Charlton M., Humberston J.W. Positron Physics

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4.3 Experimental techniques 179

The results obtained by the Texas group (Fornari, Diana and Coleman,

1983; Diana et al., 1986b) are discussed below along with those obtained

by the Bielefeld group, whose apparatus is shown schematically in Fig-

ure 4.13. In this case the manner of detecting positronium depended upon

method (iii) outlined above, i.e. detection of the ions formed without an

accompanying free positron in the ﬁnal state.

Figure 4.13 shows the diﬀerentially pumped gas cell, of length 50 cm,

used by Fromme et al. (1986) to form their scattering target; the inset

illustrates the inner construction of the cell, which consisted of a glass

tube with a tungsten helix lining. The latter was used to deﬁne the

electrical potential of the wall and, by having a potential diﬀerence along

the wire of approximately 10 V, to provide an extraction ﬁeld for the ions

created along the axis of the tube. However, this electric ﬁeld also had the

detrimental eﬀect of accelerating the positrons as they passed through the

scattering region. The beam was transported in an axial magnetic ﬁeld

with a strength in the scattering region of approximately 35 mT, which

also helped to conﬁne the ions and facilitate their detection. Later (e.g.

Kruse et al., 1991) a ﬁeld of 0.12 T was used to increase the eﬃciency of

ion transport.

The beam was formed using a 70 MBq

Na source, held oﬀ the axis of

the beamline to remove it from the line of sight of the detectors, together

with a tungsten plate moderator held at 45

◦

with respect to the same

axis. The beam diameter was set at 4 mm by the scattering-cell entrance

aperture, and its energy was varied by varying the voltage applied to

the moderator. Positrons were conﬁned by the magnetic ﬁeld to reach

detector 1, a channel electron multiplier array. An E ×B mass separator

was located after the scattering region, and the ions and positrons were

both accelerated into this region, the ions being deﬂected by the ﬁeld

combination to detector 2. After appropriate allowance for background

counts in each detector and for the subtraction of any ions originating

outside the scattering region, a measurement consisted of recording the

count rates from detectors 1 and 2, their ratio being proportional to the

combined cross section σ

+ σ

, where σ

is the total single-ionization

cross section. By recording also the (background-subtracted) coincidences

between detectors 1 and 2 and selecting the appropriate time of ﬂight for

the ions required, a relative measurement of σ

was obtained.

The absolute scale of the cross sections was obtained by making mea-

surements with the secondary electron beam produced by β

bombard-

ment of the moderator. Comparing the ion count rates measured at

detector 2 obtained from both electron and positron impact gave the

ratio [σ

+ σ

)]/σ

−

), which converged to σ

)/σ

−

)above

a positron energy of 300 eV. Above 600 eV this ratio remained constant,

180 4 Positronium formation

Fig. 4.14. Layout of the positron–atomic hydrogen scattering experiment de-

veloped by the Bielefeld–Brookhaven collaboration.

Reprinted from

Physical

Review Letters 68, Sperber et al., Measurement of positronium formation in

American Physical Society.

indicating that the cross sections for the two projectiles had merged (see

section 5.4 for further discussion). Values above 750 eV were used for

normalization, which introduced an estimated ±8% systematic error into

all the ﬁnal results. Finally, σ

was obtained by subtracting σ

from

+ σ

Although there have been many theoretical studies of positronium for-

mation in positron scattering by atomic hydrogen (see subsection 4.2.1),

the only experimental studies have been those of Sperber et al. (1992),

Weber et al. (1994), Zhou et al. (1997) and Kara (1999). The apparatus,

shown in Figure 4.14, is a variant of that used by Spicher et al. (1990),

see section 5.3, to measure the positron impact-ionization cross section

for atomic hydrogen. One of the main diﬃculties in that work was the

low ion signal rate, caused by a combination of the low density of atomic

hydrogen in the gas beam and a relatively weak positron beam. In order

to overcome the latter restriction, the apparatus was moved to the high

intensity positron beam facility at Brookhaven National Laboratory, USA,

where the positronium formation experiments were performed.

The principle of the experiment was similar to that used by the Bielefeld

group in their earlier measurements of positronium formation in helium

and H

in that the positronium signal was identiﬁed by the production of

an ion (proton) without an accompanying free positron in the ﬁnal state.

The apparatus consisted of a target gas beam, which was a mixture of

atomic and molecular hydrogen, crossed by an electrostatically conﬁned

positron beam. Ions were extracted using a d.c. electric ﬁeld of 8 V cm

−1

and passed through a quadrupole mass analyser (QMA), set to transmit

protons, before being detected.

4.3 Experimental techniques 181

The atomic hydrogen was produced in a radio-frequency Slevin-type

discharge, whose operating conditions were held stable to facilitate data-

taking over long periods, and the degree of dissociation of the hydrogen

was found to be ≈55% in the region where the scattering took place. The

CEM detector shown in Figure 4.14 could only detect positrons scattered

into a cone with a half-angle of 30

◦

, as well as the unscattered beam, the

eﬀect of which will be described below.

The measurement consisted of recording the background-corrected

QMA count rate for particles with unit mass, which, since both positro-

nium formation and ionization produce protons, is proportional to σ

, and also monitoring the QMA–CEM coincidence rate, which should

be proportional to the ionization cross section alone. The positronium

formation cross section can then be determined by subtraction, once

appropriate normalization has taken place at an energy where σ



Several systematic eﬀects were considered by Sperber et al. (1992)

and Weber et al. (1994), but the most serious potential source of er-

ror arose from the limited angular acceptance of the CEM for scattered

positrons. In general terms it is expected that at higher positron energies

the ionization cross section will be predominantly forward peaked, so that

the scattered positrons can be detected by the CEM. At lower energies,

however, some positrons may be scattered through angles > 30

◦

, leading

to loss of the positron and the signal thus being wrongly classiﬁed as

due to positronium formation. Although this eﬀect becomes potentially

serious as the positron energy is lowered, it is oﬀset by the fact that σ

decreases relative to σ

in this region. In an attempt to correct for this

eﬀect, angular distributions obtained from the ﬁrst Born approximation

were used to estimate the fraction of ionizing positrons scattered through

angles > 30

◦

. Sperber et al. (1992) found that their measured values of

had to be lowered by an amount which varied between 4% and 20%

in the energy range 15–65 eV.

Investigations of the relative behaviour of the QMA and QMA–CEM

coincidence signals showed that their ratio remained constant within the

experimental errors at energies greater than approximately 70 eV. This

was taken to imply that in this energy region σ

≈ 0 and as a result both

signals were proportional to σ

. Thus, relative eﬃciencies of the two

signals could be determined, so that a relative measure of σ

could be

subtracted from the QMA signal, the residue being directly proportional

to σ

. Weber et al. (1994) described how this was done in detail and

presented some new data and corrections to the original values of Sper-

ber et al. (1992). They also included a discussion of the normalization

procedure and tabulated the ﬁnal cross sections. We note, however, that

in correcting the ionization work of Spicher et al. (1990), Hofmann et al.

182 4 Positronium formation

Fig. 4.15. Illustration of the apparatus developed by Zhou et al. (1994b) for

studies of positronium formation in positron–alkali metal collisions.

(1997) pointed out that there are large changes to the values of σ

determined by Weber et al. (1994).

Positronium formation cross sections for atomic hydrogen have also

been reported by Zhou et al. (1997), who used the special hydrogen

scattering cell described in subsection 2.5.4 with reference to their mea-

surements of total scattering cross sections. Details were given by Zhou

et al. (1997) and descriptions can be found below, since the same methods

were used by the Detroit group (e.g. Zhou et al., 1994b) in their alkali

metal work.

The apparatus employed by Zhou et al. (1994b) and Surdutovich et al.

(1996) for measurements of positronium formation in the alkali metals

is shown in Figure 4.15. The experiment used more than one method

to estimate σ

, which resulted in the setting of both upper and lower

limits on the cross section. Comparisons with the results of the Arlington

group (Fornari, Diana and Coleman, 1983) for argon gas were made, to

test the reliability of the methods. In order to investigate the eﬀects of

diﬀerent geometries on the apparent value of σ

derived by a gamma-

ray technique, two gas cells with diﬀerent length-to-diameter ratios were

employed for the argon studies, and both are shown in Figure 4.15 along

with the oven used for the alkalis.

After the scattering region, the beam passed a retarding grid assembly,

which could be used to monitor the longitudinal energy spread of the beam

and also to provide angular discrimination against forward elastically

scattered projectiles when the apparatus was used to determine total cross

sections. These measurements were undertaken to check the product of

4.3 Experimental techniques 183

the gas density and the path length as seen by the beam, which was mon-

itored at the end of the ﬂight path using a CEM detector. Annihilations

occurring in the gas-cell region could be monitored using the two NaI(Tl)

detectors set to count coincident pairs of 511 keV gamma-rays. These

could arise either from annihilations of para-positronium or scattered

positrons, as they strike one of the apertures, or from ortho-positronium

which quenched on collision with the walls of the cell. Full details of the

procedure were given by Zhou et al. (1994b). In short, they were able

to set upper limits to σ

by using the Arlington technique, described

above, to conﬁne the scattered positrons and ascribing any lost positrons

to positronium formation. Care was taken to ensure that all forward-

scattered positrons passed through the relevant apertures to reach the

CEM; the retarder in front of the CEM was, of course, set to zero bias.

However, positrons scattered into the backward hemisphere could also

fail to reach the detector and so contribute to the apparent positronium

formation cross section, thus rendering these results upper limits to the

true values.

The apparent para-positronium formation cross section, obtained using

the 511 keV coincidence signal,

was used as a lower limit on

.If

signiﬁcant ortho-positronium quenching takes place at the chamber walls

within sight of the gamma-ray detectors, then this cross section may be

close to the true value. In practice, experiments with the two diﬀerent

gas-cell geometries found this to be the case, the quenching eﬀect being

highest in the scattering cell having the smallest inner diameter. In a

study of argon, Zhou et al. (1994b) found that most of the experimental

results of Fornari et al. (1983), which were considered to be the most

reliable, lay between their measured upper and lower limits.

The ﬁnal system we describe here is that used by Moxom and coworkers

(1993, 1994) for studies of the near-threshold behaviour of the positronium

formation cross section. Their apparatus, which is described in detail by

Moxom, Laricchia and Charlton (1995a), actually measured the total ion

yield and thus gave values of σ

+ σ

. The scattering cell they employed

is shown in Figure 4.16. The positron beam was guided by an axial

magnetic ﬁeld through various electrostatic elements located prior to the

cell, which enabled unwanted secondary electrons to be eliminated and

the energy spread of the beam to be reduced for the near-threshold work.

This property of the beam could be analysed using the grid arrangement

in front of the ceratron detector. A Wien ﬁlter was also used to help

prevent unwanted particles from reaching the scattering cell and also to

chop the beam.

The scattering took place in the hemispherical chamber. The base of the

hemisphere consisted of a series of electrodes which, when appropriately

biassed, created a radial electric ﬁeld throughout the interaction region.

Fig. 4.16. Layout of the apparatus of Moxom, Laricchia and Charlton (1995a) for studies of positronium formation and

ionization (not to scale). Reprinted from Applied Surface Science 85, Moxom et al., A gated positron beam incorporating a

4.4 Results 185

Thus, ions from all points in the cell were focussed onto a hole in the centre

of the extractor, through which they could pass to be detected by a second

ceratron. The resultant eﬃcient ion extraction and detection facilitated

measurements near threshold, where the cross sections are small. The

measurement cycle was as follows: the beam was allowed to pass through

the scattering chamber for a period of around 40 µs, before being chopped

by grounding the Wien ﬁlter; after a 1 µs delay to allow all the beam to

leave the cell, a voltage of −180 V was applied to the extraction electrodes

for a duration of 5 µs and the total number of ions created was counted.

This cycle was repeated at each desired beam energy until the required

statistical accuracy on the ion counts was achieved. It was found that

the lifetime of the ions in the scattering cell was suﬃciently long that the

majority were extracted before they were lost by collision with the cell

wall.

4.4 Results

1 Noble gases and molecules

In Figure 4.17 experimental and theoretical data for σ

for helium gas are

shown for the energy range from threshold up to around 300 eV. (A closer

look at the data near threshold is given later in this section.) Whereas

the experimental results of Fornari, Diana and Coleman (1983) are in

good agreement with those of Fromme et al. (1986), the higher energy

extensions of the Texas work by Diana et al. (1986b) are not in such good

accord. In particular, the oscillatory behaviour observed by Diana et al.

(1986b) is not found in the results of Fromme et al., which appear to vary

smoothly up to the highest energy of approximately 300 eV. This is also

true of the data of Overton, Mills and Coleman (1993), who tentatively

attributed the phenomenon observed by the Texas group to beam optic

eﬀects (see below).

Among the theoretical data shown in Figure 4.17 are the polarized-

orbital approximation results of Khan and Ghosh (1983) and Khan,

Mazumdar and Ghosh (1985), which together give the contributions from

positronium formation into the n

= 1 and n

= 2 states. Also shown

are the results of the distorted-wave method used by Mandal, Guha

and Sil (1980), the coupled-static (plus second order optical potential)

approximation of McAlinden and Walters (1992) and the coupled-state

calculations of Hewitt, Noble and Bransden (1992a). Although some

of these theories are rather crude, nonetheless inspection of Figure 4.17

reveals that the magnitude and position of the observed maximum are

reasonably well reproduced. A comparison up to 140 eV of the experi-

186 4 Positronium formation

Fig. 4.17. Total positronium formation cross sections for positron–helium scat-

tering. Experiment: •, Overton, Mills and Coleman (1993); ◦, Fornari, Diana

and Coleman (1983) and Diana et al. (1986b);

, Fromme et al. (1986). The error

bars have been omitted from the latter two sets of data for clarity. Theory: ——,

McAlinden and Walters (1992); ·····, Schultz and Olson (1988); -- -- -- --, Mandal,

Guha and Sil (1980); — · —, Hewitt, Noble and Bransden (1992a); — ··—,

Igarashi and Toshima (1992); — —, Deb, Crothers and Fromme (1990). See

also Figure 4.8(b).

mental data with the 27-coupled-state approximation of Campbell et al.

(1998a) was shown in Figure 4.8(b).

It is notable that the experimental results of Diana et al. (1986b) and

Fromme et al. (1986) exceed the theoretical results at energies above

approximately 70–80 eV. Indeed, as Schultz and Olson (1988) pointed out,

the energy dependences of the measured and calculated cross sections do

not agree, even allowing for the size of the experimental errors. Schultz

and Olson (1988) noted that above 100 eV the energy dependence of the

experimental data is intermediate between E

−1

and E

−1.5

, whilst that

of the theoretical results is closer to E

−3.5

. These authors have argued

further that the discrepancy may be an experimental error caused by an

incomplete collection of positrons scattered at large angles in the process

of direct ionization. If this were so, it would lead to an overestimate of

in both experiments. This error would be most pronounced where σ

is small, but the resulting diﬀerence in the ionization cross section would

be negligible in view of the size of σ

at these energies. This argument

was taken further by direct calculations of Schultz, Reinhold and Olson

(1989), although the basis of their argument was criticized by Deb et al.

(1990) from both experimental and theoretical standpoints.

4.4 Results 187

Fig. 4.18. Positronium formation cross sections plotted as functions of the

positronium kinetic energy for the following gases: (a) helium, (b) neon, (c)

argon, (d) krypton, (e) xenon. The ionization threshold in each case is indicated

by ‘ion’. Reprinted from Physical Review A50, Moxom et al., Threshold eﬀects

Physical Society.

More recent experiments by Overton, Mills and Coleman (1993) may

have gone some way towards resolving this dispute in favour of theory.

As mentioned above, this group used the Texas technique to measure σ

but paid particular attention to obtaining a well-controlled positron beam,

188 4 Positronium formation

particularly at intermediate energies. Based on their previous experience,

they concluded that small E ×B eﬀects introduced by grid misalignments

etc. could have deleterious energy-dependent eﬀects on positron trajecto-

ries, leading to the possible loss of scattered particles and perhaps to

the oscillatory behaviour reported by Diana et al. (1986b). The data of

Overton, Mills and Coleman, shown in Figure 4.17, are found to be lower

than the previous experimental measurements above ≈100 eV, and in

better accord with theory.

Close to threshold, the most accurate experimental data are those

obtained by Moxom, Laricchia and Charlton (1993) and Moxom et al.

(1994), whose system, as described in section 4.3, measured the total

ion yield (see e.g. Moxom, Laricchia and Charlton, 1995b, for a study

at intermediate energies) but gave σ

at energies below the ionization

threshold. The noble-gas data reported by Moxom

et al. (1994) are

plotted in Figure 4.18 as a function of E



, the kinetic energy of the

positronium. In section 3.3 it was described how these data were used

in an analysis of the threshold behaviour of the elastic scattering cross

section. The plots show broken and solid lines ﬁtted by Moxom et al.

(1994) in an attempt to determine which partial waves contribute most

to σ

. In the case of helium, the data appeared to suggest that σ

is dominated by the l



= 1 partial wave, where l



is the orbital angular

momentum of the outgoing positronium and is related to that of the

incident positron by angular momentum conservation. However, as has

been described in sections 3.3 and 4.2, it is now known from the theoretical

calculations of Van Reeth et al. (1997) that in fact several partial waves

contribute signiﬁcantly to the overall positronium formation cross section,

but they add in such a way as to produce an eﬀect similar to that of a

dominant partial wave with l



= 1. A similar eﬀect may also be the case

for the heavier noble gases, where σ

appears to be dominated by l



=0.

The positronium formation cross sections for these gases rise more rapidly

above the threshold than does that for helium, and the eﬀect of this on the

behaviour of the elastic scattering cross sections is discussed by Moxom

et al. (1994).

Results for the heavier noble gases were obtained by the Arlington

group (Fornari, Diana and Coleman, 1983; Diana and coworkers,

1986a, c). As in the case of helium, the oscillations observed at higher

energies (shown most clearly for argon in Figure 4.19) have yet to ﬁnd a

plausible physical explanation. The data of Zhou et al. (1994b) for argon,

obtained, as described in section 4.3, in an investigation into positronium

formation from the alkali metals, do not agree with the data of Diana et al.

(1986c) and reveal no structure. The only calculations of positronium

formation for argon have been, as described in subsection 4.2.4, those

of McAlinden and Walters (1992), who used a truncated coupled-static