Prediction of a roton-type feature in warm dense hydrogen

In a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 121, 255001 (2018)], it was predicted on the basis of ab initio quantum Monte Carlo simulations that, in a uniform electron gas, the peak ${\omega }_0$ of the dynamic structure factor $S(q,\omega )$ exhibits an unusual nonmonotonic wave number dependence, where $d{\omega }_0/dq<0$, at intermediate $q$, under strong coupling conditions. This effect was subsequently explained by the pair alignment of electrons [T. Dornheim et al., Commun. Phys. 5, 304 (2022)]. Here we predict that this nonmonotonic dispersion resembling the roton-type behavior known from superfluids should be observable in a dense, partially ionized hydrogen plasma. Based on a combination of path integral Monte Carlo simulations and linear response results for the density response function, we present the approximate range of densities, temperatures and wave numbers and make predictions for possible experimental observations.

Figure 1

Sketch of the negative dispersion [peak of $S(q,\omega )$] and notation for characteristic $q$ and $\omega$ values. At $r_s=10$ and $\theta =1.0$, Ref. [34] predicts $q_{\text{1}}\approx 1.1q_F, q_{\text{min}}\approx 1.8q_F, {\omega }_{\text{max}}\approx 1.25{\omega }_p$, and ${\omega }_{\text{min}}\approx 0.90{\omega }_p$, with ${\omega }_p={[n_e{e}^2/\left({\epsilon }_0{m}_e\right)]}^{1/2}$ denoting the electron plasma frequency. The difference ${\omega }_{\text{max}}-{\omega }_{\text{min}}$ amounts to 0.52 eV for the uniform electron gas model.

Figure 2

Phase diagram for the predicted negative dispersion of the peak, ${\omega }_0\left(q\right)$, of the dynamic structure factor, $S(q,\omega )$, of the uniform electron gas in the WDM regime. Green (red) symbols: observed (not observed) negative dispersion in PIMC simulations. ESA: predictions based on effective static approximation for the uniform electron gas of Ref. [50]. The negative dispersion, $d{\omega }_0\left(q\right)/dq<0$, is predicted to exist to the right of the dashed black line, see also Fig. 1.

Figure 3

Top: Dynamic structure factor of the warm dense electron gas and wave number dispersion of its peak position for the case of weak (left) and moderate (right) coupling at the electronic Fermi temperature, $\mathrm{\Theta }=1$. The roton feature is visible in (d) for the PIMC data and for the static approximation for the LFC (static). The RPA does not exhibit a nonmonotonic dispersion.

Figure 4

Dependence of the static collision frequency on the free electron density at different temperatures.

Figure 5

Static ion-ion structure factor and pair distribution function (inset) for $r_s=5$ and four different temperatures obtained from first principles path-integral Monte Carlo simulations of a hydrogen plasma. The peak of $g_{ii}$ at the lowest temperature is due to hydrogen molecules. The Brueckner parameter $r_s^{*}$ that refers to the free electron density is different for each curve and can be obtained using the degree of ionization, cf. Fig. 7.

Figure 6

Dispersion of the peak of the dynamic structure factor of a two-component electron-proton plasma using the Mermin dielectric function for different densities and temperatures. Shown are results for jellium without (RPA) and with (LFC) correlations, the two other curves are for a two-component hydrogen plasma without (Born-Mermin) and with ($\text{Born-Mermin}+\text{LFC}$) electron-ion collisions. The density parameter $r_s^{*}$ refers to the unbound electrons.

Figure 7

Selected isotherms of the degree of ionization of hydrogen from B. Militzer et al. [56]. For comparison, the $30000$ K isotherm from D. Kremp et al. [55] was added. At the largest density, $n_e^{\text{tot}}\approx {10}^{24}{\text{cm}}^{-3}$ corresponding to $r_s\approx 1.2$, the plasma is expected to be nearly fully ionized [77]. In addition, full ionization is also expected at temperatures above $150000$ K.

Figure 8

Illustration of the density shift along isotherms, from a uniform electron gas (black curve: fully ionized plasma with density $n_e^{*}$, Mermin results) to a partially ionized hydrogen plasma (red) with total electron density $n_e^{\text{tot}}=n_e^{*}+n_e^{\text{bound}}.$ The vertical axis is scaled to the Fermi temperature of the free electron component, ${\mathrm{\Theta }}^{*}=k_{B}T/E_F^{*}$.

Figure 9

“Phase diagram” of the negative plasmon dispersion for jellium and hydrogen. Negative dispersion is predicted to exist to the left of the curves. Black curve: Mermin results for jellium (full ionization). Colored curves: results for partially ionized hydrogen plasma with the degree of ionization taken either from D. Kremp et al. [55] (red), or from B. Militzer et al. [56] (blue). The green curve is the arithmetic mean of the red and blue curves. The yellow stripe corresponds to the parameters accessible with hydrogen jets around the density of solid hydrogen (vertical line) $\pm 20\%$ [78]. Symbols in the left part refer to data points listed in Table 1: “$+$” refers to $r_s^{*}=5$ and “$*$” to $r_s^{*}=7$. Numbers next to the symbol correspond to $\mathrm{\Delta }\omega$ (in eV) and $\mathrm{\Delta }{\theta }_s$ (for a photon energy of 6 keV, cf. Fig. 11).

Figure 10

$\text{Mermin}+\text{LFC}$ results for the peak position of the dynamic structure factor vs wave number. Shown are results for two densities and five temperatures each.

Figure 11

Temperature dependence of the Thomson scattering angles required to detect the roton feature in hydrogen for two photon energies $k_i=3{\text{Å}}^{-1}$ (6 keV, orange) and $k_i=4.6{\text{Å}}^{-1}$ (9 keV, blue) and two densities $r_s^{*}=5$ ($\approx 1.29·{10}^{22}{\text{cm}}^{-3}$, solid lines) and $r_s^{*}=7$ ($\approx 0.47·{10}^{22}{\text{cm}}^{-3}$, dashed lines). Upper (lower) lines of each pair of lines with the same color and style correspond to the minimum (maximum) of the dispersion ${\omega }_0\left(q\right)$, cf. Fig. 10. The angles were computed using Eq. (6).

Figure 12

Dynamic structure factor at $r_s^{*}=5,T=62000K$, for different approximations, as explained in the inset. Blue curve: $\text{Mermin}+\text{LFC}$ result which is additionally convolved with a Gaussian instrument function with $\sigma =3.65\text{eV}\approx 0.1378{\omega }_p^{*}$. The vertical dotted line indicates the local maximum ${\omega }_{\max }$ of the peak position ($\text{Mermin}+\text{LFC}$).