Space distribution and energy straggling of charged particles via Fokker-Planck equation

Abstract

The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-known Bohr or Bethe formula is found as a first-order approximation of the Fokker-Planck equation.PACS 51.10 -Kinetic and transport theory. -IntroductionIn this paper we consider the slowing down and diffusion of a beam of charged particles in a homogeneous medium. In the frame of the Fokker-Planck equation we will find an analytical expression of the spectrum in energy and space. We will then be able to calculate some important parameters such as the range and the loss of energy per unit space (which will be referred to as Space-Dependent Stopping Power or SDSP to distinguish it from the usual definition of stopping power found in the literature). It is worthwhile to underline that these parameters are calculated here taking into account also the energy straggling effect described by the diffusion-in-velocity term (a second-order term) in the Fokker-Planck equation. Actually the energy loss process of a monoenergetic particles beam has a statistical nature. Thus, after the particles have passed through a medium, the energy distribution does not remain a delta function as in the continuous slowing-down (*) The authors of this paper have agreed to not receive the proofs for correction.

Investigation on energy straggling of protons via Fokker–Planck equation

“…However, it is notable that some articles deal with the analysis of the Fokker-Planck equations (FPEs) as a standard mathematical tool for the problem of energy straggling of protons passing through a medium, which is intrinsically a stochastic process [6][7][8]. Simplicity and flexibility of the FPE cause it a popular kinetic equation both for theoreticians and for experimentalists.…”