Charlton M., Humberston J.W. Positron Physics

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6.2 Experimental details 279

Fig. 6.6. Schematic illustration of the electrode structure of the positron trap of

Greaves, Tinkle and Surko (1994). The variation of the electrical potential along

the trap, together with the gas pressure in the various regions, is also shown.

The letters A, B and C indicate energy-loss collisions of the positrons with the

buﬀer gas. Reprinted from Phys. Plasmas 1, Greaves et al., Creation and

of Physics.

It is based on a trap in which positrons from a low energy beam are con-

ﬁned in a region of low density gas. As will be shown in subsection 6.3.2

below, this approach has been particularly valuable in elucidating positron

annihilation on large molecules, where some form of temporary attach-

ment of a positron to the molecule may play an important role.

The operation of the trap was discussed at length by Murphy and Surko

(1992), with additional details provided by Greaves, Tinkle and Surko

(1994). A schematic illustration of the cylindrically symmetric electrode

structure of the trap and the variation of the electrical potential and

pressure along the trap are shown together in Figure 6.6. Slow positrons

enter the electrode arrangement from the right, pass over an electrostatic

barrier (whose height can be adjusted depending upon the initial kinetic

energy of the positrons) and into region I, which contains molecular

nitrogen gas at a pressure of around 10

−3

torr. The gas is introduced

at the centre of the ﬁrst electrode, and diﬀerential pumping between this

region and the remainder of the trap gives rise to the pressure gradient

identiﬁed in the ﬁgure. The positrons are conﬁned radially by an axial

magnetic ﬁeld, which is typically 0.1–0.2 T.

280 6 Positron annihilation

Fig. 6.7. (a) The accumulation of positrons in the trap shown in Figure 6.6.

(b) The storage of positrons: •,N

gas pressure of 5 × 10

−7

torr; ◦,N

gas oﬀ,

base pressure of 7 × 10

−10

torr. , electrons stored under the same conditions.

Reprinted from Phys. Plasmas 1, Greaves, Tinkle and Surko, Creation and uses

Physics.

Positrons pass through the regions I, II and III before being reﬂected

by the electrical potential barrier at the end of region III, after which

they return towards the entrance of the trap. During the transit there is

a reasonable chance, around 30%, that a positron will lose kinetic energy

by electronic excitation of the gas. Such positrons are then trapped

and eventually lose further energy by exciting vibrational and rotational

transitions of the molecule. Thus, they are cooled to room temperature

after approximately one second, whereupon they reside in region III at a

pressure of 10

−6

torr.

In this manner positrons can be continuously accumulated, the number

entrapped having a time dependence N(t)=Rτ[1 − exp(−t/τ)], where

6.3 Results – positron annihilation 281

R is the trapping rate and τ is the lifetime in the trap. As shown in

Figure 6.7(a), over 10

positrons were accumulated in a time of the order

of 100 s. If desired, the positrons can then be ‘shuttled’ into region IV of

the trap, which has approximately hyperbolic electrodes, by lowering the

potential barrier. In the absence of any added gas, the positron lifetime

in region IV is governed largely by annihilation on the N

buﬀer gas.

Figure 6.7 shows that under these conditions the lifetime is around 60 s,

though this is increased to 30 minutes if the N

gas is pumped out. This

time is still limited by annihilation on the remaining gas molecules in the

trap (at a pressure of 7 × 10

−10

torr), since electrons held in the same

environment can be conﬁned with a time constant of approximately three

hours.

Once positron accumulation has been completed, the gas to be investi-

gated can be added to the system and its eﬀect on the number of trapped

particles measured. Thus, positron lifetimes, and the value of the asso-

ciated Z

eﬀ

 parameter, can be deduced under conditions which ensure

single positron–molecule interactions. Information can also be obtained

by measuring the Doppler broadening of the annihilation radiation. In

addition, Greaves, Tinkle and Surko (1994) showed how the energy of the

trapped positrons can be raised in a controlled manner by the application

of an abrupt radio-frequency pulse to one of the electrodes of the trap.

This has been used to determine the energy dependence of the annihilation

rates for the noble gases (Kurz, Greaves and Surko, 1996), as described

in subsection 6.3.2 below.

6.3 Results – positron annihilation

In this section we review the results from positron annihilation exper-

iments, predominantly those performed using the lifetime and positron

trap techniques described in section 6.2. Comparisons are made with

theory where possible. The discussion includes positron thermalization

phenomena and equilibrium annihilation rates, and the associated values

of Z

eﬀ

, over a wide range of gas densities and temperatures. Some stud-

ies of positron behaviour in gases under the inﬂuence of applied electric

ﬁelds are also summarized, though the extraction of drift parameters (e.g.

mobilities) is treated separately in section 6.4. Positronium formation

fractions in dense media were described in section 4.8.

1 Thermalization phenomena

Some positrons undergo annihilation prior to thermalization, provided

the slowing down process is not too rapid; this is illustrated by features

present in the lifetime data for the noble gases shown in Figures 6.2, 6.5

282 6 Positron annihilation

Fig. 6.8. Time dependence of the Z

eﬀ

 parameter for the noble gases from

neon through to xenon, for experiment (solid lines) and theory (various types

of broken-line curve). Neon: ——, Coleman et al. (1975b); — —, Campeanu

(1981), both curves calculated using the data of McEachran, Ryman and Stauf-

fer (1978) but with diﬀerent approximations to the positron–neon scattering.

Argon (the two lowest curves): ——, Coleman et al. (1975b); — —, Campeanu

(1981). Krypton (upper three curves): ——, Wright et al. (1985); — · —,

Campeanu (1982), using data of McEachran, Stauﬀer and Campbell (1980);

— ··—, Campeanu (1982), using momentum transfer cross sections calculated

by Schrader (1979). Xenon: ——, Wright et al. (1985); — · —, Campeanu

(1982), using data of McEachran, Stauﬀer and Campbell (1980); — ··—,

Campeanu (1982), using the momentum transfer cross sections of Schrader

(1979).

6.3 Results – positron annihilation 283

Table 6.1. Positron thermalization times and shoulder widths for a number of

gases. Original references can be found in Charlton (1985a)

Shoulder Thermalization Thermalization

width τ

ρ time time

Gas (ns amagat) Gas (ns amagat) Gas (ns amagat)

He 1700 ± 50 H

3.3 C

0.05

Ne 2300 ± 200 D

4.4 CCl

<0.05

Ar 362 ± 5N

14 ± 2 CCl

<0.05

Kr 325 ± 6CO

0.09

Xe 178 ± 3CH

0.25

and 6.8. Thermalization shoulders have been observed directly for the

noble gases and for N

, but most molecular gases have thermalization

times which are too short for such phenomena to be resolved directly.

A list of measured positron thermalization times is given for a variety

of gases in table 6.1, along with the shoulder-width parameter, whose

deﬁnition, for the noble gases, is encompassed in equation (6.23). The

molecular thermalization times, except that for N

, were obtained using

the method of Paul and Leung (1968), who noted the reduction in the

shoulder width of argon following the addition of small quantities of

molecular impurities.

For helium, the momentum transfer and annihilation cross sections

calculated using the accurate elastic scattering wave functions described

in section 3.2 have enabled detailed comparisons to be made between

theory and experiment for the variation of Z

eﬀ

(t) across the shoulder.

Campeanu and Humberston (1977b) solved the time-dependent diﬀusion

equation (6.12) for zero electric ﬁeld and at a temperature of 300 K

and obtained the results shown in Figure 6.2. The agreement between

theory and the room temperature experiment of Coleman et al. (1975b)

is good, except at very short times. In the experiment, processes other

than direct annihilation of the positrons in the gas contribute to the

spectrum close to t = 0 where, theoretically, uncertainties relating to

the choice of initial speed distribution are most signiﬁcant. Conﬁrmation

that the diﬀusion and annihilation of positrons in a gas is accurately

described by the diﬀusion equation approach was provided by Farazdel

and Epstein (1977, 1978), using a Monte Carlo technique. They used the

diﬀerential scattering cross sections and annihilation rates calculated by

Campeanu and Humberston (1975) and obtained time-dependent annihi-

284 6 Positron annihilation

lation rates in excellent agreement with those of Campeanu and Humber-

ston (1977b).

Similar theoretical investigations of positron diﬀusion were made by

Campeanu (1981, 1982) for the heavier noble gases using momentum

transfer and annihilation cross sections obtained in the polarized-orbital

approximation. The results are summarized in Figure 6.8, where they

may be compared with the experimental data of Coleman et al. (1975b),

for neon and argon, and Wright et al. (1985), for krypton and xenon.

In general, the lack of accurate data for σ

(v) and Z

eﬀ

(v) precludes the

attainment of accurate theoretical lifetime spectra. For neon, however,

Campeanu (1981) showed that the shoulder length predicted using the

theoretical cross sections of McEachran, Ryman and Stauﬀer (1978) is

much longer than that found experimentally, and he concluded that the

calculated momentum transfer cross sections were too low at energies

below approximately 1 eV. The situation in argon is somewhat better,

and Campeanu found that the cross sections of McEachran, Ryman and

Stauﬀer (1979) gave reasonable agreement with experiment.

The theoretical shoulders for krypton and xenon were computed using

values of σ

(v) calculated by McEachran, Stauﬀer and Campbell (1980)

and Schrader (1979) and values of Z

eﬀ

(v) calculated by the former. In

the case of krypton, the experimental shoulder length and shape are

reproduced well using the data of McEachran, Stauﬀer and Campbell

but the results of Schrader give a much longer shoulder than is observed.

For xenon, poorer agreement between theory and experiment has been

found for both the shape and the magnitude of Z

eﬀ

(t).

In addition to the noble gases, N

has also been found to possess an

observable shoulder. The most complete study of this system was reported

by Coleman, Griﬃth and Heyland (1981). These workers, using a method

to be described below, analysed the Z

eﬀ

(t) measurements of Coleman

et al. (1976a) at a density of 0.84 amagat and at room temperature, and

those of Sharma and McNutt (1978) at densities below 2 amagat and

at 77 K, to estimate the cross sections for momentum transfer and for

rotational excitation and de-excitation. Less detailed analyses for H

and

were also performed using the thermalization times given in table 6.1,

even though shoulders have not been observed directly for either of these

molecules. The long thermalization times in H

and N

, which were

also reported by Paul and Leung (1968) and Tao (1970), are caused by

low energy-loss rates below the thresholds for vibrational excitation, E

vib

= 0.516 eV, 0.360 eV and 0.290 eV respectively. Below these thresholds

the energy loss is due to rotational excitation and momentum transfer col-

lisions, with rotational de-excitation providing a heating mechanism that

competes when there is a signiﬁcant population of rotationally excited

states. This applies to N

, which, according to Coleman, Griﬃth and

6.3 Results – positron annihilation 285

Heyland (1981), has 33 rotational levels occupied at room temperature

and 16 at 77 K.

Coleman, Griﬃth and Heyland (1981) used a simple slowing-down

equation to analyse the data, rather than the full diﬀusion equation (6.12).

The positrons were assumed to start their slowing down at E

vib

and then

lose energy at a rate

−

= −











1 −



, (6.24)

where −(dE/dt)

and −(dE/dt)

are the energy-loss rates for rotational

and elastic collisions respectively and the factor 1 − 3k

T/2E allows, to

ﬁrst order, for the energy gained from the gas molecules at each collision.

The individual energy-loss rates were deduced from

−









1/2

(E) (6.25)

and

−









1/2



[σ

J,J +2

(E)(E

J+2

− E

)

−σ

J,J −2

(E)(E

− E

J−2

)]; (6.26)

here the number density of gas molecules in the Jth rotational state is

and σ

J,J +2

(E) and σ

J,J −2

(E) are the cross sections for transitions

between the energy levels E

and E

J±2

Further details of these calculations are given by Coleman, Griﬃth and

Heyland (1981), although it is not diﬃcult to see that equations (6.25)

and (6.26) can be used to derive the time variation of the positron energy.

This was then employed, with an expression for Z

eﬀ

(E) taken from the

work of Darewych and Baille (1974) but ﬁtted to the results of Sharma

and McNutt (1978) and Coleman et al. (1976a), at 77 K and 300 K

respectively, to deduce values of Z

eﬀ

(t). The results compare favourably

with experiment. The values of σ

(E) obtained from diﬀerent calculations

did not agree, and therefore Coleman, Griﬃth and Heyland (1981) varied

this parameter to produce the best ﬁt to the data of Coleman et al.

(1976a), which were, statistically, the more accurate set. The reader

is referred to the original work for further details and for information,

derived from the study, regarding positron behaviour in H

and D

Thermalization in N

gas has also been studied using the positron-trap

apparatus developed by Surko and coworkers and described in subsec-

tion 6.2.2. By storing positrons in the trap at a known pressure for

various lengths of time before ejecting them and measuring their mean

286 6 Positron annihilation

Table 6.2. Experimental and theoretical values for Z

eﬀ

 at low densities and

room temperature for the noble gases. See the work of Griﬃth and Heyland

(1978) for a review of earlier results

Z

eﬀ

, Z

eﬀ

,

Gas experiment theory

He 3.94 ± 0.02 3.88

Ne 5.99 ± 0.08 7.0 ± 0.3

Ar 26.77 ± 0.09 27.6

Kr 65.7 ± 0.3 57.6 ± 2.9

Xe 320 ± 5 217 ± 11

energy, the rate of slowing down could be obtained (Murphy and Surko,

1992). Analysis reveals the thermalization time to be in fair accord with

the results of Coleman and coworkers (1976a, 1981) and Paul and Leung

(1968), given the very diﬀerent nature of the experiments.

2 Equilibrium phenomena at low gas densities

Measurements of the equilibrium annihilation-rate parameter, Z

eﬀ

, have

been made for a plethora of gases over a wide range of densities. In this

section we conﬁne ourselves to the low density and high temperature

region, in which many-body eﬀects can be ignored and where the results

may be compared with those obtained from scattering theory, as outlined

in section 6.1.

Table 6.2 presents theoretical and experimental values of Z

eﬀ

 for the

noble gases helium to xenon. Traditional lifetime experiments have found

Z

eﬀ

 to be density dependent, particularly for argon, krypton and xenon;

the values quoted in table 6.2 correspond to low gas densities. The results

obtained from theory and experiment are in reasonable agreement, the

theoretical results for the heavier targets having been derived from the

results of McEachran and coworkers (1978, 1979, 1980), who performed

a systematic study of all the noble gases, using the polarized-orbital

approximation. For helium, the most accurate theoretical value, 3.88,

obtained by Van Reeth et al. (1996), is in good accord with the accurate

experimental result, 3.94 ± 0.02, of Coleman et al. (1975b). A more

detailed survey of the experimental work on helium gas was given by

Griﬃth and Heyland (1978).

The theoretical values for Z

eﬀ

 quoted in table 6.2 have been speed-

averaged to correspond to the room-temperature positron speed distri-

6.3 Results – positron annihilation 287

bution. This is particularly important, as pointed out in section 6.1,

for xenon, where the theoretical results vary rapidly at low energies. In

their work on xenon, Wright et al. (1985) reported that their measured

‘equilibrium’ value, Z

eﬀ

 = 320 ± 5, may be an underestimate, owing

to incomplete thermalization of the positrons in this gas and the high

probability of epithermal annihilation. This tentative conclusion was

reached on the basis of measurements with small quantities of H

,He

or N

added to the xenon, which eﬀected more rapid thermalization;

Z

eﬀ

 was found to increase to around 400. This value was corroborated

by Murphy and Surko (1990), who used the positron-trap method to

determine the annihilation rate in xenon at very low pressures (around

−4

Pa). The accord between the results obtained using the traditional

lifetime and positron-trap methods is also reasonable for argon, although,

in the case of krypton, Iwata et al. (1995) found Z

eﬀ

≈90, previous

values being in the range 65–66, as given in table 6.2. Evidence presented

by Wright et al. (1985) from studies of krypton–gas mixtures suggests

that this cannot be due to the eﬀects of incomplete thermalization in the

traditional lifetime experiments.

Many positron annihilation experiments have also been performed on

molecular gases using the traditional lifetime technique. Table 6.3 gives

a selection of values of Z

eﬀ

 for a range of molecular species (Heyland

et al., 1982; Wright et al., 1983). Heyland et al. (1982) grouped these

gases into two broad categories: those for which Z

eﬀ

≈Z and those for

which Z

eﬀ

Z. It is now thought that the formation of temporary

bound states is responsible for the very large values of Z

eﬀ

 in the latter

category (see below).

In the case of H

, the low density measurements of Z

eﬀ

 by McNutt,

Sharma and Brisbon (1979) are in accord with those of Wright et al.

(1983), though until fairly recently all the theoretical results were much

closer to Z = 2 (see Wright et al., 1983, for a detailed discussion). How-

ever, as mentioned in section 6.1, Armour, Baker and Plummer (1990)

investigated annihilation as part of their general study of positron–H

scattering using the Kohn variational method, see subsection 3.2.1, and

by using increasingly elaborate trial wave functions, which allow explicitly

for positron–electron correlations, they obtained the zero energy value

eﬀ

= 10.2. This is in much better agreement with the room temperature

experimental value, 14.8; it is likely that further theoretical advances will

lead to even better accord.

Theory and experiment can also be compared in the case of N

Darewych and Baille (1974) calculated Z

eﬀ

= 22 and 20 at 9 MeV and 38

MeV respectively, in reasonable accord with the experiments of Coleman

et al. (1976a) and Tao (1970) at 297 K, and of Sharma and McNutt (1978)

at 77 K. Methane is also of interest, with a measured value for Z

eﬀ



288 6 Positron annihilation

Table 6.3. Measured values of Z

eﬀ

 at low gas densities and 297 K, for various

molecular gases. Values of Z are included for comparison

Gas Z Z

eﬀ



2 14.7

14 30.5

CO 14 38.5

NO 15 34

16 26

22 53

O22 78

70 97

10 140

18 660

26 3 500

34 15 000

Cl 26 15 000

CCl

58 700

23 1 090

10 1 300

of 140, which has been considered to be too large to arise from direct

scattering alone (see e.g. McNutt et al., 1975, and references therein).

However, the only calculation of this quantity, by Jain and Thompson

(1983), found Z

eﬀ

≈ 100 at 297 K, in fair accord with the experimental

value considering the complexity of the target.

The positron-trap technique has been used to measure the annihilation

rate of positrons interacting with a wide variety of molecules. The species

investigated by Iwata et al. (1995) include many hydrocarbons, substi-

tuted (e.g. ﬂuorinated and chlorinated) hydrocarbons and aromatics; as

mentioned in section 6.1, large values of Z

eﬀ

 (in excess of 10

) were

found for some molecules. Several distinct trends are exhibited in the

data of Iwata et al. (1995). Though much of the detailed physics involved

in the annihilation process on these large molecules is still unclear, the

model of Laricchia and Wilkin (1997), described in section 6.1, may oﬀer

a qualitative explanation of the observations.

The temperature, or energy, dependence of the annihilation rate, or

Z

eﬀ

, has also been investigated using a positron trap. In this technique

positrons are ﬁrst accumulated at room temperature, and then their