The distribution of energy deposition in electromagnetic showers
generated by high-energy electrons and photons is summarized by the
Particle Data Group [1]. In their review is given the
following expression for the shower profile as a function of
penetration depth into a material.
The quantity of interest in reconstructing the polar angle from cluster
measurements in the LGD is the average depth of the Cerenkov emission
in the LGD. This is not exactly the quantity in Eq. 2
but the two are closely related. From Eq. 1 one can see that
the quantity
in Eq. 2 is just the depth at which the
energy deposition rate reaches its maximum. Using Eq. 2 to
define
allows us to go back to Eq. 1 and calculate the
average deposition depth
which one finds is simply
.
This leads to a formula that gives
for gamma showers in
terms of the shower energy
and physical constants.