The electron source Re and the energy loss due to inelastic collisions Rε are defined later. If a strong DC magnetic field is present then the electron mobility can be a tensor:

where the inverse of the mobility has been used because the actual expression for the electron mobility cannot be written in a compact form. The quantity μdc is the electron mobility in the absence of a magnetic field. The electron diffusivity, energy mobility, and energy diffusivity are then calculated using:

The source coefficients in the above equations are determined by the plasma chemistry and are written using either rate or Townsend coefficients. Suppose that there are M reactions that contribute to the growth or decay of electron density and P inelastic electron-neutral collisions. In general P >> M. In the case of rate coefficients, the electron source term is given by

where xj is the mole fraction of the target species for reaction j, kj is the rate coefficient for reaction j (SI unit: m3/s), and Nn is the total neutral number density (SI unit: 1/m3). When Townsend coefficients are used, the source term becomes

where αj is the Townsend coefficient for reaction j (SI unit: m2) and Γe is the electron flux (SI unit: 1/(m2·s)). Townsend coefficients can increase the stability of the numerical scheme when the electron flux is field driven as is the case with DC discharges.

where Δεj is the energy loss from reaction j (SI unit: V). In the case of Townsend coefficients, the energy loss is given by

The electron source and inelastic energy loss are automatically computed by the multiphysics interface. The rate coefficients can be computed from cross section data by the following integral:

The space charge density, ρ is automatically computed based on the plasma chemistry specified in the model using the formula: