### 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 M_{n}. The quantities M_{n} 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 language | English (US) |
---|---|

Pages (from-to) | 349-363 |

Number of pages | 15 |

Journal | The Journal of Chemical Physics |

Volume | 45 |

Issue number | 1 |

State | Published - 1966 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*The Journal of Chemical Physics*,

*45*(1), 349-363.

**Cluster expansion for the ising model.** / Farrell, R. A.; Morita, T.; Meijer, P. H E.

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 45, no. 1, pp. 349-363.

}

TY - JOUR

T1 - Cluster expansion for the ising model

AU - Farrell, R. A.

AU - Morita, T.

AU - Meijer, P. H E

PY - 1966

Y1 - 1966

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=36849108443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36849108443&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:36849108443

VL - 45

SP - 349

EP - 363

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 1

ER -