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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 semi-empirical potentials (up to 100 GPa). Zero-point vibrations have been treated in the Debye approximation assuming $^4$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 ($\triangle P \approx 1.14$, $\triangle S_1 \approx 1.7$, $\triangle S_2 \approx 0.93$ at low pressures). Under terapascal 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 ($V_{mol}=0.228$ 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.