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Nonlinear elasticity of 
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 -Fe: The pressure effect

Description of elasticity of iron at the ultrahigh pressures is a challenging task for physics, with a potential strong impact on other branches of science. In the present work, we calculate the elastic properties of hcp iron in the pressure range of 50–340 GPa beyond the linear elasticity approximation, conventionally assumed in theoretical studies. We deﬁne the higher order elastic constants and present expressions for the long-wave acoustic modes Grüneisen parameters of a compressed hcp crystal. We obtain the second and third order elastic constants of the hcp Fe in the considered pressure interval, as well as its Grüneisen parameters for the high-symmetry directions. The latter are directly compared with the Grüneisen parameters derived from the volume dependences of the vibrational frequencies calculated in the quasiharmonic approximation. The obtained results are used for the stability analysis of the hcp phase of iron at high pressures.