Microfield dynamics in dense hydrogen plasmas with high-$Z$ impurities

We use large-scale classical molecular dynamics to determine microfield properties for several dense plasma mixtures. By employing quantum statistical potentials (QSPs) to regularize the Coulomb interaction, our simulations follow motions of electrons as well as ions for times long enough to track relaxation phenomena involving both types of particles. Coulomb coupling, relative to temperature, of different pairs of species in the hot, dense matter being simulated ranges from weak to strong. We first study the effect of such coupling differences, along with composition and QSP differences, on the roles of electrons and various mixture components in determining probability distributions of instantaneous, total microfields experienced by the ions. Then, we address two important dynamical questions: (1) How is the quasistatic part of the total field to be extracted from the time-dependent simulation data? (2) Under what conditions does the commonly used approximation of ions with fixed Yukawa-like screening by free electrons accurately describe quasistatic fields? We identify a running, short-time average of the total field at each ion as its slowly evolving, quasistatic part. We consider several ways to specify the averaging interval, and note the influence of ion dynamics in this issue. When all species are weakly coupled, the quasistatic fields have probability distributions agreeing well with those we obtain from simulations of Yukawa-screened ions. However, agreement deteriorates as the coupling between high-$Z$ ions increases well beyond unity, principally because the Yukawa model tends to underestimate the true screening of close high-$Z$ pairs. Examples of this fact are given, and some consequences for the high-field portions of probability distributions are discussed.

Figure 1

Scaled probability distributions $P_{\mathrm{Kr}}\left({\beta }_{\mathrm{tot}}\right)$ and $P_H\left({\beta }_{\mathrm{tot}}\right)$ at $\mathrm{K}{\mathrm{r}}^{+36}$ and at ${\mathrm{H}}^{+}$ in plasmas. Panels (a) and (d), (b) and (e), and (c) and (f) correspond to simulation cases Kr1, Kr3, and Kr4, respectively. [CT] denotes the result when the QSP of Eq. (12) is used. The corresponding Holtsmark distribution, Eq. (15), is plotted for each case. The scaled microfield is ${\beta }_{\mathrm{tot}}=F_{\mathrm{tot}}/F_0$, and $F_0$ is the atomic unit of field strength.

Figure 2

Scaled probability distributions $P_{\mathrm{Kr}}\left({\beta }_x\right)$ and $P_H\left({\beta }_x\right)$ for the total microfield, and for its respective component species, at $\mathrm{K}{\mathrm{r}}^{+36}$ and at ${\mathrm{H}}^{+}$ ions, for simulation case Kr3. Each arrow indicates the ordering of curves corresponding to that panel's legend: top label belongs to first curve intersected, etc.

Figure 3

Scaled probability distributions $P_{\mathrm{Ar}}\left({\beta }_x\right)$ and $P_H\left({\beta }_x\right)$ for the total microfield at $\mathrm{A}{\mathrm{r}}^{+18}$ and at ${\mathrm{H}}^{+}$ ions. The label [CT] is defined as in Fig. 1, and the arrows, as in Fig. 2.

Figure 4

Scaled probability distributions $P_{\mathrm{Ar}}\left({\beta }_x\right)$ and $P_H\left({\beta }_x\right)$ for the total microfield, and for its respective component species, at $\mathrm{A}{\mathrm{r}}^{+18}$ and at ${\mathrm{H}}^{+}$ ions. Panels (a) and (d), (b) and (e), and (c) and (f) correspond to simulation cases Ar1, Ar2, and Ar3, respectively. Arrows connect the ordering of the curves and the legend.

Figure 5

Scaled probability distributions of the slow microfield for different values of the dimensionless interval parameter ${\omega }_e\tau$. Panels (a)–(c) correspond to the simulation cases C1, Ar3, and Kr3, respectively. Arrows connect the ordering of the curves and the legend.

Figure 6

Autocorrelation functions $A_{\nu }\left(t\right)$ of the total field at various ions $\nu$, for different plasma conditions. Arrows connect the ordering of the curves and the legend.

Figure 7

Scaled second moment of the slow microfield as a function of the dimensionless interval parameter ${\omega }_e\tau$, for high-$Z$ ions. Dots mark parameter values of Table 1, and the arrow connects the ordering of the curves and the legend. The dashed curve marked [h] Kr3 denotes results when ${\mathrm{H}}^{+}$ ions are given a mass equal to that of krypton.

Figure 8

Properties of scaled slow microfields at $\mathrm{K}{\mathrm{r}}^{+36}$ ions, for simulation case [h] Kr3 in a plasma of ${\mathrm{H}}^{+}$ ions whose mass equals that of krypton. Arrows connect the ordering of the curves and the legend's values of ${\omega }_e\tau$.

Figure 9

Ratios of ion-ion pair distribution functions $g_{s{s}^{\prime }}\left(r\right)$ as determined by Yukawa HNC and all-particle HNC. (a) Krypton-hydrogen plasmas with ${\xi }_{\mathrm{Kr}}=0.01, T=6.7\mathrm{keV}$ and two different ion densities; (b) argon-hydrogen plasmas with $N_{\mathrm{ion}}={10}^{24}\mathrm{c}{\mathrm{m}}^{-3}, T=2.4\mathrm{keV}$, and three different argon concentrations. Arrows connect the ordering of the curves and the legend.

Figure 10

Top row: comparisons of small sets of scaled slow-field probability distributions at $\mathrm{K}{\mathrm{r}}^{+36}$ with Yukawa MD and Holtsmark results, for (left to right) simulation cases Kr1, Kr2, Kr3. Bottom row: comparison of probability distributions at $\mathrm{A}{\mathrm{r}}^{+18}$, for (left to right) simulation cases Ar1, Ar2, and Ar3. Shown here are the slow field and the field of plasma ions only, both for all-particle MD, and the Yukawa MD result. Arrows connect the ordering of the curves and the legend.

Figure 11

Scaled probability distributions of the slow microfield for different values of the interval parameter ${\omega }_e\tau$ are compared with Yukawa MD results, in mixtures with high $\mathrm{A}{\mathrm{r}}^{+18}$ concentration, ${\xi }_{\mathrm{Ar}}=\frac{1}{3}$. Panels (a) and (b) show near-peak values for simulation cases Ar5 and Ar4, respectively. Arrows connect the ordering of the curves and the legend.

Figure 12

Scaled second moments of the slow microfield, plotted as in Fig. 7, are compared with the corresponding fixed values calculated from Yukawa MD (horizontal dashed lines). Dots mark curve intersection points, and shaded bars denote the range of plausible ${\omega }_e\tau$ values. (a) At ${\mathrm{C}}^{+6}$ impurities. (b) At $\mathrm{K}{\mathrm{r}}^{+36}$ impurities. (c) At $\mathrm{A}{\mathrm{r}}^{+18}$ impurities. Each panel's order of its curves is the same as the legend.

Figure 13

Comparisons of cumulative microfield distribution moments $Q_{\mathrm{Ar}}^{\left[\lambda \right]}\left(\beta \right), \lambda =0$ or $\lambda =2$, for Yukawa and all-particle screening models in selected argon-hydrogen mixtures: simulation case Ar4 for panels (a)–(c), and simulation case Ar1 for panels (d)–(f). Panels (b) and (e) show upper end of data in panels (a) and (d), respectively, on a much expanded scale. Arrows connect the ordering of the curves and the legend.