Simulations

In Ref. [3] a detailed Monte Carlo simulation of the LGD was described, based upon GEANT-3.21. The simulation tracks all shower particles down to a kinetic energy of 100KeV, where they stop emitting Cerenkov light and are stopped. The Cerenkov light is then propagated down the block, undergoing reflections until it is either absorbed or exits the block and is absorbed. Just downstream of the block is a phototube; those Cerenkov photons that are transmit into the glass surface of the phototube are evaluated according to the quantum efficiency of the tube at the given wavelength, and either discarded or counted in the photoelectron sum $p_e$. The measured attenuation spectrum of clean lead glass is used by the simulation inside the block. Rays that strike the surface of the block are stochastically transmitted or reflected according to the familiar Fresnel formulae.

The latter requires the knowledge of how the blocks are wrapped. In Ref. [3] I assumed that the block is surrounded by a layer of air followed by a non-reflecting surface. This permits total internal reflection at large incidence angles, but discards all photons that exit the glass. Under these conditions it was seen that the mean emission depth of the light observed at the phototube is not appreciably shifted by the presence of attenuation.

To check the sensitivity of this result to the wrapping, this study was repeated [4] with an alternative wrapping model. In this second model, the air gap was removed and a reflecting surface put in its place with a reflectivity of 90%. As described in that report, this change did make a noticeable difference, shifting the effective shower centroid deeper into the glass by about 2cm, independent of shower energy. Believing that this second model is a better approximation to the actual LGD than the former, I update Eq. 3 of Ref. [3] to read

$$z_0 = 102.9 + X_0\left[ ln\left(\frac{E}{E_c}\right) + 3.0 \right]$$ (5)

where $X_0$ is the radiation length of lead glass, about 3.1cm, and $E_c$ is the critical energy of lead glass, about 14.5MeV. The 0 subscript in Eq. 5 is there to remind the reader that this equation describes showers at normal incidence to the glass.

To go beyond the restriction of normal incidence, a new simulation was performed. In this simulation the reflective wrapping model was used for the LGD. Photons were generated uniformly in energy from 0 to 2GeV. Runs of 1000 showers each were generated starting at $\theta=1^0$ up to $\theta=28^0$ in steps of $1^0$. Each gamma ray was independently tracked from inside the target to when all of its energy was either absorbed or it exited the alcove area. All phototubes were summed for each event, i.e. there was no clustering performed, but only the blocks right around the shower core contributed appreciably to the $p_e$ sum. For each detected Cerenkov photon a record was kept of the $z$ coordinate where it originated, for purposes of calculating the third coordinate of the shower centroid.

The showers were then binned in $\theta$ and $E$, and the quantity $p_e$ and the mean $z$ were histogrammed on this grid. The results can be seen in Figs. 1-2. The error bars in these figures represent the r.m.s. fluctuations in those quantities from one shower to the next; the error on the mean value itself is about a factor of 30 smaller.

Figure 1: Total photoelectron yield of showers in the LGD, as a function of polar angle $\theta$, for the given slices in $E$. The curves are the 2d-fit described in the text.

Figure 2: Third coordinate of the shower centroid inside the LGD, as a function of polar angle $\theta$, for the given slices in $E$. Only light detected in the phototubes is counted in calculating the mean. Note that $z$ is the coordinate along the beam direction, not the depth of the shower penetration along the shower axis.

In Fig. 1 one sees at all energies the expected decrease in the observed yield that appears beyond $\theta=20^0$. There are a couple of interesting things to note here. First, the transition from rising to falling gain does not show a kink, as would be obtained if one just calculated the geometrical wall thickness along the photon ray. This is because showers fan out as they develop and are not confined to the ray of the original photon. Even more interesting is the observation that the best energy resolution is obtained in a narrow angular band around $\theta=22^0$. This is seen in the figure at all photon energies: the shortest %-error is obtained in the region just beyond the turnover. This is the effect described by S. Teige (see previous section) where at a certain block length there is a magic point where the effects of attenuation and shower leakage tend to cancel and shower conversion depth fluctuations do not affect the observed yield to leading order. It is interesting to see how dramatic that effect is in the simulation results.

The corresponding results for $z$ are shown in Fig. 2. Here the decrease at $20^0$ is less abrupt, but the effect is clearly visible. One might be surprised to see that the mean depth does not equal the front face of the glass at $28^0$. If one looks at the Cerenkov $z$ distribution for these showers, one sees a sharp peak near the front of the glass corresponding to the shower core, but then a tail extending downstream into the glass. This tail comes from shower fragments that happen to split off the core up near the front and move down into the block. As can be seen from Fig. 1, only a fragment of the shower is visible to the detector so there is a bias towards the part you can see.

Overall these results are quite encouraging. They show that we obtain a fairly good energy resolution all the way out to $25^0$ and perhaps further. However to make effective use of the these high-angle showers we need to apply a $\theta$-dependent gain correction. The fits shown by the curves in Figs. 1-2 are parameterizations of these two-dimensional functions that will enable us to apply this correction to our data.