Spherical-harmonics decomposition of the Boltzmann equation for charged-particle swarms in the presence of both electric and magnetic fields

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Abstract
The Boltzmann equation for reacting charged-particle swarms in neutral gases in the presence of both electric and magnetic fields is decomposed into a hierarchy of kinetic equations by expanding the velocity dependence of the phase-space distribution function in terms of spherical harmonics. No limit is set on the number of spherical harmonics and no approximation is made concerning the mass of the charged particles related to that of the neutrals species. The space-time dependence is treated by making the hydrodynamic assumption which is taken to second order in density gradients. Spherical tensors are used throughout. The resulting heirarchy of equations has universal validity and is amenable to a range of numerical solutions. The structure of these equations is discussed and the inadequacies of a Legendre-polynomial expansion are pointed out. The special configurations of the magnetic field parallel and perpendicular to the electric field are discussed in detail.