# Single-crystal elastic moduli, anisotropy, and the B1–B2 phase transition of NaCl at high pressures

Feng Xu, Laurent Belliard, Chenhui Li, Philippe Djemia, Loïc Becerra, Haijun

Huang, Bernard Perrin, Andreas Zerr

► To cite this version:

Feng Xu, Laurent Belliard, Chenhui Li, Philippe Djemia, Loïc Becerra, et al., Single-crystal elastic moduli, anisotropy, and the B1–B2 phase transition of NaCl at high pressures. *Applied Physics Letters*, 2024, 125 (16), pp.164102. 10.1063/5.0231544. *hal-05377726*

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Single-crystal elastic moduli, anisotropy and the B1-B2 phase transition of NaCl
at high pressures

Feng Xu,<sup>1,a)</sup> Laurent Belliard,<sup>2</sup> Chenhui Li,<sup>3,4</sup> Philippe Djemia,<sup>3</sup> Loïc Becerra,<sup>2</sup> Haijun Huang,<sup>1</sup> Bernard Perrin<sup>2</sup> and Andreas Zerr<sup>3,a)</sup>

<sup>1</sup>School of Physics and Mechanics, Wuhan University of Technology, 430070 Wuhan, China

<sup>2</sup>Institut des NanoSciences de Paris, CNRS UMR 7588, Sorbonne Université, 75005 Paris, France

<sup>3</sup>Laboratoire des Sciences des Procédés et des Matériaux, CNRS UPR 3407, Université Sorbonne Paris Nord, Alliance Sorbonne-Paris-Cité, 93430 Villetaneuse, France

<sup>4</sup>State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, 100083 Beijing, China

<sup>a)</sup>Authors to whom correspondence should be addressed: xufeng@whut.edu.cn and zerr@uni-paris13.fr

**Abstract**

Pressure dependences of the highest and lowest possible longitudinal sound velocities in single crystals of the B1 and B2 phases of NaCl were extracted from examination of their polycrystalline samples using the technique of time-domain Brillouin scattering. Based on the data collected to 41 GPa, we largely extended pressure range where single-crystal elastic moduli, $C_{ij}(P)$, and elastic anisotropy of the two cubic phases are measured, especially the experimental data for the B2 phase formed upon the reconstructive phase transition. The B1 phase of NaCl exhibits a strong and growing with pressure anisotropy while that of the B2 phase is much weaker. Comparing with the previous experimental $C_{ij}(P)$ of other compounds exhibiting or expected to exhibit the B1-B2 phase transition, such as CaO, MgO and $(Mg_{1-x}Fe_x)O$, we concluded that the transition is initiated by the shear instability due to violation of the Born stability criterion ($C_{44}(P)-P>0$) and predicted the presently-not-verified transition pressures for MgO and $(Mg_{1-x}Fe_x)O$.

**Main text**

Sodium chloride (NaCl), being the prototype of ionic solids, is the material of choice for investigating specifics of properties and phase transitions of multiple alkali halides, alkaline-earth oxides (MgO, CaO etc.), and transition metal nitrides/carbides (TiN, TiC etc.) crystallizing in the cubic B1 structure (space group *Fm$\overline{3}$m). Detailed examinations of high-pressure behavior of this diversity of compounds permits extraction and evaluation of contribution of composition, bonding, and structure to physical properties (e.g. elasticity, conductivity, hardness) of solids in general, and NaCl in particular. NaCl is also remarkable because it is the most abundant nonoxide compound on the Earth's surface, if that dissolved in oceans is included,<sup>1,2</sup> and because equation of state (EOS) of NaCl-B1, calculated 50 years ago,<sup>3</sup> is recommended as the primary pressure standard.<sup>4</sup> Above ~28 GPa, NaCl-B1 transforms to NaCl-B2 having cubic CsCl-type structure (space group *Pm$\overline{3}$m),<sup>5,6</sup> also proposed as a pressure standard.<sup>7</sup>

Most of high-pressure investigations of NaCl used X-ray diffraction (XRD) to examine its structure evolution and the B1-B2 phase transition.<sup>5,6,8-15</sup> However, experimental information
on pressure dependences of single-crystal elastic moduli and elastic anisotropy of the two phases is limited.¹⁶⁻¹⁸ For a cubic crystal, its elastic response to any deformation can be described by just three independent elastic moduli: $C_{11}$, $C_{12}$ and $C_{44}$ or B, $C_{44}$ and C' where $B=(C_{11}+2 \cdot C_{12})/3$ and $C'=(C_{11}-C_{12})/2$. In the latter case, bulk modulus B is used to describe the uniform volume change, $C_{44}$ simple shear along one of the symmetry directions, and C' tetragonal shear.¹⁹ Knowledge of the three moduli permits a deeper understanding of structural stability and mechanical behavior of the cubic solid under compression. For example, two of the three Born stability criteria state that cubic structure of a solid persists as long as $(C_{44}-P)>0$ and $(C'-P)>0$.¹⁹ In contrast to NaCl-B1, reliable experimental $C_{ij}(P)$ of NaCl-B2 were not available prior to this work, and experimental $C_{ij}(P)$ of the B2 phase of any other compound, formed after the reconstructive B1-B2 phase transition, have not been reported yet. Besides, scarce reports on the average longitudinal sound velocity in compressed polycrystalline NaCl-B1, $V_{L(\text{avg})}$, obtained using classical techniques such as Brillouin light scattering (BLS) or GHz-ultrasonics⁷,²⁰⁻²² are inconclusive because the growing with pressure elastic anisotropy, quantified through the Zener ratio $A=C_{44}/C'$, induces systematic errors.²³

The straight-forward way to access $C_{ij}(P)$ is measurement of longitudinal- and transversal sound velocities, $V_L$ and $V_T$, respectively, along different crystallographic directions in a compressed single crystal.²⁴ Ultrasonic techniques provide $V_L(P)$ but require big samples limiting the measurements to $P<10$ GPa.²⁵ Spontaneous BLS and impulsive-stimulated scattering are other methods to measure $C_{ij}(P)$ of transparent solids at high pressures but only if single crystals persist on compression in a diamond anvil cell (DAC).²⁶,²⁷ However, even when single crystals persist upon compression, they inevitably degrade to polycrystals upon reconstructive phase transitions²⁸,²⁹ and the measured BLS peaks represent (due to a limited resolution along the beam path) sums of contributions from differently oriented crystallites in unknown proportion. Inelastic X-ray scattering (IXS) also permits measurement of $V_L$ in a DAC but it necessities single-crystalline samples to access $C_{ij}(P)$.³⁰ In contrast, time-domain Brillouin scattering (TDBS), based on a pump-probe method,³¹ permits measuring $V_L(P)$ of transparent samples in a DAC with a high spatial resolution not only in the lateral but also in the axial direction, up to 0.15 µm and 0.17 µm, respectively.³² Accordingly, $V_L(P)$ can be measured in a single grain/crystallite (within a polycrystalline sample) along the crystallographic direction parallel to the DAC axis. Then, the so-called envelope method³³⁻³⁵ allows extraction of extremes of the $V_L$ values in a single crystal provided a large number of independent laterally resolved TDBS-data are collected. Here, we used TDBS to access $C_{ij}(P)$ of NaCl-B1 and NaCl-B2 to 41 GPa, at which the specific volume and bond length decreased by ~42% and ~17%, respectively, the degree of compression previously not accessible for such measurements.

A symmetric-type DAC with beveled diamond anvils (culet of 300 µm in diameter) was used in this work. A hole of ~100 µm in diameter drilled in a 40 µm-thick pre-indented gasket served as the sample volume which was filled with polycrystalline NaCl containing randomly oriented grains. No additional pressure transmitting medium (PTM) was used. Detailed information about the sample preparation and its poly-crystallinity is provided in supplementary material. Pressure was measured using the ruby-fluorescence scale calibrated earlier.³⁶ An absorbing opto-acoustic transducer (OAT) was placed on one of the diamond anvils, in contact with the sample. We used as OATs thin metallic films sputtered on a 10-µm-thick single-crystal of
silicon or directly on one of the diamond anvils. Ruby grains were placed around the OATs, close to locations of the TDBS measurements. Although no additional PTM was used, the degree of non-hydrostaticity in the compressed NaCl-samples was minor because yield strength ($\sigma_y$) of NaCl is similar, within the pressure range of our measurements and beyond, to that of the softest known PTMs such as solid helium, neon or argon.6,10,37 At the highest pressure of our work, the upper limit of pressure difference, $\delta P$, between locations of our TDBS measurements and the ruby grains we estimated to be 1.1 GPa applying a well-known approach.38 Details about the estimation of $\delta P$ in our samples and choice of the ruby-fluorescence scale are given in supplementary material.

In order to measure sound velocities and their distribution in our polycrystalline NaCl-samples compressed in a DAC, we applied TDBS which principals are described elsewhere.23,34,35 Here, ultrashort acoustic pulses propagating through our samples were generated and detected using picosecond pulses of a mode-locked Ti:sapphire laser (MAI-TAI Spectra, $\lambda$=800 nm, repetition rate 80 MHz, pulse width 100 fs) which radiation was split into a pump- and probe beam. The pump beam, modulated at 1.8 MHz by an acousto-optic modulator, was doubled in frequency by a BBO crystal while the probe beam was delayed by a mechanical translation stage. Both pump and probe beams were focused to spots of ~2 $\mu$m in diameter using an objective (magnification $\times$50) with working distance adapted to geometry of our DAC. The beams passed through the transparent NaCl sample which back-side was in contact with an OAT. The pump-laser pulse generated a coherent acoustic pulse (CAP) near the OAT-surface which then propagated through the sample. Probe-laser pulses permitted measuring the CAP velocity: In particular, we measured transient reflectivity of the sample (in which the CAP was moving along the sample axis) as a function of the time delay with respect to the pump-laser pulse. Brillouin oscillations, as a function of the delay time, resulting from the interference of the probe-laser beam scattered by the CAP and by stationary interfaces (see inset in Fig. 1) were measured by a photodetector. Their frequencies, $f_B$, varied with the delay time, are proportional to the local $V_L$ of the sample material at particular pressure:31

$$f_B = 2nV_L/\lambda \quad (1)$$

where $\lambda$=800 nm is wavelength of the probe-beam in vacuum and n refractive index of NaCl at this wavelength. Frequencies corresponding to $V_T$ were not recognized in the collected TDBS signals. A CAP moving in the polycrystalline NaCl sample provided local $V_L$ along crystallographic direction in the traversed grain parallel to the direction of the CAP propagation. Temporal window $\Delta t \approx 1$ oscillation, selected for the signal-frequency evaluation using the Short-Time Fourier Transform (STFT), resulted in the axial/depth resolution of ~250 nm which didn't degrade with pressure. Consideration of the TDBS signal-segments, where the oscillation amplitudes exceeded 35% of the maximal one, permitted STFT-analysis of 7-23 depth segments providing independent local $f_B$ and, consequently, $V_L$ values. According to earlier publications, grain sizes in our NaCl samples were similar or larger than the above-stated axial resolution: In analogous TDBS-experiments on H$_2$O-ice compressed in a DAC,39 crystallites of relatively soft solids (e.g. H$_2$O or NaCl) were shown to have typical sizes $\le$450 nm. This followed from the observation of complete XRD-rings collected from polycrystalline samples of cubic H$_2$O-ice and NaCl.6,39 On the other hand, because the XRD-peaks of H$_2$O and NaCl remained relatively
sharp on compression, grain sizes of H₂O or NaCl couldn't decrease significantly below ~100 nm.<sup>6,39</sup> Therefore, size of grains in our NaCl sample (100-450 nm) was comparable with the axial resolution of our TDBS-signal processing. This permitted a reliable extraction of the maximal and minimal V<sub>L</sub> in a single crystal of NaCl applying the envelope method to a large number of spatially-resolved V<sub>L</sub> values extracted from the independent TDBS-signal segments. For each sample and at each pressure step, we collected one/two TDBS-signals, performed the STFT analysis and extracted the local frequencies. In Fig. 1, each pair of the triangles showing up and down, at each pressure step, represents the uppermost, f<sub>B(max)</sub>, and the lowermost, f<sub>B(min)</sub>, frequency among several intermediate frequencies recovered from each TDBS-signal. The intermediate frequencies are not shown in order to avoid the figure's overloading. Apparently, not each of the analyzed TDBS signals contained segments with the frequency equal to one of the two possible extreme values, f<sub>max</sub> and f<sub>min</sub>, corresponding to the fastest and slowest V<sub>L</sub> in a single crystal, respectively. In the present work, six experimental runs provided a large number of independent f<sub>B(max)</sub> and f<sub>B(min)</sub> values permitting a reliable application of the envelope method and extraction of f<sub>max</sub> and f<sub>min</sub> with a high degree of confidence. A strong deviation of f<sub>max</sub> from f<sub>min</sub> in the pressure region of thermodynamic stability of NaCl-B1 (Fig. 1) indicated its pronounced elastic anisotropy when compared with the higher-pressure NaCl-B2 phase. Here, the experimental values at P≤28 GPa we attributed to NaCl-B1, and those at P≥32 GPa to NaCl-B2. The intermediate pressures, where both phases could coexist,<sup>6,13,14</sup> were omitted (see supplementary material).

General consideration of elastic behavior of cubic solids and all available experimental and
theoretical information on C<sub>ij</sub>(P) of NaCl-B1 and NaCl-B2 indicate that the crystallographic
directions (100) and (111) are the fastest and the slowest directions of propagation of
longitudinal CAPs in single crystals of these phases, respectively. This follows, in particular,
from all published DFT calculations of C<sub>ij</sub>(P) of NaCl-B1 and NaCl-B2<sup>6,40-42</sup> (see
supplementary material for further details). Accordingly, a second-order polynomial was fitted,
for each phase, to the f<sub>max</sub> data-points to obtain a smooth dependence f<sub>max</sub>(P) and then V<sub>L(100)</sub>(P)
was recovered applying Equation (1) and our calculated n(P) (Fig. 2, insert). Similarly, a
smooth V<sub>L(111)</sub>(P) dependence was derived from the f<sub>min</sub> data-points. Because pressure
dependence of refractive index of NaCl, n(P), is not well established experimentally, we derived
it from DFT calculations using the hybrid functional GGA-HSE06 known to provide accurate
optical properties.<sup>43</sup> A detailed description of simulation of dielectric properties of solids using
this functional is given elsewhere.<sup>44</sup>
<figure><img src="image_2.png" /><figcaption>FIG. 1. Frequencies $f_{B(max)}$ and $f_{B(min)}$, for each of the TDBS signals collected from the compressed polycrystalline NaCl samples, are represented by triangles showing up and down, respectively. The extremes, $f_{max}$ and $f_{min}$, are highlighted as solid symbols with error bars. Fits of quadratic polynomials, $f_{max}(P)$ and $f_{min}(P)$, to the latter data-points are shown by solid lines. Gray bands represent experimental uncertainties. Inset: A raw TDBS signal showing transient reflectivity of the sample as a function of the pump-probe delay time recorded at $P=41$ GPa, and schematics of the sample inside a DAC, showing the pump and probe beams.</figcaption></figure>

At $P<8$ GPa, our experimental $V_{L(100)}(P)$ and $V_{L(111)}(P)$ of NaCl-B1 (Fig. 2) agreed with the previous measurements on single crystals.¹⁶⁻¹⁸ As already recognized for other cubic solids,³⁴,³⁵ the earlier measured $V_{L(avg)}(P)$ of polycrystalline NaCl²⁰⁻²² approached $V_{L(111)}(P)$ because in any cubic crystal (i) directions close to $\langle 110 \rangle$ have the highest multiplicity and thus dominate signal from a polycrystalline sample collected using conventional techniques such as BLS or ultrasonics, (ii) $V_{L(110)}$ is much closer to $V_{L(111)}$ than to $V_{L(100)}$ and the collected BLS peaks are asymmetric, and (iii) the asymmetry becomes pronounced when elastic anisotropy increases.²³ Our measurements could not be affected by sample texturing because it was reported to be weak up to the B1-B2 phase transition.⁶ Accordingly, the number of crystallites oriented with $\langle 111 \rangle$ and $\langle 100 \rangle$ along the DAC axis was significant and a possible weak texture effect was compensated by our numerous independent spatially resolved $V_L$ measurements. Our data are further supported by a good agreement with the unique BLS data-point obtained for a single crystal of NaCl-B1 at $P=26$ GPa⁴⁵ (Fig. 2). Previous IXS measurements provided $V_{L(100)}(P)$ consistent with our results, but the $V_{L(111)}(P)$ dramatically deviated, upon compression, both from our $V_{L(111)}(P)$ and the earlier data-point at $P=26$ GPa.
<figure><img src="image_3.png" /><figcaption>FIG. 2. $V_{L(100)}$ and $V_{L(111)}$ of compressed NaCl-B1 and NaCl-B2: Solid black lines indicate our experimental results and grey bands their uncertainties. They agree with the earlier measurements on single crystals performed using BLS<sup>16,45</sup> (blue solid circles and diamonds) and ultrasonics<sup>17</sup> (green solid squares). Violet solid triangles show single-crystal data obtained using IXS.<sup>18</sup> Earlier experimental and theoretical $V_{L(\text{avg})}(P)$ are shown as follows: ultrasonics - green open triangles<sup>20</sup> and circles<sup>21</sup>; BLS - blue open squares<sup>22</sup> and diamonds<sup>7</sup> (for NaCl-B2); TDBS - orange dashed lines<sup>46</sup> (for both NaCl-B1 and NaCl-B2); Eulerian calculations - magenta solid line.<sup>47</sup> Inset: $n(P)$ of NaCl, we calculated using the GGA-HSE06 functional (red solid lines), is compared with earlier shock-compression data: solid squares<sup>48</sup> and open squares.<sup>49</sup> Black dashed line shows extrapolated optical measurements,<sup>50</sup> black solid line - theoretical dependence.<sup>51</sup></figcaption></figure>

Our experimental $V_{L(100)}(P)$ and $V_{L(111)}(P)$ combined with $B(P) = (C_{11}(P)+2C_{12}(P))/3$ permitted determining $C_{ij}(P)$ (see supplementary material), and thus the principal shear moduli $C_{44}(P)$ and $C'(P)$, of both NaCl phases.<sup>23</sup> Here, $B(P)$ of NaCl-B1 and of NaCl-B2 were taken from the Decker's EOS<sup>3</sup> and Ref.<sup>52</sup>, respectively. Our experimental $C_{44}(P)$ and $C'(P)$ of NaCl-B1 agreed with the earlier low-pressure measurements,<sup>16,17</sup> the BLS data-point at $P=26$ GPa<sup>45</sup> and data in a thesis work reporting on radial-XRD measurements on polycrystalline NaCl compressed in a DAC.<sup>53</sup> The IXS technique provided $C'(P)$ agreeing with our observations but the $C_{44}(P)$ deviated significantly upon compression. Flattening of our experimental $C'(P)$ at $P>25$ GPa cannot be explained by a sample-state change because the experimental $C_{44}(P)$ increased monotonically. For NaCl-B2, $C'(P)$ was found to drop significantly while $C_{44}(P)$ changed weakly. We also derived $G(P)$ of both NaCl phases (Fig. 3b) using the Voigt, Reuss and Hill approximations.<sup>54</sup> Our experimental $G(P)$ of NaCl-B2 agrees perfectly with the unique BLS data obtained for polycrystalline NaCl-B2<sup>7</sup> (Fig. 3b). Apparently, the weaker elastic anisotropy of NaCl-B2 led to less asymmetric BLS peaks from the polycrystalline samples and thus a reasonable $G(P)$.<sup>23</sup> This agreement further supports our TDBS results. Finally, our experimental $G(P)$ confirms the radial-XRD results<sup>53</sup> for NaCl-B1 while agreement for NaCl-B2 is rather qualitative due to a strong scattering of the radial-XRD data.
<figure><img src="image_4.png" /><figcaption>FIG. 3. Our experimental and theoretical *C*'(P), *C*₄₄(P), *G*(P) and *A*(P) of NaCl-B1 and NaCl-B2 compared with earlier experiments. All data, shown in Fig. 2, are presented here in the same style. Green crosses indicate the radial-XRD data.⁵³ For *G*(*P*), solid lines show result of the Voigt-Reuss-Hill approximation and the grey band its uncertainty. Dashed and dotted lines represent results of the Voigt and Reuss approximation, respectively.</figcaption></figure>

Comparison of our experimental pressure dependences $C_{44}(P)$ and $C'(P)$ of NaCl-B1 with those of MgO, CaO, and (Mg$_{1-x}$,Fe$_x$)O permitted proposing the driving force of the B1-B2 phase transitions for all the compounds and to predict pressures of the presently not verified transitions of MgO and (Mg$_{1-x}$,Fe$_x$)O. The latter compounds are of outstanding importance for geosciences because MgO is the end-member of the solid solution (Mg$_{1-x}$,Fe$_x$)O called ferropericlase, one of the two most abundant nonconducting compounds of our planet.⁵⁵ The other compound is (Mg,Fe)SiO$_3$-rich perovskite and/or post-perovskite.⁵⁶ It appears that the B1-B2 phase transitions are initiated by a simple shear instability along one of the symmetry directions of the B1 structure manifesting itself in violation of the Born stability criterion ($C_{44}(P)-P>0$):¹⁹ According to our measurements, ($C_{44}(P)-P$) of NaCl-B1 vanishes at $P$~30 GPa (Fig. 4a), consistent with the experimental transition pressure $P_{tr}$ determined using XRD techniques. In contrast, the second Born stability criterion ($C'(P)-P$)>0 holds for NaCl-B1. Similarly, the experimental ($C_{44}(P)-P$) of CaO-B1 vanishes at $P$~58 GPa perfectly matching $P_{tr}$=57.8 GPa from XRD measurements⁵⁷ while ($C'(P)-P$) remains positive (Fig. 4b). All experimental data for (Mg$_{1-x}$,Fe$_x$)O suggest that only ($C_{44}(P)-P$) becomes negative below 1 TPa (Fig. 4c) but its $P_{tr}$ is not determined yet: Using the most recent measurements of $C_{ij}(P)$ to the highest $ρ(P)/ρ_0$,²⁶ we predict $P_{tr}$ between 200 and 300 GPa applying a linear extrapolation of the last part of the experimental ($C_{44}(P)-P$) dependence (Fig. 4c). Our recent $C_{ij}(P)$ data for MgO-B1⁵⁸ show a similar tendency (Fig. 4d): The extrapolation of our experimental ($C_{44}(P)-P$)
suggests $P_{tr} \sim 710$ GPa while the uncertainties cover the pressure region comprising $P_{tr} \sim 600$ GPa proposed from the laser-driven ramp-compression measurements.⁵⁹

<figure><img src="image_5.png" /></figure>

**FIG. 4.** Born stability criteria for B1 phases of (a) NaCl (this work), (b) CaO⁵⁷, (c) $(Mg_{1-x}Fe_x)O$²⁶ and (d) MgO⁵⁸ derived using experimental data only. $C'(P)$ and $(C'(P)-P)$ are shown by black solid lines, while $C_{44}(P)$ and $(C_{44}(P)-P)$ by red solid lines. For $(Mg_{1-x}Fe_x)O$ and MgO, $(C_{44}(P)-P)$ are shown by solid lines within the experimental pressure regions and shaded bands represent the experimental uncertainties.²⁶,⁵⁸ At higher pressures, dashed lines and shaded bands represent linear extrapolations of the last experimental data-points and their uncertainties, respectively.

Summarizing, our TDBS measurements on polycrystalline NaCl, compressed in a DAC to 41 GPa, yielded extremes of $V_L$ in single crystals of both NaCl-B1 and NaCl-B2, $V_{L(100)}(P)$ and $V_{L(111)}(P)$. From these data, we derived the dependences $C_{44}(P)$, $C'(P)$, $G(P)$ and $A(P)$, while those for NaCl-B2 could not be measured with earlier-used techniques requiring significantly big isolated single crystals not accessible after the B1-B2 phase transition. We have found that NaCl-B1 exhibits a strong and growing with pressure elastic anisotropy, while that of NaCl-B2 is weaker. This supports the earlier proposal⁷ to consider NaCl-B2 as an alternative pressure standard above ~32 GPa. Last but not least, our and earlier measurements indicate that violation of the Born stability criterion ($C_{44}(P)-P$)>0 portends the B1-B2 phase transition in NaCl, CaO, $(Mg_{1-x}Fe_x)O$ and, most probably, MgO.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional details on: (i) Sample preparation and its polycrystallinity; (ii) Estimation of pressure gradient and choice of the ruby-fluorescence pressure scale; (iii) Pressure range of the B1-B2 phase transition of NaCl; and (iv) Orientation dependence of sound velocities in cubic crystals of NaCl-B1 and NaCl-B2.
## ACKNOWLEDGEMENTS

The authors acknowledge financial support from NSFC-41504070, NSFC-42274124, ZBYY-80922010601, CSC-201606955092 and the French National Research Agency, ANR (project I2T2M, No. ANR-18-CE42-0017). We thank E. Peronne for experimental assistance, V. E. Gusev, X. Liu and Y. Wu for discussions. We acknowledge access to the cluster MAGI of University Sorbonne Paris Nord and N. Greneche for the support.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Feng Xu:** Data curation (lead); Funding acquisition (supporting); Writing – original draft (equal); Writing – review & editing (equal). **Laurent Belliard:** Conceptualization (equal); Data curation (equal); Investigation (supporting). **Chenhui Li:** Software (equal); Data curation (supporting). **Philippe Djemia:** Software (equal); Data curation (supporting). **Loïc Becerra:** Investigation (supporting). **Haijun Huang:** Investigation (supporting); Funding acquisition (lead). **Bernard Perrin:** Investigation (supporting). **Andreas Zerr:** Conceptualization (equal); Data curation (equal); Investigation (supporting); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

1. J. Lyman and R. H. Fleming, J. Mar. Res. 134 (1940).

2. D. C. Blanchard, J. Geophys. Res. Oceans 90, 961-963 (1985).

3. D. L. Decker, J. Appl. Phys. 42, 3239-3244 (1971).

4. V. E. Bean, S. Akimoto, P. M. Bell, S. Block, W. B. Holzapfel, M. H. Manghnani, M. F. Nicol, and S. M. Stishov, Physica B+C 139-140, 52-54 (1986).

5. G. J. Piermarini and S. Block, Rev. Sci. Instrum. 46, 973-979 (1975).

6. Z. Mi, S. R. Shieh, A. Kavner, B. Kiefer, H.-R. Wenk, and T. S. Duffy, J. Appl. Phys. 123, 135901 (2018).

7. M. Murakami and N. Takata, J. Geophys. Res. Solid Earth 124, 6581-6588 (2019).

8. W. A. Bassett, T. Takahashi, H. K. Mao, and J. S. Weaver, J. Appl. Phys. 39, 319-325 (1968).

9. N. Sata, G. Shen, M. L. Rivers, and S. R. Sutton, Phys. Rev. B 65, 104114 (2002).

10. T. Sakai, E. Ohtani, N. Hirao, and Y. Ohishi, J. Appl. Phys. 109, 084912 (2011).

11. W. Zhang, A. R. Oganov, A. F. Goncharov, Q. Zhu, S. E. Boulfelfel, A. O. Lyakhov, E. Stavrou, M. Somayazulu, V. B. Prakapenka, and Z. Konôpková, Science 342, 1502-1505 (2013).

12. L. G. Liu and W. A. Bassett, J. Appl. Phys. 44, 1475-1479 (1973).

13. S. Ono, T. Kikegawa, and Y. Ohishi, Solid State Commun. 137, 517-521 (2006).

14. D. L. Heinz and R. Jeanloz, Phys. Rev. B 30, 6045-6050 (1984).
15 Y. Sato-Sorensen, J. Geophys. Res. Solid Earth **88**, 3543-3548 (1983).

16 C. H. Whitfield, E. M. Brody, and W. A. Bassett, Rev. Sci. Instrum. **47**, 942-947 (1976).

17 H. Kinoshita, N. Hamaya, and H. Fujisawa, J. Phys. Earth **27**, 337-350 (1979).

18 H. Fukui, A. Yoneda, S. Kamada, H. Uchiyama, S. Tsutsui, N. Hirao, and A. Q. Baron, High Pressure Res. **40**, 465-477 (2020).

19 J. Wang, J. Li, S. Yip, S. Phillpot, and D. Wolf, Phys. Rev. B **52**, 12627-12635 (1995).

20 J. Frankel, F. J. Rich, and C. G. Homan, J. Geophys. Res. **81**, 6357-6363 (1976).

21 M. Matsui, Y. Higo, Y. Okamoto, T. Irifune, and K.-I. Funakoshi, Am. Mineral. **97**, 1670-1675 (2012).

22 A. J. Campbell and D. L. Heinz, Science **257**, 66-68 (1992).

23 F. Xu, P. Djemia, L. Belliard, H. Huang, B. Perrin, and A. Zerr, J. Appl. Phys. **130**, 035903 (2021).

24 Z. Mao, D. Fan, J.-F. Lin, J. Yang, S. N. Tkachev, K. Zhuravlev, and V. B. Prakapenka, Earth Planet. Sci. Lett. **426**, 204-215 (2015).

25 B. Li, J. Kung, and R. C. Liebermann, Phys. Earth Planet. Inter. **143-144**, 559-574 (2004).

26 J. Yang, X. Tong, J.-F. Lin, T. Okuchi, and N. Tomioka, Sci. Rep. **5**, 17188 (2015).

27 J. C. Crowhurst, J. M. Brown, A. F. Goncharov, and S. D. Jacobsen, Science **319**, 451-453 (2008).

28 B. Zhao, F. Xu, L. Belliard, H. Huang, B. Perrin, P. Djemia, and A. Zerr, Phys. Status Solidi-R **13**, 1900173 (2019).

29 A. D. Rosa, N. Hilairet, S. Ghosh, J.-P. Perrillat, G. Garbarino, and S. Merkel, J. Geophys. Res. Solid Earth **121**, 7161-7176 (2016).

30 D. Antonangeli, J. Siebert, C. M. Aracne, D. L. Farber, A. Bosak, M. Hoesch, M. Krisch, F. J. Ryerson, G. Fiquet, and J. Badro, Science **331**, 64-67 (2011).

31 C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Phys. Rev. B **34**, 4129-4138 (1986).

32 S. Sandeep, T. Thréard, E. De Lima Savi, N. Chigarev, A. Bulou, V. Tournat, A. Zerr, V. E. Gusev, and S. Raetz, J. Appl. Phys. **130**, 053104 (2021).

33 M. Grimsditch, P. Loubeyre, and A. Polian, Phys. Rev. B **33**, 7192-7200 (1986).

34 M. Kuriakose, S. Raetz, Q. M. Hu, S. M. Nikitin, N. Chigarev, V. Tournat, A. Bulou, A. Lomonosov, P. Djemia, V. E. Gusev, and A. Zerr, Phys. Rev. B **96**, 134122 (2017).

35 S. Raetz, M. Kuriakose, P. Djemia, S. M. Nikitin, N. Chigarev, V. Tournat, A. Bulou, A. Lomonosov, V. E. Gusev, and A. Zerr, Phys. Rev. B **99**, 224102 (2019).

36 A. Dewaele, M. Torrent, P. Loubeyre, and M. Mezouar, Phys. Rev. B **78**, 104102 (2008).

37 S. Klotz, J. C. Chervin, P. Munsch, and G. Le Marchand, Journal of Physics D: Applied Physics **42**, 075413 (2009).

38 M. I. Eremets, *High Pressure Experimental Methods*. (Oxford University Press, 1996).

39 S. M. Nikitin, N. Chigarev, V. Tournat, A. Bulou, D. Gasteau, B. Castagnede, A. Zerr, and V. E. Gusev, Sci. Rep. **5**, 9352 (2015).

40 L. Liu, Y. Bi, J. Xu, and X. Chen, Physica B **405**, 2175-2180 (2010).

41 X.-H. Xiao, J. Zhu, X.-R. Chen, and W.-Q. Yang, Chinese Physics Letters **23**, 2845-2847 (2006).

42 F. Xu, L. Belliard, C. Li, P. Djemia, L. Becerra, H. Huang, B. Perrin, and A. Zerr, arXiv (2022). https://doi.org/10.48550/arXiv.2209.10899

43 A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. **125**, 224106
(2006).

44 M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, Phys. Rev. B **73**, 045112 (2006).

45 S. Sinogeikin, J. Bass, V. Prakapenka, D. Lakshtanov, G. Shen, C. Sanchez-Valle, and M. Rivers, Rev. Sci. Instrum. **77**, 103905 (2006).

46 W.-P. Hsieh, Sci. Rep. **11**, 21321 (2021).

47 F. Birch, J. Geophys. Res. Solid Earth **83**, 1257-1268 (1978).

48 J. Wackerle and H. Stacy, in *Shock Waves in Condensed Matter*, edited by S. C. Schmidt and N. C. Holmes (Elsevier, Amsterdam, 1988).

49 S. B. Kormer, Sov. Phys. Usp. **11**, 229 (1968).

50 R. Waxler and C. Weir, J. Res. Nat. Bur. Stand. **69**, 325 (1965).

51 P. G. Johannsen, Phys. Rev. B **55**, 6856-6864 (1997).

52 Y. Fei, A. Ricolleau, M. Frank, K. Mibe, G. Shen, and V. Prakapenka, Proc. Natl. Acad. Sci. **104**, 9182-9186 (2007).

53 Z. Mi, Ph.D. thesis, University of Western Ontario, 2013.

54 R. Hill, Proc. Phys. Soc. **65**, 349 (1952).

55 B. Romanowicz and H.-R. Wenk, Phys. Earth Planet. Inter. **269**, 58-90 (2017).

56 M. Murakami, K. Hirose, K. Kawamura, N. Sata, and Y. Ohishi, Science **304**, 855-858 (2004).

57 S. Speziale, S. R. Shieh, and T. S. Duffy, J. Geophys. Res. Solid Earth **111**, B02203 (2006).

58 X. Zhang, C. Li, F. Xu, J. Zhang, C. Gao, Z. Cheng, Y. Wu, X. Liu, A. Zerr, and H. Huang, J. Geophys. Res. Solid Earth **128**, e2022JB026311 (2023).

59 F. Coppari, R. F. Smith, J. H. Eggert, J. Wang, J. R. Rygg, A. Lazicki, J. A. Hawreliak, G. W. Collins, and T. S. Duffy, Nat. Geosci. **6**, 926-929 (2013).