Approximate, analytic solutions of the Bethe equation for charged particle range
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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 be integrated analytically to obtain the range of charged particles in the continuous deceleration approximation. Ranges match reference data to the expected accuracy of the Bethe model. In the non-relativistic limit, the energy deposition rate was also found analytically. The analytic relations can be used to complement and validate numerical solutions including more detailed physics.
Work performed in the process of developing (p,n) neutron sources. It seems surprising that such solutions have not been found previously, but we have not so far found any reported. We would be grateful for comments from anyone more knowledgeable in the field. V2: More elegant scaling factors, several typos fixed
Work performed in the process of developing (p,n) neutron sources. It seems surprising that such solutions have not been found previously, but we have not so far found any reported. We would be grateful for comments from anyone more knowledgeable in the field. V2: More elegant scaling factors, several typos fixed