THE EGS5 CODE SYSTEM

at low energies. The latter, coupled with the Compton process, gives rise to a lateral spread in
the shower. The net effect in the forward (longitudinal) direction is an increase in the number of
particles and a decrease in their average energy at each step in the process.
As electrons slow, radiation energy loss through bremsstrahlung collisions become less prevalent,
and the energy of the primary electron is dissipated primarily through excitation and ionization
collisions with atomic electrons. The so-called tail of the shower consists mainly of photons with
energies near the minimum in the mass absorption coefficient for the medium, since Compton
scattered photons predominate at large shower depths.
Analytical descriptions of this shower process generally begin with a set of coupled integrodifferential
equations that are prohibitively difficult to solve except under severe approximation.
One such approximation uses asymptotic formulas to describe pair production and bremsstrahlung,
and all other processes are ignored. The mathematics in this case is still rather tedious[141], and
the results only apply in the longitudinal direction and for certain energy restrictions. Three
dimensional shower theory is exceedingly more difficult.
The Monte Carlo technique provides a much better way for solving the shower generation
problem, not only because all of the fundamental processes can be included, but because arbitrary
geometries can be modeled. In addition, other minor processes, such as photoneutron production,
can be modeled more readily using Monte Carlo methods when further generalizations of the shower
process are required.
Another fundamental reason for using the Monte Carlo method to simulate showers is their
intrinsic random nature. Since showers develop randomly according to the quantum laws of probability,
each shower is different. For applications in which only averages over many showers are of
interest, analytic solutions of average shower behaviors, if available, would be sufficient. However,
for many situations of interest (such as in the use of large NaI crystals to measure the energy
of a single high energy electron or gamma ray), the shower-by-shower fluctuations are important.
Applications such as these would require not just computation of mean values, but such quantities
as the probability that a certain amount of energy is contained in a given volume of material. Such
calculations are much more difficult than computing mean shower behavior, and are beyond our
present ability to compute analytically. Thus we again are led to the Monte Carlo method as the
best option for attacking these problems.
2.2 Probability Theory and Sampling Methods—A Short Tutorial
There are many good references on probability theory and Monte Carlo methods (viz. Halmos [69],
Hammersley and Handscomb [70], Kingman and Taylor [89], Parzen[130], Loeve[99], Shreider[156],
Spanier and Gelbard[157], Carter and Cashwell[41]) and we shall not try to duplicate their effort
here. Rather, we shall mention only enough to establish our own notation and make the assumption
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