Spin Effects on the Long-Wavelength Oscillations of a Quantum Plasma in a Magnetic Field

The results of our earlier work on the quantum theory of electron-gas plasma oscillations in a magnetic field are extended here to take account of the difference in masses associated with the orbital and spin parts of the individual electronic motions in the presence of a lattice, and allowance is made for an anomalous electronic $g$ factor. Tractable expressions for the complete plasmon dispersion relation and damping constant (at arbitrary temperature and arbitrary magnetic field strength), which are obtained using a Green's-function formulation of the random-phase approximation, are reported. The low-wave-number ($p$) approximation of the dispersion relation is investigated in detail. For $p=0\mathrm{}$ the usual result obtains, $\frac{1}{{{\omega }_p}^2}=\frac{{sin}^2\theta }{{{\Omega }_o}^2}+\frac{{cos}^2\theta }{({{\Omega }_o}^2-{{\omega }_c}^2)}$, with two plasma modes; these are shifted to order $p^2$ by terms that are oscillatory in the de Haas-van Alphen (DHVA) sense in the degenerate case. Another plasma resonance in the vicinity of $2{\omega }_c$, which is active to order $p^2$, exhibits such DHVA oscillatory behavior, and the same may be said for a gap in the frequency spectrum for propogation perpendicular to the magnetic field in the interval [${\omega }_c, 2{\omega }_c$]. The relative amplitudes for the plasma modes are reported also. The terms which are oscillatory in the DHVA sense are exhibited in terms of an appropriate Fourier series, with no restriction on temperature or magnetic field strength, save that $\zeta \beta \gg 1$ (that is to say that no restriction is placed on $\hslash {\omega }_c\beta$). Moreover, the spectral composition of the DHVA oscillatory terms is thereby explicitly shown to be a sensitive function of $g\left(\frac{m}{m_o}\right)$, and this result may be useful for the experimental determination of the product of anomalous electronic $g$ factor and effective mass $m$. A considerable improvement over other recent work on this subject has been achieved through a careful and correct determination of the role of the DHVA terms as a whole in the plasma oscillation spectrum.