1. Recently Bethe has given a comprehensive treatment of the passage of fast particles through matter using Born’s collision theory. The approximation should be valid so long as the velocity v of the particle is large compared with the orbital velocities of the molecular electrons. Bethe’s result (p. 375) for the loss of energy of a heavy particle of mass M, and charge ze , in a gas containing N atoms per unit volume is - d T/ dx = 4π e 4 z 2 N/ mv 2 Z log 2 mv 2 /E, (1) where m is the mass of an electron and E is the mean excitation potential of the atoms, defined by Z log E = Σ nt f nt log A nt , (2) The summation is over all the Z electrons in the atom and A nt is the mean excitation potential of a shell and f nt is the corresponding generalised oscillation strength. If we substitute T = ½M v 2 and u = - 2 log 2 mv 2 /E we obtain for the range R between velocities v 1 and v 2 , R = ME 2 /32π e 4 z 2 Z m N ∫ - y 2 - y 1 e - u / u du = ME 2 /32π e 4 z 2 Z m N [E 1 ( y 2 ) - E 2 ( y 1 )], (3) where y 1 = 2 log 2 mv 1 2 /E, and y 2 = 2 log 2 mv 2 2 /E. The values of the exponential integral E 1 ( x ) have been tabulated by Glaisher and Bretschneider. When v is large or E Small, so that the variation of the logarithmic term can be neglected, equation (1) integrates directly to R ∝ v 4 + const. For lower v or higher E, the effective exponent n in the relation between R and v , when expressed in the form R ∝ v n , becomes smaller. This is known to be in general agreement with facts relating to the varation of stopping power both with velocity and with the mean excitation potential of the absorbing atoms.

On the loss of energy of alpha-particles and H-particles. (1932). Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 135(826), 132–142.