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Alloy Phase Diagrams From First Principles

  • J. W. D. Connolly
  • A. R. Williams
Part of the NATO ASI Series book series (NSSB, volume 113)

Abstract

The first applications of density-functional theory to disordered systems are described. The usual difficulty of solving the single-particle Schrödinger equation for systems lacking long-range order is circumvented by using the cluster-expansion method of Sanchez and deFontaine to describe the alloy, and deducing the coefficients appearing in this expansion from energy-band calculations for several ordered compounds. The cluster expansion obtained in this way describes the alloy as a continuous function of stoichiometry, short-range order and volume. We have used the technique, in combination with an approximate description of the entropy of mixing, to calculate the dominant features of the phase diagrams of 28 transition-metal alloy systems. Agreement with measured phase diagrams is generally very good. Not tested in these first applications of the technique is its ability to describe properties of disordered systems other than the total energy, properties such as the state density. Also not yet tested is the ability of the technique to directly predict the degree of short-range order present in the system, by minimization of the free energy.

Keywords

Phase Diagram Alloy Phase Diagram Cluster Expansion Diagram Type NATO Advance Study Institute 
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.

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References

  1. 1.
    W. Kohn and L.J. Sham, Phys.Rev. 140, A1133 (1965).MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    A.R. Williams and U. von Barth, “Applications of Density-Functional Theory to Atoms, Molecules and Solids” in “The Theory of the Inhomogeneous Electron Gas”, N.H. March and S. Lundqvist Eds. Plenum, New York (1982).Google Scholar
  3. 3.
    M. Schlüter and L.J. Sham, Physics Today 35, 36 (1982).CrossRefGoogle Scholar
  4. 4.
    O. Gunnarsson, J. Harris and R.O. Jones, J.Chem.Phys. 67, 3970 (1977).ADSCrossRefGoogle Scholar
  5. 5.
    B.I. Dunlap, J.W.D. Connolly and J.R. Sabin, J.Chem.Phys. 71, 4993 (1979).ADSCrossRefGoogle Scholar
  6. 6.
    V.L. Moruzzi, J.F. Janak and A.R. Williams, “Calculated Electronic Properties of Metals”, Pergamon Press, New York (1978).Google Scholar
  7. 7.
    M.T. Yin and M.L. Cohen, Phys.Rev.Lett. 45, 1004 (1980).ADSCrossRefGoogle Scholar
  8. 8.
    A.R. Williams, C.D. Gelatt and V.L. Moruzzi, Phys.Rev.Lett. 44, 429 (1980).ADSCrossRefGoogle Scholar
  9. 9.
    H. Ehrenreich and L. Schwartz, Solid State Physics 31, 149 (1976).CrossRefGoogle Scholar
  10. 10.
    F. Gautier, F. Ducastelle and.J. Giver, Phil.Mag. 31, 1373 (1975).ADSCrossRefGoogle Scholar
  11. 11.
    J.M. Sanchez and D. deFontaine, “Theoretical Prediction of Ordered Superstructures in Metallic Alloys”, in Structure and Bonding in Crystals, vol. II, M. O’Keeffe and A. Navrotsky Eds. (Academic Press 1981 ), p. 117.Google Scholar
  12. 12.
    A.R. Williams, J. Kübler and C.D. Gelatt,Jr., Phys.Rev.B 19, 6094-(1974). The aspect of reliability that is particularly important for the study of chemical trends is the ability of the computer programs that implement the Augmented-Spherical-Wave method to run unattended during the night. This means that both the self-consistent-field iteration and the iteration to find the crystal volume that minimizes the calculated total energy have been successfully automated (by V.L. Moruzzi).ADSCrossRefGoogle Scholar
  13. 13.
    If one substitutes- the values of vn, of (3) into the expression for ED one finds that the coefficients of EM (m = the structure index for the five structures of Table I) are the same as in the density of states expression used by C.B. Sommers et al., Solid State Comm. 37, 761 (1981).Google Scholar
  14. 14.
    D. deFontaine, Solid State Physics 34, 73 (1979).CrossRefGoogle Scholar
  15. 15.
    R. Kikuchi, Phys.Rev. 81, 988 (1951).MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    R. Kikuchi and D. deFontaine, NBS Publ. SP-496, 967 (1978).Google Scholar
  17. 17.
    J.M. Sanchez and D. deFontaine, Phys.Rev. B21, 216 (1980).ADSGoogle Scholar
  18. 18.
    M.E. Fisher and R.J. Burford, Phys.Rev. 156, 583 (1967)ADSCrossRefGoogle Scholar
  19. 19.
    O.G. Mouritsen, S.J. Knak-Jensen and B. Frank, Phys.Rev. B24, 347 (1981).ADSGoogle Scholar
  20. 20.
    K. Binder, J.L. Lebowitz, M.K. Phani and M.H. Kalos, Acta.Met. 29, 1655 (1981).CrossRefGoogle Scholar
  21. 21.
    M. Hansen, Constitution of Binary Alloys (McGraw-Hill 1958); R.P. Elliott, Constitution of Binary Alloys, First Supplement (McGraw-Hill 1965); F.A. Shunk, Constitution of Binary Alloys, Second Supplement (McGraw-Hill 1969 ); W.G. Moffatt, The Handbook of Binary Phase Diagrams (General Electric Co. 1977, latest update Nov. 1981 ).Google Scholar
  22. 22.
    D.G. Pettifor, Phys.Rev.Lett. 49, 846 (1979).ADSCrossRefGoogle Scholar
  23. 23.
    J. Bernholc, N.O. Lipari and S.T. Pantelides, Phys.Rev.Lett. 41, 895 (1978); Phys.Rev.. B21,1545(1980).ADSCrossRefGoogle Scholar
  24. 24.
    The subjects of the other four lectures were: Metallic Cohesion, Transition-Metal Compound Formation, The Bonding of Transition Metals to Non-Transition Metals, and Magnetism in Transition Metals and Their Compounds. Some information on all of these subjects can be found in the book chapter cited as Ref. 2. The Augmented-Spherical-Wave energy-band method, which is the fundamental tool used to obtain much of the information discussed in the lectures is described in detail in Ref.12. The qualitative aspects of pure-metal cohesion are discussed in A.R. Williams, C.D. Gelatt Jr. and J.F. Janak, in Theory of Alloy Phase Formation L.H. Bennett editor (The Metallurgical Society of AIIIE, New York, 1980). Transition-metal compound formation is discussed in Ref. 8. The bonding of transition metals to non-transition metals is discussed in C.D. Gelatt Jr., A.R. Williams and V.L. Moruzzi, Phys.Rev. B 1982, (in press-). Magnetism in transition-metal compounds is discussed in A.R. Williams, R. Zeller, V. Moruzzi, C.D. Gelatt Jr. and J. Kübler, J.Appl.Phys. 52, 2067 (1981); A.R. Williams, V:L. Moruzzi, C.D. Gelatt Jr., J. Kubler and K. Schwarz, J.Appl.Phys. 53, 2019 (1982);and A.R. Williams, V.L. Moruzzi, C.D. Gelatt Jr. and J. Kubler, J.Mag. and Mag. Mater. (proceedings of the ICM 82 Conf., Kyoto 1982).Google Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • J. W. D. Connolly
    • 1
    • 2
  • A. R. Williams
    • 1
  1. 1.IBM T.J.Watson Research CenterNew YorkUSA
  2. 2.National Science FoundationWashingtonUSA

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