Charlton M., Humberston J.W. Positron Physics

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7.6 Studies with positronium beams 359

Fig. 7.25. Compendium of positronium total scattering cross sections from

Garner,

Ozen and Laricchia (1998): •, helium; , argon;

, molecular hydrogen;

, molecular oxygen.

concerning the angular dependence of the positronium elastic scattering

cross section.

It is notable that the behaviour for all the gases is broadly similar,

though only two energies have been investigated for molecular oxygen.

The total cross section increases as the positronium energy is raised

from its lowest value and then passes through a broad maximum before

decreasing again at higher energies. The maximum is most pronounced for

argon. The rise in the cross section may be due to positronium break-up,

which is believed from theory to be an important channel; see e.g. the

summary by Charlton and Laricchia (1991).

The results for helium and argon targets are shown in more detail in

Figure 7.26, along with the relevant theory (see subsection 7.2.2). For

helium, the data of Garner, Laricchia and

Ozen (1996) supersede those

of Zafar et al. (1991), which were obtained using an indirect method.

The results of Coleman et al. (1994) and Nagashima et al. (1998) for

the average momentum transfer cross section at low energies, derived

using the ACAR technique as described in section 7.4, are of comparable

magnitude. The theoretical results of McAlinden, MacDonald and Wal-

ters (1996) are in reasonably good agreement with experiment above ap-

proximately 60 eV, but they markedly disagree at low energies, probably

owing to neglect of exchange. The semi-classical calculations of Peach

360 7 Positronium and its interactions

Fig. 7.26. Positronium total scattering cross sections for helium

and argon

gases. Experiment: Garner, Laricchia and

Ozen (1996); •, He; , Ar. Theory

for

helium: ——, Peach (1993, priv

ate communication to G. Laricchia); – – –,

McAlinden, McDonald and Walters (1996); — · —, Sarkar and Ghosh (1997),

elastic component only. Theory for argon: ·····, McAlinden, MacDonald and

Walters (1996).

(1993, private communication to G. Laricchia) are in better accord with

experiment, both in shape and magnitude, at least up to 70 eV. For argon

the results of Garner,

Ozen and Laricchia (1998) are somewhat higher that

those of Zafar et al. (1996) (not shown in Figure 7.26), owing to lack of

angular resolution in the latter. The theory of McAlinden, MacDonald

and Walters (1996) again yields results in poor accord with experiment

at lower energies, though tending towards the experimental values as the

energy is increased.

The results described in this section mark only the beginnings of

positronium–atom(molecule) collision studies. Investigations in the in-

termediate energy range and the measurement of total cross sections of

comparable, or better, accuracy should soon be available for a variety

of targets. Extensions to both higher and lower positronium energies

await developments in beam production and detection techniques. With

an eye to the future, Charlton and Laricchia (1991) identiﬁed a number

of positronium reactions which would be of interest to study, and we

reproduce their list here:

7.6 Studies with positronium beams 361

Ps + X → Ps + X elastic scattering;

→ Ps* + X projectile excitation;

→ Ps + X* target excitation;

→ e

−

+ X projectile ionization or break-up;

→ Ps + X

−

target ionization.

Exotic species involving positrons

We now consider several of the more exotic systems in which one or more

positrons may be involved, some of which were introduced in subsec-

tion 1.2.3. The positronium negative ion (e

−

), Ps

−

, has been ob-

served in the laboratory (Mills, 1981) and its lifetime against annihilation

determined experimentally (Mills, 1983b). We discuss these experiments

and the relevant theory in section 8.1. Observation of the positronium

molecule, Ps

, and other systems containing more than one positron or

positronium atom (as yet unrealized) depends upon the generation of

large instantaneous densities of positrons. The situation here is more

encouraging than might be expected, owing to progress in developing very

intense brightness-enhanced and time-focussed beams, as summarized in

subsection 1.4.4. Many-positron systems and how they may be observed

are described in section 8.2.

Antihydrogen, as discussed in subsection 1.2.3, has recently been ob-

served in the laboratory, although only at relativistic speeds. However,

progress with the trapping of cold antiprotons and positrons, and the

production of positronium in a cryogenic environment, leads us to antic-

ipate the synthesis of antihydrogen atoms with very low kinetic energies

(or temperatures); thus it may be possible to trap them, and perform

precision spectroscopy upon them. The motivation for the production

of low temperature antihydrogen is described in section 8.3, along with

the mechanisms and methodologies involved in some of the proposed

formation processes.

8.1 The positronium negative ion

An electron can bind to positronium to form Ps

−

, provided that the

two electrons are in a singlet spin state. This system, and its charge

conjugate counterpart consisting of two positrons and one electron, was

362

8.1 The positronium negative ion 363

predicted to be bound by Wheeler (1946), and was ﬁrst detected experi-

mentally by Mills (1981). One of the most accurate values of the energy

of the bound state with respect to break-up into positronium plus an

electron (or positronium plus a positron for the charge conjugate system)

is −0.326 677 eV, which was obtained by Bhatia and Drachman (1983)

using the Rayleigh–Ritz variational method with a 220-term Hylleraas

trial function of the form

−



(1 + P

) exp[−(α

+ α

)] r

, (8.1)

where r

and r

are the position vectors of the two identical particles

relative to the third one, and r

= |r

− r

|. Very similar results have

been obtained by Ho (1983, 1993), also using a Hylleraas trial function,

and by Petelenz and Smith (1987) and Frolov and Yeremin (1989), using

exponential variation expansions. No other bound states of Ps

−

exist, but

the system does possess several autoionizing states, or resonances, in the

electron–positronium continuum, the energies and widths of which were

studied extensively by Ho (1984), using the complex coordinate rotation

method.

The rate of annihilation of the positron with one of the electrons into

two gamma-rays is given in terms of the bound state wave function, Ψ

−

Γ=2πα





1 − α



−

19π



Ψ

−

|δ(r

)|Ψ

−



Ψ

−

|Ψ

−



= 100.617

Ψ

−

|δ(r

)|Ψ

−



Ψ

−

|Ψ

−



(ns

−1

). (8.2)

Using their most accurate wave function, Bhatia and Drachman (1983)

obtained Γ = 2.0861 ns

−1

, in excellent agreement with the experimental

value of 2.09 ± 0.09 ns

−1

; see below. This value is also close to the

weighted average of the annihilation rates for the singlet and triplet

states of free ground state positronium (see subsection 7.1.1), as would

be expected for a system having the conﬁguration of positronium weakly

bound to an electron. Ho (1983) obtained similar theoretical values and

also calculated the angular correlation function for the two gamma-rays

created in the annihilation process (see subsection 1.3.3), obtaining a full

width at half-maximum of 1.4 mrad.

The photodetachment of Ps

−

was suggested by Mills (1981) as a po-

tential source of a tunable-energy positronium beam. Once produced

and accelerated electrostatically to the required energy, the Ps

−

would

undergo photodetachment to form the desired beam according to

−

+ photon → Ps+e

−

. (8.3)

364 8 Exotic species involving positrons

The photodetachment cross section for a photon of angular frequency ω

is, in the length formulation,

(L) =

kωαa

Ψ

|Ψ

−

, (8.4)

where Ψ

is the wave function for p-wave electron–positronium elastic

scattering at electron wavenumber k and Q

k ·(r

+ r

) is the dipole

transition operator. Energy conservation gives the relationship between

ω and k as

ω + E

−

= E

, (8.5)

allowing for the recoil of the positronium. An alternative expression for

the photodetachment cross section, in the so-called velocity formulation,

(v)=

2kαa

9ω

Ψ

|Ψ

−

, (8.6)

where Q

=4k ·(∇

+ ∇

). The diﬀerence between the results obtained

using these two expressions, which would be zero if the exact initial and

ﬁnal wave functions could be used, provides a measure of their actual

quality.

The photodetachment process was ﬁrst investigated theoretically by

Bhatia and Drachman (1985), using a simple asymptotic form for Ψ

−

and the p-wave component of a plane wave for the electron in the ﬁ-

nal state. Much more detailed studies have since been made by Ward,

Humberston and McDowell (1987). These authors calculated accurate

singlet p-wave electron–positronium scattering wave functions using the

Kohn variational method in a similar manner to that described previ-

ously in section 3.2 for positron–hydrogen scattering; they used a very

accurate Hylleraas wave function for the ground state wave function of

−

. Their results for both the length and velocity formulations are given

in Figure 8.1, together with the results of Bhatia and Drachman (1985).

The sharp rise in the photodetachment cross section found by Ward,

Humberston and McDowell (1987) for wavelengths less than 4000

Ais

caused by a series of Feshbach resonances just below the n

= 2 excitation

threshold of positronium.

The apparatus used by Mills (1981) for the ﬁrst observation of Ps

−

was

similar to that shown in Figure 8.2. Slow positrons were guided by an

axial magnetic ﬁeld onto a 40

A thick carbon ﬁlm G

. The kinetic energy

of the positrons was adjusted so that some could penetrate the foil and

emerge bound to two electrons as Ps

−

. The geometry and method are

analogous to the production of H

−

by proton bombardment of thin foils

(see e.g. Allison, 1958).

The grid G

located behind the carbon ﬁlm was biassed positively so

as to accelerate the Ps

−

but return any transmitted positrons to the foil.

8.1 The positronium negative ion 365

Fig. 8.1. The photodetachment cross sections for Ps

−

: ——, results obtained

with very accurate variational wave functions; L, length formulation; V, velocity

form

ulation (Ward, Humberston and

McDowell, 1987); – – –, plane

wave

ap-

proximation (Bhatia and Drachman, 1985). The vertical broken line marks the

position of the n

= 2 excitation threshold of positronium.

The gamma-ray energy spectra recorded by the Ge(Li) detector, shown in

Figure 8.3, illustrate that as the voltage diﬀerence, W = V

−V

, between

the grid and the carbon foil was increased, a small peak was discerned

which was blue-shifted with respect to the 511 keV annihilation line, the

shift being dependent on W . This is the signature of the Doppler-shifted

two-gamma annihilation of Ps

−

moving towards the detector. The arrows

in Figure 8.3 indicate the expected positions of the peak, whose shift from

an energy of 511 keV is given by

∆E

=[λ +(2λ + λ

)

1/2

cos φ]mc

, (8.7)

where λ = eW/(3mc

) and φ is the angle between the direction of gamma-

ray emission and the Ps

−

velocity.

An averaged value of cos φ (φ is dependent on the detection geometry)

was used in evaluating ∆E

. A detailed analysis performed by Mills

(1981) allowed for the fact that the Ps

−

underwent acceleration for a

signiﬁcant proportion of its lifetime. From this he was able to deduce that

the shifted annihilation line arose from a system with a mass-to-charge

366 8 Exotic species involving positrons

Fig. 8.2. Apparatus used by Mills for studies of the positronium negative ion;

is a pile-up-reducing grid, G

is the Ps

−

-forming carbon ﬁlm and G

is the

acceleration grid. See the text for further details. Reprinted from Physical

Review Letters 50, Mills, Measurement of the decay rate of the positronium

ratio of approximately three, produced in the foil with an eﬃciency of

3 × 10

−4

of the incident positrons.

Following this observation, Mills (1983b) measured the decay rate of

−

using the apparatus of Figure 8.2. Positrons were guided through

the grid G

, where they were accelerated by the voltage applied to the

carbon ﬁlm G

. As before, Ps

−

was formed on the transmission side of the

foil and accelerated by the potential applied to G

, which was separated

from the carbon ﬁlm by a distance d. Again, the blue-shifted gamma-rays

from the in-ﬂight annihilation of Ps

−

moving towards the detector were

distinguished from the 511 keV gamma-rays due to positron annihilation

in the ﬁlm. The region beyond G

was ﬁeld-free.

Mills argued that the intensity of the Doppler-shifted gamma-ray peak

is proportional to the probability that the Ps

−

reaches the grid G

, and

that the time interval, t, between emission and arrival at G

is propor-

tional to d for a given potential diﬀerence W between G

and G

.It

follows that the intensity of the Ps

−

peak will be exponentially dependent

on d so that, by measuring the intensity at various values of d, the lifetime

of Ps

−

can be determined.

Figure 8.4 shows a plot of the logarithm of f

−

, which is the ratio of

the Ps

−

and 511 keV photopeaks versus the parameter t/g(λ, ). This is

essentially the time, evaluated from

t =



λc



ln[1 + λ +(2λ + λ

)

1/2

]g(λ, ), (8.8)

where λ is as deﬁned after equation (8.7),  = T/(3mc

) and the function

8.1 The positronium negative ion 367

Fig. 8.3. Annihilation gamma-ray spectra showing the blue-shifted photo-peak

due to Ps

−

at various values of W , the acceleration potential. The arrows

indicate the expected positions of these peaks. The

207

Bi line (at approxi-

mately 569.6 keV) used for calibration is also shown. Reprinted from Positron

Solid-State Physics, Proceedings of the International School of Physics ‘Enrico

Fermi’, Course 83, Mills, Experimentation with low energy positron beams,

g(λ, ) is a correction for the non-zero initial kinetic energy, T , of the Ps

−

;

this was written by Mills as

g(λ, )=1−





1/2





+ ···. (8.9)

The data are ﬁtted by straight lines which yield values of gΓ, where Γ is

the desired Ps

−

decay rate: gΓ=1.851±0.058 ns

−1

for W = 1000 eV and

1.971 ± 0.034 ns

−1

for W = 3936 eV. These values are plotted against

−1/2

in the inset of Figure 8.4; this, from equation (8.9), should give

an approximately linear dependence, according to gΓ=Γ(1−(T/W)

1/2

Extrapolation to inﬁnite W gives Γ = 2.09 ± 0.09 ns

−1

, which is in

good agreement with, but much less precise than, the calculated values of

Bhatia and Drachman (1983) and Ho (1983, 1993). The energy of emission

of the Ps

−

, T , is found to be in the range 3–32 eV, with a mean value

368 8 Exotic species involving positrons

Fig. 8.4. Plot of the logarithm of f

−

versus t/g(λ, ); see text. The inset shows

the extrapolation to inﬁnite acceleration potential to yield the decay rate of Ps

−

Here the small hatched rectangle covers the experimental error, and the value

calculated by Ho (1983, 1993) is indicated.

of 13 eV. The level of precision in this experiment was dominated by the

counting statistics, and a preliminary account of an improved experiment

(Mills, Friedman and Zuckerman, 1990) has been given, though no new

results for Γ have been reported.

8.2 Systems containing more than one positron

Although positron plasmas can be considered to be systems containing

many positrons, and as such technically fall within the scope of this

section, we will not consider them here. Rather, we will concentrate on

the theory of, and the possibilities of observing, assemblages of particles

containing both positrons and electrons. These include the positronium

molecule and a Bose–Einstein (BE) condensate of positronium atoms.