Algebraic analysis of cluster expansion method and its application to first-principles calculations of phase equilibria

Cluster Expansion Method has been extensively employed for the first principles calculation of phase equilibria. The essential mathematical procedure is to calculate total energies for K ordered phases including pure elements, which forms K × 1 energy vector, and to operate K × K inversed matrix of correlation functions on the energy vector. The method is based on the orthogonality of the correlation functions in the thermodynamic configuration space and is versatile to various alloy systems. However, when one would like to obtain (K+ 1)-st cluster interaction energy, it is not sufficient to add an arbitrary phase in the energy vector for which the total energy is to be calculated. This is because the elements of the inversed matrix of correlation functions are generally not preserved and, therefore, the net effect of the (K+ 1)-st cluster interaction energy is obscured. In the present paper, algebraic aspects of the correlation functions are examined in detail and we propose some criterion to extract the net contribution of the (K+1)-st cluster interaction energy. The criterion is applied to Fe-Pt system and the effects of 2 ^nd nearest neighbor pair interaction and three body interaction energies on the transition temperature are discussed.