Slowing down approximations to the boltzmann equation: I. Applications to energy deposition and ion implantation

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Abstract
An approximale method for solving the Boltzmann equation for fast particles slowing down in homogeneous media is described. The Boltzmann equation is used in its straight ahead form and the method of Goertzel and Greuling, developed for neutron transport calculations, is extended to cover the problem of ion slowing down and energy deposition. Solutions are obtained in a simple analytical form and compared with exact calculations. We observe that the error passes through a maximum as the index of anisotropy, m, goes from zero to unity. The limitations of simple áge theory are discussed and it is shown how these solutions become exact in the Coulomb limit. The additional problem of electronic stopping is included and its effect on the solution is assessed. Numerical examples are given to support the general conclusions; namely that the Goertzel-Greuling method provides a convenient way to obtain useful estimates of ion implantation profiles and energy deposition.