Elastic anisotropy and single-crystal moduli of solid argon up to 64 GPa from time-domain Brillouin scattering

Single-crystal elastic moduli $C_{ij}$, shear modulus $G$, and Zener anisotropic ratio $A$ of solid argon having the face-centered cubic (fcc) structure were determined at high pressures between 12 and 64 GPa using a combination of experimental and theoretical approaches. The experimental data, namely, the maximal and minimal values of the product of the longitudinal sound velocity and refractive index $n·V_L$, were obtained from the 3D scanning of elastic inhomogeneities in compressed samples of argon using the technique of time-domain Brillouin scattering. These inhomogeneities, caused by elastic anisotropy of solid argon, were revealed to be about twice as strong as those reported in the earlier experiments using classical Brillouin light scattering (BLS). To derive the $V_L$ values, we used the refractive index obtained here from ab initio calculations which also permitted us to rule out any contribution to the amplitude of the observed elastic inhomogeneities of the hexagonal close-packed phase of argon, proposed to coexist with the fcc phase at high pressures. From the measured $C_{ij}\left(P\right)$, we derived pressure dependence of shear modulus of the fcc argon $G_H\left(P\right)$ using the Voigt-Reuss-Hill approximation and found a very good agreement with the earlier $G\left(P\right)$ obtained from shear sound velocities in the classical BLS measurements. Our results agree very well with the earlier predictions based on a relatively simple many-body model employing the Buckingham pair potential. Finally, our measurements show a much weaker change of the Cauchy discrepancy $(C_{12}-C_{44}-2P)$ of the fcc argon with pressure than reported in all earlier experimental and theoretical works.

Figure 1

Experimental and theoretical data on longitudinal sound velocities of solid argon at high pressures. (a) Our results in terms of the product $n·V_L$ obtained using the TDBS technique, represented by triangles pointing up and down, correspond to the maximal and minimal values, respectively. The representation in terms of $n·V_L$ was used here because it is not influenced by the choice of the refractive index. The smaller (dark red) and bigger (red) solid triangles indicate our results for the narrower and wider temporal window of one and two-and-a-half Brillouin oscillations, respectively. Experimental uncertainties in $n·V_{L\text{max}}$ and $n·V_{L\text{min}}$ are shown for the narrower temporal window only, because for the wider one the uncertainties are smaller than the symbols. The pressure uncertainties are the same for both data sets. Thin solid (black) lines as well as the open (blue) triangles pointing up and down represent $n·V_{L\text{max}}$ and $n·V_{L\text{min}}$, respectively, measured using the classical BLS in Refs. [14, 15]. Open (green) diamonds and open (magenta) circles indicate averaged values $n·V_{L\text{av}}$ obtained using the same technique [11, 16]. To be able to compare results reported in Refs. [14, 16], we multiplied their directly measured $V_L$ values with our theoretical $n\left(P\right)$ shown below in Fig. 5. (b) Our experimental results on $V_{L\text{max}}$ and $V_{L\text{min}}$ compared with the theoretical calculations of $V_{L\langle 111\rangle }$ and $V_{L\langle 100\rangle }$ for the fcc argon as a function of pressure. Our theoretical results are shown by thick solid (red) lines. Broken lines represent results of the earlier theoretical calculations: Dashed (orange) line—Pechenik et al. [13]; dotted (dark blue) line—Aoki and Kurokawa [10]; dash-dotted (dark pink) line—Iitaka and Ebisuzaki [12]; dash-double-dotted (green) line—Grimsditch et al. [11].

Figure 2

Pressure dependencies of (a) $C_{ij}$ and of (b) the Zener ratio $A$ of the fcc argon obtained experimentally and theoretically. Except our experimental data, all other theoretical and experimental results are shown in the same style as in Fig. 1. (a) The solid (dark-red) triangles pointing up and down and the rhombuses represent our experimental values of $C_{11}\left(P\right),C_{12}\left(P\right)$, and $C_{44}\left(P\right)$, respectively. Here and below, pressure uncertainties in our experiments (the same as in Fig. 1) are not shown for clarity of the presentation. (b) Our experimental values of the Zener ratio, derived from our experimental $C_{ij}\left(P\right)$ shown in (a), are represented by solid (dark-red) squares.

Figure 3

Theoretical and experimental pressure dependencies of the Cauchy discrepancy $(C_{12}-C_{44}-2P)$. Except our experimental data represented by solid (dark-red) squares, all other results are shown in the same style as in Fig. 1. The theoretical result of Ref. [11], predicting that the Cauchy discrepancy approaches zero, is not shown.

Figure 4

Raw transient reflectivity signals $S\left(t\right)$ for pressures of (bottom to top): 11.5 GPa, 28 GPa, 46.3 GPa, and 63.6 GPa.

Figure 5

Experimental and theoretical refractive index of solid argon as a function of pressure. The experimental results of Chen et al. [15] and Grimsditch et al. [22] are shown as solid dark (blue) and light (violet) bands, respectively, embracing the scattering data with error bars. The thick solid (red) line represents our calculated $n\left(P\right)$ while the dash-double-dotted lines represent the earlier predicted $n\left(P\right)$ for the two wavelengths [11].

Figure 6

Comparison of the earlier experimental (thin solid lines) and theoretical (broken lines) $B\left(P\right)$ with our calculation (thick solid (red) line). The experimental values were obtained by Richet et al. [8] (blue line), Errandonea et al. [2] (black line), and Marquardt et al. [16] (magenta line). The earlier theoretical results are shown in the same style as in Fig. 1.

Figure 7

(a) Pressure dependence of the shear modulus $G_H$ of polycrystalline fcc argon derived from our experimental $C_{ij}\left(P\right)$ using the Voigt-Reuss-Hill approximation (solid dark-red squares) in comparison with the earlier reported $G\left(P\right)$ obtained in the classical BLS experiments by Marquardt et al. [16] from the averaged $V_{T\text{av}}\left(P\right)$ (solid magenta line). $G_H\left(P\right)$ derived from the earliest experimental $C_{ij}\left(P\right)$ reported in Ref. [11] are represented by the (green) dash-double-dotted line. Vertical bars show experimental uncertainties for our experimental data-set and for that of Marquardt et al. [16]. (b) $V_{T\langle 100\rangle }\left(P\right)$ (solid dark-red triangles pointing up) and $V_{T\langle 110\rangle }\left(P\right)$ (solid dark-red triangles pointing down) of the fcc argon derived from our $C_{ij}\left(P\right)$ shown in Fig. 2. Our theoretical $V_{T\langle 100\rangle }\left(P\right)$ and $V_{T\langle 110\rangle }\left(P\right)$ are represented by solid (red) lines. Solid (blue) circles indicate the scattering $V_T$ values measured for a polycrystalline sample by Chen et al. [15]. Other previous experimental and theoretical results are shown in the same way as in Fig. 1.

Figure 8

Comparison of the maximal and minimal products $n·V_L$ for the fcc phase (solid red lines) and hcp phase (dashed violet lines) of solid argon at high pressures calculated in this paper. The theoretical results are compared with our experimental $n\left(P\right)·V_{L\text{max}}\left(P\right)$ and $n\left(P\right)·V_{L\text{min}}\left(P\right)$ obtained for the two temporal windows applied to the TDBS signals. Meaning of all symbols is the same as in Fig. 1.