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Progress of Theoretical Physics

Magnetic Properties of Random Mixture of the One-Dimensional Ising Chain

Fumitaka Matsubara

Published Feb 1, 1974• 18 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.51.378

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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.
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.
Binary magnetic square lattice Ising system with nearest neighbour interactions were simulated using the Monte Carlo technique. Two types of ions were randomly distributed on the lattice sites, one type interacting ferromagnetic and the other antiferromagnetic. A phase diagram of the ion concentration dependent critical temperature, Tc was deduced. Combined Bethe Peierls approximation and Mean Field theory phase transition results were compared to the results of the present method. An improved accuracy of the approximations of the critical temperatures was observed. The Monte Carlo simulation is thus shown to be a more reliable method for obtaining the physical properties of the random binary two-dimensional Ising system. Keywords: Binary magnetic two-dimensional Ising system, phase transition, Monte Carlo simulations
Two properties of the Ising model are considered. Firstly, by extrapolating appropriate series expansions, the spontaneous magnetization of a ferromagnet for three-dimensional lattices is obtained. From the results accurate values for the critical temperature are found. Secondly, the high temperature susceptibility of an antiferromagnet is derived from the approximation of Kikuchi. The resulting susceptibility curves show that for loose-packed lattices a maximum value occurs at a temperature slightly above the critical point, while for the close-packed lattices no such effect is obtained.
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 two-dimensional Ising model with two kinds of bonds –ferro and antiferromagnetic– is investigated in both quenched and annealed approximations. High temperature series expansion technique is used to get the respective free energies and susceptibilities. Form the analysis of the ratio method results, the phase diagrams are drawn and critical concentrations are estimated. It is shown that in both approximations the system behaves similarly, although in the annealed case the magnetically ordered phases are more stable against increasing impurity concentration. It is also proved that for Bethe lattices initial susceptibilities for quenced and annealed systems coincide.
The critical behavior of the mixed Ising model of the type AB p C1−p ternary alloy consisting of spins σ = 1/2, S = 1, and m = 3/2 is investigated on the Bethe lattice by using the exact recursion relations. The exact expressions for the magnetizations and magnetic susceptibilities are found, and the thermal behaviors of the magnetizations and susceptibilities are studied. The magnetizations and susceptibilities have also been investigated as functions of the crystal-field interaction or single-ion anisotropy. The phase diagram has been constructed according to which the system always undergoes a second-order phase transition for the coordination number q ≤ 3 and second-and first-order phase transitions for q > 3; hence, the system has a tricritical point. The system also exhibits reentrant behaviors.