A shower propagated to 1 MeV ignores the physical processes that occur with photons, electrons and positrons below this energy. Electrons and positrons lose their remaining energy through ionization in the steel, and have a small range dictated by their energy upon entering the steel, which we will assume is 1 MeV for randomly placed electrons in steel. Photons lose their remaining energy by Compton scattering for energies 1 MeV to 100 keV, and photoelectric effect below 100 keV.
We start by asking how many photons, electrons and positrons do we
have at the end of a shower started by an electron of energy
, and tracked down to 1 MeV. Assuming that some fraction
( 
) of the initial electron energy is remaining after energy
loss to ionization along the way, then the number of 1 MeV particles
at the end of the shower is proportional to in MeV by
. Further, we assume that the separation into 
and
is defined by the ratios 
, 
and
, where the electron and positron ratios are equal
( 
), and all three sum to unity
( 
). Down to tens of MeV, the cascading shower
model with Bremsstrahlung radiation and pair production predicts
.
1 MeV Electrons and positrons placed randomly in a slab steel of a
given width have a probability (P(1 MeV)) of escaping the steel and
creating a hit in the gas, which is dependent on the width of the
steel. Upon
escaping the steel into the gas, a new hit will be created if the cell
didn't already have enough ionzing energy deposited to have a hit
scored. The probability of an escaping electron entering a cell
without a hit( 
) will be energy dependent since the
density of shower hits depends on shower energy. The number of
additional hits directly caused by electrons is then expressed in
Equation 1.
Photons below 1 MeV create hits by ejecting electrons from steel
by either Compton scattering or photoelectric effect. These
electrons can be assumed to be ejected at a random position in the
steel, and thus have an energy dependent probability of escaping the
steel(P(E)), which is again determined by
the steel width. These electrons will have an energy spectrum
determined by the physics of Compton scattering, expressed as
. This is further broken into 
where 
is the number of initial
scattering photons and 
is the differential energy
spectrum of electrons scattered from a single photon, which is
independent of shower energy. The number of additional hits from
photons is expressed in Equation 2, which doesn't have
the factor of 2 in Equation 1 due to positrons and
electrons. The factor is also not present in Eqn.
2 as we assume that the low energy photons are all far
from the shower core, and would be producing hits in cells with out a hit.
If we take to be roughly constant, then the total
increase in hits( 
) is linearly related to the
energy of the shower. Assuming that the number of hits generated by a
shower, when tracked to 1 MeV, is roughly linear with the shower energy
( 
), then we can take the ratio of the hits
increased to the hits at 1 MeV to be constant with energy(Equation
3). This ratio is the ultimate measurement goal of our
study.
We can thus predict that the fractional increase in hits is
independent of the incident shower energy, as long as the number of
hits is approximately linear with the shower energy, and the shower
density is independent of shower energy. The constant 
in
Equation 3 is determined from simulations using a 1 MeV
tracking cutoff. In establishing a simple geometry for our study one
goal is that (the number of hits/MeV for a shower with a
1 MeV tracking cutoff) should be similar to Soudan 2.
Once we found that the increase in hits was constant with shower energy,
the calculation of 
and 
was determined to be more
detailed than deemed with in the scope of this paper. The probability
function P(E) can be separated into energy dependent range of the
electron in steel, called R(E), and the probability of a particle
with that range escaping from steel L wide, Equation 5.
Now and can be expressed entirely as measurables,
or physics dependent calculable values: is determined by
ionization energy loss models, 
and are both
determined by a modified cascade model that replaces Bremsstrahlung
with ionization and pair production with Compton scattering below
tens of MeV, R(E) comes from the continuous slow down approximation
(CSDA) range combined with multiple scattering, and
is the distribution of all Compton electrons generated by a single
photon until that photon reaches energies dominated by photoelectric effect,
at which point a final electron is generated with the energy of the
photon minus the K-edge value of steel.
