Thermodynamic origin of the pressure-induced Invar effect: General criterion and experimental study of Fe₆₈⁢Pd₃₂

Synchrotron x-ray diffraction measurements were performed on ⁵⁷Fe₆₈⁢Pd₃₂ at multiple pressures and two temperatures in a diamond-anvil cell. Between 4 and 11 GPa, the thermal expansion was zero or slightly negative. This pressure-induced Invar effect was studied further with ⁵⁷Fe nuclear forward scattering and nuclear resonant inelastic x-ray scattering to obtain information on the pressure-induced changes in both the magnetization and the phonon density of states. Magnetic entropy and phonon entropy were obtained from these results, the latter with additional measurements from inelastic neutron scattering to account for the contributions from Pd atoms. The dependencies of these entropies on pressure gave the magnetic and phonon contributions to thermal expansion. These canceled in the region of pressure-induced Invar behavior, even though they individually increased by more than a factor of 2 below the Curie pressure. The behavior of phonons gives evidence for spin-phonon interactions in Fe₆₈⁢Pd₃₂. A general explanation of the pressure-induced Invar effect is presented, showing that Invar behavior is typically expected at a pressure 𝑃* below a magnetic transition at pressure 𝑃C. The difference in pressure 𝑃C−𝑃* scales with the fractional reduction in magnetic exchange interaction per fractional change in volume, divided by the average Grüneisen parameter.