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Cluster Variation and Cluster Statics

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Theory and Applications of the Cluster Variation and Path Probability Methods

Abstract

Ever since its early formulation by Ryoichi Kikuchi in 1951, the Cluster Variation Method (CVM) has become a versatile tool for Ising Model applications, particularly in complex situations such as those required by phase diagram calculations. Now that first-principles approaches are becoming feasible, the CVM has become virtually indispensable: the deficiencies of the Bragg-Williams can no longer be masked by fudging adjustable interaction parameters. Thanks to delightful tutorials given to us by Ryo Kikuchi at UCLA in the 70’s, Juan Miguel Sanchez was soon able to develop a remarkable “Cluster Algebra” which has proved to be the formalism of choice for ab initio alloy thermodynamics. I shall review this formalism and its possible applications to alloy phase stability.

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References

  1. R. Kikuchi, Phys. Rev. 79, 718 (1950).

    Article  ADS  Google Scholar 

  2. R. Kikuchi, Phys. Rev. 81, 988 (1951).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. R. Kikuchi, Prog. Theor. Phys. Suppl. 35, 1 (1966).

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. D. de Fontaine and R. Kikuchi, in Application of Phase Diagrams in Metallurgy and Ceramics, N.B.S. Special publication 496, edited by G. C. Carter (1978), p. 967.

    Google Scholar 

  6. W. Shockley, J. Chem. Phys. 6, 130 (1938).

    Article  ADS  Google Scholar 

  7. J. Kanamori, Prog. Theor. Phys. 35, 66 (1966).

    Article  Google Scholar 

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

    Google Scholar 

  9. S. M. Allen and J. W. Cahn, Scripta. Metall. 7, 1261 (1973).

    Article  Google Scholar 

  10. J. M. Sanchez and D. de Fontaine, Phys. Rev. B 17, 2926 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  11. J. M. Sanchez, F. Ducastelle, and D. Gratias, Physica 128A, 334 (1984).

    MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. T. Mohri, J.M. Sanchez, and D. de Fontaine, Acta Metall. 33, 1171 (1985).

    Article  Google Scholar 

  16. M. Sluiter, P. Turchi, F. Zezhong, and D. de Fontaine, Physica A 148, 61 (1988).

    Article  ADS  Google Scholar 

  17. R. McCormack, M. Asta, D. de Fontaine, and G. Ceder, Phys. Rev. B 48, 6767 (1993).

    Article  ADS  Google Scholar 

  18. C. Sigh and J. M. Sanchez, Acta Metall. 36, 367 (1988).

    Article  Google Scholar 

  19. M. H. F. Sluiter, D. de Fontaine, X.Q. Guo, R. Podloucky, and A. J. Freeman, Phys. Rev. B 42, 10460 (1990).

    Article  ADS  Google Scholar 

  20. J. M. Sanchez, Physica 111A, 200 (1982).

    ADS  Google Scholar 

  21. G. Ceder, Alloy Theory and its Application to Long-Period Superstructure Ordering in Metallic Alloys and High-Temperature Superconductors. Doctoral Dissertation, University of California, Berkeley (1991); University Microfilms Int’l. Ann Arbor, MI (# 9203517).

    Google Scholar 

  22. J. M. Sanchez and D. de Fontaine, Structure and Bonding in Crystals (Academic Press, New York, 1981), Vol. II, p. 117.

    Google Scholar 

  23. F. Ducastelle, Order and Phase Stability in Alloys, (North-Holland, New York, 1991).

    Google Scholar 

  24. G. Inden and W. Pitsch, “Atomic Ordering,” in Phase Transformations in Materials, edited by P. Haasen, (VCH Press, Weinheim and New York, 1991), p. 497.

    Google Scholar 

  25. J. M. Sanchez, V. Pierron-Bohnes, and F. Mejía-Lira, Phys. Rev. B 51, 3429 (1995).

    Article  ADS  Google Scholar 

  26. D. de Fontaine, in Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull, (Academic Press, 1979), Vol. 34, p. 74.

    Google Scholar 

  27. D. de Fontaine, Solid State Physics, edited by H. Ehrenreich and D. Turnbull, (Academic Press, 1994), Vol. 47, p. 33

    Google Scholar 

  28. A. Finel, Contribution à l’Etude des Effets d’Ordre dans le Cadre du Modèle d’Ising: Etats de Base et Diagrammes de Phase, doctoral dissertation, Université Pierre et Marie Curie, Paris, 1987 (unpublished).

    Google Scholar 

  29. K. Kawasaki, in Phase Transitions and Critical Phenomena, Vol. 2, Ed. C. Domb and M. S. Green, (Academic Press, New York, 1973), p. 465.

    Google Scholar 

  30. M. Asta, C. Wolverton, D. de Fontaine, and H. Dreyssé, Phys. Rev. B 44, 4907 (1991)

    Article  ADS  Google Scholar 

  31. C. Wolverton, M. Asta, H. Dreyssé, and D. de Fontaine, Phys. Rev. B 44, 4914 (1991).

    Article  ADS  Google Scholar 

  32. D. de Fontaine, A. Finel, H. Dreyssé, M. Asta, R. McCormack, and C. Wolverton, in Proc. NATO Advanced Research Workshop on Metallic Alloys: Experimental and Theoretical Perspectives, Boca Raton, FL, July 16–21, 1993, (1994).

    Google Scholar 

  33. J. M. Sanchez, Phys. Rev. B 48, 14013 (1993).

    Article  ADS  Google Scholar 

  34. S. Ouannasser and H. Dreyssé, in preparation.

    Google Scholar 

  35. A. Gonis, P. P. Singh, P. E. A. Turchi, and X. -G. Zhang, Phys. Rev. B 51, 2122 (1995).

    Article  ADS  Google Scholar 

  36. G. Ceder, Compt. Mater. Sci. 1, 144 (1993).

    Article  Google Scholar 

  37. H. Dreyssé, A. Berera, L. T. Wille, and D. de Fontaine, Phys. Rev. B 39, 2442 (1989)

    Article  ADS  Google Scholar 

  38. C. Wolverton, H. Dreyssé, and D. de Fontaine, Mat. Res. Soc. Symp. Proc. 193, 183 (1991)

    Article  Google Scholar 

  39. C. Wolverton, G. Ceder, D. de Fontaine, and H. Dreyssé, Phys. Rev. B 48, 726 (1993)

    Article  ADS  Google Scholar 

  40. C. Wolverton and D. de Fontaine, Phys. Rev. B (1994)

    Google Scholar 

  41. C. Wolverton and A. Zunger, Phys. Rev. B 50, 10548 (1994).

    Article  ADS  Google Scholar 

  42. J. W. D. Connolly and A. R. Williams, Phys. Rev. B 27, 5169 (1983).

    Article  ADS  Google Scholar 

  43. Z. W. Lu, S. -H. Wei, A. Zunger, S. Frotta-Pessoa, and L.G. Ferreira, Phys. Rev. B 44, 512 (1991)

    Article  ADS  Google Scholar 

  44. A. Zunger, in Statics and Dynamics of Alloy Phase Transformations, edited by P. E. A. Turchi and A. Gonis, (NATO ASI Series, Kluwer, Dordrecht, 1994), p. 361 and references cited therein.

    Chapter  Google Scholar 

  45. J. S. Faulkner, Prog. Mater. Sci. 27, 1 (1982).

    Article  Google Scholar 

  46. F. Ducastelle and F. Gautier, J. Phys. F6, 2039 (1976).

    Article  ADS  Google Scholar 

  47. B. L. Györffy and G. M. Stocks, Phys. Rev. Lett. 50, 374 (1983).

    Article  ADS  Google Scholar 

  48. A. Gonis and J. W. Garland, Phys. Rev. B 16, 1495, 2424 (1977); 18, 3999 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  49. A. Chiolero, Doctoral Dissertation, University of Fribourg, Switzerland (1995).

    Google Scholar 

  50. D. B. Laks, L. G. Ferreira, S. Froyen, and A. Zunger, Phys. Rev. B 46, 12587 (1992).

    Article  ADS  Google Scholar 

  51. C. Amador, J.J. Hoyt, B.C. Chakoumakos, and D. de Fontaine, Phys. Rev. Lett. 74, 4955 (1994).

    Article  ADS  Google Scholar 

  52. M. H. F. Sluiter, Y. Watanabe, D. de Fontaine, and Y. Kawazoe, submitted for publication.

    Google Scholar 

  53. C. Amador, W. R. L. Lambrecht, M. van Schilfgaarde, and B. Segall, Phys. Rev. B 47, 15276 (1993).

    Article  ADS  Google Scholar 

  54. C. Amador and G. Bozzolo, Phys. Rev. B 49, 956 (1994).

    Article  ADS  Google Scholar 

  55. P. Xie and D. de Fontaine, unpublished work at UC Berkeley.

    Google Scholar 

  56. A. Chiolero and D. Baeriswyl, J. Stat. Phys. 76, 347 (1994).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  57. C. Wolverton and A. Zunger, Phys. Rev. Lett., in press.

    Google Scholar 

  58. V. L. Moruzzi, J. F. Janak, and K. Schwarz, Phys. Rev. B 37, 790 (1988)

    Article  ADS  Google Scholar 

  59. J. M. Sanchez, J. P. Stark, and V. L. Moruzzi, Phys. Rev. B 44, 5411 (1991).

    Article  ADS  Google Scholar 

  60. A. Finel, Progr. Theor. Phys. 115, 59 (1994).

    Article  Google Scholar 

  61. A. Finel and R. Tétot, preprint.

    Google Scholar 

  62. C. Wolverton and A. Zunger, Phys. Rev. B 49, 16058 (1994) and Phys. Rev. B, in press.

    Article  ADS  Google Scholar 

  63. M. Asta, R. McCormack, and D. de Fontaine, Phys. Rev. B 48, 748 (1993).

    Article  ADS  Google Scholar 

  64. W. Hume-Rothery and G. V. Raynor, Proc. Roy. Soc. A174, 471 (1940).

    ADS  Google Scholar 

  65. P. J. Craievich, M. Weinert, J. M. Sanchez, and R. E. Watson, Phys. Rev. Lett. 72, 3076 (1994).

    Article  ADS  Google Scholar 

  66. D. A. Vul and D. de Fontaine, Mat. Res. Soc. Symp. Proc. 291, 401 (1993).

    Article  Google Scholar 

  67. M. Sluiter, Comp. Mat, Sc. 2, 293 (1994).

    Article  Google Scholar 

  68. A. G. Schlijper, A. R. D. Vandebergen, and B. Smit, Phys. Rev. A 41, 1175 (1990).

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Materials Science Department and Lawrence Berkeley Laboratory, University of California, Berkeley, USA

    Didier de Fontaine

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  1. Didier de Fontaine
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Editor information

Editors and Affiliations

  1. Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico

    J. L. Morán-López

  2. The University of Texas at Austin, Austin, Texas, USA

    J. M. Sanchez

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© 1996 Plenum Press, New York

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de Fontaine, D. (1996). Cluster Variation and Cluster Statics. In: Morán-López, J.L., Sanchez, J.M. (eds) Theory and Applications of the Cluster Variation and Path Probability Methods. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0419-7_8

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  • DOI: https://doi.org/10.1007/978-1-4613-0419-7_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-8043-6

  • Online ISBN: 978-1-4613-0419-7

  • eBook Packages: Springer Book Archive

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