Synchrotron x-ray diffraction measurements were performed on ${}^{57}\mathrm{Fe}_{68}{\mathrm{Pd}}_{32}$ 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 ${}^{57}\mathrm{Fe}$ 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 ${\mathrm{Fe}}_{68}{\mathrm{Pd}}_{32}$. 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 $P_{\mathrm{C}}$. The difference in pressure $P_{\mathrm{C}}-P^{*}$ scales with the fractional reduction in magnetic exchange interaction per fractional change in volume, divided by the average Grüneisen parameter.

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• High-pressure studies
• Magnetic interactions
• Phase transitions
• Phonons
• Spin-phonon coupling
• Thermal expansion
• Transition metal alloys