Positron and Positronium Detection Efficiencies
Chapter 4- Positron and Ps Detection Efficiency 110 Annihilations may now be resolved into, (a) those emanating from positrons at the
channel plates, (b) those from Ps formed by the backscattered positrons, either through inflight decay or annihilating on the grids and (c) from the elastically scattered positron beam annihilating on the grids. The Csl detector may measure all these annihilations with an effective efficiency given by the place of annihilation and the respective Csl shielding profile. Therefore, the total number of Csl counts, , detected when measuring the positron detection efficiencies may be written as:
A.W = + iv, + w K + SaridfeK + + rpos (4.34)
where the primed symbol refers to the incident positron beam. The counts recorded by the Csl detector which can give rise to a coincidence are
{ A Nc s/ = C C S ^ N ^ + ( ^ 2 ^ 3 ) ^ / ^ P s { ^ i n f l ig h t ^ G r i d ^ i (4.35)
Given that the energy of the incident positrons is below lOOeV, the inflight annihilation o f the Ps atoms was determined assuming that the reflected positrons formed lOeV Ps (Howell et al, 1986). By comparison vdth Equation 4.34 we can rewrite Equation 4.35 as
C sl) c = y total - ï a m b - y P o s - ^ R \ ^ i - (4.36)
Equations 4,35 and 4.36 may now be used to find the incident positron beam,
_ ______________________ y to tal y am b ^ _P o s______________________________________________________ _ 37)
{ a s ^ + S Q r id s + inflight ^ P s ^ i n f l i g h t ^ G r id s G rids ) )
This may be substituted back into Equation 4.36 in order to calculate the number of annihilations measured by the Csl detector, which may give rise to a coincidence. The modified Csl rate is then used in Equation 4.7 to give the measured detection efficiency.
Unlike in the Ps case, when detecting positrons the grid R l is grounded and any positron, which elastically scatters from the surface o f the channel plates and does not annihilate on one of the three grids infront o f CEMA2, will be lost to the Csl
Chapter 4~ Positron and Ps Detection Efficiency____________________________ UJ. detector leading to an underestimate of the incident positron beam. The true positron detection efficiency may then be obtained by scaling the measured value with the fraction of positrons reflected away from the Csl detector, i.e.
(4.38)
where Piost is the fraction o f positrons lost to the Csl detector. As a fraction of the elastically scattered positrons will annihilate at the grids infront o f CEMA2, Phst is given by
0.73^;. (4.39)
where 0.73 accounts for the beam attenuation through the grids and Fe is given by Equation 4.33.
Knights and Coleman (1996) have determined the fraction o f epithermal positrons from SiOi/Si to be -0.30, decreasing to -0.17 for Sn. From measurements of the branching ratio for positrons at a Cu (110) surface, Baker et al (1988) have determined a re-emitted positron fraction, Fe, in the energy range 0-40eV o f 0.18 to 0.35. Baker et al also observed a decrease in the Ps formation fraction, Fps, in this energy range of 0.33 to 0.25. Assuming that the total branching ratio for all other channels is negligible, the fraction of positrons annihilating at the surface is given by
\-Fps-F+, i.e. 0.47-0.40. By employing a one-dimensional Boltzmann equation, Kong and Lynn (1990) have determined the fraction of positrons, re-emitted as positrons or Ps as 0.40. Howell et al (1986) have measured the Ps produced through reflected positrons from Ni, below lOOeV incident energy, to be 0.20. Table 4.3 shows a summary o f the re-emitted positron fraction and Ps formation fractions determined from solid-state studies. Also shown in table 4.3 are the elastic scattering and Ps formation fractions for high energy positrons, i.e. E=250eV, used to correct the high energy positron detection efficiency value determined in section 4.4.
Chapter 4- Positron and Ps Detection Efficiency 112 Energy (eV) Fe Eps Ee+Fps Low Energy Howell et al (1986) <100 020
Knights and Coleman <100 0.17-0.3 (1996)
Baker et al (1988) 0-40 0.18-0.35 0.35-0.25
Kong and Lynn <100 0.40
(1990)
High Energy
Knights and Coleman 200-300 0.07-0.20 (1996)
Howell et al (1986) 250 0.14-0.15
Table 4.3 Summary o f re-emitted positron fraction and Ps formation fractions.
The limits for Fg and Fps in table 4.3 are given by the different materials investigated and the variance across the energy range shown.
Figure 4.15 shows the ratio of the elastic to total cross-sections for a number of atomic targets, as calculated by Baluja and Jain (1992). As can be seen from the plot, the value for the positron re-emission fractions, determined from solid-state surface studies, are more consistent with the higher elastic scattering fraction of positrons from atomic targets.
Chapter 4- Positron and Ps Detection Efficiency 113 0. 6 P o s i t r o n - N e P o s i t r o n - A r P o s i t r o n - X e P o s i t r o n - H e 0 .4 0 .3 O 0. 2 0. 0 0 5 0 1 0 0 1 5 0 2 00 2 5 0 E n e r g y ( e V )
Figure 4.15 Ratio o f elastic to total cross-sections fo r positron-Atom scatterings as calculated by Baluja and Jain (1992).
In figure 4.16, the equivelocity and corrected (Fg=0.20 and F/>.^=0.20) positron detection efficiencies are compared with the raw and corrected Ps detection efficiency values. As can be seen from figure 4.16, the corrected positron detection efficiencies are 17% lower than the equivelocity positron values. A much better agreement is seen between the two methods used to determine the Ps detection efficiency, after correction for elastic scattering, Ps formation and Ps break-up from the surface of the channel plates. The good agreement between the corrected positrons and the corrected Ps detection efficiencies suggests that at least for these two projectiles, the dominant
0 . 2 0 -I --- --- 0 .1 8 - 0 .1 6 - I 0 .1 4 - £ § 0. 1 2 -
I
c 0. 10 - 0 .0 8 - 0 .0 6 - 10 1 5 Raw Ps • E q u iv e lo c ity p o s itro n • C o rre c te d Ps • C o rre c te d p o sitro n 20 2 5 3 0 3 5 4 0 E nergy (e V )Figure 4.16 Comparison o f the corrected positron detection efficiencies to the measured positron and the measured and corrected Ps efficiencies.
Chapter 4- Positron and Ps Detection Efficiency 114
factor in determining the detection efficiency is their impact velocity.
In order to determine the uncertainty associated with the correeted positron detection efficiencies, the values of Fg and Fps have been varied within the ranges given in table 4.3 and the results are plotted in figure 4.17.
0 . 2 0 - [ --- 0 .1 8 - 0 .1 6 - ^ 0 .1 4 - Î I _ u 0.12 H 0 . 1 0 - 0 .0 8 - 0 .0 6 10 F e = 0 ,3 5 , F p s = 0 .2 5 F e = 0 .1 8, F p s= 0 .3 5 F e = 0 .2 . F p s= 0 .2 1 5 20 2 5 3 0 3 5 4 0 E nergy (e V )
Figure 4.17 Variation o f the corrected positron detection efficiencies with elastic scattering and Ps formation probability.
As can be seen in figure 4.17, the correction to the equivelocity detection efficiency values are dominated by Equation 4.38, whereby the fraction of the incident beam lost to elastic scattering is corrected for. Therefore, the lowest value for the detection efficiencies corresponds to the largest elastic scattering fraction. With Fg=0.18 an increase of 2% is observed, whilst for Fe=0.35, the detection efficiencies are reduced by 17%, as expected from Equation 4.38.
Chapter 4- Positron and Ps Detection Efficiency 115 0 .5 5 0 .5 0 - 0 .4 5 - 0 .4 0 - ’5 0 .3 5 - % 0 .3 0 - 0 .2 5 - a 3 Q 0 .2 0 - • E q u iv e lo c ity p o sitro n • C o rre c te d Ps • C o rre c te d p o s itro n 0. 10 0 .0 5 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 E nergy (e V )
Figure 4.18 Comparison o f the corrected positron detection efficiencies^ compared with the equivelocity positron and corrected Ps.
In figure 4.18 the corrected Ps detection efficiencies are shown along with the equivelocity positron and the corrected positron detection efficiencies. Also included in the plot is the high-energy (~250eV) positron value, determined in section 4.4 as 0.51+0.01, also corrected for elastic scattering and Ps formation using the average of the 250eV positron values for Fg and Fps given in table 4.3, to give a value of 0.46+0.01. 0 .5 0 0 .4 9 - 0 .4 8 0 .4 7 - UJ 0 .4 6 - Q 0 .4 5 - 0 .4 4 - 0 .4 3 - 0 .4 2 0 .0 6 0 .0 8 0 . 1 0 0 . 1 2 0 .1 4 0 .1 6 0 .1 8 0 . 20 0. 22 Fe
Figure 4.19 Variation o f the 250eVpositron detection efficiency with elastic scattering fraction.
Chapter 4- Positron and Ps Detection Efficiency____________________________ 116