Charlton M., Humberston J.W. Positron Physics

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2.7 Partitioning of the total cross section 89

2.3. It is immediately apparent that the total cross section does not show

a dramatic rise or change of slope as the impact energy is raised through

the threshold for positronium formation. This is in marked contrast to

the results for the other gases discussed so far in this section and for the

noble gases, discussed in subsections 2.5.1 and 2.5.2. In fact, following

an initial decline from approximately 2 ×10

−15

at1eVto7×10

−16

at 5 eV, σ

remains nearly constant up to 20 eV (there is a gradual

decline at higher energies). The apparent dearth of positronium formation

may, however, be an illusion resulting from the behaviour of the elastic

scattering cross section near this threshold (see sections 3.3 and 4.4 and

the discussion of Meyerhof et al., 1996). There have, thus far, been no

direct studies of positronium formation in water, although a study of low

energy positron impact with an ice surface found abundant positronium

emission (Eldrup et al., 1985). In positron lifetime studies in other gases

with high dipole moments (e.g. NH

,CH

Cl), high positronium formation

probabilities have been found (Heyland et al., 1982).

2.7 Partitioning of the total cross section

In this, the concluding section of this chapter, we present a discussion of

the partitioning of the positron and electron total cross sections for helium

gas, based upon the work of Campeanu et al. (1987) for positron scattering

and of de Heer and Jansen (1977) for electron scattering. This target has

been chosen because it has been the subject of extensive theoretical and

experimental study. A similar exercise has also recently been undertaken

by Zhou et al. (1997) in connection with their measurements of σ

for

atomic hydrogen, and by Sueoka and Mori (1994) for helium, neon and

argon gases.

The presentation of positron and electron data in sections 2.5 and

2.6 has illustrated that there are often large diﬀerences between the two

total cross sections, both in magnitude and energy dependence, although

sometimes the converse is true, with the behaviour being remarkably

similar. In an eﬀort to understand some of these eﬀects it is necessary

to probe deeper into the total cross section by splitting it up into its

partial contributions, taking the most reliable data from both theory

and experiment. In so doing, one may also hope to gain some degree

of self-consistency. Detailed discussions of the processes which make the

major contributions to σ

are contained in Chapters 3–5, and one of our

purposes here is to set the scene for these.

The earliest attempts to partition σ

for positron–helium scattering

were those of Griﬃth et al. (1979b), before the advent of any partial

cross section measurements, and of Coleman et al. (1982), when the ﬁrst

inelastic scattering data became available. The most detailed study is

90 2 Total scattering cross sections

that of Campeanu et al. (1987), who deﬁned

no ion

= σ

− σ

, (2.19)

where σ

is the positronium formation cross section, σ

is the positron

impact-ionization cross section and σ

no ion

is that part of the total positron

cross section in which there is no ion left in the ﬁnal state. The right-hand

side of equation (2.19) may be rewritten as

no ion

= σ

∞



n=2

(nLS), (2.20)

where σ

is the elastic scattering cross section and the summation is

over the angle-integrated excitation cross sections for various speciﬁc

bound electronic states of the target system. Similar equations would

of course apply to electron scattering, but without the positronium for-

mation term. Campeanu et al. (1987) justiﬁed the partitioning in this

way by noting that σ

no ion

, and the corresponding quantity for electrons, is

relatively easy to obtain from various experimental measurements through

equation (2.19). This can be seen from the discussion in this chapter

and in Chapters 4 and 5. However, the measurement of σ

no ion

from

equation (2.20) is not straightforward because σ

can only be easily

determined below the positronium formation threshold

energy and mea-

surements of the total excitation cross section are diﬃcult (see e.g. the

discussion of electron impact data given by Heddle, 1979). In a theoretical

partitioning of the total scattering cross section the situation is reversed,

and σ

no ion

is easier to determine via equation (2.20) than via equation

(2.19) because the cross sections for positronium formation and ionization

are the most diﬃcult to compute accurately. Thus, by diﬀerent routes

it is possible to extract reliable values for σ

no ion

from both theory and

experiment.

In order to test the general utility of their approach, Campeanu et al.

(1987) performed an analysis of electron–helium scattering, for which

several sets of theoretical and experimental data were available, relating

to all the major partial cross sections. Their approach was similar to that

of de Heer and Jansen (1977), except that these latter authors aimed

to construct a semi-empirical σ

for electrons by summing all available

partial cross sections, many of which (e.g. the elastic and excitation

contributions) were determined by integrating published diﬀerential cross

sections. The cross sections obtained by de Heer and Janson agreed well

with what were then the most recent experimental results of Blaauw et al.

(1977), from 30 eV up to the experimental limit of 700 eV. Campeanu

et al. (1987) used the data assembled and evaluated by de Heer and Jansen

(1977), constructed σ

no ion

for electrons according to equation (2.20) and

2.7 Partitioning of the total cross section 91

Fig. 2.24. The integrated cross sections for positron scattering from helium

atoms without ionization (σ

no ion

), obtained from experiment according to equa-

tion (2.19), and theoretical predictions derived from equation (2.20). The corre-

sponding cross sections for electrons (superscript minus) are also shown on the

high energy side. Curves H and L represent the highest and lowest theoretical

estimates of σ

no ion

respectively. The σ

(APO) curve is from Campeanu et al.

(1987) – see the main text for the origins of the other theoretical and exper-

imental data. The broken curves at the higher energies are for electron (−)

and positron (+) elastic scattering cross sections from the distorted-wave second

Born (DWSB) calculations of Dewangan and Walters (1977).

then compared ‘theory’ with experiment. They found reasonably good

agreement between the two when the lowest theoretical values of σ

no ion

were used, and this led to some conﬁdence being placed in the correspond-

ing positron analysis.

Data for positrons are presented in Figure 2.24, which shows some of

the major contributions to σ

. The ‘experimental’ values of σ

no ion

for

positrons were deduced from the σ

measurements of Kauppila et al.

(1981) by subtracting the cross sections for positronium formation and

ionization as given by Fromme et al. (1986). The theoretical results, as

for electrons, have been given with higher and lower values (curves H

and L). For the excitation cross sections the distorted-wave approach of

Parcell, McEachran and Stauﬀer (1983, 1987) was extended to higher

impact energies and to higher values of the principal quantum number

of helium, n

, for the excited channels. (Note that since exchange

92 2 Total scattering cross sections

between the projectile and the electrons in the target atom is absent

in positron collisions, the collisional excitation of triplet states is strictly

forbidden.) In particular, Campeanu et al. (1987) reported cross sections

for excitation to the 3

P and 4

P states and showed that their ratio obeys

the familiar 1/n

scaling law, so that σ

P )/σ

P )=(3/4)

. They

therefore assumed that it was acceptable to sum the contributions from

all higher

P channels to give

∞



P) = 1.56σ

P) (2.21)

for each impact energy.

Inspection of Figure 2.24 reveals that the ‘experimental’ values of σ

no ion

for positrons, as calculated from equation (2.19), lie between the higher

and lower theoretical estimates at all energies. Above 100 eV they are in

better agreement with curve L, in view of the error bars on the experi-

mental values. Campeanu et al. (1987) noted that the main diﬀerences

between the two theoretical estimates deriv

e entirely from diﬀerences in

the cross sections for processes leaving the target atom in the ﬁnal states

S (elastic scattering), 2

S and 2

P. The upper curve, H, comes from the

work of Willis and McDowell (1982); their results are markedly higher

than those of Dewangan and Walters (1977), Campeanu et al. (1987) and

Parcell, McEachran and Stauﬀer (1983, 1987), and are thought to be

overestimates at all energies.

As noted above, the lower curve, L, is in good agreement with experi-

ment above 100 eV but falls signiﬁcantly below the experimental values

at lower energies. This is attributed by Campeanu et al. (1987) to a

failure of Parcell’s polarized-orbital method. The experimental work of

Coleman et al. (1982) and of Sueoka (1982) is cited in support of this, and

the later results of Mori and Sueoka (1994), which supersede the earlier

data of Sueoka (1982), do not alter this conclusion. Further discussion of

elastic scattering can be found in Chapter 3; positron impact excitation

is treated in Chapter 5.

Before leaving this section it is worth noting some other points raised

in the work of Campeanu et al. (1987). One pertains to the established

merging of the total cross sections for positrons and electrons at approxi-

mately 200 eV for helium, as previously mentioned in section 2.5. It can

be seen from Figure 2.24, by subtracting the ionizing channels from the

relevant σ

, that the merging of the remainder does not occur until 600 eV.

Moreover, since the distorted-wave Born approximation calculations of

Dewangan and Walters (1977) reveal that σ

for electrons exceeds σ

for positrons even out to 2 keV impact energy, the preponderance of

the excitation cross section for positrons over that for electrons must be

2.7 Partitioning of the total cross section 93

responsible for the merging of the no-ionization cross sections. As noted

by Campeanu et al. (1987), theoretical veriﬁcation of this has not been

made using the same model for both projectiles.

Campeanu et al. (1987) also discussed the behaviour of the ionization

cross sections for positrons and electrons near to the ionization threshold,

but our treatment of this topic is deferred until subsection 5.4.5. Fur-

thermore, in subtracting σ

from σ

Campeanu et al. (1987) obtained

an estimate of the behaviour of σ

between the positronium formation

threshold and the ﬁrst excitation threshold of the helium atom. Their

derived cross section appeared to contain a cusp or threshold anomaly

around E

, but more recent experimentation and theoretical analysis

has cast some doubt on the existence of a feature of this size in helium.

Further discussion of these interesting phenomena is given in Chapters 3

and 4.

Elastic scattering

3.1 Introduction

In this chapter we describe the elastic scattering of positrons

by atoms

and molecules over the kinetic energy range from zero to several keV,

concentrating mainly on the angle-integrated cross section, σ

. However,

reference is also made to diﬀerential cross sections, dσ

/dΩ, which have

recently become amenable to experimental measurement using crossed

gas and positron beams.

Particular attention is given to relatively simple targets, e.g. atomic

hydrogen, helium, the alkali and heavier rare gas atoms and small molecules,

and some comparisons are made with the corresponding data for electron

impact. This again highlights the diﬀerences and similarities in the

scattering properties of the two projectiles, which have already been

mentioned in subsection 1.6.1 and in Chapter 2.

At energies below the lowest inelastic threshold, elastic scattering is

the only open channel (except for electron–positron annihilation, which

is always possible but which usually has a negligibly small cross section).

For all atoms, the lowest inelastic threshold is that for positronium for-

mation, at an energy E

, but for the alkali atoms positronium formation

is possible even at zero incident energy. Molecular targets usually have

thresholds for rotational and vibrational excitation at energies below E

although the elastic scattering cross section is nevertheless expected to

dominate over the cross sections for these inelastic channels.

We continue this chapter with a detailed description of the theoreti-

cal models applied to the elastic scattering of positrons by atoms and

molecules.

3.2 Theory 95

Fig. 3.1. The coordinates of the positron–hydrogen system.

3.2 Theory

A discussion of elastic positron–atom scattering is most conveniently in-

troduced in the context of positron–hydrogen

scattering, and we therefore

describe this system in considerable detail and use it to illustrate some of

the more important methods of approximation used in positron collision

physics.

1 Positron–hydrogen scattering

The Hamiltonian of the positron–hydrogen system may be written, with

reference to the nomenclature of Figure 3.1, as

H = −

∇

−

+ H

, (3.1)

where

= −

∇

−

(3.2)

is the Hamiltonian of the hydrogen atom. If elastic scattering is the only

open channel, the total wave function when the positron is far from the

target atom has the product form

Ψ(r

, r

) ∼

→∞

F (r

)Φ

), (3.3)

where Φ

) is the hydrogen target wave function, with form

√

exp(−r

), (3.4)

and

F (r

) ∼

→∞

ik·r

+ f

(θ)

ikr

. (3.5)

96 3 Elastic scattering

A partial-wave expansion of the scattering function F (r

) gives

F (r

) ∼

→∞

∞



l=0

(2l +1)i



sin(kr

−

lπ + η

)



(cos θ), (3.6)

where η

is the lth partial-wave phase shift and P

(cos θ) is the corre-

sponding Legendre function. The elastic scattering amplitude is then

(θ)=

2ik

∞



l=0

(2l + 1)(e

2iη

− 1)P

(cos θ) (3.7)

and the elastic scattering cross section is

=2π



(θ)|

sin θdθ=

4π

∞



l=0

(2l + 1) sin

. (3.8)

The diﬀerential scattering cross section is

dσ

dΩ

(θ)=|f

(θ)|

= A

+ B

, (3.9)

where

A =

∞



l=0

(2l + 1)(cos 2η

− 1)P

(cos θ) (3.10)

and

B =

∞



l=0

(2l + 1) sin 2η

(cos θ). (3.11)

It readily follows from equations (3.7) and (3.8) that

4π

Im f

(θ =0). (3.12)

This is the optical theorem, and it expresses the conservation of the

number of particles in the scattering process. As already mentioned in

section 2.2, it is valid even when inelastic processes can occur, although

is then replaced by the total scattering cross section σ

, which includes

contributions from all open scattering channels.

As the positron approaches the target system it interacts with and

distorts it, so that the total wave function no longer has the separable

form of equation (3.3). Nevertheless, an equivalent Schr¨odinger equation

can be derived for the positron, the solution to which is a function of the

positron coordinate r

only, with the correct asymptotic form but at the

cost of introducing a non-local optical potential.

3.2 Theory 97

If one introduces the projection operator (Feshbach, 1962)

P = |Φ

Φ

|, (3.13)

which projects onto the ground state of the target atom, and

Q =1− P, (3.14)

which projects on to all other states, then

P + Q =1,P

= P, Q

= Q, P Q = QP =0. (3.15)

The Schr¨odinger equation for the total system can be expressed as

(H − E) |Ψ =(H − E)(P + Q) |Ψ =0, (3.16)

where E = −

is the total energy of the positron–hydrogen system.

Applying the projection operators P and Q separately to equation (3.16)

and eliminating Q|Ψ by substituting from one equation into the other,

the following equation for P |Ψ is obtained:

{P (H − E)P + PV

int

Q[Q(E − H)Q]

−1

int

P }P |Ψ =0, (3.17)

where V

int

is the interaction potential between the positron and the par-

ticles in the target. For the positron–hydrogen

system

int



−



. (3.18)

In conﬁguration space, P |Ψ has the representation

)Φ

|Ψ =Φ

)F (r

), (3.19)

and the equation satisﬁed by the positron function F (r

) is then



−

∇

+ V

−



F (r



opt

, r



)F (r



) dr



=0, (3.20)

where

= e

−2r





(3.21)

is the static potential between the positron and the undistorted hydrogen

atom and V

opt

is the non-local optical potential. Thus, the Schr¨odinger

equation for a positron–hydrogen system has been converted to an equa-

tion for the positron alone moving in a potential ﬁeld. This may seem

a signiﬁcant simpliﬁcation of the scattering problem, but the diﬃculties

have merely been transferred to the task of calculating the optical poten-

tial. However, this formalism provides a convenient and consistent means

98 3 Elastic scattering

of generating approximation schemes for solving the scattering problem;

diﬀerent approximate trial wave functions will correspond to diﬀerent

choices of the optical potential.

An approximation to the exact optical potential may be written as

opt





|PHQΦ

Φ

QHP |

E − E



, (3.22)

where the functions Φ

and energies E

are the eigenfunctions and asso-

ciated eigenvalues of the matrix eigenvalue equation

QHQ |Φ = E |Φ (3.23)

in whatever basis is being used.

It has been shown by Gailitis (1965) that the optical potential deﬁned

by equation (3.22) is less attractive than the exact optical potential for

energies below the lowest eigenvalue of QHQ, and the resulting phase

shifts are therefore lower bounds on the exact values. Furthermore, as

the number of basis functions used in the matrix representation of the

operator QHQ is increased, the optical potential becomes more attractive

and the resulting phase shift therefore also increases, becoming closer to

the exact value.

The simplest approximation is to retain only the local static potential

; see equation (3.20). This deﬁnes the static approximation, in which

there is assumed to be no distortion of the target by the incident positron,

so that the total wave function is

Ψ(r

, r

)=F (r

)Φ

) (3.24)

for all values of r

. For positrons, the static potential is always repulsive,

and all partial-wave phase shifts in the static approximation are therefore

negative. Although these phase shifts are rigorous lower bounds, they are

in poor agreement with the exact results, as may be seen in Figures 3.2

and 3.3, because the attractive polarization potential, which arises from

the distortion of the target, has been neglected.

Account must therefore be taken of the distortion of the target if more

accurate results are to be obtained. A relatively easy means of doing

so, which has been quite widely used for investigating positron scattering

by a variety of atoms, is the polarized-orbital method (Temkin, 1957,

1959; McEachran et al., 1977). The method proceeds in two stages.

First, the wave function of the distorted atom in the ﬁeld of a stationary

positron is expressed as [1+G(r

, r

)]Φ

), and G(r

, r

) is determined

using either perturbation theory, as for hydrogen (Drachman, 1965), or a

variational method in which the energy of the stationary positron–atom