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 $\alpha \gg \sqrt{Z{m}_{\mathrm{e}}/m_{\mathrm{i}}}$ where $m_{\mathrm{e}}$ and $m_{\mathrm{i}}$ 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 $\varphi$ is set by the electron temperature $T_{\mathrm{e}}$, $e\varphi \sim T_{\mathrm{e}}$, where $e$ is the proton charge. An asymptotic scale separation between the Debye length, ${\lambda }_{\mathrm{D}}=\sqrt{{\epsilon }_0{T}_{\text{e}}/e^2{n}_{\text{e}}}$, the ion sound gyroradius ${\rho }_{\mathrm{s}}=\sqrt{m_{\mathrm{i}}(Z{T}_{\mathrm{e}}+T_{\mathrm{i}})}/(ZeB)$, and the size of the collisional region $d_{\mathrm{c}}=\alpha {\lambda }_{\mathrm{mfp}}$ is assumed, ${\lambda }_{\mathrm{D}}\ll {\rho }_{\mathrm{s}}\ll d_{\mathrm{c}}$. Here ${\epsilon }_0$ is the permittivity of free space, $n_{\mathrm{e}}$ is the electron density, $T_{\mathrm{i}}$ is the ion temperature, $B=|\vec{B}|$ and ${\lambda }_{\mathrm{mfp}}$ 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 ${\rho }_{\mathrm{s}}\ll x\ll d_{\mathrm{c}}$. A self-consistent solution of $\varphi (x)$ 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 $\varphi (x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $\tau =T_{\mathrm{i}}/(Z{T}_{\mathrm{e}})\gg 1$, 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.

30 pages, 7 figures

References in corpus (2)

• Gyrokinetic treatment of a grazing angle magnetic field
• Reply to Comment on "Kinetic Theory of the Presheath and the Bohm Criterion"