Exact calculation of the energy and heat capacity for the triangular lattice with three different coupling constants

Abstract

Calculation of the internal energy and heat capacity of the general anisotropic triangular Ising lattice is derived from the double integral form of the partition function. The principal result is the reduction of the elliptic integrals to a standard form for three arbitrary coupling constants. Both the standard form and the method of reduction are due to Legendre. The method of reduction is one involving two linear transformations. A straightforward reduction of elliptic integrals to standard form could not be used in this application. This is because of the functional dependence of the two linear transformations on the many combinations and permutations of the signs and relative magnitudes of the coupling energies of the lattice. A relatively simple formulation is presented in which the many combinations and permutations previously mentioned are reduced to only two distinct cases. An independent numerical solution was calculated directly from the partition function as a means of verifying the formulation presented in this paper.