Anisotropy of the electron component in a cylindrical magnetron discharge. I. Theory of the multiterm analysis

Abstract

A general multiterm representation of the phase space electron distribution function in terms of spherical tensors is used to solve the Boltzmann kinetic equation in crossed electric and magnetic fields. The problem is formulated for an axisymmetric cylindrical magnetron discharge with the homogeneous magnetic field being directed axially and the electric field between the coaxial cathode and anode varying in radius only. A spherical harmonic representation of the velocity distribution function in Cartesian coordinates becomes especially cumbersome in the presence of the magnetic field. In contrast, the employment of a spherical tensor representation leads to a compact hierarchy of equations that accurately take into account the spatial inhomogeneities and anisotropy of the plasma in crossed fields. To describe the spatially inhomogeneous plasma the hierarchy of the kinetic equations is formulated in terms of the total energy and the radial coordinate. Appropriate boundary conditions at the electrodes for the tensor expansion coefficients are obtained.