Space Dispersive Properties of Plasma

The "space dispersion," i.e., the occurrence of the term k in the dielectric constant $\epsilon (\omega , \mathrm{k})$ can be attributed either to the Doppler effect or to the magnitude of the term $\mathrm{ak}$ that may appear in the formulation of the problem. ($a$ is a characteristic distance such as the Debye length.) Using an approach based on the Doppler effect, the macroscopic parameters of a plasma have been represented in the form of four-dimensional tensors of the fourth order (similar to those introduced by Mandelstam and Tamm). The phenomenological description of plasma has also been formulated in a three-dimensional space by means of two macroscopic parameters: the electric susceptibility ${\chi }_e$ and the "proper magnetic susceptibility" $\frac{{\chi }_{\mu }}{\mu }$. Expressions for these parameters have been given for the general case of a plasma having an electron velocity distribution $f\left(\mathrm{v}\right)d\mathrm{v}$ and for a few typical specific cases. Both parameters are functions of the frequency and of the wave vector. This formulation brings into evidence the fact that a plasma is a magnetically polarizable medium and the term $\frac{{\chi }_{\mu }}{\mu }$ vanishes only if the electron velocity distribution is isotropic. In the current literature on the subject the existence of the term $\frac{{\chi }_{\mu }}{\mu }$ has been taken into account, since, by using a "modified representation" of the dielectric constant, the magnetic properties of plasma have not been brought into evidence. In the "modified representation" the dielectric constant ${\epsilon }_M$ is defined by the relationship $\mathrm{k}\times \mathrm{B}=-\left(\frac{\omega }{c}\right){\epsilon }_M\mathrm{E}$, whereas in the conventional representation the same relationship has the form $\mathrm{k}\times \mathrm{B}=-\left(\frac{\omega }{c}\right)\epsilon \mathrm{E}+4\mathrm{}\pi \left(\frac{{\chi }_{\mu }}{\mu }\right)\mathrm{k}\times \mathrm{B}$ (where $\epsilon =1+4\mathrm{}\pi {\chi }_e$). A general formalism has been developed for deriving the electric and magnetic plasma parameters directly from the Boltzmann-Vlasov equations.