Equilibration dynamics and conductivity of warm dense hydrogen

We investigate subpicosecond dynamics of warm dense hydrogen at the XUV free-electron laser facility (FLASH) at DESY (Hamburg). Ultrafast impulsive electron heating is initiated by a $\le 300$-fs short x-ray burst of 92-eV photon energy. A second pulse probes the sample via x-ray scattering at jitter-free variable time delay. We show that the initial molecular structure dissociates within $(0.9\pm 0.2)$ ps, allowing us to infer the energy transfer rate between electrons and ions. We evaluate Saha and Thomas-Fermi ionization models in radiation hydrodynamics simulations, predicting plasma parameters that are subsequently used to calculate the static structure factor. A conductivity model for partially ionized plasma is validated by two-temperature density-functional theory coupled to molecular dynamic simulations and agrees with the experimental data. Our results provide important insights and the needed experimental data on transport properties of dense plasmas.

Figure 1

Schematic of the experimental setup (figure from Ref. [7]). Soft-x-ray FEL pulses are split and delayed before they are focused onto a cryogenic hydrogen jet. Scattering at 90${}^{\circ }$ is observed with an XUV spectrometer. For further details refer to the text.

Figure 2

Relative geometrical areas $A_f$ and $A_v$ of the split-and-delay unit.

Figure 3

Simulation of the temporal evolution of the plasma parameters and the structure factor. The temperatures of the electronic and atomic-ionic subsystems are calculated with the one-dimensional radiation-hydrodynamics code helios using propaceos 4.2 [21], which calculates the ionization via a Saha model. Here $T_e$ and $T_i$ are the electron and ion temperatures, respectively. Also shown is the structure factor at $k=0$ (the lines connecting the data points are a guide to the eye). No temporal evolution in the initial ion density is observed. The ionization ranges between 4% and 6%, temporally following the electron temperature dependence.

Figure 4

Mean ionization $Z_{\mathrm{free}}$ as a function of plasma temperature $T$, comparing Saha-like and QEOS-like models for the EOS, as well as the predictions made by the Saha equation itself and the code comptra [23]. QEOS predicts significant intrinsic pressure ionization, while all other models agree on less than 10% ionization for $T<2$ eV.

Figure 5

Experimental scattering spectra as a function of pump-probe delay, from Ref. [7]. The inset shows two 16-bit raw spectra, $25\times 50$ pixels of 13.5 $\mu$m${}^2$ in size, with a horizontally oriented spectral axis corresponding to the graph, where the spectral profiles are integrated along the vertical axis.

Figure 6

Calculated pair distribution function, from Ref. [7]. Shown is the pair distribution function $g\left(r\right)$ as a function of the radial interparticle distance $r$ for different time delays and temperatures. For cryogenic hydrogen ($T=20$ K), a peak at $r\sim 0.75$ Å indicates the distance in the H${}_2$ molecule. With increasing temperature the molecular bonds break and the intensity of the peak drops.

Figure 7

Two-dimensional illustration of the hydrogen atoms. Shown is the transition from initial cryogenic hydrogen (left) to an atomic ideal plasma (right).

Figure 8

Dissociation degree of hydrogen molecules as a function of delay for 2T DFT ($T_e\ne T_i$) and 1T DFT ($T_e\equiv T_i$).

Figure 9

Calculated structure factor $S\left(k\right)$ for different time delays, extracted from the DFT simulations. For $k<1.25/$Å the DFT simulations are not trustable, but for $k=0$ (points) they are valid.

Figure 10

Structure factor $S(k=0)$ calculated with 2T DFT MD, with the CHNC, and in the Salpeter approximation.

Figure 11

Temporal evolution of the measured (points) and simulated (lines) x-ray scattering, from Ref. [7]. A total of 300 exposures were grouped together by delay and their errors resemble their root mean square deviations. For the blue diamonds, ranging from $\Delta t=0$ to 1 ps only, the more intense 27-TW/cm${}^2$ pulse pumps the target, while it is probed by the 19-TW/cm${}^2$ pulse. The red points, ranging from 0 to 5 ps, resemble the inverse case where the pump was less intense (19 TW/cm${}^2$) than the probe (27 TW/cm${}^2$). In both cases the signal increases within 1 ps and for the more intense pump the amplitude is larger. The lines represent the simulated scattered fraction via Eq. (3), the red and blue (lower) ones using a Saha-like ionization model and the gray (upper) one using QEOS.

Figure 12

Temporal evolution of the plasma parameters derived from DFT MD. The electron temperature $T_e$ is an input parameter from helios. The ionization degree $Z_{\mathrm{free}}$ and the plasma conductivity $\sigma$ follow the temporal evolution of $T_e$ (left axis). On the right axis, the equilibration time is plotted, while the collision time is ${10}^{-3}$ times the equilibration time.