Range-Energy Relations for Electrons and the Determination of Beta-Ray End-Point Energies by Absorption

It is shown that for aluminum absorbers, a single range-energy equation $R=412$ ${{E}_{0}}^{1.265\ensuremath{-}0.0954\mathrm{ln}{E}_{0}}$ (mg/${\mathrm{cm}}^{2}$) will fit the most reliable published values of practical ranges of monoenergetic electrons and the maximum ranges of nuclear beta-rays in the energy region $0.01<~{E}_{0}<~2.5$ Mev. The average deviation of 59 monoenergetic measurements from this equation is +0.08 percent in energy, and -0.05 percent in energy for 35 beta-ray measurements. The mean deviation is 4.1 percent in each case. There are few ranges for energies above 2.5 Mev. All the higher energy values found in the literature and four new measurements on monoenergetic electrons are presented and are shown to be consistent with the range-energy equation $R=530$ ${E}_{0}\ensuremath{-}106$ (mg/${\mathrm{cm}}^{2}$) for ${E}_{0}>~2.5$ Mev. It is shown that the curve ($\frac{d{E}_{0}}{\mathrm{dR}}$) is nearly parallel to the theoretical curve for the rate of energy loss by ionization in the region between 0.01 and 20 Mev and is about 25 percent larger. The reason for this discrepancy is not known. All the methods commonly used to determine the ranges of beta-rays from absorption curves are discussed and a new method developed by the authors is presented.