In recent years, *α*-quartz has been used prolifically as an impedance matching standard in shock wave experiments in the multi-Mbar regime (1 Mbar = 100 GPa = 0.1 TPa). This is due to the fact that above ∼90–100 GPa along the principal Hugoniot *α*-quartz becomes reflective, and thus, shock velocities can be measured to high precision using velocity interferometry. The Hugoniot and release of *α*-quartz have been studied extensively, enabling the development of an analytical release model for use in impedance matching. However, this analytical release model has only been validated over a range of 300–1200 GPa (0.3–1.2 TPa). Here, we extend this analytical model to 200–3000 GPa (0.2–3 TPa) through additional *α*-quartz Hugoniot and release measurements, as well as first-principles molecular dynamics calculations.

## I. INTRODUCTION

With the advent of high-energy density facilities, such as large lasers or pulsed power accelerators, shock wave studies have become routine in the multi-Mbar regime (1 Mbar = 100 GPa = 0.1 TPa). The vast majority of these studies rely on an impedance matching (IM) technique, where the shock response of the material of interest is determined through comparison of the shock response of that material with the shock response of a known material standard.

In recent years, *α*-quartz has been used prolifically as an IM standard. This is due to the fact that above ∼90–100 GPa along the principal Hugoniot—the locus of end states achievable through compression by large-amplitude shock waves—*α*-quartz melts into a conducting fluid with appreciable reflectivity.^{1–3} This enables the use of velocity interferometry (VISAR^{4}) techniques to directly measure the shock velocity to high precision, significantly improving the precision of inferred results using the IM method. However, the accuracy of the inferred shock response of the sample depends on both the Hugoniot and either the release or reshock response of *α*-quartz, depending on the sample's relative shock impedance.

This paper builds upon previous work^{5} that utilized *α*-quartz Hugoniot and release measurements to develop an analytical release model for use in the IM technique. The previous analytical model was validated over a range of 300–1200 GPa (0.3–1.2 TPa). Here, we utilize additional *α*-quartz Hugoniot and release measurements to extend the region of validation to lower pressure (*P*), and first-principles molecular dynamics (FPMD) calculations to constrain the extrapolation of the model to higher *P*.

Section II describes the FPMD calculations of the Hugoniot and release in the few TPa regime. The results of additional *α*-quartz Hugoniot and release experiments are described in Sec. III. The extension of the Hugoniot and release model for *α*-quartz is presented in Sec. IV. The main findings are summarized in Sec. V.

## II. FIRST-PRINCIPLES MOLECULAR DYNAMICS CALCULATIONS OF *α*-QUARTZ

To extend the Hugoniot and release model of *α*-quartz to higher *P*, FPMD calculations were performed using Vienna *ab-initio* simulation package (VASP), a plane-wave density functional theory code developed at the Technical University of Vienna.^{6} We used the same general method that was reported to be in excellent agreement with plate-impact shock wave experiments on *α*-quartz using the Z machine,^{2} and that used in the development of the recent release model.^{5}

Specifically, for the Hugoniot calculations in this work, the silicon and oxygen atoms were represented with projector augmented wave (PAW) potentials.^{7,8} For both silicon and oxygen, we used PAW potentials with only the 1 s electrons in the frozen core. Because of the very high energy thermally excited states for the electrons, we used the “GW” PAW potentials which are built with added consideration for accurate high energy scattering properties. The exchange and correlation were modeled with the PBEsol functional.^{9} A total of 72 atoms were included in the supercell. The plane wave cutoff energy was 1200 eV and the cutoff energy for the augmentation charge was set to 2400 eV. Molecular dynamics simulations were performed in the NVT ensemble, with simple velocity scaling as a thermostat, and typically covered on the order of one picosecond of real time with time steps of 0.1 fs. As a test of the consequence of the choice of valence versus core electrons, simulations were performed with both 1 s and 2 s electrons in the silicon frozen core and these simulations showed negligible differences.

The Rankine-Hugoniot jump conditions,^{10} which are derived by considering conservation of mass, momentum, and energy across a steady propagating wave, provide a set of equations relating the initial energy, volume, and pressure with steady state, post-shock values

where *E*, *P*, *V*, *ρ*, *U _{s}*, and

*u*denote the energy, pressure, volume, density, shock velocity, and particle velocity, respectively, and the subscript 0 denotes initial values. The first of these equations, derived from the conservation of energy, provides a prescription for calculation of the Hugoniot. Temperatures and densities are iterated to obtain final states close to satisfying Eq. (1) for the pressure range of interest with the final Hugoniot state obtained through interpolation on those values. Hugoniot points at ∼2.5 and ∼3 TPa calculated in this way are listed in Table I.

_{p}P (TPa)
. | ρ (g/cm^{3})
. | $Usquartz$ (km/s) . | $upquartz$ (km/s) . |
---|---|---|---|

2.462 | 8.38 | 36.87 | 25.24 |

3.025 | 8.70 | 40.53 | 28.22 |

P (TPa)
. | ρ (g/cm^{3})
. | $Usquartz$ (km/s) . | $upquartz$ (km/s) . |
---|---|---|---|

2.462 | 8.38 | 36.87 | 25.24 |

3.025 | 8.70 | 40.53 | 28.22 |

A release path from high *P* was calculated by taking advantage of the fact that at the initial reference state the isentrope and the Hugoniot have a second order contact,^{10} which is most easily seen by considering a Taylor series expansion of the entropy as a function of volume. Thus for small volume changes, the isentrope is well approximated by the Hugoniot. We therefore approximated the release path as a series of small Hugoniot jumps, where each calculated Hugoniot state along the approximated release path served as the initial reference state for the subsequent Hugoniot calculation. Volume jumps were of the order of 6% (0.94 multiplier on the density for each successive step), resulting in pressure drops of ∼13% for each of a total of 16 individual Hugoniot solutions along the release path. For the release path calculations, we took advantage of the negligible difference observed when including the 2 s electrons in the silicon frozen core. This allowed for a reduction in the plane wave cutoff energy to 700 eV with a comparable level of accuracy and substantially more efficient simulations.

A release path calculated in this way from ∼3 TPa is shown as the green line in Fig. 1. Also shown for comparison (black line) is a reflection of the *α*-quartz principal Hugoniot about the particle velocity of the shocked state. Initially, the release path drops below the reflected Hugoniot (RH), due to the higher sound speed at high *P*; however at lower pressures, the release path crosses above the RH. This is due to the fact that at a given volume, the release path has significantly higher entropy, and therefore increased thermal pressure, than the corresponding state on the RH. For reference, shown as gray lines in Fig. 1, are Hugoniots for several materials that have recently been studied with *α*-quartz as a standard. For moderate impedance materials, such as CO_{2}, GDP, and H_{2}O, the difference between the release path and the RH, which has in the past been used as an approximation to the release path, is ∼2% to lower *u _{p}*, while for low impedance materials, such as D

_{2}, He, and H

_{2}, the difference can be as large as ∼5% to higher

*u*.

_{p}In accordance with the recent release model for *α*-quartz,^{5} we compared the FPMD calculated release path with that from a Mie-Grüneisen (MG) model holding Γ constant, with a linear $Us\u2212up$ Hugoniot response as the reference curve for the MG model; this model is referred to as the MG, linear reference (MGLR) model. The MGLR model has two parameters; Γ and the slope, *S*, of the linear $Us\u2212up$ Hugoniot ($Us=C0+Sup$) used for the reference curve. Note that for a given value of *S*, there is a unique value of *C*_{0} that will produce $(P1,up1)$ along the Hugoniot

where the notation *C*_{01} explicitly denotes that *C*_{0} is a function of *P* along the Hugoniot. The values of Γ and *S* can be simultaneously optimized to minimize the integral

where $uprel$ and $upFPMD$ are the particle velocities along the MGLR and FPMD release paths, respectively.

The optimal release path for the MGLR model is shown as the dashed red line in Fig. 1, with $\Gamma =0.601$ and *S* = 1.213. The MGLR release path with these values of Γ and *S* agrees quite well with the calculated FPMD release path, as can be seen by the particle velocity residual with respect to the FPMD release path shown in the right panel of Fig. 1. Note that the value of *S* obtained from the optimization is similar to that found at lower *P* (see Table II). It was also found that there exists a broad, shallow minimum in the evaluated integral [Eq. (5)] along a line in Γ-*S* space, as illustrated in Fig. 2. This broad minimum is what enabled the simplification of the reported MGLR model,^{5} allowing *S* to be held constant, thereby reducing the model to a single free parameter, Γ. Using the value of *S* = 1.197 (the same as that used in the recent release model^{5}) results in an optimized value of $\Gamma =0.582$. The corresponding release curve is shown in Fig. 1 as the dashed black line. Note that there is a negligible degradation in agreement between the MGLR and FPMD release paths with *S* = 1.197 (see also Fig. 2), suggesting that the previous analytical model with *S* = 1.197 can be suitably extended to *P* in the several TPa range.

P (TPa)
. _{H} | U (km/s)
. _{s} | $\Gamma ,S$ optimized . | Γ optimized . | ||
---|---|---|---|---|---|

Γ . | S
. | Γ . | S
. | ||

0.306 | 14.492 | 0.205 | 1.189 | 0.220 | 1.197 |

0.408 | 16.486 | 0.356 | 1.198 | 0.355 | 1.197 |

0.537 | 18.508 | 0.447 | 1.190 | 0.457 | 1.197 |

0.805 | 22.126 | 0.578 | 1.211 | 0.558 | 1.197 |

1.048 | 25.034 | 0.592 | 1.205 | 0.580 | 1.197 |

3.007 | 40.530 | 0.601 | 1.213 | 0.582 | 1.197 |

P (TPa)
. _{H} | U (km/s)
. _{s} | $\Gamma ,S$ optimized . | Γ optimized . | ||
---|---|---|---|---|---|

Γ . | S
. | Γ . | S
. | ||

0.306 | 14.492 | 0.205 | 1.189 | 0.220 | 1.197 |

0.408 | 16.486 | 0.356 | 1.198 | 0.355 | 1.197 |

0.537 | 18.508 | 0.447 | 1.190 | 0.457 | 1.197 |

0.805 | 22.126 | 0.578 | 1.211 | 0.558 | 1.197 |

1.048 | 25.034 | 0.592 | 1.205 | 0.580 | 1.197 |

3.007 | 40.530 | 0.601 | 1.213 | 0.582 | 1.197 |

## III. EXPERIMENTAL *α*-QUARTZ MEASUREMENTS

A series of planar, plate-impact, shock wave experiments were performed at the Sandia Z machine^{11} to obtain additional Hugoniot data for *α*-quartz and to extend the experimental release measurements of *α*-quartz to lower *P*. The experimental configuration used is the same as that described in Ref. 5. A silica aerogel with an initial density of ∼190 mg/cm^{3} was used as a low-impedance standard. The shock response of the aerogel has been previously investigated on the Z machine through plate-impact, shock wave experiments.^{12,13} Since the aerogel is solid, it could be directly impacted by the flyer-plate, and thus, the Hugoniot states could be inferred through simple IM with aluminum under compression, to relatively high-precision. The linear $Us\u2212up$ coefficients and covariance matrix elements for the aerogel, which were used in the analysis of the release experiments described here, are listed in Table III.

. | C_{0} (km/s)
. | S
. | $\sigma C02$ (x10^{−2})
. | $\sigma S2$ (x10^{−4})
. | $\sigma C0\sigma S$ (x10^{−3})
. |
---|---|---|---|---|---|

$\u223c190$ mg/cm^{3} aerogel | –0.385 | 1.248 | 2.631 | 2.710 | –1.493 |

Aluminum | 6.322 | 1.189 | 5.358 | 4.196 | –4.605 |

Copper | 4.384 | 1.382 | 1.344 | 6.084 | –2.689 |

. | C_{0} (km/s)
. | S
. | $\sigma C02$ (x10^{−2})
. | $\sigma S2$ (x10^{−4})
. | $\sigma C0\sigma S$ (x10^{−3})
. |
---|---|---|---|---|---|

$\u223c190$ mg/cm^{3} aerogel | –0.385 | 1.248 | 2.631 | 2.710 | –1.493 |

Aluminum | 6.322 | 1.189 | 5.358 | 4.196 | –4.605 |

Copper | 4.384 | 1.382 | 1.344 | 6.084 | –2.689 |

The *α*-quartz (single-crystal, *z*-cut, obtained from Argus International) and ∼190 mg/cm^{3} silica aerogel (fabricated by General Atomics) samples were all nominally 5 mm in lateral dimension. The thickness of the *α*-quartz was nominally 300 *μ*m, while the thickness of the silica aerogel was nominally 1000 *μ*m. The aerogel samples were metrologized using a measuring microscope to determine sample diameters and an interferometer to measure the thickness to a precision of ∼5 *μ*m and less than 1 *μ*m, respectively. Density of the silica aerogel was inferred from high-precision mass measurements and inferred volume assuming that the samples were right-circular cylinders. Slight departure from the right-circular cylinder assumption resulted in a density uncertainty of ∼2%.

The *α*-quartz samples and silica aerogel were glued together to form experimental “stacks” using the techniques described in Ref. 5. The flyer-plates and experimental “stacks” were diagnosed using a velocity interferometer (VISAR^{4}). Since all of the materials in the “stacks” are transparent, the 532 nm laser light could pass through the “stack” and reflect off the flyer-plate surface. This allowed an in-line measurement of the flyer-plate velocity from initial motion to impact. Upon impact, a shock wave of several 100 GPa was sent through the *α*-quartz sample. This shock was of sufficient magnitude that the shocked *α*-quartz became weakly reflective in the visible range. This immediate onset of reflectivity allowed for direct measurement of the shock velocity within the *α*-quartz using the VISAR diagnostic. Upon traversal of the *α*-quartz sample, the shock was transmitted into the silica aerogel and a substantial release wave was reflected back into the *α*-quartz sample. The resulting 10's of GPa shock in the silica aerogel was of sufficient magnitude that it also became weakly reflecting, allowing direct measure of the shock velocity in the silica aerogel with the VISAR diagnostic.

The measured apparent velocity, *v _{a}*, of the shock in the

*α*-quartz and silica aerogel was reduced by a factor equal to the refractive index of the unshocked material: $v=va/n0$. The values of

*n*

_{0}used in this study for

*α*-quartz and the silica aerogel were 1.547 and 1.038, respectively.

^{14–16}Ambiguity in the fringe shift upon both impact and transition of the shock velocity measurement from the

*α*-quartz sample to the silica aerogel was mitigated through the use of three different VISAR sensitivities, or velocity per fringe (vpf) settings, at each measurement location, included a high sensitivity vpf setting of 0.2771 km/s/fringe. We conservatively estimate the resolution of the VISAR system at one tenth of a fringe, resulting in uncertainty in the flyer-plate and shock velocities of a few tenths of a percent.

The flyer velocity immediately before impact and the *α*-quartz shock velocity immediately after impact enabled a Hugoniot measurement through the IM method described in Ref. 2. The linear $Us\u2212up$ coefficients and covariance matrix elements for the aluminum and copper, which were used in the analysis of the Hugoniot experiments described here, are listed in Table III.

The *α*-quartz release experiments were analyzed within the framework of the MGLR model. The measured $Usquartz$ and known *α*-quartz Hugoniot^{2,5} defined the initial state in the $P\u2212up$ plane, $(P1,up1)$. The measured shock velocity and the known Hugoniot of the silica aerogel^{13} defined the release state $(Pr,upr)$ along the *α*-quartz release path. The MGLR model, with *S* = 1.197, was then used to determine the value of $\Gamma eff$ such that the release path emanating from $(P1,up1)$ went through the point $(Pr,upr)$. Uncertainties in the inferred quantities were determined using the Monte Carlo method described in Ref. 5. Note that the uncertainty in *u _{pr}* that arises from both the uncertainty of the silica aerogel Hugoniot

^{13}and the measured shock velocity is less than 1%, and provides a tight constraint on the value of $\Gamma eff$ that connects $(P1,up1)$ and $(Pr,upr)$. This translates into an uncertainty in $\Gamma eff$ of between 0.05 and 0.1 for the individual measurements. We note that because (

*i*)

*n*

_{0}for the aerogel samples is common to both the direct impact experiments and the release experiments, and (

*ii*) the shock impedance of the silica aerogel is so much lower than the shock impedance of

*α*-quartz, $\Gamma eff$ is only weakly dependent on

*n*

_{0}and the estimated 1% uncertainty in

*n*

_{0}for the aerogel does not contribute significantly to the uncertainty in $\Gamma eff$.

A total of 9 *α*-quartz Hugoniot points were obtained in this study. The pertinent parameters for these measurements are displayed in Table IV. Additionally, four *α*-quartz release measurements were performed using ∼190 mg/cm^{3} silica aerogel as the standard to extend the empirical release model to lower *P*. The pertinent parameters for these experiments are listed in Table V. Finally, we note that in finalizing the TPX Hugoniot publication^{17} it was discovered that in the analysis of experiment Z2332 an incorrect number of fringe jumps was used for both the Hugoniot measurement (correct values listed in Ref. 17) and the release measurement (compare Table V in Ref. 5 with Table VI here). Also, a more precise value for the refractive index of TPX (full density polymethylpentene) was used ($n0=1.461$) resulting in slightly higher inferred values of $UsTPX$ in the release experiments. The revised values for $\Gamma eff$ for TPX are listed in Table VI.

Expt . | flyer . | v_{f}
. | $Usquartz$ . | $upquartz$ . | $\sigma Us2$ . | $\sigma up2$ . | $\sigma Us\sigma up$ . | P . | ρ . |
---|---|---|---|---|---|---|---|---|---|

. | . | (km/s) . | (km/s) . | (km/s) . | (×10^{−3})
. | (×10^{−3})
. | (×10^{−4})
. | (GPa) . | (g/cm^{3})
. |

Z2877 | Cu | 8.89 ± 0.05 | 12.01 | 6.16 | 1.600 | 1.586 | −1.914 | 195.9 ± 1.4 | 5.44 ± 0.04 |

Z2858 | Al | 14.59 ± 0.05 | 14.02 | 7.54 | 1.600 | 1.424 | −3.176 | 280.2 ± 1.5 | 5.73 ± 0.04 |

Z2858 | Al | 14.77 ± 0.05 | 14.16 | 7.63 | 1.600 | 1.411 | −3.262 | 286.0 ± 1.5 | 5.74 ± 0.04 |

Z2858 | Al | 15.93 ± 0.05 | 14.96 | 8.19 | 1.600 | 1.367 | −3.323 | 324.7 ± 1.5 | 5.86 ± 0.04 |

Z2586 | Al | 16.72 ± 0.05 | 15.51 | 8.57 | 1.600 | 1.343 | −3.340 | 352.4 ± 1.6 | 5.93 ± 0.04 |

Z2690 | Al | 26.97 ± 0.05 | 22.23 | 13.61 | 1.600 | 1.437 | −3.601 | 801.8 ± 2.4 | 6.84 ± 0.04 |

Z2690 | Al | 28.91 ± 0.05 | 23.52 | 14.56 | 1.600 | 1.564 | −3.601 | 907.3 ± 2.7 | 6.95 ± 0.04 |

Z2577 | Al | 31.59 ± 0.05 | 25.02 | 15.93 | 1.600 | 1.835 | −3.654 | 1056.0 ± 3.1 | 7.29 ± 0.04 |

Z2577 | Al | 31.84 ± 0.05 | 25.34 | 16.01 | 1.600 | 1.881 | −3.782 | 1075.3 ± 3.2 | 7.20 ± 0.04 |

Expt . | flyer . | v_{f}
. | $Usquartz$ . | $upquartz$ . | $\sigma Us2$ . | $\sigma up2$ . | $\sigma Us\sigma up$ . | P . | ρ . |
---|---|---|---|---|---|---|---|---|---|

. | . | (km/s) . | (km/s) . | (km/s) . | (×10^{−3})
. | (×10^{−3})
. | (×10^{−4})
. | (GPa) . | (g/cm^{3})
. |

Z2877 | Cu | 8.89 ± 0.05 | 12.01 | 6.16 | 1.600 | 1.586 | −1.914 | 195.9 ± 1.4 | 5.44 ± 0.04 |

Z2858 | Al | 14.59 ± 0.05 | 14.02 | 7.54 | 1.600 | 1.424 | −3.176 | 280.2 ± 1.5 | 5.73 ± 0.04 |

Z2858 | Al | 14.77 ± 0.05 | 14.16 | 7.63 | 1.600 | 1.411 | −3.262 | 286.0 ± 1.5 | 5.74 ± 0.04 |

Z2858 | Al | 15.93 ± 0.05 | 14.96 | 8.19 | 1.600 | 1.367 | −3.323 | 324.7 ± 1.5 | 5.86 ± 0.04 |

Z2586 | Al | 16.72 ± 0.05 | 15.51 | 8.57 | 1.600 | 1.343 | −3.340 | 352.4 ± 1.6 | 5.93 ± 0.04 |

Z2690 | Al | 26.97 ± 0.05 | 22.23 | 13.61 | 1.600 | 1.437 | −3.601 | 801.8 ± 2.4 | 6.84 ± 0.04 |

Z2690 | Al | 28.91 ± 0.05 | 23.52 | 14.56 | 1.600 | 1.564 | −3.601 | 907.3 ± 2.7 | 6.95 ± 0.04 |

Z2577 | Al | 31.59 ± 0.05 | 25.02 | 15.93 | 1.600 | 1.835 | −3.654 | 1056.0 ± 3.1 | 7.29 ± 0.04 |

Z2577 | Al | 31.84 ± 0.05 | 25.34 | 16.01 | 1.600 | 1.881 | −3.782 | 1075.3 ± 3.2 | 7.20 ± 0.04 |

Expt . | $UsQ$ . | $Usgel$ . | $\rho 0gel$ . | $\Gamma eff$ . |
---|---|---|---|---|

. | (km/s) . | (km/s) . | (mg/cm^{3})
. | . |

Z2877N | 11.07 ± 0.03 | 10.97 ± 0.03 | 194 ± 4 | −0.182 ± 0.097 |

Z2877S | 12.02 ± 0.03 | 12.20 ± 0.03 | 194 ± 4 | −0.135 ± 0.076 |

Z2858N | 14.02 ± 0.03 | 15.06 ± 0.03 | 190 ± 4 | 0.060 ± 0.051 |

Z2858S | 15.10 ± 0.04 | 16.70 ± 0.04 | 190 ± 4 | 0.175 ± 0.059 |

Expt . | $UsQ$ . | $Usgel$ . | $\rho 0gel$ . | $\Gamma eff$ . |
---|---|---|---|---|

. | (km/s) . | (km/s) . | (mg/cm^{3})
. | . |

Z2877N | 11.07 ± 0.03 | 10.97 ± 0.03 | 194 ± 4 | −0.182 ± 0.097 |

Z2877S | 12.02 ± 0.03 | 12.20 ± 0.03 | 194 ± 4 | −0.135 ± 0.076 |

Z2858N | 14.02 ± 0.03 | 15.06 ± 0.03 | 190 ± 4 | 0.060 ± 0.051 |

Z2858S | 15.10 ± 0.04 | 16.70 ± 0.04 | 190 ± 4 | 0.175 ± 0.059 |

Expt . | $UsQ$ . | $UsTPX$ . | $\rho 0TPX$ . | $\Gamma eff$ . |
---|---|---|---|---|

. | (km/s) . | (km/s) . | (g/cm^{3})
. | . |

Z2436 | 15.69 ± 0.03 | 17.67 ± 0.03 | 0.83 ± 0.004 | 0.264 ± 0.085 |

Z2450N | 16.30 ± 0.03 | 18.47 ± 0.03 | 0.83 ± 0.004 | 0.377 ± 0.077 |

Z2450S | 17.45 ± 0.03 | 19.94 ± 0.03 | 0.83 ± 0.004 | 0.476 ± 0.068 |

Z2345N | 20.45 ± 0.03 | 23.84 ± 0.03 | 0.83 ± 0.004 | 0.577 ± 0.051 |

Z2345S | 21.69 ± 0.03 | 25.50 ± 0.03 | 0.83 ± 0.004 | 0.599 ± 0.046 |

Z2333N | 22.00 ± 0.03 | 25.92 ± 0.03 | 0.83 ± 0.004 | 0.604 ± 0.045 |

Z2333S | 22.97 ± 0.03 | 27.21 ± 0.03 | 0.83 ± 0.004 | 0.595 ± 0.041 |

Z2375 | 25.19 ± 0.03 | 30.12 ± 0.03 | 0.83 ± 0.004 | 0.530 ± 0.039 |

Z2332 | 25.45 ± 0.03 | 30.63 ± 0.03 | 0.83 ± 0.004 | 0.607 ± 0.040 |

Expt . | $UsQ$ . | $UsTPX$ . | $\rho 0TPX$ . | $\Gamma eff$ . |
---|---|---|---|---|

. | (km/s) . | (km/s) . | (g/cm^{3})
. | . |

Z2436 | 15.69 ± 0.03 | 17.67 ± 0.03 | 0.83 ± 0.004 | 0.264 ± 0.085 |

Z2450N | 16.30 ± 0.03 | 18.47 ± 0.03 | 0.83 ± 0.004 | 0.377 ± 0.077 |

Z2450S | 17.45 ± 0.03 | 19.94 ± 0.03 | 0.83 ± 0.004 | 0.476 ± 0.068 |

Z2345N | 20.45 ± 0.03 | 23.84 ± 0.03 | 0.83 ± 0.004 | 0.577 ± 0.051 |

Z2345S | 21.69 ± 0.03 | 25.50 ± 0.03 | 0.83 ± 0.004 | 0.599 ± 0.046 |

Z2333N | 22.00 ± 0.03 | 25.92 ± 0.03 | 0.83 ± 0.004 | 0.604 ± 0.045 |

Z2333S | 22.97 ± 0.03 | 27.21 ± 0.03 | 0.83 ± 0.004 | 0.595 ± 0.041 |

Z2375 | 25.19 ± 0.03 | 30.12 ± 0.03 | 0.83 ± 0.004 | 0.530 ± 0.039 |

Z2332 | 25.45 ± 0.03 | 30.63 ± 0.03 | 0.83 ± 0.004 | 0.607 ± 0.040 |

## IV. EXTENSION OF HUGONIOT AND RELEASE MODEL FOR *α*-QUARTZ

The experimental Hugoniot measurements from this study (red diamonds) are plotted along with the previous plate-impact experimental results^{2,5} (blue crosses) and fit^{5} (dashed black line) in Fig. 3. These results are in good agreement with both the previously published data and fit. Also plotted in Fig. 3 are the two FPMD calculated Hugoniot points at ∼2.5 and ∼3 TPa (green diamonds). In contrast, the FPMD results exhibit shock velocities that are systematically higher than the extrapolation of the previous fit, suggesting that the previous extrapolation is too compressible, with a slope that is slightly too low. This observation that the previous fit is too compressible at *P* beyond the range of the experimental data is consistent with that of Ref. 18 on the basis of their orbital free FPMD calculations above 2 TPa.

Comparison of the FPMD calculations with experiment over the *P* range of 100–1200 GPa (0.1–1.2 TPa) demonstrates that the FPMD calculations are within 1% throughout this entire range (see Fig. 2 in Ref. 2), with the largest difference being in the *P* range where the molecular fluid undergoes dissociation into an atomic fluid. This level of agreement suggests that the FPMD calculations accurately describe the hot dense fluid, particularly at higher *P* where the effects of disorder and dissociation of the molecular fluid become less significant, and that the FPMD results in the few TPa range can be used to constrain the extrapolation of the fit to the experimental *U _{s}*–

*u*data.

_{p}We therefore performed a new least squares, weighted fit using the same functional form as that in Ref. 2

A weighting factor of 1/300 (fractional uncertainty of a few tenths of a percent, similar to that of the experimental data) was chosen such that the percent uncertainty in the fit at high *P* was of the same order as that of the previous fit in the *P* range (below about 1 TPa) constrained by the experimental data (see Fig. 4). The coefficients and covariance matrix elements for the new fit are listed in Tables VII and VIII, respectively. The difference between the previous fit and the new fit is less than 0.5% over the particle velocity range for which plate-impact experimental data exist, as can be seen in the bottom panel of Fig. 3. However, at higher *P*, in the few TPa range, the difference grows to over 1% due to the difference in their asymptotic slopes (1.193 and 1.242 for the previous^{5} and new fit, respectively). When used for IM in the few TPa regime, this behavior would tend to result in an inferred response that is systematically too compressible when using the previous fit (the inferred *u _{p}* would be too high for a given

*U*).

_{s}. | a (km/s)
. | b
. | c
. | d (km/s)^{−1}
. |
---|---|---|---|---|

α-quartz | 5.477 | 1.242 | 2.453 | 0.4336 |

. | a (km/s)
. | b
. | c
. | d (km/s)^{−1}
. |
---|---|---|---|---|

α-quartz | 5.477 | 1.242 | 2.453 | 0.4336 |

. | $\sigma a2$ . | $\sigma a\sigma b$ . | $\sigma a\sigma c$ . | $\sigma a\sigma d$ . | $\sigma b2$ . | $\sigma b\sigma c$ . | $\sigma b\sigma d$ . | $\sigma c2$ . | $\sigma c\sigma d$ . | $\sigma d2$ . |
---|---|---|---|---|---|---|---|---|---|---|

. | (×10^{–3})
. | (×10^{−4})
. | (×10^{–3})
. | (×10^{−4})
. | (×10^{−6})
. | (×10^{−4})
. | (×10^{−5})
. | (×10^{−2})
. | (×10^{–3})
. | (×10^{−4})
. |

α-quartz | 3.028 | –1.490 | –3.715 | –6.275 | 7.839 | 1.448 | 2.752 | 1.729 | 1.605 | 1.907 |

. | $\sigma a2$ . | $\sigma a\sigma b$ . | $\sigma a\sigma c$ . | $\sigma a\sigma d$ . | $\sigma b2$ . | $\sigma b\sigma c$ . | $\sigma b\sigma d$ . | $\sigma c2$ . | $\sigma c\sigma d$ . | $\sigma d2$ . |
---|---|---|---|---|---|---|---|---|---|---|

. | (×10^{–3})
. | (×10^{−4})
. | (×10^{–3})
. | (×10^{−4})
. | (×10^{−6})
. | (×10^{−4})
. | (×10^{−5})
. | (×10^{−2})
. | (×10^{–3})
. | (×10^{−4})
. |

α-quartz | 3.028 | –1.490 | –3.715 | –6.275 | 7.839 | 1.448 | 2.752 | 1.729 | 1.605 | 1.907 |

The experimental $\Gamma eff$ values from the ∼190 mg/cm^{3} aerogel (Table V) and the revised $\Gamma eff$ values from the previous TPX release experiments (Table VI) are plotted along with the previously reported^{5} $\Gamma eff$ values for both ∼110 and ∼190 mg/cm^{3} silica aerogel in Fig. 5. Also shown in Fig. 5 are the $\Gamma eff$ values determined from the FPMD release calculations. In general, there is good agreement between the FPMD results and experiment, with the possible exception being at low *P* (∼300 GPa) where the FPMD results appear to exhibit a slightly lower slope than experiment. However, this *P* range corresponds to the region where the effects of disorder and dissociation are the most significant^{2} and where the largest difference is seen between the experimental and FPMD Hugoniots. As was the case for the Hugoniot, at high *P* the agreement between the experiment and FPMD becomes much better. In particular, the $\Gamma eff$ value determined from the FPMD release calculation from ∼3 TPa suggests a saturation in $\Gamma eff$ at high *P*. The experimental data from this study at low *P* (below 300 GPa) provide a much needed constraint on the dependence of $\Gamma eff$ at lower *P* in the region of dissociation.

The experimental data and the highest *P* FPMD datum were fit to a piecewise function that was constrained to have a second order contact at the breakpoint; see Eq. (7). These data were adequately fit with a linear function at lower $UsQ$ and the same exponential function as in Ref. 5 at higher $UsQ$

We note that the fit was essentially unchanged with and without inclusion of the FPMD value at ∼3 TPa. The uncertainty in $\Gamma eff$ was determined through an analysis of the standard deviation of the measured values with respect to the value given by Eq. (7); this analysis resulted in an uncertainty in $\Gamma eff$ of 0.036, as shown in Fig. 5. Note that the previous fit for $\Gamma eff$ from Ref. 5 (gray line) is within the uncertainty of the new fit.

## V. CONCLUSION

The previously published Hugoniot^{2,5} and release model^{5} for *α*-quartz has been extended for use over the *P* range of 0.2–3 TPa. This was accomplished through additional experimental Hugoniot and release measurements (to extend the release model to lower *P*) and FPMD calculations of the Hugoniot and release of *α*-quartz in the few TPa range (to constrain the Hugoniot fit at higher *P*). The FPMD Hugoniot calculations indicated that the asymptotic slope of the previous fit to the plate-impact experimental *U _{s}*–

*u*data was too low, and were used to constrain the extrapolation of the fit to the few TPa range. The

_{p}*α*-quartz release measurements at lower

*P*(between 200 and 300 GPa) provided a much needed constraint on the dependence of $\Gamma eff$ at lower

*P*, in the region of dissociation.

The extension to the analytical model will result in negligible differences in inferred quantities with respect to the previous model when used for IM in the 0.3–1.2 TPa range. However, when used for IM in the few TPa range the new model will result in lower inferred *u _{p}* for a given

*U*. This difference is expected to be ∼1%–2% in

_{s}*u*which corresponds to ∼3%–8% lower inferred

_{p}*ρ*, given that the error in

*ρ*scales as roughly $(\rho /\rho 0\u22121)$ times the error in

*u*(in this

_{p}*P*regime $\rho /\rho 0$ is ∼4–5). While this model can now be used to ∼3 TPa, we anticipate that the model can be extrapolated to higher

*P*with some confidence; in this regime the

*P*is sufficiently high that the effects of disordering and dissociation in the shocked fluid are becoming much less significant and the behavior of the system is approaching that of an ideal gas.

## ACKNOWLEDGMENTS

The authors would like to thank the large team at Sandia that contributed to the design and fabrication of the flyer-plate loads and the fielding of the shock diagnostics. M.P.D. would like to thank Richard Kraus and Amalia Fernandez Panella for stimulating conversations and motivating the extension of the *α*-quartz Hugoniot and release model to higher pressures. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-NA0003525.