The incomplete plasma dispersion function: Properties and application to waves and collisions near plasma boundaries

Abstract

Summary form only given. The incomplete plasma dispersion function <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1, 2</sup> is a generalization of the plasma dispersion function in which the defining integral spans a semi-infinite, rather than infinite, domain. Like the plasma dispersion function, it arises in the linear dielectric function and corresponding wave dispersion relations. However, the plasma dispersion function describes Maxwellian distributions, whereas the incomplete plasma dispersion function describes non-Maxwellian distributions so long as they can be approximated as Maxwellian in finite, or semi-infinite, intervals of velocity phase-space. Each interval may have different characteristic densities, flow speeds, and temperatures associated with them. It is particularly useful in the presence of potential barriers, such as sheaths near material walls, probes, or double layers, which create a trapped-passing boundary for electrons. In this poster, several properties of the incomplete plasma dispersion function that are useful for applying it to the linear dielectric response and wave dispersion relations in piecewise Maxwellian plasmas are provided. The depleted Maxwellian, as found near a conventional sheath, is used as an example to demonstrate the utility of using this function to compute modifications to common wave dispersion relations (ion-acoustic and Langmuir waves). It is also used to show that modifications to the dielectric function in the depleted region of velocity-space can significantly affect the local (in velocity-space) Coulomb collision rate. Here, we compare predictions of the Landau collision operator, which does not account for the plasma dielectric response, with predictions of the Lenard-Balescu equation, which does. Near equilibrium, these solutions converge.