Charlton M., Humberston J.W. Positron Physics

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2.5 Results and discussion – atoms 69

energy range, although the data of Griﬃth et al. (1979a) are 15% lower

than the other results in the range 200–400 eV, and they exhibit a marked

drop above 800 eV. This latter eﬀect is almost certainly caused by failure

to resolve at high energies the diminishing diﬀerence between the times

of ﬂight of positrons which have been scattered through small angles and

those which are unscattered.

The early measurements of Costello et al. (1972) are included in Fig-

ure 2.8 even though they have now been superseded, but the results of

Jaduszliwer and Paul (1973) are not because, as discussed by Charlton

(1985a), they were subject to a substantial systematic error in the eﬀective

path length of the positrons in the gas. Neither are the intermediate

energy data of Canter et al. (1973) and of Coleman et al. (1976b) in-

cluded in Figure 2.9, because they suﬀered from large forward scattering

errors.

Also shown in Figures 2.8 and 2.9 are results from various theoreti-

cal investigations. The methods used, and the results obtained, below

the positronium formation threshold relate only to elastic scattering and

are described in more detail in subsection 3.2.2. The most accurate

results in this energy region have been obtained using variational methods

(Humberston, 1979), and these are in good agreement with the experi-

mental measurements throughout this energy range. Similar techniques

have recently been used to obtain accurate total cross sections in the

Ore gap, the energy range between the positronium formation threshold

and the lowest positron-impact excitation threshold of the helium target

17.8–20.6 eV. The results of the polarized-orbital method, as used by

Massey, Lawson and Thompson (1966), by Montgomery and LaBahn

(1970) and, particularly, by McEachran et al. (1977) are in rather less

good agreement but the method is of special signiﬁcance because it is

one of the few to have been applied to all the noble gases. A much

simpler method of calculating the elastic scattering cross sections, which

yields surprisingly good results, has been used by Schrader (1979). The

positron–target interaction is represented by a simple central polarization

potential containing an adjustable range parameter, and the scattering

problem then reduces to solving a one-dimensional Schr¨odinger equation

for each partial-wave phase shift. At intermediate and higher energies

(> 100 eV), total cross sections have been calculated using the optical

theorem in conjunction with various approximation methods for deter-

mining the forward scattering amplitude. Among the most important of

these studies have been those by Byron and Joachain (1977a), who used

the eikonal Born series, and Dewangan and Walters (1977), who used the

distorted-wave second Born approximation. Calculations have also been

made by Byron and Joachain (1977b) using an optical potential formalism

70 2 Total scattering cross sections

Fig. 2.10. Low energy positron–neon total scattering cross sections. Experi-

ment:

, Stein et al. (1978); , Coleman et al. (1979); , Sinapius et al. (1980);

•, Charlton et al. (1984);

, Kauppila et al. (1981); ×, Tsai, Lebow and Paul

(1976); , Brenton, Dutton and Harris (1978).

Theory: —

· —, Campeanu and

Dubau (1978); ——, McEachran, Ryman and Stauﬀer (1978); – – –, Schrader

(1979).

in which the imaginary part of the potential represents absorption into

channels not explicitly included in the formulation. The results of De-

wangan and Walters (1977) are in particularly good agreement with the

measured values of Kauppila et al. (1981) at all energies beyond 100 eV,

but all methods agree well beyond 700 eV. Virtually no theoretical studies

have been made of the total cross section between the upper limit of

the Ore gap and 100 eV, although several calculations have been made

of individual partial cross sections, details of which are given in later

chapters.

2 Heavier noble gases

The values of σ

for neon at low and intermediate energies are shown in

Figures 2.10 and 2.11, which include experimental results obtained by a

number of authors in the energy range 0–1000 eV. For the same reasons

as outlined above when discussing helium, the data of Coleman et al.

2.5 Results and discussion – atoms 71

(1976b) and Jaduszwiler and Paul (1974) are not included. Figure 2.10

shows that only the data of Stein et al. (1978) extend to suﬃciently low

energies to observe the deep, narrow Ramsauer minimum which exists at

around 0.6 eV. All groups ﬁnd that the cross section rises sharply with

energy and then levels out around 10 eV before rising dramatically again

above E

. This latter eﬀect is similar to that found in positron–helium

scattering and is again due to positronium formation.

Although the data from all the experiments exhibit similar qualitative

features, closer inspection reveals discrepancies which, according to the

analysis of Kauppila and Stein (1982), cannot be explained by assuming

that the main systematic error is due to the neglect of forward scattering.

The results of Stein et al. (1978) are mostly higher than those of Coleman

et al. (1979) above 6 eV, but they fall signiﬁcantly below them at lower

energies. Both measurements are lower than those of Charlton et al.

(1984), except perhaps at the higher part of the energy range covered by

the latter authors. As mentioned earlier, the system used by Sinapius

et al. (1980), which has the lowest value of θ

disc

, should produce the

most reliable results. It is noticeable from Figure 2.10 that their data are

signiﬁcantly lower than those of the other groups. The agreement between

the results of the Bielefeld and Detroit groups can be improved somewhat

since the latter (Kauppila and Stein, 1982) have reported that their energy

scale should be increased by approximately 0.2 eV. However, in the range

4–6 eV the data of Sinapius et al. (1980) are still approximately 7% lower

than those of Stein et al. (1978) whereas, according to Stein and Kauppila

(1982), they should agree to within 1%–2% at these energies.

Theoretical results are also included in Figure 2.10. Nothing compa-

rable to the accurate variational calculations for positron–helium elastic

scattering has been attempted for the other noble gases, and simpler

methods of approximation have had to be used instead. The most notable

of these has been the polarized-orbital method, which has been used by

McEachran and coworkers (1978, 1979, 1980) for all the heavier noble

gases. Their results are in reasonably good agreement with the exper-

imental measurements for all the noble gases except argon, where the

method fails to reproduce the ﬂat energy dependence between 2 eV and

the positronium formation threshold at 9 eV. The simple model-potential

method of Schrader (1979) has also been applied to the heavier noble

gases and, as with helium, the results are surprisingly good, particularly

for neon and argon. Very few theoretical results have been obtained at

energies beyond 100 eV; optical model calculations have been performed

by Byron and Joachain (1977b) for neon and by Joachain et al. (1977)

for argon, and the distorted-wave Born approximation has been used by

Dewangan and Walters (1977) for neon.

72 2 Total scattering cross sections

Fig. 2.11. Intermediate energy positron–neon total scattering cross sections.

Experiment: , Coleman et al. (1979); , Brenton, Dutton and Harris (1978); ×,

Tsai, Lebow and Paul (1976); ◦, Griﬃth et al. (1979a);

, Kauppila et al. (1981).

Theory: ——, Byron and Joachain (1977b); — · —, Dewangan and Walters

(1977); — ··—, Bethe–Born calculation of Inokuti and McDowell (1974). The

experimental electron data (– – –) are shown for comparison.

Theoretical and experimental data at intermediate energies are shown

in Figure 2.11. The experimental measurements are in tolerable agree-

ment with one another, but are everywhere lower than the theoretical

results.

The available experimental and theoretical data for positron–argon

scattering at low energies are shown in Figure 2.12, where it can be

seen that the data of Kauppila, Stein and Jesion (1976) and Coleman

et al. (1980a) are in reasonable agreement with each other, although both

sets are lower than the results of Sinapius et al. (1980) and Charlton

et al. (1984). As was the case with helium and neon, it is unlikely that

these discrepancies are the result of forward scattering errors alone. Of

particular note are the large diﬀerences between the data of Charlton et al.

(1984), Kauppila, Stein and Jesion (1976) and Coleman et al. (1980a) at

energies above E

, where it is expected that forward elastic scattering will

contribute only a small fraction to σ

. Furthermore, the measurements

2.5 Results and discussion – atoms 73

Fig. 2.12. Low energy positron–argon total scattering cross sections. Experi-

ment:

, Kauppila, Stein and Jesion (1976); ×, Tsai, Lebow and Paul (1976);

, Sinapius, Raith and Wilson (1980); , Coleman et al. (1980a); •, Charlton

et al. (1984);

, Kauppila et al. (1981); ◦, Griﬃth et al. (1979a). Theory: ——,

McEachran, Ryman and Stauﬀer (1979);–––,Schrader (1979).

of Charlton et al. (1984) exhibit some structure in the 12–18 eV energy

range, whereas none can be discerned in the results from the other groups.

The calculations of McEachran et al. (1979) and Schrader (1979) for

elastic cross sections are also included, but they do little to help resolve

the experimental discrepancies.

Experimental data at intermediate energies are presented in Figure 2.13,

together with the results of the optical model calculations of Joachain

et al. (1977). The results of Kauppila et al. (1981), of Tsai, Lebow

and Paul (1976) in the energy range 25–270 eV and of Brenton, Dutton

and Harris (1978) in the energy range 200–1000 eV are in reasonable

agreement, whereas the data of Griﬃth et al. (1979a) are 7%–13% higher

below 150 eV and around 12% lower above this energy. The optical

model results of Joachain et al. (1977) are somewhat higher than the

experimental data.

Most experimental groups have performed measurements of σ

for kryp-

ton and xenon although, as described by Stein and Kauppila (1982),

the results are not in good agreement with each other. At the lowest

energies the magnitude and the trend of σ

, as observed by Dababneh

74 2 Total scattering cross sections

Fig. 2.13. Intermediate energy positron–argon total scattering cross sections.

Experiment: ×, Tsai, Lebow and Paul (1976); , Coleman et al. (1980a); ◦,

Griﬃth et al. (1979a);

, Kauppila et al. (1981); , Brenton, Dutton and Harris

(1978). Theory: Joachain et al. (1977). The experimental electron data (– – –)

are shown for comparison.

et al. (1980), appear to be closest to the polarized-orbital calculations

of McEachran, Stauﬀer and Campbell (1980) and the model potential

approach of Schrader (1979). It is, however, notable that, in contrast to

the case of neon, but like that for argon, the data of Sinapius, Raith and

Wilson (1980) are much higher than the Detroit results (Dababneh et al.,

1980). It should be remembered that the Bielefeld data are expected

to be the least prone to forward scattering errors. Stein and Kauppila

(1982) argued that if the Detroit and Bielefeld results are each corrected

for their respective forward scattering errors, as calculated using the

diﬀerential cross sections of McEachran, Stauﬀer and Campbell (1980),

then the two data sets are in much better agreement, leaving the results of

Canter et al. (1973) and Coleman et al. (1980a) apparently too low, by as

much as a factor of two around 2 eV. However, in view of the theoretical

diﬃculty of treating such complex target atoms accurately, and the lack

of overall consistency in the experimental data of the noble gases when

taken together, this conclusion should be regarded as tentative.

2.5 Results and discussion – atoms 75

Fig. 2.14. Compendium of total cross section data for positron–noble gas and

electron–noble gas scattering. The arrows refer to thresholds for (in order of

increasing energy) positronium formation (positrons only), excitation and ion-

ization. (From Kauppila and Stein, 1982.)

It is also instructive to compare the positron and electron scattering

cross sections for the heavier noble gases, as was done for helium in

subsection 2.5.1. Rather than present all the available data for both

projectiles, we show in Figure 2.14 a compendium based upon the results

of the Detroit group. This concentrates on the low energy range, where

most of the structure in the cross sections can be found. The arrows

on the curves indicate the major inelastic thresholds, E

(positronium

formation), E

(excitation) and E

(ionization) for positrons, and E

and E

for electrons. At higher impact energies the cross sections for

both projectiles fall monotonically, with those for electrons tending to

the positron values from above. The merging of these two cross sections

for helium has been described previously, but it has not been observed for

the heavier noble gases even at the highest energies investigated so far,

typically 1 keV; nevertheless, it is expected to occur at suﬃciently high

energies.

76 2 Total scattering cross sections

Fig. 2.15. Positron–sodium and electron–sodium total (and also for positrons

partial) scattering cross sections. Experiment (positrons): ,Kwanet al. (1991);

, Kauppila et al. (1994). Experiment (electrons): ◦, Kauppila et al. (1994).

Theory: •, Hewitt, Noble and Bransden (1993); ——, Kernoghan (1996). Var-

ious partial cross sections from the work of Kernoghan (1996) are also shown:

— —, elastic scattering; — · —, positronium formation; -- -- -- --, excitation of

sodium to the 3D level; ·····, resonant excitation (3S–3P).

3 Alkali metals

The apparatus and technique used by Stein et al. (1985), Kwan et al.

(1991), Parikh et al. (1993) and Kauppila et al. (1994) has already been

discussed in section 2.3. Kwan et al. (1991) stated that their data for

positron and electron scattering by potassium superseded those obtained

by Stein et al. (1985), and consequently the earlier results are not de-

scribed here.

In Figure 2.15 we present the total cross sections for positron–sodium

scattering. Shown here are the experimental data of Kwan et al. (1991)

and Kauppila et al. (1994), and the theoretical coupled-state results of

2.5 Results and discussion – atoms 77

Hewitt, Noble and Bransden (1993) and Kernoghan (1996). As described

in section 2.2, these latter authors are believed to have produced the most

accurate theoretical results at low energies. The experimental results

have been corrected by Kernoghan (1996) to account for the failure to

discriminate completely against small-angle elastic scattering. Angular

discrimination values given by Kwan et al. (1991) were used for this

purpose, the largest corrections arising at impact energies below 10 eV.

The theoretical results of Kernoghan (1996) and the corrected experi-

mental data (which were further scaled upwards by 4% compared to the

estimated systematic error of ±20% in order to achieve the best agreement

in shape) are in reasonable accord, although less so below 10 eV. The two

sets of theoretical results shown are in good agreement, except at the

very lowest energies when the predictions of Hewitt, Noble and Bransden

(1993) greatly exceed those of Kernoghan (1996). This discrepancy has

yet to be explained satisfactorily.

The total cross section for electron–sodium scattering is also shown

in Figure 2.15, where it can be seen to decrease monotonically with

increasing energy. Considering the diﬃculty in obtaining an absolute

scale for these cross sections (see section 2.3), the data of Kwan

et al.

(1991) are in good agreement with most of the other direct measurements

and also with theoretical estimates derived from the integrated forms of

several diﬀerential elastic scattering cross sections obtained by Srivastava

and Vuˇskovi´c (1980), to which were added excitation and ionization cross

sections obtained by other workers. The behaviour of the electron scat-

tering cross sections for other alkali atoms is similar to that for sodium

(see e.g. Kwan et al., 1991; Parikh et al., 1993).

The total cross sections for positron scattering by potassium and ru-

bidium, shown in Figures 2.16 and 2.17, display a markedly diﬀerent be-

haviour at low positron energies from the corresponding data for positron

and electron scattering from sodium. Of particular note is the broad

maximum in the cross sections around 6 eV, in both theory and experi-

ment (after corrections for forward scattering errors, and an overall slight

upward rescaling), for both targets. The behaviour of the partial cross

sections as calculated in the coupled-state approximation by McAlinden,

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

(1996) is also illustrated in Figures 2.16 and 2.17. Again, the dominant

contributions are resonant excitation, 4S–4P for potassium and 5S–5P for

rubidium, at the higher energies and elastic scattering at lower energies,

except in the limit of zero incident energy when positronium formation

becomes dominant. The fall in σ

at low energies is caused by a more

rapid decrease of the resonant excitation cross section than is the case for

sodium, and also the behaviour of the positronium formation cross section.

78 2 Total scattering cross sections

Fig. 2.16. Total (and partial) cross sections for positron–potassium scattering.

Experiment:

, Parikh et al. (1993). Theory: •, Hewitt, Noble and Bransden

(1993); upper solid line, McAlinden, Kernoghan and Walters (1996). Various

partial cross sections from the work of the latter are also shown: — —, elastic

scattering; ·····, resonant excitation (4S–4P); lower solid curve, 4S–3D excita-

tion; -- -- -- --, excitation to the n = 5 levels of potassium; — · —, positronium

formation.

This latter cross section peaks at a higher positron energy for potassium

and rubidium than for sodium, and the calculations of Hewitt, Noble

and Bransden (1993), McAlinden, Kernoghan and Walters (1996) and

Kernoghan, McAlinden and Walters (1996) have established that this is

due to large contributions from positronium formation into excited states,

outweighing the contribution from ground state positronium formation.

This situation is unique in targets studied so far and is discussed further

in Chapter 4.

A further noteworthy feature of the total scattering cross sections for

the alkali atoms, which is clearly illustrated in the data presented by Kwan

et al. (1991), is the merging of the data for electrons and positrons at a

projectile energy of approximately 50 eV. This is much lower than the

energy at which the corresponding two cross sections merge for other

atomic targets, most notably the noble gases. The energy at which