Rodrigues-Frank Representations of Crystallographic Texture

Foundations for Misorientation Imaging Microscopy
  • Krishna Rajan


Electron backscatter diffraction (EBSD) as a technique has achieved a great deal of usage with the apparent ease by which one can associate grain specific diffraction information from a polycrystalline sample. This enabled a coupling of microstructural features within the diffraction based resolution volume of the electron beam with the local or microtexture characteristics. The development of the experimental technique of EBSD has been aided by image processing tools and automation techniques that has been significant in the establishment of orientation mapping. Using electron beam based diffraction techniques offers a distinct advantage over x-ray diffraction techniques in many cases. For example, one may be interested in characterizing texture gradients over small regions of the microstructure below the spatial resolution of x-ray diffraction techniques; or the sample dimension may be too small (such as thin films and nanostructures) to even contain many grains. Yet, while the issues of local texture are still important to assess and interpret in these cases, and while EBSD may provide an experimental means of obtaining texture information, the predominant means of representing texture is still through orientation space as discussed in the last chapter.


Texture Component Crystallographic Texture Stereographic Projection Coincidence Site Lattice Inverse Pole Figure 
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  1. Ashbee, K.H.G. and Sargent, J.P., 1990, Stereoscopic presentation of Rodrigues vector representation of the full three dimensional disorientation of iron crystals by rolling, Metallurgical Transactions 21 A:253.CrossRefGoogle Scholar
  2. Becker, R. and Panchadeeswaran, S., 1989, Crystal rotations represented as Rodrigues vectors, Textures and Microstructures, 10:167.CrossRefGoogle Scholar
  3. Dawson, P.R. and Kumar, A., 1995, Polycrystalline modeling with finite elements over orientation space, in: Computational Plasticity, D.R.J. Owen and E. Onate, eds., Pineridge Press, Swansea, pp. 853.Google Scholar
  4. Egan, M.A., Krishnamoorthy, M., and Rajan, K., 1998, FLCUST: “A visualization tool for fuzzy clustering, SIGSCE Bulletin, 30:227.Google Scholar
  5. Field, D.P., 1995, On the asymmetric domain of cubic misorientations, Scripta Metall et. Mater. 32:67.CrossRefGoogle Scholar
  6. Frank, F.C, 1988, Orientation mapping, in: 8 th Intl Conf. on Textures of Materials, J.S. Kallend, and G. Gottstein, eds., TMS , Warrendale PA, p. 3.Google Scholar
  7. Frank F.C., 1991, Orientation mapping, Metallurgical Transactions 19A:403.Google Scholar
  8. Heinz, A. and Neumann, P., Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals, Acta Crystall. A47:780.Google Scholar
  9. Mackenzie, J.K., 1958, Second paper on statistics associated with the random disorientation of cubes, Biometrika, 45:229.Google Scholar
  10. Neumann, P., 1991, Representation of orientations of symmetrical objects by Rodrigues vectors, Textures and Microstructures, 14–18:53.Google Scholar
  11. Neumann, P., 1991, Graphical representations of orientations and ODFs by Rodrigues vectors, Steel Research, 62:560.Google Scholar
  12. Petkie, R., 1996, Ph.D. Thesis, Rensselaer Polytechnic Institute.Google Scholar
  13. Petkie, R, Rajan, K., and Tu, K-N., 1994, Crystallographic evolution of micro structures in thin film processing II- grain boundary structure, Journal of Electronic Materials, 23:893.CrossRefGoogle Scholar
  14. Rajan, K., 1998, Rodrigues-Frank mapping of interface crystallography, in: Boundaries and Interfaces in Materials, R.C. Pond, W.A.T. Clark, A.H. King and D.B. Williams, eds., TMS, Warrendale.Google Scholar
  15. Rajan, K., Gao, W., and Bhattacharyya, D., 1994, Analytical descriptions of deformation texture gradients in ceramic superconductors, in: Processing of Long Lengths of Superconductors, U. Balachandran, E.W. Collings and A. Goyal, eds., TMS Warrendale PA, p.311.Google Scholar
  16. Rajan, K. and Petkie, R., 1996, Rodrigues-Frank space representation of fiber texture, in: Polycrystalline Thin Films II, H. Frost, M.A. Parker, C.A. Ross and E. Holm eds., 403:207 MRS, Pittsburgh.Google Scholar
  17. Rajan, K. and Petike, R., 1998, Microtexture and anisotropy in wire drawn copper, Materials Science and Engineering, 4,46:185.Google Scholar
  18. Randle, V., 1993, The Measurement of Grain Boundary Geometry, Institute of Physics, Bristol.Google Scholar
  19. Randle, V., 1990, Representation of grain misorientations (mesotexture) in Rodrigues-Frank space, Proc. Royal Society, A431:61.CrossRefGoogle Scholar
  20. Randle, V and Day, A., 1993, Use of Rodrigues-Frank space for representation of microtexture and grain boundary parameters, Materials Science and Technology, 9:1069.CrossRefGoogle Scholar
  21. Santoro, A. and Mighell, A.D., 1973, Coincidence site lattices, Acta Crystallographica, A29:160.Google Scholar
  22. Schläfer, U. and Bunge, HJ., 1972, Cyclic textures in aluminum wires, Texture, 1:31.CrossRefGoogle Scholar
  23. Trogolo, J., 1997, Ph.D. Thesis, Rensselaer Polytechnic Institute.Google Scholar
  24. Trogolo, J. and Rajan, K., 1998, Crystallographic evolution in directionally solidified microstructures, in: Solidification 1998, S.P. Marsh et al., TMS, Warrendale, PA p. 39–47.Google Scholar
  25. Warrington, D., and Lück, R., 1995, The use of the Wiering roof to examine coincidence site quasilattice in icosahedral quasicrystals, in: Proc. Intl Conf. on Aperiodic Crystals, G. Chapuis and W. Packiorek, eds., World Scientific p. 30.Google Scholar

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© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Krishna Rajan
    • 1
  1. 1.Department of Materials Science and EngineeringRensselaer Polytechnic InstituteTroyUSA

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