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Nano Science and Nano Technology An Indian Journal

Heat capacity of the simple-cubic Ising model

Igor A. Stepanov

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Earlier, the isochoric heat capacity of the cubic lattice for the Ising model was calculated approximately. In the present paper, using the exact solution of the three-dimensional Ising model, this heat capacity is calculated exactly.

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The partition function of a three-dimensional Ising model was calculated by a high-speed computer for a cubic array of twenty-seven points. The periodicity condition was imposed. The specific heat of the 3×3×3 lattice was calculated as a function of temperature. The critical temperature estimated from the specific heat curve seems slightly lower that obtained from the Padé approximation. The zeros of the partition function of the complex temperature plane were also obtained. The critical temperature estimated from the zeros near the positive real axis seems slightly higher than both of the above values. Similar calculations were carried out for the case of the two-dimensional Ising model. Several results are compared with those obtained from Onsager's exact solution.
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.
The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.