Vacancies and antisite defects in ordered alloys
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Equations for the concentrations of vacancies and antisite defects in ordered alloys in thermodynamic equilibrium at and near stoichiometry have been derived as functions of defect energies and a Lagrangian parameter. While the resulting equations cannot be solved analytically and in general require iterative calculations, an approximation is given that permits simple numerical evaluation with just a minor loss of accuracy. Using defect energies obtained from an embedded-atom method calculation for Cu3Au, it is found that the adjustment for off-stoichiometric compositions is accounted for primarily by the creation of antisite defects rather than vacancies, and the vacancy concentration on Au sites is orders of magnitude less than that on Cu sites. There is a significant increase in the Au vacancy concentration but a slight decrease in the net vacancy content with increasing Cu fraction.
I. INTRODUCTION Recent articles by Foiles and Daw1 and Kim2 have contributed to the understanding of thermodynamic concentrations of vacancies and antisite atoms in binary intermetallic compounds. Foiles and Daw used a thermodynamic approach with a grand potential involving vacancy and antisite energies and the chemical potentials of the constituent atom types. The problem they implicitly address is, "For a particular structure (e.g., A 3 B in an LI 2 lattice, possibly off stoichiometry), given a constant number of total lattice sites and ratio of A to B type atoms, what are the concentrations of A type atoms, B type atoms, and vacancies on A type sites and on B type sites." The energies in their equations for vacancies involve removing an atom from the interior of the crystal to infinity, not to the surface, and the energies for the antisite defects involve removing one atom type to infinity and bringing the other atom type from infinity to the empty site. The energies are obtained from calculations using the embedded-atom method (EAM) for Ni 3 Al. In the approach used by Kim, the underlying problem is, "For a particular structure, given a constant number of A type atoms and B type atoms, what are the concentrations of A type atoms, B type atoms and vacancies in A type sites and on B type sites." The crystal energy is written in terms of nearest-neighbor A-A, A-B, B-B, A-V, and B-V "bond" energies. Effective bond energies are then defined that eliminate the A-V and B-V energies, and these effective energies are estimated as 1/6 the pure-metal vacancy formation energy for the A-A and B-B energies and from the temperature dependence of the ordering parameter for the A-B energy. In the present development, the same problem that was studied by Kim is analyzed, i.e., the numbers of A type and B type atoms are held constant and the total J. Mater. Res., Vol. 7, No. 12, Dec 1992 http://journals.cambridge.org
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number of lattice sites is varied. Defect energies rather than bond energies are used to define the Gibbs energy of the system. This Gibbs energy is then minimized subject to the condition that the ratio of
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