Investigation of Two-Dimensional Triangular Ising Lattice in the Absence and Presence of a Magnetic Field

S. Ranjbar

《Chinese Journal of Physics》 41卷4期 (2003/08) Pp. 383-398

並列摘要

An example of a two-dimensional crystal in which each lattice point interacts with six neighbors is the so-called ”triangular Ising lattice”. In this work the transfer matrix method is used to find the thermodynamic properties of the triangular Ising model with a limited number of rows, in the absence or presence of a magnetic field. In the absence of a magnetic field the results have been used to find an analytical expression for the partition function of a model with any number of rows, from which the thermodynamic properties of the triangular planar lattice model are obtained as a polynomial in terms of x = e-2j tanh 2j/2 cosh 2j, where J = jkT is the interaction energy of the two nearest neighbor spins. The degree of the polynomial depends on the number of rows in the model. The results are compared with the exact values. The calculated free energy is found to be very accurate, but the calculated internal energy and heat capacity show some small deviations from the exact values. In the presence of a magnetic field the solution to the matrices is too complicated for a general analytical expression to be proposed for a model with any number of rows. However, our exact solution for the model with n = 4, 5, and 7 reveals an important point: that such results are independent of n when the field-spin interaction energy is almost equal to or larger than that of the spin-spin interaction.