Charlton M., Humberston J.W. Positron Physics

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3.4 Angle-resolved elastic scattering 139

contribution from one partial wave, this being l = l



= 1 for helium

and l =1,l



= 0 for the other gases, although small admixtures from

the partial wave with l =0,l



= 1 were also believed to be present for

krypton and argon. Thus, in all cases the dominant contribution to the

positronium formation cross section seemed to be provided by an incoming

positron wave with l =1.

Using this theoretical and experimental information, in conjunction

with equations (3.107) and (3.108), Moxom et al. (1994) predicted the

energy dependence of the elastic and total scattering cross sections on

either side of the positronium formation threshold. Comparisons of the

predicted and measured values are shown in Figure 3.13. Although the

quantitative agreement is not very good, there are important qualitative

similarities which show that for helium there is no discernible cusp in the

elastic scattering cross section but that a progressively more prominent

cusp develops as the atomic number of the target atom increases.

The conclusion of Moxom et al. (1994) that the positronium forma-

tion process in helium is dominated by the p-wave contribution over an

energy range of a few electron volts above the threshold is at variance

with accurate theoretical results obtained by Van Reeth and Humberston

(1997) and described in detail in Chapter 4. The l = 0 contribution to σ

is indeed small, but the l = 1 contribution, although rapidly exceeding

the l = 0 contribution, is itself quite rapidly overtaken by the l =2

contribution as the positron energy is increased. However, these various

partial-wave contributions to σ

are found to add in such a way as to

give an overall energy dependence of σ

very similar to that which would

be expected on the assumption that the p-wave contribution to σ

dominant (Van Reeth and Humberston, 1997).

3.4 Angle-resolved elastic scattering

There has been growing experimental activity in the area of angle-resolved

elastic scattering since the mid-1980s. It is well known from electron

collision physics that the angle-resolved elastic scattering cross section,

dσ

/dΩ, contains more detailed information than the integrated cross

section, and therefore its accurate determination provides a more strin-

gent test of collision theory. As emphasized by McDaniel (1989), the

ﬁrst observations of this quantity for electron–atom scattering provided

convincing evidence in favour of the validity of a quantum mechanical

description of scattering phenomena. More recently, comparisons be-

tween phase shift analyses of experimental diﬀerential cross sections and

theoretical predictions of these phase shifts have led to a comprehensive

understanding of the behaviour of electrons undergoing elastic collisions

140 3 Elastic scattering

Fig. 3.13. Total (tot, upper solid line) and elastic (el, lower solid line) cross sec-

tions for positron–noble gas scattering near the positronium formation threshold

from the R-matrix analysis of Moxom et al. (1994). Graphs (a)–(e) correspond

to helium through to xenon. The data points shown are total cross section

measurements from the literature (see Chapter 2 and Moxom et al., 1994, for

details) except for the solid diamonds for helium, which are the σ

−σ

results

of Coleman et al. (1992) (see Figure 3.12). The curves for σ

, which is the

elastic scattering cross section calculated without the inclusion of positronium

formation, are from the work of McEachran and collaborators. Reprinted from

Physical Review A50, Moxom et al., Threshold eﬀects in positron scattering on

3.4 Angle-resolved elastic scattering 141

with atoms (see e.g. Newell, Brewer and Smith, 1981; Brewer et al., 1981,

and references therein).

For the elastic scattering of a beam of unpolarized projectiles by an un-

polarized target, the cross section has axial symmetry about the incident

beam direction and therefore no dependence upon the azimuthal angle

φ, so that the diﬀerential elastic cross section is related to its integral

counterpart by



dσ

dΩ

dΩ=2π



dσ

dθ

sin θdθ. (3.109)

All the physical information is then contained in dσ

/dθ, which is deter-

mined by measuring the probability that a particle of ﬁxed kinetic energy

is elastically scattered through an angle θ.

Over the years there have been several attempts to measure dσ

/dθ

for positron–atom scattering. This work culminated in the experiments

of Hyder et al. (1986), Floeder et al. (1988) and Smith et al. (1990),

who reported the ﬁrst measurements obtained using the crossed positron

beam–gas beam geometry. Such a conﬁguration

is similar to that fre-

quently encountered in electron scattering. These workers investigated

positron–noble gas scattering, at ﬁrst using neon and argon targets rather

than more theoretically tractable targets, because their larger elastic

scattering cross sections made the measurements practicable using the

available ﬂuxes of low energy positrons. Extensions to other targets are

now, however, becoming possible.

The very ﬁrst attempts to derive information concerning the angular

distribution of positron–atom scattering were made in the transmission-

type experiments of Jaduszliwer and Paul (1973, 1974). These authors

found that as the strength of the axial magnetic guiding ﬁeld in their scat-

tering chamber was increased the number of detected scattered positrons

also increased. The extent of this increase depended on the form of the

diﬀerential cross section, and it proved possible, using a Monte Carlo tra-

jectory analysis, to extract s-, p- and d-wave phase shifts at energies below

for positron scattering by helium (Jaduszliwer and Paul, 1973) or by

neon and argon (Jaduszliwer and Paul, 1974). These phase shifts were

in reasonable agreement with the results of existing theories, although

the p-wave phase shifts for helium were subsequently found to be much

larger than the accurate theoretical values of Humberston and Campeanu

(1980). Although these data have been superseded by the results of later

work, the principle underlying this experiment is sound and the work

represents an ingenious attempt to extract angular information using a

feeble positron beam.

Similar ingenuity was also displayed by Coleman and coworkers (Cole-

man and McNutt, 1979; Coleman et al., 1980b). In these experiments a

142 3 Elastic scattering

beam of low energy positrons was timed over a ﬂight path 250 mm long in

an axial magnetic guiding ﬁeld of 14 mT, which constrained the positrons

scattered through almost 90

◦

to follow a helical trajectory of 1 mm radius

at the highest energy investigated (8.7 eV). The scattering was conﬁned

to a gas cell 10 mm long located close to the positron moderator, and,

owing to their delayed time of arrival at the detector, scattered particles

could be distinguished from those which had not been scattered using

a time-of-ﬂight (TOF) technique. Coleman and McNutt (1979) showed

how to extract absolute values for dσ

/dθ from the observed diﬀerences

between the TOF spectra obtained with and without the gas present. A

full discussion of the possible systematic eﬀects in this experiment was

given by Coleman et al. (1980b).

Results for positron–argon collisions are shown in Figure 3.14 at vari-

ous mean positron impact energies and over the angular range 20

◦

–60

◦

Also shown in Figure 3.14 are the polarized-orbital results of McEachran

et al. (1979) and those of Schrader (1979), who used a semi-empirical

polarization-potential method. There is good agreement between the

shapes of the theoretical and experimental data,

although the results of

McEachran et al. (1979) have been scaled down by an energy-dependent

factor which reﬂects the deviations of their calculated total cross sections

from the experimental values. The validity of this procedure is open to

question.

Coleman et al. (1980b) also performed electron–argon scattering exper-

iments using the same apparatus and method by utilizing the secondary

electrons emitted from their moderator after high energy β

bombard-

ment. They reported results over the same 40

◦

angular range, at energies

up to 20 eV, which were in good agreement with those of Williams (1979).

This test established the reliability of their procedure and measurements.

Following this early work, groups in Detroit and Bielefeld initiated

measurements of dσ

/dθ using a crossed-beam geometry. The two groups

used rather similar apparatus, and in Figure 3.15 we show the layout of

Hyder et al. (1986), which, with some small modiﬁcations, has been used

for many subsequent measurements.

The atomic beam was formed by a multichannel capillary array, placed

perpendicular to the positron beam, with a 2.5 mm

eﬀusing area and a

length-to-diameter ratio of 25 : 1. The head pressure behind the array

was kept at 9 torr (≈10

Pa) in the initial measurements. An an-

nealed tungsten moderator was used to provide a beam of more than 10

positrons per second at 200 eV. A much more intense beam of electrons

could also be obtained by reversing the electrostatic potentials on the

various elements which made up the transport system. Channel electron

multipliers (CEM1 and CEM2 respectively) were used to monitor the

incident and scattered beams. In later versions of the apparatus, a third

3.4 Angle-resolved elastic scattering 143

Fig. 3.14. Diﬀerential cross sections for positron–argon elastic scattering at the

following energies: (a) •, 2.2 eV; (b) •, 3.4 eV; (a) ◦, 6.7 eV; (b) ◦, 8.7 eV.

(The corresponding k-values are respectively 0.4, 0.5, 0.7 and 0.8 a.u.) The solid

curves give the theory of Schrader (1979) whilst the broken curves are the scaled

results of McEachran, Ryman and Stauﬀer (1979).

CEM was added to monitor the scattered beam at a diﬀerent angle. Note

that CEM1 was oﬀset from the direct line of sight with the

Na source,

in order to reduce background counts from β

particles and γ-rays. The

angular acceptance of CEM2, which was deﬁned by the geometry and the

collimators, was estimated to be ±8

◦

. In order to reduce the noise counts

in this detector, a non-reﬂective surface composed of a stack of knife-edge

plates was located directly opposite it in the chamber.

144 3 Elastic scattering

Fig. 3.15. Schematic illustration of the crossed-beam apparatus developed by

Hyder et al. (1986) for the measurement of positron elastic diﬀerential scattering

cross sections. Reprinted from Physical Review Letters 57, Hyder et al., Positron

diﬀerential elastic scattering cross section measurements for argon, 2252–2255,

The grid retarding elements located before CEM2 were used to separate

the elastically scattered signal from the background. This was done by

making measurements of the count rate of CEM2 both with and without

gas present, the retarder voltage being set just above and just below

the beam energy (see Hyder et al., 1986, for details). Relative values

of dσ

/dθ at diﬀerent scattering angles were obtained by dividing the

scattered signal by that for the incident beam measured at CEM1. This

ratio was found to be in the range 10

−5

–10

−7

, the signal-to-noise ratio

for positrons in CEM2 being typically 10

−1

–10

−2

for angles greater than

◦

. This entailed the use of long run times and computer-controlled

automated data taking. The demanding nature of such experiments also

explains why a large acceptance angle was tolerated for CEM2 and a large

drive pressure was used for the capillary-array gas source.

Some of the data of Hyder et al. (1986) for positron and electron impact

on argon gas at 100 eV and 200 eV are shown in Figures 3.16(a)–(d).

The data were independently normalized, at 90

◦

for both projectiles,

to the existing experimental data of Srivastava et al. (1981) and DuBois

and Rudd (1975) for electrons and to the theoretical results of McEachran

3.4 Angle-resolved elastic scattering 145

Fig. 3.16. Elastic diﬀerential cross sections for positrons and electrons scatter-

ing from argon. (a) Electrons at 100 eV: ◦, Hyder et al. (1986); •, Srivastava

et al. (1981). (b) Positrons at 100 eV: Hyder et al. (1986); ——, theory of

McEachran and Stauﬀer (1986); ·····, theory of Nahar and Wadehra (1987).

Positrons at 200 eV, same key as (b). Reprinted from Physical Review Letters 57,

Hyder et al., Positron diﬀerential elastic scattering cross section measurements

and Stauﬀer (1986) and Nahar and Wadehra (1987) (which are in good

agreement at this angle) for positrons. As stated by Hyder et al. (1986),

their electron measurements compare favourably with those of other

workers except at small angles; the latter eﬀect may be attributed to

the geometry of their gas beam. A similar situation arises when their

positron data are compared with theory. Joachain and Potvliege (1987)

also performed calculations for this system; their results are somewhat

lower than the other theoretical values and exhibit markedly diﬀerent

behaviour at small angles, lacking the structures found by McEachran

and Stauﬀer (1986) and Nahar and Wadehra (1987). It should be

noted that if the data of Hyder et al. (1986) are normalized to the

theoretical results of Joachain and Potvliege (1987) at 90

◦

, for both

146 3 Elastic scattering

100 eV and 200 eV, then the experimental results still exceed the calcu-

lated values at small angles in a similar way to that seen in the ﬁgure.

Joachain and Potvliege (1987), in extending the theory of Joachain

et al. (1977) to lower energies, discussed the diﬀering trends of the

theories at small angles (< 30

◦

at 100 eV). They showed that similar

behaviour to that found by McEachran and Stauﬀer (1986) and Nahar

and Wadehra (1987) can also be obtained from their calculations if the

absorption potential is neglected. Joachain (1987) argued that the full

optical model results are the most reliable since, as required by unitarity,

proper account is taken of inelastic channels by means of the absorption

potential.

Measurements have also been made at lower energies, where the pre-

dicted structure in dσ

/dθ moves to larger, experimentally amenable

angles. Theoretical and experimental results

for positron–argon collisions

at 8.7 eV and 30 eV are shown in Figure 3.17. At small angles the

experimental results agree best with the optical potential calculations

of Bartschat, McEachran and Stauﬀer (1988), which take account of

absorption, whilst other theoretical approaches (Nahar and Wadehra,

1987; McEachran and Stauﬀer, 1986; also Nakanishi and Schrader, 1986b)

apparently produce spurious structure at small angles. It should be noted

that there is a discrepancy between the shape of the normalized data of

Smith et al. (1990) at 30 eV and all calculations, even when normalization

at diﬀerent angles is attempted; see Figure 3.17.

At lower energies, below the threshold for positronium formation in ar-

gon (8.9 eV), the results of both polarized-orbital calculations (McEachran

and Stauﬀer, 1986) and model-potential calculations (Nakanishi and

Schrader, 1986a) are in better agreement with the shape of the experi-

mental data. The results shown in Figure 3.17 are at 8.7 eV, an energy

also investigated earlier by Coleman and McNutt (1979), as shown in

Figure 3.14. It should be noted that the data of Floeder et al. (1988)

(which were actually taken at a beam energy of 8.5 eV) were normalized

to those of Coleman and McNutt (1979) in a manner which produced

the best overall comparison of the two sets of values. The shape of these

data are in good agreement with one another and with the theories of

McEachran et al. (1979) and Montgomery and LaBahn (1970) and also

with that of Nakanishi and Schrader (1986a) (not shown). However, the

data of Smith et al. (1990) display a signiﬁcantly shallower minimum

than all the other results in the vicinity of a scattering angle of 40

◦

.Itis

unlikely that this discrepancy can be explained by the diﬀering angular

resolutions of the experiments.

The above discussion shows that some useful information has been

obtained on the need to include absorption eﬀects into the theory. In-

deed, as predicted by Joachain and Potvliege (1987), it appears that any

3.4 Angle-resolved elastic scattering 147

Fig. 3.17. Positron–argon elastic diﬀerential scattering cross sections. Experi-

ment: •, ◦, Smith et al. (1990); , Coleman and McNutt (1979);

, Floeder et al.

(1988). Theory: ——, McEachran, Ryman and Stauﬀer (1979) and McEachran

and Stauﬀer (1986); ·····, Bartschat, McEachran and Stauﬀer (1988); – – –,

Montgomery and LaBahn (1970). The number in parentheses following 30 eV

indicates the power of ten by which the cross sections have been multiplied.

Normalization to theory was done at 30

◦

and 120

◦

by Smith et al. (1990) and

at 30

◦

and 60

◦

by Floeder et al. (1988). Reprinted from Physical Review Letters

64, Smith et al., Evidence for absorption eﬀects in positron elastic scattering by

structure which could appear in the calculated positron diﬀerential elastic

scattering cross sections, at small angles and in the intermediate energy

region, is washed out when these eﬀects are properly taken into account.

At lower energies the experimental data begin to show features similar

to those present in the polarized-orbital (without absorption) results of

McEachran et al. (1979), and theory and experiment are in reasonable

agreement below the positronium formation threshold. The trend of the

148 3 Elastic scattering

Fig. 3.18. Experimental elastic positron–argon diﬀerential scattering cross sec-

tions in the range 5–50 eV from Smith et al. (1990). The theoretical data

are from McEachran and Stauﬀer (1986) and McEachran et al. (1979) (——)

and from Bartschat, McEachran and Stauﬀer (1988) (·····). The numbers in

parentheses following the energy values indicate the power of ten by which the

cross sections have been multiplied for clarity of display. Reprinted from Physical

Review Letters 64, Smith et al., Evidence for absorption eﬀects in positron

Society.

data as the positron energy is lowered is nicely illustrated in Figure 3.18,

which shows a compendium of data by Smith et al. (1990) for positrons

at 5–50 eV scattering from argon. The data are all normalized to theory

at 120

◦

. As summarized by Kauppila et al. (1996), similar conclusions

can be drawn from studies of the other inert gases at energies below their

respective positronium formation thresholds.