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History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance

Abstract

This is the second in a series of three papers that charts the history of the Lenz–Ising model (commonly called just the Ising model in the physics literature) in considerable detail, from its invention in the early 1920s to its recognition as an important tool in the study of phase transitions by the late 1960s. By focusing on the development in physicists’ perception of the model’s ability to yield physical insight—in contrast to the more technical perspective in previous historical accounts, for example, Brush (Rev Modern Phys 39: 883–893, 1967) and Hoddeson et al. (Out of the Crystal Maze. Chapters from the History of Solid-State Physics. Oxford University Press, New York, pp. 489–616, 1992)—the series aims to cover and explain in depth why this model went from relative obscurity to a prominent position in modern physics, and to examine the consequences of this change. In the present paper, which is self-contained, I deal with the development from the early 1950s to the 1960s and document that this period witnessed a major change in the perception of the model: In the 1950s it was not in the cards that the model was to become a pivotal tool of theoretical physics in the following decade. In fact, I show, based upon recollections and research papers, that many of the physicists in the 1950s interested in understanding phase transitions saw the model as irrelevant for this endeavor because it oversimplifies the nature of the microscopic constituents of the physical systems exhibiting phase transitions. However, one group, Cyril Domb’s in London, held a more positive view during this decade. To bring out the basis for their view, I analyze in detail their motivation and work. In the last part of the paper I document that the model was seen as much more physically relevant in the early 1960s and examine the development that led to this change in perception. I argue that the main factor behind the change was the realization of the surprising and striking agreement between aspects of the model, notably its critical behavior, and empirical features of the physical phenomena.

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References

  • Ashrafi, B., Hall, K., and Schweber, S. S. (n.d.): “The Solution to the Phase Transition Problem: 1965–1975” (unpublished).

  • Atkins K.R., Edwards M.H. (1955) “Coefficient of Expansion of Liquid Helium II,”. Physical Review 97: 1429–1434

    Article  Google Scholar 

  • Bagatskii, M. I., Voronel’, A. V., and Gusak, V. G (1963): “Measurement of the Specific Heat C V of Argon in the Immediate Vicinity of the Critical Point”, Soviet Physics JETP-USSR 16, 517–518. For Russian original, see (1962): Zhyrnal Eksperimental’noi i Teoreticheskoi Fiziki 43, 728

  • Baker G.A. Jr. (1961) “Application of the Padé Approximant Method to the Investigation of Some Magnetic Properties of the Ising Model,”. Physical Review 124: 768–774

    Article  Google Scholar 

  • Baker G.A. Jr., Gammel J.L. (1961) “The Padé Approximant,”. Journal of Mathematical Analysis and Applications 2: 21–30

    MATH  Article  MathSciNet  Google Scholar 

  • Benedek, G. B. (1966): “Equilibrium Properties of Ferromagnets and Antiferromagnets in the Vicinity of the Critical Point,” in Green and Sengers (1966), pp. 42–48.

  • Binney, J. J, Dowrick, N. J., Fisher, A. J., and Newman, M. E. J. (1992): The Theory of Critical Phenomena. Clarendon, Oxford

  • Brooks J.E., Domb C. (1951) “Order–Disorder Statistics. III. The Antiferromagnet and Order– Disorder Transitions,”. Proceedings of the Royal Society [A] 207: 343–358

    MATH  Article  MathSciNet  Google Scholar 

  • Brush, S. G. (1964): “History of the Lenz–Ising Model,” UCRL-7940, Lawrence Radiation Laboratory, Livermore, California, June.

  • Brush S.G. (1967) “History of the Lenz–Ising Model,”. Reviews of Modern Physics 39: 883–893

    Article  Google Scholar 

  • Brush S.G. (1983) Statistical Physics and the Atomic Theory of Matter. Princeton University Press, Princeton

    Google Scholar 

  • Buckingham M.J., Fairbank W.M. (1961) “The Nature of the Lambda Transition,”. Progress in Low Temperature Physics 3: 80–113

    Article  Google Scholar 

  • Burley D.M. (1960) “Some Magnetic Properties of the Ising Model,”. Philosophical Magazine 5: 909–919

    Article  Google Scholar 

  • Chase C.E., Williamson R.C., Tisza L. (1964) “Ultrasonic Propagation near the Critical Point in Helium,”. Physical Review 13: 467–469

    Google Scholar 

  • Cooke, A. H., Edmonds, D. T., McKim, F. R., and Wolf, W. P. (1959): “Magnetic Dipole Interactions in Dysprosium Ethyl Sulphate. 1. Susceptibility and Specific Heat between 20-Degrees-K and 1-Degrees-K,” Proceedings of the Royal Society [A] 252, 246–259.

  • Cushing J.T. (1994) Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony. University of Chicago Press, Chicago

    MATH  Google Scholar 

  • Domb C. (1949a) “Order–Disorder Statistics. I.,”. Proceedings of the Royal Society[A] 196: 36–50

    MATH  Article  MathSciNet  Google Scholar 

  • Domb C. (1949b) “Order–Disorder Statistics. II. A Two-Dimensional Model,”. Proceedings of the Royal Society[A] 199: 199–221

    MATH  Article  MathSciNet  Google Scholar 

  • Domb C. (1949c) “Statistical Mechanics of Some Co-operative Phenomena,”. Nature 163: 775–776

    MATH  Article  Google Scholar 

  • Domb, C. (1952), “L’influence de la structure du réseau sur l’anomalie de la chaleur spécifique du modèle d’Ising,” in Changements de Phases, Comptes Rendus de la deuxième Réunion Annuelle tenue en commun avec la Commission de Thermodynamique de l’Union Internationale de Physique, Société de Chimie Physique, Presse Universitaires de France, Paris, pp. 8–18.

  • Domb C. (1955) “Statistical Physics and its Problems,”. Science Progress 43: 402–417

    Google Scholar 

  • Domb C. (1960) “On the Theory of Cooperative Phenomena in Crystals,”. Advances in Physics 9: 149–295

    Article  Google Scholar 

  • Domb, C. (1966): “Critical Properties of Lattice Models,” in Green and Sengers (1966), pp. 29–41.

  • Domb, C. (1971): “The Curie Point,” in E. G. D. Cohen, ed., Statistical Mechanics at the Turn of the Decade, Marcel Dekker, New York, pp. 81–128.

  • Domb, C. (1974a): “Ising Model,” in Domb and Green (1974a), pp. 357–484.

  • Domb C. (1974b) “Configurational Studies of the Potts Models,”. Journal of Physics A: Mathematical, Nuclear and General 7: 1335–1348

    Article  Google Scholar 

  • Domb C. (1990a) “Some Reminiscences about My Early Career,”. Physica A 168: 1–21

    Article  MathSciNet  Google Scholar 

  • Domb C. (1990b) “On Hammersley’s Method for One-Dimensional Covering Problems,”. In: Grimmett G.R., Welsh D.J.A. (eds) Disorder in Physical Systems: Essays in Honour of John M. Hammersley on the occasion of his 70th Birthday. Oxford University Press, New York, pp 33–53

    Google Scholar 

  • Domb C. (1996) The Critical Point. Taylor and Francis, London

    Google Scholar 

  • Domb, C. with Schweber, Silvan S. (2002): Interview, June 2002, History of Recent Science & Technology, The Dibner Institute for the History of Science and Technology, MIT, Massachusetts, available at (accessed February 18, 2008). http://authors.library.caltech.edu/5456/01/hrst.mit.edu/hrs/renormalization/Domb/

  • Domb C. (2003) “Some Observations on the Early History of Equilibrium Statistical Mechanics,”. Journal of Statistical Physics 110: 475–496

    MATH  Article  MathSciNet  Google Scholar 

  • Domb, C. and Green, M. S., eds. (1972a): Phase Transitions and Critical Phenomena, Academic Press, London, New York, vol. 1.

  • Domb, C. and Green, M. S. (1972b): “Preface to Volume 1,” in Domb and Green (1972a), pp. ix-xii.

  • Domb, C. and Green, M. S., eds. (1974a): Phase Transitions and Critical Phenomena, Academic Press, London, New York, vol. 3.

  • Domb C., Sykes M.F. (1956) “On Metastable Approximations in Co-operative Assemblies,”. Proceedings of the Royal Society of London [A] 235: 247–259

    Article  Google Scholar 

  • Domb C., Sykes M.F. (1957a) “On the Susceptibility of a Ferromagnetic above the Curie Point,”. Proceedings of the Royal Society of London[A] 240: 214–228

    MATH  Article  Google Scholar 

  • Domb C., Sykes M.F. (1957b) “Specific Heat of a Ferromagnetic Substance above the Curie Point,”. Physical Review 108: 1415–1416

    Article  Google Scholar 

  • Domb C., Sykes M.F. (1957c) “The Calculation of Lattice Constants in Crystal Statistics,”. Philosophical Magazine 2: 733–749

    MATH  Article  Google Scholar 

  • DombC. Sykes M.F. (1962) “Effect of Change of Spin on Critical Properties of Ising and Heisenberg Models,”. Physical Review 128: 168–173

    Article  Google Scholar 

  • Dyson F.J. (1956) “General Theory of Spin-Wave Interactions,”. Physical Review 102: 1217–1230

    MATH  Article  MathSciNet  Google Scholar 

  • Dyson F.J. (1995) “The Coulomb Fluid and the Fifth Painlevé Transcendent,”. In: Liu C.S., Yau S.-T (eds) Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Boston, pp 131–146

    Google Scholar 

  • Elcock E.W. (1956) Order–Disorder Phenomena. Methuen, London

    Google Scholar 

  • Essam J.W., Fisher M.E. (1963) “Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point,”. Journal of Chemical Physics 38: 802–812

    Article  Google Scholar 

  • Fairbank, W. M., Buckingham, M. J., and Kellers, F. (1957): “Specific Heat of Liquid He4 Near the Lambda Point,” in J. R. Dillinger, ed., Proceedings of the Fifth International Conference on Low-Temperature Physics and Chemistry, University of Wisconsin Press, Madison, pp. 50–52.

  • Fisher M.E. (1959) “The Susceptibility of the Plane Ising Model,”. Physica 25: 521–524

    Article  Google Scholar 

  • Fisher M.E. (1964a) “Specific Heat of a Gas near the Critical Point,”. Physical Review 136: A1599–A1604

    Article  Google Scholar 

  • Fisher M.E. (1964b) “Correlation Functions and the Critical Region of Simple Fluids,”. Journal of Mathematical Physics 5: 944–962

    Article  MathSciNet  Google Scholar 

  • Fisher, M. E. (1965): “The Nature of Critical Points,” in W. E. Brittin, ed., Lectures in Theoretical Physics VII C, University of Colorado Press, Boulder, pp. 1–159.

  • Fisher M.E. (1967) “The Theory of Equilibrium Critical Phenomena,”. Reports in Progress in Physics 30: 615–730

    Article  Google Scholar 

  • Fisher, M. E. (1996): “Foreword: About the Author and the Subject,” Domb (1996), pp. xiii-xviii.

  • Fisher M.E., Sykes M.F. (1962) “Antiferromagnetic Susceptibilities of the Simple Cubic and Body-Centered Cubic Ising Lattices,”. Physics 28: 939–956

    Google Scholar 

  • Gaunt D.S., Fisher M.E., Sykes M.F., Essam J.W. (1964) “Critical Isotherm of a Ferromagnet and of a Fluid,”. Physical Review Letters 13: 713–715

    Article  Google Scholar 

  • Gelfert A. (2005) “Mathematical Rigor in Physics: Putting Exact Results in Their Place,”. Philosophy of Science 72: 723–738

    Article  MathSciNet  Google Scholar 

  • Goldenfeld N. (1992): Lectures on Phase Transitions and the Renormalization Group, Perseus Books, Reading, Massachusetts.

  • Green, M. S. (1966): “Introduction,” in Green and Sengers (1966), pp. xi–xi.

  • Green, M. S. and Sengers, eds., J. V. (1966): Critical Phenomena. Proceedings of a Conference Held in Washington, D. C., April 1965, National Bureau of Standards

  • Habgood H.W., Schneider W.G. (1954) “PVT measurements in the Critical Region of Xenon,”. Canadian Journal of Chemistry 32: 98–112

    Article  Google Scholar 

  • Hill T.S. (1956) Statistical Mechanics. McGraw-Hill, New York

    MATH  Google Scholar 

  • Hoddeson L., Schubert H., Heims S.J., Baym G. (1992) “Collective Phenomena,”. In: Hoddeson L., Braun E., Teichmann J., Weart S. (eds) Out of the Crystal Maze. Chapters from the History of Solid-State Physics. Oxford University Press, New York, pp 489–616

    Google Scholar 

  • Hunt K.L. (1953) “Collective Electron Ferromagnetism: A Generalization of the Treatment and an Analysis of Experimental Results,”. Proceedings of the Royal Society of London [A] 32: 103–117

    Google Scholar 

  • Ising, E. (1924): “Beitrag zur Theorie des Ferro- und Paramagnetismus,” Ph.D. Thesis, University of Hamburg.

  • Ising E. (1925) “Beitrag zur Theorie des Ferromagnetismus,”. Zeitschrift für Physik 31: 253–258

    Article  Google Scholar 

  • Jaeger G. (1998) “The Ehrenfest Classification of Phase Transitions: Introduction and Evolution,”. Archive for History of Exact Sciences 53: 51–81

    MATH  Article  MathSciNet  Google Scholar 

  • Kaufman B. (1949) “Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis,”. Physical Review 76: 1232–1243

    MATH  Article  Google Scholar 

  • Kaufman B., Onsager L. (1949) “Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice,”. Physical Review 76: 1244–1252

    MATH  Article  Google Scholar 

  • Kirkwood J.G. (1950) “Critique of the Free Volume Theory of the Liquid State,”. Journal of Chemical Physics 18: 380–382

    Article  Google Scholar 

  • Kouvel J.S., Fisher M.E. (1964) “Detailed Magnetic Behavior of Nickel near its Curie Point,”. Physical Review 136: A1626–A1632

    Article  Google Scholar 

  • Krieger M.H. (1996) Constitutions of Matter. University of Chicago Press, Chicago

    MATH  Google Scholar 

  • Landau L.D. (1937) “Zur Theorie der Phasenumwandlungen. I.,”. Physikalische Zeitschrift der Sowjetunion 11: 26–47

    MATH  Google Scholar 

  • Landau L.D., Lifshitz E.M. (1958) Statistical Physics. Pergamon, London

    MATH  Google Scholar 

  • Lasheen M.A., Vanden Broek J., Gorter C.J. (1958) “The magnetic Susceptibility and Relaxation of a MnCl 2·H 2 O Single Crystal in the Paramagnetic and Antiferromagnetic States,”. Physica 24: 1061–1075

    Article  Google Scholar 

  • Lebowitz J.L (1995) “Lars Onsager November 27, 1903 - October 5, 1976: In Memoriam,”. Journal of Statistical Physics 78: 1–3

    MATH  Article  MathSciNet  Google Scholar 

  • Lee T.D., Yang C.N. (1952) “Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising model,”. Physical Review 87: 410–419

    MATH  Article  MathSciNet  Google Scholar 

  • Lenz W. (1920) “Beitrag zum Verständnis der magnetischen Erscheinungen in festen Köpern,”. Physikalische Zeitschrift 21: 613–615

    Google Scholar 

  • Linder B. (1954) “Order–Disorder Phenomena,”. Journal of Chemical Physics 22: 970–973

    Article  Google Scholar 

  • Lipa J.A., Swanson D.R., Nissen J.A., Chui T.C.P., Israelsson U.E. (1996) “Heat Capacity and Thermal Relaxation of Bulk Helium very near the Lambda Point,”. Physical Review Letters 76: 944–947

    Article  Google Scholar 

  • LonguetHiggins, H. C. and Fisher, M. E. (1996): “Lars Onsager: 27 November, 1903–5 October, 1976, “in Hemmer, P. C., Holden, H. and Kjelstrup Ratkje, S., eds. (1996): The Collected Works of Lars Onsager.World Scientific, Singapore, pp. 9–34.

  • Moldover M.R., Little W.A. (1965) “Specific Heat of He 3 and He 4 in the Neighborhood of their Critical Points,”. Physical Review Letters 15: 54–56

    Article  Google Scholar 

  • Montroll E.W., Potts R.B., Ward J.C. (1963) “Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model,”. Journal of Mathematical Physics 4: 308–323

    Article  MathSciNet  Google Scholar 

  • Moser H. (1936) “Messung der wahren spezifischen Wärme von Silber, Nickel, β-Messing, Quartzkristall und Quartzglas zwischen + 50 und 700° C nach verfeinerten Methode,”. Physikalische Zeitschrift 37: 737–753

    Google Scholar 

  • Muto T., Takagi Y. (1955) “The Theory of Order–Disorder Transitions in Alloys,”. Solid State Physics 1: 194–284

    Google Scholar 

  • Nielsen, A. I. and Timmermann, S. (2002): En historisk undersøgelse af udviklingen af L. D. Landaus teori for kontinuerte overgange.Unpublished report, Department of Mathematics and Physics, Roskilde University, Roskilde, Denmark.

  • Niss M. (2005) “History of the Lenz–Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena,”. Archive for History of Exact Science 59: 267–318

    MATH  Article  MathSciNet  Google Scholar 

  • Onsager L. (1944) “Crystal Statistics. I. A Two-Dimensional Model with an Order–Disorder Transition,”. Physical Review 65: 117–149

    MATH  Article  MathSciNet  Google Scholar 

  • Onsager L. (1949) discussion remark. Nuovo Cimento, Suppl. 6: 261

    Article  MathSciNet  Google Scholar 

  • Pelissetto A., Vicari E. (2002) “Critical Phenomena and Renormalization-group Theory,”. Physics Reports 368: 549–727

    MATH  Article  MathSciNet  Google Scholar 

  • Pais, A. (1958): “The Scientific Work of T. D. Lee and C. N. Yang,” Nuclear Physics 5, 297–300.

  • Potter H.H. (1934) “The magneto-caloric effect and other magnetic phenomena in iron,”. Proceedings of the Royal Society of London [A] 146: 262–387

    Google Scholar 

  • Robinson, W. K. and Friedberg, S. A. (1960): “Specific Heats of NiCl 2· 6H 2 O and CoCl 2·6H 2 O between 1.4° and 20° K,” Physical Review Letters 117, 402–408.

  • Rowlinson, J. S. (1966): “Critical States of Simple Fluids and Fluid Mixtures: a Review of the Experimental Position,” in Green and Sengers (1966), pp. 9–12.

  • Ruelle D. (1969) Statistical Mechanics: Rigorous Results. Benjamin, New York

    MATH  Google Scholar 

  • Rushbrooke G.S., Wood P.J. (1958) “On the Curie Points and High Temperature Susceptibilities of Heisenberg Model Ferromagnetics,”. Molecular Physics 1: 257–283

    Article  Google Scholar 

  • Sauer T. (2001) “The Feynman Path Goes Monte Carlo,”. In: Janke W., Pelster A., Scmidt H-J., Bachmann M. (eds) Fluctuating Paths and Fields. Festschrift Dedicated to Hagen Kleinert on the Occasion of His 60th Birthday. World Scientific, Singapore, New Jersey, London, Hong Kong, pp 29–42

    Google Scholar 

  • Schweber S.S. (1994) QED and the Men Who Made It. Princeton University, Princeton

    MATH  Google Scholar 

  • Skalyo J., Friedberg S.A. (1964) “Heat Capacity of the Antiferromagnet CoCl 2·H 2 O near its Néel Point,”. Physical Review Letters 13: 113–135

    Article  Google Scholar 

  • Stanley H.E. (1971) Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, New York, Oxford

    Google Scholar 

  • Sucksmith W., Clark C.A., Oliver D.J., Thompson J.E. (1953) “Spontaneous Magnetization— Techniques and Measurements,”. Reviews of Modern Physics 25: 34–41

    Article  Google Scholar 

  • Suchsmith W., Pearce R.R. (1938) “The Paramagnetism of the Ferromagnetic Elements,”. Proceedings of the Royal Society of London[A] 167: 189–204

    Article  Google Scholar 

  • Sykes M.F., Fisher M.E. (1962) “Antiferromagnetic Susceptibility of Plane Square and Honeycomb Ising Lattices,”. Physica 28: 919–938

    MATH  Article  Google Scholar 

  • Sykes C., Wilkinson H. (1937) “The Transformation in the Beta Brasses,”. Journal of the Institute of Metals 61: 223–240

    Google Scholar 

  • Sykes C., Wilkinson H. (1938) “The Specific Heat of Nickel from 100 Degrees C. to 600 Degrees C,”. Proceedings of the Physical Society of London 50: 834–851

    Article  Google Scholar 

  • Temperley H.N.V (1956) Changes of State. Cleaver-Hume, London

    MATH  Google Scholar 

  • Tisza L. (1951) “On the General Theory of Phase Transitions”. In: Smoluchowski R., Mayer J.E., Weyl W.A. (eds) Phase Transformations in Solids [Symposium held at Cornell University, 1948]. Wiley, New York, pp 1–37

    Google Scholar 

  • Van Vleck J.H. (1945) “A Survey of the Theory of Ferromagnetism,”. Reviews of Modern Physics 17: 27–47

    Article  Google Scholar 

  • Van Vleck J.H. (1953) “Models of Exchange Coupling in Ferromagnetic Media,”. Reviews of Modern Physics 25: 220–228

    MATH  Article  Google Scholar 

  • Wannier G.H. (1945) “The Statistical Problem in Cooperative Phenomena,”. Reviews of Modern Physics 17: 50–60

    Article  Google Scholar 

  • Widom B. (1964) “Degree of the Critical Isotherm,”. Journal of Chemical Physics 41: 1633–1634

    Article  MathSciNet  Google Scholar 

  • Widom B., Rice O.K. (1955) “Critical Isotherm and the Equation of Liquid-Vapor Systems,”. Journal of Chemical Physics 23: 1250–1255

    Article  Google Scholar 

  • Wolf W.P. (2000) “The Ising Model and Real Magnetic Materials,”. Brazilian Journal of Physics 30: 794–810

    Article  Google Scholar 

  • Yang C.N. (1952) “The Spontaneous Magnetization of a Two-Dimensional Ising Model,”. Physical Review 85: 808–816

    MATH  Article  Google Scholar 

  • Yang, C. N. (1972): “Introductory Note on Phase Transitions and Critical Phenomena” in Domb and Green (1972a), pp. 1–5.

  • Yang, C. N. (1983): “Commentary,” in C. N. Yang, Selected Papers 1945–1980, Freeman, San Francisco, pp. 1–82.

  • Yang C.N. (1995) “Remarks about Some Developments in Statistical Mechanics,”. AAPPS Bulletin 5: 2–3

    Google Scholar 

  • Yang C.N., Yang C.P. (1964) “Critical Point in Liquid-Gas Transitions,”. Physical Review Letters 13: 303–305

    Article  Google Scholar 

  • Ziman J.M. (1965) “Mathematical Models and Physical Toys,”. Nature 206: 1187–1192

    Article  Google Scholar 

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Niss, M. History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance. Arch. Hist. Exact Sci. 63, 243 (2009). https://doi.org/10.1007/s00407-008-0039-5

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Keywords

  • Phase Transition
  • Ising Model
  • Physical Review
  • Critical Behavior
  • Critical Phenomenon