Ion potential in warm dense matter: Wake effects due to streaming degenerate electrons

The effective dynamically screened potential of a classical ion in a stationary flowing quantum plasma at finite temperature is investigated. This is a key quantity for thermodynamics and transport of dense plasmas in the warm-dense-matter regime. This potential has been studied before within hydrodynamic approaches or based on the zero temperature Lindhard dielectric function. Here we extend the kinetic analysis by including the effects of finite temperature and of collisions based on the Mermin dielectric function. The resulting ion potential exhibits an oscillatory structure with attractive minima (wakes) and, thus, strongly deviates from the static Yukawa potential of equilibrium plasmas. This potential is analyzed in detail for high-density plasmas with values of the Brueckner parameter in the range $0.1\le r_s\le 1$ for a broad range of plasma temperature and electron streaming velocity. It is shown that wake effects become weaker with increasing temperature of the electrons. Finally, we obtain the minimal electron streaming velocity for which attraction between ions occurs. This velocity turns out to be less than the electron Fermi velocity. Our results allow for reliable predictions of the strength of wake effects in nonequilibrium quantum plasmas with fast streaming electrons showing that these effects are crucial for transport under warm-dense-matter conditions, in particular for laser-matter interaction, electron-ion temperature equilibration, and stopping power.

Figure 1

Contours of constant electron-electron collision frequency, $\nu$, Eq. (4), in units of ${\tilde{\omega }}_p$.

Figure 2

Temperature dependence of static properties. (a) Screening parameter $k_S$ (in units of $a_B^{-1}$) in the potential (9) for various densities given in the figure. (b) Quantum kinetic energy ${\langle K\rangle }_Q$ of electrons in units of the Fermi energy compared to the streaming kinetic energy ${\langle K\rangle }_S$ for various streaming velocities. Note that ${\langle K\rangle }_Q/E_F$ is independent of $r_s$, cf. Eq. (11).

Figure 3

Potential ${\left|\mathbf{r}\right|}^3\Phi \left(\mathbf{r}\right)$ illustrating the behavior of Friedel oscillations in the presence of electrons streaming in positive $z$ direction with $u_e=0.5v_F$ and $\theta =0.01$. (a) $r_s=4.52$; (b) $r_s=1$. The result agrees well with Fig. 7 of Ref. [28]. Note the strongly increased amplitude of the potential in the lower figure (different color scale).

Figure 4

Effective dynamically screened ion potential $\Phi \left(z\right)$ and its variation with the streaming velocity $M$ (a), density parameter $r_s$ (b), and temperature $\theta$ (c). The potential is shown along the streaming direction $z$ at the ion position $(r=0)$.

Figure 5

Absolute depth (a) and location (b) of the first minimum of $\Phi$ behind the ion versus streaming parameter $M$ for $\theta =0.01$ and different density parameters $r_s$ shown in the figure.

Figure 6

Absolute value (a) and location (b) of the first minimum of $\Phi$ behind the ion as a function of temperature for $r_s=0.5$ and fixed $M$ values indicated in the figure.

Figure 7

Temperature effect on the dynamically screened ion potential $\Phi \left(z\right)$ in streaming direction at the ion position $(r=0)$. (a) At $M=1.0$; (b) at $M=0.6$. Temperatures are indicated in the figure.

Figure 8

Effect of collisions on the dynamically screened ion potential $\Phi \left(z\right)$ for $r_s=0.5$, $\theta =0.01$ and (a) $M=0.4$, (b) $M=0.6$, and (c) $M=1.0$. The potential is shown in streaming direction at the ion position $(r=0)$.

Figure 9

Effect of collisions on the depth of the first minimum of $\Phi$ behind the ion as a function of $M$ for $r_s=0.5$ and $\theta =0.01$. Note the nonmonotonic effect of collisions leading either to a reduction $(M>0.65)$ or an enhancement $(M<0.65)$ of the potential minimum compared to the collisionless case (full black curve).

Figure 10

Combined effect of finite temperature and collisions on the dynamically screened ion potential $\Phi \left(z\right)$. The result of the Mermin dielectric function for a plasma at $\theta =1$ and $r_s=0.5$ is shown for the example $M=1$ and compared to the result using the RPA dielectric function in the zero-temperature limit (full black line).

Figure 11

Dynamically screened ion potential $\Phi (r,z)$ for $r_s=0.5$ and $M=1$ and three different temperatures: (a) $T=2.32\times {10}^{7}K(\theta =10)$, (b) $T=2.32\times {10}^{6}K(\theta =1)$, and (c) $T=2.32\times {10}^{5}K(\theta =0.1)$. The ion is located at $\{r,z\}=\{0,0\}$. The electrons stream in $z$ direction from left to right.

Figure 12

Dynamically screened ion potential $\Phi (r,z)$ for $\theta =0.1$ and $M=1$ and three different densities: (a) $r_s=0.1$, (b) $r_s=0.3$, and (c) $r_s=1.0$. The ion is located at $\{r,z\}=\{0,0\}$. The electrons stream in the $z$-direction from left to right.

Figure 13

Dynamically screened ion potential $\Phi (r,z)$ for $\theta =0.1$ and $r_s=0.3$ for three different streaming velocities: (a) $M=0.4$, (b) $M=0.6$, and (c) $M=1.0$. The ion is located at $\{r,z\}=\{0,0\}$. The electrons stream in the $z$ direction from left to right.

Figure 14

Electron density perturbation (in units of the unperturbed density $n_0$) produced by the focusing effect of the ion for $\theta =0.1$ and $r_s=0.3$ for three different streaming velocities: (a) $M=0.4$, (b) $M=0.6$, and (c) $M=1.0$. Same parameters as in Fig. 13.

Figure 15

Electron density perturbation (in units of the unperturbed density $n_0$) produced by the focusing effect of the ion for $\theta =0.1$ and $M=1$ for three different electron densities: (a) $r_s=0.1$, (b) $r_s=0.3$, and (c) $r_s=1.0$. For better comparison, in (c) the density is reduced by a factor 10. Same parameters as in Fig. 12.

Figure 16

Function $g\left(x\right)$ defined by Eq. (A5) for different values of $\theta$. Comparison of results of numerical integration and the implementation in mathematica (symbols).

Figure 17

Dynamically screened potential $\Phi \left(z\right)$ of an ultrarelativistic plasma in streaming direction. (a) Potential for different values $M=v/c$. (b) Absolute depth and location of the first minimum versus streaming parameter $M=v/c$.

Figure 18

Dynamically screened potential $\Phi (r,z)/(e/{\lambda }_D)$ of an ultrarelativistic plasma for three different streaming velocities $M=v/c$: (a) $M=0$, (b) $M=0.55$, and (c) $M=0.99$.