First-principles equation of state calculations of warm dense nitrogen

Using path integral Monte Carlo (PIMC) and density functional molecular dynamics (DFT-MD) simulation methods, we compute a coherent equation of state (EOS) of nitrogen that spans the liquid, warm dense matter (WDM), and plasma regimes. Simulations cover a wide range of density-temperature space, $1.5–13.9\mathrm{g}{\mathrm{cm}}^{-3}$ and ${10}^3–{10}^9$ K. In the molecular dissociation regime, we extend the pressure-temperature phase diagram beyond previous studies, providing dissociation and Hugoniot curves in good agreement with experiments and previous DFT-MD work. Analysis of pair-correlation functions and the electronic density of states in the WDM regime reveals an evolving plasma structure and ionization process that is driven by temperature and pressure. Our Hugoniot curves display a sharp change in slope in the dissociation regime and feature two compression maxima as the K and L shells are ionized in the WDM regime, which have some significant differences from the predictions of plasma models.

Figure 1

Pressure-temperature phase diagram of nitrogen. The lower panel displays solid phases, molecular, polymeric, and atomic fluid phases, and the plasma regime. Phases well characterized by experiments are outlined with solid black lines, while others are outlined with a dashed line. Circles represent a subset of our DFT-MD isochore data used to compute the Hugoniot (thick, short-dashed curve) and dissociation curves. The latter changes from a dashed to solid curve to indicate the change to a first order liquid-liquid transition region. The upper panel is a magnified view of the molecular dissociation region, showing a larger subset of our DFT-MD calculations. The thick and thin dashed curves are our predicted Hugoniot curves for two different initial densities of 0.808 and 1.035 g ${\mathrm{cm}}^{-3}$, respectively. Here, we also compare our dissociation curve with previous DFT-MD simulations by Boates et al. [9] (blue line). The green shaded area marks the region from the onset of dissociation, where the isochores begin to show that ${(\partial P/\partial T)}_V<0$, to the point at which pressure returns to its value before the onset of dissociation.

Figure 2

Temperature-pressure isochores computed with DFT-MD (circles) and PIMC (squares) at densities of 2.5, 3.7, 7.8, and 13.9 g ${\mathrm{cm}}^{-3}$. The blue dash-dotted line shows the Hugoniot curve for an initial density of ${\rho }_0=1.035$ g ${\mathrm{cm}}^{-3}$.

Figure 3

Isochores computed with PIMC (squares), DFT-MD (circles), and the Debye-Hückel model (dashes) at four densities. The high-temperature relativistic correction is shown as a dotted line. To improve visibility on a log scale, the energies of the four isochores have been shifted by the ${\mathrm{N}}_2$ molecule energy, $-54.614969$ Ha/atom, and multiplied by factors of 10, 100, 1000, and 10 000 as indicated in the labels. The original energies are given in the Supplemental Material [83].

Figure 4

Differences in PIMC and DFT-MD energies and pressures. The top panel shows energy differences, while the bottom panel shows the absolute relative error of pressure in percent. One-$\sigma$ errors in the differences are shown in black.

Figure 5

Nuclear pair-correlation functions for nitrogen from PIMC over a wide range of temperatures and densities.

Figure 6

Comparison of PIMC and DFT nuclear pair-correlation functions for nitrogen at a temperature of $1\times {10}^6$ K and a density of 13.946 $\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 7

$N\left(r\right)$ function representing the number of electrons contained in a sphere of radius $r$ around a nitrogen nucleus. PIMC data at four temperatures is compared with the analytic $1s$ core state.

Figure 8

The electron-electron pair-correlation functions for electrons with opposite spins computed with PIMC.

Figure 9

The electron-electron pair-correlation functions for electrons with parallel spins computed with PIMC.

Figure 10

Temperature dependence of the total (all) and occupied (occ) electronic DOS of dense, fluid nitrogen at densities of 2.53 and 13.95 $\mathrm{g}{\mathrm{cm}}^{-3}$. Each DOS curve has had the relevant Fermi energy for each temperature subtracted from it.

Figure 11

Shock Hugoniot curves for different initial densities ranging from 0.75- to 2.5-fold the density of solid ${\mathrm{N}}_2$, 1.035 $\mathrm{g}{\mathrm{cm}}^{-3}$, at ambient pressure.

Figure 12

Hugoniot curves as a function of the shock compression ratio for different initial densities as plotted in Fig. 11. The 1-fold curve is shown with (dashed line) and without (solid line) the relativistic correction. The dark shaded marks the temperature range of highest compression.

Figure 13

Comparison of our combined PIMC and DFT-MD Hugoniot curve with predictions of ACTEX plasma model calculations by Ross and Rogers [8]. The dashed line portion of the plasma model curve indicates where the ACTEX results were interpolated to match experimental data below 100 GPa. The initial density was ${\rho }_0=0.8076\mathrm{g}{\mathrm{cm}}^{-3}$ ($V_0=28.80 {\AA }^3$/atom).

Figure 14

Comparison of the liquid DFT-MD Hugoniot with the experiments of Nellis et al. [7] and Zubarev et al. [40] and the theory of Ross et al. [49] (variational fluid theory) and Kress et al. [58] (DFT-MD). The blue shaded region indicates the region of dissociation with ${(\partial P/\partial T)}_v<0$ in the phase diagram of Fig. 1. Our Hugoniot passes through this region, but there is no evidence of cooling along the principal Hugoniot curve. The green dashed lines show isotherms from our DFT-MD simulations.

Figure 15

Comparison of the shock Hugoniot curves for different materials. The initial volume $V_0$ has been chosen such that the density of the electrons is the same for all materials ($V/N_e=3.586 {\AA }^3$). The various maxima in compression corresponds to excitations of electrons in the first and second electron shells.

Figure 16

Pressure vs temperature is shown for isochores of different materials. The pressure of a fully-ionized, noninteracting plasma $P_0$ has been removed in order to compare the excess pressure due to interactions. The densities have been chosen such that electronic density is the same for all materials ($V/N_e=0.8966$ ${\AA }^3$). This electronic density corresponds to the high-temperature limit of fourfold compression of the shock Hugoniot curves in Fig. 15. The Debye model has been included for helium and silicon.

Figure 17

Internal energy vs temperature is shown for the isochores in Fig. 16. The energy contribution from a fully-ionized, noninteracting plasma $E_0$ has been removed in order to compare only the interaction effects.