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Group theoretical analysis of structural instability, vacancy ordering and magnetic transitions in the system troilite (FeS) - pyrrhotite (Fe$_{1-x}$S)

A group-theoretical framework to describe vacancy ordering and magnetism in the Fe$_{1-x}$S system is developed. This framework is used to determine the sequence of crystal structures consistent with the observed magnetic structures of troilite (FeS), and to determine the crystallographic nature of the low-temperature Besnus transition in Fe$_{0.875}$S. We conclude that the Besnus transition is a magnetically driven transition characterised by the rotation of the moments out of the ac-plane, accompanied by small atomic displacements that lower the symmetry to triclinic at low temperatures. Based on our phase diagram, we predict related magnetically driven phase transitions at low temperatures in all the commensurate superstructures of pyrrhotite. The exact nature of the transition is determined by the symmetry of the vacancy ordered state Based on this we predict spin-flop transitions in 3C and 5C pyrrhotite and a transition akin to the Besnus transition in 6C pyrrhotite. Furthermore, we clarify that 3C and 4C pyrrhotite carry a ferrimagnetic moment whereas 5C and 6C are antiferromagnetic.