The free energy of the one-dimensional Ising model of a dilute magnet

• Semkin S.V.

  Sergey V. Semkin. Vladivostok State University. Vladivostok. Russia

• Smagin V.P.

  Sergey V. Semkin. Vladivostok State University. Vladivostok. Russia

Abstract. The Ising model with non-magnetic dilution is used for the theoretical description of many
objects and phenomena in condensed matter physics and nuclear physics. The effect of non-magnetic dilu-
tion on the critical behavior of magnets, including those described by the Ising model, is of con¬siderable
scientific interest. For the Ising model with non-magnetic dilution, it is not possible to con¬struct the exact
solution for any crystal lattice. The properties of this model are investigated either numerically or in one
approximation or another. In this paper, the exact solution is obtained for a one-dimensional Ising model
with stationary, randomly arranged non-magnetic impurities. This exact solution is based on the represen-
tation of the statistical sum of the diluted chain as a product of the statistical sums of isolated segments of
the chain of different lengths. To calculate the statistical sums of these segments, the method of an asym-
metric transfer matrix is used. By this method, the exact value of free energy of the impurity chain is found
as a function of impurity concentration, tempera¬ture and external magnetic field. Then, we compare the
exact solution obtained by us with the solu¬tion obtained in the pseudo-chaotic approximation. It turns out
that in the zero external field, the exact solution completely coincides with the solution obtained in the
pseudo-chaotic approximation. In the presence of an external field, the difference between the exact and
approximate value of the free energy depends on the concentration of impurities and temperature. This dif-
ference was investigated in this paper.
Keywords: phase transitions, Ising model, dilute magnet