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Canadian Journal of Physics

High-temperature properties of the Ising model on the octahedral lattice

J. Oitmaa, C. J. Elliott

Published Oct 15, 1970• 2 citations
Publisher

Abstract

The high-temperature initial susceptibility and specific heat of the spin 1/2 Ising model on the octahedral lattice are investigated by the method of exact series expansions. From the susceptibility series the critical temperature is found to be νc = tanh J/kTc = 0.1613 ± 0.0001. By using a method due to Gibberd the specific-heat series is calculated to 15 terms but a reliable estimate of the exponent a is not obtained, although the results do support the presently believed value α = 1/8.

DOI
10.1139/p70-298

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Journal of Physics C Solid State Physics•Apr 20, 1980•6 citations
The high-temperature initial susceptibility and specific heat of the spin 1/2 Ising model on the octahedral lattice are investigated by the method of exact series expansions. From the susceptibility series the critical temperature is found to be νc = tanh J/kTc = 0.1613 ± 0.0001. By using a method due to Gibberd the specific-heat series is calculated to 15 terms but a reliable estimate of the exponent a is not obtained, although the results do support the presently believed value α = 1/8.
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.
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. 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. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.
The high-temperature series expansions for the locus of the maximum susceptibility are calculated from the high-temperature series for the magnetic field for various two- and three-dimensional lattices of the spin- 1 / 2 Ising model. The authors examine their critical behaviour and determine the critical temperature nu e and the exponent Delta (= beta delta ) using the ratio method and the Pade approximant method. The values obtained for nu c are in good agreement with exact values and values obtained from standard series method and the Delta also agrees with beta delta obtained from other estimates. These results are another new supporting evidence for the universality hypothesis.