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The elastic moduli, elastic anisotropy coefficients, sound velocities andPoisson's ratio of hcp solid helium have been calculated using densityfunctional theory in generalized gradient approximation (up to $30$ TPa), andpair+triple semi-empirical potentials (up to 100 GPa). Zero-point vibrationshave been treated in the Debye approximation assuming $^4$He isotope (weexclude 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 lowpressures). Under terapascal pressures helium becomes more elasticallyisotropic. At the metallization point there is a sharp feature in the elasticmodulus $C_S$, which is the stiffness with respect to the isochoric change ofthe $c/a$ ratio. This is connected with the previously obtained sharp minimumof the $c/a$ ratio at the metallization point.  Our calculations confirm the previously measured decrease of the Poisson'sratio with increasing pressure. This is not a quantum effect, as the same signof 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 theoreticalmetallization point ($V_{mol}=0.228$ cm$^3$/mol, $p=17.48$ TPa) and $0.29$ at30 TPa. For $p=0$ we predict a Poisson's ratio of $0.38$ which is in excellentagreement with the low-$p$-low-$T$ experimental data.

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