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Journal of Physics A Mathematical and General

The gap exponent Δ for the spin-s Ising model on the FCC lattice

Sati McKenzie

Published Sep 11, 1983• 3 citations
Publisher

Abstract

High-temperature series expansions are presented for the magnetisation M(K,h) of the spin-s Ising model on the face-centred cubic lattice. Series coefficients are derived to order 13 and 5 respectively in the variables K=J/kT and h=mH/kT, for 1 / 2 <or=s<or= 9 / 2 . The data are analysed for 2 Delta - gamma , where Delta and gamma denote the gap exponent and the susceptibility exponent respectively. The estimate 2 Delta - gamma =1.890+or-0.003 satisfies the data for all s.

DOI
10.1088/0305-4470/16/13/038

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Journal of Physics C Solid State Physics•Nov 1, 1969•32 citations
High-temperature series expansions are presented for the magnetisation M(K,h) of the spin-s Ising model on the face-centred cubic lattice. Series coefficients are derived to order 13 and 5 respectively in the variables K=J/kT and h=mH/kT, for 1 / 2 <or=s<or= 9 / 2 . The data are analysed for 2 Delta - gamma , where Delta and gamma denote the gap exponent and the susceptibility exponent respectively. The estimate 2 Delta - gamma =1.890+or-0.003 satisfies the data for all s.
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.
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 .
Extended series expansions are derived for the high-temperature susceptibility of the classical Heisenberg model, on three-dimensional lattices. Series coefficients are presented to twelfth order for the simple cubic (SC) and face centred cubic (FCC) lattices and to eleventh order for the body centred cubic (BCC) lattice. The results are in agreement with earlier calculations apart from a small discrepancy at the tenth order on the FCC lattice. Extrapolation studies on the extended series are used to obtain revised estimates for the critical points (K c ) and the susceptibility exponent ( gamma ). On the FCC lattice, the authors also investigate the possibility of a confluent non-analytic correction to the dominant singularity. While the coefficients are consistent with the presence of such a correction term with an exponent ( Delta 1 ) of 0.55, as predicted by renormalisation group (RG) calculations, the amplitude of the correction term appears to be very small compared with that of the first analytic correction term.
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.
Low temperature Ising model series in the usual variable z = exp(−4 J/kT ) are analysed. The positions of the singularities in the disk|z| <or=|z c | and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0·312 ± 0·002 for the simple cubic and face-centred cubic lattices, and β = 0·312 ± 0·004 for the body-centred cubic lattice. γprime = 1·31 ± 0·02 for the simple cubic lattice, γprime = 1·28 ± 0·04 for the body-centred cubic lattice, and γprime = 1·3 ± 0·1 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.