Oscillations of a three-component assembly in the presence of a magnetic field using the generalized BGK collision model

Abstract

The BGK collision model for one-component assembly of neutral particles has been extended to two-component assembly of charged particles by Gross and Krook (1956) and later on modified by Bhatnagar (1962). Following the lines of the latter, the model has been generalized to N-component assembly of both charged and neutral particles. This model is further applied to the study of small amplitude plasma oscillations in an assembly consisting of ions, electrons and neutral particles in the direction perpendicular to a uniform magnetic field. The dispersion relation splits up into two, one determining the transverse oscillations and the other longitudinal oscillations. In the transverse oscillations for small wave numbers k, it has been shown that apart from the Gross-gaps occurring at the multiples of gyro-frequencies of electrons and ions, if the magnetic energy density M is greater than one-third the kinetic energy density K of charged particles, and terms only up to k² are retained, five more forbidden ranges of frequencies occur. If M < ± ½ K, the number of additional gaps reduces to three. When M = 0, Oster's (1960) result is obtained as a particular case. The oscillations of neutral particles excited by collisions are strong at low frequencies, whereas for high frequencies they are mostly damped out. Exact analytical and graphical discussion of the transverse dispersion relations is given. Longitudinal propagation has been studied under very restricted circumstance numerically and it is shown that, unless the magnetic field is very high, propagation is possible for all frequencies. For a sufficiently high magnetic field, when the Alfven velocity is comparable with the velocity of light, there is one forbidden range but, for the discussion of such high velocities, one should work with the relativistic equations.