A low-current electron beam may be used as a diagnostic tool to measure the concentrations of heavy metals (Pb, Hg, Sb, etc.) present in the flue gas particulates produced by smelters or cement kilns. A small electron accelerator is being constructed as part of a prototype emissions monitoring system. The electron beam energy has a design energy of 1 MeV, a peak current of 5 mA, and a duty factor of 0.1 percent. In this paper, we discuss the results of a set of EGS4 calculations used to model the transport properties of a 1 MeV electron beam passing through a thin vacuum window and the flue gas.