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Progress in a long-standing objective in computational materials science, the calculation of alloy phase diagrams from first-principles, has been paced by the associated problems of accurate yet efficient total-energy calculations and statistical approximations of the alloy partition function. The former is well understood, and the evolution of Density-Functional Theory (DFT) to computationally suitable methods using the Local-Density Approximation (LDA) is a renowned (though, arguably, a qualified) success. The high precision required of the total-energy band-structure methods for phase-equilibrium calculations is attainable for most chemical systems. The connection between the total-energy calculations and a general solution to the Ising problem is provided by a cluster expansion in a series of multi-atom interaction energies. This rigorous expansion my be inverted to calculate the total energy of any lattice configuration using the multi-atom interactions. Two main aspects of the cluster expansion method remain unresolved: the convergence of the expansion and the effect of strain energy on the convergence. These two issues are central to this thesis, and a major portion of this study is devoted toward a resolution.