A plain derivation of birch-murnaghan equations of state, and comparison with other equations of state

Abstract

The Eulerian finite strain of an elastically isotropic body is defined by taking the state after compression as the reference state and expanding the squared length. The second-, third- and fourth-order Birch-Murnaghan equations of state are plainly derived based on the Eulerian finite strain. The key for the plain derivation is no use of differenital or tensor because of isotropic, uniform and finite change in length. For better understanding, the finite strain in the Lagrangian scheme is defined by taking the state before compression as the reference state, and the Lagrange equations of state are derived in this scheme. In this scheme, pressure increases less significantly with compression than the Eulerian scheme. The different Eulerian strains are also defined by expanding the linear and cubed lengths instead of the squared length, and the first- and third-power Eulerian equations of state are derived in these schemes. Fitting of pressure-scale-free data to these equations indicates that the Lagrangian scheme is inappropriate to describe P-V-T relations of MgO, whereas three Eulerian equations of state have equivalent significance, and the Birch-Murnaghan equations of state does not have special meaning compared to the other Eulerian equations of state.