Archive for History of Exact Sciences

, Volume 59, Issue 3, pp 267–318 | Cite as

History of the Lenz-Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena

  • Martin Niss


I chart the considerable changes in the status and conception of the Lenz-Ising model from 1920 to 1950 in terms of three phases: In the early 1920s, Lenz and Ising introduced the model in the field of ferromagnetism. Based on an exact derivation, Ising concluded that it is incapable of displaying ferromagnetic behavior, a result he erroneously extended to three dimensions. In the next phase, Lenz and Ising’s contemporaries rejected the model as a representation of ferromagnetic materials because of its conflict with the new quantum mechanics. In the third phase, from the early 1930s to the early 1940s, the model was revived as a model of cooperative phenomena. I provide more detail on this history than the earlier accounts of Brush (1967) and Hoddeson, Schubert, Heims, and Baym (1992) and question some of their conclusions. Moreover, my account differs from these in its focus on the development of the model in its capacity as a model. It examines three aspects of this development: (1) the attitudes on the degree of physical realism of the Lenz-Ising model in its representation of physical phenomena; (2) the various reasons for studying and using it; and (3) the effect of the change in its theoretical basis during the transition from the old to the new quantum mechanics. A major theme of my study is that even though the Lenz-Ising model is not fully realistic, it is more useful than more realistic models because of its mathematical tractability. I argue that this point of view, important for the modern conception of the model, is novel and that its emergence, while perhaps not a consequence of its study, is coincident with the third phase of its development.


Quantum Mechanic Theoretical Basis Physical Phenomenon Realistic Model Major Theme 
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  1. 1.
    Anderson, P. W. (1984): “Gregory Wannier,” Physics Today 40(5), 100–102.Google Scholar
  2. 2.
    Ashkin, J. and Teller, E. (1943): “Statistics of Two-Dimensional Lattices with Four Components,” Physical Review 64, 179–184.Google Scholar
  3. 3.
    Berlin, T. H. and Kac, M. (1952): “The Spherical Model of a Ferromagnet,” Physical Review 86, 821–835.Google Scholar
  4. 4.
    Bethe, H. A. (1935): Statistical Theory of Superlattices,” Proceedings of the Royal Society [A] 150, 552–575.Google Scholar
  5. 5.
    Bethe, H. A. and Kirkwood, J. G. (1939): “Critical Behavior of Solid Solutions in the Order- Disorder Transformation,” Journal of Chemical Physics 7, 578–582.Google Scholar
  6. 6.
    Bethe, H. A. with Hoddeson, Lillian (1981): Interview, April 29, 1981, Niels Bohr Library, American Institute of Physics, College Park, Maryland.Google Scholar
  7. 7.
    Bhattacharjee, S. M and Khare, A. (1995): “Fifty Years of the Exact Solution of the Two-dimensional Ising Model by Onsager,” Current Science (India) 69, 816–821.Google Scholar
  8. 8.
    Bitter, F. (1937): Introduction to Ferromagnetism. McGraw-Hill, New York.Google Scholar
  9. 9.
    Born, M. (1915): Dynamik der Kristallgitter. B. G. Teubner, Leipzig.Google Scholar
  10. 10.
    Bragg, W. L. and Williams, E. J. (1934): “The Effect of Thermal Agitation on Atomic Arrangement in Alloys,” Proceedings of the Royal Society [A] 145, 699–730.Google Scholar
  11. 11.
    Brush, S. G. (1966): Kinetic Theory, Vol 1. Pergamon Press, Oxford.Google Scholar
  12. 12.
    Brush, S. G. (1967): “History of the Lenz-Ising Model,” Reviews of Modern Physics 39, 883–893.Google Scholar
  13. 13.
    Brush, S. G. (1976): The Kind of Motion We Call Heat: A History of Kinetic Theory of Gases in the 19th Century. 2 Vols, North-Holland, Amsterdam.Google Scholar
  14. 14.
    Brush, S. G. (1983): Statistical Physics and the Atomic Theory of Matter. Princeton University Press, Princeton.Google Scholar
  15. 15.
    Cat, J. (2001): “On Understanding: Maxwell on the Methods of Illustration and Scientific Metaphor,” Studies in History and Philosophy of Modern Physics 32, 395–442.Google Scholar
  16. 16.
    Cipra, B. (2000): “Mathematics: Statistical Physicists Phase Out a Dream,” Science 288, 1561–1562.Google Scholar
  17. 17.
    Courant, R. (1930): Vorlesungen über Differential- und Integralrechnung, Second Edition. Springer, Berlin.Google Scholar
  18. 18.
    Dalitz, R. H. and Peierls, R. E., eds. (1997): Selected Scientific Papers of Sir Rudolf Peierls. With Commentary. World Scientific, Singapore and Imperial College Press, London.Google Scholar
  19. 19.
    Dirac, P. A. M. (1929): “Quantum Mechanics of Many-Electron Systems,” Proceedings of the Royal Society [A] 123, 714–733.Google Scholar
  20. 20.
    Domb, C. (1949): “Order-Disorder Statistics. I,” Proceedings of the Royal Society [A] 196, 36–50.Google Scholar
  21. 21.
    Domb, C. (1996): The Critical Point. Taylor and Francis, London.Google Scholar
  22. 22.
    Dresden, M. (1987): H. A. Kramers: Between Tradition and Revolution. Springer, New York.Google Scholar
  23. 23.
    Dresden, M. (1988): “Kramers’s Contribution to Statistical Mechanics,” Physics Today 41(9), 26–33.Google Scholar
  24. 24.
    Eckert, M., Schubert, H. and Torkar, G. with C. Blondel and P. Quédec (1992): “The Roots of Solid-State Physics before Quantum Mechanics,” in Hoddeson, Braun, Teichmann, and Weart (1992), pp. 3–87.Google Scholar
  25. 25.
    Ehrenfest, P. (1921): “Note on the Paramagnetism of Solids,” Verhandlingen der Koninklijke Akademie van Wetenschappen (Amsterdam) 29, 793–796.Google Scholar
  26. 26.
    Fowler, R. H. (1934): “Quelques remarques sur la théorie des métaux liquides de Mott et sur les points de transition des métaux et d’autres solides,” Helvetica Physica Acta Supplementum 7, 72–80.Google Scholar
  27. 27.
    Fowler, R. H. (1936): “Adsorption Isotherms. Critical Conditions,” Proceedings of the Cambridge Philosophical Society 32, 144–151.Google Scholar
  28. 28.
    ter Haar, D. (1998): Master of Modern Physics. The Scientific Contributions of H. A. Kramers”. Princeton University Press, Princeton.Google Scholar
  29. 29.
    ter Haar, D. and Martin, B. (1950): “Statistics of the 3-Dimensional Ferromagnet,” Physical Review 77, 721–722.Google Scholar
  30. 30.
    Heisenberg, W. (1928a): “Zur Theorie des Ferromagnetismus,” Zeitschrift für Physik 49, 619–636.Google Scholar
  31. 31.
    Heisenberg, W. (1928b): “Zur Quantentheorie des Ferromagnetismus,” in P. Debye, ed.: Probleme der modernen Physik: Arnold Sommerfeld zum 60. Geburtstage gewidmet von seinen Schülern. S. Hirzel, Leipzig, pp. 114–122.Google Scholar
  32. 32.
    Heller, G. and Kramers, H. A. (1934): “Ein Klassisches Modell des Ferromagnetikums und seine nachträgliche Quantisierung im Gebiete tiefer Temperaturen,” Verhandlingen der Koninklijke Akademie van Wetenschappen (Amsterdam) 37, 378–385.Google Scholar
  33. 33.
    Hemmer, P. C., Holden, H. and Kjelstrup Ratkje, S., eds. (1996): The Collected Works of Lars Onsager. World Scientific, Singapore.Google Scholar
  34. 34.
    Hermann, A., von Meyenn, K., and Weisskopf, V. F. (1979): Wolfgang Pauli. Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u. a. Scientific Correspondence with Bohr, Einstein, Heisenberg, a. o.. Vol. 1: 1919–1929. Springer, New York.Google Scholar
  35. 35.
    Herzfeld, K. F. (1925): “Molekular- und Atomtheorie des Magnetismus,” Physikalische Zeitschrift 26, 825–832.Google Scholar
  36. 36.
    Hoddeson, L., Baym, G., and Eckert, M. (1992): “The Development of the Quantum Mechanical Electron Theory of Metals, 1926–1933,” in Hoddeson, Braun, Teichmann, and Weart (1992), pp. 88–181.Google Scholar
  37. 37.
    Hoddeson, L., Braun, E., Teichmann, J. and Weart, S. (1992): Out of the Crystal Maze. Chapters from the History of Solid-State Physics. Oxford University Press, New York.Google Scholar
  38. 38.
    Hoddeson, L., Schubert, H., Heims, S. J., and Baym, G. (1992): “Collective Phenomena,” in Hoddeson, Braun, Teichmann, and Weart (1992), pp. 489–616.Google Scholar
  39. 39.
    Hofstadter, D. R. (1984): “A Nose for Depth: Gregory Wannier’s Style in Physics,” Physics Reports 110, 273–278.Google Scholar
  40. 40.
    Huang, K. (1963): Statistical Mechanics. Wiley, New York.Google Scholar
  41. 41.
    Hughes, R. I. G. (1999): “The Ising model, Computer Simulation, and Universal Physics,” in Morgan and Morrison (1999), pp. 97–145.Google Scholar
  42. 42.
    Hulthén, L. (1938): “Über das Austauschproblem eines Kristalles,” Arkiv för Matematik, Astronomi och Fysik 26A, 1–106Google Scholar
  43. 43.
    Ising, E. (1924): “Beitrag zur Theorie des Ferro- und Paramagnetismus,” Ph.D. Thesis, University of Hamburg.Google Scholar
  44. 44.
    Ising, E. (1925): “Beitrag zur Theorie des Ferromagnetismus,” Zeitschrift für Physik 31, 253–258.Google Scholar
  45. 45.
    Jammer, M. (1966): The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York.Google Scholar
  46. 46.
    Kac, M. (1964): “The work of T. H. Berlin in Statistical Mechanics...A Personal Reminisence,” Physics Today 17(10), 40–42.Google Scholar
  47. 47.
    Kac, M. (1971): “The Role of Models in Understanding Phase Transitions,” in Mills, Ascher, and Jaffee (1971), pp. 23–39.Google Scholar
  48. 48.
    Kaufman, B. (1949): “Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis,” Physical Review 76, 1232–1243.Google Scholar
  49. 49.
    Keith, S.T. and Quedec, P. (1992): “Magnetism and Magnetic Materials,” in Hoddeson, Braun, Teichmann, and Weart (1992), pp. 359–442.Google Scholar
  50. 50.
    Kikuchi, R. (1951): “A Theory of Cooperative Phenomena,” Physical Review 81, 988–1003.Google Scholar
  51. 51.
    Kirkwood, J. G. (1938): “Order and Disorder in Binary Solid Solutions,” Journal of Chemical Physics 6, 70–75.Google Scholar
  52. 52.
    Kobe, S. (1997): “Ernst Ising - Physicist and Teacher,” Journal of Statistical Mechanics 88, 991–995.Google Scholar
  53. 53.
    Kobe, S. (2000): “Ernst Ising 1900–1998,” Brazilian Journal of Physics 40, 649–653.Google Scholar
  54. 54.
    Kramers, H. A. (1929): “La rotation paramagnétique du plan de polarisation dans les cristaux uniaxes de terres rares,” Communications from the Physical Laboratory of the University at Leiden 18, Supplement 68b, 19–36.Google Scholar
  55. 55.
    Kramers, H. A. (1936): “Zur Theorie des Ferromagnetismus,” in 7e Congres international du froid: La Haye-Amsterdam juin 1936. Rapports et Communications. Also in: Communications from the Physical laboratory of the University of Leiden 22, Supplement 83, 1–22.Google Scholar
  56. 56.
    Kramers, H. A. and Becquerel, J. (1929): “La rotation paramagnétique du plan de polarisation dans les cristaux de tysonite et de xénotime,” Communications from the Physical Laboratory of the University at Leiden 18, Supplement 68c, 39–50.Google Scholar
  57. 57.
    Kramers, H. A. and Wannier, G. H. (1941a): “Statistics of the Two-Dimensional Ferromagnet Part I,” Physical Review 60, 252–262.Google Scholar
  58. 58.
    Kramers, H. A. and Wannier, G. H. (1941b): “Statistics of the Two-Dimensional Ferromagnet Part II,” Physical Review 60, 263–277.Google Scholar
  59. 59.
    Krieger, M. H. (1996): Constitutions of Matter: Mathematically Modeling the Most Everyday of Physical Phenomena. The University of Chicago Press, Chicago.Google Scholar
  60. 60.
    Lacki, J., H. Ruegg, V. L. Telegdi (1999): “The Road to Stueckelberg’s Covariant Perturbation Theory as Illustrated by Successive Treatments of Compton Scattering,” Studies in History and Philosophy of Modern Physics 30, 457–518.Google Scholar
  61. 61.
    Langevin, P. (1905): “Magnétisme et théorie des électrons,” Annales de Chimie et de Physique 8e série 5, 70–127.Google Scholar
  62. 62.
    Lassettre, E. N. and Howe, J. P. (1941): “Thermodynamic Properties of Binary Solid Solutions on the Basis of the Nearest-Neighbor Approximation,” Journal of Chemical Physics 9, 747–754.Google Scholar
  63. 63.
    Lenz, W. (1920): “Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern,” Physikalische Zeitschrift 21, 613–615.Google Scholar
  64. 64.
    Liu, C. (1999): “Explaining the Emergence of Cooperative Phenomena,” Philosophy of Science 66, S92-S106.Google Scholar
  65. 65.
    Longuet-Higgins, H. C. and Fisher, M. E. (1996): “Lars Onsager: 27 November, 1903–5 October, 1976,” in Hemmer, Holden, Kjelstrup Ratkje (1996), pp. 9–34.Google Scholar
  66. 66.
    Mattis, D. C. (1985): The Theory of Magnetism, Vol. 2. Springer, Berlin.Google Scholar
  67. 67.
    Maxwell, J. C. (1867): “On the Dynamical Theory of Gases,” Philosophical Transactions of the Royal Society of London 157, 49–88.Google Scholar
  68. 68.
    McCoy, B. M. and Wu, T. T. (1973): The Two-Dimensional Ising Model. Harvard University Press, Cambridge, Mass.Google Scholar
  69. 69.
    Mehra, J. and Rechenberg, H. (1982a): The Historical Development of Quantum Theory, Vol. 1. Springer, New York.Google Scholar
  70. 70.
    Mehra, J. and Rechenberg, H. (1982b): The Historical Development of Quantum Theory, Vol. 3. Springer, New York.Google Scholar
  71. 71.
    Mills, R. E., Ascher, E., and Jaffee, R. I., eds. (1971): Critical Phenomena in Alloys, Magnets and Superconductors [Battelle Institute Materials Science Colloquia, Geneva and Gstaad, September, 1970], McGraw-Hill, New York.Google Scholar
  72. 72.
    Montroll, E. W. (1941): “Statistical Mechanics of Nearest Neighbor Systems,” Journal of Chemical Physics 9, 706–721.Google Scholar
  73. 73.
    Morgan, M. S. and Morrison, M., eds. (1999): Models as Mediators. Cambridge University Press, Cambridge.Google Scholar
  74. 74.
    Morrison, M. (1999): “Models as Autonomous Agents,” in Morgan and Morrison (1999), pp. 38–65.Google Scholar
  75. 75.
    Nambu, Y. (1949): “A Note on the Eigenvalue Problem in Crystal Statistics,” Progress in Theoretical Physics 5 1–13.Google Scholar
  76. 76.
    Newell, G. F. (1950): “Crystal Statistics of a Two-Dimensional Triangular Ising Lattice,” Physical Review 79, 876–882.Google Scholar
  77. 77.
    Newell, G. F. and Montroll, E. W. (1953): “On the Theory of the Ising Model of Ferromagnetism,” Reviews of Modern Physics 25, 353–389.Google Scholar
  78. 78.
    Nix, F. C. and Shockley, W. (1938): “Order-Disorder Transformations in Alloys,” Reviews of Modern Physics 10, 1–71.Google Scholar
  79. 79.
    Nordheim, L. (1934): “Quantentheorie des Magnetismus”. In Müller-Poillet, ed.: Lehrbuch der Physik, Vol. 4. Vieweg, Braunschwieg, pp. 798–876Google Scholar
  80. 80.
    Onsager, L. (1944): “Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition,” Physical Review 65, 117–149.Google Scholar
  81. 81.
    Onsager, L. (1971): “Autobiographical Commentary of Lars Onsager,” in Mills, Ascher, and Jaffee (1971), pp. xix–xxiv.Google Scholar
  82. 82.
    Onsager, L. and Kaufman, B. (1947): “Transition Points,” in Report International Conference on Fundamental Particles and Low Temperatures, Cambridge, July 1946, Vol. 2, The Physical Society, London, pp.137–144.Google Scholar
  83. 83.
    Pauli, W. (1932): “Les théories quantiques du magnétisme: l’électron magnètique,” in Le Magnétisme. Rapports et discussion du sixièmes conseil de physique tenu à Bruxelles du 20 au 25 Octobre 1930. Insitut International de Physique Solvay. Gauthier-Villars, Paris.Google Scholar
  84. 84.
    Peierls, R. E. (1934): “Remarks on Transition Temperatures,” Helvetica Physica Acta Supplementum 7 (Suppl. 2), 81–83. Translation by G. Ford from “Bemerkungen über Umwandlungstemperaturen,” in Dalitz and Peierls (1997), pp. 137–138.Google Scholar
  85. 85.
    Peierls, R. E. (1936a): “Statistical Theory of Adsorption with Interaction between the Adsorbed Atoms,” Proceedings of the Cambridge Philosophical Society 32, 471–476.Google Scholar
  86. 86.
    Peierls, R. E. (1936b): “On Ising’s Model of Ferromagnetism,” Proceedings of the Cambridge Philosophical Society 32, 477–481.Google Scholar
  87. 87.
    Peierls, R. E. (1985): Bird of Passage. Recollections of a Physicist. Princeton University Press, Princeton.Google Scholar
  88. 88.
    Peierls, R. E. with Hoddeson, Lillian (1981): Interview, July 1981, Niels Bohr Library, American Institute of Physics, College Park, Maryland.Google Scholar
  89. 89.
    Purrington, R. D. (1997): Physics in the Nineteenth Century. Rutgers University Press, New Brunswick, NJ.Google Scholar
  90. 90.
    Schottky, W. (1922): “Über die Drehung der Atomachsen in festen Körpern. Mit magnetischen, thermischen und chemischen Beziehungen,” Physikalische Zeitschrift 23, 448–455.Google Scholar
  91. 91.
    Siegel, S. (1951): “Order-Disorder Transitions in Metal Alloys” in R. Smoluchowski, J. E. Mayer and W. A. Weyl, eds., Phase Transformations in Solids [Symposium held at Cornell University, August, 1944], Wiley, New York, pp. 366–387.Google Scholar
  92. 92.
    Shlesinger, M. F. and Weiss, G. H. (1985): “Elliott W. Montroll (May 4, 1916-December 3, 1983),” in Shlesinger, M. F. and Weiss, G. H., eds., The Wonderful World of Stochastics, North-Holland, Amsterdam, pp 1–15.Google Scholar
  93. 93.
    Smith, C. and Wise, M. N. (1989): Energy and Empire: A Biographical Study of Lord Kelvin. Cambridge University Press, Cambridge.Google Scholar
  94. 94.
    Sommerfeld, A. (1948): “Wilhelm Lenz zum 60. Geburtstag am 8. Februar 1948,” Zeitschrift für Naturforschung 3A, 186.Google Scholar
  95. 95.
    Stern, O. (1920): “Zur Molekulartheorie des Paramagnetismus fester Salze,” Zeitschrift für Physik 1, 147–153.Google Scholar
  96. 96.
    Stoner, E. C. (1926): Magnetism. Methuen, London.Google Scholar
  97. 97.
    Stoner, E. C. (1934): Magnetism and Matter. Methuen, London.Google Scholar
  98. 98.
    Stutz, C. and Williams, B. (1999): “Ernst Ising,” Physics Today 52 (3), 106–108.Google Scholar
  99. 99.
    Temperley, H. N. V. (1956): Changes of State. Cleaver-Hume, London.Google Scholar
  100. 100.
    Van Vleck, J. H. (1932): The Theory of Electric and Magnetic Susceptibilies. Oxford University Press, New York.Google Scholar
  101. 101.
    Van Vleck, J. H. (1945): “A Survey of the Theory of Ferromagnetism,” Reviews of Modern Physics 17, 27–47.Google Scholar
  102. 102.
    Van Vleck, J. H. (1947): “Quelques aspects de la théorie du magnétisme,” Annales de l’Institut Henri Poincaré 10, 57–190.Google Scholar
  103. 103.
    Wannier, G. H. (1945): “The Statistical Problem in Cooperative Phenomena,” Reviews of Modern Physics 17, 50–60.Google Scholar
  104. 104.
    Weiss, G. H. (1994): “Elliott Waters Montroll,” Bibliographical Memoirs of the National Academy of Sciences 63, 364–381.Google Scholar
  105. 105.
    Weiss, P. (1905): “Les Propriétés magnétiques de la pyrrhotine,” Journal de Physique Théorique et Appliquée 4e série 4, 469–508, 829–846.Google Scholar
  106. 106.
    Weiss, P. (1907): “L’Hypothèse du champ moléculaire et la propriété ferromagnétique,” Journal de Physique et le Radium 6, 661–690.Google Scholar
  107. 107.
    Weiss, P. (1911): “Sur la rationalité des rapport des moments magnétique moléculaires et la magnéton,” Journal de Physique 5e série 1, 900–912, 965–988.Google Scholar
  108. 108.
    Whittaker, E. T. and Watson, G. N. (1927): A Course of Modern Analysis, Fourth Edition. Cambridge University Press, London.Google Scholar
  109. 109.
    Wolf, W. P. (2000): “The Ising Model and Real Magnetic Materials”. Brazilian Journal of Physics 30, 794–810.Google Scholar
  110. 110.
    Yang, C. N. (1952): “The Spontaneous Magnetization of a Two-Dimensional Ising Model,” Physical Review 85, 808–816.Google Scholar
  111. 111.
    Zwicky, F. (1933): “On Cooperative Phenomena,” Physical Review 42, 270–278.Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsRoskilde UniversityRoskildeDenmark

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