A Novel Approach to Calculate the Range of High-Energy Charged Particles Within a Medium

The determination of energy loss of charged particles as they pass through a medium and consequently the calculation of their range within the medium are of tremendous importance in various areas of physics from both theoretical and practical perspectives. Previous works have derived approximate equations regarding the range of ions within a medium, focusing on providing simple solutions for practitioners in the radiotherapy field that do not require significant computational cost, unlike traditional Monte Carlo methods. These solutions focus on radiotherapy and are limited to specific ions’ initial speeds, which should be up to 0.65c (where c is the speed of light in vacuum). In this paper, solutions for significantly larger initial velocities are explored. A new analytical equation for determining the range of charged particles within a medium for initial velocities between 0.6c and 0.9c is presented. This equation provides excellent results when compared to the accurate numerical solution. Beyond its theoretical and mathematical interest, this solution is also reliable for radiotherapy applications. It provides excellent results for protons with initial energies between 200 MeV and 350 MeV and has the major advantage of being expressed in terms of elementary functions, making its use more straightforward compared to other approaches.