Phase Diagram

Figure: The phase diagram of deuterium computed with PIMC simulations using free particle nodes is shown in the temperature-density plane. $(\times,\diamond,\bigtriangleup)$ indicate the different simulations and distinguish between different degrees of degeneracy of the electrons ($\times$ less than 10% exchanges, $\diamond$ more 10%, and $\bigtriangleup$ over 80%). The thin dashed lines indicates the boundary of the metallic regime from simulations with VDM nodes shown below.

Figure: Phase diagram of deuterium as in Fig.  but calculated with PIMC using VDM nodes.

Figure: Pressure vs. temperature from PIMC simulations of deuterium at $r_s=2$ with free particle nodes, $n_{\rm A}=1$, $n_{\rm E}=2$, and different time steps. The negative slope $\left.\frac{dP}{dT}\right\vert _{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}}$ indicates a first order phase transition.

Figure: Pressure vs. temperature as in Fig.  but calculated with VDM nodes. Within the error bars, no regions of $\left.\frac{dP}{dT}\right\vert _{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}}<0$ occur.

We used PIMC simulations with 32 protons and 32 electrons, a time step $\tau^{-1}=10^6\,$K, $n_{\rm A}=1$, and $n_{\rm E}=2$ to generate the phase diagram shown in Fig. . In the low density and low temperature regime, we find a molecular fluid. In the proton-proton pair correlation functions (see section , Fig. ), one finds a clear peak at the bond length of $1.4$. The integral under the peak is proportional to the number of molecules. This criterion works well for low densities where the peak is well separated from the remaining contributions. Alternatively, one can estimate the number of molecules as well as other compound particles by a cluster analysis, in which the individual path configurations from PIMC simulations are analyzed. As described in (Militzer et al., 1998), we consider two protons as belonging to one cluster if they are less than 1$\rm {\AA}$ apart. An electron belongs to one particular cluster if it is less than 0.75$\rm {\AA}$ away from any proton in the cluster. The two cut-off radii were chosen from the molecular and atomic ground state distribution. This analysis give reasonable estimates for the molecular and atomic fractions at low temperatures. At high temperature, it typically overestimated the number of bound states because in a collision, two particles are close but this is not a bound state. We corrected for this artifact by applying an additional criterion. A particle can only be considered as bound if the difference in action (or energy) to remove it from the cluster is positive. This method leads to the expected corrections at high temperature. The regime boundaries in Fig.  are hardly affected. Summarizing one can say that PIMC simulations provide good estimates for the number of atoms and molecules at low density but a rigorous quantum-mechanical definition of what a bound state is and how to identify it remains to be given. Several ideas are discussed in the work by Girardeau (1990).

Starting in the molecular regime, one finds that increasing temperature at constant density leads to the gradual process of thermal dissociation of molecules, which results into a regime with a majority of atoms. The atoms are then gradually ionized at even higher temperatures leading to a plasma of free protons and electrons. Lowering the density at constant temperature leads to a decrease in the number of molecules, or atoms respectively. We call these processes entropy dissociation of molecules and entropy ionization of atoms because both processes are driven by the increased entropy of the unbound states due to the larger volume.

If the density is increased at constant temperature, pressure dissociation diminishes the molecular fraction. This transition was described by Magro et al. (1996). In simulations with the time step $\tau^{-1}=10^6\,\rm K$, it was found that the number of molecules drops significantly within a small density interval. Secondly, a region with $\left.\frac{\partial P}{\partial T}\right\vert _{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}}<0$ was found as shown in Fig. . Both results are consistent with a first order plasma phase transition (PPT). In this case, one expects to find a coexistence region indicated by thick red line in Fig. , which ends in a critical point.

Since the work by Magro et al. (1996), we were able to obtain simulation results with better convergence, smaller time steps, for larger systems and different nodal surfaces. It should be emphasized that the type of nodal surface has a significant effect on the thermodynamic properties in the region of the PPT, which will be discussed in detail in section . First, we verified that simulations using free particle nodes and smaller time steps ( $\tau_B^{-1}=\tau_F^{-1}=2 \cdot 10^6\,\rm K$ and $\tau_B^{-1}=2 \cdot 10^6\,\rm K$, $\tau_F^{-1}=8 \cdot 10^6\,\rm K$) also predict $\left.\frac{\partial P}{\partial T}\right\vert _{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}}<0$ as shown in Fig. . Using a smaller time step makes the pressure drop less pronounced but it can still be clearly identified.

Figure: Hydrogen in molecular state as it appears in a PIMC simulation at $T=5\,000\,\rm K$ and $r_S=4.0$. The salmon-colored spheres denote the protons. The bonds (white lines) were put in a guide to the eye. The electron paths are shown in red and blue depending on their spin state.

Figure: Deuterium in molecular state with significantly degenerate electrons as it appears in a PIMC simulation at $T=5\,000\,\rm K$ and $r_S=1.86$ similar to Fig. . Due to the higher density, the electron paths permute with a higher probability (shown as yellow lines) but are localized enough to form a bond between the two protons in the molecule. The electron density average over many electron configurations is indicated in gray color on the blue rectangles.

Figure: Deuterium in metallic state with unbound protons and a gas of degenerate electrons as it appears in a PIMC simulation at $T=6\,250\,\rm K$ and $r_s=1.6$ similar to Fig. . The electron paths are delocalized and permute frequently.

Figure: Histogram of the fraction of permuting electrons from PIMC simulations with free particle nodes at different temperatures and densities. The colors are chosen as in Fig. : blue for an average of less than 10% exchanges, green for between 10% and 80%, and red for over 80%.

As a next step, we looked at the number of permuting electrons, which is one of the key properties to understand this transition. We determined the fraction of electrons involved in a permutation as an indication of electronic delocalization. Permuting electrons are required to form a Fermi surface (see chapter 5), which means that a high number of permutations indicates a high degree of degeneracy of the electrons. If, in average, over 80% of the electrons are involved in a permutation we label this state metallic. Permuting electrons form long chains of paths and therefore occupy delocalized states. This delocalization destabilizes the hydrogen molecules. Before all bonds are broken, one finds a molecular fluid with some permuting electrons as visualized in Fig. . It still shows a significant molecular signature but differs from the molecular fluid at lower density in Fig.  by the increased fraction of permuting electrons. If the density is increased further, the majority of the electron become delocalized, all bonds are broken, which leads to a metallic fluid of unbound proton and the degenerate electron gas as shown in Fig. .

Fig.  shows histograms of the number of permuting electrons. At low density, the permutation probability is small and the peak in the histogram is on the left. The peak position shifts to the right with increasing density indicating the higher fraction of permuting electrons. Eventually, one finds a sharp peak near 1, which corresponds to degenerate electronic states where almost all electrons permute. The histogram also provides information on how this transition occurs in simulations with free particle nodes. Near the critical density, we found that the simulation exhibits a switching behavior between a less degenerate (and presumably more molecular) and highly degenerate (with unbound protons) state. The bimodal distribution can be seen best in the simulation at $r_s=1.86$ and $T=6944\,$K in Fig.  and to a lesser extent for $r_s=1.93$ and $T=7812\,$K as well as for $T=6944\,$K. This switching behavior indicates that the transition occurs as a collective effect, which is required for a first order phase transition.

The boundaries of the metallic regime in Fig.  are determined by two effects. With increasing temperature, the degree of degeneracy of the electrons is reduced and one finds a gradual transition to a less degenerate plasma state. If the temperature is lowered, the attraction to the protons becomes more relevant, which localizes the electrons and decreases the degree of degeneracy as also can be seen in Fig. .