THE EGS5 CODE SYSTEM

at low energies. The latter, coupled with the Compton process, gives rise to a lateral spread inthe shower. The net effect in the forward (longitudinal) direction is an increase in the number ofparticles 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 ionizationcollisions with atomic electrons. The so-called tail of the shower consists mainly of photons withenergies near the minimum in the mass absorption coefficient for the medium, since Comptonscattered photons predominate at large shower depths.Analytical descriptions of this shower process generally begin with a set of coupled integrodifferentialequations 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], andthe results only apply in the longitudinal direction and for certain energy restrictions. Threedimensional shower theory is exceedingly more difficult.The Monte Carlo technique provides a much better way for solving the shower generationproblem, not only because all of the fundamental processes can be included, but because arbitrarygeometries 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 showerprocess are required.Another fundamental reason for using the Monte Carlo method to simulate showers is theirintrinsic 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 ofinterest, 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 energyof 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 quantitiesas the probability that a certain amount of energy is contained in a given volume of material. Suchcalculations are much more difficult than computing mean shower behavior, and are beyond ourpresent ability to compute analytically. Thus we again are led to the Monte Carlo method as thebest option for attacking these problems.2.2 Probability Theory and Sampling Methods—A Short TutorialThere 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 efforthere. Rather, we shall mention only enough to establish our own notation and make the assumption21