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By either performing a Taylor expansion or making a polynomial approximation,the Bethe equation for charged particle stopping power in matter can beintegrated analytically to obtain the range of charged particles in thecontinuous deceleration approximation. Ranges match reference data to theexpected accuracy of the Bethe model. In the non-relativistic limit, the energydeposition rate was also found analytically. The analytic relations can be usedto complement and validate numerical solutions including more detailed physics.

Approximate, analytic solutions of the Bethe equation for charged particle range