Approximate analytical expression for knock-on tail due to large-angle elastic scattering

We have derived approximate analytical expressions for the knock-on tail, a high-energy component in the ion velocity distribution function, formed by nuclear elastic scattering (NES) and large-angle Coulomb scattering (CS). We assume a steady-state, uniform, isotropic plasma and a monoenergetic source for fast ions, which causes a knock-on tail in the bulk ion velocity distribution function. We use the cold plasma approximation to derive approximate analytical expressions for the knock-on source of fast ions, a source term in the Fokker–Planck equation generated by nuclear elastic scattering and large-angle Coulomb scattering. We discuss the dependence of the magnitude of the knock-on tail on the plasma density and temperature. We compare the derived expressions and numerical results for the knock-on source and knock-on tail. For the knock-on source, the difference between the approximate expression and the numerical calculation result increases with the ion temperature because of the effect of ion thermal motion in two-body scattering, especially near the maximum possible energy. For the knock-on tail, the difference between the approximate and numerical solutions increases with decreasing electron temperature because of the effect of the energy diffusion term in the Fokker–Planck collision term, which is not considered when deriving the approximate analytical solution. The expressions derived are beneficial for examining the large-angle scattering effect in burning plasmas qualitatively and serve as a convenient formula for plasma diagnostics.