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Journal of Physics A Mathematical and General

Random mixture of Ising systems of different spin values

Seiichiro Katsura

Published Nov 1, 1979• 9 citations
Publisher

Abstract

The magnetic properties of the Ising mixture of a site model of A and B atoms, of which the magnetic moments, spins, concentrations and exchange energies are mu A , mu B , S A , S B , p A , p B , J AA , J AB and J BB , are investigated. As examples, mixtures with S=1 and S= 1 / 2 , the former of which has the anisotropy D, are studied by the distribution function method in the Bethe approximation. The critical temperature (phase boundary between the paramagnetic (P)-ferromagnetic (F) and paramagnetic-antiferromagnetic (AF) phases), the energy and the zero-field susceptibility in the paramagnetic phase are obtained. They are expressed in terms of generalised Brillouin functions as a natural extension of the S= 1 / 2 Ising model and the classical Heisenberg model.

DOI
10.1088/0305-4470/12/11/019

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The magnetic properties of the Ising mixture of a site model of A and B atoms, of which the magnetic moments, spins, concentrations and exchange energies are mu A , mu B , S A , S B , p A , p B , J AA , J AB and J BB , are investigated. As examples, mixtures with S=1 and S= 1 / 2 , the former of which has the anisotropy D, are studied by the distribution function method in the Bethe approximation. The critical temperature (phase boundary between the paramagnetic (P)-ferromagnetic (F) and paramagnetic-antiferromagnetic (AF) phases), the energy and the zero-field susceptibility in the paramagnetic phase are obtained. They are expressed in terms of generalised Brillouin functions as a natural extension of the S= 1 / 2 Ising model and the classical Heisenberg model.
The specific heat, the susceptibility, and the correlation function at zero field above the critical temperature of the random mixture (quenched site and bond problems) of the classical Heisenberg spins with nearest neighbor interaction were obtained exactly for the linear chain and for an infinite Bethe lattice (Bethe approximation of the two and three dimensional lattices) above the critical temperature. The results are simply expressed by the replacements of 2 cosh K → 4π (sinh K)/K and tanh K → L(K) (L(K) = Langevin function) for K = KAA, KAB, KBA, and KBB in the corresponding expressions of the random mixture of the Ising spins, and qualitative properties of both models are similar.
The three-dimensional site-random Ising model consisting of two types of magnetic ions, A and B, is investigated by means of the exchange Monte Carlo method. The nearest-neighbor interactions are antiferromagnetic (AF) for the B-B pair and ferromagnetic (FM) otherwise. A uniform FM or AF ordering is found to exist in a wide range of concentration c of A ions, in contrast with the bond-random Ising model such as the ± J model. The phase diagram is found to be symmetric on the c – T plane, though the ratio of the FM bonds to AF ones is asymmetric with respect to c =1/2. Spin configurations of degenerate ground states and their cluster structure are also studied. Even in the system with c =0.55, there exists only a unique rigid cluster which includes about 90% of spins and that cluster shows nonvanishing uniform magnetization. These results strongly suggest that the present model has neither a spin glass phase nor an FM-AF mixed phase, though their existence in a quite narrow region around c =1/2 has not been completely excluded yet.
Magnetic properties of the annealed quenched systems each in the bond and site models are investigated by the Kikuchi pair approximation. In the annealed system, phase diagrams and susceptibilities are obtained. Two critical temperatures in some case in the bond system and no critical concentration in the site system are obtained. The physical interpretation of them is given. In the quenched system, two different approximate conditions are considered for the complete randomness. The case of the first condition is equivalent to the cluster variation method proposed by Sato, Arrott and Kikuchi, and the results are in agreement with those obtained by Katsura and Matsubara. The second is essentially equivalent to the average coordination number method by Sato, Arrott and Kikuchi for the dilute ferromagnetism. However, the results for both cases are unreliable at low temperatures, because of the appearance of negative entropy. This is inevitable under the pair condition.
In this work we study a disordered binary Ising model on the square lattice. The model system consists of two different particles with spin-1/2 and spin-1, which are randomly distributed on the lattice. It has been considered only spin nearest-neighbor exchange interactions with J>0J>0J>0. This system can represent a disordered magnetic binary alloy AxB1−xA_{x}B_{1-x}Ax​B1−x​, obtained from the high temperature quenching of a liquid mixture. The results were obtained by the use of Monte Carlo simulations for several lattice sizes LLL, temperature TTT and concentration xxx of ions AAA with spin-1/2. We found its critical temperature, through the reduced fourth-order Binder cumulant for the several values of the concentration xxx of particles (spin-1/2, spin-1), and also the magnetization, the susceptibility and the specific heat as a function of temperature TTT.
In this paper we investigate the effect of a random transverse field, distributed according to a trimodal distribution, on the phase diagram and magnetic properties of a two-dimensional lattice (square with z=4),  ferromagnetic Ising system consisting of magnetic atoms with spin-1. This study is done using the effectivefield theory (EFT) with correlations method. The equations are derived using a probability distribution method based on the use of exact Van der Waerden identities. We present our numerical results, such as the phase diagrams, the thermal variations of the transverse magnetization, the internal energy, the magnetic specific heat as a function of different values of p, the concentration of the random transverse field. As a result, the critical values of transverse field , temperature and concentration were obtained for the square lattice.
The susceptibilities of both the one-dimensional random Ising chain and the random Ising Bethe lattice composed of plural kinds of magnetic atoms of S = 1/2 and non-magnetic atoms are studied based on a new method, i.e., by using simultaneous linear equation of order n, where n is the number of kinds of magnetic atoms. The susceptibilities of the binary and ternary systems, which correspond to n = 1 and 2, respectively, are exactly obtained for both the one-dimensional chain and the Bethe lattice. The critical temperature of the random Bethe lattice is also obtained. It is shown that the critical temperature of the binary Bethe lattice is in agreement with that of the ordinary two- or three-dimensional lattice in the cluster variation method and that the critical temperature of the ternary Bethe lattice is independent of the signs of exchange integrals. It is also conjectured that the critical temperature of the ordinary ternary system does not depend on the signs of exchange integrals when two kinds of magnetic atoms are of the same type and the lattice structure is loose-packed. The problem of the random exchange integrals is also discussed.