Anode Sheath and Double Layer Solutions with Ionization

When an electrode in a plasma is biased more positive than the plasma potential it attracts electrons and repels ions forming a region of negative space charge (electron sheath). Ballistic electrons moving towards this anode gain energy equal to the difference in electrostatic potential energy, $\Delta \phi=\phi(x) -\phi_{plasma}$, with a maximum of $\phi_{anode}-\phi_{plasma}$. When $\phi_{anode}$ is large enough, electrons can gain enough energy to ionize neutral atoms through electron impact ionization. This leads to a layer of increased ion density near the anode, which can exceed the local electron density at large enough anode biases forming a double layer. We model the sheath potential profile using Poisson's equation with a fluid model for the electron density in the case without ionization and formulate an integral equation for the case with ionization where the ion density depends on an integral from $\phi(x)$ to $\phi_{anode}$. An analytic form of the sheath electric field is obtained for the case without ionization and we demonstrate that it asymptotically agrees with the Child-Langmuir solution. We numerically obtain double layer solutions when including ionization and show that the potential profile expands beyond that of the Child-Langmuir solution.