Comparison of calculational methods for straggling in thin absorbers

The solution of the transport equation by the Laplace-transform method, introduced by Landau and modified by Vavilov and others, is compared with the solution of that equation by what we term a convolution method. An energy-loss cross section determined for solid aluminum has been used as input to the calculations. For very small values of absorber thickness, the straggling distribution is dominated by peaks at multiples of the Al plasmon frequency. At intermediate values, these peaks are a small structure superimposed on a broad energy-loss distribution which is not described well by any of the Laplace-transform theories. At larger values, the straggling distribution consists of one peak which is well described by the Vavilov theory.