Pressure-Induced Irreversible Disorder in $\beta^{\prime}$-Mn$_3$(PO$_4$)$_2$: A High-Pressure X-ray Diffraction and Density-Functional Theory Study

The high-pressure structural behavior of $\beta^\prime$-Mn$_3$(PO$_4$)$_2$ was investigated using synchrotron X-ray diffraction up to 20 GPa combined with density-functional theory calculations. At ambient conditions, $\beta^\prime$-Mn$_3$(PO$_4$)$_2$ crystallizes in a monoclinic structure that exhibits strongly anisotropic compression. The pressure dependence of the unit-cell volume was described using a third-order Birch--Murnaghan equation of state, and the principal axes of compressibility were determined. Above 14.1 GPa, significant broadening and weakening of the diffraction peaks are attributed to the onset of irreversible pressure-induced structural disorder associated with the loss of long-range crystallographic order. The disordered state persists after decompression to ambient pressure, demonstrating the irreversible nature of the transformation. The calculations accurately reproduce the experimental compressional behavior and provide insights into the microscopic structural evolution under pressure. Compression is mainly accommodated through distortions of the Mn--O polyhedra, whereas the PO$_4$ tetrahedra behave as comparatively rigid units. Several initially penta-coordinated Mn sites progressively evolve toward octahedral coordination under compression, while selected MnO$_6$ polyhedra exhibit anomalous distortions and elastic softening preceding the onset of disorder. Elastic constant calculations further reveal that the crystalline phase becomes mechanically unstable near the experimentally observed transition pressure. The combined experimental and computational results suggest that the HP response of $\beta^\prime$-Mn$_3$(PO$_4$)$_2$ is influenced by the interplay between framework complexity, anisotropic polyhedral compressibility, and elastic instability, providing new insight into pressure-induced structural degradation in structurally complex phosphate frameworks.