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Composition Modulations in Solid Solutions

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Modulated Structure Materials

Part of the book series: Nato ASI Series ((NSSE,volume 83))

Abstract

Most crystalline solids contain various atomic or molecular species which may substitute for one another on lattice sites. Kence, in any solid solution, site occupation can be regarded as composition-modulated. To describe the state of order of a crystalline phase, one must therefore find a suitable mathematical formulation for its composition modulation. It is further desirable not only to describe the state of order, but also to predict it, or at least to determine theoretically the change of order, long-range (LRO) or short-range (SRO), with respect to temperature, pressure, or average composition (or chemical potential). In this paper, a technique will be presented which accomplishes both tasks rather well, i.e. those of description and of prediction: the Cluster Variation Method (CVM). The CVM was initially proposed by Kikuchi(1) as a hierarchy of statistical mechanical approximations to solve the classical Ising model. The mathematical apparatus was subsequently improved upon by Kikuchi and co-workers(2,3), by Barker(4), Hijmans and De Boer(5) and more recently by Sanchez and co-workers(6–10)

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References

  1. R. Kikuchi, Phys. Rev. 81, 998 (1951).

    Google Scholar 

  2. R. Kikuchi, J. Chem. Phys. 60, 1071 (1974).

    Article  CAS  Google Scholar 

  3. M. Kurata, R. Kikuchi, and T. Watari, J. Chem. Phys. 21, 434(1953).

    Article  CAS  Google Scholar 

  4. J. A. Barker, Proc. Roy. Soc.A216, 45 (1953).

    Google Scholar 

  5. J. Hijmans and J. De Boer, Physica 21, 471, 485, 499 (1955); 408 (1956).

    Article  CAS  Google Scholar 

  6. J. M. Sanchez and D. de Fontaine, Phys. Rev. B L7, 2926 (1978).

    Article  Google Scholar 

  7. J. M. Sanchez and D. de Fontaine, Phys. Rev. B 21, 216 (1980).

    Article  CAS  Google Scholar 

  8. J. M. Sanchez and D. de Fontaine, Phys. Rev. B 25, 1759 (1982).

    Article  CAS  Google Scholar 

  9. J. M. Sanchez, D. de Fontaine, and W. Teitler, Phys. Rev. B 26, 1465 (1982).

    Article  CAS  Google Scholar 

  10. J. M. Sanchez, D. Gratias, and D. de Fontaine, Acta Cryst. A 38, 214 (1982).

    Article  Google Scholar 

  11. D. de Fontaine, Solid State Physics 34 (1979).

    Google Scholar 

  12. D. de Fontaine, Physica (Utrecht) 103B, 57 (1981).

    Google Scholar 

  13. D. de Fontaine, Course on Mechanical and Thermal Behavior of Metallic Materials, Enrico Fermi School of Physics, G. Cagliotti and A. Ferro Milone, Eds., Vol. 82 (1981).

    Google Scholar 

  14. D. de Fontaine in the Proceeding of the International Conference on Solid-Solid Phase Transformation, Pittsburgh, PA (1981).

    Google Scholar 

  15. D. de Fontaine, to be published in the Proceedings of the MRC Annual Meeting (1982).

    Google Scholar 

  16. R. Kikuchi, Acta Metall. 25, 195 (1977).

    Article  CAS  Google Scholar 

  17. R. Kikuchi, J. M. Sanchez, D. de Fontaine and H. Yamauchi, Acta Metall. 28, 651 (1980).

    Article  CAS  Google Scholar 

  18. T. Morita, J. Phys. Soc. Japan 12, 753 (1957); J. Math. Phys. 13, 115 (1972).

    Article  Google Scholar 

  19. J. M. Sanchez, unpublished.

    Google Scholar 

  20. F. Ducastelle, private communication.

    Google Scholar 

  21. A. Bieber, F. Gautier, G. Treglia, and F. Ducastelle, Solid State Comm., 39, 149 (1981).

    Article  CAS  Google Scholar 

  22. J. M. Sanchez, unpublished research at UCLA and U.C. Berkeley.

    Google Scholar 

  23. J.M. Sanchez and D. de Fontaine in Structure and Bonding in Crystals, M. O’Keeffe and A. Navrotsky, Eds., (Academic, N.Y. 1981) Vol. II, pp. 117–132.

    Google Scholar 

  24. J. Kanamori, Progr. Theor. Phys. 35, 66 (1966); J. Kanamori and Y. Kakehashi, J. Phys. (Paris) 38, C7-274 (1977).

    Article  Google Scholar 

  25. M. J. Richards and J. W. Cahn, Acta Metall. 19, 1263 (1971).

    Article  CAS  Google Scholar 

  26. S. M. Allen and J. W. Cahn, Acta Metall. 20, 423 (1972); Scripta Metall. 7, 1261 (1973).

    Article  CAS  Google Scholar 

  27. A. G. Khachaturyan in Progress in Materials Science, J. W. Christian, P. Haasen, and T. B. Massalski, Eds. (Pergamon, Oxford 1978).

    Google Scholar 

  28. D. de Fontaine, Acta Metall. 22, 553 (1975).

    Google Scholar 

  29. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, 3rd Edition, Part 1 (Oxford 1980).

    Google Scholar 

  30. International Tables for X-ray Crystallography, Symmetry Groups, edited by N. F. M. Henry and K. Lonsdale, Vol. 1, Kynoch Press, Birmingham (1952).

    Google Scholar 

  31. J. M. Sanchez, D. Gratias, and D. de Fontaine, Acta Cryst. A 38, 214 (1982).

    Article  Google Scholar 

  32. L. D. Landau and E. M. Lifshitz, Statistical Physics, Chapter XII, Addison-Wesley, Reading, Massachusetts (1958).

    Google Scholar 

  33. J. M. Sanchez, Physica U1A, 200 (1982).

    Google Scholar 

  34. R. Kikuchi and D. de Fontaine in NBS Publication SP-496, 967 (1978).

    Google Scholar 

  35. D. de Fontaine and R. Kikuchi in NBS Publication SP-496, 999 (1978).

    Google Scholar 

  36. R. Kikuchi and C. M. van Baal, Scripta Metall. 8, 425 (1974).

    Article  CAS  Google Scholar 

  37. C. M. van Baal, Physica (Utrecht) 64, 571 (1973).

    Article  Google Scholar 

  38. K. Binder, J. L. Lebowitz, M. K. Phani and M. H. Kalos, Acta Metall. 29 1655 (1981).

    Article  CAS  Google Scholar 

  39. R. A. Bond and D. K. Ross, J. Phys. F, 12, 597 (1982).

    Article  CAS  Google Scholar 

  40. J. M. Sanchez, T. Mohri, and D. de Fontaine, to be published.

    Google Scholar 

  41. R. P. Okamoto and G. Thomas, Acta Metall., 19, 825 (1971).

    Article  CAS  Google Scholar 

  42. S. M. Allen and J. W. Cahn, to be published in the Proceedings of the MRC Annual Meeting (1982).

    Google Scholar 

  43. J. W. Cahn, private communication to D. de Fontaine.

    Google Scholar 

  44. P. L. Martin, Ph.D. Dissertation, Dept. of Metallurgical Eng. and Materials Sc., Carnegie-Mellon University, Pittsburgh, PA (1982); P. L. Martin and J. C. Williams, to be published.

    Google Scholar 

  45. W. A. Soffa and D. E. Laughlin, pp. 159–183, Procedings of an International Conference on Solid-Solid Phase Transformations, H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman, Eds., The Metallurgical Society of AIME, Warrendale, Pennsylvania (1982).

    Google Scholar 

  46. S. K. Das, P. R. Okamoto, P. M. J. Fisher, and G. Thomas, Acta Metall. 21, 913 (1973).

    Article  CAS  Google Scholar 

  47. R. Kikuchi, Progr. Theor. Phys. 535, 1 (1966).

    Google Scholar 

  48. G. Van Tendeloo and S. Amelinckx, pp. 305–309, Proceedings of an International Conference on Solid-Solid Phase Transformations, H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman, Eds., The Metallurgical Society of AIME, Warrendale, Pennsylvania (1982).

    Google Scholar 

  49. S. Banerjee, U. Urban, and W. Wilkens, pp. 311–315, Proceedings of an International Conference on Solid-Solid Phase Transformations, H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman, Eds., The Metallurgical Society of AIME, Warrendale, Pennsylvania (1982).

    Google Scholar 

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Authors and Affiliations

  1. Department of Materials Science and Mineral Engineering, University of California, Berkeley, California, 94720, USA

    D. de Fontaine

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  1. D. de Fontaine
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Editors and Affiliations

  1. Department of Mechanics and Materials Science, Rutgers University, Piscataway, NJ, 08854, USA

    T. Tsakalakos

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© 1984 Martinus Nijhoff Publishers, Dordrecht

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de Fontaine, D. (1984). Composition Modulations in Solid Solutions. In: Tsakalakos, T. (eds) Modulated Structure Materials. Nato ASI Series, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6195-1_3

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  • DOI: https://doi.org/10.1007/978-94-009-6195-1_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-6197-5

  • Online ISBN: 978-94-009-6195-1

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