Parallel propagating electromagnetic waves in magnetized quantum electron plasmas with finite temperature

We studied parallel propagating electromagnetic waves in a magnetized quantum electron plasma of finite temperature, as an extension of our previous study on a zero temperature plasma. We obtained simple analytic dispersion relations in the long wavelength limit that included the thermal effect as correction terms to the zero temperature results. As in the zero temperature case, the lower branch of the R wave showed significant damping and became ill-defined at short wavelengths. Quantum effects seemed to give qualitative changes, such as the appearance of anomalous dispersion regions, to the classical dispersion relations when $v_F/v_{th}\le 0.2$ for a set of exemplary parameters of $v_F=0.1c$ and ${\omega }_{ce}/{\omega }_{pe}=0.05$ was used. We also noted that introduction of the Planck constant in the quantum Vlasov equation changed the shape of the anomalous dispersion region qualitatively, by forming a normal dispersion region in the middle of the original single broad anomalous dispersion region.

Figure 1

The dispersion relations for the case of $v_F=0.1c,v_{th}=0.07c$, and ${\omega }_{ce}/{\omega }_{pe}=0.05$. (a) The upper branch of the R wave and the L wave. (b) The lower branch of the R wave. The dotted line in (b) corresponds to the large values of ${\epsilon }_i(>5)$, implying severe damping.

Figure 2

Temperature effect on the dispersion relation for the lower R wave with $v_F=0.1c$ and ${\omega }_{ce}/{\omega }_{pe}=0.05$. No anomalous dispersion is seen when $v_{th}=0.02c$ (red, dashed) whereas the cases of $v_{th}=0$ (black, solid) and $v_{th}=0.01c$ (green, dot-dashed) exhibit anomalous dispersion.

Figure 3

Change in the anomalous dispersion region according to the values of $\hslash$ for $v_F=0.1c,v_{th}=0$, and ${\omega }_{ce}/{\omega }_{pe}=0.05$. The $\hslash$ values in the figure are normalized by the true Planck constant.