Electron dose calculation using multiple-scattering theory: thin planar inhomogeneities

In this article in our series on electron dose calculation using multiple-scattering theory, we apply the Fermi-Eyges theory to the problem of a thin planar inhomogeneity present in an otherwise-layered medium. We derive expressions for the distribution function P and the location distribution L (which multiplied by the restricted mass collision stopping power is the dose directly deposited by the primary electrons) for various types of incident beams: a completely arbitrary distribution, a Gaussian point source, a pencil beam, an isotropic point source, and a broad parallel beam. We show how divergent-beam dose distributions can be determined from parallel-beam calculations, through use of equivalent configurations dependent upon the depth of dose calculation. Also, we indicate how this work can be applied to the design of wedges (or "compensators") for beam shaping to provide desired dose distributions or to match juxtaposed radiation fields. Explicit formulas for thin plates are then worked out, and we examine the appearance of hot and cold spots distal to the edge of a localized inhomogeneity, for thin half-slabs and for narrow strips. Finally, considering the case of a thin straight wedge-shaped inhomogeneity, we theoretically discover the phenomenon of a "focused hot spot" without an accompanying cold spot, and suggest the design of a "multiple-scattering lens".