Ion transport in inhomogeneous media based on the bipartition model for primary ions

Abstract

a b s t r a c tThe present paper is focused on the mathematical modeling of the charged particle transport in nonuniform media. We study the energy deposition of high energy protons and electrons in an energy range of ≈50-500 MeV. This work is an extension of the bipartition model; for high energy electrons studied by Luo and Brahme in [Z. Luo, A. Brahme, High energy electron transport, Phys. Rev. B 46 (1992) 739-752] [42]; and for light ions studied by Luo and Wang in [Z. Luo, S. Wang, Bipartition model of ion transport: an outline of new range theory for light ions, Phys. Rev. B 36 (1987) 1885-1893]; to the field of high energy ions in inhomogeneous media with the retained energy-loss straggling term. In the bipartition model, the transport equation is split into a coupled system of convection-diffusion equations controlled by a partition condition. A similar split is obtained in an asymptotic expansion approach applied to the linear transport equation yielding pencil beam and broad beam models, which are again convection-diffusion type equations. We shall focus on the bipartition model applied for solving three types of problems: (i) normally incident ion transport in a slab; (ii) obliquely incident ion transport in a semi-infinite medium; (iii) energy deposition of ions in a multilayer medium. The broad beam model of the proton absorbed dose was illustrated with the results of a modified Monte Carlo code: SHIELD -HIT +.

Analytical Solutions for the Pencil-Beam Equation with Energy Loss and Straggling

“…Applying the Fourier transform, i.e., letting F in E( Z(z, ξ) := F Z(z, E)), we get Now, we invoke the NESA, (an approximate version of the fundamental theorem of calculus cf., e.g., Asadzadeh et al, 2010),…”

Spherical Harmonics and a Semidiscrete Finite Element Approximation for the Transport Equation

“…In a previous study (Asadzadeh et al, 2010a) we considered a detailed study of the bipartition model for ion transport. A related approach, based on a split of the scattering cross-section into the hard and soft parts, is given by Larsen and Liang (2007).…”

A New Explicit Four-Step Symmetric Method for Solving Schrödinger’s Equation

“…As it is seen in Equation (1), this model does not involve the first derivative explicitly. In practice, there are several mathematical models in the form of Equation (1), characterizing certain scientific problems in the fields of chemistry, quantum chemistry, physics, quantum mechanics, etc., [1]. In the literature, there are various algorithms addressing the numerical solution of boundary and initial value problems of this type.…”

“…A well known example of Equation (1), in science, is Schrödinger's equation. Much research has been conducted on numerical methods for solving such a model numerically, (see for example [16][17][18][19][20][21] and the references therein).…”