Spin and Exchange Corrections to Plasma Dispersion Relations

Abstract

Longitudinal and transverse dispersion relations including spin and exchange contributions are derived for electromagnetic waves in a fully ionized plasma. The nonrelativistic Pauli description of the electrons is used, and the electron density matrix is assumed to obey the quantum analog of the Vlasov equation. The electromagnetic fields satisfy Maxwell's equations with self-consistent source terms.The dispersion relation for longitudinal modes is unaffected by spin, and the exchange correction derived agrees with that obtained by previous investigators. The transverse dispersion relation is evaluated in the long-wavelength limit for the completely degenerate plasma at 0\ifmmode^\circ\else\textdegree\fi{}K. If the transverse frequency be written $\ensuremath{\omega}={\ensuremath{\omega}}_{0}+{\ensuremath{\omega}}_{\mathrm{sp}}+{\ensuremath{\omega}}_{\mathrm{ex}}$, where ${\ensuremath{\omega}}_{0}$ is the frequency with spin and exchange effects omitted, our results for the corrections are ${\ensuremath{\omega}}_{\mathrm{sp}}=(\frac{{{\ensuremath{\omega}}_{p}}^{2}}{8{{\ensuremath{\omega}}_{0}}^{3}})(\frac{{\ensuremath{\hbar}}^{2}}{{m}^{2}})$, ${\ensuremath{\omega}}_{\mathrm{ex}}=\ensuremath{-}\frac{3{{\ensuremath{\omega}}_{p}}^{4}{q}^{2}}{80{{\ensuremath{\omega}}_{0}}^{3}{{K}_{F}}^{2}}$, where q is the wave vector of a disturbance and $\ensuremath{\hbar}{K}_{F}$ is the momentum limit of the Fermi distribution.