Charlton M., Humberston J.W. Positron Physics

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5.5 Multiple ionization 249

with that for the former projectile being the greater. This was not in

accord with theoretical expectations at the time. McGuire (1982) ﬁrst

suggested that interference between the ﬁrst and second order amplitudes

(or equivalently between mechanisms which involve the projectile in one-

and two-electron encounters) would lead to a term in the cross section

proportional to the cube of the projectile charge, giving the observed

charge dependence of σ

. Most theorists now believe, however, that

the eﬀect is caused by electron–electron correlation during the collision,

although, as summarized by McGuire (1992), there is still disagreement

about the mechanisms by which the correlation operates, with several

theories that involve diﬀerent physical assumptions each ﬁnding general

agreement with experiment.

To date there have been several experimental measurements of σ

for

positron impact. The apparatus and methods used by most of the groups

involved are closely related to techniques developed for single-ionization

studies, as described in section 5.3, and we will not describe these further.

The exception is the work of Hippler and colleagues, whose main focus

was multiple ionization. Their technique was centred around the use of a

pulsed low energy positron beam, provided by a linear accelerator source

(see subsection 1.4.4). The use of a positron pulse enabled a simple ion

time-of-ﬂight system to be used for ionic charge-to-mass discrimination.

More detailed discussions of the technique were given by Helms et al.

(1994a, 1995).

Most authors, after appropriate corrections for the relative detection

eﬃciency of the ions (see e.g. Andersen et al., 1987; Helms et al., 1995),

have presented their data as cross section ratios, R

(m)

= σ

/σ

; these

can then be compared with the corresponding parameter for other, singly

charged, projectiles. Values of R

(2)

for helium are shown in Figure 5.17,

where comparisons are made with data for electron, proton and antipro-

ton impact. It is clear from this ﬁgure that at high speeds (around 1

MeV a.m.u.

−1

) the positron and proton ratios are in broad agreement

but are lower by a factor of two or more than those for their negatively

charged counterparts. As the speed of the positrons is reduced, and hence

the kinetic energy lowered towards the threshold (≈ 79 eV for helium),

the positron and proton data diverge. This can be compared with the

eﬀects described in subsection 5.4.4 and can be understood if the similar

behaviour for R

(2)

−

and R

(2)

¯p

is noted; both result from the fact that less

kinetic energy is carried by the lighter particles, which causes a decrease

in the accessible phase space of the He

and a consequent drop in the

cross section. The fall in R

(2)

and R

(2)

−

occurs in an energy range in which

the value of σ

does not vary strongly for either projectile, and is rising

as E is lowered, so that this fall must be due to a decrease in σ

250 5 Excitation and ionization

Fig. 5.17. Ratio of the cross sections for double ionization and

single ionization,

for positrons, electrons, protons and antiprotons on helium gas. The positron

data (

) are shown explicitly. The broken line is the ‘ratio of ratios’ as deﬁned

by equation (5.15). Reprinted from Journal of Physics B21, Charlton et al.,

Positron and electron impact double ionization of helium, L545–L549, copyright

1988, with permission from IOP Publishing.

Charlton et al. (1988, 1989) also constructed the ‘ratio of ratios’ given

(2)

= R

(2)

−

(2)

¯p

, (5.15)

which, it was hoped, would exhibit similar behaviour to R

(2)

. R

(2)

shown as a broken curve in Figure 5.17, and it can be seen that that this

conjecture has some physical basis. Paludan et al. (1997) showed how the

expression (5.15) gives reasonably good accord also for the heavier noble

gases. They noted that this implies that any trajectory eﬀect, noted above

with respect to positron and electron single ionization, must cancel out

and thus be of equal importance in both single and double ionization. In

addition, exchange eﬀects for electron impact must be either negligible or

of roughly similar importance for the two processes.

Double-ionization studies of the heavier noble gases have also been re-

ported by Charlton et al. (1989), Kruse et al. (1991), Helms et al. (1994a,

1994b, 1995) and Kara et al. (1997a). Although there are small diﬀerences

5.5 Multiple ionization 251

between some of the data reported, overall there is reasonable accord.

Absolute values for double-ionization cross sections and comparison plots,

for both heavy and light projectiles, can be found in the work of Paludan

et al. (1997).

There are a number of points of note. As the target atomic number

increases, R

(2)

becomes larger, reaching 10% for xenon. In addition, the

diﬀerences between the electron and positron ratios become smaller and

they appear to merge around 1 keV impact energy. Also, the form of R

(2)

for both positrons and electrons, with a sharp rise from threshold followed

by a plateau or minimum and then a further rise, suggests that there is

more than one contributory process. Again, considering the behaviour of

the cross section for single ionization, the structure must be due to varia-

tions in the values of σ

for diﬀerent targets. Helms et al. (1994a, 1995)

attributed this to the inﬂuence of inner shell contributions to this cross

section, which become increasingly important as the impact energy is

increased. Thus in this picture, double and higher order ionization can be

caused by a single inner shell ionizing event followed by shake-oﬀ, Auger

and Koster–Cronig processes. These authors supported this contention

with calculations for argon, based upon a modiﬁed Born approximation

which takes account of the acceleration (deceleration) of the electron

(positron) in the nuclear ﬁeld of the target. This eﬀect also explains

the relative sizes of the electron and positron cross sections. Theoretical

results for xenon have also been derived by Helms et al. (1995) from the

semi-empirical Lotz formula, but with the inclusion of a simple energy

shift to allow for the slowing down experienced by the positron during

the collision. Further discussion of inner shell ionization can be found in

section 5.7.

Triple ionization of the heavier noble gases has been reported by Kruse

et al. (1991) and Helms et al. (1994, 1995a), and again the inﬂuence of

inner shell eﬀects is notable and quickly becomes the dominant mechanism

for this process.

The coming years will see extensions of the work reported in this section

to further reactions and cross sections for positron-induced transitions

involving more than one electron. In particular, studies of double ioniza-

tion near to the threshold seem worthwhile, particularly in the light of

the results of Helms et al. (1994, 1995b) where the electron and positron

cross sections appear to intersect. New results for helium gas by Bluhme

et al. (1998) have been analysed using a modiﬁed Rost–Pattard param-

eterization (Rost and Pattard, 1997), which has shown that a simple

uniﬁed treatment of double ionization, similar to that developed for single

ionization, is applicable. One surprising feature of the study of Bluhme

et al. (1998) was the strong suppression of the transfer-ionization channel

(double ionization involving the formation of positronium).

252 5 Excitation and ionization

5.6 Diﬀerential cross sections

Understanding the complexities of the few-body physics involving strongly

correlated particles which occurs in ionization by charged projectiles re-

mains a fundamental challenge to scattering theory. Over the last three

decades or so major advances have been made in this area, particularly

for electron impact, through the measurement of diﬀerential ionization

cross sections and detailed comparisons with ever more sophisticated

theories (see e.g. Lahmam-Bennani, 1991, and references therein). Part

of the motivation for attempting similar work for positron impact is

to gain, by comparisons with both electron data and theory, a greater

understanding of collision dynamics, though with the new feature of

strong electron–positron correlations and the eﬀects of competition with

positronium formation.

Diﬀerent layers of diﬀerential information are available in ionizing col-

lisions. The simplest quantity is the single diﬀerential cross section,

dσ

/dΩ, where all the positrons scattered, or the electrons ejected, into

a particular solid angle are measured, irrespective of their kinetic energy

and the fate of the undetected particle. McDaniel (1989, section 6.9A)

stated that no measurements of this cross section have been made for

electron impact since it has no physical content, owing to the presence of

two electrons in the ﬁnal state. However, this quantity is of interest in

positron collisions.

The collision parameters can be speciﬁed further if the double diﬀer-

ential cross section is measured. This is usually written as d

/dEdΩ,

where E and Ω refer to the energy and solid angle of either the scattered

positron or the ejected electron. Measurements of this quantity have been

made for positron impact and will be described below and compared with

data for electrons.

Measurements of dσ

/dΩ for positron–argon collisions, in which the

scattered positron was detected, have been reported by Finch et al.

(1996a). This work was undertaken in order to search for structures in

this cross section which might reﬂect those reported previously in dσ

/dΩ

by the Detroit group (see section 3.4). Accordingly, measurements were

taken at a ﬁxed angle of 60

◦

over the energy range 40–150 eV, with

particular emphasis on the 50–60 eV range where a step-like structure

in dσ

/dΩ was reputed to exist. However, no structure was found in

dσ

/dΩ, which, at 60

◦

, was found to be approximately constant between

40 eV and 60 eV and to fall monotonically at higher energies. A more

comprehensive series of measurements of dσ

/dΩ for both single and

double ionization for argon and krypton targets was reported by Falke

et al. (1997) at energies between 75 eV and 120 eV.

5.6 Diﬀerential cross sections 253

The ﬁrst reported study of the behaviour of double diﬀerential cross

sections for positron impact ionization was that of Moxom et al. (1992);

these workers conducted a search for electron capture to the continuum

(ECC) in positron–argon collisions. In this experiment electrons ejected

over a restricted angular range around 0

◦

were energy-analysed to search

for evidence of a cusp similar to that found in heavy-particle collisions

(e.g. Rødbro and Andersen, 1979; Briggs, 1989, and references therein),

which would be the signature of the ECC process.

The scattering cell and ion-extraction system used for this study have

been described by Moxom et al. (1992) and Moxom, Laricchia and Charl-

ton (1995a). The positron beam was timed by the remoderator-tagging

technique, described in section 7.6 (Laricchia et al., 1988; Zafar et al.,

1991). The beam was then passed through the scattering cell, and elec-

trons ejected in the forward direction were monitored at the end of the

ﬂight path using a ceratron detector. Thus, time-of-ﬂight spectra could

be obtained, from which an estimate of the ejected electron energy could

be derived. The signal from the ceratron was used to gate on the ion

extractor, and the electron signal was only accepted when an ion was

found in coincidence. Retarding grids between the scattering cell and the

detector were also used to energy-analyse the ejected electrons.

A major limitation of this experiment was that beam conﬁnement was

provided by an axial magnetic ﬁeld. The ﬁeld at the scattering cell, and

beyond to the detector, was deliberately kept low to reduce the angular

acceptance, although, as described by Moxom et al. (1992), this quantity

was dependent upon the energy of the ejected electrons. Experiments

at 50 eV, 100 eV and 150 eV positron impact energies revealed that the

lower electron energies are most favoured, the number of electrons ejected

dropping rapidly to close to zero at around half the impact energy. Close

inspection of the energy spectra revealed small bumps in the distributions

at 40 eV and 60 eV for 100 eV and 150 eV impact energies respectively,

whereas the ECC peak would be expected at around 42 eV and 67 eV.

Whether or not these features can be associated with the ECC process,

the experiment found that ECC makes a small contribution to the double

diﬀerential cross section for positron impact ionization. This was not in

accord with quantum mechanical calculations at the time, though in line

with results of a classical trajectory Monte Carlo calculation reported by

Schultz and Reinhold (1990). The latter workers did not ﬁnd a sharp cusp

in the double diﬀerential cross section at the ECC energy but, instead, a

broad ridge-like feature, which they attributed to the fact that the light

positron can scatter over a large angular range. Although these theoretical

data were for the positron–hydrogen system, whereas the experiment

was with an argon target, Moxom et al. (1992) attempted to compare

the shape of their data with the results of Schultz and Reinhold (1990)

Fig. 5.18. Schematic illustration of the apparatus developed by K¨ov´er and coworkers for studies of positron-impact diﬀerential

ionization cross sections. Reprinted from Journal of Physics B26,K¨ov´er, Laricchia and Charlton, Ionization by positrons and

electrons at 0

◦

5.6 Diﬀerential cross sections 255

by numerically convoluting the latter with the transmission probability

factor of their apparatus. This quantity was obtained by solving the

equations of motion for electrons ejected with various energies and angles

and over the range of possible starting coordinates in their scattering

cell. A small structure was found in this convoluted theoretical energy

spectrum located at around 40 eV, though somewhat more pronounced

than in the experiment.

In order to probe further the possible role of ECC in positron collisions

the UCL group developed the electrostatic system shown in Figure 5.18.

The positron beam was formed using a slightly modiﬁed version of the Soa

system of Canter et al. (1986); see also subsection 1.4.3. The positrons

were separated from the high-energy β and γ ﬂuxes, using a pair of

cylindrical plates, and then focussed using a ﬁve-element lens onto a gas

target formed by a narrow nozzle. An electron beam could also be derived

from this system, though for this a weaker radioactive source was used,

which resulted in primary beam intensities similar to those for positrons.

The energy distributions of the scattered positrons (electrons)

or ejected

electrons were measured using a parallel plate analyser (PPA). The ions

were extracted using an electric ﬁeld which was pulsed on when the PPA

detector registered a count. Coincidences of the ions with scattered

positrons or ejected electrons were monitored using a standard timing

arrangement, whilst a multichannel scaler system was used to ramp the

voltage applied to the PPA in order to measure the energy spectra. The

positron beam was chopped when the extraction ﬁeld was present to

prevent particles being pulled into the ion detector.

K¨ov´er, Laricchia and Charlton (1993) measured the energy distribu-

tions at impact energies of 100 eV, 150 eV and 250 eV for positrons and

electrons scattered close to 0

◦

. A surprising feature of these data, shown

in Figure 5.19, is that despite the large diﬀerence found in the integrated

cross sections for positrons and electrons at these impact energies the

energy distributions are very similar. In the positron case, no structure

was found which could be attributed to the ECC process; the latter, at

an impact energy of 100 eV, for instance, should have been present at an

energy loss of 58 eV. The level of sensitivity is set by the absolute scale on

the 100 eV data to be approximately 10

−21

−1

. These ﬁndings

are consistent with those of Moxom et al. (1992) described above and with

the classical trajectory calculations of Schultz and Reinhold (1990). The

quantum mechanical studies of Sil and coworkers (e.g. Bandyopadhyay

et al., 1994, and references therein), appear to overestimate the ECC

contribution; however, they are for an atomic hydrogen target.

Double diﬀerential cross sections for positron–argon scattering at angles

other than 0

◦

have been reported by K¨ov´er, Laricchia and Charlton (1994)

and Schmitt et al. (1994) at 100 eV impact energy and K¨ov´er et al. (1997)

256 5 Excitation and ionization

Fig. 5.19. Data for inelastically scattered positrons (•) and electrons (◦) around

◦

at impact energies of (a) 100 eV, (b) 150 eV and (c) 250 eV. The solid line

is from the calculation of Sparrow and Olson (1994) for an incident angle of

◦

. Reprinted from Journal of Physics B26,K¨ov´er, Laricchia and Charlton,

with permission from IOP Publishing.

at 60 eV. All these studies used positron–ion coincidences to distinguish

ionization from other scattering processes, though in the case of Schmitt

et al. (1994) the ions were extracted using a small d.c. electric ﬁeld rather

than the pulsed ﬁeld technique. The apparatus employed by K¨ov´er,

Laricchia and Charlton (1994) was very similar to this, except that the

energy analysis was performed using a simple retarding ﬁeld analyser

5.6 Diﬀerential cross sections 257

Fig. 5.20. Double diﬀerential cross sections (DDCS) for the single ionization of

argon gas by impact of 100 eV electrons. The solid circles on a line (the latter

serves only to guide the eye) are from the work of DuBois and Rudd (1978)

whilst the open circles are from K¨ov´er, Laricchia and Charlton (1994). (a) 30

◦

(b) 45

◦

(RFA), which, though having a poorer energy resolution than the PPA

had the virtue of a wider angular acceptance for scattered particles.

The projectile beam was monitored with a separate detector, also hav-

ing an RFA for diagnostics. The system of Schmitt et al. (1994) was

similar, except that the scattered positrons or ejected electrons were

monitored using a rotatable 90

◦

cylindrical condenser spectrometer.

The electron impact results of K¨ov´er, Laricchia and Charlton (1994)

for scattering angles of 30

◦

and 45

◦

are presented in Figure 5.20, where

the energy dependences of the data for ejected electrons are in good

accord with those found by DuBois and Rudd (1978), to which they were

normalized at high energies of ejection. The data at both angles exhibit

similar features, namely a preponderance of electrons at both low and

high energies. The low energy data are usually attributed to the liberated

electron and the high energy data to the scattered projectile, though of

course this demarcation cannot be unambiguous, owing to exchange. It

is notable that the higher energy electrons are more forward peaked (the

cross section for 30

◦

is greater than that for 45

◦

at these energies) and

that the probability that the two electrons will be emitted with roughly

the same energy is low. There are no theoretical data for comparison.

The positron impact data at 30

◦

and 45

◦

are presented in Figure 5.21.

The absolute scale was derived, as described by K¨ov´er, Laricchia and

Charlton (1994), by comparison with electron data and positron elastic

diﬀerential cross sections, and results for both the scattered positrons and

the ejected electrons are displayed. Again, the ejected electrons, which

258 5 Excitation and ionization

Fig. 5.21. Double diﬀerential cross sections for the single ionization of argon gas

by impact of 100 eV positrons. Key: , ejected electrons; , scattered positrons;

——, calculation of Sparrow and Olson (1994). (a) 30

◦

, (b) 45

◦

can now be unambiguously identiﬁed, have mainly low energies, below

(E − E

)/2; the positrons are in the higher energy range and there is no

evidence of a substantial contribution to the cross section from ECC. The

solid line is the classical trajectory Monte Carlo calculation of Sparrow

and Olson (1994), which is in good accord with experiment for the ejected

electron spectra but higher than experiment for the scattered positrons,

particularly in the low energy-loss region. Similar conclusions have been

forthcoming from a further study by K¨ov´er et al. (1997), who used the

more sensitive PPA to analyse the scattered positron energies. These

authors, guided by theoretical work on the positron–hydrogen system

by Berakder and Klar (1993), tentatively attributed this eﬀect to the

inﬂuence of close positron–electron–ion interactions during the collision.

The experiment of Schmitt et al. (1994) concentrated on the double

diﬀerential cross section at an electron ejection energy of 15 eV over

the angular range 0

◦

–90

◦

for 100 eV positron–argon collisions. This was

chosen because unpublished work for hydrogen by Klar and Berakdar,

communicated to Schmitt et al. (1994), suggested that for these circum-

stances the cross section for positron impact would greatly exceed that for

electrons. This prediction seems to have been borne out by the results of

Schmitt et al. (1994), who found e

−

ratios of 16 ±11 and 5.4 ± 3.4at

◦

and 45

◦

respectively. However, K¨ov´er, Laricchia and Charlton (1994)

obtained the values 2.7±1.0 and 1.4±0.6 for this ratio at the same two

angles.

Recently, K¨ov´er and Laricchia (1998) reported the ﬁrst measurement

of d

/dΩ

dΩ

dE, the triple diﬀerential cross section for positron col-

lisions. Molecular hydrogen was chosen as the target for positrons at