Skip to main content
SpringerLink
Log in

Mathematics Analysis in Electron Diffraction and Crystallography

  • Chapter
Characterization of Microstructures by Analytical Electron Microscopy (AEM)

  • 824 Accesses

Abstract

In this chapter the ways how to use mathematics methods to deal with issues on electron diffraction and crystallography are described. These issues include the prediction of an arbitrary zone of diffraction pattern based on the known orientation relationship (OR), prediction of the possible ORs and crystallographic features according to the crystal structure and lattice parameters of the parent phase and precipitate phase, determination of characteristics parameters of coincidence site lattice, the systematic extinction caused by crystal symmetry, etc. In these mathematics analyses, matrix analysis is widely used, and it includes transformation matrices of basis vectors, indices of direction and plane between two phases or in different coordinate systems. In theoretical prediction of ORs, three methods are introduced in detail, and they are edge-to-edge matching model, invariant line strain model, O-line model. In order to understand the systematic extinction in electron diffraction caused by crystallographic symmetries, firstly basic knowledges of crystallography are briefly introduced, such as macro-symmetry elements and their combination laws, point groups, space groups, equivalent positions (equipoints), two dimensional lattice, plane point groups and plane groups; then the relationship between the systematic extinction and the symmetry elements are analyzed, and finally an example of determining crystal structure by extinction features is given.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Liu W X, Huang X Y, Chen Y R. Electron Microscopy Analysis of Materials Structures. Tianjin University Press, Tianjin, 1989. (in Chinese)

    Google Scholar 

  2. Huang X Y. The Principles and Applications of Patterns in Electron Microscopy. China Astronautic Publishing House, Beijing, 1989. (in Chinese)

    Google Scholar 

  3. Rong Yonghua, Chen Shipu, Hu Gengxiang, et al. Prediction and characterization of variant electron diffraction patterns for γ and δ precipitates on an inconel 718 alloy. Metallurgical and Materials Transactions A, 30(9): 2297–2303, 1999.

    Article  Google Scholar 

  4. Rong Y H, Gao M, Wei R P. Identification of grain boundary precipitates of the commercial Inconel 718 alloy. Journal of Shanghai Jiao Tong University, 31: 27–33, 1997. (in Chinese)

    CAS  Google Scholar 

  5. Rong Y H, Wang M, Wang Y R, et al. Computer simulation of EDPs obtained from ε-Nitrocarbides. Physical Testing and Chemical Analysis Part A: Physical Testing, 29: 5-10, 1993. (in Chinese)

    Google Scholar 

  6. Rong Y H, Wang Y R, Chen X K, et al. TEM Investigation of the asquenched compound layer formed by austenitic nitrocarburizing. Materials Characterization, 41: 35–39, 1998.

    Article  CAS  Google Scholar 

  7. Yang Y X, Qi R. X-ray Diffraction Analysis. Shanghai Jiao Tong University Press, 1994. (in Chinese)

    Google Scholar 

  8. Gui L F, Tang R J. Handbook of Mechanical Engineering Materials and Their Testing, Volume of Physics and Metallography. Liaoning Science and Technology Publishing House, 1999. (in Chinese)

    Google Scholar 

  9. Rong Y H, Peng M, Hu G X, et al. New method on TEM determination of CSL parameters. Journal of Shanghai Jiao Tong University, 31: 34–39, 1997. (in Chinese)

    CAS  Google Scholar 

  10. Christian J W. The Theory of Transformation in Metals and Alloys, 3rd Ed. Pergamon Press, Oxford, 2002.

    Google Scholar 

  11. Warrington D H, Bufalini P. The coincidence site lattice and grain boundaries. Scripta Metallurgic, 5(9): 771–776, 1971.

    Article  CAS  Google Scholar 

  12. Bhadeshia H K D H. Worked Examples in the Geometry of Crystals. 2nd Ed. Institute of Materials, London, 2001.

    Google Scholar 

  13. Wen C S. Effect of surface mechanical attrition treatment on microstructure, transformation and properties of metal materials. Shanghai Jiao Tong University Doctoral Dissertation, 2005. (in Chinese)

    Google Scholar 

  14. Wechsler M S, Lieberman D S, Read T A. On the theory of formation of martensite. Trans. AIME, 197: 1503–1515, 1953.

    Google Scholar 

  15. Bowles J S, Mackenzie J K. The crystallography of martensite transformations I-III. Acta Metallurgica, 2: 129–234, 1954.

    Article  CAS  Google Scholar 

  16. Hall M G, Arronson H I, Kinsma K R. The structure of nearly coherent FCC: BCC boundaries in a Cu-Cr alloy. Surface Science, 31: 257–274, 1972.

    Article  CAS  Google Scholar 

  17. Rigsbee J M, Aaronson H I. A computer modeling study of partially coherent FCC: BCC boundaries. Acta Metallurgica, 27: 351–363, 1979.

    Article  CAS  Google Scholar 

  18. Liang Q, Reynolds WT. Determining interphase boundary orientations from near-coincidence sites. Metall. Mater. Trans. A, 29(8): 2059–2072, 1998.

    Article  Google Scholar 

  19. Miyano N, Ameyama K, Weatherly G C. Three dimensional near-coincidence site lattice modeling of α/β interface boundary structure in two phase Titanium alloy. ISIJ International, 40: S199–S203, 2000.

    Article  CAS  Google Scholar 

  20. Dahmen U. Orientation relationships in precipitation systems. Acta Metallurgica, 30: 63–73, 1982.

    Article  CAS  Google Scholar 

  21. Luo C P, Weatherly G C. The invariant line and precipitation in a Ni-45wt% Cr alloy. Acta Metallurgica, 35: 1963–1972, 1987.

    Article  CAS  Google Scholar 

  22. Luo C P, Dahmen U, Westmacott K H. Morphology and crystallography of Cr precipitates in a Cu-0.33wt% Cr alloy. Acta Metallurgica, 42: 1923–1932, 1994.

    Article  Google Scholar 

  23. Kelly P M, Zhang M X. Edge-to-edge matching — a new approach to the morphology and crystallography of precipitates. Materials Forum, 23: 41–62, 1999.

    CAS  Google Scholar 

  24. Zhang M X, Kelly P M. Edge-to-edge matching and its applications part I: aplication to the simple HCP/BCC system. Acta Metallurgica, 53: 1073–1084, 2005.

    CAS  Google Scholar 

  25. Zhang M X, Kelly P M. Edge-to-edge matching model for prediction orientation relationships and habit planes — the improvements. Scripta Materialia, 52: 963–968, 2005.

    Article  CAS  Google Scholar 

  26. Zhang W Z, Purdy G R. O-lattice analysis of interfacial misfit, I: general construction. Philosophical Magazine, 68A: 279–290, 1993.

    Google Scholar 

  27. Zhang W Z, Purdy G R. O-lattice analysis of interfacial misfit, II: system containing invariant lines. Philosophical Magazine, 68A: 291–303, 1993.

    Google Scholar 

  28. Qiu D, Zhang W Z. A systematic study of irrational precipitation crystallography in FCC-BCC systems with an analytical O-line method. Philosophical Magazine, 83: 3093–3116, 2003.

    Article  CAS  Google Scholar 

  29. Cao Y, Zhong N, Wang X D, et al. An edge-to-edge matching model and its application to the HCP/ FCC System. Journal of Shanghai Jiao Tong University, 41: 586–591, 2007. (in Chinese)

    CAS  Google Scholar 

  30. Ramanujan R V, Lee J K, Aaronson H I. Discrete lattice plane analysis of the interfacial energy of coherent FCC: HCP interfaces and its application to the nucleation of γ in Al-Ag alloys. Acta Metallurgica et Materialia, 40(12): 3421–3432, 1992.

    Article  CAS  Google Scholar 

  31. Zhong N. Surface nanocrystallization mechanism of eutectoid steel and prediction of orientation relationship between cementite and ferrite. Shanghai Jiao Tong University Master’s thesis, 2005. (in Chinese)

    Google Scholar 

  32. Bollmann W. Crystal Lattices, Interfaces, Matrices. Bollmann, Geneva, 1982.

    Google Scholar 

  33. Waymann C W. Introduction to the crystallography of martensitic transformations. MacMillan, New York, 1964.

    Google Scholar 

  34. Luo C P, Xiao X L, Liu J W, et al. Principles of the invariant line model and its applications in crystallography. Progress in Science, 10: 193–200, 2000. (in Chinese)

    Google Scholar 

  35. Zhang WZ, Weatherly G C. On the crystallography of precipitation. Progress in Materials Science, 50: 181–292, 2005.

    Article  CAS  Google Scholar 

  36. Bollmann W. Crystal Defects and Crystalline Interfaces. Springer, Berlin, 1970.

    Google Scholar 

  37. Ogawa K, Kajiwara S. High-resolution electron microscopy study of ledge structures and transformation lattices at the austenite-martensite interface in Fe-based alloys. Philosophical Magazine, 84: 2919–2947, 2004.

    Article  CAS  Google Scholar 

  38. Zhang W Z. Use of D-lattice for study of crystallography of phase transformations. Proceeding of the International Conference on Solid-Solid Phase Transformations, Edited by Koiwa M, Otsuka K, Miyazaki T. The Japan Institute of Metals, 581–584, 1999.

    Google Scholar 

  39. Qiu D, Shen Y X, Zhang W Z. An extended invariant line analysis for FCC/BCC precipitation systems, Acta Materialia, 54: 339–347, 2006.

    Article  CAS  Google Scholar 

  40. Ye F, Zhang W Z, Qiu D. Near-coincidence-sites modeling of the edge facet dislocation structures of α precipitates in a Ti-7.26 wt.% Cr alloy. Acta Materialia, 54: 5377–5384, 2006.

    Article  CAS  Google Scholar 

  41. Zhang W Z, Wu J. Dislocation description of martensite interfaces based on misfit analysis. Materials Science and Engineering A, 438–440: 118–121, 2006.

    Article  Google Scholar 

  42. Zhang W Z, Qiu D. An extended near-coincidence-sites method and the interfacial structure of austenite precipitates in a duplex stainless steel. Acta Materialia, 56: 2003–2014, 2008.

    Article  Google Scholar 

  43. Wu J, Zhang W Z, Gu X F. A two-dimensional analytical approach for phase transformations involving an invariant line strain. Acta Materialia, 57: 635–645, 2009.

    Article  CAS  Google Scholar 

  44. Buerger M J. Elementary Crystallography: An Introduction to the Fundamental Geometrical Features of Crystal, Revised Edition. The MIT Press, Cambridge, Massachusetts, and London, England, 1963.

    Google Scholar 

  45. Guo K X, Ye H Q, Wu Y K. Application of Electron Diffraction Pattern in Crystallography. Science Press, Beijing, 1983. (in Chinese)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China

    Prof. Yonghua Rong

Authors
  1. Prof. Yonghua Rong
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Rong, Y. (2012). Mathematics Analysis in Electron Diffraction and Crystallography. In: Characterization of Microstructures by Analytical Electron Microscopy (AEM). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20119-6_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-20119-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20118-9

  • Online ISBN: 978-3-642-20119-6

Publish with us

Policies and ethics

Search

Navigation