Charlton M., Humberston J.W. Positron Physics

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4.4 Results 189

Fig. 4.19. Positronium formation cross sections for positron–argon scattering.

Experiment:

, Fornari, Diana and Coleman (1983); •, Diana et al. (1986c);

the solid line and the broken line are drawn through the upper and lower best

estimates of Zhou et al. (1994b). Theory: ·····, distorted-wave calculation,

— · —, ﬁrst Born approximation, both from Gillespie and Thompson (1977);

— ··—, McAlinden and Walters (1992).

approximation, and those of Gillespie and Thompson (1977), who used

a distorted-wave approximation up to 8 eV above threshold. The results

of the latter are smaller than the experimental measurements by more

than an order of magnitude. Given the nature of the approximation and

the complexity of the target, the data of McAlinden and Walters (1992)

are in fair accord with experiment. The cross sections shown comprise

the capture of electrons from both the 3p and 3s shells and the feature

observed around 75 eV is caused by capture from the 3s shell.

Data also exist for H

and in this case reasonable agreement has been

found between the theoretical estimates of Bussard, Ramaty and Drach-

man (1979) and the experimental results of the Arlington and Bielefeld

groups at energies up to ≈ 20 eV. Between 20 eV and 35 eV, however,

theory exceeds experiment by up to 30%. At higher energies the calcu-

lations of Ray, Ray and Saha (1980), using the molecular Jackson–Schiﬀ

approximation, and those of Sural and Mukherjee (1970), using the Born

approximation, are consistent with experiment, the latter only slightly

exceeding the theoretical values.

The near-threshold behaviour of the positronium formation cross sec-

tion has also been investigated for O

(Laricchia, Moxom and Charlton,

190 4 Positronium formation

Fig. 4.20. Positronium formation in positron–hydrogen scattering. Experiment:

, Zhou et al. (1997); ◦, Hofmann et al. (1997); •, Kara (1999). Theory: ——

——

——,

Brown and Humberston (1985); ——, Kernoghan et al. (1996); — —, Mitroy

(1996); — · —, Igarishi and Toshima (1994); ·····, Higgins and Burke (1993);

— ··—, Janev and Solov’ev (1998).

1993) and CO

(Laricchia and Moxom, 1993). In the former case the

cross section was found to have a distinct peak a few eV above threshold,

which was linked by Laricchia and coworkers to the behaviour of the

cross section for positron impact excitation of the molecule, as will be

described in subsection 5.1.2. In the case of CO

a detailed investigation

of the energy dependence of the ion yield supported the interpretation

of Laricchia, Charlton and Griﬃth (1988) that simultaneous positronium

formation and excitation of the residual ion is an important process in

this gas. This phenomenon is discussed in more detail in section 4.5.

2 Atomic hydrogen

The results obtained by Hofmann et al. (1997), who corrected those of

Sperber et al. (1992), are shown in Figure 4.20 along with the data of

Zhou et al. (1997), Kara (1999) and the results obtained from several

recent theoretical studies. A comprehensive discussion of the theoretical

data was given in subsection 4.2.1. The observed maximum in σ

occurs

at around 15 eV and has a value of approximately 3.0 × 10

−16

4.4 Results 191

Fig. 4.21. Positronium formation in positron–sodium collisions. The experi-

mental points are from Zhou et al. (1994b): the open and solid circles give the

upper and lower limits respectively. Theory: upper solid line, the cross section

summed for n

= 1 and n

= 2, lower solid line with solid circles, cross section

for n

= 1, lower solid line with open circles, cross section for n

= 2, all from

the calculations of Hewitt, Noble and Bransden (1993); — · —, total positronium

formation cross section from Campbell et al. (1998a).

Thereafter, the cross section appears to fall monotonically to a value close

to zero at 100 eV. As explained by Hofmann et al. (1997) the re-evaluated

data of Sperber et al. (1992), which are lower than both theory and other

experiments over the entire low energy region, are now thought to be

incorrect.

The best agreement with experiment has been obtained by Kernoghan

et al. (1996) and Mitroy (1996), using the coupled-state approximation

with large basis sets (see subsection 4.2.1), and by Igarashi and Toshima

(1994), Janev and Solov’ev (1998) and Higgins and Burke (1993). The

results of several other calculations (not shown in Figure 4.20) fall below

the experimental values, particularly near the cross section maximum,

e.g. the R-matrix method as used by Higgins and Burke (1991), the

classical calculations of Ohsaki et al. (1985) and Wetmore and Olson

192 4 Positronium formation

(1986) and the polarized-orbital work of Khan and Ghosh (1983). The

ﬁrst Born approximation (Massey and Mohr, 1954) and the Fock–Tani

ﬁeld-theoretic approach of Straton (1987) both yield cross sections that

are in excess of the experimental values in the vicinity of the maximum

and have a somewhat sharper energy dependence at higher energies. Also

shown in Figure 4.20 are the accurate variational results obtained by

Brown and Humberston (1984, 1985) for positron energies between E

and the ﬁrst excitation threshold of hydrogen at 10.2 eV.

3 The alkali metals

As emphasized in subsection 4.2.3, of the alkali metals the most detailed

theoretical work has been performed on the lithium atom, though to

date there have been no experimental studies reported. The situation

for sodium is summarized in Figure 4.21, which shows the experimental

upper and lower limits obtained by Zhou et al. (1994b) (see section 4.3

for a discussion of the origins of these limits) together with the results of

various theories. The measured cross section falls steeply as the incident

positron energy is increased from 1 eV, the lowest energy investigated,

reaching almost zero by 25 eV. This behaviour contrasts sharply with that

for atomic hydrogen and helium presented in the preceding sections. For

the alkalis the outer electron is suﬃciently weakly bound that positronium

formation is an exothermic reaction, and this fact governs the shape of

the cross section. In particular, as mentioned previously and in accord

with the measurements, the cross section is expected to diverge as the

incident positron energy is lowered to zero.

The theoretical work of Hewitt, Noble and Bransden (1993) used a

seven-coupled-state approximation (four states of sodium and three of

positronium) to calculate cross sections for positronium formation into

states with n

= 1 and 2. Both results are shown in Figure 4.21,

together with the sum. Reasonable accord is found with the lower-limit

measurements of Zhou et al. (1994b). As expected, the calculations

show that positronium formation into the ground state dominates at

low energies and that the n

= 2 contribution, with its threshold at

3.4 eV, is more important above approximately 7 eV. Only limited agree-

ment is found between the experimental measurements and the results

of the distorted-wave and ﬁrst Born approximation calculations of Guha

and Mandal (1980). In particular, the ﬁrst Born approximation gives

completely the wrong energy dependence. Coupled-state calculations

on this system have also been reported by McAlinden, Kernoghan and

Walters (1994), though using a smaller set of states. The agreement with

experiment is poorer than that found by Hewitt, Noble and Bransden

(1993) and the results are not shown in Figure 4.21. The most elaborate

4.4 Results 193

Fig. 4.22. Positronium formation in positron–potassium collisions. The ex-

perimental points are from Zhou et al. (1994b); the open and solid circles are

the upper and lower limits respectively. Theory: thick solid line, cross section

summed for n

= 1 and n

= 2, thin solid line with solid circles, cross section

for n

= 1, thin solid line with open circles, cross section for n

= 2, all from

calculations of Hewitt, Noble and Bransden (1993); — · —, total positronium

formation cross section from McAlinden, Kernoghan and Walters (1996).

coupled-state calculations are those of Campbell et al. (1998a), who used

six states of positronium and 27 of sodium. Their results are in fair

agreement with experiment beyond 4 eV but are too low at lower energies.

Results for potassium and rubidium are given in Figures 4.22 and 4.23.

Note that the simpler approximations of Guha and Mandal (1980), as

shown in the summary of Stein et al. (1996), are in even poorer agree-

ment with experiment than was the case with sodium and are therefore

not presented here. The experimental data for potassium show a larger

diﬀerence between the upper and lower limits than was the case for

sodium. Indeed the two sets of measurements exhibit opposite trends

below approximately 5 eV, the lower-limit results showing a dramatic

decrease whilst the upper-limit values continue to rise. Although at ﬁrst

sight this might suggest that little of signiﬁcance could be said about σ

194 4 Positronium formation

Fig. 4.23. Positronium formation in positron–rubidium

collisions. The experi-

mental points are from Surdutovitch et al. (1996); the open and solid circles are

the upper and lower limits respectively. Theory:

—

· —, the total positronium

formation cross section from Hewitt, Noble and Bransden (1993): ——, total

positronium formation cross section; — —, the contributions from n

=1,2

and 3; -- -- -- --, the contributions from n

= 1 and 2; ·····, the contribution from

= 1. All these theoretical results are taken from the work of Kernoghan et al.

(1996).

for this target, theory suggests that the lower-limit results are closer

to the true cross sections and that the trend discovered is a genuine

eﬀect. It is notable that the lower limits on the cross sections deduced

by Zhou et al. (1994b) for argon gas are closest to the data of Fornari,

Diana and Coleman (1983), which also lends support to their alkali-metal

measurements.

The theoretical results shown are from the coupled-state calculations

of Hewitt, Noble and Bransden (1993) and McAlinden, Kernoghan and

Walters (1996) (for potassium), and Kernoghan et al. (1996) (for ru-

bidium). These authors have also studied caesium and have obtained

similar conclusions to those for rubidium. Hewitt and coworkers again

used a seven-state approximation, whereas McAlinden, Kernoghan and

4.5 Other processes involving positronium formation 195

Walters (1996) and Kernoghan et al. (1996) both employed 11 states (ﬁve

atomic and six positronium). A common feature of all the calculations is

that, although they diﬀer in the details of the magnitudes, they do show

that the formation of positronium into excited states is a very important

process for these targets and is responsible for the peak-shaped cross

sections measured by Zhou et al. (1994b). A detailed study by McAlinden,

Kernoghan and Walters (1996) shows how the breakdown into the various

positronium states occurs. These authors obtained cross sections which,

at least for positronium formed in the 2s and 2p states, were a factor

2–3 smaller than those calculated by Hewitt, Noble and Bransden (1993)

using the seven-state approximation.

Reasonable agreement is obtained by McAlinden, Kernoghan and Wal-

ters (1996) and Kernoghan et al. (1996) with the experimental measure-

ments of Zhou et al. (1994b) and Surdutovich et al. (1996), particularly

given the complex nature of the targets and the approximations inherent

in both theory and experiment. In calling for a tightening up on the cross

sections available from both sources, McAlinden, Kernoghan and Walters

(1996) issued some warnings. They highlighted the fact that yet more

atomic and positronium states may be needed in the calculation, together

with some pseudostates which could help take account of coupling to the

continuum (these were found to be of great value in the positron–atomic

hydrogen case; see subsection 4.2.1). A better representation of exchange

and inner shell eﬀects may also need to be incorporated into the theory.

4.5 Other processes involving positronium formation

The formation of excited states of positronium, usually termed Ps

∗

through reaction 4.2, can make signiﬁcant contributions to σ

. This

has already been discussed brieﬂy in the theoretical section 4.2 and in

subsection 4.4.3, where we saw that excited state positronium is thought

to dominate σ

at certain energies in some of the alkali metals. Here we

describe the only experiment to date which has directly detected excited

state positronium formation in gases, namely that of Laricchia et al.

(1985).

The principle of the experiment is similar to that of Canter, Mills

and Berko (1975), in that Ps

∗

was detected by monitoring coincidences

between an ultraviolet photon (the positronium Lyman-α line at 243

nm) and a gamma-ray from the subsequent annihilation of ground state

ortho-positronium. The apparatus employed by Laricchia et al. (1985) is

shown in Figure 4.24. A low energy positron beam passed through a hemi-

spherical aluminium scattering cell, which was coupled, via a light pipe

made from (Al + MgF

)-coated glass tubes, to an ultraviolet-sensitive

phototube; the coated glass enhanced the transmission of photons to the

196 4 Positronium formation

Fig. 4.24. Apparatus of Laricchia et al. (1985) for the detection of excited state

positronium formed in positron–gas collisions.

detector along the light guide. This arrangement allowed the phototube to

be removed from the magnetic ﬁeld used to guide the beam. A borosilicate

glass disc was used as a crude ﬁlter to prevent photons with wavelengths

≤ 285 nm from reaching the phototube. This disc could be inserted into

the light path using a retractable shaft. The phototube signals were used

to start a conventional delayed-coincidence timing arrangement; timing

was stopped by the arrival of a signal from a large NaI(Tl) gamma-ray

detector placed close to the scattering cell.

At each of the energies investigated, two coincidence spectra were ac-

cumulated, one with, and one without, the borosilicate glass shutter in

position. Examples are given in Figure 4.25(a), which shows total (shutter

out) and partial (shutter in) spectra for 17 eV positrons incident upon

argon gas. The total spectrum, oﬀset from zero by 200 ns, contains a

peak around the time t = 0, which is due to gamma-ray–gamma-ray

4.5 Other processes involving positronium formation 197

Fig. 4.25. (a) Timing spectra for Lyman-α and gamma-ray coincidence, for

17 eV positrons on argon gas: ◦, total spectrum; ×, partial spectrum. (b) The

measured excited state positronium formation eﬃciency versus positron impact

energy for H

(), Ne () and Ar (•) gases.

coincidences plus events on both sides of t = 0. The partial spectrum

contains no events at t>0 which, because of the constraints thus placed

upon the energies of the photons causing these events, could be attributed

to Ps

∗

formation into the 2

P state. The results of Laricchia et al. (1985)

for the measured Ps

∗

formation eﬃciency, α

, deduced from the t>0

events per scattered positron, are presented in Figure 4.25(b) for the tar-

gets argon, neon and H

. The measured yields exhibit deﬁnite thresholds

at energies which are in accord with the known values of 19.9 eV for neon,

14.1 eV for argon and 13.8 eV for H

By estimating the detection eﬃciency of the Lyman-α–gamma-ray co-

incidence arrangement, Laricchia et al. (1985) were able to estimate the

198 4 Positronium formation

absolute Ps

∗

production eﬃciency. Using a correction for the solid angles

of the two detectors, plus the ultraviolet-photon transport eﬃciency and

the photocathode quantum eﬃciency, it was found that the true Ps

∗

formation eﬃciency was α = (950 ± 380)α

, resulting in a maximum yield

of around 5% for H

gas. This is somewhat larger than the eﬃciencies of

−2

–10

−3

quoted by workers who have studied Ps

∗

production at surfaces

(Schoepf et al., 1992; Steiger and Conti, 1992). Unfortunately, it has not

been possible to convert these measured yields, which were found to fall

or level oﬀ a few eV above threshold, into absolute excited state formation

cross sections since the detection eﬃciencies would be strong functions of

energy, owing to the rapid motion of the ortho-positronium. To date, no

further attempts have been made to measure Ps

∗

formation cross sections

using such arrangements as ultraviolet photon–ion coincidence.

In an experiment similar to the one just described, Laricchia, Charlton

and Griﬃth (1988) reported the observation of simultaneous excitation

and positronium formation, whereby positronium is produced in the col-

lision and the residual ion is left in an excited state. This study was

performed in CO

and was prompted by the suggestion of Kwan et al.

(1984) that the rise in σ

observed above 12 eV may be due to the onset

of Ps

∗

formation. The behaviour of the total cross section for low energy

positron–CO

scattering was described in subsection 2.6.3, and it was

noted there that good agreement exists between the data of Kwan et al.

(1984) and Charlton et al. (1983a): both measurements show the apparent

onset of a new channel, although the threshold energy is not easy to

determine. As pointed out by Kwan et al. (1984), similar behaviour is

found in the total cross section for positron scattering by N

O, a molecule

structurally similar to CO

The apparatus used was that shown in Figure 4.24. Copious coinci-

dences were found between ultraviolet photons and annihilation gamma-

rays and, as seen also in the spectrum shown in Figure 4.25(a), although

now much more pronounced, they occurred on both sides of t =0. Itwas

also demonstrated that the signal at t>0 did not disappear when the

borosilicate slide was inserted into the light path. Thus, this component

did not originate from photons with energies > 4.3 eV and was not due to

excited state positronium. Laricchia, Charlton and Griﬃth (1988) ﬁtted

exponentials to the data on both sides of t = 0 to obtain yields over

the energy range 12–20 eV. These were an order of magnitude greater

than those found in their Ps

∗

studies and consistent with cross sections

of approximately 10

−16

Details of the interpretation of the features of the spectra are given

in the original work; overall the data were found to be consistent with

photons having energies around 3.5 eV, emitted from a slowly moving

source. It was proposed that the events at t<0 (i.e. gamma-rays followed