Charlton M., Humberston J.W. Positron Physics

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7.1 Fundamental studies with the positronium atom 309

agreement with the experimental value of Asai, Orito and Shinohara

(1995); see below.

In the early 1970s theory and experiment seemed to be converging on a

value for the ortho-positronium decay rate,

o-Ps

, of approximately 7.24

µs

−1

. However, this was called into doubt later in that decade by the

work of Gidley and colleagues (Gidley, Marko and Rich, 1976; Gidley and

Zitzewitz, 1978) who obtained values below 7.1 µs

−1

. This discrepancy

was partly resolved by the calculations of Caswell, Lepage and Sapirstein

(1977) and Caswell and Lepage (1979), and the experimental measure-

ment of Griﬃth et al. (1978b). However, as more accurate measurements

were performed by the Michigan group during the next ten years or so,

including some in vacuum using a positron-beam technique, a small but

signiﬁcant discrepancy with theory arose again. This would now appear

to be resolved by the aforementioned measurements of Asai, Orito and

Shinohara (1995). A similar conclusion had previously been obtained by

Hasbach et al. (1987). These authors created positronium in vacuum in a

manner similar to that developed by Gidley and Zitzewitz (1978) and lat-

terly applied by Nico et al. (1990), but they used a very diﬀerent counting

technique. Asai, Orito and Shinohara (1995) formed ortho-positronium

in low density silica powder, and derived a value for

o-Ps

using a novel

method to account for pick-oﬀ annihilation. We describe some of these

measurements below.

The apparatus used by Westbrook et al. (1987, 1989) is shown in

Figure 7.1 and consists of a cylindrical gas chamber placed between the

pole pieces of an electromagnet. The latter was used to provide a ﬁeld of

0.68 T across the interaction region to increase the signal rate by causing

all positrons with a forward momentum component to follow helical paths

through the region viewed by the gamma-ray detectors. Although this

ﬁeld mixed the m = 0 substates, it did not alter the decay rate of the

m = ±1 ortho-positronium states.

The β

particles were derived from a

Na source deposited onto a

thin plastic scintillator coupled by a light pipe to a phototube. This

provided start signals for the timing sequence with high eﬃciency. The

stop was furnished by the annihilation gamma-rays detected using two

semi-annular scintillators surrounding the gas chamber. The combined

detection eﬃciency for the three-gamma ortho-positronium decay was

found to be in the range 25%–50%. Tungsten annuli inside the chamber

shielded the stop detector from the

Na source and from annihilations

on the opposite walls of the chamber.

Gas could be admitted to the chamber through the tubing shown,

which also served for evacuation, and the static gas sample was pumped

out, ﬂushed and recharged on a daily basis. Absolute pressures and

temperatures were recorded every hour. Various other tests were made

310 7 Positronium and its interactions

Fig. 7.1. Schematic illustration of the positronium formation chamber and

detector arrangement used by Westbrook et al. (1987, 1989). Reprinted from

Physical Review A40, Westbrook et al., Precision measurement of the ortho-

positronium vacuum decay rate using the gas technique, 5489–5499, copyright

1989 by the American Physical Society.

for gas leaks, contaminants from outgassing and for the thoroughness of

preparation of gas mixtures (used in the cases of N

and neon buﬀer

gases); the gas densities were computed from the relevant measurements

with appropriate virial coeﬃcient corrections. The electronics used for

this experiment incorporated fast and slow timing systems, each with its

own deliberately imposed dead time. Details are given by Westbrook

et al. (1989).

In common with all recent measurements of

o-Ps

, the raw data were

corrected for background and then carefully analysed (see also e.g. Griﬃth

et al., 1978b; Hasbach et al., 1987). Westbrook et al. (1989) used data

from the spectrum which corresponded to times between, typically, 180

ns and 930 ns, the exponential ﬁt being repeated at regular intervals out

from the 100 ns point in steps of 8–10 ns. Two examples of the variation

of the ﬁtted decay rate with starting channel are shown in Figure 7.2 for

7.1 Fundamental studies with the positronium atom 311

Fig. 7.2. The left-hand boxes show the ﬁtted ortho-positronium decay rate

at two values of isobutane pressure, for various start times of the ﬁt to the

component. The right-hand plot shows the observed decay rates, and their

extrapolation to zero density, of Westbrook et al. (1989). The error bars on

the individual points are approximately equal to the thickness of the line.

isobutane gas. Note that the ﬁtted decay rate appears to settle down

to a value independent of the starting channel by approximately 180 ns.

The ﬁnal ﬁtted decay rates are also shown for four gases, together with

the extrapolations to zero gas density. The ﬁnal result of 7.0514 ± 0.0014

µs

−1

is more than six standard deviations above the best theoretical value.

Support for these extrapolations also came from ‘joining’ the higher

pressure points taken by Westbrook et al. (1989) for neon and N

to data

taken much earlier for the same gases by Coleman et al. (1976a), using a

completely diﬀerent apparatus and data reduction technique, and over a

range of much higher densities. Very good agreement was found with the

extrapolations to zero density deduced from the ‘low density’ results.

Westbrook et al. (1989) considered a number of possible systematic

eﬀects which may have aﬀected their data and caused the extrapolated

value of

o-Ps

to be higher than the theoretical prediction. The two most

diﬃcult possibilities to account for were those related to the production of

excited state positronium, Ps

∗

, in the gas (see section 4.5 for a discussion

of investigations of this phenomenon by Laricchia et al., 1985) and those

related to positronium thermalization (see section 7.4). At the time both

were ruled out, but subsequent work on the latter eﬀect (Skalsey et al.,

1998) has shown that the positronium was not completely thermalized, so

that the energy dependence, or equivalently the temperature dependence,

312 7 Positronium and its interactions

of the collisional quenching rate becomes an important parameter. The

implications for possible corrections to the result of Westbrook et al.

(1989) have not yet been ﬁnalized, although indications are that they will

probably bring the value into better agreement with that determined from

a study involving positronium production in vacuum, which is described

below.

Asai, Orito and Shinohara (1995) used silica powder as the positronium-

forming medium; they developed a technique to take account of pick-oﬀ

annihilation of the positronium in collisions with the grains, thereby

circumventing the usual extrapolation to zero density of the powder (or

gas) inherent in most other work. Their experimental method consisted of

the usual timing arrangement, but supplemented by a detector to measure

the energy spectrum of the gamma-rays arising from annihilation of the

positronium. As described in more detail in section 7.2 below, pick-oﬀ pro-

cesses lead to the emission of two gamma-rays, each with 511 keV energy,

in contrast to the three-gamma-ray vacuum decay of ortho-positronium,

which has the continuous energy distribution illustrated in Figure 1.3. By

using this energy distribution and a Monte Carlo simulation to allow for all

gamma-ray interactions, both in the detector and the materials making up

the apparatus, Asai, Orito and Shinohara (1995) were able to show that

the expected and measured three-gamma-ray distributions were nearly

indistinguishable. The ratio of the three-gamma-ray and two-gamma-ray

events could thus be derived from the measured energy spectra and then

time-correlated with the signal from the trigger detector to yield the ratio

(t)/

o-Ps

, where λ

(t) is the time-dependent pick-oﬀ annihilation

rate. Once this parameter was derived, it could be used to ﬁt the time

spectrum, which had the form

N(t)=N

exp



−

o-Ps





1+λ



o-Ps







. (7.3)

Background contributions and relative detection eﬃciencies had to be

taken into account also; see Asai, Orito and Shinohara (1995). The ﬁnal

result, 7.0398 ± 0.0025 (stat.) ±0.0015 (sys.) µs

−1

, obtained for

o-Ps

is in agreement with the latest theoretical value but diﬀers from the gas

extrapolation result from Michigan. Asai, Orito and Shinohara (1995) of-

fer some speculations as to potential inaccuracies in the gas extrapolation

method but, given the new results on positronium thermalization and

related phenomena (see section 7.4), it is not appropriate to add further

comment here. Furthermore, the results for

o-Ps

from the Michigan gas

experiments have, to date, been validated by experiments performed by

the same group in vacuum using a low energy positron beam.

7.1 Fundamental studies with the positronium atom 313

Fig. 7.3. Schematic illustration of the timed and gated slow positron beam used

by Nico et al. (1990) to measure the vacuum decay rate of ortho-positronium.

Reprinted from Physical Review Letters 65, Nico et al., Precision measurement

of the ortho-positronium

decay rate using the vacuum technique, 1344–1347,

The Michigan apparatus (Nico et al., 1990), shown in Figure 7.3, con-

sisted of a time ‘tagged’ positron beam, in which positronium was formed

in a cavity lined with magnesium oxide. The primary low energy positrons

were focussed onto a nickel foil remoderator, and secondary electrons

liberated in this process were detected by the CEMA and used to start

the timing sequence. This signal also opened an electrostatic gate to

allow only the timed positrons to enter the cavity, thereby eliminating

background from the untagged remoderated beam (85% of the total). A

quarter of the positrons which entered the cavity at 700 eV formed positro-

nium on the magnesium oxide surface, which also served to minimize

quenching in collisions with the wall. The annihilation gamma-rays were

detected by two semi-annular fast plastic scintillation counters arranged

around the cavity. Details of the electronics used, and analysis of the

results, were given by Nico et al. (1990).

Systematic eﬀects arising from the disappearance of ortho-positronium

through the cavity entrance aperture, and the rate of annihilation by

collisions with the cavity walls, were taken into account by expressing the

measured annihilation rate as

λ =

o-Ps

+ c



/S)ν + P

ν, (7.4)

where S is the cavity surface area, A



is the eﬀective area of the cavity

entrance, ν is the collision rate of the ortho-positronium with the wall, on

314 7 Positronium and its interactions

which it has a probability P

of annihilation, and c

is the probability that

a gamma-ray from ortho-positronium which has escaped from the cavity

will not be detected. For a uniform distribution of ortho-positronium

in the cavity, Nico et al. (1990) discussed the modiﬁcation of the actual

physical area of the aperture to give A



in terms of two parameters, one of

which accounts for the non-uniform particle density and the other for the

random disappearance of positronium through an aperture of non-zero

thickness. The collision rate with the walls is given by ν = vS/(4V ),

where v is the average speed of the positronium and V is the volume of

the cavity. Thus, equation (7.4) can be rewritten as

λ =

o-Ps



v

vS

, (7.5)

and

o-Ps

can then be obtained by a two-variable extrapolation in A



and S/V . Figure 7.4 shows the results of these extrapolations. The

two intercepts, 7.0497 ± 0.0013 µs

−1

and 7.0482 ± 0.0015 µs

−1

, are in

good accord but the theoretical value, shown on each plot for comparison,

is around ﬁve standard deviations lower. The possibility that various

systematic eﬀects inﬂuence the results, most notably the formation at

the surface of excited state positronium, was considered, though all were

discounted as having a negligible eﬀect on the ﬁnal values.

It should be clear from the preceding discussion of the experimental

situation that there are still unresolved issues that warrant further work

on the measurement of

o-Ps

and on consideration of the related experi-

mental systematic errors.

In addition to the decay rate for the triplet state of positronium, it

is important to consider the rate for the singlet state, although, as

mentioned above, its 125 ps lifetime precludes a direct determination

p-Ps

at present. This is mainly due to the diﬃculty in isolating

the para-positronium component from the signal obtained from other

rapid positron annihilation mechanisms. Other techniques, however,

have been successfully applied, the ﬁrst determination being by Theriot

et al. (1970); they derived a value 7.99 ± 0.11 ns

−1

from the width

of the radio-frequency resonance in a measurement of the positronium

ground state hyperﬁne splitting (see subsection 7.1.2 below). The latest

theoretical results are 7989.5 µs

−1

(Khriplovich and Yelkhovsky, 1990)

and 7986.7 µs

−1

(Caswell and Lepage, 1979), the diﬀerence being due to

a discrepancy in the calculated coeﬃcient of the α

ln α

−1

term.

The most recent experimental determination of

p-Ps

is that of Al-

Ramadhan and Gidley (1994). The apparatus and analysis techniques

are similar to those of Westbrook et al. (1989) and will therefore not be

described here. Their method used the eﬀect of singlet–triplet mixing

in a static magnetic ﬁeld (Gidley et al., 1982); this allowed

p-Ps

to be

7.1 Fundamental studies with the positronium atom 315

Fig. 7.4. Extrapolations of the measured ortho-positronium decay rates in A



and S/V (see text). Reprinted from Physical Review Letters 65, Nico et al.,

Precision measurement of the ortho-positronium decay rate using the vacuum

extracted from



-Ps

, the decay rate of the mixed m = 0 states. Rich

(1981, and references therein) showed that the perturbed ortho-positronium

vacuum decay rate can be written as



-Ps

=(1− b

)

o-Ps

+ b

p-Ps

, (7.6)

where b = y

/(1 + y

), y = x/[1+(1+x

)

1/2

], x =2g



B/h∆ν

hfs

≈ B/3.65 tesla, g



= g(1 − 5α

/24) and ∆ν

hfs

is the ground state, zero

316 7 Positronium and its interactions

ﬁeld, hyperﬁne frequency interval. A typical value for



-Ps

is 30 µs

−1

a ﬁeld of 0.4 T, so that

p-Ps

can be determined at a known ﬁxed ﬁeld

by measuring this quantity and assuming values for

o-Ps

and ∆ν

hfs

The experiments were performed at two values of the magnetic ﬁeld,

0.375 T and 0.425 T, and at various densities of N

gas with small

admixtures of isobutane to quench the free-positron component (see sub-

section 6.3.2). Al-Ramadhan and Gidley (1994) derived a quantity Λ(ρ)

from their measured values of λ



-Ps

and λ

o-Ps

, for the mixed and unmixed

ortho-positronium states respectively, at a gas density ρ given by

Λ(ρ)=[λ



-Ps

(ρ) − λ

o-Ps

(ρ)]/b

o-Ps

, (7.7)

where b, deﬁned above, parameterizes the degree of mixing induced by

the magnetic ﬁeld. The value of

o-Ps

was taken to be 7.0482 ± 0.0016

µs

−1

(Nico et al., 1990). In the absence of a collisional spin-exc

hange

mechanism the quenching coeﬃcients for the mixed and unmixed ortho-

positronium were expected to be identical, so that Λ(ρ) could be taken

as a measure of

o-Ps

. Thus, the measurements could be biassed towards

higher gas densities for increased statistical accuracy. A plot of Λ(ρ)

versus ρ is shown in Figure 7.5. The straight line which was ﬁtted to

these data had a slope consistent with zero and provided experimental

conﬁrmation of the equality of the perturbed and unperturbed ortho-

positronium quenching rates in the gas.

The zero-density intercept from Figure 7.5 is 7990.3 ± 3.1 µs

−1

, but

a simple weighted average of the data gave 7990.9 ± 1.0 µs

−1

, with the

error due to statistics only. This procedure was justiﬁed by obtaining

independent information on the equality of the quenching coeﬃcients

for perturbed and unperturbed ortho-positronium by searching for a dif-

ference in λ

o-Ps

(ρ) with the ﬁeld on and oﬀ. No signiﬁcant eﬀect was

found, although an extra contribution to the overall error was thereby

incorporated. Other errors included in the ﬁnal analysis were due to

time calibration and linearity of the lifetime spectra, the magnetic ﬁeld

measurement, averaging over the magnetic ﬁelds sampled by the ortho-

positronium (necessary because of slight positional variations) and a small

correction for the possibility that there was more than one positronium

atom in the chamber following the start of the timing system. The ﬁnal

result was given by Al-Ramadhan and Gidley (1994) as

p-Ps

= 7990.9

± 1.7 µs

−1

, which, with an error of 215 ppm, is of similar accuracy

to the precision measurements of

o-Ps

and capable of distinguishing

between the two calculated values of the coeﬃcient of the α

ln α

−1

term

in equation (7.2): the result ﬁnds in favour of the later work, that of

Khriplovich and Yelkhovsky (1990).

7.1 Fundamental studies with the positronium atom 317

Fig. 7.5. Plot of the measured Λ(ρ), as deﬁned by equation (7.7), in N

–

isobutane mixtures. Data were taken at magnetic ﬁelds of 0.375 T (×) and 0.425

T(◦). Reprinted from Physical Review Letters 72, Al-Ramadhan and Gidley,

New precision measurement of the decay rate of singlet positronium, 1632–1635,

2 Spectroscopic properties

Experimentally, the ﬁrst property of positronium to be investigated was

the ground state hyperﬁne structure (hfs), as described in the excellent

review of Rich (1981). The historic measurements of Deutsch and Dulit

(1951) ﬁrst veriﬁed the necessity of incorporating the virtual annihilation

term into the calculation of the energy separation, and soon afterwards

Deutsch and Brown (1952) made a relatively precise measurement of

∆ν

hfs

using a technique employed later in more accurate experiments

by groups at Brandeis and Yale Universities. The most recent values for

this quantity are 203.3875 ± 0.0016 GHz (Mills and Bearman, 1975) and

203.389 10 ± 0.000 57 ± 0.000 43 GHz (Egan, Hughes and Yam, 1977).

See also Hughes (1998), for a recent review. The above values can be

compared with the theoretical result (Karplus and Klein, 1952; Bodwin

and Yennie, 1978; Caswell and Lepage, 1979)

∆ν

hfs

4π



−



+ 2 ln 2



ln(α

−1

)+O(α

)



, (7.8)

which yields 203.400 GHz. Rich (1981) stated that uncalculated terms of

order α

would contribute around 7 MHz to this value if their coeﬃcients

were unity. This has been borne out by recent work; see e.g. Czarnecki,

Melnikov and Yelkhovsky (1999) and Pachucki and Karshenboim (1998).

318 7 Positronium and its interactions

Fig. 7.6. The energy levels of ground state positronium in a magnetic ﬁeld.

The value found is 203.392 01(46) GHz, around three standard deviations

from experiment. Here we will describe only the Yale experiment in

detail, since that of the Brandeis group is similar in many respects. A

summary of the latter can be found in the review of Berko and Pendleton

(1980). Note that the values quoted above were both corrected upwards

by Mills (1983c), who, acting on a suggestion of Rich (1981), properly

took into account the deviation, caused by the eﬀects of annihilation, of

the line shape from a true Lorentzian. Shifts of order (

p-Ps

/4π∆ν

hfs

)

≈ 10

−5

were found, resulting in corrections of 2.5 ppm and 21 ppm to

the values of Mills and Bearman (1975) and of Egan, Hughes and Yam

(1977) respectively.

An illustration of the principle behind the technique is given in Fig-

ure 7.6, which shows the ground state positronium energy levels in a

static magnetic ﬁeld. In the experiment a ﬁeld of around 0.7–1.0 T was

chosen, and transitions between the unperturbed m = ±1 states and the

m = 0 state were induced by applying a radio frequency magnetic ﬁeld

perpendicular to the static ﬁeld. For convenience, the magnitude of the

static ﬁeld was varied, rather than the radio frequency, in the search for