PEREIRA DISFRAZADA DE VICTORIA.

A detailed analysis has been performed to investigate the energy dependence of the positronium detection efficiency for the geometry o f the present gas cell. For the purpose o f the present analysis, it has been assumed that p-Ps decays instantaneously

emitting 2 y-rays and has a detection efiBciency . If o-Ps reaches the walls of the gas

cell, it is assumed to quench. Defining P as the probability of o-Ps reaching the walls of the gas cell, the probability o f o-Ps decaying in flight before reaching the walls is (1-f). Whilst quenching on the walls is associated with the same detection efficiency o f as

( lia p lc r 5 Direct mcasuremenis o f()i\ /// He, A/' anJXe

(5.2)

to account for the number o f y-rays emitted in each mode. The probability P was obtained from the exponential decay law:

P = 1 / 1q = exp(-Àt) (5.3)

where À = 7.04//s“' and the time t, needed for o-Ps o f certain energy to reach the walls of the cell, was computed using the Simion 6.0 ion optics simulation program.

Figure 5.3 illustrates schematically the trajectories of Ps atoms originating from three positions along the centre of the beam axis (i.e. at x = 0mm). As mentioned in section 2.5, ions can only be extracted from half o f the volume o f the gas cell. Thus, in this simulation only trajectories of Ps atoms originating in 5mm intervals from y = -25mm to y = 25mm were taken into consideration. The time needed for Ps atoms, emitted isotropically at varying angles, to travel from the point of origin to the walls of the gas cell was recorded. Both the azimuthal (Û) and the polar (^, not shown in the graph) were stepped in 10° intervals, where 0° < ^ < 360° andO° < ^ < 180°. Overall, a

y (mm)

atm!

To Nal _{"i To ion}

detector _{I detector}

e beam

-14 0 43 X (mm)

Figure 5.3. Illustration o f Ps trajectories originating along the centre o f the beam axis. For clarity only three starting positions are shown, the central and the two most extreme ones (to scale).

( liapter 5 Direci mcasnrcnicnls o f On, in He. Ar and Xe

total of 6754 trajectories were simulated. The extracted times where fed in to eq. 5.3, from which an average probability of o-Ps, at a given energy, reaching the walls of the gas cell was calculated. This procedure was repeated for several energies of Ps atoms in the range from leV to 80eV. The coordinates of the position where o-Ps hit the walls were recorded simultaneously with the time, therefore it was possible to correct P for the solid angle subtended by the y-detector to that point.

The present model is limited by the assumption that Ps atoms are emitted with equal probabilities at all angles. This poses a problem for example at higher positron impact energies, where Ps formation is expected to be forward collimated (Laricchia et al, 1987) and, therefore, Ps atoms may escape undetected through the apertures of the gas cell. This effect would tend to decrease the total Ps detection efficiency with increasing energy.

In figure 5.4, the energy dependence of the probability of o-Ps reaching the walls o f the gas cell is shown (blue curve). At low energies, it increases rapidly from approximately 0.55 at leV to almost 0.9 at around 30eV. The shape of the function is not modified considerably when the correction is made for the solid angle, the largest change being approximately 2% at the highest energies considered. The solid angle correction for the o-Ps portion that decays in flight in the gas cell before reaching the

0 .9

CL,

0.7

calculated from eq. 5.3 corrected for solid angle and norm alized at E„ = le V

0.6

0.5

0 10 20 30 40 50 60 70 80 90

Ps energy (eV )

Figure 5.4. The energy dependence o f the probability o f o-Ps reaching the walls o f the gas cell

( h a p tc r 5 D ireci m casurcm cnis o f O f s /// He, A r a n d Xe

walls should be even smaller and is therefore ignored.

The energy dependence o f the total detection efficiency of positronium ) for

the present geometry o f the gas cell can be written as:

(E) = 0.75[p(£)£j, +

(l

J+

where the coefficients o f 0.75 and 0.25 take into account that the relative population of p-Ps:o-Ps is 1:3. Substitution o f eq. 5.2 into eq. 5.4 with = 1 leads to:

gf, (E) = 0.75[P(£) + 3/2(1 - £(£ ))]+ 0 .2 5 . (5.5) In figure 5.5, this energy dependence is shown normalized to 1 at leV. It decreases rapidly with increasing energies but the reduction between leV and 80eV is only approximately 12%. This analysis demonstrates that although the probability of o- Ps quenching on the walls of the gas cell is greater than its probability to decay in flight, this latter mode dominates the energy dependence of the detection efficiency due to the emission o f three rather than two y-rays.

The positronium formation cross-sections for He, Ar and Xe, presented in the next

section of this chapter, have been divided by shifted by the appropriate Ps threshold

energy in order to obtain the correct energy dependence.

1.00 0 .9 8 0 .9 6 0 .94 0 .92 0 .9 0 0.88 0 10 20 30 40 50 60 70 80 90 Ps en ergy (e V )

Figure 5.5. The energy dependence o f the positronium detection efficiency fo r the geometry o f the present gas cell.

( lia p lc r 5 D ireci m eusiircnicnts o f O i■, in He. A r an d Xe

5.4. Results and discussion

In figure 5.6, the direct measurement of the positronium formation cross-section in He is compared to experimental results of Overton et al (1993) and the elaborate calculation o f Campbell et al (1998), both already discussed in chapter 3. The present results have been normalized to the data of Campbell et al (1998) in the region of the peak and errors bars of up to 6.5% arise primarily fi-om the uncertainty in the

normalization procedure. The present agrees with the energy dependence o f the

measurements of Overton et al (1993) but rises above the theoretical data from

approximately 50eV onwards by up to 30% at 80eV. Due to the fact that in the energy range from 60eV to 120eV the results o f Overton et al (1993) lie approximately 20% above the calculation of Campbell et al (1998), it had been speculated whether they might still have been affected by incomplete confinement of scattered positrons. However, in the light of the present measurements, obtained with a different method to that of Overton et al (1993) and believed to be converged with respect to scattered projectile confinement in the energy range shown in the graph (see discussion section 5.2.), further theoretical investigations would be desirable.

0.6