Elastic constants of solid ${}^4$He under pressure: Diffusion Monte Carlo study

We study the elasticity of perfect solid ${}^4$He at zero temperature using the diffusion Monte Carlo method and a realistic semiempirical pairwise potential to describe the He-He interactions. We calculate the value of the elastic constants of hcp helium $\left\{C_{ij}\right\}$ as a function of pressure from zero up to $\sim 110$ bar. It is found that the pressure dependence of all nonzero elastic constants is linearly increasing and we provide accurate parametrization of each of them. Our $C_{ij}$ results are compared to previous variational calculations and low-temperature measurements and, in general, we find notably good agreement among them. Furthermore, we report results for the Grüneisen parameters, sound velocities, and Debye temperature over a wide range of pressures. This work represents a comprehensive quantum atomistic calculation of the elastic properties of solid helium under compression.

Figure 1

(a) Representation of the hcp unit cell with primitive translational vectors ${\mathbf{a}}_{\mathbf{1}}$, ${\mathbf{a}}_{\mathbf{2}}$, and ${\mathbf{a}}_{\mathbf{3}}$, and two-atom basis set (see text). (b) Sketch of the 200-atom supercell used in the pure shear calculations, which is built in by replicating the hcp unit cell $5\times 5\times 4$ times along the primitive translational vectors ${\mathbf{a}}_{\mathbf{1}}$, ${\mathbf{a}}_{\mathbf{2}}$, and ${\mathbf{a}}_{\mathbf{3}}$, respectively.

Figure 2

Energy per particle results obtained by varying the degree of shear stress, as defined in Eq. (8), in perfect hcp ${}^4$He at density $\rho =0.480{\sigma }^{-3}$. The equilibrium value corresponding to the undistorted hcp structure is found at $\epsilon =0$ and the solid line represents a third-order polynomial fit to the energies. Statistical uncertainties in the energies are represented with error bars.

Figure 3

Zero-temperature $C_{11}$ and $C_{12}$ elastic constants of perfect hcp ${}^4$He as a function of pressure. Previous variational Monte Carlo (VMC) calculations (Ref. 23) and experimental data (Refs. 2 and3) are shown for comparison. The vertical dotted line represents the zero-temperature freezing pressure of ${}^4$He and the straight dashed lines are linear fits to the DMC results (see text).

Figure 4

Zero-temperature $C_{13}$ and $C_{44}$ elastic constants of perfect hcp ${}^4$He as a function of pressure. Previous variational Monte Carlo (VMC) calculations (Ref. 23) and experimental data (Refs. 2, 3, and 5) are shown for comparison. The vertical dotted line represents the zero-temperature freezing pressure of ${}^4$He and the straight dashed lines are linear fits to the DMC results (see text).

Figure 5

(a) Zero-temperature $C_{33}$ elastic constant of perfect hcp ${}^4$He as a function of pressure. A previous variational Monte Carlo (VMC) calculation (Ref. 23) and experimental data (Refs. 2 and3) are shown for comparison. The vertical dotted line represents the zero-temperature freezing pressure of ${}^4$He and the straight dashed line is a linear fit to the DMC results (see text). (b) Dependence of the $C_{33}$ elastic constant on volume. The dashed line represents a power-law fit to the DMC results from which the value of the corresponding Grüneisen parameter is obtained (see text).

Figure 6

Pressure dependence of the longitudinal ($L$) and transvere ($T$) sound velocities of hcp ${}^4$He along its corresponding $c$ axis and basal plane. $v_D$ represents the averaged Debye velocity (see text). Basal sound-velocity data from Refs.3 ($▵$),2 ($▴$), and1 ($▿$) are shown for comparison. The vertical dotted line represents the zero-temperature freezing pressure of ${}^4$He.

Figure 7

$T=0$ Debye temperature of hcp ${}^4$He as a function of pressure (solid line). Experimental results from Refs. 58 and59 are shown for comparison. The vertical dotted line represents the zero-temperature freezing pressure of ${}^4$He, and the thickness of the line corresponds to the uncertainty associated to our calculations ($\delta {\Theta }_D/{\Theta }_D\sim 1\%$).