We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-
s Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-
s Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.
High temperature series expansion for the critical exponents of the Ising model are reanalysed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent γ in three dimensions, close to 1.245, therefore lower than the quoted value 1.250, and much closer to the renormalization group (R.G.) value 1.241. The exponent is analysed in two ways : in one method γ is estimated directly while in the other one Tc, is calculated first. With these new values of Tc, the exponent α is recalculated and found to be very close to the R.G. value 0.110. The value of ν is not modified (0.638) and is therefore still a problem for hyperscaling, and is in disagreement with the R.G. value (0.630).
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
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.
The high-temperature, zero-field susceptibility series of the spin- 1 / 2 Ising model on the face centred cubic lattice is analysed assuming the asymptotic form A 1 t - gamma +A 2 t - gamma +1 +Bt - gamma + Delta 1 where t is the reduced temperature. Good convergence is obtained for gamma =1.241 and Delta 1 =0.496, the values predicted by renormalisation group theory. Attempts to fit the series coefficients with gamma =1.25 and B=0, do not prove as successful. The amplitudes A 2 /A 1 and B/A 1 are estimated.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.