Electron Dynamics in Partially Magnetized Plasmas

Partially magnetized plasma discharges in magnetron configurations are versatile and offer a wide range of applications

in science and industry. These applications range from space propulsion systems that utilize Hall-effect thrusters to

the deposition of thin films in physical vapor deposition technology using "high power impulse magnetron sputtering" (HiPIMS).

Their magnetic field configuration is typically described in cylindrical geometry (r, Θ, z) with the magnetic field lines

in the r-z plane. In such magnetron discharges, plasma non-uniformities are observable in the form of rotating spokes moving

in the azimuthal (Θ) direction. These features break the azimuthal symmetry of the discharge and enhance the difficulty of

analytical and numerical investigations. To describe electrons in discharges of this kind, a coordinate transformation is performed

from cylindrical geometry (r, Θ, z, v_r, v_Θ, v_z) to the so-called gyro-coordinates (ψ, θ, s, v, χ, φ),

connecting the position of an electron to its guiding center. This transformation utilizes the fact that the ratio of the gyroradius r_L

to the system dimension L is much smaller than 1, yielding the small parameter η=r_L/L that governs the physical effects

in the equations of motion. One possible advantage of describing magnetized electrons in gyro-coordinates is that the equations of motion are

simplified, and the drift motion of electrons is separated from the gyro motion. In these coordinates, the equations of motion for electrons are solved.

As an example of diagnostic applications, the magnetic moment in higher-order approximations is calculated,

which serves to understand the energization process of electrons under such conditions.