Charlton M., Humberston J.W. Positron Physics

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3.2 Theory 109

see Armour and Humberston (1991). Alternatively, techniques can be

used in which these singularities do not arise, such as the method devised

by Harris (1967). This method assumes a trial function of the same

general form as that used in the Kohn variational method, equation (3.42),

and the eigenvalues and eigenvectors of the total Hamiltonian matrix are

then calculated in the basis φ

of the short-range correlation terms. Thus

(H − EN)C =0, (3.55)

where the matrix elements of H and N are φ

|H|φ

 and φ

|φ

 respec-

tively. At one of the energy eigenvalues, E

, the trial function is expressed,

using a similar nomenclature to that in equation (3.47), as

= S + tan ηC +



i=1

= S + tan ηC +Φ

, (3.56)

where the c

are the elements of the eigenvector C corresponding to the

eigenvalue E

. The projection of (H −E

)Ψ

onto Φ

is then required to

be zero,

Φ

|H − E

|Ψ

 =0, (3.57)

whence

tan η = −

Φ

|H − E

|S

Φ

|H − E

|C

. (3.58)

Schwartz singularities are avoided using the Harris method, but results

can only be obtained at the discrete energies E

(although the values of

can be altered by changing the values of the non-linear parameters

in the trial function). Furthermore, the error in the phase shift is only

of ﬁrst order in the error in the trial wave function, and the results may

therefore be less accurate than those of a well-behaved Kohn calculation.

Probably the most accurate positron–hydrogen s-wave phase shifts

are those obtained by Bhatia et al. (1971), who avoided the possibility

of Schwartz singularities by using a bounded variational method based

on the optical potential formalism described previously. These authors

chose their basis functions spanning the closed-channel Q-space, see

equation (3.14), to be of essentially the same Hylleraas form as those used

in the Kohn trial function, equation (3.42), and their most accurate results

were obtained with 84 such terms. By extrapolating to inﬁnite ω in a

somewhat similar way to that described in equation (3.54), they obtained

phase shifts which are believed to be accurate to within 0.0002 rad.

They also established that there are no Feshbach resonances below the

positronium formation threshold.

Several other accurate calculations have been made of s-wave elastic

scattering phase shifts, the more recent ones usually being the byproduct

110 3 Elastic scattering

of studies of positronium formation and other inelastic processes at higher

energies. Among these have been the calculations of Doolen et al. (1971),

who used the extrapolated T-matrix method, and Chan and Fraser (1973),

who used the formalism of the coupled-static approximation with the

addition of several short-range Hylleraas correlation terms. Worthy of

particular mention is the moment T-matrix method developed by Winick

and Reinhardt (1978a, b), who used it to calculate the various partial-

wave elastic scattering amplitudes, t

, from the oﬀ-shell elastic scattering

T-matrix. In terms of these amplitudes, the lth partial-wave contribution

to the elastic scattering cross section is then

4π

(2l +1)|t

. (3.59)

The most accurate values of the positron–hydrogen s-wave phase shifts,

obtained using variational methods, are plotted in Figure 3.2, together

with the results of other less accurate approximations. At low energies,

the attractive polarization potential is dominant and the phase shift is

positive. As the energy increases, however, the polarization potential be-

comes less attractive and the repulsive static potential begins to dominate,

whereupon the phase shift changes sign and becomes negative.

(b) The scattering length According to Levinson’s theorem, the s-wave

phase shift tends to nπ as the energy of the projectile tends to zero, where

n is the number of bound states of the composite projectile–target system.

However,

lim

k→0



−

tan η



= a, (3.60)

where a is the scattering length, and the elastic scattering cross section

at zero positron energy is therefore

(k =0)=4πa

. (3.61)

The scattering length can be calculated using the Kohn variational

method in a similar manner to that employed for the phase shift, but

the Kohn functional then becomes

= a

+ Ψ

|L|Ψ

. (3.62)

For positron–hydrogen scattering the trial function must have the

asymptotic form

∼

→∞

√

4π



1 −



). (3.63)

If no bound states of the composite projectile–target system exist,

which has been proved by Armour (1978, 1982) to be the case for the

3.2 Theory 111

positron–hydrogen system, the Kohn variational method immediately

gives a rigorous upper bound on the scattering length (Spruch and Rosen-

berg, 1960). If, however, there are bound states of the projectile–target

system then the Kohn method will also yield an upper bound on the

scattering length, provided that the matrix representation of the total

Hamiltonian, in the basis of the short-range correlation terms in the trial

function has as many eigenvalues below the energy of the target system

as there are bound states of the composite system.

The existence of an upper bound ensures that the value of the scattering

length becomes more negative as the ﬂexibility of the trial function is

increased by the addition of more short-range correlation terms. How-

ever, as already mentioned, the convergence of the scattering length with

respect to the number of such terms is poor unless long-range correla-

tion terms with a 1/r

dependence are added to the trial wave function

(Schwartz, 1961b; Humberston and Wallace, 1972). These terms arise

from the long-range character of the interaction potential between the

projectile and the target at zero projectile energy, which, for suﬃciently

large values of r

, reduces to the adiabatic polarization potential whose

asymptotic form is given by equation (3.27). The distortion of the hydro-

gen atom in the ﬁeld of the stationary positron, which gives rise to the

polarization potential, has the dipole form (r

+ r

/2)Φ

) cos(θ

)/r

(Temkin and Lamkin, 1961). Thus, an appropriate form of zero energy

trial function incorporating these long-range features is

√

4π



1 −

[1 − exp(−λr

)] +

[1 − exp(−λr

)]

+ b





cos θ

[1 − exp(−λr

)]



)



i=1

, (3.64)

where the long-range monopole term multiplied by the linear parameter

is also included, to represent more accurately the inﬂuence of the

polarization potential on the positron.

This form of trial wave function has been used in several variational

calculations (Schwartz, 1961b; Houston and Drachman, 1971; Humber-

ston and Wallace, 1972) to obtain well-converged upper bounds on the

scattering length, the most accurate value found being −2.103a

. In its

most ﬂexible form the coeﬃcients b

and b

multiplying the long-range

correlation terms in equation (3.64) were left free, to be determined

by the variational method (Humberston and Wallace, 1972), and their

values were found to be very close to those predicted by the adiabatic

approximation, namely −2.25 and 1.0 respectively.

112 3 Elastic scattering

phase shifts require the use of trial wave functions which incorporate

the total angular momentum of the system in the most general manner.

Asymptotically, the total orbital angular momentum of the positron–

hydrogen system resides solely on the positron, but when the positron

is close to the atom, and interacting with it, the total angular momentum

is shared between the positron, with angular momentum l

, and the

electron, with angular momentum l

. In principle, all combinations of

and l

which couple together to form a total angular momentum of l

should be included in the trial function, but Schwartz (1961a) showed that

an equivalent formulation may be used which involves a summation over

just l + 1 rotational harmonics of the required parity, each one multiplied

by a suitable spherically symmetric function of the three interparticle

coordinates of the system.

For p-wave scattering, with l = 1, there are two such rotational har-

monics of odd parity, Y

1,0

(θ

,φ

) and Y

1,0

(θ

,φ

), corresponding to l

=1,

= 0 and l

=0,l

= 1 respectively; Y

1,0

(θ, φ)=



4π

cos θ.For

d-wave scattering, with l = 2, there are three rotational harmonics of

even parity, Y

2,0

(θ

,φ

) and Y

2,0

(θ

,φ

), corresponding to l

=2,l

and l

=0,l

= 2 respectively, where Y

2,0

(θ, φ)=



4π

(

cos

θ −

), and

Y (θ

,φ

; θ

,φ

4π

√

(3 cos θ

cos θ

− cos θ

corresponding to l

=1,l

= 1. A suitably ﬂexible trial function for

p-wave scattering, similar in form to equation (3.42) but now containing

short-range correlation terms of both symmetries, is then

= Y

1,0

(θ

)

√



(kr

) − tan η

(kr

)[1 − exp(−λr

)]



)

+ Y

1,0

(θ



i=1

exp [−(αr

+ βr

+ γr

)] r

+ Y

1,0

(θ



j=1

exp[−(αr

+ βr

+ γr

)]r

, (3.65)

where the summations include all short-range correlation terms with

+ l

+ m

≤ ω

and k

+ l

+ m

≤ ω

. (3.66)

This form of trial function was used in the Kohn variational method

by Armstead (1968) and Humberston and Campeanu (1980) to obtain

well-converged p-wave phase shifts.

3.2 Theory 113

Humberston and Campeanu (1980) investigated the convergence of the

p-wave phase shifts with respect to the number of short-range correlation

terms of each symmetry separately, and they showed that at low positron

energies the ﬁrst-symmetry terms are the most important but that as

the positron energy is raised the inclusion of the second-symmetry terms

becomes increasingly signiﬁcant.

Several partial-wave phase shifts have been calculated using a range

of other techniques, notable examples being the Harris method, used by

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

method (Bray and Stelbovics, 1994) and various forms of the coupled-

state method (Kernoghan et al., 1995, 1996; Higgins, Burke and Walters,

1990; Kuang and Gien, 1997; Mitroy and Ratnavelu, 1995).

At suﬃciently low positron energies the scattering in all partial waves

with l>0 is dominated by the polarization potential, which has the

asymptotic form of equation (3.27); the phase shifts are given by the

formula (O’Malley, Spruch and Rosenberg, 1962)

παk

(2l − 1)(2l + 1)(2l +3)

+ R, (3.67)

where the remainder term R is of order k

for l = 1 and of order k

for

l>1. As can be seen in Figure 3.3, at very low positron energies the

variational values of the p-wave phase shift display the linear increase

with positron energy predicted by equation (3.67), but as the energy is

increased the phase shifts fall progressively below the linear form. The

d-wave phase shifts, however, are in reasonably good agreement with the

linear energy dependence of equation (3.67) over a much wider range,

almost up to the positronium formation threshold at 6.8 eV. This equation

can therefore be expected to provide reasonably accurate phase shifts for

all higher partial waves, and this has indeed been conﬁrmed by the results

of accurate coupled-state calculations.

2 Positron–helium scattering

Until quite recently, helium was the simplest atomic target used in experi-

mental studies of positron collisions. It is also the simplest atom for which

the wave function is not known exactly. Accordingly, positron–helium

scattering has attracted considerable theoretical attention, and detailed

comparisons have been made between the experimental measurements

of the scattering parameters and the corresponding theoretical results

obtained using a wide variety of approximation methods.

The use of an inexact target wave function in a scattering calculation

inevitably introduces some inconsistencies into the formulation; this is

114 3 Elastic scattering

Fig. 3.5. The coordinates of the positron–helium system.

so even for elastic scattering, where the usual practice is to replace the

exact energy of the target system by the expectation value of the target

Hamiltonian as calculated using the chosen inexact target wave function.

However, it has been shown by several authors that this procedure can

produce quite inaccurate results unless very accurate target wave func-

tions are used (Peterkop and Rabik, 1971; Houston, 1973; Page, 1975; Van

Reeth and Humberston, 1995a). The most satisfactory means of avoiding

this problem is to use the method of models (Drachman, 1972), but it can

only be used for the elastic scattering of projectiles, such as positrons,

which are distinguishable from the electrons in the target. The exact

target wave function, Φ

, although not known, satisﬁes the eigenvalue

equation

= E

, (3.68)

where H

is the exact target Hamiltonian and E

is the exact, but also

unknown, target ground state energy. It is assumed that a known ap-

proximation to the target wave function, Φ

, is an exact eigenfunction of

a model Hamiltonian H

, so that

= E

. (3.69)

For the helium atom, with the two electron coordinates r

and r

= −

(∇

+ ∇

)+V

, (3.70)

where the model interaction potential V

and the model energy E

may

be determined by substituting the approximate wave function into the

Schr¨odinger equation for the model, equation (3.69).

The exact scattering process is now replaced by one in which the pro-

jectile is scattered by the model target system but it is assumed that the

interactions between the projectile and the constituent particles in the

3.2 Theory 115

target retain their exact Coulomb form. Thus, using the nomenclature of

Figure 3.5, the interaction potential between the positron, with position

vector r

, and the helium atom is

int



−



, (3.71)

and the total potential function is therefore

V = V

int

+ V

. (3.72)

It is important to appreciate that any results obtained using the method

of models are approximations to the exact results

for scattering by the

model target rather than the real target. This distinction is particularly

signiﬁcant when calculating a rigorous bound on a scattering parameter.

If, however, the model target wave function is a good approximation to

the exact target wave function, then an accurate scattering result for the

model system is expected to be a good approximation to the exact result

for the real system.

Although providing a self-consistent formulation for the elastic scatter-

ing of a distinguishable particle by an inexact target system, the use of

the method of models fails to avoid inconsistencies in the formulation of

rearrangement collisions. This may be readily appreciated by considering

the example of positronium formation in positron–helium scattering, dis-

cussed in detail in subsection 4.2.2, where the interactions of the electrons

with the nucleus, and with each other, in the initial helium target are given

in terms of the model potential V

, but the electron–nucleus interaction

in the residual He

ion has the exact Coulomb form.

Let us now consider further the use of the method of models in elastic

positron–helium scattering, which is the sole open channel for positron

energies below 17.8 eV, the threshold for ground state positronium for-

mation. The total Hamiltonian of the system is

H = −

(∇

+ ∇

)+V

int

+ V

(3.73)

and the total energy is

E =

+ E

. (3.74)

Because the ground state helium model wave function is nodeless, the

total positron–helium wave function may be written without any loss of

generality as the product form

Ψ(r

, r

)=F (r

, r

)Φ

, r

), (3.75)

116 3 Elastic scattering

where Φ

, r

) is the model helium wave function. Operating with

L =2(H − E) on Ψ and using equation (3.69) yields

LΨ=LF Φ

=Φ



−(∇

+ ∇

)+2V

int

− k

−2(∇

·∇

F + ∇

·∇

F ) , (3.76)

in which there is no longer any explicit reference to either the form of the

model potential, V

, or the energy eigenvalue of the model Hamiltonian,

. Thus, if the method of models is used with the product form of wave

function, equation (3.75), nothing need be known about the model target

except its wave function.

Several authors have used the method of models in conjunction with the

Kohn variational method to obtain partial-wave phase shifts for elastic

positron–helium scattering. The most detailed investigations were those

of Humberston (1973), Campeanu and Humberston (1975) and Humber-

ston and Campeanu (1980), who formulated the problem in a manner

rather similar to that described previously for elastic positron–hydrogen

scattering in subsection 3.2.1 (Humberston and Wallace, 1972). Here,

however, the short-range correlation terms in the trial function involve

all six interparticle distances between the four particles in the system

(see Figure 3.5), and there is the additional complication of ensuring the

spatial symmetry of the trial function with respect to the interchange of

the two electrons, which are in a singlet spin state. For s-wave scattering,

the total wave function used by Humberston (1973) had the product form

of equation (3.75), with

F =



4π

(kr

) − tan η

(kr

)[1 − exp(−λr

)]}

+(1+P

)



exp[−(αr

+ βr

)] r

(3.77)

where P

is the space-exchange operator for the two electrons. The

summation included all short-range correlation terms such that

+ l

+ m

+ n

+ p

+ q

≤ ω, (3.78)

where k

, l

, m

, n

, p

, q

and ω are all non-negative integers; furthermore,

for numerical convenience, only even values of q

were included. In order to

avoid the same term being generated twice by the action of the exchange

operator P

, additional constraints were imposed, namely that l

≥ n

and if l

= n

then m

≥ p

. Further details of the formulation and a

3.2 Theory 117

description of the numerical techniques developed to evaluate the various

matrix elements were given by Armour and Humberston (1991).

Systematic improvements in the trial wave function were achieved by in-

creasing the value of ω, and investigations of the convergence of the phase

shifts revealed a similar pattern to that described earlier for positron–

hydrogen scattering, equation (3.54), with extrapolation to inﬁnite ω

expected to yield essentially exact results for the particular helium model

being used.

A suitable representation of the helium wave function was taken to be

the Hylleraas form

= exp[−B(r

+ r

)]



j=1

+ r

)

− r

)

, (3.79)

where the summation included all terms with non-negative integer powers

such that

+ M

+ N

≤ ω

, (3.80)

with M

taking only even values. Again for numerical convenience, N

was also limited to even values. The values of the parameters d

(j =

1,...,n

) and B were determined using the Rayleigh–Ritz variational

method to minimize the expectation value of the ground state energy of

the helium atom subject to its dipole polarizability having the correct

value, α =1.383a

(Dalgarno and Kingston, 1960; Thomas and Humber-

ston, 1972). This requirement was imposed because of the importance of

the dipole polarizability in determining the values of the low energy phase

shifts with l>0; see equation (3.67).

With such an elaborate form for the helium wave function, the matrix

elements of the operator L between any two short-range terms in the trial

function of the product form Φ

and Φ

may be more conveniently

expressed, after integrating by parts, as

Φ

|L|Φ





[∇

·∇

+ ∇

·∇

+ ∇

·∇

+(V

int

− k

)φ

] dr

(3.81)

rather than the form derived from direct use of equation (3.76). The

helium wave function now enters only as a squared factor multiplying the

remainder of the integrand and not in terms of its gradient.

Three helium models were generated (Thomas and Humberston, 1972)

for these investigations by setting ω

= 0, 2 and 4; the corresponding

numbers of terms in the helium model wave function, according to the

scheme in equation (3.80), are 1, 5 and 14 respectively. These models are

118 3 Elastic scattering

Table 3.1. Properties of three helium wave functions used in variational calcu-

lations of positron–helium elastic scattering. Each wave function is of the form

given in equation (3.79), the number of terms n being determined by the value

of ω

according to the pattern speciﬁed in equation (3.80)

H1 H5 H14 Exact

02 4

n(ω

) 1514

Energy (a.u.) −2.840 −2.895 −2.900 −2.9037

Polarizability (a.u.) 1.376 1.372 1.387 1.3834

r

 (a

) 1.1730 1.2117 1.196 81 1.1935

therefore referred to as H1, H5 and H14 and their properties are given

in table 3.1. Increasing the value of ω

in this way makes it possible

to investigate the convergence of the scattering parameters with respect

to systematic improvements in the helium model wave function as well

as with respect to ω for a given helium model. Results which are well

converged with respect to increases in both ω and ω

are then assumed

to be very close to the exact results for positron scattering by a real helium

atom.

Well-converged (with respect to ω) s-wave phase shifts for the three

helium models deﬁned above are plotted in Figure 3.6. As in positron–

hydrogen scattering (see subsection 3.2.1), the rate of convergence with

respect to ω deteriorates as the positron energy approaches zero. An

accurate determination of the scattering length requires a zero energy

trial wave function which is a suitably modiﬁed form of equation (3.77),

with the addition of long-range polarization terms similar to those in-

troduced into the zero energy positron–hydrogen trial wave function,

equation (3.64).

Substantial diﬀerences exist between the results for the simple uncor-

related helium function H1 and those for the more elaborate correlated

helium functions H5 and H14, but the very close agreement between these

latter two sets of results strongly suggests that they are both close to

the exact values. Further evidence in support of this claim has recently

been provided by Van Reeth and Humberston (1995a), who obtained very

similar results using even more accurate helium wave functions, both with

and without the use of the method of models.

Similar variational techniques were employed by Campeanu and Hum-

berston (1975) (see also Humberston, 1979) to determine accurate values

of the p- and d-wave phase shifts. Trial wave functions were used which