The S = 1 Ising Model for 3He-4He Mixtures

  • W. M. Ng
  • J. H. Barry
  • T. Tanaka


As a preliminary study toward understanding phase transitions in 3He-4He mixtures, the S = 1 Ising model has recently been proposed1 by Blume, Emery, and Griffiths (BEG) and, within the Weiss molecular-field approximation, qualitatively good agreement was obtained in comparison with experiment. Previously the model as applied to magnetism had been studied in the Weiss approximation independently by Capel2 and Blume.3 For helium mixtures the Hamiltonian used by BEG and formulated in terms of the grand canonical ensemble is given by
$$\widetilde H = - J\sum\limits_{ < i,j > } {{S_i}{S_j}} + \Delta \sum\limits_i {S_i^2} $$
where the summation is taken over all distinct nearest-neighbor pairs, S i = 0 corresponds to a 3He atom occupying lattice site i, and S i = ±1 corresponds similarly for a 4He atom, 2J > 0 denotes the interaction strength between neighboring 4He atoms, and ∆ ≡ μ 3μ 4 is the difference in chemical potentials of the 3He and 4He atoms. There is a fixed number N of lattice sites and no vacancies are allowed. Then the thermal expectation value 〈S i 〉 simulates the superfluid order parameter and the concentration of 3He atoms is given by x ≡ 1 − 〈S i 2 〉.


Ising Model Helmholtz Free Energy Grand Canonical Ensemble Tricritical Point Chemical Potential Difference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Blume, V.J. Emery, and R.B. Griffiths, Phys. Rev. A4, 1071 (1971).ADSCrossRefGoogle Scholar
  2. 2.
    H.W. Capel, Physica 32, 966 (1966).ADSCrossRefGoogle Scholar
  3. 3.
    M. Blume, Phys. Rev. 141, 517 (1966).ADSCrossRefGoogle Scholar
  4. 4.
    T. Morita and T. Tanaka, Phys. Rev. 145, 288 (1966).MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    J. Halow, T. Tanaka, and T. Morita, Phys. Rev. 175, 680 (1968).ADSCrossRefGoogle Scholar
  6. 6.
    J. Oitmaa, J. Phys. C. Sol. St. Phys. 4, 2466 (1971).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1974

Authors and Affiliations

  • W. M. Ng
    • 1
  • J. H. Barry
    • 1
  • T. Tanaka
    • 1
  1. 1.Department of PhysicsOhio UniversityAthensUSA

Personalised recommendations