Charlton M., Humberston J.W. Positron Physics

Подождите немного. Документ загружается.

6.1 Introduction and theoretical considerations 269

so that



ion

P (n

ion

)=1.

Drachman (1966b), using a rather simple representation of positron–

helium scattering based on a modiﬁed form of the adiabatic approxi-

mation, calculated the probabilities of the residual helium ion being in

various states after annihilation and found a 95% probability of its being

in the ground state.

Traditionally, experimental values of Z

eﬀ

have been derived from mea-

surements of the lifetime spectra of positrons that are diﬀusing, and

eventually annihilating, in a gas. The lifetime of each positron is measured

separately, and these individual pieces of data are accumulated to form

the lifetime spectrum. (The positron-trap technique, to be described in

subsection 6.2.2, uses a diﬀerent approach.) An alternative but equivalent

procedure, which is adopted in electron diﬀusion studies and also in the

theoretical treatment of positron diﬀusion, is to consider the injection

of a swarm of positrons into the gas at a given time and then to inves-

tigate the time dependence of the speed distribution, as the positrons

thermalize and annihilate, by solving the appropriate diﬀusion equation.

The experimentally measured Z

eﬀ

, termed Z

eﬀ

, is the average over the

speed distribution of the positrons, y(v, t), where y(v, t) dv is the number

density of positrons with speeds in the interval v to v + dv at time t after

the swarm is injected into the gas. The time-dependent speed-averaged

eﬀ

is therefore

Z

eﬀ

(t) =

∞

eﬀ

(v)y(v, t) dv

∞

y(v, t) dv

. (6.11)

Assuming that all positrons in the swarm have energies below the

positronium formation threshold and that only elastic collisions and

annihilation are possible, the speed distribution may be derived theo-

retically as the solution of the following diﬀusion equation (Orth and

Jones, 1969):

∂y(v,t)

∂t

∂

∂v





nvσ

(v)

vnσ

(v)k



∂y(v,t)

∂v



vnσ

(v)

−



nσ

(v)

−

2nσ

(v)k



y(v, t)



−λ

(v)y(v, t), (6.12)

where  is the electric ﬁeld applied across the gas cell, T is the temperature

of the gas, k

is Boltzmann’s constant, M is the mass of each gas atom,

(v) is the momentum transfer cross section and λ

(v) is the annihilation

270 6 Positron annihilation

Fig. 6.2. The time dependence of Z

eﬀ

 for positrons diﬀusing in helium gas at

room temperature and zero electric ﬁeld. Experiment: — ··—, Coleman et al.

(1975b). Theory: ——, helium model H5; — · —, helium model H1, both due

to Campeanu and Humberston (1977b).

rate. The momentum transfer cross section is deﬁned as

(v)=2π



|f(θ)|

(1 − cos θ) sin θdθ, (6.13)

which can also be expressed in terms of the partial-wave phase shifts as

(v)=

4π

∞



l=0

(l + 1) sin

(η

− η

l+1

). (6.14)

Having calculated σ

(v) and λ

(v) from the elastic scattering wave

function and made a suitable choice of the initial speed distribution

y(v, t = 0), the diﬀusion equation can then be solved to obtain the

theoretical positron speed distribution for all subsequent times. Provided

a reasonable choice of the initial speed distribution is made, the solution

to the diﬀusion equation is not very sensitive to its form, except at very

small times after t = 0. The time dependence of Z

eﬀ

(t) obtained in

this way for positrons diﬀusing in helium gas in zero electric ﬁeld is

shown in Figure 6.2 (Campeanu and Humberston, 1977b). The input

data for equation (6.12), σ

(v) and λ

(v), were derived from the phase

shifts and scattering wave functions of Humberston (1973) and Campeanu

and Humberston (1975, 1977a).

6.1 Introduction and theoretical considerations 271

After a suﬃciently long time the positrons reach equilibrium, so that

y(v, t)=f(v) exp(−λ

t), where λ

 is the equilibrium annihilation rate

and f(v) is the associated speed distribution, which is the solution of the

time-independent equation





nvσ

(v)

vnσ

(v)k



df (v)



vσ

(v)

−



nσ

(ν)

−

2nσ

(v)k



f(v)



− λ

(v)f(v)

= −λ

f(v). (6.15)

The equilibrium value of Z

eﬀ

is then

Z

eﬀ

 =

∞

eﬀ

(v)f(v) dv

∞

f(v) dv

λ



πr

. (6.16)

If the annihilation rate, λ

(v), is approximately constant over the width of

the dominant part of the equilibrium speed distribution, so that λ

(v)f(v)

≈λ

f(v), the solution to equation (6.15) has the form

f(v)=Cv

exp







−







e

mnv



)



+ k



−1









(6.17)

where C is the normalization constant. For zero electric ﬁeld this reduces

to the Maxwell–Boltzmann form

f(v)=Cv

exp[−mv

/(2k

T )].

In cases where the annihilation rate, and hence Z

eﬀ

, is a rapidly varying

function of the positron energy, as with xenon (Schrader and Svetic,

1982), the simpliﬁcation introduced above is not valid and the solution

to equation (6.15) must be used. The functional form for f(v) given

in equation (6.17) was used by Campeanu and Humberston (1977b) to

investigate the variation of the equilibrium value of Z

eﬀ

 with electric

ﬁeld and temperature, and their results for the former are shown in

Figure 6.3.

The experimental techniques involved in measuring the angular corre-

lation and the Doppler broadening of the two annihilation gamma-rays

were introduced in section 1.3. These techniques rely on the fact that

the motion of the positron–electron pair immediately prior to annihi-

lation causes the two gamma-rays to be emitted in directions diﬀering

272 6 Positron annihilation

Fig. 6.3. Values of Z

eﬀ

 for helium gas at various density-normalized electric

ﬁelds and at (•) 35.7 amagat and () 3.5 amagat, from Davies, Charlton and

Griﬃth (1989); ◦, Leung and Paul (1969); , Lee, Orth and Jones (1969). The

results of the calculations of Campeanu and Humberston (1977b) (——) and

Shizgal and Ness (1987) (– – –) are also given.

by an angle θ from exact collinearity and with energies shifted from

511 keV. Thus, measuring the angular correlation between the gamma-

rays provides information about the momentum distribution of the anni-

hilating electron–positron pairs. In many cases it can be assumed that

the positrons have thermalized before annihilation, so that the momen-

tum of the pair is predominantly that of the electron alone. However,

the attractive positron–atom interaction causes the positron to speed

up slightly as it approaches an atom, and the electrons also have their

velocities modiﬁed by the attraction towards the positron. Therefore, the

momentum distribution of an annihilating positron–electron pair is not

identical to the momentum distribution of the electrons in an undistorted

atom.

6.1 Introduction and theoretical considerations 273

Fig. 6.4. Angular correlation function F (θ) for the two-gamma annihilation of

thermal energy positrons incident on atomic hydrogen: A, F (θ) obtained using

the accurate variationally determined wave function; B, F (θ) obtained in the

Born approximation.

The probability that the annihilating electron–positron pair has a mo-

mentum p is proportional to

Γ(p)=



|exp(−ip ·r

)Ψ(r

, r

,...,r

Z+1

)δ(r

−r

···dr

Z+1

(6.18)

In many determinations of angular correlation functions for positrons

annihilating in gases, only the angle between the two gamma-rays as

projected onto a given plane is measured; therefore, in order to make

comparison with experiment, the theoretical angular correlation function

for the two gamma-rays should be integrated with respect to two compo-

nents of the momentum to give the distribution function,

F (θ)=



Γ(p

= mcθ, p

) dp

. (6.19)

As an example, the angular distribution function for positrons annihi-

lating in atomic hydrogen, obtained using the accurate variational wave

function for zero energy positron–hydrogen scattering described in section

274 6 Positron annihilation

3.2 (Humberston and Wallace, 1972), is shown in Figure 6.4. Also shown

there is the distribution function obtained using the Born approximation,

in which neither the positron nor the atomic wave function is modiﬁed

by the interaction. This latter curve therefore represents the momentum

distribution of the electron in the undistorted hydrogen atom. The distri-

bution function for the accurate wave function is narrower than that for

the undistorted case because the positron attracts the electron towards

itself and away from the nucleus, thereby enhancing the probability of

low values of the momentum of the pair.

Instead of measuring the angular correlation between the two annihi-

lation gamma-rays, the same information concerning the momentum

distribution of the annihilating pair can be obtained by measuring the

Doppler shift ∆E

in the energy of either of the gamma-rays. This shift is

related to the angle, π − θ, between the gamma-rays by ∆E

= mc

θ/2,

so that the energy in keV of either of the gamma-rays is related to

the angle θ in milliradians by E

= 511(1 + θ/2) keV. This Doppler

shift technique has recently been successfully employed by Surko and

coworkers using thermalized positrons annihilating in a Penning trap (see

subsection 6.3.5).

6.2 Experimental details

1 The traditional positron lifetime method – analysis and

observables

The basic principle behind positron lifetime spectroscopy is simple. A

timing signature is obtained when a positron is emitted from a radioac-

tive isotope and also when it annihilates. By measuring the diﬀerence

between these times for 10

–10

positrons, a lifetime spectrum is obtained

from which various annihilation rates and associated parameters can be

derived. Schematic illustrations encapsulating the methodology of the

most commonly encountered positron lifetime spectrometry were shown

in Figure 1.4. The technical level of the discussion in subsection 1.3.1 is

suﬃcient for the present purposes. Further details can be found in the

original paper of Coleman et al. (1974), who developed a system to fa-

cilitate precision measurements of annihilation parameters in low density

gases. A concise summary of techniques relevant to positron annihilation

in gases was given by Griﬃth and Heyland (1978). The overall design of

lifetime apparatus has not changed signiﬁcantly since then, though there

have been advances in the commercially available timing electronics and

in data storage and analysis.

Lifetime spectra obtained for the noble gases argon and xenon are

shown in Figures 6.5(a) and (b) respectively. These spectra serve to

6.2 Experimental details 275

Fig. 6.5. Examples of

positron lifetime spectra for (a) argon and (b) xenon

gases. The argon data are for a density of 6.3 amagat at 297 K. The channel

width is 1.92 ns. In (a), (i) shows the raw data, (ii) sho

ws the signal with

background removed, (iii) shows the free-positron component and (iv) shows the

ﬁtted ortho-positronium componen

t. In (b), the spectrum for xenon is for room

temperature and 9.64 amagat and has a channel width of 0.109 ns. The inset

shows the fast components as extracted and discussed by Wright et al. (1985).

illustrate both the type of data analysis applied and the parameters

which can be derived. The argon spectrum shows the raw data obtained,

including the background contribution. Full discussions of the sources of

such background events can be found in the works of Coleman, Griﬃth

and Heyland (1974) and Coleman (1979).

Once the background is subtracted, the component of the spectrum

due to the annihilation of ortho-positronium is usually visible (see Fig-

ure 6.5(a), curve (ii) and the ﬁtted line (iv)). The analysis of the spectrum

can now proceed, and a number of diﬀerent methods have been applied

to derive annihilation rates and the amplitudes of the various compo-

nents. One method, introduced by Orth, Falk and Jones (1968), applies

a maximum-likelihood technique to ﬁt a double exponential function to

the free-positron and ortho-positronium components (where applicable).

Alternatively, the ﬁts to the components can be made individually, if their

decay rates are suﬃciently well separated, by ﬁtting to the longest compo-

nent (usually ortho-positronium) ﬁrst and then subtracting this from the

276 6 Positron annihilation

signal data to leave the free-positron signal. The main assumption built

into this, as in other methods of analysis, is that the ortho-positronium

component can be described by a single exponential back to zero time.

The shoulder width of the positron lifetime spectrum corresponds to

the annihilation of free positrons as they slow down through the en-

ergy interval from just below the positronium formation threshold energy

towards thermal energies. The region of the spectrum in which both

the ortho-positronium and free-positron components are exponential is

usually termed the equilibrium region, since the measured annihilation

rates no longer vary with time. From the discussion given earlier this

does not necessarily mean that the positrons (or ortho-positronium) have

thermalized, only that the associated annihilation parameters, Z

eﬀ

 for

the positrons and 

eﬀ

 for ortho-positronium, see equation (6.20) and

section 7.3, do not have an observable time dependence.

A second method used to analyse positron lifetime spectra is based on

the POSITRONFIT programme (see Kirkegaard, Pederson and Eldrup,

1989). Here, too, the individual components of the spectrum are assumed

to be of exponential form, but the ﬁt also contains a resolution function

which can be a sum of Gaussians and which is normally measured using a

well-characterized reference sample. This procedure has not been widely

applied in gas lifetime studies because, in most cases, the presence of a

large peak in the spectrum at short times, arising

from positrons annihi-

lating in the walls of the chamber, precludes a detailed analysis in this

time region. This feature, which is commonly referred to as the ‘prompt

peak’, is particularly visible in the xenon spectrum shown in Figure 6.5(b).

The analyses described above can be applied directly to the equilibrium

region of a lifetime spectrum. However, in atomic gases, where slowing

down below the positronium formation threshold is by elastic collisions

only, the positron speed distribution y(v, t) varies relatively slowly with

time. Consequently the annihilation rate also varies slowly with time.

From Figures 6.5(a) and (b) the existence of a non-exponential, or so-

called shoulder, region close to t = 0 is evident, and the analysis of this

region must be treated separately, as outlined below. Further details of

the shape and length of the shoulder can be found in subsection 6.3.1

below.

We now consider the parameters, listed below as (i)–(x) (Heyland et al.,

1982), which can be derived from analysis of a gas lifetime spectrum.

(i) The total number of signal events is N

, which includes gas events

and also, if applicable, events due to annihilation in the source and

in the chamber walls (which make up the bulk of the prompt peak).

(ii) Using the method(s) of analysis described earlier, the number of

events in each of the resolvable components of the spectrum can

6.2 Experimental details 277

be derived by back-extrapolation of the ﬁtted component to t =0.

(This is not applicable to the noble gases, which exhibit a non-

equilibrium shoulder, but it can be applied to molecular gases where

there is usually no such feature.) This can yield the numbers of

ortho-positronium and free-positron events, N

o-Ps

and N

respec-

tively, plus the number N

due to any other (faster) components,

if present. In fact N

o-Ps

may have to be corrected to allow for any

diﬀerence in the detection eﬃciencies of ortho-positronium and free

events that arises from their diﬀerent annihilation modes. This is

important because it allows absolute positronium fractions to be

determined; see (vi) below and Coleman et al. (1975a).

(iii) The number of para-positronium events which contribute to the gas

spectrum but which cannot normally be resolved because of the

presence of the prompt peak is assumed to be N

/3, based upon

the spin statistics which govern the relative formation probabilities.

Hence the total number of positronium events is N

=4N

o-Ps

/3.

(iv) The total number of gas events, N

, is then deﬁned by N

o-Ps

/3+N

+ N

(v) The quantity G = N

, the gas fraction, can be deduced and used

to compare the stopping power of diﬀerent gases for

particles.

Furthermore, an unexpectedly low measured value of G could indi-

cate the presence of fast components in the spectrum which cannot

be resolved from the prompt peak.

(vi) The positronium fraction F is one of the most widely quoted observ-

ables in positron lifetime work and it can be deﬁned as F = N

(vii) A mean value, λ

, for the ortho-positronium decay rate can be

determined from the equilibrium portion of the spectrum. This

represents the sum of the self-annihilation rate,

o-Ps

, and that due

to quenching in collisions with the surrounding medium. Thus λ



is usually written as

λ

 =

o-Ps

+ qρ

o-Ps

+0.804ρ

eﬀ

 (µs

−1

), (6.20)

where ρ was deﬁned in equation (6.3), q is the so-called quenching

coeﬃcient and, following Fraser (1968), 

eﬀ

 is a measure of the

eﬀective number of electrons per atom or molecule responsible for

quenching. All averages are over the positronium speed distribution

at annihilation. Values of 

eﬀ

 are discussed in Chapter 7, which

deals with positronium and its interactions with other systems,

278 6 Positron annihilation

along with the range of validity of equation (6.20) in various gaseous

media.

(viii) As mentioned above, the time spectrum for free-positron annihi-

lation, S

(t), can be obtained from the total gas spectrum G(t) once

the ortho-positronium component has been ﬁtted. The subtraction

of the assumed exponential ortho-positronium component can be

written as

(t)=G(t) −λ

N

o-Ps

exp(−λ

t). (6.21)

The instantaneous free-positron annihilation rate is particularly use-

ful where it is time dependent (i.e. on the shoulder region); it is

deﬁned as

λ

(t) =

(t)

∞



) dt



=0.201ρZ

eﬀ

(t) (µs

−1

), (6.22)

where Z

eﬀ

(t) is deﬁned by equation (6.11). At the end of the

shoulder region Z

eﬀ

(t) tends to Z

eﬀ

, the equilibrium value given

by equation (6.16). The experimental value of Z

eﬀ

(t) can be

compared with the theoretical speed-average of the calculated values

of Z

eﬀ

(v), equation (6.11), the speed distribution being obtained by

solving the diﬀusion equation (6.12).

(ix) The shoulder width is expressed in density-independent terms as

ρ, usually given in ns amagat, where τ

was deﬁned by Paul and

Leung (1968) as the time for which

Z

eﬀ

(τ

) = Z

eﬀ

−0.1∆Z, (6.23)

with ∆Z = Z

eﬀ

−Z

min

, where Z

min

 is the minimum value of

Z

eﬀ

(t) on the shoulder region; see Griﬃth and Heyland (1978) for

an example.

(x) If there are any fast components in the spectrum which can be

resolved from the prompt peak, as, e.g. in the case of xenon, shown

in Figure 6.5(b), then their presence can be revealed if Z

eﬀ

(t) rises

abruptly before the prompt peak has been reached. The spectrum

of such components can only be deduced by subtraction of the

free-positron component, which requires assumptions to be made

concerning the shape of the shoulder.

2 The positron-trap method for positron–gas studies

Since the mid-1980s a new and radically diﬀerent method of studying

positron annihilation in gases has been developed by Surko and coworkers.