Charlton M., Humberston J.W. Positron Physics

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7.3 Experimental studies of positronium annihilation in gases 339

introduced. The momentum resolution of the apparatus was given by

Kakimoto et al. (1987) as 0.27 × 10

−3

mc.

The ACAR data of Kakimoto et al. (1987) are shown in Figure 7.15,

which illustrates the progression from vacuum up to an O

pressure of

0.8 atmospheres and also includes, for comparison, a spectrum taken at 1

atmosphere of N

, a gas for which only pick-oﬀ quenching can occur. The

curves for N

and the vacuum could be decomposed into two components,

and those for O

into three. The broader or broadest component, B(p),

shown in each case by a broken line, was independent of gas pressure and

was attributed to positron annihilation within the silica grains, whilst the

lower momentum components were due to the annihilation of positronium

between the grains. The momentum distribution of these latter compo-

nents, F (p), was obtained from the entire angular correlation curve N(p),

according to F (p)=−2pd[N(p) − B(p)]/dp, and these data are shown

in Figure 7.15 in the right-hand diagrams. The positronium momentum

corresponding to the ﬁrst and second electronic excitation energies of O

∆

at 0.98 eV) and E

at 1.62 eV) are indicated. The ground

state is a triplet with conﬁguration X

−

As described by Kakimoto et al. (1987), the vacuum spectrum is the re-

sult, to a good approximation, of the distribution in the para-positronium

momentum on emission from the grains. The N

result, which is only

slightly lower in momentum, shows the moderating eﬀect of a gas in which

the electronic excitation energy is much higher than the initial kinetic

energy of the positronium, so that slowing down (see section 7.4) can only

occur by elastic scattering and rotational and vibrational excitation of the

molecule. This has hardly any eﬀect on the short-lived para-positronium.

The situation for O

is markedly diﬀerent: two components, denoted by

I and II, are clearly visible, the narrower component, I, being responsible

for the lower momentum peak in Figure 7.15.

Kakimoto et al. (1987) attributed component I to the conversion

of ortho-positronium, with initial kinetic energy below E

, to para-

positronium by elastic exchange collisions. The momentum is smaller

than that of component II since the lower energy ortho-positronium has a

suﬃciently long lifetime to undergo appreciable slowing down. Analysis of

the spectra yielded an elastic conversion cross section of 1.5 ×10

−19

in accord with other measurements (Klobuchar and Karol, 1980; Kieﬂ,

1982).

Turning to component II, if the initial kinetic energy of the ortho-

positronium is above E

then it can undergo inelastic conversion by

excitation of the

∆

level of O

. However, Figure 7.15 shows that

as the O

pressure increases so the peak of component II increases at

the expense of the tail (at energies > 1 eV). This tail is due to the

340 7 Positronium and its interactions

initial momentum of the short-lived para-positronium, implying that

this process has a much higher cross section than that responsible for

component I. Kakimoto et al. (1987) therefore ascribed component II to

the slowing down of para-positronium by excitation of the O

, though

they noted that in so doing it converts to ortho-positronium owing to

the conservation of spin conﬁguration. Thus, to treat the data properly,

ortho-positronium to para-positronium conversion must be accounted

for. The shape of component II also indicates a threshold energy close

to 1 eV, in accord with that for the

∆

level. By noting that the peak

of component II is already present at 0.1 atmospheres of O

, Kakimoto

et al. (1987) estimated the inelastic ortho-positronium conversion cross

section, applicable to a positronium kinetic energy of 1.5 eV, to be

≥ 2 × 10

−17

, or more than a hundred times that for the elastic

process.

A follow-up study at O

pressures below 0.05 atmospheres (Kaki-

moto, Hyodo and Chang, 1990), where the para-positronium to ortho-

positronium conversion is suppressed because it occurs at a rate lower

than that for para-positronium annihilation, yielded cross sections in

good accord with the estimates given above.

Returning to the discussion of 

eﬀ

 from pick-oﬀ quenching, we now

turn to gas density and temperature regimes where signiﬁcant departures

from the linear rise predicted by equation (7.11) have been observed. As

reviewed by Iakubov and Khrapak (1982), the ﬁrst evidence for anoma-

lous eﬀects in the pick-oﬀ quenching rate in gases came from the low

temperature helium work of Daniel and Stump (1959). The most detailed

study of this system was performed by Hautoj¨arvi and Ryts¨ol¨a (1979),

who found that the ortho-positronium lifetime in low temperature, high

density helium gas is substantially longer than that predicted using the

value of 

eﬀ

 given in table 7.2. This eﬀect is illustrated in Figure 7.16,

which shows their data for the pick-oﬀ quenching rate qρ plotted against

gas density ρ for a variety of temperatures.

It is now considered that this type of behaviour is caused by the self-

trapping of ortho-positronium in bubbles in the low temperature gas.

The bubbles are thought to form because the ortho-positronium–atom

interaction at low energies is dominated by repulsive exchange forces,

and this eﬀect results in a lowering of the annihilation rate; the bubble is

so rariﬁed in some cases that qρ approaches zero.

The behaviour shown in Figure 7.16 is, in some respects, typical of other

gases investigated. In particular, the behaviour found for helium at 77 K

by Fox et al. (1977), in which λ

 falls gradually from the extrapolated

low density and high temperature line, has been observed for a number of

other gases at moderate densities. Examples include the work of McNutt

7.3 Experimental studies of positronium annihilation in gases 341

Fig. 7.16. The pick-oﬀ annihilation rate qρ, see equation (7.11), for ortho-

positronium in

He gas at various temperatures, observed by Hautoj¨arvi and

Ryts¨ol¨a (1979). At the lowest temperature qρ is almost independent of density,

indicating stable bubble formation. The behaviour gradually changes to that of

free ortho-positronium, indicated by the straight line whose slope corresponds

to 

eﬀ

 = 0.125 (see table 7.2). The data at 77 K are due

to Fox

et al. (1977).

and Sharma (1978, CH

), McNutt et al. (1979, H

) and Wright et al.

(1983, CO

) and the data presented for argon and krypton at 297 K by

Griﬃth and Heyland (1978).

Sharma, Eftekhari and McNutt (1982) and Sharma, Kaﬂe and Hart

(1984) found particularly striking eﬀects for C

, which has a critical

temperature just above room temperature, and some of their data are pre-

sented in Figure 7.17. McNutt and Sharma (1978) and Sharma, Eftekhari

and McNutt (1982) attempted to account for this behaviour using a model

in which the positronium preferentially annihilates in regions of lower than

average density caused by naturally occurring rarefactions in the gas. This

approach is analogous to the free-volume model of positronium annihi-

lation in condensed media and is qualitatively diﬀerent from the bubble

model in that it ignores the positronium–atom(molecule) interactions and

attributes the anomalous behaviour of λ

 solely to the thermodynamic

properties of the gas. Some support for the free-volume approach at

moderate densities of N

gas was provided by Kawaratani, Nakayama and

Mizogawa (1985). Further aspects of many-body phenomena involving

positronium can be found in the reviews of Iakubov and Khrapak (1982)

and Sharma (1988, 1992).

342 7 Positronium and its interactions

Fig. 7.17. Ortho-positronium annihilation rates at various values of ethane gas

density D, at a temperature of 305.45 K. The solid line is a weighted average

of the annihilation rates between 120 and 180 amagat. The broken line is the

prediction for free

ortho-positronium. The data are due to Sharma, Kaﬂe and

Hart (1984). Reprinted from Physical Review Letters 52, Sharma, Kaﬂe and

Hart, New features in the behaviour of ortho-positronium annihilation rates

the American Physical Society.

7.4 Slowing down

Recent experimental advances involving the use of the ACAR technique

and positronium formation in silica aerogel or powder, and time-resolved

Doppler broadening studies (TRDBS) of positronium annihilation in

gases, have meant that it has been possible to observe the eﬀects of the

slowing down of positronium in collisions with gas atoms and molecules,

allowing average momentum transfer cross sections to be derived. One

advantage of the aerogel method is that it allows the positronium to

interact with the low density gas in the large spaces between the grains,

whilst the β

particles from the source are stopped and form positronium

eﬃciently. A potential disadvantage is that interactions of the positron-

ium with the surface of the gel have to be accounted for. This feature

can, however, be circumvented by the TRDBS technique.

The ﬁrst discussion of the thermalization of positronium appears to

have been that of Sauder (1968), who derived a general (classical) expres-

sion for moderation by elastic collisions of a particle in a medium, allowing

for the thermal motion of the atoms or molecules of the medium. By

assuming that the momentum transfer cross section, σ

, is a constant he

found that the time dependence of the mean positronium kinetic energy,

7.4 Slowing down 343

E, was given by

E/E

= coth

(β + ρΓt), (7.25)

where E

is the thermal energy and Γ = σ

(2E

1/2

/M , with m the

positronium mass and M the mass of the gas atoms. The parameter

β is deﬁned at t = 0, when E = E

, as coth

−1

)

1/2

.Itis

relatively straightforward to show that this expression is equivalent to

that derived by Hyodo et al. (1989) and applied by Kakimoto et al. (1989)

and Nagashima et al. (1995), though in these cases an extra term due to

positronium interactions with the aerogel had to be introduced.

To our knowledge, the ﬁrst study of gases in powders was that of Fox

and Canter (1978), who investigated the eﬀects of adding high pressure

helium to silica and magnesium oxide powders. Behaviour inconsistent

with a simple single-exponential ortho-positronium component was ob-

served in the lifetime spectrum with the gas evacuated, and this was

substantially modiﬁed by the introduction of the gas. Indeed the ap-

proach to the exponential (and assumed equilibrium) behaviour of the

ortho-positronium component was hastened by the helium, and this was

attributed to moderation of the kinetic energy of the positronium by the

gas.

We now proceed to describe in more detail the ACAR-aerogel work of

Hyodo and coworkers, and the TRDBS measurements of Skalsey et al.

(1998). It is worth noting, following Hyodo (1992), the physical processes

which contribute to the two-gamma events registered by the ACAR tech-

nique in this type of experiment:

(i) positron and positronium annihilation in the grains of the aerogel;

(ii) self-annihilation of para-positronium between the grains;

(iii) pick-oﬀ annihilation of ortho-positronium in collisions with the grain

surface or the added gas;

(iv) two-gamma self-annihilation of the m = 0 mixed ortho-positronium

state in a magnetic ﬁeld;

(v) the self-annihilation of para-positronium resulting from exchange

quenching of ortho-positronium in a paramagnetic gas.

The key to the ACAR technique is process (iv), since by application of

a magnetic ﬁeld the lifetime of the mixed state is changed and the time-

dependence of the positronium momentum distribution can be accessed.

ACAR studies have been made of all the noble gases and some molecules

(Hyodo, 1992; Nagashima et al., 1995; Coleman et al., 1994) at a ﬁxed

magnetic ﬁeld; by Hyodo and coworkers at 0.29 T and by Coleman et al.

344 7 Positronium and its interactions

Fig. 7.18. Derived momentum distributions for the perturbed m = 0 state of

ortho-positronium, for various gases (Nagashima et al., 1995) in an applied static

magnetic ﬁeld of 0.29 T. Reprinted from Physical Review A52, Nagashima et al.,

Thermalization of free positronium atoms by collisions with silica-powder grains,

Physical Society.

(1994) at 0.8 T. The results of Nagashima et al. (1995) are shown in

Figure 7.18 and, interestingly, exhibit a progressive broadening of the

narrow component of the distribution as the atomic or molecular mass

increases. Similar eﬀects to these were observed by Coleman et al. (1994)

in their study of the noble gases, although their momentum resolution was

inferior to that of Nagashima et al. (1995), so that the narrow positronium

component could only be separated in helium, neon and argon.

The derived momentum transfer cross sections, typically in the range

50–150 × 10

−16

(dependent upon the gas) at positronium energies

below 100 meV, have been given by Nagashima et al. (1995), who also

compared their data with geometric cross sections taken from viscosity

measurements. They found that the ratio of these two cross sections

does not increase when going from atoms to molecules and they argued

that this is evidence for the near absence of low energy inelastic (ro-

tational and vibrational) eﬀects in the interactions of positronium with

molecules.

Hyodo (1992) and Nagashima et al. (1995) justiﬁed this by noting that

there is a mismatch between the period, ν

−1

, of internal excitation of the

molecule and the positronium collision time. The latter is determined by

7.4 Slowing down 345

Fig. 7.19. The time dependence of the mean ortho-positronium kinetic energy

measured in silica aerogel under vacuum (◦) and with 0.92 amagat of helium gas

added (•). The two curves were simultaneously ﬁtted to the data by Nagashima

et al. (1998), to obtain estimates of the Ps–He momentum

transfer cross section.

Reprinted from Journal of Physics B31, Nagashima et al., Momentum transfer

cross section for slow p

ositronium–He scattering, 329–339,

permission from IOP Publishing.

short-range van der Waals forces and is approximately a

, where v

is the speed of the positronium and a

is an estimate of the molecular

size. That is, the excitation is only likely when ν

−1

≈ a

. Inserting

values for the spacings between the rotational and vibrational energy

levels, approximately 10

−2

eV and 0.1 eV respectively, it is found that

−1

is of order 4 × 10

−12

s and 4 × 10

−14

s respectively. However, the

positronium interaction time is typically one or two orders of magnitude

smaller, eﬀectively suppressing these processes. This information can also

be gleaned qualitatively from Figure 7.18, where it can be seen that H

a much more eﬃcient positronium moderator than the heavier and more

complex molecules CO

and C

Nagashima et al. (1998) re-investigated helium using a variable ﬁeld,

from 0.16 to 1.5 T, resulting in a lifetime of the m =01

state in

the range 86–3.2 ns. The ACAR spectra of the mixed component were

isolated by subtracting the suitably normalized B = 0 spectrum from

those for B = 0. Broadening of the the mixed component was evident

as the magnetic ﬁeld was increased, a clear manifestation that the ortho-

positronium had had less time to moderate before conversion (and rapid

annihilation) occurred.

346 7 Positronium and its interactions

Nagashima et al. (1998) performed an analysis improved over that

used in their previous work by using a Boltzmann-equation approach in

which an average momentum transfer cross section was considered as an

adjustable parameter; this was used to ﬁt the average positronium kinetic

energy derived from the ACAR spectra. However, they had to build

into this approach an energy-dependent, and therefore time-dependent,

term to allow for the eﬀects of collisions with the grains. Values for the

time dependence of the average positronium kinetic energy are shown in

Figure 7.19, which gives separate results for the aerogel when evacuated

and with the helium added. The lines show simultaneous ﬁts to the data

for the two cases, which yielded a value of σ

equal to (11±3)×10

−16

below 0.3 eV. This is much larger than the value derived from a lifetime

experiment by Spektor and Paul (1975) (see Charlton and Laricchia, 1991,

for a discussion of that work) but is in much closer accord with the

available theoretical estimates (see section 7.2). Coleman et al. (1994)

reported a lower-bound value of 7.9×10

−16

from a simple analysis of

their ACAR data. This is similar in magnitude to the values obtained at

somewhat higher positronium kinetic energies using the beam technique

(see section 7.6).

The TRDBS technique (Skalsey et al., 1998) involves measuring the

Doppler broadening of the 511 keV photons emitted as a result of Ps

annihilation using a high-resolution Ge detector (see subsection 1.3.2 and

Figure 1.5) in a particular time interval. The γ-ray energy information,

which is a measure of the longitudinal positronium momentum upon

annihilation, was recorded in the time range 30–50 ns for the perturbed

m = 0 ortho-positronium component, which had a vacuum lifetime of

52 ns in the magnetic ﬁeld of 0.285 T applied to the gas. Skalsey et al.

(1998) discussed how this component of the γ-ray spectrum was isolated

from other contributions, notably that due to free-positron annihilation.

The essence of the technique is that the ortho-positronium energy is

measured in the ﬁxed time interval at diﬀerent gas densities, so that

a measure of the thermalization, and hence the momentum transfer cross

sections, can be obtained. It is noteworthy that only collisions with the

gas need to be considered; thus the thermalization model is independent

of eﬀects due to the aerogel, which was used in most of the ACAR

work.

Accordingly, from equation (7.25) of Sauder (1968), Skalsey et al.

(1998) plotted arccoth [(E/E

)

1/2

] against ρt, where t =38nsisthe

weighted average of the delayed time window. Data for several gases are

shown in Figure 7.20, where qualitative accord with the elastic scattering

model in the energy range from approximately 0.3 eV to 2 eV is evident.

All gases approach an intercept as ρt→0 of around 2–4 eV, which is

7.4 Slowing down 347

Fig. 7.20. Fits performed by Skalsey et al. (1998) to Sauder’s positronium

thermalization model using their TRDBS data. The slopes yield the positronium

thermalization rate, whilst the intercepts give the average initial energy. (For

clarity only the ﬁtted lines are shown for He, Ne and iso-C

.) Reprinted

from Physical Review Letters 80, Skalsey et al., Thermalization of positronium

consistent with the Ore model of positronium formation in dense gases

described in section 4.8. The value of Γ and hence σ

can be derived

from Figure 7.20, and Skalsey et al. (1998) found that σ

/M varies by a

factor of around 20 for the gases investigated. Their values of σ

range

from 2.3±0.4

for He to 208±17

for C

. Thus, the value for He,

though at a somewhat higher average energy than that for the study of

Nagashima et al. (1998), is not in accord with the latter. It is also evident

that the conclusions of Skalsey et al. (1998) are not in accord with those

of Nagashima et al. (1995), noted above; these workers found that the

thermalization rate, as determined in their ACAR study, was the same

for He, Ne and iso-C

Although simplifying assumptions have had to be made in analysing the

data obtained from both types of experiment described in this section, it is

clear that valuable information on positronium scattering and interactions

at low energies can be obtained, particularly from the TRDBS technique.

It is hoped that this work will complement the experimental data which

are becoming available at higher kinetic energies from positronium-beam

studies and which oﬀer a stimulus to renewed theoretical eﬀorts.

348 7 Positronium and its interactions

7.5 Bound states involving positronium

A bound system containing a positron may be considered either as positro-

nium bound to a positive ion or an atom or as a positron bound to the

corresponding atom or negative ion. It is therefore appropriate to consider

here all bound states containing positrons. Bound systems containing

more than one positron, however, such as e

−

(the anti-system of

the positronium negative ion Ps

−

) and the positronium molecule Ps

, are

considered in more detail in sections 8.1 and 8.2.

Until recently there was no ﬁrm theoretical evidence that a positron

could bind to any atom other than positronium; it had been rigorously

proved by Armour (1978, 1982) that it cannot bind to atomic hydrogen,

and the evidence that it cannot bind to helium is overwhelming. The most

likely candidates were the highly polarizable alkali atoms, and states of the

positron–atom system below the positron–atom scattering threshold do

indeed exist. However, they were all believed to lie above the threshold

for positronium scattering by the corresponding positive ion, and were

therefore not true bound states.

Evidence of true bound states then began to emerge, possible candi-

dates being magnesium, zinc, cadmium and mercury (Dzuba et al., 1995)

and the HF molecule (Danby and Tennyson, 1988); nevertheless doubts

remained, in view of the fact that the models of the systems

to which

positrons were supposed to be bound were not exact. Recently, however,

it has been rigorously established by Ryzhikh and Mitroy (1997) that a

true bound state of the positron–lithium atom exists below the threshold

for positronium–Li

scattering. This state is therefore more appropriately

described as positronium bound to the Li

ion, and the binding energy

with respect to break-up into positronium and Li

was calculated to be

0.065 eV. Ryzhikh and Mitroy used the Rayleigh–Ritz variational method

with the full non-relativistic Hamiltonian and a suitably antisymmetrized

trial wave function; this consisted of the sum of a large number of Gaussoid

basis functions, each containing several non-linear variational parameters

and of the form

= exp

−

N−1



µ,ν=1

µν

, (7.26)

where the x

are Jacobi coordinates of the system and the A

µν

are

variational parameters. Minimization of the lowest energy eigenvalue with

respect to these parameters was achieved using the stochastic variational

method. Because the variational method yields an upper bound on the

energy of the system, any value obtained for this quantity that is below

the lowest scattering threshold provides rigorous proof of binding.

Similar methods were used by Ryzhikh, Mitroy and Varga (1998a, b)