Remarks on Texture Coefficients of Polycrystals with Improper Crystallite Symmetry

  • Chi-Sing ManEmail author
  • Ding Zhao
Research Note


The orientation distribution function (ODF) in classical texture analysis is defined on the rotation group SO(3). For polycrystalline aggregates with crystallite symmetry defined by a crystallographic point group \(G_{\mathrm{cr}}\) which is not a subgroup of SO(3), the improper group \(G_{\mathrm{cr}}\) is routinely replaced by its proper peer (i.e., a subgroup of SO(3)) in the same Laue class. In this note we examine how the texture coefficients obtained from such a practice are related to their counterparts that pertain to the corresponding ODF defined on the orthogonal group O(3) as it should.


Crystallographic texture Orthogonal group Texture coefficients Improper point groups Laue class 

Mathematics Subject Classification (2010)

74A99 74E10 74E15 74E25 60B05 



  1. 1.
    Rousseau, J.-J.: Basic Crystallography. Wiley, Chichester (1988) Google Scholar
  2. 2.
    Bunge, H.-J.: Texture Analysis in Materials Science: Mathematical Methods. Butterworths, London (1982) Google Scholar
  3. 3.
    Roe, R.-J.: Description of crystallite orientation in polycrystalline materials, III: general solution to pole figures. J. Appl. Phys. 36, 2024–2031 (1965) ADSCrossRefGoogle Scholar
  4. 4.
    Kocks, U.F., Tomé, C.N., Wenk, H.-R. (eds.): Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effects on Material Properties. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  5. 5.
    Randle, V., Engler, O.: Introduction to Texture Analysis: Macrotexture, Microtexture and Orientation Mapping. Gordon and Breach, Amsterdam (2000) Google Scholar
  6. 6.
    Adams, B.L., Kalidindi, S.R., Fullwood, D.T.: Microstructure-Sensitive Design for Performance Optimization. Butterworth-Heinemann, Waltham (2013) Google Scholar
  7. 7.
    Newnham, R.E.: Properties of Materials: Anisotropy, Symmetry, Structure. Oxford University Press, Oxford (2005) Google Scholar
  8. 8.
    Du, W., Man, C.-S.: Material tensors and pseudotensors of weakly-textured polycrystals with orientation distribution function defined on the orthogonal group. J. Elast. 127, 197–233 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Du, W.: Material tensors and pseudotensors of weakly-textured polycrystals with orientation measure defined on the orthogonal group. Doctoral dissertation, University of Kentucky, Lexington (2015) Google Scholar
  10. 10.
    Esling, C., Bunge, H.J., Muller, J.: Description of the texture by distribution functions on the space of orthogonal transformations. Implications on the inversion centre. J. Phys. Lett. 41, 543–545 (1980) CrossRefGoogle Scholar
  11. 11.
    Bunge, H.J., Esling, C., Muller, J.: The role of the inversion centre in texture analysis. J. Appl. Crystallogr. 13, 544–554 (1980) CrossRefGoogle Scholar
  12. 12.
    Bunge, H.J., Esling, C., Muller, J.: The influence of crystal and sample symmetries on the orientation distribution function of the crystallites in polycrystalline materials. Acta Crystallogr., Sect. A 37, 889–899 (1981) ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics: Theory and Application. Addison-Wesley, Reading (1981) zbMATHGoogle Scholar
  14. 14.
    Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific Publishing, Singapore (1988) CrossRefGoogle Scholar
  15. 15.
    Man, C.-S.: On the constitutive equations of some weakly-textured materials. Arch. Ration. Mech. Anal. 14, 77–103 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2006) Google Scholar
  17. 17.
    Lee, J.M.: Manifolds and Differential Geometry. American Mathematical Society, Providence (2009) CrossRefzbMATHGoogle Scholar
  18. 18.
    Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic, New York (1975) zbMATHGoogle Scholar
  19. 19.
    Bauer, H.: Measure and Integration Theory. Walter de Gruyter, Berlin (2001) CrossRefzbMATHGoogle Scholar
  20. 20.
    Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (1999) zbMATHGoogle Scholar
  21. 21.
    Tjur, T.: Probability Based on Radon Measures. Wiley, Chichester (1980) zbMATHGoogle Scholar
  22. 22.
    Dieudonné, J.: Foundations of Modern Analysis (Enlarged and Corrected Printing). Academic, New York (1969) zbMATHGoogle Scholar
  23. 23.
    Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, New York (2000) Google Scholar
  24. 24.
    Roe, R.-J.: Inversion of pole figures for materials having cubic crystal symmetry. J. Appl. Phys. 37, 2069–2072 (1966) ADSCrossRefGoogle Scholar
  25. 25.
    Knightly, A., Li, C.: Traces of Hecke Operators. American Mathematical Society, Providence (2006) CrossRefzbMATHGoogle Scholar
  26. 26.
    Paroni, R.: Optimal bounds on texture coefficients. J. Elast. 60, 19–34 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Man, C.-S., Noble, L.: Designing textured polycrystals with specific isotropic material tensors: the ODF method. Rend. Semin. Mat. 58, 155–170 (2000) MathSciNetzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

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