Existing treatment

The Radphi analysis code that calculates the penetration depth of the showers in the LGD is found in the file makehits/makePhotonPatch.c. The relevant part of the code, as it exists now, is shown below.

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#define TARGET_TO_LGD_DISTANCE 108.0
#define ZMAX_MULT 1.06
#define ZMAX_COEFF 4.52
#define E_CRITICAL 0.01455
...
  zMax = ZMAX_MULT*ZMAX_COEFF*log(cluster->energy/E_CRITICAL);
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Figure 3: Various forms for the relation between the $z$ of the shower centroid and the shower energy, plotted for showers incident at 20$^o$ from the normal: the phenomenological formula recommended in this note (red), a simplified approximation to the former (black), the treatment implemented in the present code (blue), and the present treatment with the corrected target-LGD distance (green). Figure courtesy of D. Armstrong

Figure 4: Various forms for the relation between the $z$ of the shower centroid and the shower energy, plotted for showers incident at 20$^o$ from the normal: the phenomenological formula recommended in this note (red), a simplified approximation to the former (black), the treatment implemented in the present code (blue), and the present treatment with the corrected target-LGD distance (green).

Figure 5: Comparison between the shower centroid from the Monte Carlo generator (data points with error bars indicating r.m.s. fluctuations) with that calculated by the recommended makePhotonPatch code.

The meaning of the parameter zMax is shown in Fig. 3. Further down in the code it is used to project the measured $(x,y)$ of the cluster onto the plane defined by $z$=TARGET_TO_LGD_DISTANCE which nominally corresponds to the front face of the glass. The calculated value of zMax may be used to find the $z$ coordinate (beware the notation) of the shower centroid according to Fig. 3. This is shown in Fig. 4. The blue curve represents the code listed above. The data point and error bar represent the Monte Carlo distribution from Fig. 2 rotated to 20$^o$. The blue curve misses the data because the value of TARGET_TO_LGD_DISTANCE is too large. Survey measurements give 102.9cm for this parameter. Just changing that parameter gives the green curve. The red curve is just Eq. 3 applied to an incidence angle of 20$^o$.