Exact and approximate solutions for the one-dimensional Ising model of a diluted magnet

• Semkin S.V.

  S.V. Semkin. Vladivostok State University of Economics and Service. Vladivostok. Russia

• Smagin V.P.

  V.P. Smagin Vladivostok State University of Economics and Service. Vladivostok. Russia

The Ising model with nonmagnetic dilution is used for the theoretical description of many objects and phenomena in the physics of condensed matter and nuclear physics. The effect of nonmagnetic dilution on the critical behavior of magnets, including those described by the Ising model, is of considerable scientific interest. For the Ising model with nonmagnetic dilution, it is not possible to construct an exact solution for any crystal lattice. The properties of this model are investigated either numerically or in one or another approximation. In this paper, an exact solution is obtained for the one-dimensional Ising model with fixed, randomly located nonmagnetic impurities. This exact solution is based on the compilation of recurrent equations, through whose solutions the magnetization and correlation of neighboring spins are expressed. It is shown that in the limit when the diluted magnet goes into a pure one, the solution obtained by us goes into the well-known exact solution for the one-dimensional Ising model. Then, we compare the exact solution obtained by us with several approximate solutions. The following approximate methods were considered: the mean field method, the averaging method over exchange fields both with and without correlations, and the pseudochaotic approximation. The roughest, as one would expect, is the mean field method. The solutions obtained by the averaging method over exchange fields, especially taking into account the correlation, are much better consistent with the exact solution. But the solution obtained in the pseudochaotic approximation turns out to be the closest to the exact solution. Moreover, this approximation exceeds the other approximations in accuracy in the entire range of concentrations of magnetic atoms.

Keywords: phase transitions, Ising model, dilute magnet.