Simulations

The phenomenology described above contains a few assumptions that can be checked with a detailed Monte Carlo simulation. For this purpose I adapted the Gradphi program to write out detailed information regarding the tracking within a single shower in the LGD. GEANT was instructed to track each electron down to 100keV of kinetic energy. Any bremsstrahlung photons over 100keV generated during tracking were individually simulated. A shower from a 1GeV gamma ray at normal incidence contained about 160 separate $e^{\pm}$ tracks in the glass. Each track was broken up into segments of length about 1mm and entered vs depth into a histogram weighted by track segment length. This histogram represents the Cerenkov emission rate as a function of depth. The result for 10000 1GeV showers at normal incidence is shown in Fig. 1.

Figure 1: Simulated Cerenkov emission intensity em vs penetration depth for 1.0GeV gamma-ray showers at normal incidence. The front face of the LGD is located at 102.9cm. The simulation contains 10000 showers.

Fig. 1 shows the average longitudinal profile of the showers but does not indicate the scale of the fluctuations from one shower to the next. In Fig. 2 is shown the mean depth from Fig. 1 calculated individually for each event. As expected, the mean is in the same place and the standard deviation has decreased. Note that the standard deviation does not decrease as $1/\sqrt{160}$ because the tracks are not independent. Most of the fluctuations are tied to the variations in the conversion depth of the primary gamma ray.

Figure 2: Mean depth of the Cerenkov emission profile, calculated separately for each event and then histogrammed, for the 10000 showers shown in Fig. 1. The fluctuations are mainly determined by the conversion depth of the primary gamma ray.

The fit in Fig. 1 is of the form Eq. 1 with $b$ allowed to vary. The best value for $b$ of 0.54 is in good agreement with expectations [1]. The fit was done with 2 parameters, $b$ and a vertical scale factor. It is clear from the figure that a significant amount of shower energy escapes out the back of the block. This fact was known by the I.U. team that designed the calorimeter. The choice of 15 radiation lengths was made to compromise between shower containment and attenuation of Cerenkov light in the glass. In Radphi the shower leakage gets somewhat better as the polar angle of the shower increased from normal incidence up to about 20$^o$ beyond which it abruptly increases due to leakage out the sides. Correcting the shower energy for leakage is an important matter for a future study. Only showers up to 20$^o$ will be considered in this study.