Charlton M., Humberston J.W. Positron Physics

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3.4 Angle-resolved elastic scattering 149

All the values of dσ

/dθ described here so far have been relative, the

absolute scale usually having been obtained by normalization to theory.

Eﬀorts have been made by the Detroit group to make direct absolute

diﬀerential cross section measurements for positrons, where the only com-

parison was between the positron data and their own normalized electron

data. Absolute values of dσ

/dθ for positrons were reported by Dou

et al. (1992a, b), but Kauppila et al. (1996) were subsequently unable to

reproduce these data. Clearly, the unambiguous determination of absolute

positron diﬀerential cross sections remains a task for the future.

One innovation introduced by Dou et al. (1992a, b) was to measure

dσ

/dθ not at a ﬁxed energy and varying angle but at a ﬁxed angle and

varying energy. In so doing, they discovered apparent structure, most

notably in the energy range 55–60 eV, at various angles for positron scat-

tering in argon gas. In particular, they found that at 60

◦

the diﬀerential

cross section appeared to fall abruptly by a factor of approximately two

in this energy region. They tentatively raised the possibility that this

eﬀect was linked to channel coupling but, as mentioned above, subsequent

work reported by Kauppila et al. (1996) failed to reproduce the observed

structure as also, independently, did that of Finch et al. (1996a, b) in a

study of single diﬀerential cross sections for elastic scattering, positronium

formation and ionization at an angle of 60

◦

in argon gas.

Positronium formation

4.1 Introduction

Positronium formation involves the capture by an incident positron of

one of the target electrons, to form the bound state Ps. This is one of

the simplest examples of a rearrangement collision and accordingly it has

attracted considerable attention, both experimental and theoretical.

Although, as described in section 1.5, positronium can be formed when

positrons interact with many diﬀerent media, in this chapter we are

mainly concerned with the reaction

+ X → Ps + X

, (4.1)

where the target atom or molecule, X, is in its ground state. Reaction

(4.1) can be generalized to

+ X → Ps(n

)+X

), (4.2)

where the quantities in parentheses are the relevant principal and orbital

angular momentum quantum numbers of the positronium and of the

residual ionic state. In a few instances we shall present data in which the

and n

values are not those of the appropriate ground states. The

total cross section for positronium formation, which includes all possible

states of the positronium and of the residual ion, is denoted by σ

Whilst reactions (4.1) and (4.2) are unique to positrons, they do have

close counterparts in the charge-transfer reactions of heavy positively

charged projectiles. Most notable are those involving protons, p

, namely,

+ X → H+X

, (4.3)

and it is instructive in some cases to compare results for the two projec-

tiles. This is attempted in section 4.6.

150

4.2 Theory 151

Positronium formation in gases has been the subject of investigation

since the pioneering works of Ore (1949) and Deutsch (1951) (see section

1.5), who studied the slowing down of β

particles, and the consequential

positronium formation, in dense gases. We describe in section 4.8 the

current situation in this ﬁeld.

After discussing the theory of positronium formation, with applications

to several relatively simple systems, we shall describe various techniques

used to measure positronium formation cross sections and present the

results so obtained, comparing them with theoretical predictions wher-

ever applicable. The chapter also includes a discussion of the angular

dependence of the positronium formation cross section. As well as being

of intrinsic interest as a test of theory, the diﬀerential formation cross sec-

tion, dσ

/dΩ, is also relevant for the production of energy-tunable beams

of positronium atoms. This topic is treated more fully in section 7.6.

4.2 Theory

Positronium formed in a positron–atom collision can be in a state with

principal quantum number up to n

provided that the kinetic energy of

the incident positron, E, exceeds the diﬀerence between the ionization

energy of the target, E

, and the binding energy of the positronium in

that state, i.e.

E ≥ E

− 6.8/n

, (4.4)

where energies are measured in eV. This inequality follows from the

deﬁnition of the threshold E

given in equation (1.13). All states of

positronium with principal quantum numbers <n

can also be formed

in such a collision. If the ionization energy of the target atom is less

than 6.8 eV, as it is for all the alkali metal atoms, then positronium

formation into the ground state is possible even at zero incident positron

energy. More commonly, however, the positronium formation threshold is

at some positive energy, for example, it is at 6.8 eV for atomic hydrogen

and 17.8 eV for helium, the highest value for any atom.

If the ﬁrst inelastic threshold of the target is at a higher energy than

that of the ground state positronium formation threshold, there is an

energy interval between these two thresholds, in which the only two

scattering processes are elastic scattering and ground state positronium

formation. It is in this energy interval, the so-called Ore gap, that the

most detailed theoretical investigations of positronium formation have

been made, as will now be described.

152 4 Positronium formation

1 Positron–hydrogen scattering

Because of its relative simplicity, particular attention has been devoted

to positronium formation in positron collisions with atomic hydrogen.

Within the Ore gap the two open channels (other than direct annihilation)

are

+H → e

→ Ps + p.

(4.5)

The total wave function therefore has two components:

Ψ=





, (4.6)

where Ψ

represents positron–hydrogen elastic scattering plus positron-

ium formation, and Ψ

represents positronium–proton elastic scattering

plus hydrogen formation. Using real wave functions, the asymptotic forms

of the two components for the lth partial wave are, using the nomenclature

of Figure 3.1,

∼

→∞

l,0

(θ

,φ

)

√

k [ j

(kr

) − K

(kr

)] Φ

)

∼

ρ→∞

−Y

l,0

(θ

,φ

)

√

2κK

(κρ)Φ

)

(4.7)

∼

ρ→∞

l,0

(θ

,φ

)

√

2κ [ j

(κρ) − K

(κρ)] Φ

)

∼

→∞

−Y

l,0

(θ

,φ

)

√

(kr

)Φ

(4.8)

where K

, K

and K

are the elements of the K-matrix, φ

) and

) are the ground state wave functions of hydrogen and positronium

respectively and ρ =(r

)/2 is the position of the centre of mass of the

positronium relative to the proton. Conservation of the total energy of

the entire system of positron plus target atom, E

, gives the relationship

between the wave numbers of the positron and positronium, k and κ

respectively, as

−

. (4.9)

The relationships between the K-, S- and T-matrices (Mott and Massey,

1965, Chapter 13) are

S =(1 + iK)/(1 − iK), (4.10)

T = 1 − S = −2iK/(1 − iK). (4.11)

In terms of the K-matrix, the cross section for scattering between an

initial channel i and a ﬁnal channel f is expressed as

4π(2l +1)



1 − iK



, (4.12)

4.2 Theory 153

where the subscripts i and f, with values 1 or 2, refer to the positron–

hydrogen and the positronium–proton channels respectively; also, k

= k

and k

= κ. Thus, σ

is the positron–hydrogen elastic scattering cross

section σ

and σ

is the positronium formation cross section σ

Among the most detailed and accurate investigations of positronium

formation in the Ore gap are those of Humberston (1982, 1984) and

Brown and Humberston (1984, 1985), who used an extension of the Kohn

variational method described previously, see section 3.2, to two open

channels. The single-channel Kohn functional, equation (3.37), is now

replaced by the following stationary functional for the K-matrix:



−



Ψ

|L|Ψ

Ψ

|L|Ψ



Ψ

|L|Ψ

Ψ

|L|Ψ





, (4.13)

where Ψ

and Ψ

are trial functions representing the two components of

the total wave function, as in equation (4.6). These trial functions must

satisfy the asymptotic forms given in equations (4.7) and (4.8), and they

should also provide adequate representations of the short-range correla-

tions between all the particles in the system. Suitable trial functions for

s-wave scattering, similar in form to those used in the elaborate variational

calculations of the s-wave positron–hydrogen elastic scattering phase shift,

equation (3.42), are as follows:



4π



(kr

) − K

(kr

)[1 − exp(−λr

)]



)

−



2κ

4π

(κρ)



1 − exp(−µρ)



µρ



)



i=1

exp[−(αr

+ βr

+ γr

)]r

(4.14)



2κ

4π



(κρ) − K

(κρ)



1 − exp(−µρ)



µρ



)

−



4π

(kr

)[1 − exp(−λr

)]Φ

)



j=1

exp[−(αr

+ βr

+ γr

)]r

. (4.15)

154 4 Positronium formation

These trial functions may be written in a more concise form as

= S

+ K



i=1

(4.16)

= S

+ K



i=1

, (4.17)

where



4π

(kr

)Φ

) (4.18)



4π

(kr

)[1 − exp(λr

)]Φ

) (4.19)



2κ

4π

(κρ)Φ

) (4.20)



2κ

4π

(κρ)



1 − exp(−µρ)



µρ



). (4.21)

Now, with two types of open-channel function, two forms of the kinetic

energy operator T are required:

T = −



∇

+ ∇



(4.22)

and

T = −



∇

+ ∇



, (4.23)

the latter form being appropriate when operating on the positronium

open-channel functions S

and C

.IfT operates on a function with

spherical symmetry, representing a system with zero total orbital angular

momentum, it can be expressed in terms of ρ, r

and α (the angle between

ρ and r

)as

T = −



4ρ

∂

∂ρ



∂

∂ρ



∂

∂r



∂

∂r





4ρ



sin α

∂

∂α



sin α

∂

∂α



. (4.24)

Its form in terms of the interparticle coordinates r

, r

and r

has been

given in equation (3.46).

The variational K-matrix is symmetric, as is required by unitarity, but

the trial K-matrix is not: K

= K

but K

= K

. This identity may

be readily proved by applying Green’s theorem to

− K

= K

− K

− [Ψ

|L|Ψ

−Ψ

|L|Ψ

] (4.25)

4.2 Theory 155

and noting that S

→ 0 and C

→ 0 suﬃciently rapidly on the surface

→∞, and that S

→ 0 and C

→ 0 suﬃciently rapidly on the surface

ρ →∞.

The stationary property of the Kohn-matrix functional (3.37) requires

that the partial derivatives of all the elements of the variational K-matrix,

, K

(= K

) and K

, with respect to the linear parameters in Ψ

and Ψ

, i.e. K

, K

(j =1,...,N) and d

(j =1, ...,N),

be zero.

As with single-channel scattering (see section 3.2), the resulting set of

linear simultaneous equations can be written as

AX = −B, (4.26)

but now

A =







C

|L|C

C

|L|C

 ... C

|L|φ

 ...

C

|L|C

C

|L|C

 ... C

|L|φ

 ...

φ

|L|C

φ

|L|C

 ... φ

|L|φ

 ...







B =







C

|L|S

C

|L|S



C

|L|S

C

|L|S



φ

|L|S

φ

|L|S









, X =













This is a rather obvious and natural extension, to two open channels,

of the set of linear simultaneous equations for a single open channel,

equation (3.53). The formulation can be readily extended to accom-

modate more open channels. Once all the matrix elements have been

calculated, the determination of the variational K-matrix proceeds in a

similar manner to that described previously for single-channel scattering.

Further details were given by Armour and Humberston (1991).

The multichannel Kohn variational method does not yield rigorous

lower bounds on any of the scattering parameters. However, the addition

of more short-range correlation terms to the trial wave function usually

produces an increase in the values of the diagonal K-matrix elements

and the eigenphase shifts; there is a similar pattern of convergence with

respect to increasing the number of terms in the trial wave function, as

described previously for single-channel scattering, see equation (3.54).

The values of the diagonal K-matrix elements obtained in this way are

156 4 Positronium formation

Fig. 4.1. The results of various calculations of the l = 0 partial-wave contribu-

tion to the positronium formation cross section in positron–hydrogen scattering

in the Ore gap: A, Archer, Parker and Pack (1990); B, Humberston (1982);

C, Stein and Sternlicht (1972); D, Chan and Fraser (1973); E, Wakid (1973);

F, Dirks and Hahn (×10) (1971); G, Wakid and LaBahn (1972); H, Khan and

Ghosh (×10

−1

) (1983); I, Born approximation (×10

−3

therefore probably lower bounds on the exact values, but they do not

translate into bounds on σ

The most accurate values of the s-wave positronium formation cross

section calculated by Humberston (1982) and Humberston et al. (1997)

are shown in Figure 4.1. (The latter results are more accurate but there

is no diﬀerence between the two sets of results on the scale of this ﬁgure.)

This cross section is much smaller than the s-wave elastic scattering cross

section and also, as we shall see, much smaller than other contributions

to σ

of low orbital angular momentum. It has recently been shown by

Ward, Macek and Ovchinnikov (1998), using hidden crossing theory, that

the small magnitude of the s-wave contribution to σ

is a consequence

4.2 Theory 157

of the near-total destructive interference of two terms in the S-matrix

element for positronium formation.

The results of several other calculations of the s-wave contribution to

are also shown in Figure 4.1, and they reveal extreme sensitivity to

the method of approximation being used. The range of results spans

several orders of magnitude, with the values obtained using the ﬁrst Born

approximation being a factor 200 times too large and those obtained using

the coupled-static approximation (see Cody et al., 1964, for a description

of this approximation) being approximately a factor of ten too small.

However, several sets of results obtained using variational methods with

ﬂexible trial functions are in reasonable agreement with each other. Stein

and Sternlicht (1972) used a similar technique to that of Humberston,

with a somewhat similar but rather less ﬂexible trial function, and the

two sets of results are indeed in rather good agreement except close to

the positronium formation threshold.

The only other calculation dating

from that time which yielded moderately accurate results was that of

Chan and Fraser (1973). These authors used a formulation based on the

coupled-static approximation, with the addition of several short-range

Hylleraas correlation terms, and they obtained rigorous lower bounds on

the diagonal elements of the K-matrix and the eigenphases, but their

values of σ

are little more than half those of Humberston. Winick

and Reinhardt (1978b) used a moment T-matrix method to determine

the elastic scattering amplitude, from which they then calculated the

elastic scattering cross section σ

and also, using the optical theorem,

equation (2.3), the total scattering cross section σ

. In the Ore gap the

diﬀerence between these two cross sections is then σ

. It is perhaps sur-

prising that, although their wave functions contained many Hylleraas cor-

relation terms, their results, not shown in Figure 4.1, are approximately

ﬁve times larger than those of Humberston. The probable reason for this

discrepancy is that the subtraction procedure involved in obtaining σ

rather inaccurate, the magnitudes of σ

and σ

being very similar.

More recent, detailed investigations by Archer, Parker and Pack (1990),

who used the reactive scattering method of Pack and Parker (1987),

yielded results, shown in Figure 4.1, which, within the Ore gap, are

approximately 15% lower than those of Humberston. These authors

also found two resonances just below the n

= 2 excitation threshold

of hydrogen, as well as other resonances just below higher excitation

thresholds, which Humberston failed to ﬁnd. Similar results to those of

Archer et al. were obtained by McAlinden, Kernoghan and Walters (1994)

and Kernoghan, McAlinden and Walters (1995) using the coupled-state

approximation with the following 18 states: H(1s, 2s,

3s, 4s, 2p, 3p, 4p,

3d, 4d), Ps(1s, 2s, 3s, 4s, 2p, 3p, 4p, 3d, 4d), where a bar implies a

pseudostate. However, Kvitsinsky, Carbonell and Gignoux (1995) and

158 4 Positronium formation

Kvitsinsky, Wu and Hu (1995), using two diﬀerent methods to solve the

Fadeev equations in conﬁguration space, obtained well-converged results

in excellent agreement with Humberston’s, as also did Mitroy, Berge and

Stelbovics (1994), Zhou and Lin (1994, 1995b) and Gien (1997).

Several other recent studies of positronium formation have been based

on some form of the coupled-state approximation. Basu, Mukherjee and

Ghosh (1990) used an integral form of the coupled equations in momentum

space with the expansion H(1s, 2s,

2p, 3d), Ps(1s) and obtained s-wave

positronium formation cross sections in moderately good agreement with

those of Humberston, although their elastic scattering cross sections are

rather less accurate. Higgins and Burke (1993) used the R-matrix method

with a more elaborate six-state approximation of the form H(1s,

2s, 2p),

Ps(1s,

2s, 2p), but their s-wave contribution to σ

is more than twice as

large as Humberston’s, and with a rather diﬀerent energy dependence.

An interesting feature of the accurate s-wave positronium formation

cross section (curve B of Figure 4.1) is the very rapid rise just above

threshold, from zero at the threshold, k =1/

√

2=0.707 12a

−1

,toa

value of 0.004πa

at k =0.71a

−1

and increasing slowly thereafter up to

the n

= 2 threshold. As mentioned in section 3.3, when discussing the

eﬀect of the opening of the positronium formation channel on the elastic

scattering cross section, Wigner’s threshold law predicts that the positro-

nium formation cross section for a given partial wave l should increase just

above threshold according to σ

∝ E

l+1/2

, where E

= E − E

is the kinetic energy of the positronium. Consequently σ

should in-

crease linearly with κ and have an inﬁnite derivative with respect to the

positron energy at the positronium formation threshold. The results of

Humberston (1982), and more recently those of Archer, Parker and Pack

(1990), Kvitsinsky and coworkers (1995), Mitroy, Berge and Stelbovics

(1994), Zhou and Lin (1995b) and Humberston et al. (1997) all conﬁrm

this law, but the linear increase with κ is restricted to a very narrow range

of positron energies just above the threshold.

Brown and Humberston (1984, 1985) used a similar variational method

to that of Humberston (1982) to calculate the p- and d-wave contributions

to σ

in the Ore gap. Their trial functions were obvious modiﬁcations to

higher partial waves of the s-wave trial functions of Humberston, equa-

tions (4.14) and (4.15), but, as with the trial functions for elastic scatter-

ing, there were now l + 1 groups of short-range correlation functions,

each group being associated with a diﬀerent rotational harmonic; see

equation (3.65). The results of these and some other calculations are

given in Figures 4.2 and 4.3. There is still signiﬁcant sensitivity to the

method of calculation, and to the form of the trial function, in the p-wave

results, although much less so than with the s-wave. The results of the