# Low-frequency response functions of random magnetic systems

## Abstract

The frequencies of long-wavelength spin waves in random magnets are studied through their relation to the static magnetic elastic constants A, the domain-wall stiffness, and (for antiferromagnets), the perpendicular susceptibility. We treat the classical limit of large spin and low temperature. In the case of random dilution A and are evaluated numerically as a function of magnetic concentration p for common lattices. Exact analytic results for the static susceptibility, (q), where q is the wave vector, are given for some models of disorder in one dimension and, for higher dimensionality, in the limit of low concentrations of vacancies. One general conclusion is that local fluctuations in the spin magnitude significantly affect, causing it to diverge for isotropic random systems in two or fewer dimensions. If critical exponents are defined for ppc by A|p-pc|, |p-pc|-, P|p-pc|, and |p-pc|-, where pc is the percolation threshold, P is the percolation probability, and is the correlation length, then our numerical results in three dimensions yield =1.60.1 and =0.50.2. A simple physical argument shows that -+(2-d). Our data are consistent with the possibility that this is an equality. Using mean-field-theory values for the exponents in this relation leads to a critical dimensionality dc=6. We study pc, A, and in diluted YIG and mixed garnets and give a detailed discussion of the regime near angular momentum compensation, where a low-frequency optical mode with both q and q2 regimes occurs. Our work contradicts the common assumption of a concentration-independent relationship between Tc and A or D, the spin-wave stiffness. We also present nonlinear calculations which allow us to study the dependence of on magnetic field. Our calculations agree with the experimental results on diluted KMnF3 and K2MnF4 and show that the observed nonlinearity is largely the result of local ferrimagnetic fluctuations. A novel configuration for elastic neutron scattering in the presence of a transverse magnetic field is proposed to permit direct observation of the magnitude and characteristic length scale of these fluctuations. © 1977 The American Physical Society.