Charlton M., Humberston J.W. Positron Physics

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Fig. 7.7. Schematic of the Yale experiment to measure the hyperﬁne interval of ground state positronium. On the left there is

an overall view of the apparatus, whilst the right-hand ﬁgure shows the microwave cavity and gas cell in which the positronium

is formed.

320 7 Positronium and its interactions

Fig. 7.8. (a) An example of the Yale data for the determination of the positro-

nium hyperﬁne structure resonance. (b) Linear extrapolation to zero gas density

of the measured hyperﬁne interval.

the resonance. The location of this resonance is governed by the energy

diﬀerence between the perturbed and unperturbed triplet levels and is

given by

∆ν

hfs

[(1 + x

)

1/2

− 1], (7.9)

where x was deﬁned in subsection 7.1.1.

Schematic illustrations of the Yale experiment are given in Figure 7.7,

which shows an overall view and also a view of the microwave cavity and

the source and gas region. Positrons emitted from a

Na source form

positronium in a suitable low-pressure gas (e.g. helium or neon). The

volume in which the gas is conﬁned, which is also the microwave cavity, is

located in the ﬁeld produced by an electromagnet. The quantity measured

is the number of 511 keV gamma-rays detected using pairs of oppositely

situated scintillation counters. Rich (1981) noted that on passing through

the resonance, the number of two-gamma-ray events increased by around

10%, and an example of the Yale data is shown in Figure 7.8(a). The

rise in the ‘background’ with increasing magnetic ﬁeld strength is due to

the small increase in oﬀ-resonance quenching of the m =01

state of

ortho-positronium.

Inspection of Figure 7.8(a) also reveals one of the major diﬃculties in

determining ∆ν

hfs

to a few ppm, namely that the natural line width is

large, approximately 6200 ppm, and extremely ﬁne line splitting must

be accomplished. This topic, and related systematics, were discussed in

detail by Rich (1981), who noted that the accord between the two groups,

who used diﬀerent data analysis techniques, is important conﬁrmation of

the reported accuracy.

7.1 Fundamental studies with the positronium atom 321

The other major uncertainty in the measurements arises from collisions

of the ortho-positronium with the buﬀer gas, the main eﬀect of which is

to lead to a shift in the value of ∆ν

hfs

. In fact, the dominant mechanism

involved is the attractive long-range van der Waals force, which tends to

increase the positron–electron separation in positronium, thus lowering

the splitting. This is shown explicitly in Figure 7.8(b) (Egan, Hughes

and Yam, 1977), where a linear extrapolation to zero gas density was

made (see also Mills and Bearman, 1975).

Laser spectroscopy of the 1S–2S transition has been performed by

Mills and coworkers at Bell Laboratories (Chu, Mills and Hall, 1984; Fee

et al., 1993a, b) following the ﬁrst excitation of this transition by Chu

and Mills (1982). Apart from various technicalities, the main diﬀerence

between the 1984 and 1993 measurements was that in the latter a pulse

created from a tuned 486 nm continuous-wave laser with a Fabry–P´erot

power build-up cavity, was used to excite the transition by two-photon

Doppler-free absorption, followed by photoionization from the 2S level

using an intense pulsed YAG laser doubled to 532 nm. Chu, Mills and Hall

(1984), however, employed an intense pulsed 486 nm laser to photoionize

the positronium directly by three-photon absorption from the ground

state in tuning through the resonance. For reasons outlined by Fee et al.

(1993b), it was hoped that the use of a continuous-wave laser to excite the

transition would lead to a more accurate determination of the frequency

interval than the value 1 233 607 218.9 ± 10.7 MHz obtained in the pulsed

486 nm laser experiment (after correction by Danzmann, Fee and Chu,

1989, and adjustment consequent on a recalibration of the Te

reference

line by McIntyre and H¨ansch, 1986).

The experimental arrangement of Fee et al. (1993a, b) is shown schemat-

ically in Figure 7.9. A 15 ns burst of approximately 10

low energy

positrons, produced by the bunched output of a microtron-based beam

(Mills et al., 1989b), was incident upon a heated single crystal of alu-

minium. Here it produced positronium in vacuum, mainly by the thermal

desorption of surface-trapped positrons, as described in subsection 1.5.3.

The background at the CEMA detector, which registered the laser-ionized

positrons, was reduced by pre- and post-skimmers. The pre-skimmer was

positioned so that ionized positrons would be extracted only from the

region below the target to where the neutral 2S positronium could migrate

and still interact with the YAG laser. The ionized positrons were drifted

in the magnetic ﬁeld to the CEMA, where they could be distinguished

from background by their time of ﬂight relative to the pulsed YAG laser.

Figure 7.10 shows the positronium 1S–2S resonance curve, along with

the Te

resonance line used for calibration. A ﬁt to these data yielded

a value 1 233 607 216.4 ± 3.2 MHz for the frequency interval (Fee et al.,

1993a, b). This is in good accord with the value 1 233 607 221.7 MHz

322 7 Positronium and its interactions

Fig. 7.9. The positronium source and laser interaction region used by Fee and

coworkers. A magnetic ﬁeld of 100 G (0.01 T) guides the incident beam onto

the target and also transports the positrons liberated from the photoionized 2S

positronium to the detector. Reprinted from Physical Review Letters 70,Fee

et al., Measurement of the positronium 1

–2

interval by continuous wave

Society.

(Fell, 1992; Khriplovich, Milstein and Yelkhovsky, 1992) obtained from

the theoretical expression for the frequency:

=[E(2

) − E(1

)]h

−1

= cR

∞



−

719

1536



161

960

−

ln 2 −

ln α

−1

−

ln 2.811 77 +

ln 2.984 129



−

ln α

−1



(7.10)

The term of order α

has been calculated by Pachuki and Karshenboim

(1998) and has shifted ν

downwards by 0.7 MHz (with an estimated

error of 1.0 MHz) from the value quoted above.

The ﬁrst crude spectroscopic measurement performed on excited state

positronium was the identiﬁcation of the 243 nm Lyman-α radiation

emitted in the 2P–1S transition. The ﬁrst observation of this line was

7.1 Fundamental studies with the positronium atom 323

Fig. 7.10. The positronium 1

–2

resonance curve, together with the Te

calibration line. Details of the model ﬁt to the experimental data can be found

in Fee et al. (1993b).

obtained in the slow-positron-beam studies of Canter, Mills and Berko

(1975), which followed two decades of unsuccessful attempts, by a variety

of workers, based upon traditional methods, e.g. positrons stopping in

solids and high density gases. These measurements were catalogued by

Day (1993) and need not concern us further here.

The basic principle of the experiment of Canter, Mills and Berko (1975)

was to collide low energy positrons with a surface and to look for coin-

cidence between a Lyman-α photon and a delayed gamma-ray arising

from the subsequent annihilation of a 1

S positronium. The presence of

the Lyman-α signal was veriﬁed by the use of three interference ﬁlters

with pass bands centred on, just above, and just below, 243 nm. An

enhanced coincidence rate was found with the 243 nm ﬁlter in place. A

similar Lyman-α gamma-ray technique has been adopted by all subse-

quent workers in this ﬁeld (e.g. Laricchia et al., 1985; Hatamian, Conti

and Rich, 1987; Ley et al., 1990; Schoepf et al., 1992; Steiger and Conti,

1992; Hagena et al., 1993; Day, Charlton and Laricchia, 2000).

To date, it has been found that the positronium formation potential,

see equation (1.14), is positive for n

> 1, so that excited state positro-

nium emission by a work function process is forbidden. Only positrons

which reach a surface epithermally (which occurs predominantly at low

impact energies where they do not penetrate far into the solid) may

324 7 Positronium and its interactions

Table 7.1. Summary of the available theoretical and experimental data for

the 2

–2

transitions of positronium. Key: experiment, (a) Hagena et al.

(1993), (b) Hatamian, Conti and Rich (1987) (the ﬁrst-quoted errors are statis-

tical whilst the second-quoted are systematic); theory, (c) Pachuki and Karshen-

boim (1998)

Transition Transition frequencies (MHz)

(a) (b) (c)

→ 2

8 624.38 ± 0.54 ± 1.40 8 619.6 ± 2.7 ± 0.9 8 626.87

→ 2

13 012.42 ± 0.67 ± 1.54 13 001.3 ± 3.9 ± 0.9 13 012.58

→ 2

18 499.65 ± 1.20 ± 4.00 18 504 ± 10.0 ± 1.7 18 498.42

form positronium with n

> 1. Such positronium atoms are probably

emitted from the surface, with a range of kinetic energies which reﬂects

the energy of the positrons as they return to the surface, and also the

energy dependence of the capture process; it is expected to be several eV

wide. This energy spread inﬂuences spectroscopic investigations, owing

to the Doppler eﬀect. Nevertheless, frequencies for the 2

–2

(J =

0, 1, 2) transitions have been measured, and table 7.1 summarizes the

available experimental and theoretical data.

Since the principle of the experiments is the same in all cases, we will

restrict the discussion to the work of Hagena et al. (1993); a schematic

illustration of the apparatus they used to determine the 2

–2

tran-

sitions is shown in Figure 7.11(a). Positrons produced by pair production

at the Giessen Electron Linac (Faust et al., 1991) were, after moderation,

incident at 100 eV upon a molybdenum surface inside a microwave guide.

To reduce motional Stark eﬀects, this part of the apparatus was removed

from the axial magnetic guiding ﬁeld used at the linac beam, so that the

residual ﬁeld was around 0.3 ± 0.1 mT. Lyman-α photons were observed

through the grids using a solar blind phototube, located at the end of a 25

cm light guide, to reduce background from the detection of annihilation

gamma-rays.

The signal, S, for the ﬁne structure transition was obtained as a ratio

involving the numbers of counts with the microwave power on and oﬀ:

S =(R

− R

oﬀ

)/R

oﬀ

. The value of R

oﬀ

included contributions from

annihilation photons and from Lyman-α photons produced from direct 2P

de-excitation, with a maximum increase of around 7% due to the 2

–

microwave-induced transition (followed by Lyman-α emission). A

representative example, shown in Figure 7.11(b), is that for the 2

–2

transition at 18 500 MHz. Discussion of the observed width of the lines

was given by Hagena et al. (1993).

7.1 Fundamental studies with the positronium atom 325

Fig. 7.11. (a) Schematic illustation of the apparatus of Hagena et al. (1993)

used to observe the 2

–2

transitions in positronium. (b) Resonance curve

of the 2

–2

transition.

3 Non-spectroscopic laser studies of positronium

Ziock et al. (1990a), using the pulsed positron source at the Lawrence

Livermore Laboratory, USA, optically saturated the one-photon (1

S–

P) transition. One notable feature of this work was the use of a dye laser

modiﬁed to obtain light with a bandwidth suﬃcient to cover a signiﬁcant

fraction of the Doppler proﬁle of the positronium, which also allowed all

states to be accessed. This accomplishment paved the way for Ziock

et al. (1990b) to make the ﬁrst observation of the resonant excitation of

a high-n

Rydberg state of positronium. This was achieved by shining a

second laser, again with a large bandwidth, onto the pumped atoms. The

laser could be tuned into the red region of the spectrum, and evidence for

excitation of some of the levels with n

= 13–19 was obtained. Finally,

it is noted that optical saturation of the 1S–2P transition is also the basis

326 7 Positronium and its interactions

for a scheme to laser-cool positronium. The desirability and feasibility of

so doing were discussed by Liang and Dermer (1988), who also considered

appropriate cooling rates; see also section 8.2.

4 Exotic tests involving positronium

Positronium, being a readily available purely leptonic system and also a

particle–antiparticle pair, has attracted considerable experimental inter-

est over the years as a testing ground for the existence of exotic particles

or couplings. The latter may perhaps manifest themselves in the decay

properties of positronium, so that attempts have been made to observe

forbidden modes. In particular, the longstanding discrepancy between

the Michigan experimental value for

o-Ps

and the results from QED

calculations, described in subsection 7.1.1, has acted as a spur to such

investigations.

Although an in-depth survey of all these measurements is beyond the

scope of the present discussion, we present a partial list to which the

interested reader may refer. A summary of the situation up to around

1980 was given by Rich (1981). Later

work includes symmetry tests (Arbic

et al., 1988; Conti et al., 1993), searches for the forbidden two-gamma

(Asai et al., 1991; Gidley, Nico and Skalsey, 1991; Nico et al., 1992)

and four-gamma (Yang et al., 1996) annihilations of ortho-positronium,

a search for spatial anisotropy in ortho-positronium annihilation (Mills

and Zuckerman, 1990), and various studies and searches for the emission

of particles other than gamma-rays in the decay of ortho-positronium

(Gninenko et al., 1990; Gninenko, 1994; Orito et al., 1989; Mitsui et al.,

1993; Adachi et al., 1994; Asai et al., 1994, and references therein). It is

suﬃcient for the present purpose to say that no evidence for any forbidden

decay modes or new particles has been forthcoming within the limits set

by the relevant experiments.

7.2 Theoretical aspects of annihilation and scattering in gases

1 Annihilation

During the collision of positronium with an atom or molecule, the positron

ﬁnds itself in close proximity to the target electrons as well as to its

companion electron. This results in an enhanced probability of annihi-

lation with an electron in a relative spin singlet state, leading to the

emission of two gamma-rays. This is termed quenching, and a number of

diﬀerent processes may occur, depending upon the chemical nature of the

atom or molecule. For ground state para-positronium, with its lifetime

of only 125 ps, the enhancement is very small, but this is not the case

7.2 Theoretical aspects of annihilation and scattering in gases 327

for ortho-positronium. As outlined in section 6.2, where the observable

parameters in a lifetime experiment were discussed, the annihilation rate

of ortho-positronium, λ

, is often written as

λ

 =

o-Ps

+ qρ =

o-Ps

+4ωρ

eﬀ

, (7.11)

where q is the quenching coeﬃcient, which may be a function of the

density and temperature of the gas, and 

eﬀ

 is conventionally inter-

preted as the eﬀective number of electrons per atom or molecule available

for annihilation with the positron by pick-oﬀ quenching (see below). The

angle brackets denote that in an experiment the measured value is an

average over the energy distribution of the positronium at the time of

annihilation, and the factor of four arises (Fraser, 1968) from the fact

that only one quarter of the target electrons are in a singlet spin state

relative to the positron.

The three mechanisms which dominate quenching in collisions with an

atom or molecule, denoted by X, can be summarized as follows.

(i) Pick-oﬀ quenching via the reaction

ortho-Ps + X → X

−

+ two γ-rays

for atoms or molecules which have ﬁlled outer shells of electrons. The

cross section for this annihilation process, which is typically less than

−25

, can be expressed as

=4πr

(c/v)

eﬀ

. (7.12)

The parameter

eﬀ

is a measure of the probability that the positron is

at the same position as one of the target electrons and in a singlet spin

state with it. Its value can be determined by projecting the wave function

representing ortho-positronium scattering by the target system onto the

singlet spin state representing the positron and a target electron. Thus,

if the total wave function is Ψ(r

; r

; ···; r

Z+2

where r

are the position and spin coordinates of the positron and

are the corresponding coordinates of the ith electron, the projection

onto the singlet spin state of the positron and the ith electron, χ

= r

, r

; ···; r

Z+2

)=χ

)|Ψ(r

= r

),

(7.13)

and so

eﬀ

Z+2



i=2



spins



|Φ

···dr

Z+2

. (7.14)

328 7 Positronium and its interactions

The summation over spins in equation (7.14) does not, of course, in-

clude the spins of the positron and the ith electron. The form of Ψ for

positronium–hydrogen scattering in the static-exchange approximation is

given in equation (7.18) below.

Comparisons of calculated and measured quenching rates provide a

useful measure of the accuracy of the wave function used for the system.

As an example, the value of

eﬀ

for helium calculated from the zero

energy static-exchange wave function of Barker and Bransden (1968) is

0.0347, or 0.0445 when the van der Waals potential is added to the static-

exchange equation; however, the experimental value obtained by Coleman

et al. (1975b) at room temperature is 0.125 ± 0.002 (see section 7.3).

This rather large discrepancy, a factor of three, shows that the static-

exchange wave function provides a poor representation of the electron–

positron correlations in this system.

(ii) Exchange, or conversion, quenching occurs according to

ortho-Ps + X(↑) → para-Ps + X(↓),

where the electron in the ortho-positronium exchanges with an atomic

electron to produce para-positronium, which then usually undergoes two-

gamma annihilation before the reverse process can occur. The ﬁnal state

of the target, represented above by X(↓), may be an excited state but if

the target atom has an unpaired electron, as in hydrogen, the conversion

of ortho-positronium to para-positronium can occur at all energies and

no excitation of the atom need take place. If, however, the atom has

no unpaired electrons, as in helium, exchange quenching can only occur

at positronium energies above the ﬁrst triplet excitation threshold of the

atom, which for helium is at 20.58 eV.

Calculations of the ortho-para conversion cross section have been made

for positronium scattering by electrons (Ward, Humberston and McDow-

ell, 1987) and by hydrogen atoms (Hara and Fraser, 1975; Drachman and

Houston, 1976), as described in the next subsection. In both cases the

conversion cross section is only a few per cent of the elastic positronium

scattering cross section, but it is several orders of magnitude larger than

the direct pick-oﬀ annihilation cross section. As an example, the elas-

tic and conversion cross sections for positronium–hydrogen scattering in

the static-exchange approximation (Hara and Fraser, 1975) are shown in

Figure 7.12.

(iii) Chemical quenching. Here the ortho-positronium is quenched after

forming a chemical complex, which may occur by means of the addition

or substitution reactions

ortho-Ps + X → ortho-PsX + energy,