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Canadian Journal of Physics

High-temperature specific heat series of the Ising model on the crystobalite lattice

R. W. Gibberd

Published Feb 1, 1970• 5 citations
Publisher

Abstract

Betts and Ditzian have recently published the first 11 coefficients of the exact high-temperature series for the specific heat of the spin 1/2 Ising model on a crystobalite lattice. In this paper the exact coefficients for the next 8 terms are derived by making use of an approximate transformation between the Ising partition function of the crystobalite and diamond lattices. The series is analyzed by using the ratio and Padé approximant methods, but a reliable estimate for α has not been obtained.

DOI
10.1139/p70-042

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Betts and Ditzian have recently published the first 11 coefficients of the exact high-temperature series for the specific heat of the spin 1/2 Ising model on a crystobalite lattice. In this paper the exact coefficients for the next 8 terms are derived by making use of an approximate transformation between the Ising partition function of the crystobalite and diamond lattices. The series is analyzed by using the ratio and Padé approximant methods, but a reliable estimate for α has not been obtained.
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.
The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.
Pad\'e-approximant techniques are applied to the high-temperature series for the temperature derivative of the specific heat of the Ising model. This method indicates a critical exponent $\ensuremath{\alpha}$ in the neighborhood of 18\frac{1}{8}81​ for those Pad\'e approximants which yield an estimate of the critical temperature in substantial agreement with that determined from the susceptibility series.
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.
Describes the development of high-temperature series expansions for general mixed spin-S-spin-S' Ising models on simple loose-packed lattices. It extends previous work on the simpler special case S'=1/2. Coefficients of the series for the initial susceptibility and specific heat of these models on the square, SC and BCC lattices are presented. A preliminary analysis of the results for the particular case S=1, S'=3/2 is given, and the critical ratio K c and the susceptibility exponent gamma are estimated.