Charlton M., Humberston J.W. Positron Physics

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5.2 Ionization – theoretical considerations 229

Again, the value of the constant of proportionality and the range of

validity are not given but Klar estimated that this power law was valid

over an energy range of a few eV above threshold. As in Wannier’s original

theory, the energy is predicted to be shared equally between the positron

and the electron. Rost and Heller (1994) conﬁrmed this power law using

a semi-classical treatment of ionization and they too concluded that the

range of validity was probably more than 3 eV above the threshold.

A rather diﬀerent theory of electron impact ionization was developed by

Temkin (1982); it was based on the assumption that one of the electrons

remains closer to the core than the other, so that the outer electron

moves in the dipole ﬁeld produced by the inner electron and the core.

According to this Coulomb-dipole theory the ionization cross section has

a modulated quasi-linear energy dependence of the form

∝

(ln E

)





sin(α

ln E

+ µ

)



, (5.9)

where α

and µ

are constants, and this should apply equally to positrons

and electrons. In contrast to Wannier’s, this theory predicts an un-

equal division of the energy between the two emerging light particles,

the positron energy being the greater. Even if the theory is correct, its

range of validity may be too small to be amenable to experimental test.

Ashley, Moxom and Laricchia (1996) measured the positron impact-

ionization cross section in helium and found that its energy dependence

up to 10 eV beyond the threshold was quite accurately represented by a

power law, as in equation (5.8), but with the exponent having the value

2.27 rather than Klar’s value of 2.651. This discrepancy prompted Ihra

et al. (1997) to extend the Wannier theory to energies slightly above

the ionization threshold using hidden crossing theory. They derived a

modiﬁed threshold law of the form

∝ E

2.640

exp(−0.73E

1/2

), (5.10)

which reduces essentially to Klar’s power law for suﬃciently low energies.

As will be shown in subsection 5.4.5, this latter form provides a good ﬁt

to the experimental data of Ashley, Moxom and Laricchia (1996) up to

10 eV above the ionization threshold, although the ﬁt is no better than

that of the simple power law with a modiﬁed exponent.

Although the assumptions made when developing these various theories

of threshold ionization may be valid at energies very close to the threshold,

there must be some doubt that the range of validity really does extend as

far as 10 eV above a threshold.

A variety of approximation methods have been applied to positron

impact ionization and, as with other scattering processes, the most at-

tention has been given to atomic hydrogen and helium. The ﬁrst Born

230 5 Excitation and ionization

approximation becomes accurate at suﬃciently high projectile energies,

typically 1 keV for atomic hydrogen, the results for single ionization being

the same for both electrons and positrons. In this approximation the

scattered positron is assumed to be screened from the residual ion by

the slower electron, and its wave function is therefore represented by a

plane wave, as also is the wave function of the incident positron. The

wave function of the emitted electron, however, has usually been taken to

be a Coulomb wave. Improvements over this simple approximation have

been made by representing the wave function of the scattered positron

in the distorted-wave approximation (Basu, Mazumdar and Ghosh, 1985;

Ghosh, Mazumdar and Basu, 1985; Mukherjee, Singh and Mazumdar,

1989; Mukherjee, Basu and Ghosh, 1990). The ionization cross section

then has quite a sensitive dependence on the form of the ﬁnal state, par-

ticularly at energies not far above the ionization threshold. Distortion of

the wave function of the incident positron from its plane wave form should

also be incorporated into the formulation of the ionization process, but

Basu, Mazumdar and Ghosh (1985) found that the results are relatively

insensitive to this feature.

A more complete formulation of positron and electron impact ionization

of hydrogen at intermediate energies was given by Brauner, Briggs and

Klar (1989); in this formulation three-body Coulomb functions were used

to represent the ﬁnal state in the asymptotic region. These authors

calculated the triple diﬀerential cross section for various energies of the

incident projectile and the ejected electron. At a given incident energy

the magnitude of the cross section decreases rapidly with increasing mo-

mentum transfer of the positron; attention was therefore given to the

asymmetric kinematics in which the angle of scattering of the positron

(or the incident electron) is small and the energy of the ejected electron

is much less than that of the scattered projectile. Examples of the results

obtained for both positrons and electrons, expressed as functions of the

angle of ejection of the electron relative to the incident beam direction,

are given in Figure 5.7. Here the incident energy of the projectile is

150 eV, its angle of scattering is 4

◦

and the energy of the ejected electron

is 3 eV. There are two maxima in the angular distribution. The binary

peak at positive angles (which corresponds to emergence of the initially

bound electron on the opposite side of the incident direction to that of the

scattered projectile) arises from a direct collision between the projectile

and the electron, with the nucleus as a spectator. The recoil peak at

negative angles is the result of double scattering, where the bound electron

is ﬁrst struck by the projectile and then scattered by the nucleus.

If the energy of the scattered positron is very similar to that of the

ejected electron the two particles may emerge in almost the same direction

and in a highly correlated state, which can be considered as a continuum

5.2 Ionization – theoretical considerations 231

Fig. 5.7. The triple diﬀerential cross section for the ejection of electrons from

the ionization of atomic hydrogen by positron and electron impact, expressed

as a function of the angle made by the ejected electron to the incident beam

direction. The energy of the incident projectile is 150 eV, its angle of scattering

is 4

◦

, the energy of the ejected electron is 3 eV and coplanar geometry is adopted

(Brauner, Briggs and Klar, 1989). Positive angles correspond to the ejected

electron

emerging on the opposite side of the incident beam direction

to that of

the scattered projectile, and negative angles correspond to the electron emerging

on the same side. ——, theoretical results for electron impact ionization; – – –,

theoretical results for positrons; ·····, results of the ﬁrst Born approximation,

which are the same for positrons and electrons; the experimental points are

for electrons (Ehrhardt et al., 1985). Reprinted from Journal of Physics B20,

Brauner, Briggs and Klar, Triple diﬀerential cross sections for ionization of

permission from IOP Publishing.

state of positronium. This so-called ‘electron capture to the continuum’

(ECC) gives rise to a sharply peaked structure in the triple diﬀerential

ionization cross section when the momenta (and thus the kinetic energies)

of the two emerging particles are equal. This feature was ﬁrst revealed

in studies of positron impact ionization by Brauner and Briggs (1986)

that used the ﬁrst Born approximation to the T-matrix element for the

transition, T

ﬁ

= Φ

|V |Φ

, where V is the projectile–target interaction

potential. For positron impact ionization of atomic hydrogen, the initial

state is Φ

=(2π)

−3/2

) exp(ik ·r

), i.e. the product of the hydrogen

target wave function and a plane wave for the incident positron; the ﬁnal

232 5 Excitation and ionization

Fig. 5.8. The triple

diﬀerential cross section for positron impact ionization of

atomic hydrogen, expressed as a function of the energy of the ejected electron.

The scattered positron and electron both emerge in the direction of the incident

positron beam, the energy of which is 1 keV: ——, theoretical result obtained

using a Coulomb wave to describe the relative motion of the positron and elec-

tron; – – –, result obtained using a plane wave to describe the relative motion

(Brauner and Briggs, 1986). Reprinted from Journal of Physics B19, Brauner

and Briggs, Ionization to the projectile continuum by positron and electron

IOP Publishing.

state was taken to be Φ

=(2π)

−3/2

−

) exp(iK



·ρ), i.e. the product

of a Coulomb function representing the motion of the positron relative to

the electron and a plane wave for the motion of the centre of mass of the

unbound positronium relative to the nucleus. An example of the structure

found by Brauner and Briggs (1986) is shown in Figure 5.8, which gives

the cross section for the ionization of atomic hydrogen by 1 keV positrons

when both particles emerge in the direction of the incident beam. The

cross section exhibits a singularity when the energy of the ejected electron

is equal to that of the scattered positron. Similar singular structure in the

triple diﬀerential cross section is found at all angles when the momenta

of the ejected electron and the scattered positron are equal.

As the energy of the incident positron is increased beyond several

keV it becomes increasingly valid to consider the mechanism of electron

capture into either bound or continuum states of positronium in terms of

5.2 Ionization – theoretical considerations 233

double-binary collisions (see Chapter 4), as a result of which the positron

and electron are expected to emerge preferentially at a critical angle of

◦

to the incident beam direction. There are two such types of double

collision, which were illustrated in Figure 4.6, for the case of capture

into a bound state; interference occurs between the amplitudes for the

two processes, the interference being destructive for electron capture into

even-parity states of positronium. Brauner and Briggs (1991, 1993) found

that the eﬀect of this interference was to introduce a sharp dip in the triple

diﬀerential cross section on the low electron energy side of the the ECC

singularity structure when the angle of emergence of both particles was

◦

. The dip and the singularity tend to coincide as the incident positron

energy is increased beyond 10 keV, but at such high energies the ionization

cross section is very small.

Some of the most detailed studies of positron impact ionization of

hydrogen in the intermediate energy range, 20–120 eV, were made by

Kernoghan, McAlinden and Walters (1995) and Kernoghan et al. (1996)

in the course of their elaborate coupled-state calculations of positron–

hydrogen scattering, although no continuum ionization states were ex-

plicitly included in their expansion of the wave function. In addition to

eigenstates of hydrogen and positronium, their wave functions included

several pseudostates, each of which has a non-zero overlap with all the

bound and continuum target states. The fraction of a particular pseu-

dostate |φ

 that represents the ionization continuum is

=1−



φ

n,l

|φ

, (5.11)

where the summation is over all bound states. The following ansatz was

then used to estimate the ionization cross section:



(1 − a

)[σ

(m)+σ

(m)], (5.12)

where σ

(m) and σ

(m) are the cross sections for excitation of the mth

pseudostate of hydrogen and for the formation of positronium into this

mth pseudostate respectively. These are calculated in the same way, and

at the same time, as are the cross sections for excitation of the target

and for positronium formation into the various eigenstates represented

in the coupled-state expansion. This procedure, although not completely

rigorous, is expected to provide reasonably accurate results if a large

number of states and pseudostates are included in the wave function. A

similar technique was used by Campbell et al. (1998a) to determine the

ionization cross section in positron–helium scattering.

The convergent-close-coupling (CCC) method, which was originally

developed for electron–hydrogen scattering, has also been applied to

234 5 Excitation and ionization

positron impact ionization of atomic hydrogen over the energy range

from the threshold up to 700 eV (Bray and Stelbovics, 1994). Positronium

formation is not explicitly included in this formulation and therefore the

results should be interpreted as an approximation to the total break-up

cross section, i.e. the sum of the cross sections for ionization and positro-

nium formation. Only at energies beyond 100 eV, where the positronium

formation cross section is known to be small, should the CCC results

be considered as relating solely to ionization. Beyond 200 eV the CCC

results are in good agreement with those of the Born approximation,

but this is almost certainly fortuitous and does not imply that the Born

approximation is valid at such a low energy. In the energy interval

100–700 eV the CCC results are in good agreemen

t with the experimental

measurements; these are summarized in subsection 5.4.1.

Double ionization has been investigated experimentally for positron,

electron, proton and antiproton impact but, although the process has

received much theoretical attention for electron and proton impact, few

theoretical studies have yet been made for positrons.

5.3 Ionization – experimental techniques for integrated cross

sections

The ﬁrst studies of positron impact ionization were based around the

TOF systems originally developed for total cross section measurements;

see section 2.3 and, for example, Griﬃth et al. (1979b), Coleman et al.

(1982), Mori and Sueoka (1984, 1994), though only the latter workers,

as described in section 5.1, claimed to be able to distinguish between

ionization and excitation using the TOF technique.

The ﬁrst attempts to make direct measurements of ionization cross

sections were performed by the Arlington group (Diana et al., 1985) using

an apparatus similar to that described in section 4.3 but set to count

electrons liberated from helium gas. The results obtained have now been

superseded by those obtained using the methods described below.

Detailed studies of σ

for helium and H

were ﬁrst reported by the

Bielefeld group (Fromme et al., 1986, 1988). The arrangement used for

this work was described in detail in section 4.3. Here we recall that

ionization was distinguished from positronium formation, for which there

is also a remnant ion, by detection of the scattered positron and the ion

in coincidence. Normalization to electron-impact-ionization cross sections

at intermediate energies was used to set the absolute scale since, for

helium, convergence to the ﬁrst Born values had been obtained. This

approximation, as described in section 5.2, is independent of the sign of

the projectile charge and thus predicts equal cross sections for positrons

and electrons.

5.3 Ionization – experimental techniques 235

Fig. 5.9. Schematic illustration of the apparatus of Knudsen et al. (1990) to

measure ionization cross sections. Reprinted from Journal of Physics B23,

Knudsen et al., Single ionization of H

, He, Ne and Ar by positron impact,

In addition to this work, the Bielefeld group have also measured σ

for

atomic hydrogen. The apparatus used by Spicher et al. (1990) and later by

Hofmann et al. (1997) is similar to that employed by Sperber et al. (1992)

to measure positronium formation cross sections and was described in

section 4.3. Coincidences between scattered positrons and ions were used,

by virtue of a TOF analysis, to distinguish ions of diﬀerent charge-to-mass

ratios. It is also worth noting here that the scattered positron, which was

crucial in identifying the ionization event, had an angular acceptance

limited to be less than ±30

◦

with respect to the incident beam axis as

ﬁxed by the geometry of the detection system.

The apparatus developed for investigations of positron impact ion-

ization by the Aarhus–UCL collaboration (Knudsen et al., 1990), and

subsequently used in several adaptations, is illustrated in Figure 5.9. The

positron beam was guided by an axial magnetic ﬁeld of around 5 mT.

This was increased to 7.5 mT just in front of the scattering cell and was

maintained at this level from the cell to the detector in order to provide

more eﬃcient conﬁnement of the scattered positrons. The exit aperture

of the scattering cell was chosen to be 25 mm in diameter, and a 40 mm

active diameter channel electron multiplier array (CEMA) was used to

count the beam, both dimensions having been chosen to ensure that as

large a fraction as possible of the scattered positrons struck the detector.

A detailed discussion of the transport of scattered particles was given by

Knudsen et al. (1990).

236 5 Excitation and ionization

The gas cell contained a pair of parallel plates (40 mm × 40 mm) sep-

arated by 20 mm, which served as ion extraction electrodes when appro-

priate voltages, ±V

extr

were applied. The upper plate had a grid-covered

aperture located centrally and through which ions could be extracted.

Such ions passed along a ﬂight tube, where they were further accelerated

by a voltage 4.5 ×V

extr

and focussed onto the cone of a ceratron detector.

This arrangement was also designed to achieve the time focussing of ions

produced at diﬀerent points between the plates, a feature of some use

given the relatively large (5 mm) diameter of the positron beam. Ions

of diﬀerent charge-to-mass ratios could again be distinguished by their

diﬀerent times of ﬂight to the ceratron, whose output, together with

that of the CEMA, was used in a conventional delayed-coincidence timing

arrangement.

One of the features which distinguished the positron method of Knud-

sen et al. (1990) from earlier work with heavy pro

jectiles (see e.g. An-

dersen et al., 1987) was the use of pulsed extraction voltages, in order to

prevent deleterious eﬀects due to beam deﬂection. The pulsing system

was triggered on detection of a positron at the CEMA. Each pulse had a

10 ns rise time and was held at V

extr

for a period of between 1 µs and 3 µs,

depending upon the target. The voltages were applied as soon as possible

after the detection of the positron, the intrinsic delay between ﬁring of

the CEMA and application of the pulse to the plate being approximately

220 ns. Added to this was the extra delay caused by the ﬂight time

of the scattered positron from the gas cell to the CEMA, which varied

with impact energy. Checks were performed by deliberately adding extra

delay into the pulsing system, using H

gas, the lightest species, which

established that the ion extraction eﬃciency was practically unity. This

was also in accord with an estimate of the distance moved by a H

ion

in its random thermal motion.

In order to determine absolute cross sections the ion yield I

, deﬁned

= N

/(PN

vac

), (5.13)

was measured, where N

is the number of ions of a particular charge-

to-mass ratio created in a gas held at a pressure P by a total number

of slow positrons N

vac

. The yield of singly charged ions was found to be

pressure dependent, owing to resonant charge-transfer interactions, but

extrapolation to P = 0 produced a value I

(0) which was independent of

ionic collisional eﬀects. The ionization cross section could then calculated

according to

= I

(0) [P/(nl

)] , (5.14)

where nl is the target areal density and 

is the eﬃciency of the ion

extraction and detection system. Note that the eﬃciency of the CEMA

5.3 Ionization – experimental techniques 237

detector does not enter this expression; because a pulsed extraction

system was used, N

and N

vac

are directly proportional to one an-

other. The quantity in square brackets in equation (5.14) was deter-

mined by performing a normalization of I

(0) to the quoted value of

the electron-impact-ionization cross section for the same target at an

energy of 1000 eV, where the ﬁrst Born approximation was expected to

be valid.

A later version of the Aarhus apparatus, used for an extensive study of

the ionization of atoms and molecules (Jacobsen et al., 1995a, b; Poulsen,

Frandsen and Knudsen, 1994) was similar to the apparatus just described.

The important diﬀerences were: (i) the inclusion of a pair of E ×B plates

in front of the scattering cell to reduce background; (ii) the inclusion of

movable apertures in front of, and behind, the scattering cell to achieve

precise beam alignment; (iii) the use of a retarding element in front of the

scattering cell to narrow the energy width of the beam and also control

the beam intensity; and (iv) the use of a ‘unit accelerator’ behind the

scattering region. The purpose of this last device, which consisted of

a series of resistively coupled biassed rings with the centre ring held at

a high negative potential, was to prevent secondary electrons liberated

from the front plate of the CEMA (which was held at −2 kV) from

returning down the beamline and causing ionization in the scattering

region. This unit replaced a high transmission mesh which performed the

same function in the work of Knudsen et al. (1990). In this later Aarhus

experiment it was also found that by varying the primary beam intensity

(thereby altering the mean time between pulses applied to the extraction

plates) the measured ion yield, which contained ions extracted randomly

by unrelated pulses and mostly originating from positronium formation,

could be extrapolated to zero beam intensity in order to give the true

yield due to ionization alone. This had the eﬀect of removing the necessity

for an empirical correction, as applied by Knudsen et al. (1990), which

relied upon a detailed knowledge of the behaviour of the positronium

formation cross section. Further details were given by Jacobsen et al.

(1995a).

The ﬁnal system described here is that of Jones et al. (1993), which

was developed for positron–hydrogen ionization studies and is illus-

trated in Figure 5.10. Similar apparatus has been used by Ashley,

Moxom and Laricchia (1996) (see subsection 5.4.5 below), Kara et al.

(1997a, b) and Kara (1999). Several of the basic features, including the

pulsed ion extraction and ion transport systems, are similar to those

developed by Knudsen et al. (1990). E × B plates were introduced

by Jones et al. (1993) to remove the slow positrons from the fast β

particles, secondary electrons and gamma-ray ﬂux emanating from the

source.

Fig. 5.10. Apparatus, not to scale, of Jones et al. (1993) for positron impact ionization of atomic hydrogen. Squares with

crosses, Helmholtz coil; shaded rectangles, stainless steel shielding; black rectangles in beam line, lead shielding.