Double Antisymmetry Space Group(DASG) introduced new symmetry operation, so-called rotation-reversal symmetry 1* with previous well-known time-reversal symmetry 1’ [1-3]. In addition, those direct-products give rise to rotation-time-reversal symmetry 1’*. With respect to this new crystallographic approach, magnetic moment can be reversed theoretically by well-known time-reversal symmetry and new rotation-time-reversal symmetry. It causes the question which one is general magnetic symmetry in the materials.

On the other hand, Shull and his colleagues had carried out powder neutron diffraction on transition metal monoxides, i.e. MnO, FeO, CoO and NiO [4]. They proved type-2 antiferromagnetic spin distribution in FCC structure. This pioneer work is direct experimental evidence of antiferromagnetism which was proposed theoretically by Néel. However, precise spin directions are still debate due to ferroelastic domain problem caused by antiferromagnetic transition [5].

High resolution time-of-flight neutron diffraction and group theory resolved not only the long-standing problems of magnetic structure and phase transition mechanism in MnO, CoO, and NiO but also the issue of general symmetry in magnetic moment [6, 7]. Exchangestriction and new magnetostriction, so-called dipolar-magnetostriction and roto-magnetostriction, that induce phase transition from paramagnetic rocksalt structure to common C 2/ m monoclinic structure with magnetic propagation vector k _m =(0 1 0.5). We classify those into two groups by Γ ₁ ⁺ (A _g )=(0, M _y , 0) and Γ ₂ ⁺ (B _g )=( M _x , 0, M _z ) spin direction in the order-parameter vector diagram. Time-reversal symmetry breaking causes discontinuous phase transition with linear-quadratic coupling Δβ ~ M ² between monoclinic angle Δβ and magnetic moment M . However, CoO and NiO show continuous phase transition with linear-cubic coupling Δβ ~ M ³ forbidden by time-reversal symmetry. To the best of our knowledge, this is first experimental evidence that both magnetic moment and structural distortion can belong to same irreducible representation. Time-reversal symmetry prohibits same irreducible representation between them. The experimental results open a way to find new physics forbidden by time-reversal symmetry in condensed matter physics and material science.

[1] V. Gopalan and D. B. Litvin, Nat. Mater. 10 , 376 (2011)

[2] B. K. VanLeeuwen, V. Gopalan, and D. B. Litvin, Acta Cryst. A 70 , 24 (2014)

[3] M. Huang, B. K. VanLeeuwen, D. B. Litvin, and V. Gopalan, Acta Cryst. A 70 , 373 (2014)

[4] C. G. Shull, W. A. Strauser, and E. O. Wollan, Phys. Rev. 83 , 333 (1951)

[5] A. Schrön, C. Rödl, and F. Bechstedt, Phys. Rev. B 86 , 115134 (2012)

[6] S. Lee, Y. Ishikawa, P. Miao, S. Torii, and T. Kamiyama, JPS Conf. Proc. 8 , 034007 (2015)

[7] S. Lee, Y. Ishikawa, P. Miao, S. Torii, T. Ishigaki, and T. Kamiyama, submitted

ACKNOWLEDGEMENT: The neutron diffraction experiments using SuperHRPD were the result of Proposal No. 2014S05