Cross Section for Energy Transfer between Two Moving Particles

Abstract

The classical cross section ${\ensuremath{\sigma}}_{\ensuremath{\Delta}E}$, for producing a specified energy transfer $\ensuremath{\Delta}E$ in the collision of two particles 1,2 having arbitrary masses and velocities ${\mathbf{v}}_{1}$,${\mathbf{v}}_{2}$ in the laboratory system, is derived. The effective average (for fixed speeds ${v}_{1}$,${v}_{2}$) of ${\ensuremath{\sigma}}_{\ensuremath{\Delta}E}$ over all directions of the particle velocities ${\mathbf{v}}_{1}$ and/or ${\mathbf{v}}_{2}$ is then computed. These results are required in the classical calculations of atomic-collision cross sections via the procedures recently proposed by Gryzinski. The method will yield the average of any function $F(v,V,cos\ensuremath{\theta})$ over all directions of the particle velocities, where v=${\mathbf{v}}_{1}$-${\mathbf{v}}_{2}$, V is the velocity of the center of mass, and $\ensuremath{\theta}$ is the angle between v and V.