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The Clapeyron slope is the slope of a phase boundary in P–T space and is essential for understanding mantle dynamics and evolution. The phase boundary is delineating instead of balancing a phase transition's normal and reverse reactions. Many previous high pressure–temperature experiments determining the phase boundaries of major mantle minerals experienced severe problems due to instantaneous pressure increase by thermal pressure, pressure drop during heating, and sluggish transition kinetics. These complex pressure changes underestimate the transition pressure, while the sluggish kinetics require excess pressures to initiate or proceed with the transition, misinterpreting the phase stability and preventing tight bracketing of the phase boundary. Our recent study developed a novel approach to strictly determine phase stability based on the phase equilibrium definition. Here, we explain the details of this technique, using the post-spinel transition in Mg2SiO4 determined by our recent work as an example. An essential technique is to observe the change in X-ray diffraction intensity between ringwoodite and bridgmanite + periclase during the spontaneous pressure drop at a constant temperature and press load with the coexistence of both phases. This observation removes the complicated pressure change upon heating and kinetic problem, providing an accurate and precise phase boundary. [ABSTRACT FROM AUTHOR]

Eulerian finite strain of an elastically isotropic body is defined using the expansion of squared length and the post-compression state as reference. The key to deriving second-, third- and fourth-order Birch–Murnaghan equations-of-state (EOSs) is not requiring a differential to describe the dimensions of a body owing to isotropic, uniform, and finite change in length and, therefore, volume. Truncation of higher orders of finite strain to express the Helmholtz free energy is not equal to ignoring higher-order pressure derivatives of the bulk modulus as zero. To better understand the Eulerian scheme, finite strain is defined by taking the pre-compressed state as the reference and EOSs are derived in both the Lagrangian and Eulerian schemes. In the Lagrangian scheme, pressure increases less significantly upon compression than the Eulerian scheme. Different Eulerian strains are defined by expansion of linear and cubed length and the first- and third-power Eulerian EOSs are derived in these schemes. Fitting analysis of pressure-scale-free data using these equations indicates that the Lagrangian scheme is inappropriate to describe P-V-T relations of MgO, whereas three Eulerian EOSs including the Birch–Murnaghan EOS have equivalent significance. [ABSTRACT FROM AUTHOR]