/n

(Omidvar, 1975). The resulting

total σ

, which for hydrogen is dominated by formation into the ground

state, is shown in Figure 4.8(a). Its peak value of 3.5πa

is attained at

an incident positron energy of 15 eV, after which it declines fairly rapidly

with increasing positron energy, so that at an energy of 80 eV it is already

less than one quarter of the elastic scattering cross section, whose value

is 0.25πa

at this energy. These theoretical values for σ

are compared

with the experimental results in subsection 4.4.2.

2 Positron–helium scattering

The calculation of accurate cross sections for positronium formation is

a particularly challenging task when the target is helium or some other

complex atom. As we have already seen with hydrogen (subsection 4.2.1),

the simple methods of approximation used for positronium formation at

low positron energies can be very unreliable for the ﬁrst few partial waves,

and the results obtained may be seriously in error. For helium there is the

additional problem of having to use an inexact target wave function. This

can be conveniently avoided in elastic scattering by the use of the method

of models (subsection 3.2.2), but no such self-consistent formulation is

possible for a rearrangement collision: the model potential describing the

interaction between each electron and the nucleus is inconsistent with the

Coulomb interaction between the electron and the nucleus in the residual

ion. Consequently, the exact Hamiltonian should be used throughout the

formulation.

Because of its complexity, most calculations of positronium forma-

tion in positron–helium scattering have been made using relatively crude

methods of approximation with rather simple uncorrelated helium wave

4.2 Theory 167

Fig. 4.9. The total positronium formation cross section, and the various partial-

wave contributions to it, for positron–helium scattering in the Ore gap. The

contributions with l ≤ 3 are determined variationally whilst the sum of all higher

partial waves is calculated in the Born approximation.

functions, and the results are therefore of somewhat uncertain accuracy.

Massey and Moussa (1961) used the ﬁrst Born approximation but their

results were subsequently found to be in error, and were corrected by Man-

dal, Ghosh and Sil (1975), who also applied the coupled-static approxi-

mation to the problem. The ﬁrst use of the coupled-static approximation

was by Kraidy and Fraser (1967), who also introduced extra potential

terms to take some account of the polarization of both the positronium

and the helium atom. Their results, which were obtained in the energy

range of 10 eV above the positronium formation threshold, suggested that

the d-wave contribution to σ

is dominant, as has been conﬁrmed by Van

Reeth and Humberston (1999b).

The only detailed investigations of low energy positronium formation

in positron–helium scattering in the Ore gap have been made by Van

Reeth and Humberston (1995b, 1997, 1999b); see also Humberston and

Van Reeth (1996) and Van Reeth and Humberston (1999a). These authors

used the Kohn variational method in a similar manner to that described in

168 4 Positronium formation

Fig. 4.10. Comparison, on a log–log plot, of the theoretical results of Van Reeth

and Humberston (1999b) (——) with the experimental measurements of Moxom

et al. (1994) () for the positronium formation cross section in positron–helium

collisions in the Ore gap. The broken line is a linear ﬁt to the experimental data;

is the kinetic energy of the positronium.

subsection 4.2.1 for positronium formation in positron–hydrogen scatter-

ing, but with the short-range correlation terms in the trial wave function

having a form like those used in the detailed investigations of elastic

positron–helium scattering, described in subsection 3.2.2. A very accurate

correlated Hylleraas-type helium wave function was used, similar in form

to that described by equation (3.79) but containing 22 terms, so as to

minimize any problems that might be associated with the use of an inexact

target wave function.

The most accurate results obtained by these authors for the s-, p-, d-

and f-wave contributions to σ

are displayed in Figure 4.9. Also given

there is the total σ

obtained by summing the ﬁrst four partial-wave

contributions and adding the ﬁrst Born results for all higher partial waves.

The s-wave contribution exhibits a very steep rise from zero at the thresh-

old to a gently rising plateau of very similar magnitude to that obtained

for hydrogen, although in helium it constitutes a larger fraction of what

is a much smaller total positronium formation cross section. The p-wave

contribution rapidly exceeds that of the s-wave, and this in turn is ex-

ceeded by that of the d-wave, which forms the largest single partial-wave

contribution to σ

throughout the upper half of the Ore gap. A com-

parison of the accurate theoretical results of Van Reeth and Humberston

(1999b) with the low energy experimental measurements of Moxom et al.

(1994) is shown in Figure 4.10. The diﬀerential positronium formation