Kinetic study of quantum two-stream instability by Wigner approach

Classical plasma are typically of low density and/or high temperature. Two of its basic properties are Landau damping and two-stream instabilities. When increasing the plasma density, quantum effects appear and beam-plasma interactions show behavior different from the classical cases. We revisit Landau damping and two-stream instabilities under conditions when quantum hydrodynamic and quantum kinetic theory can be applied, the latter accounting for wave-particle interactions. We find that the instability growth rate behaves as pure two-stream instability without Landau damping when the countering stream velocity exceeds a certain threshold, which differs from the classical case.

Figure 1

Growth rates of different conditions at countering drift $u_0=1.15v_{\mathrm{F}}$, with parameters $T_{\mathrm{e}}=0$ and $n_{\mathrm{p}}=n_{\mathrm{b}}={10}^{24}{\mathrm{cm}}^{-3}$.

Figure 2

Real dispersion relation of longitudinal oscillations of a degenerate electron gas by QKT with $T_{\mathrm{e}}=0$ and $n_{\mathrm{e}}={10}^{24}{\mathrm{cm}}^{-3}$. Here $a$ is the optical mode and $b$ is the acoustic mode. $k_c$ and $k_d$ represent, respectively, damping turning points of two modes. ${\omega }^{\pm }=\hslash k/2m(2k_F\pm k)$ is the kinetic resonance frequency for particles on Fermi sphere [20].

Figure 3

The real part of longitudinal dispersion relation as the density changes shown by the solid lines (blue). The dashed lines represent the theoretical approach of optical mode ${\omega }_{\mathrm{LW}}\left(k\right)$ (red).

Figure 4

(a) Various values of quantum undamping region limit $k_{\mathrm{c}}$ of optical mode with $T_{\mathrm{e}}=0$ as the density changes using QKT and QHD. Here, $a_{\mathrm{B}}=52.9\mathrm{pm}$ is the hydrogen Bohr radius. (b) The ratio of plasmon energy and Fermi energy of degenerate system decreases as the density increases. In the density range of WDM, ${10}^{21}\sim {10}^{27}{\mathrm{cm}}^{-3}$ we can find the ratio is between 3.2 and 0.32. And in the density range of ICF, ${10}^{24}\sim {10}^{26}{\mathrm{cm}}^{-3}$, the ratio is between 1 and 0.45.

Figure 5

The solid lines (blue) represent the real part of dispersion relation as the temperature $\theta =k_{\mathrm{B}}T/{\epsilon }_{\mathrm{F}}$ changes. The dashed line (gray) is classical Langmuir waves $\omega ={({\omega }_{\mathrm{p}}^2+\langle v_{\mathrm{th}}^2\rangle k^2)}^{1/2}$. The dotted line (red) is classical EAWs $\omega =1.31k{v}_{\mathrm{th}}$ [38].

Figure 6

The undamping region limit of optical mode ($n={10}^{24}{\mathrm{cm}}^{-3}$) at finite temperature according to the approximate resonance condition is shown by the dashed line. The solid line represents the theoretical approach of imaginary part of dielectric function.

Figure 7

The same as Fig. 2, but for countering-stream case ($T_{\mathrm{e}}=0,n_{\mathrm{p}}=n_{\mathrm{b}}={10}^{24}{\mathrm{cm}}^{-3}$). Solid lines (blue) represent the numerical solution of QKT, and dotted lines represent the real (red) and imaginary (gray) solution of QHD. The enclosed dashed curves (light blue) form a new dissipative undamping region.

Figure 8

the value of countering velocity threshold with the change of electron density.

Figure 9

The value of countering stream velocity threshold with the change of temperature ($n={10}^{24}{\mathrm{cm}}^{-3}$).