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Journal of Physics A Mathematical and General

High-temperature reduced susceptibility of the Ising model

Sati McKenzie

Published Oct 1, 1975• 25 citations
Publisher

Abstract

Two new coefficients are given for the high-temperature series expansion of the zero-field susceptibility of the Ising model for the face-centred cubic lattice. The new coefficients are analysed and it is estimated that the critical temperature nu c =tanh K c =0.10173+or-0.00001 and the critical index gamma =1.250+or-0.005.
DOI
10.1088/0305-4470/8/10/003

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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, 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 expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.
We have extended the series for the zero-field susceptibility of the spin-SSS Ising model to eighth order in the reduced temperature KKK, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series hn(S){h}_{n}(S)hn​(S) are expressed as simple polynomials in X=S(S+1)X=S(S+1)X=S(S+1). For the face-centered-cubic lattice, an accurate polynomial fit to the critical point Kc(S){K}_{c}(S)Kc​(S) is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all SSS. However, the large-SSS series seems to exhibit more rapid apparent convergence.
Extended high-temperature series expansions are presented for the spin-s Ising model on the face-centred cubic lattice. Series coefficients for the zero-field internal energy (U 0 ) and the second moment ( mu 2 ) of the correlation function are derived, for 1 / 2 <or=s<or= 9 / 2 , to order 14 in the high-temperature expansion variable K(=J/kT). For s= 1 / 2 , the fourth field derivative of the free energy is calculated to order K 14 and the magnetisation series is given to order 13 and 27 in the temperature and magnetic field variables respectively. For s=1 to 9 / 2 , the zero-field susceptibility series are derived to order K 14 .