Charlton M., Humberston J.W. Positron Physics

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4.8 Dense gases 209

The simple Ore model described above has been used extensively in

discussing positronium formation in dense gases, though there is no

a priori reason to assume that the uniform-energy TD distribution is

valid. Indeed, Paul and coworkers (e.g. Paul and B¨ose, 1982) found, in an

investigation of positron drift in dilute gases, see section 6.4, that their

data could best be ﬁtted using a phase-space Ore model in which the TD

distribution below E

is uniform in positron momentum, not energy. The

modiﬁed Ore predictions for F

max

and F

min

are similar to those given in

equations (4.38) and (4.39), except with the energies raised to the power

3/2. The band of allowed F -values thus diﬀers substantially from those

given by equations (4.38) and (4.39). Comparisons with experiment can

be found below.

Another model of positronium formation, the so-called spur model,

was originally developed by Mogensen (1974) to describe positronium

formation in liquids, but it has found some applications to dense gases.

The basic premise of this model is that when the positron loses its last few

hundred eV of kinetic energy, it creates a track, or so-called spur, in which

it resides along with atoms and molecules (excited or otherwise), ions and

electrons. The size of the spur is governed by the density and nature

of the medium since these, loosely speaking, control the thermalization

distances of the positron and the secondary electrons. It is clear that

electrostatic attraction between the positron and electron(s) in the spur

can result in positronium formation, which will be in competition with

other processes such as ion–electron recombination, diﬀusion out of the

spur and annihilation.

A semi-quantitative picture of positronium formation in a spur in a

dense gas was developed by Mogensen (1982) and Jacobsen (1984, 1986).

If the separation of the positron from an electron is r, and there is assumed

to be only one electron in the spur (a so-called single-pair spur), then

the probability of positronium formation in the spur, in the absence of

other competing processes, can be written as [1 − exp(−r

/r)]; here r

the critical, or Onsager, radius (Onsager, 1938), given for a medium of

dielectric constant  by

= e

/(4πk

T ), (4.40)

which is the separation at which the attractive Coulomb energy is equal

to k

T . Thus, if the pair separation upon thermalization, R,is r

positronium formation is unlikely.

Jacobsen (1984) gave a full discussion of the eﬀect of thermalization and

concluded that positronium formation by the spur mechanism is unlikely

in atomic gases, since R  r

irrespective of density. This is not the case

for molecular gases, where R can be of a similar order of magnitude to r

at high densities. Thus, the positronium formation fraction in molecular

210 4 Positronium formation

gases is expected to be both density and temperature dependent, and it

has been semi-quantitatively expressed by Jacobsen (1986), in a combined

Ore-plus-spur approach, as

F = F

+(1− F

)[1 − exp(−r

/R)] exp(−λ

), (4.41)

where F

is the low density, or Ore, value and is assumed to be con-

stant. The factor exp(−λ

) is the probability that the positron has

not annihilated before positronium has formed; τ

is the time taken for

positronium to form in the spur. In many cases this factor appears to be

close to unity, though it can be important at high gas densities and if the

particles are immobile (see subsection 6.3.3). Jacobsen (1986) provided

estimates of τ

Finally, Zhang and Ito (1990) proposed the resonant model, which they

claim is more successful than the Ore and spur approaches in explaining

data for a wide variety of systems. This model seems to incorporate

various aspects of these other models, although one crucial diﬀerence is the

proposal that positronium may be created via the resonant formation of

intermediate excited complexes. However, at the time of writing, there is

no evidence from positron–gas scattering experiments that the Ore model

needs to be modiﬁed for atomic species by the addition

of an intermediate

stage involving positronic complexes. For dense molecular gases, only the

conventional spur model has been applied so far and it can oﬀer qualitative

explanations of the data, although an extension of the type suggested by

Zhang and Ito (1990) cannot be ruled out.

2 Positronium formation fraction – results

This section is devoted to experimental values of F and comparisons with

predictions from the models described above. The data discussed have

mainly been obtained since 1975 because, as outlined by Coleman et al.

(1975a), earlier results, especially for the noble gases, are thought to be

prone to systematic errors.

We ﬁrst consider the noble gases, together with various mixtures con-

taining them. Our discussion is based largely around the work of Coleman

et al. (1975a), Griﬃth and Heyland (1978) and Wright et al. (1985), and

the Ore model is used to interpret the results since, as described above,

there is no contribution to F from spur processes. A selection of values

of F

min

and F

max

, see equations (4.37)–(4.39), is shown in table 4.1 along

with the observed fractions. The values for helium, neon and argon, which

are found experimentally to be independent of gas density, lie between

min

and F

max

. Thus, positronium formation in these gases is usually

regarded as being in accord with the Ore model. Surprisingly, though,

4.8 Dense gases 211

Table 4.1. Ore model parameters for both standard and modiﬁed Ore ap-

proaches (F

min

, F

max

, F

mod

min

and F

mod

max

), and the experimental fractions F for the

noble gases and a variety of molecules. Note that when E

, the minimum

predictions have been set to zero; see equation (4.38). See Charlton (1985a) for

the origin of the measurements. In general, the fractions for the molecular gases

have been found to be both density and temperature dependent. The value

quoted here is for low densities and is thus expected to be the Ore contribution

to the overall positronium fraction in these gases at higher densities

Gas F

min

mod

min

max

mod

max

He 0.14 0.19 0.28 0.38 0.23

Ne 0.09 0.11 0.32 0.43 0.26

Ar 0.17 0.20 0.43 0.57 0.33

Kr 0.20 0.24 0.49 0.63 0.11

Xe 0.26 0.29 0.56 0.71 0.03

0.18 0.21 0.44 0.58 0.32

0 0 0.44 0.58 0.19

0 0 0.49 0.64 0.40

0 0 0.56 0.71 0.32

CO 0.06 0.06 0.49 0.63 0.28

0.18 0.20 0.52 0.67 0.40

the values of F for krypton and xenon, based upon the measured intensity

of the ortho-positronium component, fall below F

min

Experiments on these two gases, reported by Griﬃth and Heyland

(1978), showed that a fast component, with a density-dependent decay

rate, was present in the lifetime spectra, and this was tentatively linked

to the dearth of long-lived ortho-positronium. Furthermore, it was found

for mixtures of krypton with helium that the maximum value of F , which

was observed at a concentration of around 0.01% of krypton, was in excess

of the sum of the individual F -values for the two gases when pure.

Similar features were later found by Wright et al. (1985), whose exper-

iments were an attempt to shed further light on the low measured values

of F in krypton and xenon by a detailed study of the lifetime spectra

of the pure gases and mixtures. The spectrum obtained for xenon at a

density of 9.64 amagat at room temperature is shown in Figure 6.5, where

the paucity of long-lived ortho-positronium is apparent, particularly when

compared with the accompanying spectrum for argon. Also, clearly re-

solved from the prompt peak are the so-called fast components, which

Wright et al. (1985) were able to ﬁt to two overlapping exponentials.

212 4 Positronium formation

These authors also found that adding gases to krypton and xenon

markedly speeded up the fast components and also raised the measured

value of F . Their conclusion was that the fast components were indeed

due to the formation of positronium but that the subsequent interactions

of the positronium led to rapid annihilation. When this contribution was

included in the evaluation of F , its value was found to be in accord with

the Ore model. The model proposed by Wright et al. (1985) and later

supported by the work of Kakimoto and Hyodo (1988), who used the

ACAR technique (see sections 7.3 and 7.4), is that positronium moder-

ation plays an important role, i.e. once formed, with a distribution of

kinetic energies, the ortho-positronium moderates only slowly in these

heavy gases. During this time the ortho-positronium can undergo col-

lisions with the krypton and xenon atoms; this leads to quenching and

the appearance of fast components. The addition of small quantities of

impurities, particularly of low mass, speeds up the moderating process,

making quenching collisions less likely and increasing the apparent inten-

sity of long-lived ortho-positronium. Wright et al. (1985) discussed their

data in terms of some kind of temporary attachment of the positronium

to krypton and xenon, though the physical nature of this state remains

unclear. Later, an alternative interpretation was oﬀered (Tuomisaari,

Ryts¨ol¨a and Hautoj¨arvi, 1988), based on a suggestion of Manninen (1987),

in which the fast components were a natural consequence of the time

dependence of the pick-oﬀ collision rate. This rate varies as the product

of the quenching cross section and the speed of the positronium. At

higher energies the increased speed clearly enhances the quenching rate,

but the cross section may also be higher since the positronium is then

more likely to overcome the exchange repulsion and penetrate further into

the the electron cloud of the atom. Some quantitative support for this

view comes from early theoretical work of Barker and Bransden (1968)

on the positronium–helium system. This, and other work concerning

positronium interactions, is reviewed in Chapter 7.

The work of Jacobsen (1984, 1986) and Mogensen (1982), described in

subsection 4.8.1 above, pointed to the potential importance of positro-

nium formation as a consequence of spur processes in dense molecular

species. Table 4.1 drew attention to the fact that the positronium frac-

tions for many molecular gases have been found to be both density and

temperature dependent. We will not attempt a detailed compilation of

these data here, but examples of the density and temperature variations

observed are shown in Figure 4.31 for SF

and CO

gases. The lines are

ﬁts to equation (4.41), and the reader is referred to the work of Jacobsen

(1986) for a full discussion of the ﬁtting procedures and assumptions. The

fact that a reasonable ﬁt to the data can be produced is strong supporting

evidence for positronium formation in spurs.

4.8 Dense gases 213

Fig. 4.31. Measured and calculated (see text and Jacobsen, 1986) positronium

fractions F for SF

(left) and CO

(right) gases. The values of the fractions at

297 K and 350 K are displayed oﬀset by +0.3 and +0.6

respectively, for clarity

of presentation.

Two other pieces of experimental evidence also add weight to this

contention. Curry and Charlton (1985), in a study which shadowed

work on liquids performed by Wikander and Mogensen (1982), noted the

eﬀect of introducing small quantities of the electron scavengers CCl

and

CCl

into high density CO

. Brieﬂy, the scavenger can remove free

electrons from the spur and prevent positronium formation, the detailed

behaviour being contingent upon the electron capture cross section, which

governs the capture rate. A deﬁnite scavenging eﬀect was observed and

such behaviour can occur only if spur processes are an important source

of positronium in molecular species. An applied electric ﬁeld can also

deplete the positronium yield in the spur by separating the electron and

the positron before they can unite. Early observations of this eﬀect were

made by Marder et al. (1956) and Obenshain and Page (1962), though

at the time its origin was not clear. More recent work to establish the

mechanism fully was reported by Charlton and Curry (1985) for CO

, and

by Sharma et al. (1985) and Jacobsen, Charlton and Laricchia (1986) for

Excitation and ionization

In this chapter we consider inelastic collisions of positrons with an atomic

or molecular target X which result in electronic excitation or ionization

(without positronium formation) of the target system. These processes

can be summarized as

(E)+X → e

(E − ∆E)+X

∗

, (5.1)

(E)+X → e

(E − ∆E)+X

+ me

−

, (5.2)

where X

∗

is the excited neutral target, E is the kinetic energy of the

positron, ∆E is the amount of energy lost by it in the collision and m

is the degree of ionization. The value of ∆E is ﬁxed, in the case of

reaction (5.1), to be one of the discrete excitation energies of the target

levels, termed E

, but is variable in (5.2), the minimum value being

the ionization energy E

. The ‘total’ cross sections for these processes

will be referred to as σ

and σ

, with the addition of other notation as

necessary to deﬁne cross sections for other projectiles, e.g. electrons. The

total ionization cross section is the sum of the integrated partial-ionization

cross sections, σ

, i.e. σ

Except for section 5.6, the discussion will centre around these integrated

cross sections because for positron impact we cannot as yet distinguish

between the various possibilities in reaction (5.2), where X

may be

left in an excited state. Ignorance of the true ﬁnal state may be par-

ticularly serious for collisions with molecules, where fragments may be a

combination of charged and neutral species.

In the section on excitation we shall treat only electronic transitions;

thus rotational and vibrational processes in molecules are excluded. As

will be described in Chapter 6, information on these latter processes

has been derived from positron lifetime and other experiments. Our

theoretical discussion will mainly concern excitation of the lower levels of

214

5.1 Excitation 215

hydrogen and helium, which have been the most comprehensively studied

target systems. The sparse experimental work is conﬁned to estimates of

angle-integrated cross sections, which, owing to poor energy resolution,

may contain contributions from more than one channel. Very little is

known from either theory or experiment concerning the excitation of

molecules by positrons, although some data exist for O

The main body of this chapter deals with ionization, including some

of the theoretical approaches and associated results for the simpler tar-

gets. There is a growing body of experimental data on various aspects

of positron impact ionization, and this will be reviewed in some depth.

Comparisons with data for other simple projectiles will be made where

appropriate, in an attempt to shed light on the mechanisms involved

in ionizing collisions. Work has begun on measurements of energy- and

angle-resolved cross sections, and this will be described in section 5.6.

5.1 Excitation

1 Theory

Positron impact excitation of a target system from an initial state (usually

the ground state) with energy E

to a ﬁnal excited state with energy E

only possible if the energy of the incident positron exceeds the threshold

value E

= E

− E

. The magnitudes of the initial and ﬁnal momenta

of the positron, k

and k

respectively, are then related through energy

conservation by

+ E

. (5.3)

In electron scattering, where there is exchange between the projectile

and the electrons in the target, all energetically accessible states of the

target can be excited, but in positron collisions only those transitions

in which the total spin of the target does not change are allowed. As

an example, the lowest excited state of helium that can be reached by

positron impact is the 2

S state, with an energy of 20.58 eV, and not the

ﬁrst excited state, 2

S, with an energy of 19.8 eV.

Consider a transition from an initial state, in which the target system

is in its ground state, to a ﬁnal state, in which the target system has been

excited to state f . In terms of the T-matrix element, the diﬀerential cross

section for scattering through the angle between k

and k

dσ

dΩ





4π

, (5.4)

where T

= φ

|V |ϕ

. Here, ϕ

is the exact total wave function of the

positron–target system, φ

is the product wave function for a positron

216 5 Excitation and ionization

with momentum k

and the target system in the ﬁnal state and V is

the positron–target interaction potential. The total cross section is then

obtained by integration over all directions of k

The simplest approximation is to replace ϕ

by φ

, the product wave-

function for a positron with momentum k

and the target system in its

ground state; this yields the ﬁrst Born approximation, T

= φ

|V |φ

.

The positron–target interaction potential is equal in magnitude but op-

posite in sign to the corresponding potential for electrons and therefore, if

electron exchange is ignored, the cross sections for electron and positron

excitation in the ﬁrst Born approximation are equal. At suﬃciently high

energies, several hundred eV for hydrogen, the ﬁrst Born approximation

yields accurate results for the angle-integrated excitation cross sections.

However, except for a small angular range about the forward direction,

which becomes progressively smaller as the projectile energy is increased,

the diﬀerential excitation cross section in this approximation falls away

much too rapidly with increasing angle of scattering. This deﬁciency can

only be remedied by including some form of second Born amplitude into

the calculation of the diﬀerential cross section, as was done by Byron,

Joachain and Potvliege (1985).

It was shown by Omidvar (1975) that the ﬁrst Born approximation to

the cross section for excitation of atomic hydrogen from its ground state

to an excited state with principal quantum number n

is proportional to

1/n

, where the constant of proportionality depends on the value of the

orbital angular momentum quantum l. Strictly, this scaling law is only

valid at high incident positron energies and for large values of n

, but

it has been used at relatively low energies to estimate the cross sections

for excitation to higher excited states when more accurate methods have

been used to calculate the cross sections for excitation to a few speciﬁc

low-lying states (see e.g. the discussion for helium in section 2.7).

Improvements over the ﬁrst Born approximation at intermediate en-

ergies can be made by taking account of the distortion of the positron

wave function in both the initial and the ﬁnal states, as is done in the

distorted-wave approximation. The T-matrix element then takes the form

= ϕ

|V |ϕ

, where ϕ

and ϕ

are wave functions representing elastic

scattering of the positron in the potential ﬁelds of the target in the initial

and ﬁnal states respectively. In addition to the static interaction, these

potentials usually incorporate some allowance for polarization of the tar-

get. This approximation has been used by several authors, most notably

by McEachran and his collaborators, to determine various excitation cross

sections for the noble gases (for helium, Parcell, McEachran and Stauﬀer,

1983, 1987; for neon, Parcell, McEachran and Stauﬀer, 1990). Particular

attention has been given to calculating the cross section for resonant

excitation from the ground state to the lowest optically allowed excited

5.1 Excitation 217

Fig. 5.1. Cross sections for the excitation of atomic hydrogen obtained using

the coupled-state

approximation with 30 hydrogen states and pseudostates

plus

three positronium states: (a) 1S–2S; (b) 1S–2P (Kernoghan et al., 1996).

state; this provides the dominant contribution to the total excitation cross

section σ

In a complete treatment of positron scattering, all the open chan-

nels, including excitation, are coupled together and the cross sections

for all the various scattering processes are determined together. Such an

approach has been adopted for positron–hydrogen scattering using the

coupled-state method, most notably by Kernoghan, McAlinden and Wal-

ters (1995), Kernoghan et al. (1996) and Mitroy and Ratnavelu (1995).

In the intermediate energy region the most accurate results are probably

those obtained by Kernoghan et al. (1996), using a 33-term expansion

of the total wave function which included three positronium states (1s,

2s, 2p) and 30 hydrogen states, some of which were pseudostates. These

authors determined the cross sections for excitation to the 2S and 2P

states and also for elastic scattering, positronium formation into vari-

ous states and ionization. The 1S–2S excitation cross section, shown in

Figure 5.1(a), exhibits quite a steep rise from the threshold, at 10.2 eV,

up to a value of 0.33πa

at approximately 15 eV. Thereafter it falls

steadily, with a roughly exponential shape, to a value of approximately

0.06πa

at 100 eV. In contrast, the resonant 1S–2P excitation cross sec-

tion, Figure 5.1(b), rises somewhat less steeply from the threshold but

reaches a signiﬁcantly larger maximum, 0.9πa

, at 20 eV; this is followed

by a broad plateau region, after which the cross section falls slowly to

0.7πa

at 100 eV. At energies beyond 40 eV it is similar in value to the

total ionization cross section, and these two cross sections become equally

dominant contributions to the total scattering cross section. Calculations

of excitation cross sections for hydrogen at higher energies were made by

Byron et al. (1985). There have been no experiments on the excitation of

atomic hydrogen to date.

218 5 Excitation and ionization

The coupled-state approximation has also been used by Hewitt,

Noble and Bransden (1992b, 1993, 1994) and Kernoghan, McAlinden

and Walters (1996) to investigate positron impact excitation of alkali

atoms, these being treated as equivalent one-electron atoms. The most

detailed investigations of positron–lithium scattering in the energy range

0–60 eV have been probably those of McAlinden, Kernoghan and Wal-

ters (1997), who used the three lowest states of positronium and 29

states and pseudostates of lithium: these investigations reveal that the

cross section for 2S–2P excitation becomes the dominant contribution to

the total cross section just a few eV above the threshold and remains

thus throughout the energy range considered. Other applications of

the coupled-state approximation to lithium were made by Ward

et al.

(1989), McEachran, Horbatsch and Stauﬀer (1991) and Khan, Dutta and

Ghosh (1987) but these authors excluded all positronium terms from the

expansion of the wave function. As stated previously in section 3.2, see

equation (3.28), an expansion in terms of the target eigenstates alone is

formally complete, but the convergence of the results is poor at energies

where positronium formation is signiﬁcant. At higher energies, however,

where positronium formation is less probable, the eﬀect on the elastic

scattering and excitation cross sections of neglecting positronium terms in

the expansion of the wave function becomes insigniﬁcant. These features

are clearly illustrated in Figure 5.2, in which various results for the 2S–2P

excitation cross section for lithium are displayed. At positron energies

below 30 eV an expansion without positronium terms yields results which

are signiﬁcantly larger than those obtained when positronium terms

are included, presumably in an attempt to compensate for the lack of

ﬂux into the positronium formation channels. This eﬀect is particularly

pronounced at energies just a few eV above the excitation threshold.

Similar coupled-state methods, both with and without the inclusion

of positronium terms, have been applied to the excitation of other alkali

atoms. The results of McAlinden, Kernoghan and Walters (1994, 1997)

and Hewitt, Noble and Bransden (1994) for the dominant resonant excita-

tion cross sections for sodium, rubidium and caesium all exhibit a similar

energy dependence to that for lithium. Also, the neglect of positronium

terms in the expansion, as in the work of McEachran, Horbatsch and

Stauﬀer (1991), again has the eﬀect of increasing the low energy excitation

cross sections over those obtained when such terms are included.

Several diﬀerent approximation methods have been used to investigate

excitation in positron–helium scattering, but in all cases rather simple

uncorrelated wave functions have been used to represent the ground and

excited states of helium. All the reported results, which relate almost

exclusively to the excitation of 2

S and 2

P states, exhibit a steady rise

from the threshold followed by a gentle fall, which continues up to a few