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

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, $ α\ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $α\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 $ϕ$ is set by the electron temperature $T_{\rm e}$, $eϕ\sim T_{\rm e}$, where $e$ is the proton charge. An asymptotic scale separation between the Debye length, $λ_{\rm D}=\sqrt{ε_0 T_{\text{e}}/e^2 n_{\text{e}}}$, the ion sound gyroradius $ρ_{\rm s}=\sqrt{ m_{\rm i}(ZT_{\rm e}+T_{\rm i})}/(ZeB)$, and the size of the collisional region $d_{\rm c} = αλ_{\rm mfp}$ is assumed, $λ_{\rm D} \ll ρ_{\rm s} \ll d_{\rm c}$. Here $ε_0$ is the permittivity of free space, $n_{\rm e}$ is the electron density, $T_{\rm i}$ is the ion temperature, $B= |\vec{B}|$ and $λ_{\rm mfp}$ 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 $ρ_{\rm s} \ll x \ll d_{\rm c}$. A self-consistent solution of $ϕ(x)$ 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 $ϕ(x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $τ=T_{\rm i}/(ZT_{\rm e})\gg 1$, and ignores the electric force on the ion trajectory until close to the target. For $τ\gtrsim 1$, the model provides a fast approximation to energy-angle distributions of ions at the target. These can be used to make sputtering predictions.