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Canadian Journal of Physics

Magnetic Properties of the Random Mixture of the Classical Heisenberg Spins

Shigetoshi Katsura

Published May 1, 1975• 16 citations
Publisher

Abstract

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.

DOI
10.1139/p75-106

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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 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 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.
In this work, we simulated a quenched random mixed-spin Ising model on the square lattice. The model system consists of two different particles with spins σ = 1/2 (states ±1/2) and S = 1 (states ±1, 0). These particles are randomly distributed on the lattice, and we considered only nearest-neighbor interactions. This model can represent a random magnetic binary alloy AxB1-x, obtained from the high-temperature quenching of a liquid mixture. We performed Monte Carlo simulations for several lattice sizes and temperatures, and we found its critical temperature through the reduced fourth-order cumulant. We also determined the magnetization, the susceptibility, and the specific heat as a function of temperature. We used finite-size scaling arguments to estimate the critical exponents β, γ, and ν of the model. We showed that the quenched model is in the same universality class of the two-dimensional pure Ising model. We also investigated the sample to sample fluctuations that occur in the values of the thermodynamic quantities in order to describe the self-averaging properties at the critical point.
We start this article by recalling rigorous results previously obtained for 2d square lattices composed of (2N+1)2classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings), in the thermodynamic limit (N→+∞) [4]: (i) the zero-field partition function ZN(0), (ii) the spin correlation which vanishes in the zero-field limit, except at T=0 K so that the critical temperature is TC=0 K, in agreement with Mermin-Wagner's theorem, (iii) the spin-spin correlation between any two lattice sites, (iv) the correlation length and (v) the static susceptibility. We exclusively focus on the low-temperature behaviors of the correlation length and the static susceptibility. This leads to the determination of a diagram characterized by three magnetic phases. Moreover we show that all the behaviors are in perfect agreement with the corresponding ones derived by using a renormalization method. Finally we give criterions allowing to directly determine the magnetic phases characterizing 2d magnetic compounds described by our microscopic model. An experimental test is given for illustrating this theoretical study.
Generalizing the recent work of Collins, the wave-vector-dependent susceptibility and the static correlation function for a randomly diluted Heisenberg paramagnet on a Bravais lattice with quenched-site, or nearest-neighbor exchange-bond, dilution is studied by high-temperature series-expansion techniques. The first five coefficients for the spin correlation function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ and susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ are calculated for arbitrary $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, the sign of the exchange and the spin magnitude and the magnetic site ccc and bond ppp concentrations. Numerical results for these functions are presented for various spins and for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ along the symmetry directions (1,1,1) and (1,0,0). Because of the generality of the present series, their length is rather limited, and as such they are not suitable for a careful analysis of the critical region. Nonetheless, to get a qualitative feel for the accuracy of the present results near the transition region, we have used Pad\'e approximants for determining the correlation length $\ensuremath{\xi}(T)$ as a function of the reduced temperature $\ensuremath{\epsilon}=\frac{[T\ensuremath{-}{T}_{C(N)}(\mathrm{random})]}{{T}_{C(N)}(\mathrm{random})}$. In this manner, we estimate that the numerical accuracy of our results should be reasonable as long as the system is moderately above the transition temperature, e.g., $\ensuremath{\epsilon}\ensuremath{\gtrsim}0.5$.