Charlton M., Humberston J.W. Positron Physics

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4.2 Theory 169

Fig. 4.11. The angular distribution of positronium formation in positron–

helium collisions at various energies in the Ore gap (Van Reeth and Humberston

(1999b). The results for E =17.8 eV, just above the positronium formation

threshold, have been multiplied by a factor of 30.

cross sections obtained by Van Reeth and Humberston (1999b) at three

energies in the Ore gap are displayed in Figure 4.11.

Most other calculations of positronium formation in positron–helium

scattering have employed much simpler methods of approximation, but

results have usually been obtained over energy ranges extending well be-

yond the Ore gap. It must therefore be borne in mind that the experimen-

tal results include contributions from positronium formation into excited

states as well as into the ground state. The Born approximation, used

ﬁrst by Massey and Moussa (1961) and subsequently by Mandal, Ghosh

170 4 Positronium formation

and Sil (1975), gives a more sharply peaked maximum, at a rather lower

energy, but more than three times larger, than that in the experimental

measurements of Fornari, Diana and Coleman (1983) and Fromme et al.

(1986), which have a maximum at approximately 50 eV. Only beyond

an energy of 80 eV does the Born approximation give reasonably good

agreement with experiment. The distorted-wave approximation, used by

Mandal, Guha and Sil (1975), and the coupled-static approximation, used

by McAlinden and Walters (1992), both give results in reasonably good

agreement with experiment in the vicinity of the maximum and up to

a positron energy of 80 eV; however, they fall below the experimental

values as the energy is increased further. A more elaborate calculation was

performed by Hewitt, Noble and Bransden (1992a) using the coupled-state

method with ﬁve states of helium and three states of positronium, but

an equivalent one-electron model was used for the helium atom. Their

results for ground state positronium formation fall signiﬁcantly below the

experimental cross sections, although rather better agreement is obtained

when contributions from formation into n

= 2 states, and estimates

based on the 1/n

scaling law for formation into even higher excited

positronium states, are included.

The most accurate theoretical results for positronium formation in

positron–helium collisions in the energy range 20–150 eV are probably

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

with the lowest three positronium states and 24 helium states, each of

which was represented by an uncorrelated frozen orbital wave function

, r

)=N

(1 ± P

)φ

)

φ(r

), (4.28)

where

φ is the frozen orbital and P

is the exchange operator for the

two electrons. This form provides a reasonably good representation of

the various excited states of helium, but the ground state is much less

accurate than that used by Van Reeth and Humberston (1999b). Cross

sections for positronium formation into states with n

= 1 and 2 were

evaluated explicitly and the 1/n

scaling formula was used to estimate the

contributions from all higher states. As in hydrogen, the total positronium

formation cross section is dominated by formation into the ground state.

The results obtained by Campbell et al. (1998a) agree reasonably well

with several sets of experimental data up to 60 eV, and with the data of

Fornari, Diana and Coleman (1983) and Diana et al. (1986b) up to 90 eV,

but thereafter they fall some way below all sets of experimental data; see

Figure 4.8(b). Further discussion and comparison with experiment can

be found in subsection 4.4.1.

4.2 Theory 171

3 Positron–alkali atom scattering

As already mentioned in subsection 3.2.3, in discussing elastic scattering,

the positronium formation channel is open even at zero incident positron

energy for all the alkali atoms. Furthermore, because this reaction is

exothermic, the positronium formation cross section is inﬁnite at zero

incident positron energy. It is a reasonably good approximation to con-

sider an alkali atom as an equivalent one-electron atom; consequently

the formulation of positronium formation in positron scattering by alkali

atoms is rather similar to that in positron–hydrogen scattering. All such

theoretical investigations of positron-alkali atom scattering have assumed

this three-body structure.

Lithium is the simplest alkali atom; in this case representation of the

atom as an electron in the central ﬁeld of the core is particularly good,

and more theoretical attention has been given to positron scattering by

lithium than by any other alkali atom, despite the complete lack of ex-

perimental data with which to compare. Detailed investigations of low

energy positron–lithium scattering in the energy region from zero up to

the ﬁrst excitation threshold have been made by Humberston and Watts

(1994), using the Kohn variational method with similar forms of trial wave

function to those used by Humberston (1982), and Brown and Humber-

ston (1985) for positron–hydrogen scattering (see subsection 4.2.1). The

representation of the lithium atom was based on an electron–core potential

devised by Peach, Saraph and Seaton (1988) and of the form

−

(r)=−

− 2

−γr

(1 + δ

r + δ

) −

ω(r), (4.29)

where the ﬁrst two terms represent the static interaction and the last

represents the polarization of the core, which has a dipole polarizability α.

The values of the parameters δ

, δ

and γ were obtained by ﬁtting to the

energy spectrum of lithium. Changing the sign of the static component

of the potential, but retaining the sign for the core polarization term,

yields the positron–core potential V

. The wave function of the valence

electron in the lithium atom should have the radial character of a 2s

orbital, which means that the electron in this equivalent one-electron

model is not in its ground state but in the ﬁrst excited state. The

1s ground state, however, is merely an artifact of the model with no

real physical signiﬁcance, and its energy, ≈−50 eV, is so far below the

energy range under consideration in the scattering calculation that it can

probably be ignored. Nevertheless, in order to test the validity of this

assumption, another model of the electron–core potential was also used by

Humberston and Watts (1994) in which the required valence electron is in

its ground state. The corresponding wave function is then of 1s character

172 4 Positronium formation

and nodeless, but the diﬀerences between the two wave functions are

conﬁned to small radial distances of little more than a

from the nucleus.

Accurate approximations to both lithium wave functions of the form

(r) = exp(−Ar)



j=0

, (4.30)

where A and e

(j =0,...,N

) are variational parameters, were generated

using the Rayleigh–Ritz variational method.

Elastic and positronium formation cross sections for l = 0, 1 and 2

were calculated in a similar manner to that previously described in sub-

section 4.2.1 for positron–hydrogen scattering. However, the convergence

of the results with respect to increasing the number of correlation terms in

the trial functions is now worse, particularly at very low positron energies;

the reason is probably that the short-range character of the correlation

terms used in these trial wave functions makes them rather less suitable

for representing the long-range distortions associated with the high dipole

polarizability of the lithium atom. Better convergence could be achieved

by adding longer-range terms to the trial wave function, as was done to

improve the convergence of the positron–hydrogen scattering length (see

subsection 3.2.1). Despite these reservations, these contributions to σ

are believed to be accurate to within 20%. From its inﬁnite value at k =0,

the s-wave contribution σ

rapidly falls to be several orders of magnitude

smaller than σ

so that again, as with hydrogen and helium, the s-wave

only makes a small contribution to σ

All other calculations for lithium, and all those for the other alkali

atoms, have used some form of the coupled-state approximation, and

results have usually been obtained over a much wider energy range, typi-

cally 0–60 eV. The simplest approximation of this type, the coupled-static

approximation, was used by Guha and Ghosh (1981) and Abdel-Raouf

(1988), but more recent calculations have included several states of both

lithium and positronium. In such calculations, cross sections have been

obtained not only for elastic scattering and ground state positronium

formation but also for positronium formation into various excited states

and for excitation, and sometimes ionization, of the target atom. Among

the most important of these calculations are those of Hewitt, Noble and

Bransden (1992b), who included the states Li (2s, 3s, 2p, 3p) and Ps (1s,

2s, 2p), Kernoghan, McAlinden and Walters (1994a), who included Li (2s,

2p, 3p, 3d) and Ps (1s, 2s,

3s, 4s, 2p, 3p, 4p, 3d, 4d), and McAlinden,

Kernoghan and Walters (1997), who included Li (39 states and pseu-

dostates) and Ps (1s, 2s, 2p). The low energy results of Kernoghan,

McAlinden and Walters (1994a) are in good agreement with the varia-

tional results of Humberston and Watts (1994), whereas those of Hewitt,

4.2 Theory 173

Noble and Bransden (1992b) are signiﬁcantly larger. The most extensive

investigations were made by McAlinden, Kernoghan and Walters (1997),

who determined the total cross section for positronium formation into all

possible states. They calculated directly the cross sections for positronium

formation into its ground state and the n

= 2 excited states and esti-

mated the contributions from all other excited positronium states using

the 1/n

scaling law. Thus

= σ

=1)+σ

=2)+σ

≥ 3), (4.31)

where

≥ 3) = 8σ

=2)

∞



1/n

=0.6165σ

=2). (4.32)

The total positronium formation cross section obtained in this way falls

steadily with increasing positron energy, so that at an energy of 25 eV it is

already very small compared with the cross sections for elastic scattering

and excitation of the target atom. Formation is mainly into the ground

state at energies below 10 eV, but formation into excited states becomes

more probable at higher energies.

McAlinden, Kernoghan and Walters (1994) also calculated the diﬀer-

ential cross sections for the formation of positronium into its various

states. The cross section for ground state formation, Ps(1S), is partic-

ularly strongly peaked in the forward direction (angles < 20

◦

) as also,

to a slightly less extent, is that for Ps(2S) formation. The cross section

for Ps(2P) formation has a rather broader angular spread, being signiﬁ-

cant out to 40

◦

, although the forward peaking increases with increasing

incident positron energy.

Similar coupled-state methods have been applied to the other alkali

atoms by Hewitt, Noble and Bransden (1993), McAlinden, Kernoghan

and Walters (1994, 1996) and Kernoghan, McAlinden and Walters (1996).

These results reveal that, except at those low energies where only ground

state positronium formation is energetically possible, the ground state

formation cross section σ

= 1) becomes a progressively smaller

fraction of the total positronium formation cross section as the atomic

number of the target alkali atom is increased. The contributions to σ

from positronium formation into excited states other than those explicitly

represented in the coupled-state expansion were again estimated using the

1/n

scaling law, in a somewhat similar manner to that described above

for lithium.

The total positronium formation cross sections obtained in this way

are, as will be described in section 4.4, in reasonably good agreement

174 4 Positronium formation

with the lower-bound measurements of Kwan et al. (1994), Zhou et al.

(1994b) and Surdutovich et al. (1996) for energies below 30 eV. Beyond

this energy, the total positronium formation cross sections for all the alkali

atoms become very small.

4 Other atoms and molecules

Theoretical studies of positronium formation have been made for several

other atoms, particularly the rare gases and the alkaline earth atoms,

magnesium, calcium and zinc, although in most cases only simple approx-

imation methods have been used. In the ﬁrst such investigation for any

atom other than hydrogen and helium, Gillespie and Thompson (1977)

applied the ﬁrst Born approximation and two forms of distorted-wave

approximation to positronium formation in neon and argon, but they

restricted the energy range of the calculations to within 10 eV of the

positronium formation threshold. For both atoms, the results of the

Born approximation exceed the experimental measurements, although by

a smaller factor for argon than for neon. The results of the distorted-wave

approximation are much lower than the experimental values. McAlinden

and Walters (1992) used the coupled-static approximation to calculate the

elastic scattering and positronium formation cross sections for all the rare

gases, and these same authors also calculated the diﬀerential positronium

formation cross sections in the same approximation for all the rare gases

except helium (McAlinden and Walters, 1994). The results in this approx-

imation compare reasonably well with the experimental measurements,

particularly in the vicinity of the maximum in the positronium formation

cross section. At higher energies these theoretical results fall signiﬁcantly

below the experimental data, although the latter also include positronium

formation into excited states. However, McAlinden and Walters (1992)

expressed doubt that the sum of the partial cross sections for positronium

formation into all excited states is suﬃciently large to account fully for

the discrepancy.

Very few theoretical investigations of positronium formation have been

made for molecular targets, and virtually all these relate to hydrogen.

The ﬁrst such study was made by Sural and Mukherjee (1970), who used

the ﬁrst Born approximation in the energy range 50–544 eV. A rather

simple hydrogen molecular wave function was used, consisting of a sum

of products of single electron wave functions of the form exp(−1.193r),

centred on one or other of the two protons. Similarly, the wave function

of the H

molecular ion was taken to be the sum of two terms, each one

being the same function as that given above and centred on one or other of

the protons. Despite the simplicity of these various approximations, the

results obtained are in moderately good agreement with the experimental

4.3 Experimental techniques 175

measurements, bearing in mind that the theoretical results relate only

to ground state positronium formation. Rather similar results for the

same system were also obtained by Biswas, Mukherjee and Ghosh (1991)

using essentially the same method of approximation, but these latter

authors performed their calculations over a wider energy range and found

that their results began to exceed the experimental measurements by an

increasingly large factor as the incident positron energy was reduced below

50 eV. Both groups of authors only considered positronium formation with

the H

ion left in the gerade state. Ray, Ray and Saha (1980), however,

using a simpliﬁed form of the full Born approximation known as the

molecular Jackson–Schiﬀ approximation, also investigated positronium

formation with the ion left in the ungerade state but found that this

made only a small contribution to the overall positronium formation cross

section, particularly at low positron energies.

If the hydrogen molecule is considered as two hydrogen atoms it might

naively be assumed that the positronium formation cross section for the

molecule should be approximately double that for the atom. Sural and

Mukherjee (1970) tested the validity of this assumption by comparing

their Born results for the molecule with the Born results for atomic

hydrogen and they found that, instead of the results for the molecule

being approximately double those for the atom, they were approximately

three times as large. However, a comparison of the experimental data

reveals that the positronium formation cross sections for the two target

systems are in fact rather similar in magnitude throughout the energy

range in which this process makes a signiﬁcant contribution to the total

scattering cross section.

Positronium formation into n

= 2 excited states in positron–H

scat-

tering was investigated in the ﬁrst Born approximation by Ray, Ray and

Saha (1980) and also by Biswas et al. (1991b).

4.3 Experimental techniques

Experimentally, positronium formation may be identiﬁed by a number of

diﬀerent signals, and these can be summarized as follows (Raith, 1987):

(i) Positrons are lost from the beam: positronium formation is the only

channel which eﬀectively removes positrons, apart from annihilation

in ﬂight, which usually has a very small cross section.

(ii) After formation, ortho-positronium will annihilate either via three

gamma-rays in vacuum or, perhaps, with the emission of two

gamma-rays upon striking part of the experimental apparatus,

whereas para-positronium, with its characteristic lifetime of around

125 ps, will annihilate in ﬂight into two gamma-rays.

176 4 Positronium formation

(iii) The formation of positronium can be monitored because the resid-

ual ion can be extracted from the scattering region and detected.

Positronium formation can be distinguished from other processes

which produce ions, e.g. impact ionization (see Chapter 5), since in

these cases the positron remains as an isolated particle in the ﬁnal

state, in addition to the ion.

As will be shown, all these methods have been used in investigations of

positronium formation.

The ﬁrst experiment to identify positronium production directly in

positron–gas collisions using a low energy beam was that of Charlton et al.

(1980c). These workers used a magnetically conﬁned beam similar to that

described in subsection 1.4.2 except that a curved solenoid was used to

remove the scattering cell from the line of sight with the

Na source.

The collision gas cell was surrounded by three large NaI(Tl) gamma-ray

detectors set to monitor triple coincidences. Thus, the signal used was

the three-gamma-ray annihilation of ortho-positronium described above

under heading (ii). Several gases were investigated, and in each case,

as the positron energy was increased beyond the positronium formation

threshold the three-gamma-ray signal was found to rise sharply before

starting to fall after a few eV.

Later experiments by other workers, described below, showed that this

energy dependence was not that of σ

and that the cross sections which

were published using this technique (Charlton et al., 1983b) were in error.

This was caused by the combination of the following eﬀects. First, most

of the fast ortho-positronium was quenched on the walls of the scatter-

ing cell, with a resultant loss in the three-gamma-ray signal. Second,

the natural forward collimation of the diﬀerential cross section dσ

/dΩ,

discussed in sections 4.2 and 4.7, meant that a signiﬁcant fraction of the

ortho-positronium formed at kinetic energies greater than a few eV either

escaped from the gas cell or moved to a region of lower three-gamma-ray

detection eﬃciency. In any event, the true σ

was much underestimated.

Later, Clark (1984) investigated a two-gamma-ray arrangement for mea-

suring σ

and, although this gave results much closer to others obtained

at that time using diﬀerent methods, there was still the suspicion that

not all systematic errors had been eliminated. A similar two-gamma-ray

method has since been employed by the Detroit group (Zhou et al., 1994b;

Surdutovich et al., 1996) in the ﬁrst attempts to measure σ

for the alkali

metals.

Reliable values of σ

were ﬁrst obtained by Fornari, Diana and Cole-

man (1983), using a technique based upon the detection method (i). The

apparatus, shown in Figure 4.12, consisted of a ﬂight path 2.3 m long

immersed in a 10

−2

T magnetic ﬁeld, the entire beamline having been

4.3 Experimental techniques 177

Fig. 4.12. Schematic of the apparatus developed by Fornari, Diana and Cole-

man (1983) for studies of positronium formation. Reprinted from Physical

Review Letters 51, Fornari et al., Positronium formation in collisions of positrons

with He, Ar and H

, 2276–2279, copyright by the American Physical Society.

shielded from the Earth’s magnetic ﬁeld. The positron beam originated

from an annealed tungsten mesh moderator which was held at a potential

to ﬁx the kinetic energy of the positrons (found to be at a mean energy

of 1.3 eV above eV

). In the ﬁrst experiments of this type carried out at

Texas, the slow positrons were timed using the thin-scintillator technique,

as shown in Figure 4.12; see section 2.3. This arrangement also allowed

the total attenuation of the beam to be measured. At the end of the ﬂight

path the beam was detected using a channel electron multiplier.

For cross section measurements, gas could be admitted to the cham-

ber, its pressure being recorded at each end of the ﬂight path using an

ionization gauge. A correction for the pressure gradient along the ﬂight

path was necessary owing to the pumping arrangement, which dispensed

with the need for gas-conﬁning apertures which might have absorbed

some of the scattered beam. This was essential to the technique, which

relied upon detecting all scattered positrons except those lost through

positronium formation. Full detection was aided by reducing the initial

beam spot to 5 mm diameter, which may be compared with the 10 mm

opening of the detector, and using a high transmission grid placed near the

178 4 Positronium formation

Fig. 4.13. Schematic illustration of the apparatus of the Bielefeld group for

studies of positronium formation and ionization (see section 5.3 for a discussion

of the latter).

source/moderator and held at a potential V

. This reﬂected practically all

back-scattered positrons which, along with those scattered in the forward

direction, were conﬁned by the axial magnetic ﬁeld to reach the detector.

The channel electron multiplier count rate and the time-of-ﬂight spectra

were recorded simultaneously. When gas was admitted into the system

the decrease in the count rate gave the fractional attenuation, f

, due

to incident positrons which formed positronium and the time-of-ﬂight

spectrum yielded the total fraction, f

, of scattered positrons, some cor-

rections having been made at higher energies to compensate for the eﬀects

of elastic scattering at small angles. The positronium formation cross

section was then given simply by σ

= f

, where the values of σ

were taken from the literature.

Fornari, Diana and Coleman (1983) investigated positronium formation

in helium, H

and argon gases over the energy ranges from their respective

formation thresholds up to 76 eV. The number of targets and the energy

range were extended by Diana and his collaborators. In a later variant

of their apparatus, Diana et al. (1986b) dispensed with the time-of-ﬂight

arrangement and instead deduced f

using a retarding tube located just

before the detector; this allowed the use of stronger radioactive sources

and higher positron ﬂuxes. This instrument is similar to that used by

Coleman et al. (1992) in their investigations of the behaviour of σ

impact energies near E

(see section 3.3).