Large gyro-orbit model of ion velocity distribution in plasma near a wall in a grazing-angle magnetic field

Abstract: A model is presented for the ion distribution function in a plasma at a solid target with a magnetic field $\vec{B}$ inclined at a small angle, $ \alpha \ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $\alpha\gg\sqrt{Zm_{\rm e}/m_{\rm i}} $ where $m_{\rm e}$ and $m_{\rm i}$ are the electron and ion mass respectively, and $Z$ is the charge state of the ion. An electric field $\vec{E}$ is present to repel electrons, and so the characteristic size of the electrostatic potential $\phi$ is set by the electron temperature $T_{\rm e}$, $e\phi \sim T_{\rm e}$, where $e$ is the proton charge. An asymptotic scale separation between the Debye length, $\lambda_{\rm D}=\sqrt{\epsilon_0 T_{\text{e}}/e^2 n_{\text{e}}}$, the ion sound gyroradius $\rho_{\rm s}=\sqrt{ m_{\rm i}(ZT_{\rm e}+T_{\rm i})}/(ZeB)$, and the size of the collisional region $d_{\rm c} = \alpha \lambda_{\rm mfp}$ is assumed, $\lambda_{\rm D} \ll \rho_{\rm s} \ll d_{\rm c}$. Here $\epsilon_0$ is the permittivity of free space, $n_{\rm e}$ is the electron density, $T_{\rm i}$ is the ion temperature, $B= |\vec{B}|$ and $\lambda_{\rm mfp}$ is the collisional mean free path of an ion. The form of the ion distribution function is assumed at distances $x$ from the wall such that $\rho_{\rm s} \ll x \ll d_{\rm c}$. A self-consistent solution of $\phi (x)$ is required to solve for the ion trajectories and for the ion distribution function at the target. The model presented here allows to bypass the numerical solution of $\phi (x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $\tau=T_{\rm i}/(ZT_{\rm e})\gg 1$, and ignores the electric force on the ion trajectory until close to the target. For $\tau \gtrsim 1$, the model provides a fast approximation to energy-angle distributions of ions at the target. These can be used to make sputtering predictions.