For Pt.1 see ibid., vol.8, no.8, p.1236 of 1975. Amplitudes for a number of three-dimensional Ising model series have been estimated from recently extended series expansions. The amplitudes have then been used to re-estimate the field and temperature scaling parameters that are fundamental to the Generalized Law of Corresponding States. These new scaling parameters have been used (i) to estimate the amplitudes of the next most singular term in the high-temperature susceptibility for the four common three-dimensional lattices, and (ii) to show that an earlier conjecture, that the field scaling parameter n x is the same for the Ising and spherical models, is unlikely to hold.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.
For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.
For pt.I see ibid., vol.8, p.1920 of 1975. The authors have carried out a further analysis of the recently derived 16 term high temperature susceptibility series for the spin 1/2 Ising model on the tetrahedron lattice (crystobalite or B-site spinel lattice). They have found convincing evidence for the asymptotic form chi =C 0 (1-K/K c -5 4/+C 1 (1-K/K c -1 4/+X 0 +C 2 (1-K/K c ) 3 4/+... and have obtained estimates for the amplitudes C 0 , C 1 , C 2 and X 0 . They have also used recent lattice-lattice scaling hypotheses to calculate the specific heat and magnetization amplitudes.
We have extended the series for the zero-field susceptibility of the spin-
s Ising model through tenth order in the reduced inverse temperature
K on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients
hn(s) are expressed as simple polynomials in
X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of
s. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For
s=21 only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of
s, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
Three regular three-dimensional lattices of coordination numbers, q = 3, 4, and 6 are introduced. Exact relations are derived among the specific-heat singularity amplitudes and among the susceptibility singularity amplitudes. Exact high-temperature series expansions for the partition function and the susceptibility are derived for the q = 3 and q = 6 lattices. Precise values of the critical temperature, susceptibility amplitude, critical energy, and critical entropy are obtained for all three lattices. The variation of Ising critical parameters with coordination number is discussed.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.