# 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]