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Equivalence of the Critical Concentrations in the Ising and Heisenberg Models of Ferromagnetism

Equivalence of the Critical Concentrations in the Ising and Heisenberg Models of Ferromagnetism HEISENBERG MODELS OF FERROMAGNETISM R. J. Elliott and B. R. Heap Clarendon Laboratory, Oxford, England and D. J. Morgan and G. S. Rushbrooke King's College, Newcastle upon Tyne, England (Received September 19, 1960) If a magnetic crystal is diluted but otherwise unchanged by the addition of nonmagnetic atoms randomly arranged in the lattice, the transition temperature falls. There is a certain concentra- tion of magnetic atoms below which the system will not exhibit a transition to magnetic order. At this critical concentration the transition tem- perature is at 0K. It is the purpose of this Letter to show that in ferromagnetic crystals the critical concentration is a topological property of the lattice and is the same in the Helsenberg and Ising models and is independent of the value of the spin S. Nearest neighbor interactions only are assumed so that the Hamiltonians are -2Jj(ilj)Si*Sj and -2JZ, OXSi S. for the Heis- enberg and Ising models, respectively. The result may be derived from a consideration of the behavior of the susceptibility X when ex- panded in a power series in the concentration p of magnetic atoms. In the random dilute system let there be N1 isolated magnetic atoms, N2 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review Letters American Physical Society (APS)

Equivalence of the Critical Concentrations in the Ising and Heisenberg Models of Ferromagnetism

Elliott, R. J; Heap, B. R
Physical Review Letters , Volume 5 (8) – Oct 15, 1960
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Publisher
American Physical Society (APS)
Copyright
Copyright © 1960 The American Physical Society
ISSN
1079-7114
DOI
10.1103/PhysRevLett.5.366
Publisher site
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Abstract

HEISENBERG MODELS OF FERROMAGNETISM R. J. Elliott and B. R. Heap Clarendon Laboratory, Oxford, England and D. J. Morgan and G. S. Rushbrooke King's College, Newcastle upon Tyne, England (Received September 19, 1960) If a magnetic crystal is diluted but otherwise unchanged by the addition of nonmagnetic atoms randomly arranged in the lattice, the transition temperature falls. There is a certain concentra- tion of magnetic atoms below which the system will not exhibit a transition to magnetic order. At this critical concentration the transition tem- perature is at 0K. It is the purpose of this Letter to show that in ferromagnetic crystals the critical concentration is a topological property of the lattice and is the same in the Helsenberg and Ising models and is independent of the value of the spin S. Nearest neighbor interactions only are assumed so that the Hamiltonians are -2Jj(ilj)Si*Sj and -2JZ, OXSi S. for the Heis- enberg and Ising models, respectively. The result may be derived from a consideration of the behavior of the susceptibility X when ex- panded in a power series in the concentration p of magnetic atoms. In the random dilute system let there be N1 isolated magnetic atoms, N2

Journal

Physical Review Letters – American Physical Society (APS)

Published: Oct 15, 1960

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