Diagrams for the Free Energy and Density Weight Factors of the Ising Models.
JOHNS HOPKINS UNIV LAUREL MD APPLIED PHYSICS LAB
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This report describes the evaluation of the free energy of the nearest neighbor Ising model through tenth order in the inverse temperature variable JkT. The free energy is expressed in terms of diagrams composed of interaction bonds and vertices. An explicit list of the diagrams with 10 or fewer bonds is presented, and the unrestricted lattice constants free multiplicities of these diagrams are given for the cubic lattices. We employ a theorem that states that a certain sum of diagrams is zero in order to obtain the density-dependent weight factors associated with the vertices. These factors are polynomials in the density and depend on the kind of articulated vertex under consideration. Explicit denumeration of the diagrams is given for those articulated vertices of the fourth and fifth kind that are needed in the tenth order expansion. These methods reproduce the known results for the cubic lattices.
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