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Journal of Physics C Solid State Physics

Determination of critical behaviour from series expansions in lattice statistics. IV

Anthony John Guttmann, Colin J. Thompson, Barry William Ninham

Published Aug 1, 1970• 14 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/3/8/002

Related papers

Determination of critical behaviour in lattice statistics from series expansions III

Anthony John Guttmann

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.

Journal of Physics C Solid State Physics•Nov 1, 1969•32 citations

Determination of critical behaviour in lattice statistics from series expansions II

Colin J. Thompson, Anthony John Guttmann, Barry William Ninham

Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| 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 = 0312 ± 0002 for the simple cubic and face-centred cubic lattices, and β = 0312 ± 0004 for the body-centred cubic lattice. γprime = 131 ± 002 for the simple cubic lattice, γprime = 128 ± 004 for the body-centred cubic lattice, and γprime = 13 ± 01 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.

Journal of Physics C Solid State Physics•Nov 1, 1969•27 citations

High temperature series for the susceptibility of the Ising model. II. Three dimensional lattices

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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.

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Journal of Physics A Mathematical and General•Oct 1, 1975•25 citations

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Journal of physics. A, Proceedings of the Physical Society. General•May 1, 1972•111 citations

Estimation of critical indices for the three-dimensional Ising model

David S. Gaunt, M F Sykes

New series expansion data for the three-dimensional Ising model are analysed. A special study of the diamond and face-centred cubic lattices is made. Below the critical temperature convergence is found to be slow. It is concluded that all available data are consistent with beta = lambda 5/16 (magnetization) and not inconsistent with the 'scaling' value gamma '=5/4 (susceptibility) and that with the data and methods available at present it is not possible to draw more precise conclusions.

Journal of Physics A Mathematical Nuclear and General•Oct 1, 1973•49 citations
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.
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.
Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| 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 = 0312 ± 0002 for the simple cubic and face-centred cubic lattices, and β = 0312 ± 0004 for the body-centred cubic lattice. γprime = 131 ± 002 for the simple cubic lattice, γprime = 128 ± 004 for the body-centred cubic lattice, and γprime = 13 ± 01 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.
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. I see abstr. A37614 of 1967. High temperature series expansions for the specific heat of the Ising model of a ferromagnet are given for the face-centred cubic, body-centred cubic, and simple cubic lattices. From a numerical study it is concluded that the critical index ( alpha ) is lattice independent and that in three dimensions alpha approximately=1/8. A numerical representation of the specific heat in the range T C <or=T<or= infinity is given in each case, together with estimates of the critical energy, entropy, and free energy. Asymptotic forms for the specific heat and energy are given.
New series expansion data for the three-dimensional Ising model are analysed. A special study of the diamond and face-centred cubic lattices is made. Below the critical temperature convergence is found to be slow. It is concluded that all available data are consistent with beta = lambda 5/16 (magnetization) and not inconsistent with the 'scaling' value gamma '=5/4 (susceptibility) and that with the data and methods available at present it is not possible to draw more precise conclusions.