Elastic anisotropy and Poisson's ratio of solid helium under pressure

The elastic moduli, elastic anisotropy coefficients, sound velocities and Poisson's ratio of hcp solid helium have been calculated using density functional theory in generalized gradient approximation (up to 30 TPa), and pair + triple semiempirical potentials (up to 100 GPa). Zero-point vibrations have been treated in the Debye approximation assuming ${}^4\mathrm{He}$ isotope (we exclude the quantum-crystal region at very low pressures from consideration). Both methods give a reasonable agreement with the available experimental data. Our calculations predict significant elastic anisotropy of helium ($▵P\approx 1.14,▵S_1\approx 1.7,▵S_2\approx 0.93$ at low pressures). Under terapascal (TPa) pressures helium becomes more elastically isotropic. At the metallization point, there is a sharp feature in the elastic modulus $C_S$, which is the stiffness with respect to the isochoric change of the $c/a$ ratio. This is connected with the previously obtained sharp minimum of the $c/a$ ratio at the metallization point. Our calculations confirm the previously measured decrease of the Poisson's ratio with increasing pressure. This is not a quantum effect, as the same sign of the pressure effect was obtained when we disregarded zero-point vibrations. At TPa pressures, Poisson's ratio reaches the value of $0.31$ at the theoretical metallization point ($V_{mol}=0.228{\mathrm{cm}}^3$/mol, $p=17.48$ TPa) and $0.29$ at 30 TPa. For $p=0$, we predict a Poisson's ratio of $0.38$, which is in excellent agreement with the low-$p$-low-$T$ experimental data.

Figure 1

Equation of state of hcp helium. (Inset) High-pressure region. The vertical line indicates the GGA metallization point.

Figure 2

Five elastic constants of hcp helium vs pressure.

Figure 3

Elastic moduli (GGA + ZPV only) of hcp helium under terapascal pressures. Symbols are the calculated data points. The vertical line indicates the GGA metallization point. $C_{66}\equiv (C_{11}-C_{12})/2$ and $C_S/6$ are also presented.

Figure 4

Elastic anisotropy parameters of hcp helium vs pressure.

Figure 5

Elastic anisotropy parameters (top) and Cauchy violations (bottom) of hcp helium under terapascal pressures. Symbols are the calculated data points. The vertical line indicates the GGA metallization point.

Figure 6

Cauchy violations of hcp helium vs pressure.

Figure 7

Aggregate properties of hcp helium vs pressure: bulk modulus $K$ (top), shear modulus $G$ (middle), and Debye temperature (bottom). $K$ and $G$ are obtained using the Voigt-Reuss-Hill averaging.

Figure 8

Aggregate sound velocities of hcp helium vs pressure.

Figure 9

Aggregate properties of hcp helium in the TPa-pressure range: sound velocities (top), bulk, and shear moduli (bottom). Symbols are the calculated data points. The vertical line indicates the GGA metallization point.

Figure 10

Debye temperature (upper panel) and Poisson's ratio (bottom) of hcp helium in the terapascal-pressure range. Symbols are the calculated data points. The vertical line indicates the GGA metallization point.

Figure 11

Poisson's ratio of hcp helium vs pressure.