The Electronic Structure of Complex Systems pp 581-592 | Cite as

# Alloy Phase Diagrams From First Principles

## 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

Entropy Librium CuAu## Preview

Unable to display preview. Download preview PDF.

### References

- 1.W. Kohn and L.J. Sham, Phys.Rev.
*140*, A1133 (1965).MathSciNetADSCrossRefGoogle Scholar - 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.M. Schlüter and L.J. Sham, Physics Today
*35*, 36 (1982).CrossRefGoogle Scholar - 4.O. Gunnarsson, J. Harris and R.O. Jones, J.Chem.Phys.
*67*, 3970 (1977).ADSCrossRefGoogle Scholar - 5.B.I. Dunlap, J.W.D. Connolly and J.R. Sabin, J.Chem.Phys.
*71*, 4993 (1979).ADSCrossRefGoogle Scholar - 6.V.L. Moruzzi, J.F. Janak and A.R. Williams, “Calculated Electronic Properties of Metals”, Pergamon Press, New York (1978).Google Scholar
- 7.M.T. Yin and M.L. Cohen, Phys.Rev.Lett.
*45*, 1004 (1980).ADSCrossRefGoogle Scholar - 8.A.R. Williams, C.D. Gelatt and V.L. Moruzzi, Phys.Rev.Lett.
*44*, 429 (1980).ADSCrossRefGoogle Scholar - 9.H. Ehrenreich and L. Schwartz, Solid State Physics
*31*, 149 (1976).CrossRefGoogle Scholar - 10.F. Gautier, F. Ducastelle and.J. Giver, Phil.Mag.
*31*, 1373 (1975).ADSCrossRefGoogle Scholar - 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.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.If one substitutes- the values of v
_{n}, of (3) into the expression for E_{D}one finds that the coefficients of E_{M}(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.D. deFontaine, Solid State Physics
*34*, 73 (1979).CrossRefGoogle Scholar - 15.R. Kikuchi, Phys.Rev.
*81*, 988 (1951).MathSciNetADSMATHCrossRefGoogle Scholar - 16.R. Kikuchi and D. deFontaine, NBS Publ. SP-496, 967 (1978).Google Scholar
- 17.J.M. Sanchez and D. deFontaine, Phys.Rev.
*B21*, 216 (1980).ADSGoogle Scholar - 18.M.E. Fisher and R.J. Burford, Phys.Rev.
*156*, 583 (1967)ADSCrossRefGoogle Scholar - 19.O.G. Mouritsen, S.J. Knak-Jensen and B. Frank, Phys.Rev.
*B24*, 347 (1981).ADSGoogle Scholar - 20.K. Binder, J.L. Lebowitz, M.K. Phani and M.H. Kalos, Acta.Met.
*29*, 1655 (1981).CrossRefGoogle Scholar - 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.D.G. Pettifor, Phys.Rev.Lett.
*49*, 846 (1979).ADSCrossRefGoogle Scholar - 23.J. Bernholc, N.O. Lipari and S.T. Pantelides, Phys.Rev.Lett.
*41*, 895 (1978); Phys.Rev..*B21*,1545(1980).ADSCrossRefGoogle Scholar - 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