The goal was the reconstruction of the impact point while the computed shower centroid would determine a point at most 10 cm within the calorimeter. The shower centroid only will have the same angular coordinates as the point of incidence when the shower develops parallel to the main crystal axis. Hence, this point has to be calculated where a connection line between the reaction centre and the shower centroid intersect with a crystal surface.
For a box-shaped crystal, as the used NaI, the relevant equation to calculate the point of incidence x would be x=l*sin( f ), where l is the shower depth and f the angle of incidence. In order to keep the computing time short rather than applying corrections for each obtained centroid, the crystal coordinates were transformed. Now instead of a crystal's front face the prospected mean shower depth for each crystal was used. This means that the crystal virtually was shifted such that a point s within each crystal had the same distance to the reaction centre as demonstrated in Figure 7-8.
After applying the position correction the angular resolution for 129 MeV photons uniformly impinging the NaI-wall improved from 1.22° to 1.09° (with an accuracy of ±0.1°). The earlier mentioned weighting factor a was determined by optimizing the angular resolution. The optimal value was 1.0 in both cases.
A similar correction was calculated for the CsI array, which was necessary especially since the CsI sphere centre did not coincide with the centre of the target. This caused errors in the calculation of the incident point as can be seen in Figure 7-9.
A correction for the actual angle of incidence therefore was applied. As in the case of the NaI-wall all CsI coordinates were shifted to reach an identical radius such that each crystal front face was about 8 cm from the shower centroid. In order to obtain the correction, the calculation of the virtual crystal positions was optimized for angular resolution. By this the angular resolution after corrections was 1.28°±0.1° compared to 2.05°±0.2° for 129 MeV photons without the correction. Table 7-1 summarizes the results[30]. There the angular resolution and the obtained position corrections are given for three energies for both calorimeters. These values are correlated with the obtained shower depths for the given energies. The positions of the calorimeters to the LH2-target centre were 100.2 cm for the NaI-wall and 55.9 cm for the centre of the CsI sphere.
These results demonstrate two facts that have been noted before.
I) The angular resolution improves with higher energies. Due to a higher momentum secondary shower particles are more likely to follow the direction of incidence of the initial particle.
II) The smaller difference of the corrections between 129 MeV and 55 MeV photons for CsI is due to a shorter radiation length compared to NaI.
The shower centroid and radial spread in NaI and CsI have been studied using GEANT. To this end the vector of each track has been histogrammed after the angle of incidence was calculated. With elementary trigonometrical relations the propagation along the crystal axis, along the position of incidence and perpendicular hereto can be computed. The obtained mean values for the three energies of interest are shown in Table 7-2. The results are in agreement with the obtained position correction and thus support the above interpretations.