Representations of Texture

  • Jeremy K. Mason
  • Christopher A. Schuh

The field of materials science and engineering is fundamentally concerned with manipulating the microstructure of materials in order to control their properties. Electron backscatter diffraction (EBSD) dramatically enhances our abilities in this regard, by providing extensive crystallographic orientation information of a given two-dimensional section of a microstructure. As this technique has been developed and combined with chemical analysis and serial sectioning methods, it has become possible to access complete three-dimensional chemistry, phase, and crystal orientation information; in short, the microstructural state of a polycrystal may now be completely quantified.


Spherical Harmonic Pole Figure Euler Angle Orientation Distribution Function Stereographic Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Altmann SL (1986) Rotations, quaternions, and double groups. Clarendon Press, OxfordMATHGoogle Scholar
  2. Altmann SL, Cracknell AP (1965) Lattice harmoncs i. Cubic groups. Rev Mod Phys 37:19–32MATHCrossRefMathSciNetADSGoogle Scholar
  3. Altmann SL, Herzig P (1994) Point-group theory tables. Clarendon Press, New YorkGoogle Scholar
  4. Bateman H, Erdélyi A (1953) Higher transcendental functions. McGraw-Hill, New YorkGoogle Scholar
  5. Bunge HJ (1993) Texture analysis in materials science: mathematical methods. Cuvillier Verlag, GottingenGoogle Scholar
  6. Domokos G (1967) Four-dimensional symmetry. Phys Rev 159:1387–1403CrossRefADSGoogle Scholar
  7. Euler L (1776) Formulae generales pro translatione quacunque corporum rigidorum. Novi Comm Acad Sci Imp Petrop 20:189–207Google Scholar
  8. Frank FC (1988) Orientation mapping. Metall Trans A 19: 403–408CrossRefADSGoogle Scholar
  9. Funda J, Taylor RH, Paul RP (1990) On homogeneous transforms, quaternions, and computational-efficiency. IEEE T Robotic Autom 6: 382–388CrossRefGoogle Scholar
  10. Gertsman VY (2001a) Geometrical theory of triple junctions of csl boundaries. Acta Crystallogr A 57:369–377CrossRefPubMedGoogle Scholar
  11. Gertsman VY (2001b) Coincidence site lattice theory of multicrystalline ensembles. Acta Crystallogr A 57:649–655CrossRefPubMedGoogle Scholar
  12. Gilmore R (1974) Lie groups, Lie algebras, and some of their applications. Wiley, New YorkMATHGoogle Scholar
  13. Glez JC, Driver J (2001) Orientation distribution analysis in deformed grains. J Appl Crystallogr 34:280–288CrossRefGoogle Scholar
  14. Gradshtein IS, Ryzhik IM, Jeffrey A (2000) Table of integrals, series, and products. Academic Press, San DiegoGoogle Scholar
  15. Grimmer H (1973) Coincidence rotations for cubic lattices. Scripta Metall Mater 7:1295–1300Google Scholar
  16. Grimmer H (1974) Disorientations and coincidence rotations for cubic lattices. Acta Crystallogr A 30:685–688MATHCrossRefADSGoogle Scholar
  17. Grimmer H (1980) A unique description of the relative orientation of neighboring grains. Acta Crystallogr A 36:382–389CrossRefMathSciNetADSGoogle Scholar
  18. Heinz A, Neumann P (1991) Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals. Acta Crystallogr A 47:780–789CrossRefMathSciNetGoogle Scholar
  19. Hicks HR, Winternitz P (1971) Relativistic two-variable expansions for three-body decay amplitudes. Phys Rev D 4: 2339–2351CrossRefADSGoogle Scholar
  20. Hopf H (1940) Systeme symmetrischer bilinearformen und euklidische modelle der projektiven räume. Vierteljschr Naturforsch Ges Zürich 85:165–177MathSciNetGoogle Scholar
  21. Humbert M, Gey N, Muller J, Esling C (1996) Determination of a mean orientation from a cloud of orientations. Application to electron back-scattering pattern measurements. J Appl Crystallogr 29:662–666CrossRefGoogle Scholar
  22. Lewis AC, Jordan KA, Geltmacher AB (2008) Determination of critical microstructural features in an austenitic stainless steel using image-based finite element modeling. Metall Mater Trans A 39A:1109–1117CrossRefADSGoogle Scholar
  23. Mason JK, Schuh CA (2008) Hyperspherical harmonics for the representation of crystallographic texture. Acta Mater, DOI: 10.1016/j.actamat.2008.08.031Google Scholar
  24. Morawiec A (2004) Orientations and rotations: computations in crystallographic textures. Springer, BerlinMATHGoogle Scholar
  25. Morawiec A, Field DP (1996) Rodrigues parameterization for orientation and misorientation distributions. Philos Mag A 73:1113–1130CrossRefADSGoogle Scholar
  26. Mueller FM, Priestley MG (1966) Inversion of cubic de hass-van alphen data, with an application to palladium. Phys Rev 148, 638–643CrossRefADSGoogle Scholar
  27. Neumann P (1992) The role of geodesic and stereographic projections for the visualization of directions, rotations, and textures. Phys Status Solid A 131:555–567CrossRefADSGoogle Scholar
  28. Siemens M, Hancock J, Siminovitch D (2007) Beyond Euler angles: Exploiting the angle-axis parametrization in a multipole expansion of the rotation operator. Solid State Nucl Mag 31:35–54CrossRefGoogle Scholar
  29. Stuelpnagel J (1964) On the parameterization of the three-dimensional rotation group. SIAM Rev 6:422–430MATHCrossRefMathSciNetGoogle Scholar
  30. Varshalovish DA, Moskalev AN, Khersonskii VK (2008) Quantum theory of angular momentum. World Scientific, SingaporeGoogle Scholar
  31. Vilenkin NJ (1968) Special functions and the theory of group representations. American Mathematical Society, ProvidenceGoogle Scholar
  32. Wenk HR, Kocks UF (1987) The representation of orientation distributions. Metall Trans A 18:1083–1092Google Scholar
  33. Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New YorkMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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