Thermodynamics of atomic and ionized hydrogen: Analytical results versus equation-of-state tables and Monte Carlo data

We compute thermodynamical properties of a low-density hydrogen gas within the physical picture, in which the system is described as a quantum electron-proton plasma interacting via the Coulomb potential. Our calculations are done using the exact scaled low-temperature (SLT) expansion, which provides a rigorous extension of the well-known virial expansion—valid in the fully ionized phase—into the Saha regime where the system is partially or fully recombined into hydrogen atoms. After recalling the SLT expansion of the pressure [A. Alastuey et al., J. Stat. Phys. 130, 1119 (2008)], we obtain the SLT expansions of the chemical potential and of the internal energy, up to order $\exp (-|E_{\mathrm{H}}|/kT)$ included ($E_{\mathrm{H}}\simeq -13.6$ eV). Those truncated expansions describe the first five nonideal corrections to the ideal Saha law. They account exactly, up to the considered order, for all effects of interactions and thermal excitations, including the formation of bound states (atom $\mathrm{H}$, ions ${\mathrm{H}}^{-}$ and ${{\mathrm{H}}_2}^{+}$, molecule ${\mathrm{H}}_2,\cdots$) and atom-charge and atom-atom interactions. Among the five leading corrections, three are easy to evaluate, while the remaining ones involve well-defined internal partition functions for the molecule H${}_2$ and ions H${}^{-}$ and H${}_2$${}^{+}$, for which no closed-form analytical formula exist currently. We provide accurate low-temperature approximations for those partition functions by using known values of rotational and vibrational energies. We compare then the predictions of the SLT expansion, for the pressure and the internal energy, with, on the one hand, the equation-of-state tables obtained within the opacity program at Livermore (OPAL) and, on the other hand, data of path integral quantum Monte Carlo (PIMC) simulations. In general, a good agreement is found. At low densities, the simple analytical SLT formulas reproduce the values of the OPAL tables up to the last digit in a large range of temperatures, while at higher densities ($\rho \sim {10}^{-2}$ g/cm${}^3$), some discrepancies among the SLT, OPAL, and PIMC results are observed.

Figure 1

Plot of Ebeling direct and exchange functions $Q\left(x\right)$ (solid line) and $E\left(x\right)$ (dashed line).

Figure 2

Comparison of the Brillouin-Planck-Larkin partition function (dashed line) to the exact virial function $Q\left(x_{pe}\left(T\right)\right)$ (solid line) for temperatures up to 100 000 K. Truncating the sum in $Z_{\mathrm{BPL}}$ after the first term yields a quite good approximation (dot-dashed line). The dotted line (constant value 1) corresponds to keeping only the ground-state contribution $\exp (-E_{\mathrm{H}}/kT)$.

Figure 3

Plot of functions $h_k\left(\beta \right)$ for $k=1,2,3,4$. Note that $\beta |E_{\mathrm{H}}|=T_{\mathrm{Rydberg}}/T$.

Figure 4

Logarithmic plot of deviations to Saha pressure for pure hydrogen along isotherm $T=6000$ K (${\rho }^{*}=2.12\times {10}^{15}$ m${}^{-3}$). Crosses correspond to tabulated points of the OPAL equation of state [1] (with corrected ground-state energy [34]).

Figure 5

Logarithmic plots of deviations to Saha pressure along isotherms from $10000$ K up to $30000$ K according to the SLT EOS (solid and dashed lines) and to the tabulated OPAL EOS (symbols).

Figure 6

Deviations to Saha internal energy for isotherms between $10000$ K and $30000$ K according to the SLT EOS (solid and dashed lines). Points of the (corrected) OPAL EOS are shown by symbols (up to density 0.1 g/cm${}^3$).

Figure 7

Deviations to Saha pressure along isochores ${10}^{-8}$ g/cm${}^3$ (black line) and ${10}^{-3}$ g/cm${}^3$ [red (light gray) line] according to the SLT EOS. Crosses correspond to values of the (corrected) OPAL EOS. The red (light gray) dashed line shows the effect of neglecting term $P_4$ in the SLT EOS.

Figure 8

Internal energy per proton along isochore ${10}^{-8}$ g/cm${}^3$ according to the SLT EOS (black line), OPAL EOS (crosses), and Saha EOS (dashed line). For the difference between the two former and the latter curves, see Fig. 9. All energies are shifted upwards by $|E_{{\mathrm{H}}_2}|/2$. The plot shows also curves $|E_{{\mathrm{H}}_2}|/2+3kT$ (red dot-dashed line) and $3kT$ (dotted line).

Figure 9

Deviations to Saha internal energy for isochores ${10}^{-8}$ g/cm${}^3$ (black line) and ${10}^{-3}$ g/cm${}^3$ [red (light gray) line] according to the SLT EOS. Crosses denote tabulated points of the (corrected) OPAL EOS.

Figure 10

Pressure (a) and internal energy (b) per electron-proton pair as a function of temperature along isochore ${10}^{-3}$ g/cm${}^3$, according to SLT EOS [red (light gray) line], OPAL EOS (dashed line), and ideal Saha EOS (dotted line). Points with error bars are simulation results of Ref. [4]. Plots (a${}^{\prime }$) and (b${}^{\prime }$) show deviations to the ideal Saha values along the isochore.

Figure 11

Deviations (with respect to ideal Saha values) of pressure (a) and internal energy per electron-proton pair (b) as a function of temperature along isochore $0.0125$ g/cm${}^3$, according to the SLT EOS [red (light gray) line] and OPAL EOS (dashed line). Points are simulation results of Ref. [4].

Figure 12

Phase diagram of a pure hydrogen gas at low densities. The crosses denote state points where simulation results are available [4]. The crossover density ${\rho }^{*}\left(T\right)$ between the plasma and the atomic phase (dot-dashed line) gives essentially the borderline of the validity domain of the standard virial expansion, which applies only in the plasma phase. The dashed line corresponds to a plasma coupling parameter $\Gamma =0.5$. The SLT expansion is valid in both the plasma and atomic phases, up to the solid (blue) line ${\rho }_c\left(T\right)$ [Eq. (57)], which locates the crossover to the molecular phase. The state point of the Sun photosphere and the track of the Sun adiabat (dotted line) are also shown.

Figure 13

The truncated trace $Z_{\mathrm{eff}}=\mathrm{Tr}[\exp (-\beta H_{\mathrm{eff}})-\exp (-\beta H_0)]$ for a quantum particle in the effective proton-proton potential of the ${\mathrm{H}}_2$ molecule (shown in the inset), as obtained from a numerically exact path integral Monte Carlo calculation (crosses), from the rigid rotor approximation (A4) (solid line) and from Irwin's partition function [53] (dashed curve). The dissociation temperature of the ${\mathrm{H}}_2$ molecule is $T_{\mathrm{diss}}=V_0/k=55459$ K.