Charlton M., Humberston J.W. Positron Physics

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2.5 Results and discussion – atoms 79

Fig. 2.17. Cross

sections for positron–rubidium scattering. Experiment:

,

Parikh et al. (1993), total cross sections. Theory: upper solid curve, total cross

sections from the work of Kernoghan, McAlinden and Walters (1996). Various

partial cross sections from the latter are also shown: -- -- -- --, elastic scattering;

— —, 5S–5P excitation; — · —, 5S–4D excitation; uneven solid curv

e, at bot-

tom of ﬁgure, sum of 5S–6S and 5S–6P excitation cross sections; −•−•−•−,

positronium formation.

merging occurs is expected to be related to the mean kinetic energy of

the electrons in the target. If the projectile energy is large compared to

this mean energy, exchange eﬀects will be small for electron projectiles

and the conditions for merging outlined in section 2.2 will be satisﬁed.

In the case of an alkali atom, probably only the mean kinetic energy of

the weakly bound valence electron is relevant, and this is much smaller

(typically a few electron volts) than the corresponding value for a noble

gas. Merging is therefore expected to occur at signiﬁcantly lower energies

for the alkali atoms than for other atoms, as observed.

4 Atomic hydrogen

The total cross sections measured by Zhou et al. (1997) for positron and

electron impact are illustrated in Figure 2.18. As noted by these authors,

the positron total cross sections are in excess of those for electrons

in the energy range 15–100 eV, a situation similar to that found for

the alkali atoms, as described in subsection 2.5.3. The experimental

electron–hydrogen data agree well with the semi-empirical values derived

80 2 Total scattering cross sections

Fig. 2.18. Positron–atomic hydrogen and electron–atomic hydrogen total scat-

tering cross sections. Experiment: •, positrons; ◦, electrons; both sets of data

are from Zhou et al. (1997). Theory: ——, Kernoghan, McAlinden and Walters

(1996); — —, Mitroy (1996); — · —, Higgins, Burke and Walters (1990); ·····,

Walters (1988); -- -- -- --, Winick and Reinhardt (1978a, b).

by de Heer, McDowell and Wagenaar (1977) and also with the results of

various theories; the positron data agree with the T-matrix calculations of

Winick and Reinhardt (1978a, b) throughout the energy range 8–300 eV

and those of Walters (1988) in the energy range 50–300 eV. At lower

energies, and at least down to the positronium formation threshold, the

experimental results of Zhou et al. (1997) are in very good accord with the

results of the elaborate coupled-state calculations of Kernoghan, McAlin-

den and Walters (1995, 1996). At energies below 8 eV the experimental

results fall below the calculated values of Kernoghan, McAlinden and

Walters (1995, 1996) and the accurate variational results of Brown and

Humberston (1985), though Stein et al. (1996) have attributed this to

forward scattering errors.

2.6 Results and discussion – molecules 81

It is interesting to note that the total cross sections, both theoretical

and experimental, for the two projectiles are close to one another above

about 12 eV, although true merging does not occur until above 100 eV.

This energy, intermediate between the energies at which merging occurs

for the alkali atoms and helium, is what would be expected, given the

relationship previously stated in subsection 2.5.3 between the energy at

which merging occurs and the mean kinetic energy of the target electrons.

As emphasized by Stein et al. (1996) and Zhou et al. (1997), the merging

occurs despite the large diﬀerences in the partial cross sections (e.g. the

cross section for elastic scattering is estimated to be four times higher for

electrons than for positrons at an energy of 30 eV, where the positronium

formation cross section is almost 50% of the total positron cross section).

Zhou et al. (1997) have speculated that this is due to some form of

coupling between the various channels which results in the total cross

sections for the two projectiles being very similar. These authors also

alluded to the argument of Dewangan (1980) as summarized in section

2.2.

2.6 Results and discussion – molecules

Total scattering cross sections have been measured for a wide variety of

molecules using low energy positron beams, but in this section we consider

just four targets, H

,CO

and H

O, each of which exhibits intriguing

and complex behaviour. Relevant theoretical work is also described, al-

though it is in shorter supply than for the atomic targets described above.

General reviews of positron–molecule theory were given by Armour (1988)

and Ghosh, Sil and Mandal (1982) and references to other measurements

of positron–molecule total scattering cross sections have been given by

Kimura et al. (2000).

1 Hydrogen

Total cross sections for low energy positron and electron scattering by H

are presented in Figures 2.19(a), (b) respectively, and the experimental

data at intermediate energies are shown in Figure 2.20. The results display

a somewhat similar energy dependence to that found for helium, the

electron data falling monotonically with increasing energy from a value

of 2 × 10

−15

at 2 eV and the positron data displaying a Ramsauer

minimum of approximately 10

−16

at 3 eV. As the positron energy is

reduced below this value the cross section rises steeply, and there is also a

pronounced rise as the energy is increased above the positronium forma-

tion threshold, indicating once again the importance of the positronium

formation channel. It should be noted, however, that other channels, e.g.

82 2 Total scattering cross sections

Fig. 2.19. Positron–H

and electron–H

total scattering cross sections

at low

energies. (a) Positron scattering. Experiment:

, Hoﬀman et al. (1982); •,

Charlton et al. (1983a); , Deuring et al. (1983). Theory: ·····, Baille,

Darewych and Lodge (1974); — · —, Hara, (1974); ——, Bhattacharyya and

Ghosh (1975); -- -- -- --, Armour (1984): — ··—, Morrison, Gibson and Austin

(1984) (two such curves for diﬀerent approximations used, labelled 1 and 2).

Other recent theoretical work (e.g. Armour, Baker and Plummer (1990), Danby

and Tennyson (1990) and Gianturco and Mukherjee (1997) is not shown – see the

main text for a discussion. (b) Electron scattering, experiment only:

, Hoﬀman

et al. (1982); , Deuring et al. (1983); ×, Dalba et al. (1980); ——, Ferch, Raith

and Schr¨oder (1980).

vibrational and rotational excitation and molecular dissociation, may also

have a minor inﬂuence on the behaviour of the total cross section in this

energy region. Of these processes, one might expect molecular dissocia-

tion, which is energetically possible at a projectile energy of 4.48 eV, to

be the most signiﬁcant, but Massey (1969, Chapter 13) has shown, by

appealing to the Frank–Condon principle, that the cross section for this

process is very small for energies less than 8.8 eV. The total cross sections

for the two projectiles merge at an energy of approximately 100 eV.

Above 9 eV all sets of experimental data are in reasonable agreement,

although those of Deuring et al. (1983) are on average 15% higher than

2.6 Results and discussion – molecules 83

Fig. 2.20. Intermediate energy experimental results

for positron–H

and

electron–H

total scattering cross sections. Positrons: •, Charlton et al. (1980b);

, Hoﬀman et al. (1982); , Deuring et al. (1983). Electrons: ◦, Van Wingerden

et al. (1980);

, Hoﬀman et al. (1982); , Deuring et al. (1983).

the Detroit and London results. At energies below approximately 5 eV,

however, the results of Charlton et al. (1983a) fall below those of Hoﬀman

et al. (1982) and exhibit a shallow minimum around 3.5 eV. This feature

is not observed by Hoﬀman et al. (1982) nor is it present in any of the

theoretical results.

The results of several calculations are also shown in Figure 2.19. Hara

(1974) obtained fair agreement with experiment using a ﬁxed-nuclei ap-

proximation for elastic scattering and rotational excitation. However,

Baille, Darewych and Lodge (1974), who applied an adiabatic nuclei

approximation for elastic scattering, obtained cross sections which are

much lower than the experimental values at all energies. Better agreement

with experiment was obtained by Bhattacharyya and Ghosh (1975) who

used an eikonal approximation to calculate elastic cross sections in the

energy range 2–32 eV. The most detailed investigations of low energy

positron scattering by molecular hydrogen have been made by Armour,

Baker and Plummer (1990), who used the Kohn variational method with

elaborate trial functions. This method is similar to that used to obtain

84 2 Total scattering cross sections

accurate results for elastic scattering by atomic hydrogen and helium,

further details of which are given in section 3.2. The results of Armour,

Baker and Plummer (1990) are in good agreement with the experimental

values of Hoﬀman et al. (1982) up to an energy of 5 eV, but at higher

energies they fall progressively below the experimental values. Armour,

Baker and Plummer (1990) extended their calculations to an energy of

14 eV, but did not include positronium formation. Cross sections for

elastic positron scattering by molecular hydrogen have also been obtained

using the R-matrix method (Danby and Tennyson, 1990) and a variety

of model potential methods (Hara, 1974; Morrison, Gibson and Austin

1984; Morrison, 1986). Recently, good accord with experiment has also

been found by Gianturco and Mukherjee (1997) using

a model in which

dynamical vibrational coupling eﬀects are taken into account.

Cross section measurements in the intermediate energy range have been

reported by Charlton et al. (1980b), Hoﬀman et al. (1982) and Deuring

et al. (1983), and their results are shown in Figure 2.20. These data are in

reasonable agreement above 150 eV, although the London results

are as

much as 20% higher than those from Bielefeld and Detroit in the energy

range 20–100 eV. Also shown are recent measurements of the total cross

section for electrons, from which it can be seen that between approxi-

mately 30 eV and 150 eV the electron cross sections fall slightly below

those for positrons. This may be due to the contribution from positronium

formation in the positron case and to the fact that the ionization cross

section for positrons has been found to exceed that for electrons in this

energy range (see Chapter 5).

2 Nitrogen

The total cross sections for positron and electron scattering by N

low energies are shown in Figure 2.21. In this case, as also for CO

and

O, see the discussion below, only low energy data are presented. The

only groups to have published reliable low energy total cross sections for

positron–N

scattering are those in London (Charlton et al., 1983a) and

Detroit (Hoﬀman et al., 1982), and their results are shown together with

a few points from the intermediate energy data of Charlton et al. (1980b)

and Dutton, Evans and Mansour (1982).

The data of Charlton et al. (1983a) and Hoﬀman et al. (1982) are in

reasonable agreement over the entire energy range, the latter showing

that the total cross section rises sharply below 2 eV. Close inspection of

the points reveals some interesting features. Both experiments indicate

that the cross section grows steadily at higher energies, although Charlton

et al. (1983a) ﬁnd the rise beginning at approximately 7 eV, whilst the

Detroit group ﬁnd the rise starting at 8 eV. The reason for this diﬀerence

Fig. 2.21. Positron–N

and electron–N

total scattering cross sections at low energies. (a) Positrons. Experiment: , Hoﬀman

et al. (1982); •, Charlton et al. (1983a); ◦, Charlton et al. (1980b); , Dutton, Evans and Mansour (1982). Theory: – – –,

Darewych and Baille (1974); ——, Gillespie and Thompson (1975); — · —, Darewych (1982). (b) Simpliﬁed experimental

situation for low energy electron–N

scattering. The discrete points are those of Hoﬀman et al. (1982) whilst the full curve is

taken from the detailed work of Kennerly (1980).

86 2 Total scattering cross sections

is not clear since neither energy coincides with the positronium formation

threshold at 8.8 eV. The possibility that the increase is due to the onset

of electronic excitation was suggested by Charlton et al. (1983a), though

Schrader and Svetic (1982) noted that the lowest threshold for excitation

of the N

molecule without a spin-ﬂip transition is at an energy of 8.6 eV.

Also shown in Figure 2.21 are the results of the calculations of Darewych

and Baille (1974), which have an incorrect energy dependence in the range

3–10 eV. These investigations were extended and improved by Darewych

(1982) to include more partial waves, and reasonable agreement with

the measured values of σ

below E

was then obtained. Gillespie and

Thompson (1975), using a polarized-orbital method, obtained lower cross

sections than the experimental values, with a minimum near 1 eV which is

not present in the data of Hoﬀman et al. (1982). Gianturco and Mukherjee

(1997) (data not shown on Figure 2.21) have found good accord with

experiment, particularly with the data of Charlton et al. (1983a).

The positron results, which have a broad minimum in the energy range

2–7 eV, are in marked contrast to the low energy electron cross sec-

tions measured by Hoﬀman et al. (1982) and Kennerly (1980), shown

in Figure 2.21(b). The striking feature with the latter projectile is the

existence of a prominent shape resonance centred around 2 eV, which

is due to the temporary formation of a negative molecular ion during

the collision. This type of phenomenon has been discussed by many

authors (e.g. Schulz, 1973) and, although observed in many low energy

electron–molecule systems, is absent from all positron–molecule systems

studied hitherto.

3 Carbon dioxide

Total cross sections for low energy positron and electron scattering by CO

are presented in Figures 2.22(a), (b) respectively. As noted by Hoﬀman

et al. (1982), both sets of data are similar in shape except that, as with N

the electron data display a large shape resonance in the vicinity of 4 eV

whilst the positron results show a ‘bump’ at the positronium formation

threshold at 7.0 eV. This abrupt rise is evident in the measurements of

Hoﬀman et al. (1982) and Charlton et al. (1983a) and is attributed to the

onset of positronium formation. Furthermore, both experiments ﬁnd a

plateau or a slight dip in the total cross section above approximately

8 eV, followed by a second rise. Kwan et al. (1984) noted that this

rise starts close to the threshold for the formation of positronium in

its ﬁrst excited state, although subsequent work by Laricchia, Charlton

and Griﬃth (1988) and Laricchia and Moxom (1993) has found that the

process responsible is the formation of ground state positronium with

Fig. 2.22. Positron–CO

and electron–CO

total scattering cross sections at low energies. (a) Positrons. Experiment: ,

Hoﬀman et al. (1982); •, Charlton et al. (1983a); ◦, Charlton et al. (1980b). Theory: Horbatsch and Darewych (1983). The

two curves correspond to the use of a ﬁxed cut-oﬀ (– – –) and an energy-dependent cut-oﬀ (——) parameter in the polarization

potential. (b) Experimental situation for electrons:

, Hoﬀman et al. (1982); , Ferch, Masche and Raith (1981); ——,

Szmytkowski and Zubek (1978).

88 2 Total scattering cross sections

Fig. 2.23. Positron–H

O total scattering cross sections at low energies (Sueoka,

Mori and Katayama, 1987).

the residual molecular ion in an excited state, this threshold being at

approximately 10.5 eV (see section 4.5).

The data of Charlton et al. (1983a) and Hoﬀman et al. (1982) are

in reasonable agreement over most of the energy range presented here,

except that the former are lower than the latter values below 4 eV. Com-

parison is also made with the theoretical work of Horbatsch and Darewych

(1983). By using a ﬁxed cut-oﬀ parameter in their polarization potential,

these authors obtained better agreement with the London data. When,

however, a variable-energy cut-oﬀ was used, better agreement with the

Detroit results was found. Curves corresponding to each calculation are

given in Figure 2.22. Theoretical work by Gianturco and Paioletti (1997)

(not shown in Figure 2.22) are in reasonable accord with experiment.

4 Water

We have chosen H

O for detailed comment rather than other molecules

because it was, until recently, one of the few with a large permanent

dipole moment to have been studied. The positron scattering data of

Sueoka, Mori and Katayama (1987) are shown in Figure 2.23, and they

were obtained using a similar TOF system to that described in section