Charlton M., Humberston J.W. Positron Physics

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4.5 Other processes involving positronium formation 199

in time by an ultraviolet photon) could be explained by the single-collision

reaction

+CO

−→ para-Ps

0.125 ns ↓ prompt

two γ-rays

+CO

+∗

↓ delayed

hν +CO

(4.33)

This reaction is consistent with the known energy levels of CO

;in

particular, a 3.5 eV photon could arise from the

–

transition

(Turner, 1969), whose rate, 9.23 ± 0.71 µs

−1

(Smith, Read and Imhof,

1975) is close to that found by Laricchia, Charlton and Griﬃth (1988).

It was also proposed that the events at t>0 were due to the related

reaction

+CO

−→ ortho-Ps

142 ns ↓ delayed

three γ-rays

+CO

+∗

↓ prompt

hν +CO

(4.34)

but now complications arise due to the fast-moving, long-lived ortho-

positronium and the possibility that events are distributed on both sides

of t = 0. In general, at low positronium speeds the events at t<0 will

have a decay rate equal to that of CO

+∗

, whilst on the opposite side of

t = 0 the rate is that due to the annihilation of ortho-positronium. The

measured yields, accounting also for the reaction (4.33) at t<0, were in

broad agreement with these expectations.

Laricchia, Charlton and Griﬃth (1988) also reported similar eﬀects

in N

O gas, with an energy threshold consistent with a rise in the total

cross section around 11 eV and with the energetics of N

O and N

.As

reported by Charlton and Laricchia (1986), a variety of other molecular

targets were also investigated in the same study, but none displayed the

same features as found for CO

and N

O. In a later investigation of

the total ion yield in CO

, Laricchia and Moxom (1993) found that the

threshold energies for the reactions (4.33) and (4.34) were located in the

range 10–11 eV; they postulated that the relatively high cross section

for these processes arose from an accidental quasi-resonant mechanism.

Here, the energy of an excited state of the parent molecule (i.e. CO

∗

)was

very close to that required to form positronium and leave the ion in an

excited state. Thus, the positron could be pictured as virtually exciting

the molecule but then emerging from the collision as positronium, leaving

a residual excited ion.

Finally, Bluhme et al. (1998) have reported a study of near-threshold

transfer ionization (in which the target is twice ionized and an electron

and positronium are liberated) in collisions of positrons with helium and

neon atoms. In contrast to the case for simple positronium formation,

200 4 Positronium formation

process (4.1), which usually has a rapid onset near threshold, the transfer-

ionization channel was found to be strongly suppressed. These data are

described further in section 5.5.

4.6 Comparisons with protons

Once reliable data for electron capture by positrons became available it

was natural to compare the behaviour of the cross sections for this process

with those for the analogous capture process in heavy positive particle im-

pact, in particular, the cross section for the formation of atomic hydrogen

in collisions of protons with various atoms and molecules, reaction (4.3).

It is also pertinent to note that comparisons between the behaviour of

protons and positrons are usually made at equal projectile speeds v rather

than at equal energies.

McGuire (1986) ﬁrst pointed out, on the basis of a calculation for an

atomic hydrogen target, that the cross section for positronium formation,

, is much greater than the cross section for atomic hydrogen formation,

, at intermediate and high speeds (v ≥ 5 a.u.). He attributed this to

the fact that the positron must share its kinetic energy with an electron

in order to capture it and must therefore slow down during the collision.

Far above the formation threshold, the positronium emerges with a speed

of approximately 1/

√

2 times that of the incident positron. In contrast,

the initial speed of the proton and the ﬁnal speed of the hydrogen atom

are almost identical. Since the capture cross section declines rapidly with

increasing speed (see section 4.2 and the discussion in subsection 4.4.1),

is expected to be larger than σ

at a given initial projectile speed.

We note in passing that, at the speeds under consideration, for either

projectile charge transfer is a relatively minor constituent of the total

scattering cross section.

The ratio σ

/σ

for helium, covering the entire range of speeds for

which experimental positronium formation data exists, is shown in Fig-

ure 4.26 (Schultz and Olson, 1988). The data are those of the Bielefeld

and Arlington groups, divided by the accepted proton results; also shown

are the classical trajectory Monte Carlo calculations. At low speeds, less

than 2 a.u., electron capture is less likely by positrons than by protons.

This can be attributed to a ‘threshold eﬀect’, whereby the cross section

is lowered by virtue of the low kinetic energy of the positron and the

fact that it must expend a signiﬁcant fraction of this in capturing the

electron. A similar eﬀect has been observed in scattering involving single

and double ionization (see sections 5.4 and 5.5).

As the speed of the projectile is raised, experiment and theory both pre-

dict that σ

/σ

becomes greater than unity, although, as outlined above,

there is a discrepancy between the results of the Arlington and Bielefeld

4.7 Diﬀerential cross sections 201

Fig. 4.26. Plot of the ratio σ

/σ

for positron and proton collisions with

helium, versus the velocity of the projectiles (following Schultz and Olson, 1988).

The curve is the result of their classical trajectory Monte Carlo calculation. The

triangles are based on the positron data of Fromme et al. (1986) whilst the circles

are from Diana et al. (1986b).

groups and the more recent data of Overton, Mills and Coleman (1993).

Overall, theory tends to predict a lower value of σ

/σ

and, since the

values of σ

are well established from both theory and experiment, Schultz

and Olson (1988) argued that the divergence is due to discrepancies in σ

Schultz and coworkers have proposed, as was outlined in subsection 4.4.1,

that the cause of this discrepancy is the diﬃculty in making accurate

measurements of the small values of σ

at these speeds. This situation

was somewhat remedied by the work of Overton et al. (1993).

4.7 Diﬀerential cross sections

In common with all angle-resolved cross sections, the diﬀerential positro-

nium formation cross section, dσ

/dΩ, contains more information than

its integrated counterpart. In particular, it sheds light on the dynamics

and detailed mechanisms involved in this unique capture process. This

is apparent at the higher kinetic energies, where striking eﬀects related

202 4 Positronium formation

to the Thomas double-scattering mechanism are predicted, which are

markedly diﬀerent from those found for proton collisions. The latter are

described in section 4.2 along with other examples of the behaviour of

dσ

/dΩ.

The measurement of dσ

/dΩ is not an easy task and, as we shall see,

there is little experimental information presently available. We note here

that many experimental arrangements which could be used to measure

diﬀerential positronium formation cross sections sum over all the possible

quantum states (n

) of the positronium, whereas calculations usually

refer to one particular state. Some diﬀerentiation between positronium

states with diﬀerent values of n

can be achieved if the time of ﬂight

of the positronium, and hence its kinetic

energy, is measured, and such

a technique has been used to investigate beams of positronium atoms

produced in positron–gas collisions (see section 7.6).

The ﬁrst experimental study of diﬀerential positronium formation cross

sections was made by Laricchia et al. (1987b) during their positronium

beam development. This work was guided by the theoretical distorted-

wave results of Mandal, Guha and Sil (1979), which show the cross section

to be forward peaked. Laricchia et al. (1987b), using a simple detection

system, measured the yield of ortho-positronium atoms

produced in a

narrow angular range about the incident positron direction for several

impact energies. Their data were in reasonable agreement with the small-

angle behaviour predicted by Mandal, Guha and Sil (1979).

Further information was forthcoming at small angles from the studies

of Tang and Surko (1993) on molecular hydrogen. A schematic view

of their apparatus is shown in Figure 4.27(a). The various grids shown

were used to prevent all charged particles from reaching the channel plate

detector. The positronium signal was identiﬁed as a channel plate count

in coincidence with a gamma-ray detected by the photodiode detector,

which incorporated a CsI scintillator. The entire detector assembly, which

was located approximately 175 mm from the exit of the gas cell, with a 25

mm diameter aperture in front to deﬁne the solid angle, could be moved

vertically through the positronium ﬂux as indicated by the large solid

arrows on the ﬁgure.

The fraction of the positron beam forming positronium at detector

angles in the range ±14

◦

at impact energies of 50 eV, 80 eV and 100 eV is

shown in Figure 4.27(b), which also gives the theoretical results of Biswas,

Mukherjee and Ghosh (1991) and Biswas et al. (1991); these yielded

values for dσ

/dΩ for positronium formation into the 1S, 2S and 2P

states using the ﬁrst Born approximation. Given the crude nature of this

approximation, the accord between theory and experiment is reasonable

and conﬁrms the forward-peaked nature of the positronium formation

process.

4.7 Diﬀerential cross sections 203

Fig. 4.27. (a) Side view of the apparatus of Tang and Surko (1993), which was

used for measurements of the angular dependence of positronium formation. (b)

The positronium formation fraction versus the angle between the direction of

the incident positron beam (at the kinetic energies shown) and

the line joining

the centre of the gas cell to the centre of the detector aperture. The solid lines

are predictions

of this quantity derived from the theoretical work of Biswas,

Mukherjee and Ghosh (1991)

and Biswas

et al. (1991).

Measurements of dσ

/dΩ for positron–argon scattering have also been

made by Finch et al. (1996a, b) and Falke et al. (1995, 1997) and for

positron–krypton scattering by Falke et al. (1997). The principle of the

experiments, involving the detection of the positronium in coincidence

with an atomic ion, is illustrated schematically in Figure 4.28. More

details of the system used by Finch et al. (1996a), which has also been

used to study the diﬀerential ionization cross section, can be found in

section 5.6.

204 4 Positronium formation

Fig. 4.28. Schematic illustrating the principle of angle-resolved measurements

of positronium formation in positron–gas collisions (following Falke et al., 1997).

Reprinted from Journal of Physics B30, Falke et al., Diﬀerential Ps-formation

and impact-ionization cross sections for positron scattering on Ar and Kr atoms,

The data of Falke et al. (1997) for argon and krypton are shown in

Figure 4.29. Falke et al. (1997) measured dσ

/dΩ at various angles

between 0

◦

and 120

◦

at impact energies of 75 eV, 90 eV and 120 eV.

Their data are plotted as the ratio of the cross section at a particular

angle to that at 0

◦

; also shown are the calculations of McAlinden and

Walters (1994). Good agreement is found between theory and experiment

at 75 eV, particularly when allowance is made for the angular resolution

of the experiment (stated as ±2

◦

). Falke et al. (1995) investigated positro-

nium formation at 30 eV impact energy at angles in the range 0

◦

–50

◦

.In

this case the theoretical prediction by McAlinden and Walters (1994) of a

minimum in dσ

/dΩat0

◦

was found to be in marked disagreement with

the experimental measurements, which revealed a distinct maximum at

this angle.

Finch et al. (1996a) measured dσ

/dΩ at a ﬁxed scattering angle of

◦

, but at energies in the range 40–150 eV. Their results show a steady

decline as the impact energy is increased, in contrast to the calculations of

McAlinden and Walters (1994), to which they were normalized at 60 eV.

It is notable that the detection eﬃciency of the channeltron used in this

work was not known for positronium, but it can be presumed to have been

energy dependent and would most likely increase with the kinetic energy

of the positronium in the range investigated. Thus, this eﬀect cannot be

responsible for the discrepancy between theory and experiment.

The ﬁnal experimental study of diﬀerential positronium formation

summarized here is that of Falke et al. (1995, 1997), who reported

Fig. 4.29. Angle-resolved positronium formation cross sections, relative to those at 0

◦

, for positron–argon and positron–

krypton collisions at impact energies of 75 eV, 90 eV and 120 eV (Falke et al., 1997). The curves are from the theoretical

work of McAlinden and Walters (1994). Reprinted from Journal of Physics, B30, Falke et al., Diﬀerential Ps-formation and

IOP Publishing.

Fig. 4.30. Angle-resolved transfer-ionization cross sections, reaction (4.35), relative to those at 0

◦

, for positron–argon and

positron–krypton collisions at impact energies of 75 eV, 90 eV and 120 eV (Falke et al., 1995, 1997). Reprinted from Journal of

Physics B30, Falke et al., Diﬀerential Ps-formation and impact-ionization cross sections for positron scattering on Ar and Kr

4.8 Dense gases 207

measurements on transfer ionization in positron–argon and positron–

krypton collisions. The process can be written as

+ Ar(Kr) → Ps + e

−

+Ar

(Kr

), (4.35)

and again the positronium was detected at each angle (with an angular

deﬁnition of around 2.5

◦

) in coincidence with the Ar

ion. The results

are shown in Figure 4.30. It is notable that the minimum in the cross

section observed by Falke et al. (1995) at 75 eV in the positron–argon

system and tentatively linked to a Thomas-scattering-like process (see

sections 4.2 and 5.2 for discussion relevant to this topic), does not seem

to persist at higher energies and is not present in the krypton data. This

would seem to rule out a connection to Thomas scattering, though the

data are highly suggestive and further work in this area is to be expected.

4.8 Dense gases

1 Models of the positronium fraction

Before the advent of low energy positron beams the only method of

studying positronium formation in gases was by allowing β

particles to

stop in dense samples. (Typically gas densities were ≥ 10

−3

.) Such

experiments were useful in elucidating the basic mechanisms by which

positronium can be formed, and in this section we brieﬂy review this

body of work.

As will be described in section 6.2, one of the parameters which can

be derived from traditional positron lifetime experiments in dense gases

is the positronium fraction F , the fraction of positrons which stop in the

medium and form positronium. The simplest approach aimed at deriving

values of F to be expected in such experiments was proposed by Ore

(1949), and the so-called Ore model has since been expounded several

times in essentially its original form (e.g. Massey, 1975; Griﬃth and Hey-

land, 1978; Schrader and Svetic, 1982; Charlton, 1985a). It involves no

other physical input than the threshold energies for positronium formation

), excitation (E

) and ionization (E

) and assumptions concerning

the touch-down energy distribution, TD(E) (Schrader and Svetic, 1982),

of the positrons as their kinetic energies E are ﬁrst moderated below E

The simplest cases to consider are atomic species in which there are no

low-lying excited states, so that positronium formation via reaction (4.1)

is the ﬁrst inelastic channel to open. The positronium fraction can then

be written as

F =

TD(E){σ

(E)/[σ

(E)+σ

(E)]}dE

TD(E) dE

, (4.36)

208 4 Positronium formation

where σ

(E) and σ

(E) are the cross sections for positronium formation

and excitation. The touch-down distribution is determined largely by the

details of how positrons slow down as they lose the last hundred eV or

so of their kinetic energy. Whilst some of the relevant processes are now

relatively well understood, other factors, for example the role played by

the formation and break-up of energetic positronium, remain unclear.

It should be noted that the upper positron energy limit considered

in this model is E

. Above this energy, neglecting small target-recoil

eﬀects, any positronium that is formed has a kinetic energy in excess of its

binding energy and is assumed to break up in a subsequent collision. Thus,

positronium formed in the ground state with a kinetic energy above 6.8 eV

is regarded as not contributing to the measured positronium fraction.

This assumption is only valid if the kinetic energy of the positronium is

not reduced to below its binding energy in subsequent collisions and at

suﬃciently high gas densities that it does not self-annihilate before break-

up can occur. This latter restriction is most serious for para-positronium

because of its high (8 ns

−1

) decay rate. Assuming a break-up cross section

of 10

−16

leads to a gas density of approximately 10

−3

, above

which break-up will dominate.

It has been commonly assumed that the TD distribution is uniform in

energy, in which case equation (4.36) reduces to

F =

− E



(E)

(E)+σ

(E)

dE, (4.37)

where the ﬁrst term arises because, for all atomic systems undergoing

positron impact, E

. This equation, and the two following, are

often used for both atoms and molecules, though for the latter there are

also low-lying vibrational and rotational levels and, in some cases (e.g.

), electronic levels to consider. By setting the contribution to F from

positrons with kinetic energies above E

to be zero, the minimum fraction

min

predicted by the Ore model is just

min

=(E

− E

)/E

. (4.38)

However, if electronic excitation is unimportant compared to positronium

formation, σ

(E)  σ

(E), the maximum Ore model prediction is

obtained from equation (4.36) as

max

=(E

− E

)/E

=6.8eV/E

. (4.39)

Without extra input from either theory or experiment for the relevant

cross sections, we are left with predictions for F

max

and F

min

which deﬁne

a band within which the experimental value is expected to fall.