Zhiguo Li,Xiang Wang,Yong Hou,Yuying Yu,Guojun Li,Long Hao,Xuhai Li,Huayun Geng,Chengda Dai,Qiang Wu,Ho-Kwang Mao,Jianbo Hu
Quantifying the partial ionization effect of gold in the transition region between condensed matter and warm dense matter
warm dense matter,gold,ultra-high pressure,shock-induced ionization
Significance Insights into the ionization of solids under extreme conditions play a critical role in the construction of the unified models for the pressure and temperature phase space covering from condensed matter (CM) to warm dense matter (WDM). In this study, by combining a unique three-stage gas gun experiments and state-of-the-art ab-initio simulations, we observed the softening behavior of the Hugoniot in gold and quantitatively identified the ionization conditions of 5d electrons for the shocked gold. These results provide significant information for the change of macroscopic physical properties and microscopic electronic properties in the transition region, which is of great value for improving our understanding on the transition from CM to WDM.
Quantifying the partial ionization effect of gold in the transition region between condensed matter and warm dense matter

Insights into the ionization of solids under extreme conditions play a critical role in the construction of the unified models for the pressure and temperature phase space covering from condensed matter (CM) to warm dense matter (WDM). In this study, by combining a unique three-stage gas gun experiments and state-of-the-art ab-initio simulations, we observed the softening behavior of the Hugoniot in gold and quantitatively identified the ionization conditions of 5d electrons for the shocked gold. These results provide significant information for the change of macroscopic physical properties and microscopic electronic properties in the transition region, which is of great value for improving our understanding on the transition from CM to WDM.

Knowledge about the thermodynamic properties of solids in a wide pressure–temperature (P–T) phase space spanning from the condensed matter, the intermediate warm dense matter (WDM), and the hot plasma regimes is of great significance in the fields of geophysics, planetary science, and inertial confinement fusion (ICF) (1–3). Although suitable theoretical models may exist for specific regions of interest, by far, no unified theory is able to cover the entire regime, particularly the key transition regions that connect and link different regimes. Its progress is not only hindered by the lack of suitable theories but also limited by the lack of high-quality data in the key regions. Shock compression generating both high pressure and high temperature, therefore, has been widely used to drive solids into different regions in the phase space, depending on the magnitude of shock waves. Significant gaps remain unaccessed, especially in the key region for the transition from condensed matter to WDM, although various techniques, such as gas guns (4, 5), high explosives (6, 7), pulsed-power loading (8, 9), and high-power lasers (10, 11), have been developed to produce extreme states. Such that, how the transition occurs from condensed matter to WDM has not yet been well explored. Moreover, theoretical models proposed for this transition regime have not been evaluated yet, leading to unknown uncertainty to predict the thermodynamic properties of matter. Thus, it is of significance to obtain high-precision shock compression data in the transition regime, which is crucial to gain insights into the condensed matter-to-WDM transition and to evaluate or establish theoretical models. Gold (Au) is a widely used pressure standard for diamond anvil cell experiments, owing to its chemical inertness, efficient X-ray scattering property, and stability under extreme conditions (12). It has also been used as an optimized wall material in hohlraum experiments in the context of indirectly laser-driven ICF (13). Properties of gold under extreme conditions are thus of practical importance. In the long-term efforts devoted to the study of Au’s properties, a great amount of shock Hugoniot data up to 580 GPa and 10 TPa have been obtained by two-stage gas gun (2sGG) (14–16) and laser-driven shock (17, 18), respectively. However, there exists a gap between ~0.6 TPa and ~1 TPa. Various theoretical models have been proposed to bridge over this gap, including the SESAME equations of state (EOSs) (Nos. 2700, 2705, and 2710) (19–21), the quotidian EOS (QEOS) (22), and the recent multiphase EOS by Atomic Weapons Establishment (AWE) (23). Yet, these models are divergent beyond the scope of 2sGG. The lack of high-precision shock wave data above 0.6 TPa significantly limits us to provide strict experimental constraints for validating theoretical models. It is worth noting that, most recently, some newly emerging physics were observed in shock-compressed Au. In-situ X-ray diffraction (XRD) measurements reveal that shocked Au undergoes the transition from face-centered cubic to body-centered cubic phases around 200 GPa (24–26), indicating that Au is not as stable as expected. The shockless (i.e., ramp) compression experiments compressed Au up to 1.1 TPa but with a much lower temperature (12, 27); thus, Au remains in the condensed matter region. In this letter, hypervelocity (up to 10 km/s) impact experiments have been carried out to study the dynamic response of Au in the transition regime of 0.5 TPa to 0.9 TPa and 15 kK to 32 kK, by using a three-stage gas gun (3sGG) launcher technique. High-precision results obtained experimentally not only fill the gap between the data provided by 2sGG and laser-shock experiments but also identify a unique softening behavior beyond ~560 GPa along the Hugoniot. The state-of-the-art ab-initio molecular dynamics (AIMD) calculations reveal that such softening results from the ionization of 5d electrons, which quantifies the transition from condensed matter to WDM in Au under extreme conditions and provides the key constraint to deal with the ionization effect in the transition region for theoretical models.

The experiments were performed on the three-stage gas gun developed recently at the Institute of Fluid Physics, CAEP (28). A schematic illustration of the three-stage gas gun launcher is displayed in Fig. 1 A–C, in which a high-Z tantalum (Ta) flyer plate driven by the graded density impactor (GDI) impacts the target to generate extreme states. More experimental details are provided in Materials and Methods. Fig. 1D shows the representative Doppler pins system (DPS) (29) signals for the impact velocity history of the Ta flyer. The history shows that the Ta flyer experiences an initial shock and then quasi-isentropic compression, finally reaching the terminal velocity within an accelerating duration of ~600 ns. This compression path can maximize the conversion efficiency of the kinetic energy from the GDI to the Ta flyer plate by reducing the shock-induced heating. And then, the Ta flyer impacts the target with the steadiness of the velocity better than 1%. The deviation of the terminal velocities measured by each DPS is less than 0.5%. The flatness of the shock front is better than 1 ns inside the central region of 8 mm in diameter, as examined by our previous planar target experiments (28). The shock wave transit time in the Au sample (Δt) is determined by the difference of the shock arriving times of the three electrical-shorting pins (ESPs) at the sample rear surface and the twelve ESPs at the baseplate rear surface, as illustrated in Fig. 1E. The shock velocity is obtained by Us = d/Δt, where d is the sample thickness. Then, the shock states are deduced using the impedance-matching method. The experimental uncertainties were determined by a Monte Carlo technique (30). The details of the inferences of the shock states, along with the uncertainty analysis, are presented in SI Appendix. The obtained Au Hugoniot data of seven-shot experiments are given in SI Appendix, Table S1, covering a pressure range from 0.5 TPa to 0.9 TPa, and the estimated temperature ranges from 15 kK to 32 kK. Fig. 2A shows the Us–Up data of our experiments, along with the results obtained by 2sGG and laser shocks. It is clear that, for Us below 8.5 km/s (corresponding to ~560 GPa in Au), our results are consistent with 2sGG data and the Us–Up shows an excellent linear relationship, while for Us > 8.5 km/s, the Us–Up data of our work obviously deviate from the linear relation (Us = 2.995 + 1.621Up) of Yokoo et al. (16) and exhibit a lower slope. A linear fitting to our data gives a slope value of 1.283, which approaches the universal Hugoniot of fluid metal at sufficiently high pressures, e.g., Us = 5.8 + 1.2Up (31). We can also fit the entire gas gun data (including our 3sGG and previous 2sGG data) with a quadratic Us–Up relation, which gives Us = 2.910 + 1.775Up − 0.051Up2. Extrapolating this nonlinear relation to the higher Us and Up regime, one finds good agreement with the data of laser-shock experiments (17) as shown in the inset of Fig. 2A, although the latter are scattered. Fig. 2B shows the same experimental data in the P–ρ space. The softening behavior at P > ~560 GPa (i.e., higher compressibility η = ρ/ρ0) becomes more visible. In order to experimentally evaluate various theoretical models mentioned above, Fig. 2 presents the comparison of experimental data with models. The SESAME 2700 and the AWE multiphase EOS obviously deviate from the experiments at Us > ~6 km/s. The QEOS model shows agreement with the 3sGG data but fails at lower shock velocities. Only the SESAME 2710 reasonably reproduces all the gas gun data up to Us = 10 km/s. The percentage difference between the EOS models and the quadratic relation can be found in the lower panel of Fig. 2A. The differences of various EOS models at high shock pressures primarily arise from their treatments for the ionic and electronic contributions. The electronic contribution in SESAME 2700 and 2710 is described by the INFERNO model (32), while that for AWE and QEOS is the Thomas–Fermi model. The ionic contribution for the fluid phase was treated with the corrected rigid ion sphere model (33) in SESAME 2710 and AWE, the Cowan fluid model (34) in QEOS, and a smooth interpolation between Debye and ideal gas models in SESAME 2700. To better understand the softening behavior observed here and to theoretically validate the physics model, we performed AIMD simulations using the Vienna ab-initio simulation package (VASP) (35), with the exchange–correlation functional of local density approximation (LDA) (36). Details of AIMD calculations are given in the Materials and Methods. We find that the AIMD simulations are in excellent agreement with the Hugoniot data in a wide P–T range (Fig. 2), indicating that the LDA approximation can reasonably represent the exchange–correlation interaction for gold. We further calculate the electronic properties including the electronic density of states (DOSs), electron localization function (ELF), electrical conductivity, and free electron number density to quantify the mechanisms driving the change in Hugoniot slope and the corresponding softening manner of the P–ρ data. The calculated DOSs for the shocked Au, as presented in Fig. 3A, reveal that the 5d electrons are highly localized at the lower shock state of 25 g/cm3 and an obvious energy gap (~1.92 eV) exists from the top of 5d band to the Fermi level. With increasing shock pressures and temperatures, the 5d electronic band broadens and the band top gradually approaches the Fermi level (37). At the onset of the Hugoniot softening (the density is ~32.5 g/cm3 and the temperature is over 19,000 K), the electrons near the top of 5d band gain enough energy to overcome the energy gap (~1.63 eV) and then become free electrons. The ELF provides a more intuitive picture for electron delocalization and chemical bonding. The overlap of strong electron-localized regions around gold atoms under such extreme conditions implies the overlap of electron clouds. Fig. 3 B–E shows the calculated ELFs at 27 g/cm3, 30 g/cm3, 32 g/cm3, and 35.8 g/cm3 along the Au Hugoniot. The obvious delocalization of the electrons occurs above 32 g/cm3 (SI Appendix, Fig. S5). The delocalization arises exactly from the ionization of 5d electrons. The ionization of 5d electrons under shock compressions is driven by both pressure and temperature effects. The increase of pressure narrows the band gap, while the temperature effect provides the energy needed for the electron to cross the band gap. This process contributes substantial free (ionization) electrons, which consume partially the system energy that should increase the shock pressure, therefore resulting in the softening manner of the Hugoniot. This softening behavior can thus be regarded as an indicator of the transformation of gold from condensed matter to WDM. Such behavior of the Hugoniot has also been observed in other materials, such as polystyrene, silicon, lithium hydride, helium, neon, aluminum, and magnesium, by the laser-shock experiments (38–41) or the first-principles simulations (42–45). Depending on the element type, the ionization of electrons could originate from different shells. Note that, at the lower shock state of 27 g/cm3 (~180 GPa), an obvious lattice structure still exists, illustrating that the melting has not yet taken place. With the increase of the density to 30 g/cm3 (~350 GPa), the ELF evidently demonstrates the loss of lattice structures, which agrees with the shock melting pressure identified by in situ XRD technique in recent laser-shock experiments (24, 25). For a quantitative characterization of the variation of free electrons, we derive the free electron number density ne through fitting the real part of the frequency-dependent electrical conductivity with a Smith–Drude model (46). Fig. 3 F and G shows the derived ne and DC conductivity σ0 along the Au Hugoniot, respectively. The ne presents a superlinear trend along with the increasing shock pressure with an inflection at ~32 g/cm3, which coincides with the onset of the softening in Hugoniot. Beyond the inflection point, the ionization of 5d electrons drives the ne to undergo a faster increase up to the highest pressure. In contrast, the DC conductivity exhibits a sublinear behavior with the same inflection point. We attribute this phenomenon to the competition between the free electrons and the electron–ion collisions. When Au is compressed beyond the inflection point, the ionization of 5d electrons increases the conductivity, while the electron–ion collision becomes extremely violent to reduce the electron transport (47). As a consequence, the increases in the DC conductivity slow down. This prediction thus provides theoretical guidance for X-ray spectroscopy experiments (48–50) to find further experimental evidence for the shock-driven partial ionization in Au, which is critical for constructing the EOS model in the ionized dense plasma regime (51, 52). Fig. 4 summarizes all EOS data of Au by shock and ramp compression experiments with the calculated temperatures, together with the theoretical Hugoniots by SESAME2700, QEOS, and AIMD in the P–T phase diagram. Comparisons of three theoretical results show that the QEOS model obviously underestimates the Hugoniot temperature at low pressures compared to SESAME 2700 and AIMD. While at high pressures, the difference between SESAME 2700 and AIMD/QEOS is small compared to that in P–ρ plot. It is visible that the experimental Hugoniot data cover wide-range states from solids to WDM with the transition regime filled by 3sGG data. With the help of the data in the transition region together with the first-principles electronic structure analysis, we quantitatively identify the transition from condensed matter to WDM for Au under extreme shock compressions at about 560 GPa, arising from the ionization of 5d electrons. Through enormous DOS calculations in a wide P–T range, we explore various conditions for the ionization of Au 5d electrons and quantitatively settle down the boundary condition for the onset of 5d electron ionization (the solid magenta line in Fig. 4). The calculation details are presented in SI Appendix. The coupling (Γ) and degeneracy (Θ) are two key parameters widely used for describing the WDM. Generally, the WDM is associated with strongly coupled ions (1 < Γ < 100) and partially degenerate electrons (0.1 < Θ < 10) (54). We calculated the Γ and Θ values of Au for temperature above 2,000 K and pressure ranging from 100 GPa to 4,000 GPa by using an average atom (AA) model (55), in which the band-broadening effect has been considered. The boundary for the onset of 5d electrons ionization determined in this work is close to the lower limit of WDM regime defined by Γ and Θ for pressures below ~1 TPa. This fact indicates that the partial ionization is also a key characteristic of WDM in addition to the Γ and Θ, especially for mid- to high-Z elements. While at higher pressures, due to the strong effect of pressure-induced ionization (band broadening), the ionization of 5d electrons occurs at temperatures far below the WDM regime. Therefore, our results provide significant information for the change of macroscopic physical properties and microscopic electronic properties in the transition region, which is of great value for improving our understanding on the transition from condensed matter to WDM. The well-defined conditions for the ionization of 5d electrons also set up a sensible limit of current models for condensed matter and WDM and provide the quantitative constraint to deal with the ionization effect in the transition region. In summary, high-precision shock data for gold in an unexplored range from 0.5 TPa to 0.9 TPa have been obtained via a unique three-stage gas gun technique. These data fill the gap inaccessible by the two-stage gas gun and laser shock experiments and provide important constraints for various bridging models in the transition region from the condensed matter to the WDM. The results reveal an obvious softening behavior of Au Hugoniot for pressure beyond ~560 GPa, arising from the ionization of 5d electrons under extreme conditions. Our findings are of importance to build a high-precision pressure standard up to TPa and provide a fundamental input for radiation hydrodynamic simulations of ICF, which has far-reaching impact in the fields of high-pressure science, ICF, astrophysics, and planetary physics.

Optimization of the add-on part connected to the 2sGG launch tube and the graded density impactor allowed a high-Z tantalum (Ta) flyer plate, with a diameter of 18 mm and a thickness of 0.5 mm, to be accelerated up to 10 km/s. The target assembly was composed of a Ta baseplate and a high-purity Au sample (99.99%). The Ta baseplate was 38 mm in diameter and 0.4 mm in thickness, while the Au sample was 5 mm in diameter and 1.1 mm in thickness. Shock waves were generated through the impact of the hypervelocity Ta flyer on the Ta baseplate and then transmitted into the Au sample across the baseplate. A DPS was used to monitor the impact velocity history of the Ta flyer plate accelerated by the GDI, through the pinhole on the baseplate. ESPs located at the rear surfaces of the Ta baseplate and the Au sample were used to detect the arrival times of the shock wave.

The AIMD simulations of Au were performed using the VASP package (35). The Born–Oppenheimer approximation was adopted to decouple the ion and electron. A cubic simulation box containing 32 atoms was used as an initial cell. In electronic structure calculations, the projector-augmented wave pseudo-potential (56, 57) was used to describe the ion–electron interaction and the LDA exchange–correlation functional (36) to account for the electron–electron interaction. The valence states of 5d106s16p0 (11 electrons) were treated. The plane-wave cutoff energy of 500 eV and a 2 × 2 × 2 Monkhorst–Pack k-point grid (58) were adopted. In the AIMD simulation, the canonical ensemble (NVT) and the Nose–Hoover thermostats (59, 60) were used to control the simulation temperature. The simulation time step was set to 0.5 fs. After reaching equilibration within 2 ps, we continued to perform 5000-step simulations to get EOS data. The simulation densities ranged from 25 to 40 g/cm3, and temperatures were between 2 kK and 50 kK. Details of derivation of shock Hugoniot and electronic properties for gold from AIMD simulation results are given in SI Appendix.

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