Crystal Statistics. I. A TwoDimensional Model with an OrderDisorder Transition
Abstract
The partition function of a twodimensional "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 twoway infinite crystal has an orderdisorder transition at a temperature T=T_{c} given by the condition (2JkT_{c}) (2J'kT_{c})=1. The energy is a continuous function of T; but the specific heat becomes infinite as  TT_{c}. For strips of finite width, the maximum of the specific heat increases linearly with n. The orderconverting 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) (2J'kT))^{n}.
 Publication:

Physical Review
 Pub Date:
 February 1944
 DOI:
 10.1103/PhysRev.65.117
 Bibcode:
 1944PhRv...65..117O