An accurate energy-range relationship for high-energy electron beams in arbitrary materials

A general analytical energy-range relationship has been derived to relate the practical range, Rp, to the most probable energy, Ep, of incident electron beams in the range 1 to 50 MeV and above, for absorbers of any atomic number. The expression is cubic in energy and requires as input parameters the total stopping power, So, the ratio of the scattering power and the total specific stopping power, T0/epsilon0, both taken at 10 MeV, and the radiation length for the material involved, X0. In addition to these parameters, five of the derived parameters are used to 'fine tune' the equation and minimize the mean square deviation from experimental and/or Monte Carlo data by means of non-linear regression. In the present study only Monte Carlo data determined with the new ITS.3 code have been employed. The standard deviations of the mean deviation from the Monte Carlo data at any energy are about 0.10, 0.12, 0.04, 0.11, 0.04, 0.03, 0.02 mm for Be, C, H2O, Al, Cu, Ag and U, respectively, and the relative standard deviation of the mean is about 0.5% for all materials. The fitting program gives some priority to water-equivalent materials, which explains the low standard deviation for water. A small error in the fall-off slope can give a different value for Rp. We describe a new method which reduces the uncertainty in the Rp determination, by fitting an odd function to the descending portion of the depth-dose curve in order to accurately determine the tangent at the inflection point, and thereby the practical range. An approximate inverse relation is given expressing the most probable energy of an electron beam as a function of the practical range. The resultant relative standard error of the energy is less than 0.7%, and the maximum energy error deltaEp is less than 0.3 MeV.