Multiple scattering of proton via stochastic differential equations

Multiple scattering of protons through a target is explained by a set of coupled stochastic differential equations. The motion of protons in matter is calculated by analytical random sampling from Moliere and Landau probability density functions (PDF). To satisfy the Vavilov theory, the moments for energy distribution of a 49.1 MeV proton beam in aluminum target are obtained. The skewness for the PDF of energy demonstrates that the energy distribution of protons in thin thickness becomes a Landau function, whereas, by increasing the thickness of the target it does not follow a Gaussian function completely. Afterwards, the depth-dose distributions are calculated for a 60 MeV proton beam traversing soft tissue and for a 160 MeV proton beam travelling through water. The results prove that when elastic scattering is taken into account, the Bragg-peak position is decreased, while the dose deposited in the Bragg region is increased. The results obtained in this article are benchmarked by comparison of our results with the experimental data reported in the literature.