Charlton M., Humberston J.W. Positron Physics

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3.2 Theory 99

Fig. 3.2. The s-wave phase shift for positron–hydrogen scattering: A, static

approximation; B, result for six-term coupled state (1s, 2s, 2p, 3s, 3p, 3d of

H) (McEachran and Fraser, 1965); C, exact variational result (Schwartz, 1961b;

Bhatia et al., 1971).

system is minimized using some suitable trial function for each electron

orbital (McEachran et al., 1977). Usually, only the dipole component

of the positron–atom interaction potential is considered. The total wave

function is then expressed as

Ψ(r

, r

) = [1 + G(r

, r

)]Φ

)F (r

), (3.25)

and this is substituted into the Schr¨odinger equation. After projecting

onto the undistorted target wave function, Φ

), the following equation

is obtained for the scattering function, F (r



−



+ V



F (r

), (3.26)

100 3 Elastic scattering

Fig. 3.3. The p-wave phase shift for positron–hydrogen scattering: A, static

approximation; B, result for six-term coupled state (1s, 2s, 2p, 3s, 3p, 3d of

H) (McEachran and Fraser, 1965); C, exact variational result (Armstead, 1968;

Bhatia, Temkin and Eiserike, 1974).

where V

is the static potential and V

is the polarization potential arising

from the distortion of the atom. In the investigations of Drachman (1965),

who ﬁrst applied the polarized-orbital method to positron–hydrogen scat-

tering, V

was taken to be the second order adiabatic potential of Dalgarno

and Lynn (1957). This potential has the correct long-range behaviour,

namely

∼

→∞

−

(3.27)

where α is the dipole polarizability of the hydrogen atom (= 4.5a

), but

it is too attractive for small values of r

. In the adiabatic approximation

the positron is assumed to be stationary whereas in reality it is moving, so

3.2 Theory 101

that the target atom does not have unlimited time in which to adjust to

the changing ﬁeld created by the moving positron. Consequently the true

positron–hydrogen interaction is less attractive than the adiabatic inter-

action, even for positrons with zero incident energy. Drachman (1965)

attempted to compensate for this excessive attraction by reducing the

monopole component of the distortion function G(r

, r

), thereby modify-

ing the form of the potential V

. He found that almost total suppression of

the monopole component gave excellent agreement with the very accurate

variational results of Schwartz (1961b) for s-wave scattering. It has since

been found that accurate variationally determined wave functions for low

energy s-wave positron–hydrogen scattering (Humberston and Wallace,

1972) do indeed contain very little monopole distortion of the target.

Drachman (1965) also applied the polarized-orbital method to higher-

partial-wave scattering, but his phase shifts are rather less positive than

the accurate p-wave results of Armstead (1968) and the d-wave results

of Register and Poe (1975), the discrepancies becoming greater

as the

positron energy is increased. In partial waves with l>0 the adia-

batic potential is less attractive than the exact interaction because the

polarized-orbital method cannot adequately represent the most general

forms of distortion of a system in which the orbital angular momentum

is shared between the positron and the electron in the target.

One of the most commonly used approximation schemes in the study

of both electron and positron collisions with atoms is the coupled-state,

or close-coupling, approximation. As applied to positron–hydrogen scat-

tering, the total wave function may be formally expanded in terms of

complete sets of the discrete and continuum states of the hydrogen atom,

, and those of positronium, Φ

. Such an expansion is doubly com-

plete, and in principle the total wave function could be expressed solely in

terms of a complete set of either hydrogen or positronium states. Higgins,

Burke and Walters (1990) considered an expansion of the wave function

in terms of states of the hydrogen atom alone, but the rate of convergence

of the elastic scattering phase shifts with respect to adding more states is

very slow. More usually, a truncated expansion involving states of both

hydrogen and positronium is used, so that

Ψ(r

, r



i=1

)Φ



j=1

(ρ)Φ

), (3.28)

where, using the nomenclature of Figure 3.1, ρ is the position vector of the

centre of mass of the positronium relative to the proton. Operating with

H − E on Ψ, and requiring that the projection of the resulting function

on each of the hydrogen and positronium eigenfunctions, Φ

and Φ

102 3 Elastic scattering

be zero, we obtain

Φ

|(H − E) |Ψ =0 (i =1,...,n) (3.29)

and

Φ

|(H − E) |Ψ =0 (j =1,...,p). (3.30)

Then the following equations are obtained for the functions F

), i =

1,...,n, and G

(ρ), j =1,...,p:



−

∇

−







)









(ρ, r



(ρ) dρ, (3.31)



−

∇

−



(ρ)=





(ρ)G



(ρ)









(ρ, r



) dr

, (3.32)

where









−









, (3.33)



(ρ)=







−









, (3.34)

and



−

∇

[Φ

)Φ

)]



−



−



)Φ

)



. (3.35)

Energy conservation gives the relationships

E =

+ E

(i =1,...,n; j =1,...,p), (3.36)

where E

and E

are the energy eigenvalues of hydrogen and positro-

nium respectively.

If a positron channel labelled by i is open, then k

is positive and the

corresponding positron function F

) is oscillatory for large values of r

However, if the channel is closed, because the positron energy is below

3.2 Theory 103

the relevant threshold, then k

is negative and the function F

) decays

exponentially for large values of r

. Similarly, if a positronium channel

labelled by j is open, then κ

is positive and G

(ρ) is oscillatory for large

values of ρ, whereas for a closed positronium channel κ

is negative and

(ρ) decays exponentially for large values of ρ.

When elastic scattering is the only open channel, k

is positive but all

other values of k

, and all values of κ

, are negative. Consequently, all

the functions F

) and G

(ρ), except for F

), decay exponentially

for large values of r

and ρ. The resulting equation for F

) is similar

in form to equation (3.20), in which the optical potential V

opt

was intro-

duced; indeed a truncated coupled-state expansion essentially deﬁnes an

approximation to the optical potential which satisﬁes the conditions for

the phase shifts to be lower bounds on the exact values.

Including only the ground state of the target atom in the coupled-state

expansion deﬁnes the static approximation, which has been mentioned

previously as poor. Adding the ground

state of positronium creates the

coupled-static approximation (Cody et al., 1964; Bransden and Jundi,

1967; Chan and Fraser, 1973) and introduces some positron–electron

correlation into the wave function, but the results, although showing

signiﬁcant improvements over the static results, still diﬀer substantially

from the exact, variationally determined, values.

The rate of convergence of the phase shifts with respect to increasing the

number of eigenstates in the close-coupling expansion is rather low, as may

be seen in Figures 3.2 and 3.3, but faster convergence may be achieved

by replacing some of the terms in the expansion by pseudostates, each

of which is chosen so as to incorporate the important features of several

eigenstates into one algebraic function. The projection of (H −E)|Ψ onto

each pseudostate is assumed to be zero, and the energy of the pseudostate

is taken to be the expectation value of the relevant Hamiltonian, either

of hydrogen or positronium, as calculated with the pseudostate.

Many diﬀerent formulations of positron–hydrogen scattering based on

the coupled-state approximation have been made, particularly where sev-

eral channels are open. One of the most detailed studies was that of

Kernoghan et al. (1995), who used 18 states, H(1s, 2s,

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

3d, 4d) and Ps(1s, 2s, 3s, 4s, 2p, 3p, 4p, 3d, 4d), where the bar above a

term signiﬁes a pseudostate, and solved the resulting integro-diﬀerential

equations using an R-matrix technique (Burke and Robb, 1975) in the

energy range 0–100 eV. Kernoghan et al. (1996) increased the number of

states in the expansion to 33 and these data are very similar to the 18-state

results shown in Figure 3.11. The elastic scattering cross sections change

only slightly when going from 18 to 33 states, implying that they are

then close to the exact values. Other calculations using similar methods

104 3 Elastic scattering

have been made by Hewitt, Noble and Bransden (1990, 1991), Higgins and

Burke (1991, 1993), Gien (1994, 1997) and Archer et al. (1990). A recently

developed variant of the standard close-coupling method, which employs

an expansion of the wave function in terms of states of the hydrogen target

alone, is the convergent close-coupling method of Bray and Stelbovics

(1993). These results are in good agreement with those of Kernoghan

et al. (1995, 1996) at energies beyond 40 eV, although somewhat smaller

in the energy interval 10–30 eV.

The formulation outlined above is in conﬁguration space, but several

authors, notably Ghosh and his collaborators (Chaudhury, Ghosh and

Sil, 1974) and Mitroy (1993), also Mitroy, Berge and Stelbovics (1994)

and Mitroy and Ratnavelu (1995), have preferred to work in momentum

space with a set of coupled integral equations rather than the coupled

integro-diﬀerential equations (3.31) and (3.32).

An alternative means of incorporating the elaborate correlations be-

tween the constituent particles into the formulation of positron–atom

scattering is to use a ﬂexible algebraic trial wave function in a varia-

tional method, somewhat similar to the way in which the Rayleigh–Ritz

variational method is used for determining the energies of bound states.

One of the simplest and most convenient computational methods for

scattering problems is the Kohn variational method, which has been

used quite extensively to obtain very accurate results for low energy

positron scattering by several light atoms. Indeed, the ﬁrst accurate

values of the s-wave positron–hydrogen phase shifts were obtained in

this way (Schwartz, 1961b), and subsequently the method has been used

to obtain accurate values of the various partial-wave elastic scattering

and positronium formation cross sections for positrons colliding with

hydrogen (Armstead, 1968; Humberston and Wallace, 1972; Houston and

Drachman, 1971; Stein and Sternlicht, 1972; Humberston, 1982, 1984;

Brown and Humberston, 1984, 1985) and also for positrons colliding with

helium (Houston and Drachman, 1971; Humberston, 1973; Campeanu

and Humberston, 1975, 1977a; Van Reeth and Humberston, 1995b). Sev-

eral references will be made to this method of approximation, and it

is therefore appropriate to describe it in some detail in the context of

positron–hydrogen scattering. Initially only elastic scattering will be con-

sidered, but the way in which the method can be extended to multichannel

scattering, and particularly to positronium formation, will be outlined

later, in section 4.2.

The basis of the Kohn variational method is the functional

tan η

= tan η

−Ψ

|L |Ψ

, (3.37)

where L =2(H − E) and Ψ

is the trial wave function representing

the scattering process. For the lth partial wave, the trial wave function

3.2 Theory 105

must be chosen to have the asymptotic form (using the nomenclature of

Figure 3.1)

∼

→∞

l,0

(θ

,φ

)

√

k [ j

(kr

) − tan η

(kr

)] Φ

), (3.38)

where j

(kr

) and n

(kr

) are spherical Bessel and Neumann functions

respectively and Y

l,0

(θ

,φ

) is the corresponding spherical harmonic. The

function Ψ

diﬀers from the exact asymptotic form of the wave function

only in the value of the trial phase shift η

. The Kohn functional tan η

is stationary with respect to small variations in the trial wave function

from the exact wave function Ψ, so that if the trial wave function is

written as

=Ψ+δΨ, (3.39)

where

δΨ ∼

→∞

(tan η − tan η

(kr

)Φ

), (3.40)

η being the exact phase shift, then the error in the variationally deter-

mined phase shift η

is given by

δ(tan η) = tan η − tan η

= δΨ|L |δΨ, (3.41)

which is of second order in the error in the trial wave function. Conse-

quently, if the trial function is reasonably accurate, the error in tan η

should be much smaller than the errors in Ψ

and tan η

(a) s-wave scattering For s-wave scattering, a convenient and ﬂexible

choice of trial function, similar to that used by Schwartz (1961b), Stein

and Sternlicht (1972) and Humberston and Wallace (1972), is



4π

(kr

) − (tan η

(kr

)[1 − exp(−λr

)]}Φ

)



i=1

exp [−(αr

+ βr

+ γr

)] r

, (3.42)

where the short-range Hylleraas terms, exp[−(αr

+ βr

+ γr

)]r

represent the various interparticle correlations. Similar functions have

been used extensively to represent correlations in many diﬀerent three-

body systems. In the summation in equation (3.42) it is usual to include

all correlation terms such that

+ l

+ m

≤ ω, (3.43)

where k

, l

, m

and ω are non-negative integers. Increasing the value of

ω then provides a convenient and systematic means of improving the trial

wave function.

106 3 Elastic scattering

Because the orbital angular momentum of the positron–hydrogen sys-

tem is zero for s-wave scattering, the total wave function is spherically

symmetric and depends only on the three internal coordinates which

specify the shape of the three-body system. The kinetic energy operator

T = −



∇

+ ∇



(3.44)

can therefore be expressed solely in terms of r

, r

and the angle between

them, θ

(see Figure 3.1):

T = −



∂

∂r



∂

∂r



∂

∂r



∂

∂r







sin θ

∂

∂θ



sin θ

∂

∂θ



. (3.45)

Alternatively, it can be expressed in terms

of the three interparticle dis-

tances r

, r

and r

,as

T = −



∂

∂r



∂

∂r



∂

∂r



∂

∂r



∂

∂r



∂

∂r



+ r

− r

∂

∂r



. (3.46)

The trial function, equation (3.42), may be written in an abbreviated

form as

= S + K

C +



i=l

(3.47)

where

= tan η

, (3.48)

S =



4π

(kr

)Φ

), (3.49)

C = −



4π

(kr

)[1 − exp(−λr

)]Φ

) (3.50)

and

= exp [−(αr

+ βr

+ γr

)] r

. (3.51)

The requirement that the Kohn functional K

= tan η

, equation (3.37),

be stationary with respect to variations of the linear parameters K

and

(i =1,...,n), i.e. that

∂K

= 0 and

∂K

∂c

=0 (i =1,...,n), (3.52)

3.2 Theory 107

then results in the following set of linear simultaneous equations, which

we express in matrix form:







C|L|CC|L|φ

 ... C|L|φ



φ

|L|Cφ

|L|φ

 ... φ

|L|φ



φ

|L|Cφ

|L|φ

 ... φ

|L|φ



φ

|L|Cφ

|L|φ

 ... φ

|L|φ





















= −







C|L|S

φ

|L|S

φ

|L|S

φ

|L|S







(3.53)

Writing this matrix as AX = −B, where X is the column matrix of the

unknown linear parameters, then X = −A

−1

B and the stationary value

of tan η

is obtained by substituting the elements of X back into the

Kohn functional, equation (3.37). The optimum values of the non-linear

variational parameters in the trial function are determined by repeating

the entire calculation for a range of values of each non-linear parameter,

thereby identifying the values which yield a maximum value of tan η

Further details of this procedure were given by Armour and Humberston

(1991).

Although the Kohn variational method is not a bounded method (ex-

cept at zero energy when, subject to certain conditions, it can yield

an upper bound on the scattering length), it is found in practice that

the phase shift usually becomes more positive as the ﬂexibility of the

trial function is enhanced by increasing ω; it converges towards what

is assumed to be the exact value according to a pattern which is quite

accurately represented by

(ω)=K

(∞)+

, (3.54)

provided ω is not too small. Thus, a plot of K

(ω) against 1/ω

gives a

straight line, which can be extrapolated to inﬁnite ω to yield a good ap-

proximation to the exact result. An example of this convergence pattern

for the s-wave phase shift is given in Figure 3.4.

At very low positron energies the rate of convergence of the phase

shift with respect to ω deteriorates signiﬁcantly because the exact wave

function then resembles the form given by the adiabatic approximation,

with long-range components varying as 1/r

. Such terms are not rep-

resented very eﬃciently by a ﬁnite sum of short-range Hylleraas terms;

instead they need to be explicitly added to the trial function, as will be

described in the following account of positron scattering at zero energy

(see subsection 3.2.1(b) below). The convergence with respect to ω also

108 3 Elastic scattering

Fig. 3.4. The convergence of the positron–hydrogen s-wave phase shift (for k =

0.7a

−1

) with respect to systematic improvements in the trial wave function; see

equation (3.54).

deteriorates in the vicinity of the positronium formation threshold because

the conﬁguration of virtual positronium loosely bound to the residual ion

is not easily represented by the Hylleraas terms either. Again, the addition

to the trial function of a term explicitly representing this conﬁguration

improves the convergence.

Although most results conform to the convergence pattern described

by equation (3.54), very erratic values of the phase shift are occasionally

obtained which clearly contravene the empirical lower bound. These so-

called Schwartz singularities (Schwartz, 1961b) arise because the matrix

A deﬁned as in equation (3.53), must formally be inverted in order to

obtain the optimum values of the linear parameters in the trial function;

however, A is not positive deﬁnite and may have an eigenvalue very close

to zero, making the matrix ill-conditioned. It is quite easy to recognize

when the results are inﬂuenced by such a singularity because they do

not conform to the pattern obtained at slightly diﬀerent energies or with

slightly diﬀerent trial functions.

Various techniques have been devised for coping with Schwartz singular-

ities. They may either be ignored, or they can be avoided by using a mod-

iﬁed form of the Kohn variational method in which the asymptotic form of

the trial function has an amplitude diﬀerent from that of equation (3.42);