Charlton M., Humberston J.W. Positron Physics

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1.2 Basic properties of the positron and other positronic systems 9

Fig. 1.3. (a) The gamma-ray energy spectrum for the three-photon decay of

ortho-positronium. The broken curve is from the theoretical

work of Ore and

Powell (1949) whilst the dotted line shows the theory of Adkins (1983); the

solid line includes an O(α) QED correction. The experimental points are from

Chang et al. (1985). (b) Schematic gamma-ray energy spectra taken using a

high resolution detector under conditions where the fraction of positrons forming

positronium is 0% and 100% (e.g. Lahtinen et al., 1986).

and Ore and Powell (1949) respectively, and are given by

2γ







(1.5)

and

3γ





9π

(π

− 9)



, (1.6)

where α ≈ 1/137.036 is the ﬁne structure constant. Inspection of these

two expressions reveals that, owing to the extra power of α and to the

numerical factor in equation (1.6), the two-gamma-ray annihilation rate

is much greater than that for the three-gamma-ray process. For n

= 1, it is found that Γ(1

) ≈ 8 GHz whereas Γ(1

) ≈ 7 MHz.

The lifetimes against annihilation of the 1

and 1

states, being the

reciprocals of their annihilation rates, are therefore around 1.25 ×10

−10

and 1.4×10

−7

s respectively. Further details of higher order contributions

to the annihilation rates can be found in the review of Rich (1981) and

in Chapter 7, where relevant experimental work is also described.

Considering the n

= 2 states (2

, and 2

, with J =

0, 1, 2), the S-state lifetimes display the n

scaling law of equations (1.5)

and (1.6). For a given value of n

the probability that the positron and

electron will be found very close together is much lower, and therefore the

10 1 Introduction

lifetime against annihilation is much greater for states with L = 0 than

for states with L = 0. Alekseev (1958, 1959) calculated the lifetime

against annihilation for positronium in the 2P states to be > 10

−4

which is several orders of magnitude greater than the mean life for optical

de-excitation. The actual lifetime of an excited state against annihilation

may therefore be determined mainly by the lifetime of the atomic transi-

tion. As an example, the 2P–1S transition has a characteristic lifetime of

3.2 ns, double the value for the corresponding transition in the hydrogen

atom. Therefore, instead of the positronium annihilating directly in a

2P state, it is far more likely to make an optical transition to a 1S

state, where annihilation will then take place rapidly at a rate given by

either equation (1.5) or equation (1.6), depending on the spin state. Note

that the prediction of equation (1.4), that annihilation from the 2

P and

P states is predominantly into two and three gamma-rays respectively,

only applies to direct annihilation. If the positronium ﬁrst undergoes the

optical 2P–1S transition, then the annihilation mode in the lower state is

determined by the quantum numbers of that state.

3 Other bound states involving positrons

The next most complex bound state after positronium, and one that has

now been observed (Mills, 1981), is the positronium negative ion, Ps

−

which consists of two electrons and a positron. This entity (and its charge

conjugate counterpart, the positive ion consisting of two positrons and an

electron) has a total spin S = 1/2 and a ground state conﬁguration of

. It has no long-lived excited states, but there are several autoionizing

resonant states. The most recent calculation of its binding energy with

respect to break-up into an electron and positronium gives 0.326 68 eV

(Ho, 1993; Frolov and Yeremin, 1989), and Ho (1993) obtained a value of

2.086 1222 ns

−1

for its annihilation rate, in agreement with the experimen-

tal result of Mills (1983b). This value is also close to the spin-averaged

annihilation rate of ground state positronium, i.e. one quarter of the rate

for the 1

state. Further discussion can be found in section 8.1.

Hylleraas and Ore (1947) ﬁrst showed that the complex involving two

electrons and two positrons, the positronium molecule Ps

, is bound, and

this was conﬁrmed by the more accurate work of Ho (1986a), Kinghorn

and Poshusta (1993) and Kozlowski and Adamowicz (1993). The later

calculations gave the binding energy with respect to break-up into two

positronium atoms as 0.435 eV, but a signiﬁcantly larger value, 0.573 eV,

has recently been obtained by El-Gogary et al. (1995). The system, with

total spin S = 0, has no bound excited states, but several autodissociating

states have been found (Ho, 1989b). Thus far, Ps

has not been observed

1.3 Basic experimental techniques 11

in the laboratory because the large instantaneous positron densities re-

quired for its formation have not yet been achieved.

A related, but somewhat less exotic, complex is positronium hydride,

PsH. This entity has been the subject of many theoretical studies, with

the most recent predicting a binding energy with respect to break-up

into hydrogen and positronium of approximately 1.067 eV (Ho, 1986b,

Ryzhikh and Mitroy, 1997). Experimental evidence for its existence has

been forthcoming from a recent study of positron–CH

collisions (Schrader

et al., 1992; see also Chapter 7), although its short lifetime against

annihilation, 0.5 ns, makes further experimentation diﬃcult. Again, the

system has no bound excited states, but it possesses a Rydberg series of

autodissociating resonances. Further discussion of these and other bound

systems containing positrons is given in sections 1.6 and 7.5.

The last bound state to be introduced here is antihydrogen, consisting

of a positron and an antiproton. From the CPT theorem this entity, which

is stable in vacuum, is expected to have spectroscopic properties identical

to those of atomic hydrogen. There is no evidence for the existence

of bulk antimatter in the universe, but antihydrogen has recently been

produced (Baur et al., 1996; Blanford et al., 1998), although at such

high kinetic energies that no further investigation of its properties was

possible. However, recent advances in the slowing down and trapping of

antiprotons (Gabrielse et al., 1990; Holzscheiter et al., 1996), described in

section 8.3, lend hope that the synthesis of very low energy antihydrogen

will be achieved in the not too distant future.

Although there are rather few true bound systems containing positrons,

numerous resonances are known to exist in the scattering of positrons

and positronium by various target systems. Many of these are Feshbach

resonances associated with the degenerate thresholds for excitation of

positronium or of a hydrogenic target, but others arise from the Rydberg

series of states of a positron ‘bound’ to the residual negative ion. For

example, in PsH, excited states of the system consisting of a positron in-

teracting with the H

−

ion have energies in the continuum for positronium–

hydrogen scattering, and they therefore manifest themselves as resonances

in positronium–hydrogen scattering. All these resonances have very nar-

row energy widths, and they cannot yet be resolved experimentally.

1.3 Basic experimental techniques

In this section we introduce three techniques frequently encountered in

positron physics, namely those used to measure annihilation lifetimes

and the Doppler broadening (or Doppler shift) and angular correlation

of the annihilation radiation. These techniques, or variants thereof, are

encountered throughout the rest of this work, and here we brieﬂy describe

12 1 Introduction

Fig. 1.4. (a) Schematic of a positron lifetime spectrometer; the star indi-

cates the

Na source and only one of the 0.511 MeV annihilation photons

is shown. Key: Sc, scintillator; CFD, constant fraction discriminator; TAC,

time-to-amplitude converter; MCA, multichannel analyser. (b) Simpliﬁed level

diagram of the

Na decay scheme. The β

fraction dominates that for electron

capture for this isotope.

the principles behind them and give illustrations of the apparatus and

methodology involved (see also Hautoj¨arvi and Vehanen, 1979).

1 Annihilation lifetimes

The basic operating principle of all traditional positron lifetime systems,

schematically illustrated in Figure 1.4(a), is to measure individual life-

times of many positrons by suitably processing the timing signals derived

from their birth and from their eventual annihilation with electrons in

the medium in which the positrons are diﬀusing. In the apparatus shown,

both the birth and annihilation signals are registered using gamma-ray

1.3 Basic experimental techniques 13

scintillation counters (see e.g. Knoll, 1989 for a general discussion). One

of the most commonly used sources of positrons in lifetime studies is

the

Na radioisotope, a simpliﬁed decay scheme for which is shown in

Figure 1.4(b). The β

branching ratio for this isotope is around 90% and

the positron emission is followed promptly by a gamma-ray of energy 1.274

MeV. This gamma-ray is used to register the positron’s birth by starting

the timing sequence. In addition,

Na has a conveniently long half-life

of around 2.6 years, and is commercially available in a form suitable for

many applications.

The annihilation gamma-rays, at energies of 511 keV and below (see

section 1.2 for a discussion of the various possible annihilation modes),

are registered by a second scintillation counter. The start and stop

signals are usually processed by a pair of discriminators, and the simplest

arrangement, shown in Figure 1.4(a), consists of two constant-fraction

discriminators, which combine good timing characteristics with the ability

to set upper and lower limits on the pulse height accepted by the instru-

ment. Thus, the higher energy ‘start’ gamma-ray can easily be selected.

After the insertion of an appropriate delay, to introduce a minimum ﬁxed

time between the start and stop signals, the pulses are fed to the inputs

of a time-to-amplitude converter. The output of this module, which is

proportional to the length of time between the start and stop signals, and

thus to the individual positron lifetime, is then stored using a multichannel

analyser. A lifetime spectrum is thereby built up, frequently containing

–10

events, from which various lifetimes, principally those due to free

positron and ortho-Ps annihilation, along with several other parameters,

can be extracted. Numerous examples of results obtained using this kind

of instrumentation are described in Chapters 6 and 7.

Another useful lifetime system, though less frequently encountered, is

the so-called β

–gamma system. A variant of this technique was used

in some of the pioneering measurements of positron–atom total collision

cross sections described in Chapter 2. The annihilation gamma-ray still

provides the stop signal, but the start signal is derived via the energy

deposited by the positrons as they traverse a thin (typically 0.1–0.3 mm)

scintillator. This method of start detection has a high eﬃciency, usually

around 50%, which permits the use of a relatively weak radioactive source,

resulting in a superior signal-to-background ratio.

Numerous technical descriptions of various aspects of lifetime apparatus

can be found (e.g. MacKenzie, 1983, and references therein), and a number

of sophisticated data analysis and ﬁtting procedures have been developed

to analyse the data collected (e.g. Coleman, Griﬃth and Heyland, 1974;

Coleman, 1979; Kirkegaard, Pedersen and Eldrup, 1989), but detailed

discussion of these topics is beyond the scope of the present treatment.

14 1 Introduction

Fig. 1.5. Schematic of a gamma-ray energy spectrometer for Doppler broaden-

ing studies. The signal from the detector pre-amp is processed by spectroscopy

and biassed ampliﬁers (SA and BA respectively) before being recorded in the

multichannel analyser (MCA).

Technical advances in these areas have frequently been documented in the

proceedings of the triennial series of Positron Annihilation conferences

(see the appendix), which can serve as a useful starting point for a

literature search.

2 Doppler shift or broadening of the annihilation radiation

In the frame of reference in which the centre of mass of an electron–

positron pair is at rest, the two gamma-rays arising from their annihilation

in a spin singlet state each have an energy of 511 keV, and they emerge in

opposite directions; i.e. the angle between the two directions is π radians.

However, the motion of the centre of mass creates a Doppler shift in the

gamma-ray energies as measured in the laboratory frame of reference,

and the angle between the gamma-rays is no longer π. When slowing

down in matter, most positrons thermalize before annihilation, and the

momentum of the centre of mass motion of an electron–positron pair is

therefore predominantly that of the electron. The shift in the energy of

the 511 keV line is given by

∆E

= mcv

cos φ, (1.7)

where v

is the speed of the centre of mass of the pair and φ is the angle

between the direction of motion of the centre of mass and that of one of the

emitted gamma-rays. Inserting typical values for v

(≈ 10

−1

) and

cos φ ( 1/

√

2), we ﬁnd that ∆E

≈ 1.2 keV. This is comparable to the

energy resolution of the commercially available high purity germanium

gamma-ray detectors which, as illustrated schematically in Figure 1.5,

are used in studies of this type. The ampliﬁcation system following the

detector is standard, usually consisting of a spectroscopy ampliﬁer and a

biassed ampliﬁer; this allows the widened annihilation line to be examined

1.3 Basic experimental techniques 15

Fig. 1.6. (a) Illustration (exaggerated) of the relationship between the momenta

and p

of the two annihilation photons, the angle θ between them and the

x-component of the electron momentum prior to annihilation. (b) Schematic

showing the principle of a two-dimensional ACAR apparatus. The source is

shown in the small rectangle above the sample, from which the two 0.511 MeV

gamma-rays are emitted.

in more detail. The gamma-ray energy distribution can then be stored in

a multichannel analyser and processed in various ways depending upon

the details of the study.

The widest ﬁeld of application of this technique is in investigations of

the solid state, where, in most cases, the geometry of the experiment and

the random nature of the direction of motion of the electron–positron

pairs means that the angle φ has a continuous distribution; consequently

the 511 keV gamma-ray line is Doppler broadened by an amount related

to the momentum distribution of the annihilating pair. This technique

has only rarely been applied to the study of the behaviour of positrons

and positronium in gases, although recent measurements of the Doppler

broadening obtained from the annihilation of very low energy positrons

conﬁned in a trap containing helium gas have been found to be in excellent

agreement with theoretical predictions (Van Reeth et al., 1996; see also

Chapter 6). Other studies using positron beams have also been reported in

which the geometry allows a narrow band of values of φ to be selected and

consequently a speciﬁc Doppler shift to be observed, as has been exploited

in studies of positronium formation (Chapter 4) and in the detection of

−

(Chapter 8).

16 1 Introduction

3 Angular correlation of annihilation radiation

The ﬁnal technique we describe in this section has been used to study the

behaviour of positrons and positronium in gases, in the latter case with the

aim of understanding thermalization phenomena and momentum transfer.

The method involves measuring θ, the small deviation from π radians in

the angle between the two annihilation gamma-rays. As mentioned above,

this deviation is a consequence of the centre-of-mass momentum of the

annihilating electron–positron pair. The relationship between θ and the

component of this momentum perpendicular to the direction of one of

the gamma-rays, which can be taken to be the x-component, p

,asin

Figure 1.6(a), is

θ = p

/(mc); (1.8)

this has a typical value of a few milliradians. An example of an an-

gular correlation of annihilation radiation (ACAR) apparatus to detect

such small angular deviations is shown in Figure 1.6(b). It consists of

a pair of two-dimensional position-sensitive gamma-ray detectors, each

typically located at a distance of 5 m from a radioactive source, which

is immediately adjacent to the sample being studied. The ﬁeld of view

of each detector is limited by lead collimating slits. Using this type of

arrangement, angular resolutions below one milliradian can be achieved

routinely. In media in which there is no preferred axis of symmetry (e.g.

in gases and liquids in the absence of external ﬁelds) it is not necessary

to use a two-dimensional system, although it can sometimes be justiﬁed

in terms of the improved count rate. Instead, the one-dimensional, or

long-slit, technique can be used, in which the position-sensitive detectors

are replaced by two single detectors, each with a long-slit collimator

placed in front of it, giving integration over one of the components of

the momentum. One of the detectors is ﬁxed, whilst the other is scanned

through the angle θ.

1.4 Slow positron beams

In this section we endeavour to describe the production of low energy,

or slow, positron beams through a discussion of the following topics:

the primary sources of high energy positrons and their slowing down,

or moderation, through implantation into solids; the subsequent diﬀusion

of the moderated positrons; the emission into vacuum of those positrons

which, prior to annihilation, reach a suitable surface; and their subsequent

manipulation to form a beam. It is hoped that this discussion will be of

particular beneﬁt to the non-specialist in illustrating some of the diﬃcul-

ties, limitations and new possibilities encountered when dealing with low

1.4 Slow positron beams 17

Fig. 1.7. Comparison of the energy spectrum of β

particles from a radioactive

source with that for moderated positrons.

energy positron beams. A similar discussion has been given by Beling

and Charlton (1987).

1 Introduction to positron moderation

In nature, high energy positrons are produced either as a result of nuclear

decay, usually of an artiﬁcially produced radioactive isotope such as

Na,

or by pair production from a photon of suﬃciently high energy. Except

near the threshold for the latter process, where the production cross

section is very low, the positrons are produced with a very large energy

spread (see Figure 1.7), typically of the order of MeV, which renders them

unsuitable for most experiments in atomic collision physics.

Table 1.1 shows some of the properties of various β

emitters which

have been used to create low energy beams. We adhere to the convention

18 1 Introduction

Table 1.1. Selected properties of some of the radioisotopes commonly used in

the creation of low energy positron beams

Isotope β

Branching Endpoint Half-life Typical

ratio energy (MeV) production

mechanism

Na 0.91 0.54 2.6 yr

Mg(d, α)

Co 0.15 0.47 70.8 d

Ni(n, p)

Cu 0.19 0.65 12.7 h

Cu(n, γ)

C 0.99 0.96 20.4 min

B(p, n)

of referring to a high energy positron

emitted in a nuclear decay as a

particle. Of particular note in table 1.1 are the end-point energies

max

, since the average β

energy, and hence the implantation depth

into matter, increases with the value of this parameter. The implantation

of energetic β

particles into solids has been studied for many years. A

fraction of the β

particles striking a material are back scattered, the size

of the fraction being dependent upon the atomic number of the target (e.g.

Mackenzie et al., 1973) and the thickness of the material (up to a certain

saturation value). The implantation proﬁle P (x) for those β

particles

not back scattered is usually expressed in terms of the depth x into the

target (or moderator) as

P (x)=µ

imp

exp(−µ

imp

x), (1.9)

where µ

imp

is the absorption coeﬃcient, which can be empirically ex-

pressed in terms of E

max

and the target density (see e.g. Brandt and

Paulin, 1977; Beling et al., 1987). The positrons lose most of their

kinetic energy in the material by ionizing collisions. In metals, where

there are eﬃcient energy-loss processes which can be excited by very low

energy positrons, annihilation of the positrons occurs predominantly after

their thermalization, which is typically reached a few picoseconds after

implantation. However, this is not true of all materials, counter-examples

being the rare gas solids, which are wide-band-gap insulators.

In solids the free positron lifetime τ lies in the approximate range

100–500 ps and is dependent upon the electron density. Following implan-

tation, the positrons are able to diﬀuse in the solid by an average distance

=(D

τ)

1/2

, where D

is the diﬀusion coeﬃcient. This quantity is

usually expressed in cm

−1

and is of order unity for defect-free metallic

moderators at 300 K (Schultz and Lynn, 1988). The requirement of

very low defect concentration arises because the value of D

is otherwise

dramatically reduced owing to positron trapping at such sites.