The correct cutoff for Rutherford scattering cross-section

Abstract

The correct cutoff variable for the integrals with Rutherford scattering cross-section is established in this paper. The traditional cutoff variables in plasma physics, such as scattering angle θ and impact parameter b, are incorrect or just partial cutoff variables for lacking the necessary cutoff on relative speed g. The correct cutoff variable is the same variable that correctly describing the singularity of the integrals. The difference of the partial cutoff variables and the correct one is compared through their contour lines in the b − g plane. With the correct cutoff, many physical results become more simplified and structured for both rigid-sphere interaction and Coulomb interaction. These physical results include the arbitrary higher order of Fokker-Planck coefficients, transition moments, and energy transfer rates for both rigid-sphere and Coulomb interactions. All the physical results depending on velocity are expressed by a set of functions q n ( k ) ( y min , u ) = 2 π ∫ y min ∞ ∫ − 1 1 e − y + xu 2 y n − x k dxdy . A useful integral formula ∫ 0 ∞ q n ( k ) ( y min , u ) u k + 2 e − u 2 ξ du = ξ k + 3 2 2 1 + ξ k − n 2 ∑ j = 0 ⌊ k / 2 ⌋ k ! Γ n + k + 1 2 − j , y min 2 1 + ξ j ! ( k − 2 j ) ! 4 ξ j , which associates q n ( k ) ( y min , u ) with incomplete gamma functions Γ j , y min 2 1 + ξ is also proved. This integral formula is the key to show that the correct cutoff constants for both rigid-sphere and Coulomb interactions are all in the form of incomplete gamma functions of different orders. In particular, the so called Coulomb logarithm ln Λ should be replaced by the exact form—the zeroth order incomplete Gamma function.