Electron collision term due to reflections in electron-electron collisions in a magnetized plasma

For a electron-electron (e-e) collision with impact parameter p between the electron thermal gyro-radius ρ_th and the Debye length λ_D, ρ_th<p<λ_D, when the initial relative parallel kinetic energy is smaller than the Coulomb repulsive potential e²/(4πε₀p), reflection will occur with interchange of the parallel velocities of the two electrons after the collision. The Fokker-Planck approach is employed to derive the electron collision term C_R due to the reflection. The electron parallel velocity friction and diffusion coefficients <Δv_z>_R, <Δv_zΔv_z>_R are calculated which depend insensitively on the electron perpendicular velocity v_perp. For Maxwellian background electrons, when v_z<<v_th, <Δv_z>_R and <Δv_zΔv_z>_R are found to be, respectively, three times the corresponding results of the no magnetic field case with v<<v_th, where v, v_z, and v_th are the electron speed, parallel velocity, and the background electron thermal velocity, respectively. When v_z>>v_th, <Δv_z>_R and <Δv_zΔv_z>_R are exponentially small as the number of e-e collisions with reflections is exponentially small in this case. Through studying the time evolution of the quantity H under the effect of C_R, it is surprising to find that C_R finally makes the electron distribution f_e(v) relax to the form f_e(v)=F_e(v_perp)G_e(v_z).