Investigation on energy straggling of protons via Fokker-Planck equation

The Fokker-Planck equation (FPE) governing the behavior of a monoenergetic proton beam passing through an absorbing element was solved by using the separation of variable technique in which the energy distribution of protons obtained from our analysis satisfies the Vavilov theory. Most noteworthy, the accuracy of the method was benchmarked by comparison our results with those of Reynolds experiments and Hines in which a satisfactory agreement was found. The variation of the distribution function in time, the interval of the primary energy of the protons traversing an element as well as the density of the medium for which the energy distribution of protons follows a Gaussian function were discussed. Moreover, on the one hand, it was shown that the broadening of the distributions increases by increasing the initial energy of protons traveling through a specified element. On the other hand, we made the conclusion that the average broadening is the same for monoenergetic protons passing through the elements having different densities. It is worth noting that solving the FPE for energy straggling of protons penetrating through an element has been carried out for the first time using the analytical approach based on the separation of variable technique.