Theoretical aspects of energy–range relations, stopping power and energy straggling of protons

The Bragg–Kleeman rule RCSDA=AE0p provides a connection between the initial energy E0E0 of a proton and the range RCSDARCSDA in a medium, if the continuous-slowing-down approximation (CSDA) is assumed. The rule results from a generalized (nonrelativistic) Langevin equation; its integration also yields information on the residual energy E(z)E(z) or dE(z)/dzdE(z)/dz of a proton at position z  . A relativistic extension of the generalized Langevin equation leads to the formula RCSDA=A(E0+E02/2Mc2)p. Since the initial energy E0E0 of therapeutic protons satisfies E0⪡2Mc2E0⪡2Mc2, relativistic contributions can be treated as correction terms. Besides this phenomenological aspect, a complete integration of Bethe–Bloch equation (BBE) is presented, which provides the determination of RCSDARCSDA, E(z)E(z), dE(z)/dzdE(z)/dz and works without any empirical parameters. The results of these different methods are compared with Monte Carlo calculations (GEANT4). Since the energy transfer from proton to the environmental atomic electrons regarded in the CSDA-framework has to account for local fluctuations, an analysis of the Gaussian convolution and the Landau–Vavilov distribution function is performed on the basis of quantum-statistical mechanics. The Landau tail can be described as a Hermite polynomial correction of a Gaussian convolution.