The total Cerenkov light yield in an electromagnetic shower measures
to within a fraction of one percent the total track length in
the shower. However only a small fraction of the total light generated
actually reaches the photomultiplier tube, and that fraction depends on
the shower position. Variations in the collection efficiency with
where the shower enters the front face of a block are simply folded into
the energy resolution of the counter because they are uncorrelated with
the shower energy. But variations in the shower depth centroid are
correlated with shower energy, and produce a nonlinearity in the measured
photoelectron yield with shower energy. It is well known that the mean
depth of a shower increases as the log of the shower energy [1].
Using a simple exponential attenuation model for the loss of Cerenkov
light propagating down a block to the phototube, S. Teige showed
[2] that the measured photoelectron yield vs shower energy
is a power law.
Attenuation is not the only nonlinear effect present. Another major
effect that prior studies have not taken into account explicitly is
shower leakage from loss of particles out the back of the LGD. This
effect is also dependent on shower energy, but it enters with the
opposite sign from attenuation: instead of increasing the gain with
higher gamma energy it decreases it. S. Teige has pointed out in
discussion that the competition between these two effects can be
exploited to optimize the energy resolution at a certain block
thickness. He indicated that the Radphi LGD blocks are somewhat longer
than optimal for Radphi energies. This means that somewhere beyond
, where showers begin to leak out the sides of the detector,
one should observe the energy resolution first improving and then
getting worse as shower leakage begins to dominate over attenuation.
In a prior study [3] I proposed a formula to be used in
estimating the mean depth of showers inside the LGD, based on a
knowledge of the shower energy . This, in turn, feeds back to the
calculation of the polar angle
of the photon. Unfortunately
this can only work reliably in the region
where
shower leakage effects are small. At larger values of
a
more complicated procedure is required because there the centroid
of the observed shower is a function both of
and
.
Even worse,
itself is not known directly in this region either
because it depends on a shower leakage correction that must depend
strongly on
. So we start off with measured values
and cluster coordinates
and we want to
end up with corrected values for
and
. This
system can be solved by iteration, by seeking the solution to
the following three constraints.