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Rotation-reversal symmetries in crystals and handed structures

Nature Materials volume 10, pages 376–381 (2011)Cite this article

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Abstract

Symmetry is a powerful framework to perceive and predict the physical world. The structure of materials is described by a combination of rotations, rotation-inversions and translational symmetries. By recognizing the reversal of static structural rotations between clockwise and counterclockwise directions as a distinct symmetry operation, here we show that there are many more structural symmetries than are currently recognized in right- or left-handed helices, spirals, and in antidistorted structures composed equally of rotations of both handedness. For example, we show that many antidistorted perovskites possess twice the number of symmetry elements as conventionally identified. These new ‘roto’ symmetries predict new forms for ‘roto’ properties that relate to static rotations, such as rotoelectricity, piezorotation, and rotomagnetism. They enable a symmetry-based search for new phenomena, such as multiferroicity involving a coupling of spins, electric polarization and static rotations. This work is relevant to structure–property relationships in all materials and structures with static rotations.

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Figure 1: Rotation-reversal symmetry and other antisymmetry operations.
Figure 2: Roto groups and property classification.
Figure 3: Symmetries in antidistorted cubic perovskite lattices.
Figure 4: Symmetries in helices and spirals.

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Acknowledgements

The authors acknowledge financial support from the National Science Foundation through the MRSEC program DMR-0820404 and grant DMR-0908718. Discussions with C. J. Fennie, K. M. Rabe and A. M. Glazer are gratefully acknowledged. We thank S. Denev for help with artwork for Figure 4.

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Authors and Affiliations

  1. Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16803, USA

    Venkatraman Gopalan

  2. Department of Physics, Eberly College of Science, The Pennsylvania State University, Penn State Berks, Reading, Pennsylvania 19610, PO Box 7009, USA

    Daniel B. Litvin

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  1. Venkatraman Gopalan
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Contributions

V.G. conceived the idea of rotation-reversal symmetry, the roto groups, their influence on properties, and derived the symmetries listed using symmetry diagrams. D.B.L. critiqued these ideas, helped develop formal definitions for these concepts, and derived the symmetries listed using group theoretical methods. V.G. and D.B.L. co-wrote the article.

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Correspondence to Venkatraman Gopalan.

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Gopalan, V., Litvin, D. Rotation-reversal symmetries in crystals and handed structures. Nature Mater 10, 376–381 (2011). https://doi.org/10.1038/nmat2987

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