Charlton M., Humberston J.W. Positron Physics

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Fig. 2.2. Schematic diagram, not to scale, of the time-of-ﬂight localized scattering system used by the UCL group for total

scattering cross section measurements.

50 2 Total scattering cross sections

Fig. 2.3. The form of the TOF spectra obtained at 300 eV, 30 eV and 3 eV by

the UCL group for positron–helium collisions. The solid lines are the vacuum

spectra, whilst the broken lines are the spectra with gas present; they are

discussed further

in the text.

where l

is the geometric length of the scattering cell. Thus, from equation

(2.1), σ

= k

ln A/(n

Figure 2.3 shows TOF spectra obtained both with and without gas

admitted to the scattering cell for incident positron energies of 300 eV,

30 eV and 3 eV. It is clear that extra events are visible in the spectrum

on the long TOF side of the gas peak at 300 eV. These are caused by

the detection of positrons which have been scattered, both elastically and

inelastically, through small forward angles with respect to the initial di-

rection of the positron beam. Inelastic scattering could easily be resolved

from the unscattered beam, and so introduced a negligible error into the

determination of A and, hence, of σ

In the case of elastic scattering, which involves nearly zero energy loss

for the positron, the increase in the TOF, ∆t, of a positron of speed v

scattering through an angle θ is given by

∆t = l

(sec θ −1)/v, (2.11)

where l

is the geometric distance between the point of scattering and

the CEM detector. Clearly ∆t → 0asθ → 0, and at some point any

positron elastically scattered through a small angle will be indistinguish-

able from the unscattered beam. Depending on the detailed behaviour of

the diﬀerential elastic scattering cross section, this may lead to systematic

overestimates in the determination of I, the beam intensity with gas

present.

2.3 Experimental techniques 51

In most of the total cross section measurements made by the UCL

group a magnetic ﬁeld gradient existed along the positron beam ﬂight

path, which facilitated the distinction between scattered and unscattered

particles. In particular, the magnetic ﬁelds in the scattering and detection

regions were held in the ratio 1 : 8 respectively, thereby producing a mag-

netic mirror. Consequently, from the adiabatic conservation law B/ sin

= constant (see e.g. Griﬃth et al., 1978a; Charlton et al., 1984, for a more

complete discussion of the experiment and e.g. Jackson, 1975, Chapter 12,

for a treatment of the underlying principle), a positron initially travelling

parallel to the axis of the magnetic ﬁeld and then scattered through an

angle greater than 20

◦

would have been unable to reach the detector.

A combination of the magnetic mirror and TOF techniques was used in

an attempt to discriminate fully against small-angle positron scattering.

Further details of the methods used to extract the true attenuation from

the TOF spectra can be found in the works of Charlton et al. (1980a,

1984) for energies below approximately 20 eV, and Coleman et al. (1976b)

and Griﬃth et al. (1979a) at higher energies. The apparatus shown in

Figure 2.2 has also been used for determinations of σ

for electron impact

in the energy range 2–50 eV. The targets studied were helium, neon, argon

and a number of simple molecules (Charlton et al., 1980a, 1984; Griﬃth

et al., 1982). Very reasonable agreement (usually to better than 10%) was

found with previously available cross sections, particularly for the noble

gases.

Another system for measuring total cross sections, which has also been

used for both positrons and electrons, is that developed by the Detroit

group (see e.g. Stein, Kauppila and Roellig, 1974). Their apparatus

and method embody many features not present in the widely used TOF

technique. In addition, this group has made the most comprehensive

survey of targets using both projectiles.

In the Detroit apparatus, illustrated in Figure 2.4, the β

source is

created in situ by bombarding a boron target with a 4.75 MeV proton

beam emanating from a van der Graaf accelerator. An unstable carbon

isotope is then produced via the reaction

B+p→

C, (2.12)

the β

being emitted with a 20.3 minute half-life via the reaction

C →

B+e

+ ν

. (2.13)

It was found that the boron target itself acted as a moderator with a

low eﬃciency of 10

−7

, but the emitted positrons had a low energy, and

therefore a narrow energy width, of approximately 0.1 eV.

Referring again to Figure 2.4, the slow positrons emitted from the

boron were accelerated and focussed by the electrostatic lens system

52 2 Total scattering cross sections

Fig. 2.4. The apparatus used by the Detroit group for the measurement of

total cross sections for the scattering of positrons and electrons from noble and

molecular gases. Linking the two parts of the apparatus is the curved positron

energy ﬁlter.

into the magnetic guiding ﬁeld (typically 10

−3

T) produced by a long

curved solenoid. This solenoid also served to remove the detector from

the line of sight of the

C source. The positrons were detected by a CEM

after passing through a 4.8 mm aperture and an electrostatic retarder

arrangement.

The total cross section was measured by monitoring the attenuation of

the beam as it passed through the curved solenoid, which could either

be evacuated or ﬁlled with gas. The average pressure of the gas was

2.3 Experimental techniques 53

Fig. 2.5. Schematic illustration of the apparatus used by the Bielefeld group

to measure total scattering cross sections. Reprinted from Journal of Physics B

13, Sinapius, Raith and Wilson, Low-energy positrons scattering from noble gas

determined by direct measurements at either end of the solenoid using

an absolute capacitance manometer. In order to obtain an absolute value

for the product of n and L, see equation (2.1), the pressure measurement

was corrected for thermal transpiration eﬀects (section 2.4) using a direct

temperature measurement on the inner surface of the wall of the solenoid

of known geometric length. Thus, absolute values for σ

could be obtained

directly without the need for normalization. A further virtue of the

Detroit apparatus was that a tungsten ﬁlament could be substituted for

the boron moderator, thereby enabling cross sections for electrons to be

measured in the same apparatus.

The narrow energy width of the positron and electron beams meant

that a retarding electrode system located just before the CEM could be

used to provide good angular discrimination against projectiles elastically

scattered through small angles. The retarder voltage V

was, as described

by Kauppila et al. (1981), usually set at a level such that a small fraction

of the incident beam was cut oﬀ, so that a particle with kinetic energy

E>eV

, scattered through an angle greater than cos

−1

(eV

/E)

1/2

, could

not pass to the detector. The reader is referred to the aforementioned

article, where the performance and importance of this device are dis-

cussed at greater length. In addition, the aperture beyond the scattering

cell provided angular discrimination because scattered particles, having

acquired larger components of velocity transverse to the magnetic ﬁeld

in the collision process, travelled in orbits with increased Larmor radii.

These points are discussed further in section 2.4, where the systematic

54 2 Total scattering cross sections

eﬀects of forward scattering on the measured values of the total cross

sections are described.

The apparatus used by the Bielefeld group, described fully by Sinapius,

Raith and Wilson (1980), is illustrated in Figure 2.5. The system is based

upon the TOF technique developed by Coleman, Griﬃth and Heyland

(1973), in which one time signal was provided by a 0.15 mm thick scintil-

lator coupled to a photomultiplier and the other came from the detection

of a slow positron by a CEM. A novel feature of this apparatus is the

predominant use of electrostatic beam transport. Positrons emitted from

a grounded oxygen-free copper moderator, following β

bombardment,

were accelerated by applying a potential to the entire scattering chamber.

This was made from a cage of copper rods, each of 1 mm diameter,

and was itself 70 mm in diameter and 280 mm long. The chamber was

surrounded by magnetic shielding which reduced the magnetic ﬁeld inside

to approximately 35 nT. Adjustments to the positron beam could be made

by two pairs of deﬂection plates at the entrance to the scattering chamber

and by a weak magnetic lens located one third of the way along the length

of the chamber, so as to facilitate focussing onto the cell exit aperture. A

series of apertures beyond the scattering cell was used to accelerate and

focus the positrons on to the channeltron.

The scattering chamber was pumped out through the apertures so that

a pressure diﬀerential of around 80 was maintained between it and the

remainder of the ﬂight path. The pressure was recorded using a calibrated

ionization gauge. The cross section was determined from measurements

of the intensity of the beam with gas ﬂow through the scattering chamber

and with the gas ﬂow bypassing the scattering chamber (see Figure 2.5)

and directly entering the detector area. Since the gas ﬂow was kept

constant, the pressure in the detector region was equal in both cases.

Thus, by simply subtracting the two signals, this method automatically

accounted for collisions which occurred outside the scattering chamber.

Mizogawa et al. (1985) made measurements of positron–helium total

cross sections at low energies using an apparatus based on the technique

developed by the UCL group. The main diﬀerences were their use of low

magnetic ﬁelds and small (2 mm radius) apertures in their scattering cell.

This combination provided good angular resolution and also reduced the

eﬀect of spiralling. Further details are given in section 2.4.

It is also necessary to describe here the apparatus used by the Detroit

group for measurements of total cross sections for positron and electron

scattering from the alkali metals, since their special heated oven, which

also served as the scattering cell, is signiﬁcantly diﬀerent from their long

curved solenoid apparatus. The scattering cell, shown in Figure 2.6 (Kwan

et al., 1991), is located at the end of the curved solenoid shown in Fig-

ure 2.4, the continuation of the magnetic ﬁeld being provided by the two

2.3 Experimental techniques 55

Fig. 2.6. The oven, which also served as the scattering cell, used by the Detroit

group for studies of positron–alkali atom and electron–alkali atom total scattering

cross sections. The open circles indicate thermocouples.

coils shown. Using appropriate voltages, the incident positron or electron

beam could be deﬂected into a CEM at the input to the cell. This detector,

and a similar one to record the beam transmitted through the cell, was

further shielded by apertures from the alkali vapour which could emanate

from the oven. A stainless steel element located on the output side of the

cell (which became coated with alkali metal when the oven was hot) was

used to provide some discrimination against forward-scattered projectiles.

The oven was heated by coils wound in its walls, and the temperature

was monitored at three diﬀerent locations. During an experimental run

these three temperatures were kept as close to one another as possible

in order to reduce variations in the vapour pressure, which is a sensitive

function of temperature. The vapour density, which is required in the

determination of absolute values of σ

, was calculated from standard

vapour pressure tables using the average temperature. A full discussion

56 2 Total scattering cross sections

Fig. 2.7. Schematic illustration of the cooled-gas-cell arrangement used to study

positron and electron collisions with atomic hydrogen (e.g. Zhou et al., 1994a).

of the uncertainties involved in the latter procedure, which introduced the

largest errors into the assignment of absolute values for σ

(±20%), has

been given by Kwan et al. (1991).

By measuring the intensity of the beam transmitted through the cell

when it was hot, and therefore with alkali vapour present, and also when

it was cold, when the vapour pressure was negligible, σ

was deduced

using

hot

= I

cold

exp(−nLσ

), (2.14)

which is equivalent to equation (2.1).

The ﬁnal apparatus we describe brieﬂy here is that used by the De-

troit group (Zhou et al., 1994a) for the ﬁrst measurements of the total

cross section for positron scattering by atomic hydrogen, and again it

2.4 General discussion of systematic errors 57

was possible to study electron collisions with the same apparatus. The

projectile beams were derived from the solenoid apparatus (see above)

and the cross sections were measured by a transmission technique, using

the special gas cell illustrated in Figure 2.7. A radio-frequency discharge

source was used to produce hydrogen atoms, which were then fed into

a small aluminium chamber; this, when cooled to 150 K, had a low

recombination coeﬃcient so that a suﬃciently high density of hydrogen

atoms (approximately 10

−3

) could be achieved for the attenuation

measurement to be made. An estimate of the ratio of the number density

of atomic hydrogen, n(H), to the number density of atomic-plus-molecular

hydrogen, n(H) + n(H

), in the cell was obtained by passing a sample

of the output of the cell into a quadrupole mass spectrometer. In the

earlier work this ratio, f = n(H)/[n(H) + n(H

)], was measured to be

approximately 0.55. However, Zhou et al. (1994a) noted that, since some

of the hydrogen atoms entering the cell recombined before leaving, the

estimated value of f as encountered by the beam passing through the cell

was likely to have been greater than 0.55, and possibly as high as unity.

Thus, these two extreme values of f were used to calculate the cross

section for atomic hydrogen using known values of the cross section for

molecular hydrogen. The latter cross section was also measured using the

same apparatus by turning the discharge oﬀ and normalizing to the earlier

measurements by the same group (Hoﬀman et al., 1982). Improvements

reported by Stein et al. (1996) and Zhou et al. (1997), both in the degree

of dissociation achieved and in its determination, mean that the results

of the Detroit group can now be presented as deﬁnite values, rather than

as upper and lower limits.

2.4 General discussion of systematic errors

Some of the systematic errors which can inﬂuence measurements of σ

for low energy positron and electron scattering were introduced in the

preceding section. We now proceed to a more general discussion of these

errors and attempt to summarize the situation pertaining to all the groups

who have made such measurements and whose data are presented later

in this chapter. This summary is based on that given in the review of

Charlton (1985a).

Equations (2.1) and (2.2) show that, in order to obtain accurate values

of σ

, it is necessary to make precise measurements of the incident and

transmitted ﬂuxes, I

and I respectively, the path length L in the scat-

tering cell and the pressure P and temperature T of the gas. During the

1970s and 1980s it became technically feasible, using systems like those

described in the previous section, to determine σ

over a wide energy

range, with quoted statistical errors usually less than ±5%. It will become

58 2 Total scattering cross sections

apparent in the following section that much greater discrepancies than

this sometimes exist between the measurements from diﬀerent groups.

This implies that there are also systematic eﬀects present in some of the

measurements which cause erroneous determinations of one or more of

the parameters listed above and which are peculiar to the method and

apparatus employed by each of the groups. Following Charlton (1985a),

the four main sources of systematic error can be summarized as follows:

(i) errors in L due to spiralling of the positrons and electrons in an

axial magnetic ﬁeld, if employed (in systems not using a magnetic

ﬁeld, other eﬀects on the trajectories may be important);

(ii) errors in L caused by end eﬀects in the gas cell;

(iii) errors in P due to incorrect determination of the absolute pressure;

(iv) errors in I, due predominantly to the detection of projectiles which

have been scattered elastically, or with small energy loss, but which

cannot be distinguished from the unscattered beam.

Any contribution from error (i) results in an overestimate of σ

since

the eﬀective path length in the gas is greater by a factor sec θ than the

assumed value; see equation (2.11). Error (iv) also leads to an underest-

imate of σ

since not all the scattered projectiles are so identiﬁed. All

reported experimental values of σ

may be subject to one or more of the

errors (i)–(iv), and it is important to bear this in mind when making

comparisons between the results from diﬀerent groups.

Errors of type (iii) are probably less important than some of the others.

Even so, as pointed out by Coleman et al. (1979) and Charlton (1985a), an

absolute measurement of the pressure using a capacitance manometer can

be complicated by thermal transpiration eﬀects. Errors quoted by manu-

facturers on the pressure, as measured using such instruments, can be as

low as 0.15%–0.25%. However, the sensor heads are usually electronically

stabilized at a temperature T

(typically ≈350 K) which, if not equal

to the actual temperature T , can give rise to a thermal transpiration

correction factor relating the true pressure P to that measured at the

sensor, P

,byP = P

(T/T

)

1/2

. This is a 4% eﬀect at T = 297 K. Such

corrections were made by the Detroit group, but Coleman et al. (1979)

pointed out that account should also be taken of the diameter of the tube

or other arrangement which connects the scattering cell to the sensor,

according to the method of Liang (1955). These authors, using a similar

manometer to that of the Detroit group, found that a smaller adjustment

should be applied, its value being dependent upon the gas under study

(e.g. 0.25% for helium and 0.5% for neon). Mizogawa et al. (1985), who