Maxwell–Boltzmann and Druyvesteyn Distribution Functions Expressing the Particle Velocity and the Energy in Sheath Plasmas

Abstract

The energy distribution of particles in a gaseous system is well understood through the implementation of a statistical tool, namely, the Maxwell–Boltzmann distribution function in the velocity–space coordinate system. The Maxwell–Boltzmann distribution function is utilized to investigate the velocity distribution of plasma particles like electrons, assuming that their collision frequency does not depend on the velocity. However, there is a swift transition in converting the Maxwell–Boltzmann distribution function to the Druyvesteyn distribution function for the case where a collision frequency is directly proportional to the velocity. Our aim is to incorporate the frequency components to investigate the Maxwell–Boltzmann and Druyvesteyn distribution functions. Employing the equation of motion, we observe that the collisional electron velocity is equal to the equilibrium electron velocity ∼eE/m_eω multiplied by the collisional frequency over the external source frequency β = ν/ω corresponding to the externally applied electric field. We investigate the difference in the Druyvesteyn distribution function between sheath and pre-sheath regions, when a stream of electrons is traversing or effusing through the part of a pre-sheath region corresponding to the dimension of the order of mean free path. Velocity and corresponding energy distribution functions are compared for non-effusion and effusion cases in the collisional and non-collisional regimes. The Maxwell–Boltzmann and Druyvesteyn velocity and energy distributions are competitive when the collisional frequency is twice the frequency of the applied electric field.