Interaction of magnetized electrons with a boundary sheath: investigation of a specular reflection model

Dennis Krüger, Ralf Peter Brinkmann

Plasma Sources Science and Technology 26, 115009 (2017)


Abstract

This publication reports analytical and numerical results concerning the interaction of gyrating electrons with a plasma boundary sheath, with focus on partially magnetized technological plasmas. It is assumed that the electron Debye length ${lambda }_{{rm{D}}}$ is much smaller than the electron gyroradius ${r}_{{rm{L}}}$, and ${r}_{{rm{L}}}$ in turn much smaller than the mean free path λ and the gradient length L of the fields. Focusing on the scale of the gyroradius, the sheath is assumed as infinitesimally thin (${lambda }_{{rm{D}}}to 0$), collisions are neglected ($lambda to infty $), the magnetic field is taken as homogeneous, and electric fields (=potential gradients) in the bulk are neglected ($Lto infty $). The interaction of an electron with the electric field of the plasma boundary sheath is represented by a specular reflection ${bf{v}}to {bf{v}}-2{bf{v}}cdot {{bf{e}}}_{z},{{bf{e}}}_{z}$ of the velocity ${bf{v}}$ at the plane z = 0 of a naturally oriented Cartesian coordinate system $(x,y,z)$. The electron trajectory is then given as sequences of helical sections, with the kinetic energy epsilon and the canonical momenta p x and p y conserved, but not the position of the axis (base point ${{bf{R}}}_{0}$), the slope (pitch angle χ), and the phase (gyrophase phiv). A 'virtual interaction' which directly maps the incoming electrons to the outgoing ones is introduced and studied in dependence of the angle γ between the field and the sheath normal ${{bf{e}}}_{z}$. The corresponding scattering operator is constructed, mathematically characterized, and given as an infinite matrix. An equivalent boundary condition for a transformed kinetic model is derived.

[DOI]

Tags: Dynamics, electron, HiPIMS