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 $\boldsymbol {B}$ inclined at a small angle, $α ≪ 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $α ≫ \sqrt {Zm_e/m_i}$, where $m_e$ and $m_i$ are the electron and ion mass, respectively, and $Z$ is the charge state of the ion. An electric field $\boldsymbol {E}$ is present to repel electrons, and so the characteristic size of the electrostatic potential $φ$ is set by the electron temperature $T_e$, $eφ ∼ T_e$, where $e$ is the proton charge. An asymptotic scale separation between the Debye length $λ_D = \sqrt {ɛ_0 T_{{e}} / e^2 n_{{e}} }$, the ion sound gyro-radius $ρ_s = \sqrt {m_i ( ZT_e + T_i ) } / (ZeB)$ and the size of the collisional region $d_c = α λ _{\textrm {mfp}}$ is assumed, $λ_D ≪ ρ_s ≪ d_c$. Here $ɛ _0$ is the permittivity of free space, $n_e$ is the electron density, $T_i$ is the ion temperature, $B= |\boldsymbol {B}|$ and $λ _{\textrm {mfp}}$ 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 $ρ_s ≪ x ≪ d_c$, that is, collisions are not treated. A self-consistent solution of the electrostatic potential for $x ∼ ρ_s$ is required to solve for the quasi-periodic ion trajectories and for the ion distribution function at the target. The large gyro-orbit 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 $\tau =T_i/(ZT_e)≫ 1$, and ignores the electric force on the quasi-periodic ion trajectory until close to the target. For $\tau \gtrsim 1$, the model provides an extremely fast approximation to energy-angle distributions of ions at the target. These can be used to make sputtering predictions.