### Abstract

The method of developing exact power‐series expansions for the partition function *Z*_{N} and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first‐, second‐, and third‐neighbor pairs. Expansions of the spontaneous magnetization*M* _{0}(*T*) and zero field susceptibility χ_{0}(*T*) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′^{r}, b.c.c., s.c., and f.c.c. lattices, when interactions and are present between first‐ and second‐neighbor spins, respectively (second‐neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (*J* _{1} = *J* _{2} = *J* _{3}) are present between first‐, second‐, and third‐neighbor pairs (third‐equivalent‐neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′^{r}, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of *M* _{0}(*T*) and χ_{0}(*T*) for as characterized by the critical exponents β and γ′, respectively. For the second‐equivalent‐neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest‐neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably and hence the transition in χ_{0} is symmetrical about *T*_{c} (γ′ = γ). A repetition of this analysis for the third‐equivalent‐neighbor model three‐dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high‐temperature (*T* > *T*_{c} ) expansions of χ_{0}(*T*) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over‐all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second‐ and third‐equivalent neighbor models. Finally the high‐temperature expansions of *Z*_{N} in zero field, and of χ_{0}(*T*) for the second‐neighbor model are used to examine the dependence of the critical temperature *T*_{c} , the critical energy (*E* _{∞} − *E*_{c} )/*kT*_{c} , and the critical entropy (*S* _{∞} − *S*_{c} )/*k* on the relative strengths of *J* _{1} and *J* _{2} for values of *J* _{2}/*J* _{1} in the range 0 to 1. It is found that the variation of the critical point is well represented by,where α = *J* _{2}/*J* _{1} and lies in the range 0 ≤ α ≤ 1; and *T*_{c} (0) is the critical temperature of the nearest‐neighbor model. The values of *m* _{1} are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′^{r} lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.

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