Charlton M., Humberston J.W. Positron Physics

Подождите немного. Документ загружается.

6.3 Results – positron annihilation 289

Fig. 6.9. Positron annihilation rates at various temperatures (normalized to

unity at room temperature) for the noble gases, from Kurz et al. (1996). Key:

◦, He; •, Ne;

, Ar; , Kr; , Xe. The curves

are from the theoretical work of

McEachran and coworkers (1977, 1978, 1979, 1980) for helium (— —) and neon,

argon, krypton and xenon (——). Reprinted from Physical Review Letters 77,

Kurtz, Greaves and Surko, Temperature dependence of positron annihilation

Society.

energy is increased by applying radio-frequency noise to one of the trap

electrodes. The annihilation rate on the relevant gas, and the positron

temperature, are then measured with time as the positrons cool in the

gas. Figure 6.9 shows the plots of normalized annihilation rate versus

temperature obtained by Kurz, Greaves and Surko (1996) for the noble

gases. In general, excellent agreement is obtained with the theoretical

work of McEachran and coworkers. Also, the results of Van Reeth et al.

(1996) for helium cannot be distinguished from experiment at the level

of accuracy displayed in Figure 6.9. Good agreement with the older data

of Lee and Jones (1974), which were taken using the traditional lifetime

method, was found by Kurz et al. (1996) in the temperature region of

overlap.

290 6 Positron annihilation

Fig. 6.10. Mean positron annihilation rate (denoted here as

) at various gas

densities for N

and Ar gases at diﬀerent temperatures. Key: ,N

at 130 K;

, Ar at 160 K; ,N

at 297 K; , Ar at 297 K. The original sources for these

measurements are given by Heyland et al. (1986). The broken line indicates the

linear rise expected, equation (6.3), for a constant Z

eﬀ

 of 27. Reprinted from

Physics Letters A119, Heyland et al., On the annihilation rate of thermalized

Science.

3 Equilibrium phenomena – many-body eﬀects

As the density of a gas is increased and/or its temperature is lowered

towards or below the critical temperature, T

, new phenomena associated

with the trapping and localization of positrons are sometimes encoun-

tered, indicating that many-body processes aﬀect positron annihilation.

We brieﬂy describe these phenomena here, but a much more detailed

treatment can be found in the review of Iakubov and Khrapak (1982).

Two seemingly distinct types of behaviour occur. Figure 6.10 (Heyland

et al., 1986), shows how both types are manifest for N

and argon gases.

The behaviour at room temperature, when λ

 rises more slowly with

density than the linear increase predicted by equations (6.16) and (6.3),

is contrasted with that in the approximate temperature range T<2T

when λ

 rises more rapidly than expected.

The density eﬀects evident in Figure 6.10 for room temperature argon

and N

were discussed by Iakubov and Khrapak (1982), and Nieminen

(1980) carried out a similar exercise for helium. These authors attributed

this behaviour to the eﬀects of multiple scattering at high densities, when

6.3 Results – positron annihilation 291

the positron is considered to be interacting continually with the medium.

Heyland et al. (1986), however, made some simple observations concerning

this behaviour, which has also been observed in room temperature data

for O

and CO (Griﬃth and Heyland, 1978), in H

(Wright et al., 1983;

McNutt, Sharma and Brisbon, 1979), and in helium gas at 77 K and

selected lower temperatures (Fox, Canter and Fishbien, 1977; Hautoj¨arvi

et al., 1977). Heyland et al. (1982) had previously suggested that the

density dependence of Z

eﬀ

 in the high temperature regime could be

represented by the form Z

eﬀ

(ρ) = Z

eﬀ

(0)/(1 + βρ), where Z

eﬀ

(0) is

the zero density limit of Z

eﬀ

 and β is a constant. This expression can

be rearranged using equation (6.3) to yield

λ



0.201ρZ

eﬀ

(0)

λ



, (6.27)

where λ

 = β/[0.201Z

eﬀ

(0)], implying that a plot of 1/λ

 versus

1/ρ should be linear with a slope of 1/[0.201Z

eﬀ

(0)] and an intercept

of 1/λ

. Examples of such plots for a variety of gases, as given by

Heyland et al. (1986), are shown in Figure 6.11. The derived values of

Z

eﬀ

(0) are found to be in close accord with the extrapolation of Z

eﬀ



to zero density, with a non-zero intercept close to λ

 =2ns

−1

at the

high density limit in all cases. This implies a positron lifetime of the same

order as that characteristic of spin-averaged positronium and roughly that

expected for a positron bound to an atom or molecule with conﬁguration

approximating to that of a positronium bound to the corresponding ion

(see section 7.5). The lines drawn for helium and neon in Figure 6.11 are

essentially predictions from the approach of Heyland and coworkers, and

the reader is referred to the original paper for further discussion.

At lower temperatures other phenomena are found. The basic be-

haviour is shown in Figure 6.10, with a further selection given in Fig-

ure 6.12 for the important case of helium gas (the data shown are those

of Hautoj¨arvi et al., 1977) where this phenomenon was ﬁrst observed

(Roellig and Kelly, 1965; Canter and Roellig, 1970). Similar features have

been reported in a variety of other species, e.g. H

(Laricchia et al., 1987a;

McNutt, Sharma and Brisbon, 1979), CO

and SF

(Heyland et al., 1985),

argon (Canter and Roellig, 1975; Tuomisaari, Ryts¨ol¨a and Hautoj¨arvi,

1985) and CH

(McNutt et al., 1975), and it is now considered that in

almost all gases over certain density and temperature ranges positrons

annihilate after self-trapping in clusters of atoms or molecules.

In helium, at low densities λ

 increases linearly with density before

rising rapidly to an approximately constant value, the magnitude of which

is dependent upon the temperature of the gas and is characteristic of the

particular clustered state. The transition is abrupt in this case, but is

shown to be ‘softer’ for N

since the rise in λ

 occupies a much broader

Fig. 6.11. (a) Plots of the inverse of the mean positron annihilation rate versus inverse density for () CO, ()N

,(◦)O

and

(

) Ar at 297 K. The oﬀset scale on the right is for argon only. (b) Similar plots for H

(◦ and •) at room temperature. The

solid line with crosses is for helium with Z

eﬀ

(0) = 4 and λ

 =2ns

−1

and the broken line is for neon with Z

eﬀ

(0) = 6 and

λ

 =2ns

−1

. Reprinted from Physics Letters A119, Heyland et al., On the annihilation rate of thermalized free positrons in

6.3 Results – positron annihilation 293

Fig. 6.12. An example of the eﬀect of self-trapping in clusters on the free-

positron annihilation rate. The right-hand side is for

He at 5.7 K and 7.2 K,

whilst the left-hand data are for

He at 4.2 K and 5.7 K (Hautoj¨arvi et al., 1977).

range of densities. Ryts¨ol¨a, Rantapuska and Hautoj¨arvi (1984) accounted

for this diﬀerence by showing that the potential energy gained by the

positron in making the transition to the (positron + cluster) state is much

larger for helium than for N

, the value for N

being comparable to the

thermal energy and so resulting in an ill-deﬁned cluster. The behaviour

of λ

 as a result of cluster formation was reproduced by Hautoj¨arvi and

Ryts¨ol¨a (1979) and Ryts¨ol¨a, Rantapuska and Hautoj¨arvi (1984), using

a density functional formalism which incorporates a calculated static

density proﬁle of the cluster and known values of Z

eﬀ

to compute the

average annihilation rate at each density and temperature.

4 Electric ﬁelds

Several studies have been made of the behaviour of low energy positrons

in gases under the inﬂuence of a static electric ﬁeld . The broad aim

of this work has been to study the diﬀusion and drift of positrons in

order to understand better the behaviour of the momentum transfer and

annihilation cross sections at very low energies. The theoretical back-

ground has been given in section 6.1, and the diﬀusion equation with an

294 6 Positron annihilation

applied electric ﬁeld, equation (6.12), was solved accurately for positrons

in helium gas by Campeanu and Humberston (1977b) and Shizgal and

Ness (1987). Similar studies were made for other noble gases by Grover

and his collaborators (Singh and Grover, 1987; Sinha and Grover, 1987;

Singh, 1989), mainly using the scattering and annihilation data derived

from the polarized-orbital calculations of McEachran, Ryman and Stauﬀer

(1978, 1979) and McEachran, Stauﬀer and Campbell (1980).

Helium was investigated experimentally by Leung and Paul (1969) at

densities between 20 amagat and 30 amagat at 77 K, and by Lee, Orth and

Jones (1969) in the density range 30–40 amagat at room temperature. In

both cases, as shown in Figure 6.3, Z

eﬀ

 was found to decrease as /ρ was

increased. It is however evident from this ﬁgure that the zero ﬁeld value

of 3.63 for Z

eﬀ

 obtained from both studies is well below that currently

accepted (see table 6.2). The results of Campeanu and Humberston

(1977b) and Shizgal and Ness (1987) are shown for comparison.

Figure 6.3 also presents the helium data of Davies, Charlton and

Griﬃth (1989) at 35.7 amagat and 3.5 amagat in the range /ρ =

0–12 V cm

−1

amagat

−1

. From there it is apparent that (i) the zero-ﬁeld

value of Z

eﬀ

 is in much better accord with theory, and with the ex-

periment of Coleman et al. (1975b), than were the earlier measurements

and (ii) there is a density eﬀect, with the measured value of Z

eﬀ

 at

the higher density being substantially greater than theory. The lower

density data are in much better accord but are subject to much larger

errors, primarily as a result of the diﬃculty in collecting statistically

accurate helium lifetime spectra. Davies, Charlton and Griﬃth (1989)

gave some discussion of the density eﬀect (a similar phenomenon was also

found for argon gas), although no ﬁrm conclusion as to its origin was

reached.

These workers were also able to extract some information on the non-

equilibrium behaviour of positrons in helium and argon by determining

the shoulder lengths at various electric ﬁelds, and qualitative agreement

with theory was found. In addition to their work on the noble gases,

Davies, Charlton and Griﬃth (1989) also presented data for four molec-

ular species in which Z

eﬀ

 was found to decrease linearly by around

10%–15% over the /ρ range investigated.

The eﬀect of an electric ﬁeld on positrons annihilating in dense he-

lium gas at low temperatures was investigated by Canter and coworkers

(Ruttenberg, Tawel and Canter, 1985; Tawel and Canter, 1986). This

work centred on the localization phenomena responsible for the clustering

behaviour of helium atoms around the positron in dense helium gas,

as illustrated in Figure 6.12. The aim of the studies was to test the

hypothesis of Canter et al. (1980), supported by Azbel and Platzman

6.3 Results – positron annihilation 295

(1981), that the clustering phenomenon was preceded by localization

of the positron when its kinetic energy fell below a certain value, E

This energy is termed the mobility edge and marks the boundary be-

tween the extended (i.e. freely diﬀusing) states and the Anderson-localized

states (see e.g. Mott and Davis, 1979) of the positron, which can oc-

cur at densities when its de Broglie wavelength and mean free path are

comparable.

Ruttenberg, Tawel and Canter (1985) studied the behaviour of the

lifetime spectrum under the inﬂuence of a near-uniform static electric

ﬁeld. They found best agreement with the Monte Carlo calculations of

Farazdel and Epstein (1978) and a later extension

by Farazdel (1986)

in which ‘sink-like’ behaviour below a certain energy was incorporated

into the simulation. Furthermore, Ruttenberg, Tawel and Canter (1985)

established that neither the energy threshold E

for positron self-trapping

(i.e. formation of a positron–helium cluster state) nor the positron annihi-

lation rate in that state were aﬀected by an electric ﬁeld of 52 V cm

−1

at a

density of 129 amagat. Both of these observations agree with expectations

since, once trapped in a cluster, the positron is immobile. Note that there

is a subtle but very important diﬀerence between the mobility edge E

at which the positron is ﬁrst immobilized, and E

, with E

Tawel and Canter (1986) extended these investigations using a pulsed

electric ﬁeld of varying amplitude, which was applied at a minimum time

of 26 ns after the emission of a positron from the

Na radioactive source.

The ﬁeld was triggered by detecting the 1.274 MeV gamma-ray. Their

work established that the two thresholds, E

and E

, were distinct, which

led to observable diﬀerences between the lifetime spectra when the electric

ﬁeld was pulsed on at diﬀerent times after positron emission.

A lifetime spectrum obtained with a pulsed electric ﬁeld is shown in

Figure 6.13, superimposed upon a spectrum taken at zero ﬁeld. The ﬁeld

was turned on after τ

=26± 0.5 ns and removed after 56 ± 5 ns. The

two spectra agree up to τ

= 28.7 ± 0.5 ns. The part of the peak which

is missing when the ﬁeld is present appears in the heated component and

is attributed to positrons with energies greater than E

at τ

. Some of

these are slowed to E

once the ﬁeld has been removed, resulting in the

small peak observed at 50–60 ns. The unheated, or localized, group of

positrons continues to lose energy towards E

and is not aﬀected by the

electric ﬁeld.

The two-threshold model, i.e. a model with distinct values for E

and

, can account for the data, notably that the pulsed ﬁeld spectrum

is identical to that at zero ﬁeld until a time ∆τ = τ

− τ

when the

distribution begins to reach E

. Figure 6.14 shows pulsed ﬁeld spectra

(and also an example with a constant ﬁeld) taken at diﬀerent times τ

296 6 Positron annihilation

Fig. 6.13. Superimposed zero ﬁeld and pulsed ﬁeld (81 V cm

−1

peak amplitude)

positron lifetime spectra. The pulsed ﬁeld spectrum has been decomposed into

heated components (broken line) and unheated components (crosses) to illustrate

how the electric ﬁeld splits up the positron ensemble. This is also illustrated

by the inset, which shows, schematically, the energy distribution ρ(E, t)ofthe

positron ensemble in the two-threshold model (see text). Reprinted from Physical

Review Letters 56, Tawel and Canter, Observation of a positron mobility thresh-

Society.

covering the range from below 26.0 ns, where the spectra are similar to

the constant ﬁeld case in which all the positrons are heated by the ﬁeld,

through to around 35 ns. There is then no diﬀerence between the spectra

with the ﬁeld on and oﬀ since all the positrons have had time to slow down

below E

and are thus not aﬀected by the ﬁeld. Tawel and Canter (1986)

estimated that E

= 5.5 ± 0.5 meV from the position of the peak in the

lifetime spectrum shown in Figure 6.14 and deduced that E

=15± 3

meV by relating these two energies, the observed ∆τ and the energy-loss

rate in the gas. They also pointed out the utility of studying the relatively

simple positron–helium system and the ‘striking manner in which the

mobility and the cluster formation thresholds manifest themselves’.

6.3 Results – positron annihilation 297

Fig. 6.14. Semilogarithmic pulsed ﬁeld positron lifetime spectra corresponding

to diﬀerent values of τ

(indicated by the arrows) for a mean ﬁeld of 41 V cm

−1

(A spectrum with a d.c. ﬁeld is also shown.) The numbers on the left are the

numbers of coincidences for the ﬁrst plotted point and the broken lines indicate

the cut-oﬀ times τ

where the spectra depart from the zero-ﬁeld spectrum.

Reprinted from Physical Review Letters 56, Tawel and Canter, Observation of

the American Physical Society.

5 Angular correlation and Doppler broadening studies

The ability of angular correlation and Doppler broadening techniques

to provide information concerning the momentum of an annihilating

electron–positron pair was brieﬂy discussed in section 1.3. Also, it

298 6 Positron annihilation

was shown in section 6.1 how the theoretical angular correlation func-

tion, F (θ), can be obtained from the positron-scattering wave function,

see equations (6.18) and (6.19), and its relation to the Doppler-shifted

gamma-ray energy spectrum was described. Early work identifying some

basic features of the angular correlation spectra obtained in the presence

of a magnetic ﬁeld, which mixes the m = 0 state of ortho-positronium

with para-positronium (see e.g. section 7.4), was undertaken by Heinberg

and Page (1957). A more detailed experimental investigation of F (θ)

for the noble gases was carried out by Coleman et al. (1994), with the

data of Stewart, Briscoe and Steinbacher (1990) for the condensed gases

available for comparison. In addition, the Doppler broadening technique

has recently been applied to positron–gas studies

using trapped positrons

by Tang et al. (1992), Iwata, Greaves and Surko (1997) and Van Reeth

et al. (1996).

Coleman et al. (1994) used a two-dimensional angular correlation ap-

paratus similar to that described by West, Mayers and Walters (1981).

Although the traditional one-dimensional method is suﬃcient to measure

the isotropic distribution in gases, a two-dimensional apparatus provides

a much greater data accumulation rate. Positrons produced from a

source were transported in a 0.8 T magnetic ﬁeld into the small central

volume of a chamber in which the gas was held at a pressure of one atmo-

sphere. The annihilation radiation was monitored by position-sensitive

gamma-ray counters located on either side of the chamber and having an

angular resolution of 3.50 ± 0.03 mrad.

The main contributions to the angular correlation spectrum in the

presence of a 0.8 T magnetic ﬁeld arose from mixing of the m =0

state of ortho-positronium with para-positronium and from free positrons.

Pick-oﬀ quenching of the m = ±1 states of ortho-positronium (see section

7.3) also contributed to the two-gamma signal, but it could be neglected

at the low gas pressure used. Unfortunately, the events arising from

para-positronium, which are due to its annihilation before appreciable

slowing down occurred, resulted in a broad component in the spectrum

which could not be distinguished from that for the free positrons. This is

the major drawback of this technique. However, the mixed-state ortho-

positronium, with a vacuum lifetime of 9.7 ns in a magnetic ﬁeld of 0.8

T, has a longer time in which to slow down in the gas (thus reducing the

centre-of-mass momentum of the annihilating pair) and this was in some

cases discernible as a narrow component in the spectra.

Figure 6.15 shows the cylindrically averaged angular distributions for

the ﬁve noble gases, helium through to xenon. The mixed-state ortho-

positronium was only clearly separable in helium, and a free ﬁt to the

data using Gaussian components was performed in this case. For neon

and argon, the mixed-state ortho-positronium was ﬁtted by constraining