Charlton M., Humberston J.W. Positron Physics

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3.2 Theory 129

Instead of expressing the scattering problem in terms of the one-centre

polar coordinates r

and θ, it is more appropriate to use the two-centre

prolate spheroidal coordinates deﬁned by

λ =(r

+ r

)/R, µ =(r

− r

)/R, (3.91)

where r

and r

are the distances of the positron from the two nuclei,

labelled A and B. Although the use of these variables complicates the

formulation in some respects, the two-centre character of the system can

thereby be more conveniently represented. In terms of these variables,

stat

) ∼

8QP

(µ)

(3.92)

and

pol

) ∼

−8

[α

+ α

(µ)]. (3.93)

These asymptotic forms for V

stat

) and V

pol

) must be modiﬁed at

short range, particularly to shield the singularities which would otherwise

exist at r

= 0, and this has been done in several diﬀerent ways.

In the earliest theoretical study of positron–H

scattering, by Massey

and Moussa (1958), only the static potential was included, and the po-

larization potential was ﬁrst added by Lodge, Darewych and McEachran

(1971). Since then, numerous calculations of positron scattering by var-

ious molecules have been made using diﬀerent model potentials. Among

the more signiﬁcant of these investigations for scattering by H

are those

of Baille, Darewych and Lodge (1974), using a one-centre formalism,

Hara (1974) and Morrison et al. (1984). Somewhat similar studies with

model potentials have also been made for more complicated molecules

such as CO

(Horbatsch and Darewych, 1983), CH

and NH

(Jain and

Thompson, 1983).

5 Elastic scattering beyond inelastic and rearrangement

thresholds

As the positron energy is raised beyond various rearrangement and in-

elastic thresholds, elastic scattering continues to make a signiﬁcant con-

tribution to the total scattering cross section σ

, but it is now just one

of several open channels, all of which are coupled together. Even if only

the elastic scattering cross section is required, its determination should

be considered as part of a more comprehensive multichannel calculation

in which the cross sections for all possible transitions between any two

open channels are included. Furthermore, the couplings between the open

channels give rise to structure in the elastic scattering cross section on

130 3 Elastic scattering

either side of the threshold for the opening of each new inelastic channel.

This is discussed in section 3.3 in the context of positronium formation.

When considering positron scattering at energies such that several in-

elastic channels are open, it is usually not feasible to represent every open

channel explicitly; however, the neglect of open inelastic channels in the

formulation of the collision process may produce extensive pseudoresonant

structure in the cross sections for elastic scattering and for those inelastic

processes which are explicitly included. Nevertheless, it is still possible

to extract meaningful cross sections by a process of T-matrix averaging

(Burke, Berrington and Sukumar, 1981). An interesting example of a

pseudoresonance was found by Higgins and Burke (1991, 1993) in their

calculations of s-wave positron–hydrogen scattering using the coupled-

static approximation. These authors found a prominent feature at an

energy of 35.6 eV, which revealed itself most conspicuously in the positro-

nium formation cross section. Its existence was conﬁrmed by several other

authors (Hewitt, Noble and Bransden, 1991; Sarkar, Basu and Ghosh,

1993) using similar approximations, and it was believed to be authentic,

being termed a coupled-channel shape resonance. However, subsequent

investigations by Kernoghan, McAlinden and Walters (1994b) and Zhou

and Lin (1995a), using more terms in the coupled-state expansion, estab-

lished that there are no real resonances above the ionization threshold of

the target hydrogen atom, a fact consistent with the theorem of Simon

(1974, 1978), which states that in a many-body system experiencing only

Coulomb interactions there can be no resonances above the total break-up

energy.

Among the most accurate values of the positron–hydrogen elastic scat-

tering cross sections in the intermediate energy region up to 80 eV are the

18-coupled-state results of Kernoghan, McAlinden and Walters (1995),

which, after being smoothed to remove pseudoresonance structure, are

displayed in Figure 3.11. Also given there are the results of the moment

T-matrix method of Winick and Reinhardt (1978a, b) and those of the

convergent close-coupling method of Bray and Stelbovics (1994), neither

of which explicitly includes positronium formation. Nevertheless, the

agreement between the results of these three methods is quite good for

energies greater than 30 eV.

Elastic scattering cross sections at intermediate energies have been cal-

culated for several other atoms. Important examples are the noble gases,

investigated by McEachran and Stauﬀer (1986) using the polarized-orbital

method, and the alkali atoms, investigated by McAlinden, Kernoghan and

Walters (1996), Kernoghan, McAlinden and Walters (1996) and Hewitt,

Noble and Bransden (1992b, 1993) using various forms of the coupled-

state method. The latter technique has also recently been applied by

Campbell et al. (1998a) to positron–helium scattering.

3.2 Theory 131

Fig. 3.11. Calculations of the positron–hydrogen elastic scattering cross sec-

tions: ——, 18-coupled-state approximation (Kernoghan et al., 1995) (a 33-

coupled-state calculation by Kernoghan et al., 1996, yielded very similar results);

×, intermediate energy R-matrix theory (Higgins, Burke and Walters, 1990);

•, moment T-matrix method (Winick and Reinhardt, 1978a, b); , convergent

close-coupling method (Bray and Stelbovics, 1994).

Throughout the intermediate and high energy regions, the elastic scat-

tering cross sections for all target atoms decrease steadily with increas-

ing positron energy, eventually merging with the elastic scattering cross

sections for electrons. However, for a given atom the energy at which

the elastic scattering cross sections for the two projectiles merge is much

higher than that at which the two total cross sections merge. For example,

in helium at 250 eV, which is comfortably above the energy at which

merging of the two total cross sections takes place, the elastic scattering

cross section for electrons is still more than three times greater than that

for positrons.

In addition to total elastic scattering cross sections, several workers

have calculated diﬀerential elastic scattering cross sections at intermediate

and high energies (Byron and Joachain, 1977a, for hydrogen and helium;

Kernoghan, McAlinden and Walters, 1995, for hydrogen; McEachran and

Stauﬀer, 1986, for the rare gases). McEachran and Stauﬀer (1986) made

a detailed study of the energy dependence of the diﬀerential cross sections

for the rare gases, each of which has the following rather similar structure.

From a peak in the forward direction, it falls steeply with increasing

132 3 Elastic scattering

scattering angle to a local minimum, followed by an increase to a local

maximum, after which it then falls steadily to a much lower value in

the backward direction. The angles at which the local minimum and

maximum occur decrease with increasing energy, so that at 250 eV in

helium, for example, they are at 6

◦

and 22

◦

respectively. This structure

is diﬀerent from that for electrons, where the diﬀerential cross section has

a somewhat larger value in the forward direction, followed by a steady

fall to a minimum at around 90

◦

and a pronounced rise in the backward

direction.

At high positron energies, where the speed of the incident positron

signiﬁcantly exceeds the speeds of the electrons in the target, the ﬁrst

Born approximation is expected to yield reasonably accurate elastic scat-

tering cross sections which are the same for positrons and electrons. For

atomic hydrogen, this condition is satisﬁed when the positron energy is

several hundred eV. Between the upper limit of 80 eV for the coupled-state

calculations of Kernoghan, McAlinden and Walters (1995) and the region

of validity of the ﬁrst Born approximation, various approximation schemes

based on higher order terms in the Born series have been employed.

The eikonal Born series, which introduces diﬀerences between the cross

sections for positrons and electrons, was applied to positron–hydrogen

scattering by Byron and Joachain (1973, 1977a, b), and an improved

unitarized version of this method has been used by Byron, Joachain and

Potvliege (1981, 1982, 1985). At an energy of 400 eV, the results of

the ﬁrst Born approximation are already quite accurate, exceeding the

two very similar eikonal Born results by less than 10%. At an energy

of 100 eV, however, the ﬁrst Born result exceeds the unitarized eikonel

Born result by 30%, and even the results of the two eikonal Born methods

diﬀer by more than 5%. These more accurate results match smoothly onto

the coupled-state results of Kernoghan, McAlinden and Walters (1995) at

lower energies. Somewhat similar results for positron–hydrogen scattering

have also been obtained using the third order optical model (Byron and

Joachain, 1981) and the second order potential method (Mukherjee and

Sural, 1982). Similar techniques were used by Byron and Joachain (1977a)

for helium and by Joachain and Potvliege (1987) for argon.

3.3 Threshold eﬀects

The threshold for positronium formation in collisions of positrons with

atoms and molecules is an example of a general class of thresholds in

collision processes where there is no residual long-range Coulomb inter-

action between the constituent subsystems in either the initial or ﬁnal

states. Since the original work of Wigner (1948), there has been much

discussion of the eﬀect of the opening of a new channel on those already

3.3 Threshold eﬀects 133

open (Baz 1958; Newton, 1959; Meyerhof, 1963). These studies, though

originally motivated by the need to analyse nuclear scattering data, also

apply to the onset of positronium formation, which is then in competition

with elastic scattering. The requirement of the conservation of ﬂux gives

rise to characteristic structures, such as cusps and rounded steps, in the

energy dependence of the elastic scattering cross section close to inelastic

thresholds. General discussions of these features have been given by Mott

and Massey (1965, Chapter 13), McDaniel (1989, Chapter 4) and New-

ton (1982, Chapter 17). The application to the positronium formation

threshold is outlined below.

The ﬁrst attempt to analyse the behaviour of the elastic scattering cross

section close to the positronium formation threshold energy, E

, was that

of Campeanu et al. (1987) who, as described in section 2.7, undertook a

detailed partitioning of the total cross section, σ

, for positron–helium

scattering. In the energy range between E

and the ﬁrst positron excita-

tion threshold of the target, at energy E

, these authors determined σ

from the relationship

= σ

− σ

, (3.94)

where they used the total cross sections of Stein et al. (1978) and the

positronium formation cross sections, σ

, of Fromme et al. (1986; see

also section 4.4). Although these experimental data for σ

were sparse in

this energy region, Campeanu et al. (1987) argued that the feature they

deduced in σ

, see the broken curve in Figure 3.12, was genuine. Thus,

they argued that according to these measurements the elastic scattering

cross section exhibits a cusp-like behaviour centred on E

, followed by

a steady fall, as the energy increases to E

, of approximately 20%. A

similar analysis was performed by Fromme et al. (1988), following their

determination of the positronium formation cross section in H

gas; here,

an even bigger eﬀect was found, the elastic scattering cross section at

being lower than its value at E

by around 50%. Thus, for both

helium and H

an important cusp-like feature was deduced in the elastic

scattering cross section in the vicinity of E

In an attempt to explore this interesting energy region further, an ex-

periment was undertaken by Coleman et al. (1992) to determine the elastic

scattering cross section according to equation (3.94), using measurements

of the total and positronium formation cross sections made in the same

apparatus. The apparatus and the method used were similar to those

developed by the Arlington group (e.g. Fornari, Diana and Coleman 1983;

Diana et al., 1986b; see also Figure 4.12 and accompanying discussion).

Positrons from a tungsten-mesh moderator, held at a potential V

, were

guided by an axial magnetic ﬁeld of approximately 0.01 T through a

system of grids and a localized scattering cell 0.03 m long to a channeltron

134 3 Elastic scattering

Fig. 3.12. Cross sections for positron–helium scattering in the vicinity of

the

positronium formation threshold (labelled Ps; ‘ex’ and ‘ion’ denote the respective

thresholds for excitation and ionization). •, σ

−σ

from Coleman et al. (1992);

——, σ

from Stein et al. (1978); – – –, σ

−σ

(see text) deduced by Campeanu

et al. (1987). The dotted curve is an attempt to account for the σ

−σ

results,

using other known cross sections as the various channels open (see Coleman et al.,

1992 for details). Reprinted from Journal of Physics B 25, Coleman et al.,

Elastic positron–helium scattering near the positronium formation threshold,

detector located in line of sight with, and 0.6 m distant from, the source

and moderator. The grid, located just after the moderator, was biassed

to V

+ 1.3 V in order to narrow the energy width of the beam by

preventing those positrons with an axial emission energy less than 1.3 eV

from traversing the scattering region. This conﬁguration eﬀectively set

the energy width of the beam (to approximately 0.8 eV FWHM) and thus

the energy resolution with which σ

was determined.

Biassing a grid in front of the channeltron to the voltage V

prevented

all the inelastically scattered, and nearly all the elastically scattered,

positrons from being detected, by virtue of their loss of axial kinetic en-

ergy. The gas pressure was kept suﬃciently low that the target thickness

approximated to single-collision conditions, so that the total cross section

was then determined according to

= f

/(nL), (3.95)

where f

is the total fractional attenuation of the beam when gas is

introduced into the scattering cell, and n and L are the gas number density

and the eﬀective path length in the cell.

3.3 Threshold eﬀects 135

In contrast, σ

was determined by lowering the voltage on the grid in

front of the channeltron to zero and so allowing all scattered positrons to

reach the detector except those neutralized by forming positronium. To a

good approximation, it could be assumed that only those positrons which

had formed positronium would then be lost from the beam, and σ

could

be obtained from

= f

/(nL), (3.96)

where f

is the fractional attenuation of the beam due to positronium

formation in the gas. Thus, from the last three equations,

=(f

− f

)/(nL).

In order to obtain an absolute scale for the measurements, the elastic

scattering cross section was normalized to the total cross sections of Stein

et al. (1978) at energies below E

The results obtained by Coleman et al. (1992) are shown in Figure 3.12,

where the most notable feature is that the elastic scattering cross section

remains essentially constant between E

and E

, with no evidence of

a cusp as signiﬁcant as that proposed by Campeanu et al. (1987). Re-

cent accurate theoretical results obtained by Van Reeth and Humberston

(1999a, b) conﬁrm the constancy of the elastic scattering cross section

throughout this energy range. When the positron energy exceeds E

− σ

is no longer equal to σ

but then contains contributions from

the various possible inelastic processes, namely excitation and ionization

(see Chapter 5). The eﬀects of this are seen in the ﬁgure. Also displayed

there are the values of σ

measured by Stein et al. (1978) and the cross

section σ

+ σ

as deduced by Campeanu et al. (1987), where σ

is the excitation cross section. Note that this latter cross-section sum

corresponds to σ

no ion

, equation (2.19).

Support for the conclusion of Coleman et al. (1992) was forthcoming

from the work of Moxom, Laricchia and Charlton (1993), who made a

detailed study of the positronium formation cross section near to E

, but

found no evidence of threshold cusps in σ

for helium, argon or molecular

hydrogen targets. An attempt to provide elucidation was made by Moxom

et al. (1994), who were the ﬁrst to apply Wigner’s R-matrix analysis of

threshold phenomena to positron collisions. This work followed the theory

of Meyerhof (1963), of which a brief account is now given. Our main

interest here is to consider how the opening of the positronium formation

channel aﬀects the elastic scattering cross section; the energy dependence

of the positronium formation cross section close to the threshold is con-

sidered in section 4.2.

The theory predicts that, over a small energy range above E

, the

positronium formation cross section in the lth partial wave increases from

136 3 Elastic scattering

zero according to

∝ κ



, (3.97)

where the superscripts l and l



refer to the orbital angular momenta of the

incident positron and the outgoing positronium relative to the residual ion

and κ is the wave number of the positronium. This is related to the wave

number of the incident positron, k, through energy conservation:

+ E

. (3.98)

If the orbital angular momenta of the ion and the initial target atom

are the same, as they are for atomic hydrogen and helium, then l = l



For many other atoms, including all the other rare gases, l = l



, but for

simplicity we shall consider as an example the case where l = l



. From

equation (3.97), the rate of increase of the partial positronium forma-

tion cross section with respect to k near to the positronium formation

threshold is

∝ κ

2l−1

. (3.99)

The s-wave contribution to σ

therefore has an inﬁnite gradient at the

threshold, where κ = 0, whereas the gradients of all higher partial-wave

contributions are zero at the threshold.

The anomalous energy dependence of the elastic scattering cross section

in the vicinity of E

may be understood by reference to the properties of

the S-matrix (see Mott and Massey, 1965, Chapter 13), in terms of which

the elastic scattering and positronium formation cross sections in the lth

partial wave are expressed as

(2l +1)|1 − S

, (3.100)

(2l +1)|S

, (3.101)

where the positron and the positronium channels are represented by the

labels 1 and 2 respectively. From the unitarity of the S-matrix, which

expresses the conservation of particles,

+ |S

=1, (3.102)

and therefore

=1−

π(2l +1)

, (3.103)

whence

= exp(i2η

)



1 −

π(2l +1)



1/2

(3.104)

 exp(i2η

)



1 −

2π(2l +1)



(3.105)

3.3 Threshold eﬀects 137

at energies just above E

, where σ

is still very small. At the threshold,

= 0 and S

= exp(i2η

), where η

is the elastic scattering phase shift

at that energy. If it is assumed that η

is a background phase which

retains a constant value over a small energy range on either side of E

then the energy dependence of S

close to the threshold is determined

by the energy dependence of σ

Although σ

= 0 for energies below E

, the expression for S

given

by equation (3.105) can nevertheless be analytically continued below the

threshold by replacing the positronium wave number κ by i|κ| and so

replacing σ

(κ)byi(−1)

(|κ|), see equation (3.97). Consequently,

just above the positronium formation threshold, S

is of the form given

in equation (3.105), but just below this threshold it is given by

 exp(i2η

)



1 −

(−1)

2π(2l +1)

(|κ|)



. (3.106)

The corresponding expressions for the elastic scattering cross section

above and below the threshold are therefore



4π

(2l + 1) sin

− 2 sin

(κ) (above) (3.107)



4π

(2l + 1) sin

− (−1)

l+1

sin 2η

(|κ|) (below). (3.108)

The elastic scattering cross section must fall as the positron energy is

increased above the threshold, and it will either rise or fall as the threshold

is approached from below, depending on the value of l and on the phase

shift at the threshold. Furthermore, because the s-wave contribution to

, considered as a function of the positron energy, has an inﬁnite slope at

the threshold energy E

, equation (3.99), so too does σ

, and its energy

dependence has the shape of either a cusp or a downward rounded step.

All other partial-wave contributions to σ

, however, continue through the

threshold with no discontinuity of slope.

A detailed study of the behaviour of the elastic scattering and positron-

ium formation cross sections in the vicinity of the positronium formation

threshold in positron–hydrogen scattering was made by Humberston et al.

(1997); see section 4.2. For s-wave scattering, the phase shift just below

the positronium formation threshold is −0.055 rad and therefore, accord-

ing to equations (3.107) and (3.108), the elastic scattering should display

a downward rounded step rather than a cusp. This prediction is conﬁrmed

by the accurate calculated values of the s-wave contribution to the elastic

scattering cross section; this contribution does indeed reveal a very small

downward step in passing through the threshold. The insigniﬁcant nature

of this feature in positron–hydrogen scattering is a consequence of the very

138 3 Elastic scattering

small value of the s-wave component of the positronium formation cross

section in comparison to the elastic scattering cross section.

The p-wave component of σ

continues to rise as the positron energy is

increased above E

, apparently contradicting the claim made above that

it must fall. However, a more rigorous analysis of the energy dependence

of the various partial-wave contributions to the elastic scattering cross sec-

tion, based on R-matrix theory (Breit, 1957) introduces a correction term

proportional to κ

into equations (3.107) and (3.108) for the components

of the elastic scattering cross section with l>0; this correction term is

positive above the threshold and negative below. Good agreement is then

obtained between the predictions of the threshold theory and the results

of accurate p-wave variational calculations (Humberston et al., 1997).

The threshold behaviour described above may only apply over a very

narrow energy range about the positronium formation threshold, but

attempts have been made by Meyerhof et al. (1996) and Humberston

et al. (1997), using R-matrix theory, to investigate the inﬂuence of the

opening of the positronium formation channel on the elastic scattering

cross section over a somewhat wider energy range. A moderately good

ﬁt to the accurate variational cross sections for elastic positron–hydrogen

scattering, positronium formation and positronium–proton elastic scat-

tering has thereby been achieved over an energy range of several eV on

either side of E

using only ﬁve parameters for each partial wave.

Recent detailed variational calculations of positronium formation in

positron–helium scattering, by Van Reeth and Humberston (1995b, 1997,

1999b), reveal a similar threshold structure to that found in positron–

hydrogen scattering. Again, the s-wave contribution to the positronium

formation cross section is small compared to the elastic scattering cross

section (see section 4.2), and the calculated threshold structure in the

s-wave component of σ

is therefore also small, making it diﬃcult to

observe in the experimental measurements of the total elastic scattering

cross section. Consequently, the elastic scattering cross section is expected

to continue rather smoothly through the threshold without displaying any

pronounced threshold feature. This theoretical prediction is consistent

with the aforementioned measurements of Coleman et al. (1992) and an

R-matrix analysis by Moxom et al. (1994).

Moxom et al. (1994) applied their analysis to all the rare gases, using

the polarized-orbital phase shifts of McEachran and his collaborators

(McEachran et al., 1977; McEachran, Ryman and Stauﬀer, 1978, 1979;

McEachran, Stauﬀer and Campbell, 1980) and the experimental measure-

ments of σ

made by Moxom, Laricchia and Charlton (1993) and Moxom

et al. (1994). From ﬁts of this data to the theoretical energy dependence

given in equation (3.97) (see Figure 4.18), these authors concluded that

the positronium formation process was, in each case, dominated by the