Charlton M., Humberston J.W. Positron Physics

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3.2 Theory 119

Fig. 3.6. Positron–helium s-wave phase shifts for three helium models: – – –,

H1; ——, H5; — · —, H14.

are obvious generalizations to positron–helium scattering of the trial func-

tions used in the determination of the higher-partial-wave phase shifts in

positron–hydrogen scattering; see equation (3.65). As in hydrogen, the

scattering at very low positron energies in all partial waves with l>

0 is dominated by the long-range polarization potential, and the phase

shifts are therefore given by equation (3.67). This equation provides little

more than the correct gradient of the p-wave phase shift in the limit of

zero incident energy, but it gives a reasonably good approximation to the

d-wave phase shift over quite a wide energy range. Well-converged results

for all helium models with the same dipole polarizability should therefore

agree at very low positron energies and this is clearly seen in Figures 3.7

and 3.8, where the p- and d-wave phase shifts for helium models H1 and

H5 are displayed. As the positron energy increases, however, the results

for the two models diverge, particularly for l =1.

Phase shifts for partial waves with l>2 are expected to be in even

better agreement with the values given by equation (3.67) up to the

120 3 Elastic scattering

Fig. 3.7. Positron–helium p-wave phase shifts: ——, Campeanu and Hum-

berston (1975) for model H5; — —, Campeanu and Humberston (1975) for

model H1; -- -- -- --, McEachran et al. (1977); — · —, Drachman (1966a); — ··—,

Aulenkamp, Heiss and Wichman (1974).

positronium formation threshold at 17.8 eV and, consequently, no elabo-

rate variational calculations have yet been performed for l>2.

Several other calculations of the ﬁrst few partial-wave phase shifts

for positron–helium scattering have been carried out using a variety of

approximation methods; in all cases, however, rather simple uncorrelated

helium wave functions have been used. Drachman (1966a, 1968) and

McEachran et al. (1977) used the polarized-orbital method, whereas Ho

and Fraser (1976) used a formulation based on the static approximation,

with the addition of several short-range correlation terms, to determine

the s-wave phase shifts only. The only other elaborate variational calcu-

lations of the s-wave phase shift were made by Houston and Drachman

(1971), who employed the Harris method with a trial wave function

similar to that used by Humberston (1973, 1974), see equation (3.77),

and with the same helium model H1. Their results were slightly less

positive than Humberston’s H1 values, and are therefore probably less

3.2 Theory 121

Fig. 3.8. Positron–helium d-wave phase shifts: ——, Campeanu (1977) for

model H5; — —, Campeanu (1977) for model H1; — · —, Drachman (1966a);

— ··—, Amusia et al. (1976); ·····, Aulenkamp, Heiss and Wichman (1974);

-- -- -- --, equation (3.67).

accurate. Houston and Drachman also calculated the scattering length,

using the Kohn variational method, and obtained essentially the same

value as Humberston, a = −0.524a

. A rather diﬀerent method, the

random-phase approximation derived from many-body theory, was used

by Amusia et al. (1976) to calculate the phase shifts for l ≤ 3. These

authors used a more accurate, but still uncorrelated, Hartree–Fock helium

wave function and obtained reasonably good agreement with the accurate

variational results.

The total elastic scattering cross sections for the three helium models,

calculated using the accurate variationally determined phase shifts for

l ≤ 2 and equation (3.67) for l>2, are shown in Figure 2.8, together

with several sets of experimental measurements. Excellent agreement is

obtained between the results for the two helium models H5 and H14 and

the experimental measurements of Canter et al. (1973) and, more recently,

those of Mizogawa et al. (1985). Also, the accurate theoretical results

122 3 Elastic scattering

are also in good agreement with an extrapolation of the experimental

cross sections of Canter et al. (1973) to energies below 2 eV obtained

by ﬁtting to the functional form given in equation (2.8) (Bransden, Hutt

and Winters, 1974; Humberston, 1974). Furthermore, the extrapolated

experimental cross section at zero energy, 0.88πa

, agrees quite well with

the value 0.92πa

derived from the accurate calculation of the scattering

length.

Other comparisons between experimental and theoretical values of the

scattering parameters were made by Bransden et al. (1974) and Bransden

and Hutt (1975), using techniques based on forward dispersion relations,

as described in section 2.2.

Additional, but rather less direct, evidence for the accuracy of the

variational results for models H5 and H14 is provided by the excellent

agreement between the theoretical and experimental lifetime spectra for

positrons diﬀusing in helium gas, where calculation of the theoretical

spectrum requires a knowledge of the momentum transfer and annihi-

lation cross sections, both of which are derived from the

wave functions

generated in the calculations of the elastic scattering phase shifts. A

detailed discussion of positron lifetime spectra is given in Chapter 6.

Diﬀerential cross sections, see equation (3.9), for elastic positron–

helium scattering, calculated for helium model H5 at several positron

energies are given in Figure 3.9. At low positron energies, in the vicinity of

the Ramsauer minimum, where the s-wave phase shift is zero, a signiﬁcant

fraction of the total elastic scattering cross section is seen to arise from

scattering through small angles, θ<30

◦

. It is precisely in this low energy

region that the measurements of Stein et al. (1978) and Coleman et al.

(1979) (see Figure 2.8) are signiﬁcantly lower than the most accurate

theoretical results. A discussion of diﬀerential cross sections at higher

incident energies is given in subsection 3.2.5; see also section 3.4.

3 Positron scattering by alkali atoms

It is a reasonably good approximation to consider an alkali atom as a

single electron moving in the modiﬁed Coulomb ﬁeld of the ionic core, and

this approximation has been made in almost all theoretical investigations

of positron scattering by the alkali atoms. The interaction of the electron

with the core is expressed as a local central potential of the general form

−

(r)=−

+ V

(r), (3.82)

where V

(r) is of short range. The positron–core potential has usually

been taken to be V

(r)=−V

−

(r). This form for V

(r) is not alto-

gether appropriate, because the electron–core interaction implicitly takes

3.2 Theory 123

Fig. 3.9. Angular distributions of positrons elastically

scattered by helium. The

model H5 phase shifts were used to obtain these results.

account of exchange between the valence and core electrons whereas there

is, of course, no exchange between the positron and the core electrons.

Nevertheless, such a positron–core potential is probably reasonably accu-

rate.

124 3 Elastic scattering

The scattering process can now be considered as a three-body problem,

rather similar to positron scattering by atomic hydrogen but with the

important diﬀerence that, because the ionization energy of an alkali atom

is less than the binding energy of positronium, 6.8 eV, the positronium

formation channel is open even at zero positron energy.

Low energy positron–alkali atom scattering should therefore be con-

sidered as a two-channel process, although the positronium formation

channel has been neglected in some calculations.

The most detailed theoretical studies of positron scattering by an alkali

atom have been made for lithium, although this remains the only such

atom for which no experimental results have yet been obtained. It is

also the only alkali atom so far for which scattering results have been

obtained using elaborate variational methods in addition to the diﬀer-

ent forms of coupled-state approximation employed in almost all other

calculations. The ﬁrst investigations were made by Guha and Ghosh

(1981) using the Born and coupled-static approximations, and more states

have subsequently been added to the expansion. Basu and Ghosh (1991)

included three states, Li(2s, 2p) and Ps(1s), and Hewitt, Noble and

Bransden (1992b) included seven, Li(2s, 2p, 3s, 3p) and Ps(1s, 2s, 2p), all

these calculations having been performed in momentum space. A similar

number of states was included by McAlinden, Kernoghan and Walters

(1994), working in conﬁguration space. These authors subsequently ex-

tended their calculations to include up to 29 states and pseudostates of

lithium and up to nine states of positronium (McAlinden, Kernoghan

and Walters, 1997). Most of these calculations were made over the energy

range 0–50 eV but Kernoghan, McAlinden and Walters (1994a), using the

coupled-state method with nine positronium states and ﬁve lithium states,

restricted their investigations to the energy range < 3 eV. Humberston

and Watts (1994) used a two-channel version of the Kohn variational

method, with trial wave functions containing many Hylleraas correlation

functions, to calculate the elastic scattering and positronium formation

cross sections for lithium, but only over an even narrower energy range,

0–2 eV. Their results agree well with the most elaborate coupled-state

results of McAlinden, Kernoghan and Walters, as may be seen in Fig-

ure 3.10, and it is therefore likely that the results of the latter are quite

accurate throughout the energy range they investigated. Further details

of these and other calculations in which positronium formation has been

included as an open channel are given in section 4.2.

The open positronium formation channel in positron–alkali atom scat-

tering was neglected in the coupled-state calculations of Ward et al. (1989)

and McEachran, Horbatsch and Stauﬀer (1991), and only states of the

target alkali atom were included in the expansion of their wave function.

At low positron energies the elastic scattering cross sections calculated by

3.2 Theory 125

Fig. 3.10. Partial-wave cross sections for positron–lithium elastic scattering: (a)

l = 0; (b) l = 1. The solid curves are the most accurate coupled-state results of

Kernoghan, McAlinden and Walters (1994a) and the crosses are the variational

results of Humberston and Watts (1994).

these authors diﬀer signiﬁcantly from those obtained with the inclusion of

positronium states, both in overall magnitude and in structure. Further-

more, several resonances were found in the elastic scattering cross sections,

but they are probably a consequence of neglect of the open positronium

channels, since they are not observed when positronium states are in-

cluded. At energies beyond 10 eV the eﬀect on the elastic scattering cross

section of excluding positronium states from the coupled-state expansion,

as was done by Ward et al. (1989), is slight, but at lower energies it

becomes very pronounced. Without such states, the elastic scattering

cross section continues to rise steeply to values of a few hundred πa

the positron energy approaches zero, but when positronium states are

126 3 Elastic scattering

included, either explicitly as in the coupled-state expansion of Kernoghan

et al. (1994a) or implicitly as in the variational method of Humberston

and Watts (1994), the cross section falls quite abruptly below 1 eV to a

very low value at zero energy.

The coupled-state approximation has also been applied to the other

alkali atoms, by Ward et al. (1988, 1989), McEachran, Horbatsch and

Stauﬀer (1991), Hewitt, Noble and Bransden (1993, 1994), Kernoghan,

McAlinden and Walters (1996) and McAlinden, Kernoghan and Walters

(1996). As with lithium, the neglect of positronium terms in the wave

function has a serious inﬂuence on the very-low-energy elastic scattering

cross section.

4 Positron scattering by other atoms and molecules

The complexity of atomic targets other than hydrogen, helium and the

alkali atoms (considered as equivalent one-electron atoms) precludes the

possibility of such elaborate and detailed investigations of positron scat-

tering as those described above. Instead, rather simpler methods of

approximation have had to be used. The most satisfactory of these has

been the polarized-orbital method (see subsection 3.2.1), some form of

which was used by Massey, Lawson and Thompson (1966), Montgomery

and LaBahn (1970), Gillespie and Thompson (1975) and, most notably, by

McEachran and coworkers (1977, 1978, 1979, 1980) to investigate elastic

scattering by the inert gases. The cross sections calculated by these latter

authors for the heavier inert gases are in reasonably good agreement

with the experimental measurements, although there is evidence from

the lifetime spectrum for positrons diﬀusing in the gas (see Chapter 6)

that the diﬀerential cross sections are not as accurate as are the integrated

elastic scattering cross sections.

The determination of accurate theoretical values of the cross sections

for positron scattering by molecules is considerably more complicated

than scattering by atoms because in the ﬁxed-nuclei approximation the

Hamiltonian of the target molecule no longer has the spherical symmetry

of an atomic target. In a diatomic molecule there is only axial symmetry

about the internuclear axis, and so the wave function can be expressed

only in terms of eigenfunctions of L

, not those of L

, resulting in the

mixing of spherical partial waves. It is, therefore, not surprising that most

theoretical studies of positron–molecule scattering have used rather simple

phenomenological model potentials to represent the positron–molecule

interaction (see Armour, 1988 for a comprehensive review of the subject).

In the Born–Oppenheimer approximation, in which the nuclear and

electronic motions within the molecule are considered separately, the total

3.2 Theory 127

positron–molecule wave function is written as

νν



, r, R)=Φ

, r, R)χ

νν



(R), (3.83)

where r

is the position of the positron, r represents the positions of all

the electrons in the molecule, R is the internuclear separation and ν and ν



represent all the quantum numbers required to specify the electronic and

nuclear wave functions respectively. The function Φ

, r, R) represents

the wave function of the positron and the electrons for a ﬁxed internuclear

separation R, and χ

νν



(R) is the wave function representing the internu-

clear motion. For elastic scattering, Φ

, r, R) has the asymptotic form

, r, R) ∼

→∞

F (r

)Φ

(r, R) (3.84)

where Φ

(r, R) is the wave function of the molecular target.

The only molecular target for which very elaborate ab initio calculations

of elastic scattering have been made is molecular hydrogen, the most

accurate results for which were obtained by Armour and his collabora-

tors (Armour, 1988; Armour, Baker and Plummer,

1990; Armour and

Humberston, 1991). These authors used the Kohn variational method

together with the method of models, in a somewhat similar manner to that

described previously for positron–helium scattering (subsection 3.2.2). An

accurate Hylleraas-type approximation was used for the H

target wave

function, with R ﬁxed at the equilibrium separation of 1.4a

. Many

terms of Σ

,Σ

,Π

and Π

symmetry, also of Hylleraas form and

expressed in terms of prolate spheroidal coordinates, were then included

in the total wave function. The most accurate results for σ

are given

in Figure 2.19, together with accurate experimental measurements of the

total positron–H

cross section already discussed in Chapter 2. Good

agreement is obtained between the theoretical and experimental results

up to a positron energy of 5 eV, but thereafter the experimental results

exceed the theoretical results, particularly beyond the positronium for-

mation threshold at 8.63 eV, where positronium formation is known to

make a substantial contribution to the total cross section. Even below

the positronium formation threshold, however, the experimental cross

sections include contributions from rotational and vibrational excitations

of the molecule, but these processes are not expected to contribute very

signiﬁcantly to the total cross section. Much of the diﬀerence between

theory and experiment in the energy range 5.0–8.63 eV is probably the

result of not having included all possible symmetries in the trial wave

function.

The R-matrix method, which has been used extensively in studies of

electron–molecule scattering, has also been applied to positron–molecule

scattering, notably by Tennyson and his collaborators (Tennyson, 1986;

128 3 Elastic scattering

Tennyson and Danby, 1987), merely by changing the sign of the charge on

the projectile and removing the exchange terms. However, such modiﬁca-

tions alone cannot provide a very satisfactory representation of electron–

positron correlations, because neither real nor virtual states of positro-

nium were included in the wave function. Comparisons with the elastic

scattering cross sections of Armour reveal that the R-matrix results for

are indeed signiﬁcantly less accurate. The R-matrix method has also

been applied to other diatomic molecules, including N

and CO (Tennyson

and Morgan, 1987) and HF (Danby and Tennyson, 1988). These authors

found evidence of a very weakly bound state of the positron–HF system.

Earlier studies of positron–molecule elastic scattering did not involve

such detailed descriptions of the scattering process as do the variational

and R-matrix formulations. Instead, the interaction between the positron

and the molecule was represented by a relatively simple model potential,

and the positron wave function F (r

) was assumed to satisfy the equiva-

lent single-particle Schr¨odinger equation

−

∇

F (r

)+V (r

)F (r

F (r

). (3.85)

The potential function was taken to be

V (r

)=V

stat

)+V

pol

), (3.86)

where the static potential between a positron and a homonuclear diatomic

molecule such as H

has a short-range spherically symmetric component

and a long-range non-spherical component with asymptotic form

stat

) ∼

→∞

(cos θ)

. (3.87)

Here Q is the quadrupole moment of the molecule, which has the value of

0.49 a.u. for H

at the equilibrium internuclear separation, and P

(cos θ)

is the second-degree Legendre function, θ being the angle between R

and r

. The polarization potential, also with spherical and non-spherical

components, has the asymptotic form

pol

) ∼

→∞

−

+ α

(cos θ)

, (3.88)

where α

and α

are deﬁned in terms of the polarizabilities parallel and

perpendicular to the internuclear axis, α



and α

⊥

,as

=(α



+2α

⊥

)/3 (3.89)

and

=2(α



− α

⊥

)/3. (3.90)