Stopping power in a cylindrical system by the modified Bessel function's method

On the basis of the general theory of stopping power for a system, where their electronic distributions are able to be treated as cylindrically symmetric, we derive the log-type formulae which are applied in the non-relativistic high-energy region. As well as the case of a free electron in a homogeneous system, we obtain analytically instructive solutions of stopping power in an inhomogeneous system of many-electrons, in which an effective impact parameter averaged over scattering angles is introduced. Numerical calculations are performed for fast protons into a Au〈1 0 0〉 string.