This paper determines the density weight factors needed to evaluate the Helmholtz free energy for the Ising model of the lattice gas through tenth order in the coupling constants for arbitrary density and interactions. This is accomplished by developing new generating functions for the evaluation of the density factors that are associated with the articulated vertices which result from summing over the "hard-core" part of the interaction potential. These factors are polynomials in the density, and their coefficients are obtained from recurrence relations derived from the generating functions. This procedure is used to deduce general equations for articulation vertices of the first, second, and third kind. These general formulas are supplemented with a few specific cases of vertices of higher kind in order to evaluate the free energy through tenth order in the coupling constant. Application to the nearest-neighbor model reproduces known results.
ASJC Scopus subject areas
- Condensed Matter Physics