Approximate calculation of backpropagated energy spectrum for a proton beam

In certain experimental setups used for proton irradiations at the TOP-IMPLART linear accelerator at the ENEA Frascati Center, the energy spectrum of the proton beam is measured at the end of a propagation path, which includes transmission through different materials, such as air, windows, slabs, etc. In this paper, we develop and test an approximate mathematical method to calculate the energy spectrum at the accelerator exit from such a measured transmitted spectrum. In the first experimental test application, the spectrum measurement exploits the visible photoluminescence of F2 and F3+ color centers generated in lithium fluoride crystals by the interaction of the crystal lattice with protons. In the second test application, a simulated measurement of a propagated energy spectrum along a transport line is considered. In principle, the proposed method is applicable to the energy spectra of proton beams measured in any manner.