Large gyro-orbit model of ion velocity distribution in plasma near a wall in a grazing-angle magnetic field. (15th February 2021) - Geraldini, Alessandro
Abstract : 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, $α \ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $α \gg \sqrt {Zm_{e}/m_{i}}$, 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 $T_{e}$, $eφ \sim T_{e}$, where $e$ is the proton charge. An asymptotic scale separation between the Debye length $λ _{D} = \sqrt {ε _0 T_{{e}} / e^{2} n_{{e}} }$, the ion sound gyro-radius $ρ _{s} = \sqrt { m_{i} ( ZT_{e} + T_{i} ) } / (ZeB)$ and the size of the collisional region $d_{c} = α λ _{mfp}$ is assumed, $λ _{D} \ll ρ _{s} \ll d_{c}$ . Here $ε _0$ is the permittivity of free space, $n_{e}$ is the electron density, $T_{i}$ is the ion temperature, $B= |\boldsymbol {B}|$ and $λ _{mfp}$ 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 $ρ _{s} \ll x \ll d_{c}$, that is, collisions are not treated. A self-consistent solution of the electrostatic potential for $x \sim ρ _{s}$ 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 $τ =T_{i}/(ZT_{e})\gg 1$, and ignores the electric force on the quasi-periodic ion trajectory until close to the target. For $τ \gtrsim 1$, the model provides an extremely fast approximation to energy–angle distributions of ions at the target. These can be used to make sputtering predictions.