Charlton M., Humberston J.W. Positron Physics

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1.6 The physical basis 39

At low energies, the absence of a direct static interaction between

positronium and other systems gives an enhanced importance to polar-

ization and exchange eﬀects, and the relatively high value, 72a

, for its

dipole polarizability enables positronium to bind to several other sys-

tems. Positronium can bind to a charged particle, either positive or

negative, provided the mass of the charged particle is suﬃciently small

(Armour, 1983). Thus, it can bind to an electron or a positron if the

two identical particles are in a singlet spin state, as has previously been

mentioned, and also to a positive or negative muon, but it cannot bind

to a proton or antiproton. Furthermore, positronium can bind to atomic

hydrogen, provided the two electrons are in a singlet spin

state, to form the

positronium–hydride molecule PsH, and also to itself to form the positro-

nium molecule Ps

, in which the two electrons and the two positrons

must separately be in singlet spin states. It has also been established

that positronium can bind to the lithium atom and can probably bind

to sodium (Ryzhikh, Mitroy and Varga, 1998a, b). A more detailed

discussion of the binding of positronium to various systems is given in

section 7.5.

The leading term in the interaction between positronium and a charged

particle is the usual 1/R

polarization potential, where R is the coordinate

of the centre of mass of the positronium relative to the other particle. If,

however, positronium interacts with a neutral atom or another positron-

ium, the interaction is dominated by the van der Waals potential arising

from the induced dipole–dipole and quadrupole–dipole terms, which has

the long-range form

V = −A/R

+ B/R

, (1.18)

where R is now the coordinate between the centre of mass of the positron-

ium and that of the other system. Values of the coeﬃcients A and B have

been determined by Drachman (1987) for the positronium–hydrogen and

positronium–positronium systems. Proper account must be taken of the

long-range character of the van der Waals interaction if accurate results

are to be obtained for low energy positronium–atom scattering. This topic

is discussed further in section 7.2.

Total scattering cross sections

2.1 Introduction

The total positron scattering cross section, σ

, is the sum of the par-

tial cross sections for all the scattering channels available to the projec-

tile, which may include elastic scattering, positronium formation, exci-

tation, ionization and positron–electron annihilation. Elastic scattering

and annihilation are always possible, but the cross section for the latter

process is typically 10

−20

–10

−22

, so that its contribution to σ

negligible except in the limit of zero positron energy. All these processes

are discussed in greater detail in Chapters 3–6.

Measurements of σ

have been undertaken for more than two decades

using low energy positron beams and the results, together with the theo-

retical predictions, are discussed in this chapter. The total cross section

is deﬁned by the Beer–Lambert law, in which projectiles scattered by a

target are considered as having been being removed from the incident

beam. Thus, the ﬂux of projectiles, I, surviving from an initial ﬂux I

with a projectile path length L, incident upon a target of number density

n, is given by

I = I

exp(−nLσ

). (2.1)

All measurements of σ

for positrons have been made using the so-called

attenuation method, in which a well-deﬁned beam is passed through a

gaseous target. The attenuation A is given by I

/I. At the densities

typical of this work the gases can be considered as ideal, and n can be

written as

n = P/(k

T )=0.724 × 10

P/T (m

−3

), (2.2)

where P and T are the pressure and absolute temperature of the gas in

units of Pa and K respectively, and k

is Boltzmann’s constant.

2.1 Introduction 41

Fig. 2.1. Schematic illustration of the behaviour of the positron–helium and

electron–helium total scattering cross sections. Notable are the large diﬀerences

in magnitude of the cross sections at low energies, their merging at approxi-

mately 200 eV and the onset of inelastic processes at the positronium formation

threshold E

in the positron curve.

As an example of the energy dependence of the total cross section for

positron–atom scattering, a schematic of the data for helium atoms is

shown in Figure 2.1, together with the corresponding data for electrons.

It is seen that σ

−

) is considerably larger than σ

) at low projectile

42 2 Total scattering cross sections

energies, but the two cross sections merge at energies greater than 200 eV.

Rather similar features are also found in the total cross sections for the

other rare gas atoms except that, for argon, krypton and xenon, σ

−

)

exhibits a narrow Ramsauer minimum at a very low energy, in the vicinity

of which the scattering cross section for positrons exceeds that for elec-

trons. Many other atoms and molecules also exhibit smaller cross sections

for positrons than for electrons at low projectile energies, and this feature

can be attributed directly to the partial cancellation of the attractive

polarization and the repulsive static components of the positron–atom

interaction, as discussed in section 1.6. There are, however, some excep-

tions to this pattern, most notably provided by the alkali atoms, where

the total cross section for positrons exceeds that for electrons at low

projectile energies. These atoms have lower ionization energies than the

binding energy of ground state positronium (6.8 eV), and consequently

the positronium formation channel is open even at zero incident positron

energy, contributing signiﬁcantly to the total cross section.

As the positron energy is raised above the positronium formation

threshold, E

, the total cross section undergoes a conspicuous increase.

Subsequent experimentation (see Chapter 4) has conﬁrmed that much

of this increase can be attributed to positronium formation via the

reaction (1.12). Signiﬁcant contributions also arise from target excitation

and, more importantly, ionization above the respective thresholds (see

Chapter 5). In marked contrast to the structure in σ

) associated

with the opening of inelastic channels, the electron total cross section

has a much smoother energy dependence, which can be attributed to the

dominance of the elastic scattering cross section for this projectile.

The example of helium given above illustrates the major trends in

), described in greater detail below. Our discussion is divided along

somewhat pragmatic lines into low energies and intermediate and high

energies. The loose dividing line is taken around the inelastic thresholds

since the inelastic channels make such important contributions to the

integrated cross section. As mentioned previously, the alkali atoms are a

special case and they are considered in a separate section. Comparisons

between experimental and theoretical results are made where possible,

together with some comparisons with corresponding data for electrons.

We ﬁrst, however, consider some of the general theoretical features of

total cross sections.

2.2 Theory

If elastic scattering is the only open channel (except for electron–positron

annihilation with its very small cross section), the total and elastic scat-

tering cross sections are identical, and the cross section may be calcu-

2.2 Theory 43

lated using the approximation schemes which have been developed for

single-channel elastic scattering, some of which are described in Chapter 3.

As the positron kinetic energy is increased, however, a succession of

rearrangement and inelastic processes also become possible, and a proper

formulation of the scattering process should then include the couplings

between all the open channels.

In positron–helium scattering, for example, elastic scattering is the

only open channel (apart from annihilation) up to an incident energy of

17.8 eV, the threshold for positronium formation, and thereafter positro-

nium formation remains the only other open channel up to the threshold

for the excitation of the 2

S state of helium at 20.6 eV. The 2

S state of

helium has a lower excitation threshold at 19.8 eV, but the transition to

this state from the singlet ground state is highly suppressed in positron

impact because it would require an electron spin ﬂip. Excitation of the

helium atom to other states, and positronium formation into a range of

excited states, all become possible within the next energy interval of 4 eV

until, at an energy of 24.6 eV, ionization of the helium atom can also

occur. As the energy is increased further, all these channels remain open,

to be eventually augmented by others for double excitation and double

ionization.

From such a multichannel formulation the required positron total scat-

tering cross section may, in principle, be obtained as the sum of the

partial cross sections for transitions between the input channel, describing

positrons incident on the target system, and all possible open output

channels. Details of the calculations of individual partial cross sections are

not given here but in the relevant chapters devoted to speciﬁc processes,

e.g. elastic scattering is discussed in Chapter 3, positronium formation

in Chapter 4, excitation and ionization in Chapter 5 and annihilation

in Chapter 6. In this section we are more concerned with the general

properties of total scattering cross sections, although features of some

speciﬁc systems are considered in more detail.

When only a few channels are open, it is feasible to calculate all the

partial cross sections explicitly and sum them to obtain σ

, as has been

done in some studies of positron scattering by atomic hydrogen (Humber-

ston, 1986; Kernoghan, McAlinden and Walters, 1995; Kernoghan et al.,

1996), the alkali atoms (Hewitt, Noble and Bransden, 1993; McAlinden,

Kernoghan and Walters, 1996, 1997; Kernoghan, McAlinden and Walters,

1996), and helium (Hewitt, Noble and Bransden, 1992a; Humberston and

Van Reeth, 1996; Campbell et al., 1998a). As the projectile energy contin-

ues to be raised, however, it ceases to be viable to calculate every possible

partial cross section, although Kernoghan and coworkers, in the references

cited above (and McAlinden, Kernoghan and Walters, 1997) have approxi-

mated to this task in their investigations of positron scattering by atomic

44 2 Total scattering cross sections

hydrogen and the alkali atoms using the coupled-state approximation.

These authors included up to nine states of the target atom (the alkali

atoms being considered as equivalent one-electron systems) and an equal

number of positronium states in their expansion of the wave function,

and they made some allowance for ionization by including pseudostates.

Further details of these calculations are given in subsection 4.2.3. Their

18-state results for atomic hydrogen, given in Figure 2.18 (see subsec-

tion 2.5.4), agree reasonably well with the experimental measurements of

Zhou et al. (1994a), who obtained both upper and lower bounds on the

total cross section. Even better agreement, however, is obtained between

the 33-state results of Kernoghan et al. (1996) and the experimental

results of Zhou et al. (1997) at all energies up to 100 eV. Below the

positronium formation threshold, however, the experimental results fall

below the accurate theoretical values. Investigations of positron–hydrogen

scattering using the coupled-state approximation with the inclusion of

several states of hydrogen and positronium have also been made by Mitroy

and Ratnavelu (1995) and Gien (1997), but these authors did not extend

their calculations to such high energies as did Kernoghan and coworkers.

Versions of the coupled-state method with all positronium terms omit-

ted, such as the intermediate energy R-matrix method used by Higgins,

Burke and Walters (1990) and the convergent close-coupling method used

by Bray and Stelbovics (1994), when applied to positron–hydrogen scat-

tering give similar results to those of Kernoghan et al. (1996) at the higher

energies. However, they underestimate the total cross section below 40 eV,

where positronium formation makes a signiﬁcant contribution to it.

The total cross sections calculated by Kernoghan, McAlinden and Wal-

ters (1996) and Campbell et al. (1998a) for positron scattering by the

alkali atoms sodium, potassium and rubidium are in good agreement

with the slightly modiﬁed experimental results of Parikh et al. (1993)

and Kwan et al. (1994), the modiﬁcations having been made to correct for

the neglect of small-angle scattering. Figure 2.15 shows the experimental

and theoretical results for sodium (see subsection 2.5.3). Coupled-state

calculations of σ

for the alkali atoms have also been performed by Ward

et al. (1989) and McEachran, Horbatsch and Stauﬀer (1991), but these

authors did not include any positronium states, although the positronium

formation channel is open even at zero incident positron energy, where

the cross section is inﬁnite. Excluding the positronium formation channel

has a very signiﬁcant eﬀect on the magnitude of the total cross section

at low positron energies because the calculated elastic scattering cross

section is then much larger than the true value, presumably in partial

compensation for the absence of the positronium channel.

A more common means of calculating σ

at intermediate and high

energies is to use the optical theorem, which expresses the conservation of

2.2 Theory 45

particle ﬂux in the scattering process as the following relationship between

the total cross section and the imaginary part of the elastic scattering

amplitude in the forward direction:

=(4π/k)Imf

(θ =0), (2.3)

where k is the wave number of the incident projectile. Winick and

Reinhardt (1978a, b) used this relationship to calculate σ

for positron–

hydrogen scattering over the energy range 0–34 eV after ﬁrst calculating

the elastic scattering amplitude using their moment T-matrix method.

At intermediate and higher energies it is appropriate to expand the

forward elastic scattering amplitude in a Born series:

(θ =0)=

∞



n=1

(θ =0), (2.4)

and a reasonably good approximation to the forward scattering amplitude

should then be given by the sum of the ﬁrst few terms in the expan-

sion. Because the ﬁrst term in the Born expansion

for

(θ =0)is

real and therefore, according to the optical theorem, does not make a

contribution to σ

, the lowest order contribution to the right-hand side

of equation (2.4) is from the second order term, f

(θ = 0). Calculations

of the ﬁrst few terms in the Born expansion of f

(θ = 0) have been

made by Byron and Joachain (1977a). These and other authors have also

used quite closely related approximation schemes such as the Glauber

approximation and the eikonal-Born approximation. For positron–helium

scattering these approximations are in reasonably good agreement with

each other, and they also yield values of σ

in reasonably good agreement

with the experimental values for energies greater than 300 eV (Byron,

1982).

At suﬃciently high positron energies, typically 1000 eV for helium,

the ﬁrst Born approximation yields reasonably accurate cross sections

for elastic scattering and for the various inelastic processes. Furthermore,

these cross sections have the same values for both electrons and positrons,

except for positronium formation, which is, of course, absent in electron

collisions. But at such high energies the positronium formation cross

section is negligible, and therefore the ﬁrst Born approximation to the

total scattering cross section is essentially the same for both projectiles.

This merging of the total cross sections for the two projectiles is indeed

observed experimentally, as mentioned above, but it takes place at much

lower energies than might have been expected. For helium, the two cross

sections have already merged at an energy of 200 eV, as may be seen in

Figure 2.1, and for the alkali atoms the merging occurs at energies as low

as 50 eV (Stein et al., 1990). The total cross section for positrons merges

46 2 Total scattering cross sections

with that for electrons from above for atomic hydrogen and the alkali

atoms, and from below for all the noble gases and many other atoms and

molecules.

This phenomenon may be at least partially understood by reference

to the optical theorem and the Born expansion of the imaginary part of

the forward scattering amplitude, equation (2.4). If exchange is ignored

when considering electron–atom scattering, the ﬁrst non-zero contribution

to the Born expansion is the second order term f

(θ = 0), and this

is the same for both electrons and positrons because it is quadratic in

the projectile–target interaction potential. Similar identities exist for all

the even-order terms in the Born series, and any diﬀerences between the

total cross sections for electrons and positrons therefore arise from the

odd-order terms in the series with n ≥ 3. But Dewangan (1980) has shown

that if the energy diﬀerences between atomic states are neglected in the

calculation of the higher order terms in the Born series, then f

(θ =

0) = 0 for odd n ≥ 3. Even though these odd higher order terms are not

actually zero, they are likely to be small in magnitude compared with the

adjacent even terms, as has been conﬁrmed for hydrogen and helium by

Byron and Joachain (1977a); see also Byron (1982).

Although the above argument is not rigorous, particularly in its neglect

of electron exchange, it nevertheless provides a plausible explanation

for the merging of the two total cross sections at much lower projectile

energies than those for which the ﬁrst Born approximation alone is valid.

The real part of the forward elastic scattering amplitude can be related

to the total scattering cross section by means of the following dispersion

relation:

Re f

(k,θ =0) − f

(k,θ =0)

=(1/2π)P



∞

2



)/(k

2

− k

)] dk



, (2.5)

or, in its subtracted form,

Re [f

(k,θ =0)− f

(0,θ = 0)] =



(k,θ =0)− f

(0,θ =0)



=(k

/2π)P



∞

[σ



)/(k

2

− k

)] dk



(2.6)

where P implies the principal value of the integral. This latter form,

lacking the factor k

2

in the numerator of the integrand, gives less weight

to the total cross sections at high energies.

At zero incident energy equation (2.5) reduces to

a + a

=(−1/2π)



∞



) dk



, (2.7)

2.2 Theory 47

where a is the scattering length, and a

is the ﬁrst Born approximation

to it. Calculations of the scattering length are discussed further in sub-

section 3.2.1.

These equations provide useful checks on the consistency of the exper-

imentally measured values of σ

with the calculated values of the real

part of the forward scattering amplitude, or the scattering length; they

have been used as such by Bransden and Hutt (1975). For helium, these

authors took the measured total cross sections of Coleman et al. (1976b)

in the energy range 2–800 eV; below 2 eV they used an extrapolation

based on a least-squares ﬁt of the functional form

(k)=b

+ b

k + b

ln k + b

+ b

(2.8)

to the measured elastic scattering cross sections below the positronium

formation threshold. The ﬁrst Born approximation was used to estimate

above 800 eV. Using all this total cross section data, the value of

(−1.24 ± 0.05)a

, where a

is the Bohr radius, was obtained for the

right-hand side of equation (2.7), in very satisfactory agreement with

the accurate value of −1.28a

for the left-hand side, as calculated by

Humberston (1973). A similar analysis by Bransden, Hutt and Winters

(1974), using the total cross section measurements of Canter et al. (1973),

revealed that these earlier experimental data suﬀered from systematic

errors which resulted in a serious underestimation of the true values of σ

at higher positron energies.

Throughout the energy range up to the positronium formation thresh-

old, Bransden and Hutt (1975) also obtained good agreement between

the right-hand side of equation (2.6), as obtained from the measured

total cross sections, and the left-hand side, as calculated from accurate

theoretical elastic scattering phase shifts (η

for the lth partial wave),

using the relationship

Re f

(θ =0)=(1/2k)

∞



l=0

(2l + 1) sin 2η

. (2.9)

The theoretical results depend quite sensitively on the quality of the phase

shifts, and good agreement with the values derived from the experimen-

tal cross sections was only obtained using the accurate phase shifts of

Humberston (1973).

A similar comparison of measured total cross sections and calculated

values of the elastic scattering parameters was also made by Bransden and

Hutt (1975) for positron–neon collisions, using the theoretical polarized-

orbital phase shifts of Montgomery and LaBahn (1970). These are al-

most certainly less accurate than the corresponding results for helium,

and there is poorer agreement between the values of the two sides of

equation (2.6).

48 2 Total scattering cross sections

2.3 Experimental techniques

In this section we describe the various experimental techniques which

have been used to measure σ

and we include a critical evaluation of

their limitations so as to aid in the intercomparison of the various sets of

data. Many experimental groups have contributed to this ﬁeld, but the

aim here is not to discuss every technique in great detail but rather to

select a representative subset in order to illustrate the most interesting

features. Discussions of the results obtained are given in sections 2.5 and

2.6.

Figure 2.2 shows the localized scattering system employed by the Uni-

versity College London group for their total cross measurements after

1976. The apparatus was based around a time-of-ﬂight (TOF) arrange-

ment where one timing trigger was provided by the ampliﬁed signal from a

thin plastic scintillator traversed by the β

particles from the radioactive

source and in which they deposited some of their kinetic energy before

striking the moderator. The detection of a low energy positron by the

channeltron electron multiplier (CEM) at the end of the ﬂight path pro-

vided the other timing signal. In order to prevent loss of data this latter

signal, with a low count rate of 1–10 s

−1

, was used to start the timing

sequence since the delayed scintillator output (stop-signal) rate, which

was governed by the activity of the radioactive source, was typically 10

−1

. In this system the axial magnetic ﬁeld used to transport the positrons

was generated by coils wound directly on to the beam vacuum pipeline.

Initially a smoked MgO moderator was used, but later this was replaced

by either tungsten vanes or a mesh.

Gas could be admitted to a localized region of the ﬂight path, close

to the moderator, which was diﬀerentially pumped by a system of three

diﬀusion pumps. The total length of the gas cell was 10 cm, including 6

mm diameter apertures, each 1 cm long, at either end. The gas pressure

was measured at the centre of the cell, on the side opposite to the inlet,

using a capacitance manometer, whilst the temperature measurement was

eﬀected on the outside of the beam pipe using a simple mercury-in-glass

thermometer. Thus, an estimate of the gas density n

at the centre of

the scattering cell could be obtained from equation (2.2). It was found

that the pressure gradients introduced by the use of a short cell could be

accounted for using n

and a single normalization constant k

, which was

found to be nearly independent of the nature and pressure of the gas and

to have the value 1.275 ±0.020. In terms of the gas density n(x)atpoint

x along the positron ﬂight path of length L,



n(x) dx = n

, (2.10)