Critical Phenomena and Phase Transitions

  • J. F. Nagle
Part of the Treatise on Solid State Chemistry book series (TSSC, volume 5)


The area of critical phenomena is a rather large one, both quantitatively, as measured by the number of researchers and the number of publications, and qualitatively, as judged by the percentage of “hard” results such as exact calculations and highly precise experiments. This makes it impossible to write an article or even a book which is completely self-contained and which includes all that has been accomplished.


Ising Model Critical Exponent Critical Phenomenon Heisenberg Model Dime Model 
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.
    J. W. Cahn, On spinoidal decomposition, Acta Met. 9, 795–801 (1961).Google Scholar
  2. 2.
    T. J. Tiedema, J. Bouman, and W. G. Burgers, Precipitation in Au–Pt alloys, Acta Met. 5, 310–321 (1957).Google Scholar
  3. 3.
    F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, J. Math. Phys. 12, 2259–2272 (1971).Google Scholar
  4. 4.
    P. W. Kasteleijn, Dimer statistics and phase transitions, J. Math. Phys. 4, 287–293 (1963).Google Scholar
  5. 5.
    J. F. Nagle and G. R. Allen, Models of the order–disorder transition in NaH3 (SeO3)2, J. Chem. Phys. 55, 2708–2714 (1971).Google Scholar
  6. 6.
    V. G. Vaks, A. I. Larkin, and Yu. N. Ovchinnikov, Ising model with interaction between non-nearest neighbors, Soviet Phys.—JETP 22, 820–826 (1966).Google Scholar
  7. 7.
    G. T. Rado and H. Suhl (eds.), Magnetism, Academic, New York (1963).Google Scholar
  8. 8.
    D. P. Landau, B. E. Keen, B. Schneider, and W. P. Wolf, Magnetic and thermal properties of DAG I, Phys. Rev. B3, 2310–2343 (1971).Google Scholar
  9. 9.
    W. P. Wolf, M. J. M. Leask, B. Mangum, and A. F. G. Wyatt, Ferromagnetism in gadolinium trichloride, J. Phys. Soc. (Japan) 17 (Suppl. B-1), 487–492 (1961).Google Scholar
  10. 10.
    W. Marshall and R. D. Lowde, Magnetic correlations and neutron scattering, Rep. Prog. Phys. 31, 705–775 (1968).Google Scholar
  11. 11.
    D. C. Mattis, The Theory of Magnetism, Harper and Row, New York (1965).Google Scholar
  12. 12.
    P. W. Anderson, in Magnetism (G. T. Rado and H. Suhl, eds.), Vol. I, pp. 25–81, Academic, New York (1963).Google Scholar
  13. 13.
    K. W. H. Stevens, Spin Hamiltonians, in Magnetism (G. T. Rado and H. Suhl, eds.), Vol. I, pp. 1–23, Academic, New York (1963).Google Scholar
  14. 14.
    W. P. Wolf, Anisotropic interactions between magnetic ions, J. de Physique 32, C1–26–C1–33 (1971).Google Scholar
  15. 15.
    H. A. Brown and J. M. Luttinger, Ferromagnetic and antiferromagnetic Curie temperatures, Phys. Rev. 100, 685–692 (1955).Google Scholar
  16. 16.
    T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transition II, Lattice gas and Ising model, Phys. Rev. 87, 410–419 (1952).Google Scholar
  17. 17.
    G. Stell, H. Narang, and C. K. Hall, Simple lattice gases with realistic phase changes, Phys. Rev. Letters 28, 292–294 (1972).Google Scholar
  18. 18.
    C. K. Hall and G. Stell, Phase transitions in two-dimensional lattice gases of hard core molecules with weak long-range attractions, Phys. Rev. A7, 1679–1689 (1973).Google Scholar
  19. 19.
    R. B. Griffiths, A proposal for notation at tricritical points, Phys. Rev. B7, 545–551 (1973).Google Scholar
  20. 20.
    M. E. Fisher, Lattice statistics in a magnetic field I. A two-dimensional super-exchange antiferromagnet, Proc. Roy. Soc. A254, 66–85 (1960).Google Scholar
  21. 21.
    J. H. Schelleng and S. A. Friedberg, Thermal behavior of the antiferromagnet MnBr24H2O in applied magnetic field, Phys. Rev. 185, 728–734 (1969).Google Scholar
  22. 22.
    F. Jona and G. Shirane, Ferroelectric Crystals, Pergamon Press (1962).Google Scholar
  23. 23.
    Y. Takagi, Theory of the transition in KH2PO4 II, J. Phys. Soc. Japan 3, 271–272 (1948).Google Scholar
  24. 24.
    R. Blinc and S. Svetina, Cluster approximation for order-disorder-type hydrogen bonded ferroelectrics II. Application to KDP, Phys. Rev. 147, 430–438 (1966).Google Scholar
  25. 25.
    W. Reese, Studies of phase transitions in order-disorder ferroelectrics. III. The phase transition in KH2PO4 and a comparison with KD2PO4, Phys. Rev. 181, 905–919 (1969).Google Scholar
  26. 26.
    L. W. Garland and R. Renard, First order phase transitions in Ising models due to lattice interactions, J. Chem. Phys. 44, 1120–1129 (1965).Google Scholar
  27. 27.
    G. A. Baker and J. W. Essam, Lattice compressibility and critical behavior, J. Chem. Phys. 55, 861–879 (1971).Google Scholar
  28. 28.
    B. D. Josephson, Relation between the superfluid density and order parameter for superfluid He near T e,, Phys. Letters 21, 608–609 (1966).Google Scholar
  29. 29.
    O. Penrose and L. Onsager, Bose-Einstein condensation and liquid helium, Phys. Rev. 104, 576–584 (1956).Google Scholar
  30. 30.
    M. D. Girandeau, Off-diagonal long range order and generalized Bose condensation, J. Math. Phys. 6, 1083–1098 (1965).Google Scholar
  31. L. P. Kadanoff et al.,Static phenomena near critical points, Rev. Mod. Phys. 39, 395–431 (1971).Google Scholar
  32. 32.
    M. E. Fisher, The theory of equilibrium critical phenomena, Rep. Progr. Phys. XXX (Part II), 615–730 (1967).Google Scholar
  33. 33.
    C. Kittel, Quantum Theory of Solids, p. 157, Wiley, New York (1963).Google Scholar
  34. 34.
    T. W. Stinson and J. D. Litster, Pretransitional phenomena in the isotropic phase of a nematic liquid crystal, Phys. Rev. Letters 25, 503–506 (1970).Google Scholar
  35. 35.
    B. Chu, C. S. Bak, and F. L. Lin, Coherence length in the isotropic phase of a room-room-temperature nematic liquid crystal, Phys. Rev. Letters 28, 1111–1114 (1972).Google Scholar
  36. 36.
    J. R. McColl and C. S. Shih, Temperature dependence of orientational order in a nematic liquid crystal at constant molar volume, Phys. Rev. Letters 29, 85–87 (1972).Google Scholar
  37. 37.
    M. E. Fisher, Correlation functions and the critical region of simple fluids, J. Math. Phys. 5, 944–962 (1964).Google Scholar
  38. 38.
    M. E. Fisher, The Nature of Critical Points, Lectures in Theoretical Physics, Vol. VIIC, pp. 1–159 University of Colorado Press (1965).Google Scholar
  39. 39.
    P. Heller, Experimental investigations of critical phenomena, Rep. Prog. Phys. XXX (Part II), 731–826 (1967).Google Scholar
  40. 40.
    B. M. McCoy, Theory of a two-dimensional Ising model with random impurities III, Phys. Rev. 188, 1014–1031 (1969).Google Scholar
  41. 41.
    B. M. McCoy, Incompleteness of critical exponent descriptions for ferromagnetic systems containing random impurities, Phys. Rev. Letters 23, 383–386 (1969).Google Scholar
  42. 42.
    R. B. Griffiths and J. C. Wheeler, Critical points in multicomponent systems, Phys. Rev. A2, 1047–1064 (1970).Google Scholar
  43. 43.
    J. F. Nagle and J. C. Bonner, Ising chain with interactions in a staggered field, J. Chem. Phys. 54, 729–734 (1971).Google Scholar
  44. 44.
    W. M. Fairbank and C. F. Kellers, the lambda transition in liquid helium, in Critical Phenomena (Nat. Bur. St. Misc. Publ. No 273) ( M. S. Green and J. V. Sengers, (eds.), pp. 71–78, National Bureau of Standards, Washington, D.C. (1966).Google Scholar
  45. 45.
    G. Ahlers, Heat capacity near the superfluid transition in He’ at saturated vapor pressure, Phys. Rev. A3, 696–716 (1971).Google Scholar
  46. 46.
    L. Kreps and S. A. Friedberg, Specific heat measurements of MnBr24H2O, to be published.Google Scholar
  47. 47.
    J. R. Clow and J. D. Reppy, Persistent-current measurements of the superfluid density and critical velocity, Phys. Rev. A5, 424–438 (1972).Google Scholar
  48. 48.
    J. A. Tyson, Superfluid density of He II in the critical region, Phys. Rev. 166, 166–176 (1968).Google Scholar
  49. 49.
    D. S. Greywell and G. Ahlers, Second-sound velocity, scaling and universality in He II under pressure near the superfluid transition, Phys. Rev. Letters 28, 1251–1254 (1972).Google Scholar
  50. 50.
    G. R. Brown and H. Meyer, Study of the specific heat singularity of He3 near its critical point, Phys. Rev. A6, 364–377 (1972).Google Scholar
  51. 51.
    B. Wallace and H. Meyer, Critical isotherm of He3, Phys. Rev. A2, 1610–1612 (1970).Google Scholar
  52. 52.
    B. Wallace and H. Meyer, Equation of state of He3 close to the critical point, Phys. Rev. A2, 1563–1575 (1970).Google Scholar
  53. 53.
    M. R. Moldover, Scaling of the specific-heat singularity of He4 near its critical point, Phys. Rev. 182, 342–351 (1969).Google Scholar
  54. 54.
    P. R. Roach, Pressure-density-temperature surface of He4 near the critical point, Phys. Rev. 170, 213–223 (1968).Google Scholar
  55. 55.
    B. Chu and J. S. Lin, Small angle scattering from CO2 in the vicinity of its critical point, J. Chem. Phys. 53, 4454 (1970).Google Scholar
  56. 56.
    J. Lipa, C. Edwards, and M. Buckingham, Precision measurement of the specific heat of CO2 near the critical point, Phys. Rev. Letters 25, 1086–1090 (1970).Google Scholar
  57. 57.
    J. M. H. Levelt Sengers and W. T. Chen, The vapor pressure, critical isochore and some metastable states of CO2, J. Chem. Phys. 56, 595–608 (1972).Google Scholar
  58. 58.
    C. Edwards, J. A. Lipa, and M. J. Buckingham, Specific heat of xenon near the critical point, Phys. Rev. Letters 20, 496–499 (1968).Google Scholar
  59. 59.
    I. W. Smith, M. Giglio, and G. B. Benedek, Correlation range and compressability of xenon near the critical point, Phys. Rev. Letters 27, 1556–1560 (1971).Google Scholar
  60. 60.
    M. Vicentini-Missoni, J. M. H. Levelt Sengers, and M. S. Green, Thermodynamic anomalies of CO2, Xe, and He4 in the Critical Region, Phys. Rev. Letters 22, 389–393 (1969).Google Scholar
  61. 61.
    J. Als-Nielsen, Investigation of scaling laws by critical neutron scattering from beta-brass, Phys. Rev. 185, 664–666 (1969).Google Scholar
  62. 62.
    D. R. Chipman and C. B. Walker, Long range order in β-brass, Phys. Rev. B5, 3823–3831 (1972).Google Scholar
  63. 63.
    G. Longworth, Temperature dependence of the 57Fe hfs in the ordered alloys FePd3 and FePd near the Curie temperature, Phys. Rev. 172, 572–576 (1968).Google Scholar
  64. 64.
    L. Guttmans, H. C. Schnyders, and G. J. Arai, Variation of long-range order in Fe3Al near T c. Phys. Rev. Letters 22, 517–519 (1969).Google Scholar
  65. 65.
    L. Guttmans and H. C. Snyders, Critical scattering of X-rays from Fe3Al, Phys. Rev. Letters 22, 520–522 (1969).Google Scholar
  66. 66.
    C. W. Garland and R. A. Young, Order-disorder phenomena. VI Anomalous changes in the volume of ammonium chloride, J. Chem. Phys. 48, 146–148 (1968).Google Scholar
  67. 67.
    H. C. Benski, R. C. Reno, C. Hohenemser, and C. Abeledo, New Mössbauer effect on 57Fe in a Ni host: the critical exponent for Ni, Phys. Rev. B6, 4266–4275 (1972).Google Scholar
  68. 68.
    J. E. Noakes and A. Arrott, Magnetization of nickel near its critical temperature, J. Appl. Phys. 39, 1235–1236 (1968).Google Scholar
  69. 69.
    J. S. Kouvel and J. B. Comly, Magnetic equation of state for nickel near its Curie point, Phys. Bev. Letters 20, 1237–1239 (1968).Google Scholar
  70. 70.
    D. L. Connelly, J. S. Loomis, and D. E. Mapother, Specific heat of nickel near T., Phys. Rev. B3, 924–934 (1971).Google Scholar
  71. 71.
    M. F. Collins, V. J. Minkiewicz, R. Nathans, L. Passell, and G. Shirane, Critical and spin-wave scattering of neutrons from iron, Phys. Rev. 179, 417–430 (1969).Google Scholar
  72. 72.
    J. T. Ho and J. D. Litster, Faraday rotation near the ferromagnetic critical temperature of CrBr3, Phys. Rev. B2, 4523–4532 (1970).Google Scholar
  73. 73.
    M. P. Schulhof, R. Nathans, P. Heller, and A. Linz, Inelastic neutron scattering from MnF2 in the critical region, Phys. Rev. B4, 2254–2276 (1971).Google Scholar
  74. 74.
    L. M. Corliss, A. Delapalme, J. M. Hastings, H. Y. Lau, and R. Nathans, Critical magnetic scattering from RbMnF3, J. Appl. Phys. 40, 1278 (1969).Google Scholar
  75. 75.
    W. B. Yelon and D. E. Cox, Magnetic ordering in RbNiCl3, Phys. Rev. B6, 204–208 (1972).Google Scholar
  76. 76.
    M. T. Hutchings, M. P. Schulhof, and H. J. Guggenheim, Critical magnetic neutron scattering from FeF2, Phys. Rev. B5, 154–168 (1972).Google Scholar
  77. 77.
    K. Deguchi and E. Nakamura, Critical region in ferroelectric TGS, Phys. Rev. 85, 1072–1073 (1972).Google Scholar
  78. 78.
    R. Blinc, M. Burger, and A. Levstik, Critical behavior of ferroelectric TGS and deuterated TGS, Solid State Commun. 8, 317–321 (1970).Google Scholar
  79. 79.
    Th. O. Waldkirch, K. A. Müller, W. Berlinger, and H. Thomas, Fluctuations and correlations in SrTiO3 for TT c, Phys. Rev. Letters 28, 503–506 (1972).Google Scholar
  80. 80.
    G. S. Rushbrooke, On the thermodynamics of the critical region for the Ising problem, J. Chem. Phys. 39, 842–843 (1963).Google Scholar
  81. 81.
    R. B. Griffiths, Rigorous results and theorems, in Phase Transitions and Critical Phenomena (C. Domb and M. S. Green, eds.), Vol. I, pp. 7–109, Academic, London (1972).Google Scholar
  82. 82.
    L. Onsager, Crystal statistics I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117–149 (1944).Google Scholar
  83. 83.
    G. A. Baker, Certain general order-disorder models in the limit of long range interactions, Phys. Rev. 126, 2071–2078 (1962).Google Scholar
  84. 84.
    M. Kac, G. E. Uhlenbeck, and P. C. Hemmer, On the van der Waals theory of the vapor-liquid equilibrium, J. Math. Phys. 4, 216–228 (1963).Google Scholar
  85. 85.
    J. Lebowitz and O. Penrose, Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition, J. Math. Phys. 7, 98–113 (1966).Google Scholar
  86. 86.
    L. D. Landau and E. M. Lifschitz, Statistical Physics, p. 135, Pergamon, Oxford (1958).Google Scholar
  87. 87.
    M. Blume, V. J. Emery, and R. B. Griffiths, Ising model for the transition and phase separation in He3-He4 mixtures, Phys. Rev. A4, 1071–1077 (1971).Google Scholar
  88. 88.
    R. B. Griffiths, Thermodynamics near the two fluid critical mixing point in He3-He4, Phys. Rev. Letters 24, 715–717 (1970).Google Scholar
  89. 89.
    W. K. Theumann and J. S. Hoye, Ising chain with several phase transitions, J. Chem. Phys. 55, 4159–4166 (1971).Google Scholar
  90. 90.
    P. C. Hemmer and G. Stell, Phase transitions due to softness of the potential core, Phys. Rev. Letters 24, 1284–1287 (1970).Google Scholar
  91. 91.
    G. Stell, H. Narang, and C. K. Hall, Simple lattice gases with realistic phase changes, Phys. Rev. Letters 28, 292–294 (1972).Google Scholar
  92. 92.
    J. Bernasconi and F. Rys, Critical behavior of a magnetic alloy, Phys. Rev. B4, 3045–3048 (1971).Google Scholar
  93. 93.
    J. Stephenson, Ising model spin correlations on the triangular lattice IV, J. Math. Phys. 11, 420–431 (1970).Google Scholar
  94. 94.
    R. J. Birgeneau, J. Skalyo, and G. Shirane, Critical magnetic scattering in K2NiF4, Phys. Rev. B3, 1736–1749 (1971).Google Scholar
  95. 95.
    B. Kaufman and L. Onsager, Crystal statistics III, Phys. Rev. 76, 1244–1252 (1949).Google Scholar
  96. 96.
    T. Schultz, D. C. Mattis, and E. H. Lieb, Two-dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys. 36, 856–871 (1964).Google Scholar
  97. 97.
    M. J. Stephen and L. Mittag, A new representation of the solution of the Ising model, J. Math. Phys. 13, 1944–1951 (1973).Google Scholar
  98. 98.
    M. Kac and J. C. Ward, A combinational solution of the two-dimensional Ising model, Phys. Rev. 88, 1332–1337 (1952).Google Scholar
  99. 99.
    E. Montroll, in Applied Combinatorial Mathematics ( E. F. Beckenbach, ed.), pp. 96–143, Wiley, New York (1964).Google Scholar
  100. 100.
    B. Widom, Plait points in two-and three-component liquid mixtures, J. Chem. Phys. 46, 3324–3333 (1967).Google Scholar
  101. 101.
    R. K. Clark and G. A. Neece, Decorated lattice model for ternary systems, J. Chem. Phys. 48, 2575–2582 (1968).Google Scholar
  102. 102.
    J. C. Wheeler, Behavior of a solute near the critical point of an almost pure solvent, Ber. Bunsenges. Physikal. Chem. 76, 308–318 (1972).Google Scholar
  103. 103.
    M. E. Fisher, Renormalization of critical exponents by hidden variables, Phys. Rev. 176, 257–272 (1968).Google Scholar
  104. 104.
    O. J. Heilmann and E. H. Lieb, Theory of monomer-timer systems, Commun. Math. Phys. 25, 190–232 (1972).Google Scholar
  105. 105.
    C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions II, Phys. Rev. 150, 327–339 (1966).Google Scholar
  106. 106.
    E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Letters 18, 692–694 (1967).Google Scholar
  107. 107.
    R. J. Bacter, Partition function of the eight-vertex lattice model, Ann. Phys. 70, 193–228 (1972).Google Scholar
  108. 108.
    F. Y. Wu, Exact solution of a model of an antiferroelectric transition, Phys. Rev. 183, 604–607 (1969).Google Scholar
  109. 109.
    E. H. Lieb and F. Y. Wu, Two-dimensional ferroelectric models, in Phase Transitions and Critical Phenomena (C. Domb and M. S. Green, eds.), Vol. I, 332–487, Academic, London (1972).Google Scholar
  110. 110.
    J. F. Nagle, Proof of the first order phase transition in the Slater KDP model, Commun. Math. Phys. 13, 62–67 (1967).Google Scholar
  111. 111.
    J. F. Nagle, The one dimensional KDP model in statistical mechanics, Am. J. Phys. 36, 1114–1117 (1968).Google Scholar
  112. 112.
    J. W. Benepe and W. Reese, Electronic studies of KH2PO4, Phys. Rev. B3, 30323039 (1971).Google Scholar
  113. 113.
    J. F. Nagle, Lattice statistics of hydrogen bonded crystals II, J. Math. Phys. 7, 1492–1496 (1966).Google Scholar
  114. 114.
    J. F. Nagle, Study of the F-model using low-temperature series, J. Chem. Phys. 50, 2813–2818 (1969).Google Scholar
  115. 115.
    R. J. Baxter, Spontaneous staggered polarization of the F-model, J. Phys. C6, L94 (1973).Google Scholar
  116. 116.
    R. J. Baxter, Exact isotherm for the F model in direct and staggered electric fields. Phys. Rev. B1, 2199–2202 (1970).Google Scholar
  117. 117.
    J. D. Johnson, S. Krinsky, and B. M. McCoy, Critical index of the vertical arrow correlation length in the eight-vertex model, Phys, Rev. Letters 29, 492–494 (1972).Google Scholar
  118. 118.
    M. E. Fisher and J. Stephenson, Statistical mechanics of dimers on a plane lattice II, Phys. Rev. 132, 1411–1431 (1963).Google Scholar
  119. 119.
    B. Sutherland, Two dimensional hydrogen bonded crystals without the ice rule, J. Math. Phys. 11, 3183–3186 (1970).Google Scholar
  120. 120.
    T. H. Berlin and M. Kac, The spherical model of a ferromagnet, Phys. Rev. 86, 821–835 (1952).Google Scholar
  121. 121.
    H. E. Stanley, Spherical model as the limit of infinite spin dimensionality, Phys. Rev. 176, 718–722 (1968).Google Scholar
  122. 122.
    G. S. Joyce, Spherical model with long-range ferromagnetic interactions, Phys. Rev. 146, 349–358 (1966).Google Scholar
  123. 123.
    J. F. Nagle and J. C. Bonner, Numerical studies of the Ising chain with long-range ferromagnetic interactions, J. Phys. C3, 352–366 (1970).Google Scholar
  124. 124.
    M. E. Fisher, S. Ma, and B. G. Nickel, Critical exponents for long-range interactions, Phys. Rev. Letters 29, 917–920 (1972).Google Scholar
  125. 125.
    M. Suzuki, Critical exponents for long-range interactions I—Dimensionality, symmetry, and potential range, Progr. Theor. Phys. 49, 424–441 (1973).Google Scholar
  126. 126.
    J. C. Bonner and J. F. Nagle, Phase behavior of models with competing interactions, J. Appl. Phys. 42, 1280–1282 (1971).Google Scholar
  127. 127.
    L. K. Runnels, J. P. Salvant, and H. R. Streiffer, Exact finite method of lattice statistics IV, J. Chem. Phys. 52, 2352–2358 (1970).Google Scholar
  128. 128.
    C. Domb and M. S. Green (eds.), Phase Transitions and Critical Phenomena, Vol. 3, Academic, London (1973).Google Scholar
  129. 129.
    C. Domb, On the theory of cooperative phenomena in crystals, Advan. Phys. 9, 149–361 (1960).Google Scholar
  130. 130.
    C. Domb and M. F. Sykes, Use of series expansions for the Ising model susceptibility and excluded volume problem, J. Math. Phys. 2, 63–67 (1961).Google Scholar
  131. 131.
    G. A. Baker and D. L. Hunter, Methods of series analysis II, Phys. Rev. 7, 3377–3392 (1973).Google Scholar
  132. 132.
    D. S. Gaunt, M. E. Fisher, M. F. Sykes, and J. W. Essam, Critical isotherms of a ferromagnet and of a fluid, Phys. Rev. Letters 13, 713–715 (1964).Google Scholar
  133. 133.
    D. S. Gaunt and M. F. Sykes, Reanalysis of the critical isotherm of the Ising ferro-magnet, J. Phys. C: Solid State Phys. 5, 1429–1444 (1972).Google Scholar
  134. 134.
    D. S. Gaunt and C. Domb, The specific heat of the 3-d Ising model below T„ J. Phys. C 1, 1038–1045 (1968).Google Scholar
  135. 135.
    A. J. Guttman, C. J. Thompson, and B. W. Ninham, Determination of critical behavior from series expansions in lattice statistics IV, J. Phys. C 3, 1641–1651 (1970).Google Scholar
  136. 136.
    M. E. Fisher and R. J. Burford, Theory of critical-point scattering and correlations I, Phys. Rev. 156, 583–622 (1967).Google Scholar
  137. 137.
    M. A. Moore, D. Jasnow, and M. Wortis, Spin—spin correlation function of the 3-d Ising ferromagnet above the Curie temperature, Phys. Rev. Letters 22, 940–943 (1967).Google Scholar
  138. 138.
    G. A. Baker, H. E. Gilbert, J. Eve, and G. S. Rushbrooke, High temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev. 164, 800–817 (1967).Google Scholar
  139. 139.
    R. G. Bowers and M. E. Woolf, Some critical properties of the Heisenberg model, Phys. Rev. 177, 917–932 (1969).Google Scholar
  140. 140.
    M. H. Lee and H. E. Stanley, Spin-1/2 Heisenberg ferromagnet on cubic lattices, Phys. Rev. B4, 1613–1630 (1971).Google Scholar
  141. 141.
    G. A. Baker, J. Eve, and G. S. Rushbrooke, Magnetic phase boundary of the spin-1/2 Heisenberg ferromagnetic model, Phys. Rev. B2, 706–721 (1970).Google Scholar
  142. 142.
    M. Ferer, M. A. Moore, and M. Wortis, Some critical properties of the nearest neighbor, classical Heisenberg model for the fcc lattice in finite field for temperatures greater than T c. Phys. Rev. B4, 3954–3963 (1971).Google Scholar
  143. 143.
    D. S. Ritchie and M. E. Fisher, Theory of critical point scattering and correlations II, Phys. Rev. B5, 2668–2692 (1972).Google Scholar
  144. 144.
    G. S. Rushbrooke and P. J. Wood, On Curie points and high temperature susceptibilities of Heisenberg model ferromagnets, Mol. Phys. 1, 257–283 (1958).Google Scholar
  145. 145.
    H. E. Stanley and T. A. Kaplan, Possibility of a phase transition for the 2-d Heisenberg model, Phys. Rev. Letters 17, 913–915 (1966).Google Scholar
  146. 146.
    N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models, Phys. Rev. Letters 17, 1133–1136 (1966).Google Scholar
  147. 147.
    M. E. Fisher and B. U. Felderhof, Phase transitions in one-dimensional cluster-interaction fluids, Ann. Phys. 58, 176–300 (1970).Google Scholar
  148. 148.
    K. G. Wilson, Renormalization group and critical phenomena II, Phys. Rev. B4, 3184–3205 (1971).Google Scholar
  149. 149.
    G. A. Baker, Ising model with a scaling interaction, Phys. Rev. B5, 2622–2633 (1972).Google Scholar
  150. 150.
    K. G. Wilson, Feynman-graph expansion for critical exponents, Phys. Rev. Letters 28, 548–551 (1972).Google Scholar
  151. 151.
    M. Suzuki, Critical exponents for long-range interactions, Progr. Theor. Phys. 49, 1440–1450 (1973).Google Scholar
  152. 152.
    B. Widom, Equation of state in the neighborhood of the critical point, J. Chem. Phys. 43, 3898–3905 (1965).Google Scholar
  153. 153.
    R. B. Griffiths, Thermodynamic functions for fluids and ferromagnets near the critical point, Phys. Rev. 158, 176–187 (1967).Google Scholar
  154. 154.
    H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York (1971).Google Scholar
  155. 155.
    L. P. Kadanoff, Scaling laws for Ising models near T c. Physics 2, 263–272 (1966).Google Scholar
  156. 156.
    G. Stell, Some implications of weak-scaling theory, Phys. Rev. B2, 2811–2813 (1970).Google Scholar
  157. 157.
    G. Stell, Scaling theory of the critical region for system with long-range forces, Phys. Rev. B5, 981–985 (1972).Google Scholar
  158. 158.
    W. Theumann, Phenomenological weak scaling theory, Phys. Rev. B6, 281–286 (1972).Google Scholar
  159. 159.
    D. Jasnow and M. Wortis, High-temperature critical indices for the classical anisotropic Heisenberg model, Phys. Rev. 176, 739–750 (1968).Google Scholar
  160. 160.
    R. B. Griffiths, Dependence of critical indices on a parameter, Phys. Rev. Letters 24, 1479–1482 (1970);Google Scholar
  161. R. B. Griffiths, Critical points dependent on parameters, in Critical Phenomena in Alloys, Magnets and Superconductors (R. E. Mills, E. Ascher, and R. I. Jaffee, eds.), pp. 377–391. McGraw-Hill, New York (1971).Google Scholar
  162. 161.
    M. Ferer, M. A. Moore, and M. Wortis, Universality of critical correlations in the three-dimensional Ising ferromagnet, Phys. Rev. B3, 3911–3914 (1971).Google Scholar
  163. 162.
    P. F. Fox and D. S. Gaunt, Critical isotherm of the Ising ferromagnet with spin 1/2, J. Phys. C; Solid State Phys. 5, 3085–3096 (1972).Google Scholar
  164. 163.
    J. F. Nagle, Ising chain with competing interactions, Phys. Rev. A2, 2124–2128 (1970).Google Scholar
  165. 164.
    L. P. Kadanoff and F. J. Wegner, Some critical properties of the 8-vertex model, Phys. Rev. B4, 3989–3993 (1971).Google Scholar
  166. 165.
    L. P. Kadanoff, in Proc. Enrico Fermi Summer School of Physics, Varenna, 1970 ( M. S. Green, ed.), Academic, New York (1972).Google Scholar
  167. 166.
    B. Widom, Private communication.Google Scholar
  168. 167.
    J. F. Nagle, Lipid bilayer phase transitions, Proc. Natl. Acad. Sci.,to be published.Google Scholar

Copyright information

© Springer Science+Business Media New York 1975

Authors and Affiliations

  • J. F. Nagle
    • 1
  1. 1.Department of PhysicsCarnegie-Mellon UniversityPittsburghUSA

Personalised recommendations