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Journal of Physics C Solid State Physics

Critical behaviour of the three-dimensional Ising model specific heat below Tc

Hemda Garelick, J W Essam

Published Dec 1, 1968• 11 citations
Publisher

Abstract

The magnetization exponent β D of the Syozi model of dilute ferro-magnetism is estimated for the four most common three-dimensional lattices using the Padé approximant method. It is found in all cases that 0·354 <or= β D <or= 0·380. Using the previously established relation αprime = 1 - β/β D between the specific heat exponent αprime and the magnetization exponent β for the Ising model below T c and assuming β = fraction five-sixteens, the above result implies 0·117 <or= αprime <or= 0·178. On the grounds that previous estimates of αprime have been much lower, it is concluded that most probably αprime = fraction one-eighth = 0·125 with the consequence β D = fraction five-fourteens = 0·357.... The possibility αprime = fraction one-sixteens is well excluded by this method. The specific heat amplitude of the Ising model below T c is deduced by computation of the magnetization amplitude of the Syozi model.

DOI
10.1088/0022-3719/1/6/315

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Journal of Physics C Solid State Physics•Nov 21, 1974•43 citations
The magnetization exponent β D of the Syozi model of dilute ferro-magnetism is estimated for the four most common three-dimensional lattices using the Padé approximant method. It is found in all cases that 0·354 <or= β D <or= 0·380. Using the previously established relation αprime = 1 - β/β D between the specific heat exponent αprime and the magnetization exponent β for the Ising model below T c and assuming β = fraction five-sixteens, the above result implies 0·117 <or= αprime <or= 0·178. On the grounds that previous estimates of αprime have been much lower, it is concluded that most probably αprime = fraction one-eighth = 0·125 with the consequence β D = fraction five-fourteens = 0·357.... The possibility αprime = fraction one-sixteens is well excluded by this method. The specific heat amplitude of the Ising model below T c is deduced by computation of the magnetization amplitude of the Syozi model.
Previous series expansion studies of the magnetization and susceptibility of the Ising model are extended to the higher derivatives of the free energy with respect to the magnetic field. It is found that just above the critical temperature the 2 n th derivative is given by where in two dimensions γ = 1¾, Δ = 1fraction seven-eighths and in three dimensions γ = 1¼, Δ = 1fraction nine-sixteens. In two dimensions γ is exact and Δ is correct to 0.6%; the three-dimensional indices are correct to 0.2%. Just below the critical temperature where in three dimensions 0.303 <or= β <or= 0.318 and 1·55 <or= Δprime <or= 1.65. If the even derivatives are to have the same index above and below T c , then β = fraction five-sixteens and Δprime = 1fraction nine-sixteens. The latter index lies very close to the lower limit so that, although we cannot rule out Δprime = Δ, Δprime = 1fraction five-eighths is not unlikely. In two dimensions our results are consistent with linearity of the index and, if this is assumed, then β = fraction one-eighths and Δprime = 1fraction seven-eighths exactly and there is symmetry about T c . The amplitudes C 2 n + and C n - are estimated for n = 1 to 6.
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 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.
For pt. III see abstr. A23213 of 1974. The critical exponent beta is examined for two three-dimensional Ising model systems which allow admixtures of three and four-body terms to be present in the Hamiltonian. Low-temperature series expansions, combined with a Pade approximant analysis are used to form estimates of the exponents. Strong evidence is found in support of potential sensitivity in the exponent beta , and examples of the specific heat exponent alpha ' are also discussed. The variability of the critical exponents is discussed in relation to the Universally postulate and to generalized lattice-gas models.