Charlton M., Humberston J.W. Positron Physics

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6.3 Results – positron annihilation 299

Fig. 6.15. Cylindrically averaged angular correlation of annihilation radiation

(ACAR) distributions for positron annihilation in the noble gases, (a) helium,

(b) neon, (c) argon, (d) krypton and (e) xenon, from the work of Coleman

et al. (1994). Reprinted from Journal of Physics B27, Coleman et al., Angular

correlation studies of positron annihilation in the noble gases, 981–991, copyright

1994, with permission from IOP Publishing.

the amplitude of this component, using the known positronium fraction

as deduced from lifetime studies (see section 4.8) and the known extent

of the mixing of the ortho-positronium component. The krypton and

xenon data, however, could be adequately described by a single Gaussian

and thus the results of the ﬁts cannot be solely due to free-positron

annihilation.

There is no rigorous justiﬁcation for assuming a Gaussian ﬁt to the

data, and accurate theoretical and experimental results for helium clearly

show the deviations from the Gaussian form out on the wings of the

distribution; see Figure 6.16 and the discussion at the end of this section.

Nevertheless, this form does provide a reasonably good approximation to

most of the data.

The positron-trap technique has been used by Surko and coworkers

to measure the Doppler broadening of the 511 keV line for positrons

in helium gas. This method does not have the drawback of the ex-

periment described above, in which both positronium and free-positron

events overlap on the angular distribution curves; here the positrons are

thermalized prior to the introduction of the gas and therefore cannot

form positronium. A comparison of the theoretically predicted and ex-

perimentally measured Doppler spectra (Van Reeth et al., 1996) is shown

in Figure 6.16. The theoretical results were obtained from the variational

wave functions for low energy positron–helium scattering calculated by

Van Reeth and Humberston (1995b); see equations (3.75) and (3.77).

300 6 Positron annihilation

Fig. 6.16. (a) Annihilation gamma-ray spectrum for positrons interacting with

helium atoms. The full curve is the theoretical prediction (see text) convoluted

with the detector response function, whilst the dotted curve is a Gaussian ﬁt to

the experimental data (◦). (b) Residuals from the Gaussian ﬁt and (c) residuals

from the theory. Reprinted from Journal of Physics B29, Van Reeth et al.,

Annihilation in low energy positron–helium scattering, L465–L471, copyright

1996, with permission from IOP Publishing.

Before comparing with experiment, however, the theoretical results at an

energy of 40 meV (equivalent to room temperature) were convoluted with

the energy resolution function of the detector used for the measurement.

This procedure was adopted because deconvolution of the experimental

data was found to be numerically unstable. The convoluted theoretical

data were then normalized to the experimental data at zero Doppler

shift to yield the results shown. The agreement between the convoluted

theoretical results and experiment is extraordinarily good, extending as it

does over more than three orders of magnitude. These results also reveal

6.4 Positron drift 301

that the shape of the gamma-ray spectrum is not Gaussian, in contrast to

what has frequently been assumed in the past (see above). Nevertheless,

Van Reeth et al. (1996) also ﬁtted a Gaussian to the data and derived a

Doppler FWHM of 2.53 ± 0.03 keV, corresponding to an angular width

of 9.90 ± 0.12 mrad. This is to be compared with the values 10.30 ± 0.05

mrad obtained by Coleman et al. (1994) and 9.4 ± 0.5 mrad found by

Stewart et al. (1990), for liquid helium. Iwata, Greaves and Surko (1997)

also studied Doppler broadened spectra for argon, krypton and xenon

and were able to observe contributions from annihilation with inner shell

electrons.

6.4 Positron drift

The study of positron drift in gases has been very limited, in contrast

to the vast literature available for electrons (e.g. Huxley and Crompton,

1974). The reason for this lies in the experimental diﬃculties in ob-

taining a drift time and distance such that the drift speed for positrons,

, can be computed. An ideal positron-drift experiment would involve

the injection of a burst of low energy positrons into a scattering cell

at a precisely known time; transport of the positrons through the cell

and on to a detector would then be accurately evaluated to yield the

mobility, µ

= v

/ (Charlton and Jacobsen, 1987). Unfortunately an

experiment of this kind has not yet been carried out. In this section we

give brief descriptions of two experimental arrangements which have been

employed to measure positron drift, though so far with a limited degree

of applicability.

The ﬁrst observations of positron drift were made by Rodionov, San-

nikov and Solodov (1969, 1971) for helium gas. Their method was later

adopted and improved by Paul and coworkers (Paul and Tsai, 1979;

Paul and B¨ose, 1982; B¨ose, Paul and Tsai, 1981) and we present here

a description of the latter apparatus only; this is shown schematically in

Figure 6.17 (Paul and B¨ose, 1982). Positrons from two

Co sources were

conﬁned to the axis of a 4 cm long chamber by a uniform magnetic ﬁeld

of approximately 0.12 T. A small fraction of the positrons thermalized

in the gas, uniformly throughout the length of the chamber owing to the

low pressures used (typically well below 10

Pa). These positrons could

then be drifted along the chamber by application of an electric ﬁeld,

using a ring and grid arrangement and the equipotential target foil which

served to terminate the drift. This foil, which was chosen to be a 600

A carbon ﬁlm in order to reduce annihilation of the fast positrons, was

viewed by a pair of detectors located behind the apertures in the lead

shielding.

302 6 Positron annihilation

Fig. 6.17. Schematic illustration of the positron-drift apparatus used by Paul

and coworkers.

Paul and Tsai (1979) showed that the fraction F

of positrons which

drift into the foil after stopping in the gas is given by

λ

l



1 − exp



−

λ

l



= X



1 − exp



−



, (6.28)

where l is the maximum drift distance and λ

 is the annihilation rate

in the gas. At higher ﬁelds, when positronium formation is possible, λ



will be augmented by the rate at which this latter process occurs. An

additional assumption made in arriving at equation (6.28) was that the

diﬀusion length is much smaller than the drift length. This, in general,

may not be true, and a more sophisticated analysis of the drift and

diﬀusion problem was given by Paul and Tsai (1979).

Coincidences due to positron annihilation at the foil were recorded for

various values of /ρ and at several pressures of molecular hydrogen in

the range 10–200 torr. From these the data at 100 torr were selected for

6.4 Positron drift 303

Fig. 6.18. Results from the positron-drift experiments of Paul and coworkers.

The full curve is v

/Z

eﬀ

, obtained from a ﬁt to all the data; see text for

details. Electron drift velocities (+) are shown for comparison on the left-hand

scale. The points to the right of the vertical broken line were taken from runs

at molecular hydrogen pressures of 50 torr (•), 25 torr () and 10 torr (◦).

an analytic ﬁt to F

, equation (6.28), in terms of the parameter X. This

parameter was density scaled, setting X

= Xρ/ρ

, to give the shape

of the other curves from F

= X

[1 − exp(−1/X

)]; X was then varied

to optimize the ﬁts at all the other pressures. Using values for X and

Z

eﬀ

 (the latter assumed to be independent of electric ﬁeld), a curve of

/Z

eﬀ

 was obtained, and this is shown in Figure 6.18 along with the

derived values of v

and the corresponding values for electrons (Huxley

and Crompton, 1974).

The broken-line portion of the v

/Z

eﬀ

 curve, which attains a max-

imum and then falls, was explained by B¨ose, Paul and Tsai (1981) in

terms of the formation of positronium due to positron heating in the

electric ﬁeld, so that the apparent value of Z

eﬀ

 rises as the amount

of positronium formation increases. At high electric ﬁelds nearly all the

positrons form positronium and do not annihilate at the foil.

As shown in Figure 6.18, electron drift velocities below /ρ =1Td

(≡ 10

Vcm

) are at least four times larger than those for positrons.

B¨ose, Paul and Tsai (1981) attributed this diﬀerence to higher momentum

transfer cross sections for positrons than for electrons at very low (i.e.

304 6 Positron annihilation

thermal) energies. The total cross section for electrons is approximately

9 × 10

−16

at 0.2 eV and it falls as the kinetic energy is lowered. As

described in subsection 2.6.1, the total cross section for positron scat-

tering in H

rises rapidly with decreasing energy at the lowest energies

investigated, lending some plausibility to the proposed explanation. Other

measurements have been made with this system (Paul, 1993, 1995, private

communications), though results for v

have not been extracted.

The other apparatus used to study positron drift in a range of molec-

ular gases has been described by Charlton (1985b) and Charlton and

Laricchia (1986). It consisted of two electrodes 10.35 mm apart, which

also formed the walls of the gas chamber and between which a poten-

tial diﬀerence could be applied. Positrons emitted from a

Na source

were detected using the thin plastic scintillator method (e.g. Coleman,

Griﬃth and Heyland, 1973). Most positrons entered the gas chamber

through a thin window and annihilated in the metal walls of the cell.

The annihilation gamma-rays were detected using a large plastic

scintil-

lator. Approximately 0.1% of the β

particles stopped in the gas, where

they annihilated as free positrons or after forming positronium or, in

the absence of an electric ﬁeld, after randomly diﬀusing to a wall of the

chamber.

When an electric ﬁeld was applied across the chamber some positrons

annihilated prematurely, following ﬁeld-induced drift to one of the elec-

trodes. In this case the free-positron component of the lifetime spectrum

was ﬁeld dependent; the maximum drift time, τ

, was given by the

end-point of the lifetime spectrum and was due to thermalized positrons

which had traversed the entire drift length l. The drift speed was then

= l/τ

and the mobility could be found from

= l/τ

. (6.29)

Charlton (1985b) selected O

and CO

gases for investigation. Experi-

ments had shown (e.g. Wright, 1982) that the ortho-positronium compo-

nent is rapidly quenched in the former, so that at 400 torr the lifetime

is only around 30 ns. Thus, the free-positron spectrum is well separated,

and despite the low signal-to-background ratio, τ

is relatively easy to

discern. The latter gas was selected because, even at low densities, there

is suﬃcient stopping power to ensure adequate statistical accuracy of the

data. Examples of truncated lifetime spectra are given in Figure 6.19(a).

Nevertheless, the major source of error in these measurements still lay in

the determination of τ

, which could only be evaluated with approxi-

mately 10%–15% accuracy.

The drift velocities for O

and CO

at various ﬁelds and at two diﬀerent

pressures are shown in Figures 6.19(b) and (c). The velocities were mea-

sured at values of /ρ in the ranges 1200–4500 V cm

−1

amagat

−1

for CO

6.4 Positron drift 305

Fig. 6.19. (a) The free-positron component with electric ﬁelds applied to 0.26

amagat of CO

, illustrating the truncation of the spectra. The solid curve is a

schematic illustration of this component with zero applied ﬁeld. (b) Positron

drift velocities for O

at various density-normalized electric ﬁelds: •, 600 torr;

◦, 400 torr. (c) As (b), but for CO

: •, 200 torr; ◦, 100 torr. The solid line

represents the corresponding electron drift velocities.

and 800–1700 V cm

−1

amagat

−1

for O

. It was assumed, with reference

to the corresponding work over these ranges of /ρ for electrons, that

the positrons remain in thermal equilibrium with the gas. The density-

normalized values of the mobilities were found to be (1.44 ± 0.19) × 10

−1

amagat for O

and (4.78 ±0.42) ×10

−1

amagat

for CO

. These results correspond to (3.9 ± 0.5) × 10

(V cm s)

−1

and

(1.29 ± 0.11) × 10

(V cm s)

−1

respectively.

Also shown in Figure 6.19(c) is the line which corresponds to the

average electron mobility for CO

in the same /ρ range. These data were

extracted from the results of Peisert and Sauli (1984) and Christophorou

(1971), and correspond to a density-normalized mobility of 600 cm

−1

amagat. In the case of O

this quantity for electrons is approxi-

mately 4.5×10

−1

amagat (Crompton and Elford, 1973) at low

/ρ, though this falls sharply to 2.8 × 10

−1

amagat as /ρ is

increased to 10

Vcm

−1

amagat

−1

. If one considers the /ρ-independent

values for electrons, then in each case the mobility of electrons is greater

than that of positrons. As was the case for H

, this can probably be

attributed to the relative behaviour of the momentum transfer cross

sections at very low energies.

Several other molecular gases have been investigated using this tech-

nique, and the results presented by Charlton and Laricchia (1986) are

306 6 Positron annihilation

Table 6.4. Values of density-normalized positron mobility (in cm

−1

amagat) for various molecular gases at T = 297 K. Uncertainties are around

10%–15%

Gas Mobility

1940

2200

1440

1560

CO 1430

480

370

300

given in table 6.4. In the case of H

, we can compare the value 1940

−1

amagat for the mobility with the value 1100 cm

−1

am-

agat derived from the work of B¨ose, Paul and Tsai (1981). The dis-

crepancy between the two measurements is greater

than the combined

errors, though given the very diﬀerent nature of the experiments and the

diﬃculty in extracting drift velocities for positrons it is perhaps gratifying

to ﬁnd agreement within a factor of two. Further work on positron drift

is desirable since it can complement beam measurements, particularly at

energies below approximately 1 eV.

Positronium and its interactions

In this chapter we consider the physics of the positronium atom and what

is known, both theoretically and experimentally, of its interactions with

other atomic and molecular species. The basic properties of positronium

have been brieﬂy mentioned in subsection 1.2.2 and will not be repeated

here. Similarly, positronium production in the collisions of positrons with

gases, and within and at the surface of solids, has been reviewed in section

1.5 and in Chapter 4. Some of the experimental methods, e.g. lifetime

spectroscopy and angular correlation studies of the annihilation radiation,

which are used to derive information on positronium interactions, have

also been described previously. These will be of most relevance to the

discussion in sections 7.3–7.5 on annihilation, slowing down and bound

states. Techniques for the production of beams of positronium atoms were

introduced in section 1.5. We describe here in more detail the method

which has allowed measurements of positronium scattering cross sections

to be made over a range of kinetic energies, typically from a few eV up

to 100–200 eV, and the ﬁrst such studies are summarized in section 7.6.

Important advances continue to be made in measurements of the in-

trinsic properties of the positronium atom, e.g. its ground state lifetimes

(Rich, 1981; Al-Ramadhan and Gidley, 1994; Asai, Orito and Shinohara,

1995) and various spectroscopic quantities (Berko and Pendleton, 1980,

Mills, 1993; Hagena et al., 1993). These are reviewed in section 7.1.

7.1 Fundamental studies with the positronium atom

1 Lifetimes against annihilation

Since the earliest work with positronium by Deutsch and coworkers (e.g.

Deutsch, 1951; Deutsch and Brown, 1952) its annihilation lifetimes, or

decay rates, have been studied both theoretically and experimentally.

307

308 7 Positronium and its interactions

The decay rate of ground state ortho-positronium in vacuum,

o-Ps

, has

been investigated more often than any other property of positronium,

because it is relatively straightforward to produce the atom in abundance

and the accurate measurement of its 142 ns lifetime, corresponding to

an annihilation rate of 7.04 µs

−1

, is well within technological capabil-

ities. In contrast, the much shorter 125 ps lifetime of ground state

para-positronium, corresponding to an annihilation rate of 8.0 ns

−1

, has

only been measured twice with a precision of better than 1%; this has been

achieved by utilizing the mixing of the m = 0 states of ortho-positronium

and para-positronium in a uniform magnetic ﬁeld and assuming a value

for the ground state hyperﬁne splitting. To our knowledge there have

been no measurements of the decay rates for excited state positronium.

In Chapter 1, the ﬁrst order contributions to the annihilation rates from

the dominant modes of decay of the S-states of both ortho- and para-

positronium (for arbitrary principal quantum number n

) were given

as equations (1.5) and (1.6). These contributions are included in the

following equations for the rates for the two ground states, which also

contain terms of higher order in the ﬁne structure constant, α:

o-Ps

=2α

− 9

9π



1 −

Aα

−

ln α

−1

+ B





−

3α

2π

(ln α)

+ ···



, (7.1)

p-Ps

2



1 −



5 −



ln α

−1

+ B





+ ···



. (7.2)

Note that, as can be seen from the discussion in subsection 1.2.1, the

contributions from the higher order annihilation modes are negligible at

the present levels of precision. Thus, the rate for the annihilation of ortho-

positronium into ﬁve gamma-rays is only 10

−6

of that for three gamma-

rays, with a similar value for the ratio of the rates for para-positronium

annihilation into four and two gamma-rays.

The most accurate determination of the coeﬃcient A in the ortho-

positronium decay rate has yielded the value 10.2866 (Adkins, Salahuddin

and Schlam, 1992), giving a 2.3% change in the lowest order annihilation

rate. For para-positronium, the corresponding ﬁrst order correction is

only 0.6%. The coeﬃcient B multiplying the term (α/π)

has been

determined by Mil’stein and Khriplovich (1994) to have the value 46

for ortho-positronium (and 40 for para-positronium), producing a further

change of approximately 250 ppm in the annihilation rate. Taking all

these corrections into account, the most accurate theoretical value of

the ortho-positronium annihilation rate is 7.0420 µs

−1

, in very good