Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition

Phys. Rev. 65, 117 – Published 1 February 1944
Lars Onsager


The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width (n atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case n= an integral replaces a sum. The choice of different interaction energies (±J,±J) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature T=Tc given by the condition sinh(2JkTc) sinh(2JkTc)=1. The energy is a continuous function of T; but the specific heat becomes infinite as -log |T-Tc|. For strips of finite width, the maximum of the specific heat increases linearly with log n. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is (exp (2JkT) tanh(2JkT))n.


  • Received 4 October 1943
  • Published in the issue dated February 1944

© 1944 The American Physical Society

Authors & Affiliations

Lars Onsager

  • Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut


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