Cluster expansion for the ising model

R. A. Farrell, T. Morita, P. H.E. Meijer

Research output: Contribution to journalArticlepeer-review

Abstract

A binary alloy system with nearest-neighbor interaction which forms a rigid lattice is the (ideal) Ising system. The present paper is an application of the theory of classical fluids, as developed by Morita and Hiroike, to such systems. The partition function for the Ising system is made formally equivalent to that for a binary fluid as discussed by Morita. Then, using the methods of graph theory, a high-temperature expansion is obtained for fixed long-range order parameters. By performing a partial summation, one finds that the free energy is given in terms of connected diagrams which are composed of βJ bonds, and black circles Mn. The quantities Mn are evaluated by noting that the diagrammatical expressions for these quantities appear in the expansion of the grand potential of a soluble problem. The method is illustrated in the case of an AB alloy or the ferromagnetic Ising system which form cubic lattices. Chang's result is easily obtained. The extension of this method to higher terms and its application to other alloy systems are retained for a forthcoming paper.

Original languageEnglish (US)
Pages (from-to)349-363
Number of pages15
JournalThe Journal of Chemical Physics
Volume45
Issue number1
DOIs
StatePublished - Jan 1 1966

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Physical and Theoretical Chemistry

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