Free Energy Models

There are two conceptually different approaches to describe hydrogen and related materials. One method is based on the physical picture where one treats the fundamental particles, in this case electrons and proton, individually and compound particles such as atoms and molecules are formed if the fundamental particle are bound together. In PIMC, this approach is used, which has the advantage that one can build a simulation from first principles that allows one to describe regimes where one has a mixture of different species without any additional assumptions. However, it should be noted that the computational requirements are orders of magnitude higher than in the chemical models described in the following.

In the chemical picture, one assumes different chemical species. For hydrogen one usually considers molecules (H$_2$), atoms (H), free protons (p), and electrons (e). Other species such as H$_2^+$, H$_2^-$ or H$^-$ are neglected, because their binding energies are very small compared to the thermal energy. For the chemical species under consideration, one constructs a free energy function with the particle numbers $N_{{\rm H}_2}, N_{\rm H}, N_{\rm p}$, and $N_{\rm e}$ as parameters.

$$F({V^{\!\!\!\!\!\!\:^\diamond}},T,N_{{\rm H}_2},N_{\rm H}, N_{\rm p}, N_{\rm e}) = F^{\rm id}_0({V^{\!\!\!\!\!\!\:^\diamond}},T,N_{{\rm H}_2},N_{\rm H}) + F^{\rm id}_\pm({V^{\!\!\!\!\!\!\:^\diamond}},T,N_{\rm p},N_{\rm e})$$

$$+ F^{\rm int}_0({V^{\!\!\!\!\!\!\:^\diamond}},T,N_{{\rm H}_2},N_{\rm H}) + F^{\rm int}_\pm({V^{\!\!\!\!\!\!\:^\diamond}},T,N_{\rm p},N_{\rm e})$$

$$+ F^{\rm int}({V^{\!\!\!\!\!\!\:^\diamond}},T,N_{{\rm H}_2},N_{\rm H},N_{\rm p},N_{\rm e}) \quad,$$ (5)

where the superscripts $id$ and $int$ denote the ideal contribution from non-interacting particles and the part caused by the interactions. The interaction terms are derived from known analytical expressions or from computer simulations. The subscripts $0$ and $\pm$ refer to neutral and charged. Introduce the total particle number,

$$N=2 N_{{\rm H}_2}+N_{\rm H}+N_{\rm p}\quad\quad {\rm with} \quad\quad N_{\rm p}=N_{\rm e}\quad,$$ (6)

one can define the number concentration $x_i=N_i/N$. The free energy is maximized with respect to the chemical composition under fixed external conditions, here temperature and volume. This lead to the condition for chemical equilibrium of dissociation H $_2\rightleftharpoons 2\rm H$ and ionization H $\rightleftharpoons {\rm p}+ {\rm e}$:

$$\left. \frac{\partial \tilde{F}}{\partial x_{{\rm H}_2}}\rig... ..._{x_{{\rm H}_2},{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}},T,N} = 0$$ (7)

with $\tilde{F}({V^{\!\!\!\!\!\!\:^\diamond}},T,N,x_{{\rm H}_2},x_{\rm H}) = F({V^{\!... ..._2},N x_{\rm H}, N_{\rm p}=N[1-2 x_{{\rm H}_2}-x_{\rm H}], N_{\rm e}=N_{\rm p})$. In terms of the chemical potentials,

$$\mu_i=\left. \frac{\partial F}{\partial N_i} \right\vert _{{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}},T,N_{j\neq i}}$$ (8)

chemical equilibrium is obtained from,

$$\mu_{{\rm H}_2} = 2 \mu_{\rm H} \quad,\quad \mu_{{\rm H}} = \mu_{\rm p}+ \mu_{\rm e} \quad.$$ (9)

Chemical models are known to work very well in regimes of weak interaction between the different species. This is usually called the Saha limit because the ideal Saha equation (Fowler and Guggenheim, 1965), which neglects interaction between particles, gives a reasonable approximation. The free energy models currently used to predict properties of hydrogen employ elaborate schemes to determine the interaction terms. Not all of them were constructed to describe the whole high temperature phase diagram as done by Saumon and Chabrier (1992). Ebeling and Richert (1985b) studied the plasma and the atomic regime, while models by Beule et al. (1999) and Bunker et al. (1997) were designed to the describe the dissociation transition. The Ross (1998) model was primarily developed to study the molecular-metallic transition. One difficulty common to free energy models is how to treat the interaction of charged and neutral particles. Often, this is done by introducing hard-sphere radii and additional corrections. These kinds of approximation lead to rather different predictions from various chemical models. The differences are especially pronounced in the regime of the molecular-metallic transition because of the high density and the presence of neutral and charged species. If the derivative of the free energy function is a continuous function in this region, then no PPT is predicted. If on the other hand the different components in $F$ lead to a discontinuous first derivative, a PPT is inevitably predicted.

One purpose of our PIMC calculation is to provide data so that free energy models can be fitted to it. Those can then be used to derive additional information, which can not be obtained directly from PIMC simulations.