Skip to main content
SpringerLink
Log in

Visualization of Time-Reversal Symmetry in Magnetic Point Groups

  • Published:
Metallurgical and Materials Transactions A Aims and scope Submit manuscript

Abstract

The direct visualization of crystallographic and magnetic point group symmetry by means of computer graphics simplifies the teaching of point group symmetry at the undergraduate and graduate levels. Still images and animated movies for all 32 crystallographic point groups, and for the 122 magnetic point groups, are presented. For each point group, the action of the symmetry operators on scalar and pseudo-scalar objects, as well as on polar and axial vectors, is represented as a three-dimensional rendered image. All images and movies are made available as supplementary educational material via a dedicated web site.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. A.S. Nowick: Crystal Properties via Group Theory, Cambridge University Press, Cambridge, United Kingdom, 1995, pp. 20–22.

    MATH  Google Scholar 

  2. Y.C. Akgöz and G.A. Saunders: J. Phys. C, 1975, vol. 8, pp. 2962–70.

    Article  ADS  Google Scholar 

  3. R.E. Newnham: Properties of Materials: Anisotropy—Symmetry—Structure, Oxford University Press, Oxford, United Kingdom, 2005, pp. 34–35.

    Google Scholar 

  4. The International Tables for Crystallography Vol. A: Space-Group Symmetry, T. Hahn, ed., Kluwer Academic Publishers, Dordrecht, 1996, vol. A, pp. 752–76.

  5. S.J. Joshua: Acta Cryst. A, 1974, vol. 30, pp. 353–59.

    Article  CAS  Google Scholar 

  6. D.E. Laughlin, M.A. Willard, and M.E. McHenry: Proc. Symp. on Phase Transformations and Evolution in Materials, TMS, Warrendale, PA, 2000, pp. 121–37

    Google Scholar 

  7. A.V. Shubnikov: Symmetry and Antisymmetry of Finite Figures, Nauka, Moscow, 1951.

    Google Scholar 

  8. M. Fiebig: J. Phys. D: Appl. Phys., 2005, vol. 38, pp. R123–R152.

    Article  CAS  ADS  Google Scholar 

  9. http://graphics.stanford.edu/~cek/rayshade

  10. M. De Graef: J. of Mater. Educ., 1998, vol. 20, pp. 31–42.

    Google Scholar 

  11. M. De Graef and M.E. McHenry: Structure of Materials: An Introduction to Crystallography, Diffraction, and Symmetry, Cambridge University Press, Cambridge, United Kingdom, 2007, pp. 198–229.

    MATH  Google Scholar 

  12. International Union for Crystallography, http://www.iucr.org/education/pamphlets/2

Download references

Acknowledgments

The point group representations as well as the accompanying web site (http://mpg.web.cmu.edu) were developed with financial support from the National Science Foundation, Award Nos. DMR-0404836 and DMR-0809048. The author thanks D. Laughlin, M. McHenry, A. Petford-Long, C. Phatak, and B. Clothier for stimulating discussions. In addition, the anonymous reviewers are acknowledged for helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Marc De Graef (Professor)

Authors
  1. Marc De Graef
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc De Graef.

Additional information

Manuscript submitted December 1, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

De Graef, M. Visualization of Time-Reversal Symmetry in Magnetic Point Groups. Metall Mater Trans A 41, 1321–1329 (2010). https://doi.org/10.1007/s11661-010-0171-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-010-0171-0

Keywords

Search

Navigation