Specific Heat of Electron Plasma Waves

Collective modes in a plasma, like phonons in a solid, contribute to a material’s equation of state and transport properties, but the long wavelengths of these modes are difficult to simulate with today’s finite-size quantum simulation techniques. A simple Debye-type calculation of the specific heat of electron plasma waves is presented, yielding up to $0.05k/e^{-}$ for warm dense matter (WDM), where thermal and Fermi energies are near $1\mathrm{Ry}=13.6\mathrm{eV}$. This overlooked energy reservoir is sufficient to explain reported compression differences between theoretical hydrogen models and shock experiments. Such an additional specific heat contribution refines our understanding of systems passing through the WDM regime, such as the convective threshold in low-mass main-sequence stars, white dwarf envelopes, and substellar objects; WDM x-ray scattering experiments; and the compression of inertial confinement fusion fuels.

Figure 1

Hydrogen density-temperature diagram. High-pressure shock experiments in deuterium [5] reach higher compression than predicted by theoretical models (solid red line; also see Fig. 4). This occurs in the warm-dense-matter regime where the thermal energy $kT$ is approximately and simultaneously equal to (i) the isolated-atom binding energy ($1\mathrm{Ry}=13.6\mathrm{eV}$), (ii) the Fermi energy $E_F$ (assuming full ionization), (iii) the electron plasmon energy $\hslash {\omega }_p$, and (iv) the Coulomb coupling energy $E_C$. Also shown are the standard solar model, a main-sequence star with $M=0.12M_{\odot }$, the pure-H Jupiter isentrope, and the dense fuel trajectory in an ignited inertial confinement fusion (ICF) implosion (experiment N210808 [6]).

Figure 2

For electron density $n_e=2.3\times {10}^{29}{\mathrm{m}}^{-3}$ ($E_F=1\mathrm{Ry}$): (a) dispersion relation (solid lines) and damping rate (dashed lines) of electron plasma waves at $kT=(0.2,1,\text{and}5)\mathrm{Ry}=(32,158,790)\mathrm{kK}$ in blue, yellow, and red, respectively; (b) screening cutoff wave number for $\zeta =1$ (solid curve), its approximation using Eq. (3) (dash-dotted curve), and for $\zeta =2^{\pm 1/3}$ (shaded gray region). Diamonds (open squares) show the screening (damping) cutoff wave numbers for the three temperature cases. Also shown is the Debye phonon model cutoff for solids (dotted black line).

Figure 3

(a),(b) Internal energy and (c),(d) isochoric heat capacity contributions of electron plasma waves. (a),(c) Contours of $E_F=kT$ and $\hslash {\omega }_p=kT$ are shown as dotted and dashed-dotted lines, respectively. The hydrogen shock Hugoniot starting from cryogenic liquid is in solid red, with a red diamond for the highest-pressure shock experiment that observed enhanced compression compared to theory [5]. (b),(d) Temperature lineouts (solid red line) are at $n_e=2.2\times {10}^{29}{\mathrm{m}}^{-3}$ ($E_F=0.98\mathrm{Ry}$), corresponding to the density of these shock experiments. The dashed black lines are the approximations of Eqs. (5) and (6), and the red shaded region shows cutoff wave numbers spanning $\zeta =2^{\pm 1/3}$ (see the text).

Figure 4

Comparison of selected deuterium experiments [5, 35, 36] and theoretical EOS models [28, 29, 30, 31, 32, 33] along the Hugoniot from a cryogenic liquid initial state. (a) Compression ratio. (b) Difference in internal energy of the data and models relative to Hu2011 [30]. The shaded band represents adjustment of the Hu2011 model with the addition of EPW energy using a range of cutoff multipliers $\zeta =2^{\pm 1/3}$. Shocked deuterium is fully ionized above 50 GPa.