A multi-term spherical harmonic expansion of the Boltzmann equation for application to low-temperature collisional plasmas

Abstract

The Boltzmann equation describes the evolution of the electron and ion distributions in a plasma over time through a six-dimensional phase space. For highly collisional plasmas, scattering collisions keep the distribution function nearly isotropic in velocity space with small perturbations created by the hydrodynamic and electromagnetic forces. For these plasmas, a spherical-harmonic expansion of the velocity-space distribution function is an effective technique for solving the Boltzmann equation. This paper examines each of the terms in the Boltzmann equation in detail to derive conditions where a spherical harmonic expansion is useful. Expressions for the matrix elements are presented which represent the projection of the various operators in the Boltzmann equation onto the spherical harmonics basis set. The resulting multiple-term spherical-harmonic expansion makes no assumptions about either the direction of the electric and magnetic fields or the magnitude of the spatial gradients and is appropriate for coupling with a Maxwell equation solver for the time- and spatially-dependent electromagnetic fields. When only the first two lowest-order terms are kept, it is shown that the resulting equations are very similar in form to the continuity and force-balance fluid equations. Additional kinetic terms appear in the continuity-like equation which are related to the changes in the energy distribution due to the electric field and collisions, including Ohmic heating. Two additional kinetic terms also appear in the force-balance-like equation. The collision term accounts for momentum-transfer during scattering collisions and the other accounts for the flow of energy in velocity space and is proportional to the derivative with respect to energy of the energy density.

Theoretical and experimental aspects of non-equilibrium plasmas in different regimes: fundamentals and selected applications

“…where f 0 is known as electron energy distribution function (eedf). This model is the first-order approximation of the spherical harmonic expansion, and in some conditions, multi-term extension is considered [71][72][73], at the cost of the computational time. As an alternative, Monte Carlo approaches [74] can be used, which is equivalent to calculate the full series expansion.…”

On the relevance of superelastic collisions in argon and nitrogen discharges

“…It should be pointed out that the two-term approximation introduces large errors in the presence of strong electric field where multi-term [21,63,64] or Monte Carlo [65] approaches are recommended. However, these methods need larger computational resources than the two-term approximation, which is a good compromise between computational time and accuracy.…”

Two-term Boltzmann approximation versus Monte-Carlo simulation: effect of magnetic field

“…This problem can be solved in several ways. For example, increase the number of terms in Legendre polynomials expansion [7][8][9][10][11][12][13], or use a fundamentally different approach: stochastic (Monte-Carlo Collisions, MCC).…”