Muon spin relaxation in ferromagnets: theoretical results for paramagnetic EuO and EuS

Abstract

The relaxation rate for depolarization of a positive muon implanted in an isotropic magnetic salt with ferromagnetic exchange interactions is studied theoretically, on the basis of the coupled-mode theory of critical and paramagnetic spin fluctuations and a full numerical evaluation of the dipole field experienced by the muon. The main findings from studies of realistic models of EuO and EuS are (a) a significant dependence of the relaxation rate, A, on the assumed position of the implanted muon and (b) a monotonic temperature dependence, with A -~3 12 in the approach to the critical temperature at which the correlation length, ~, diverges. In contrast, previous results for a model of an isotropic magnet with an antiferromagnetic exchange, RbMnF 3 , show that A for this magnet is not a monotonic function of the temperature, and in the precursor region to Tc A increases with decreasing temperature with a power law behaviour A.-~1 12• The calculated values of A for EuO are consistent with data from preliminary experiments on the same salt.(1.) Introduction Several experimental studies have demonstrated that measurements of the depolarization of positive muons implanted in magnetic materials have the potential to provide useful information, at an atomic level of detail, on the fluctuations of the magnetic moments; see, for example, Cox (1987) and Dalmas de Reotier et al. (1994).In a previous paper we provided a comprehensive theoretical investigation of relaxation in the paramagnetic phase of an antiferromagnetically coupled material (Lovesey et al. 1994). The present paper reports findings from a similar, comprehensive investigation of muon relaxation in isotropic, ferromagnetically coupled systems. Results for two materials, EuO and EuS, are provided.The overall plan of the work is the same as that used to study the antiferromagnetically coupled salt, RbMnF 3 , namely, a complete numerical evaluation of the spatial Fourier transform of the dipole field between the muon moment and the atomic moments, and a description of critical and paramagnetic fluctuations of the atomic moments from a full version of the coupled-mode theory. In view of the strong similarities in the work for the two types of magnetic salts the background to the methods given here is very brief; the reader interested in these matters is referred to Lovesey et al. (1994).Turning to the results of our work based on realistic models of EuO and EuS -an isotropic Heisenberg magnet with exchange interactions out to the second shell of neighbouring spins -we find that the magnitude and temperature dependence of the relaxation rate, A, depend on the position assumed for the implanted muon. A similar finding was obtained in the study of RbMnF 3 • For the latter material, "A is not a monotonic function of the temperature. In contrast, the relaxation rates for EuO and EuS are found to be monotonic functions of the temperature, cf. Table (1). In the approach to the critical temperature ' A increases, and the temperature dependence, expressed in term...

μSR investigation of magnetism and magnetoelectric coupling in Cu2OSeO3

“…An internal magnetic field B int (T = 0) = 85.37 (25) mT was detected below T c = 57.0(1) K, consistent with a ferrimagnetic state. 9 The effective critical exponent for the temperature dependence of B int was found to bẽ β 0.39(1), in fair agreement with the critical exponent β 1/3 expected for 3D magnetic systems.…”

“…yields T c = 57.0(1) K,α = 2.00(9),β = 0.39(1), and B 1 (0) = 85.37 (25) and 85.57 (25) mT for E = 0 and 500/3 V/mm, respectively. The value of the effective critical exponent 29β lies quite close to the critical exponent β 1/3 expected for a 3D magnetic system.…”

“…In the critical state the following relations hold: λ ∝ ξ 3/2 for ferromagnets and λ ∝ ξ 1/2 for antiferromagnets. [24][25][26] Note that these relations are strictly valid only in the critical regime very close to T c . However, it was found empirically, that these relations describe experimental data rather well for EuO, EuS, and RbMnF 3 up to ∼1.3T c .…”

“…However, it was found empirically, that these relations describe experimental data rather well for EuO, EuS, and RbMnF 3 up to ∼1.3T c . 25,26 Above T c the correlation length ξ follows the power law ξ ∝ (T /T c − 1)…”

“…24,25 ZF μSR measurements of the microscopic internal-field distribution with and without applied electric field E = 500/3 V/mm indicate a small electric-field effect on the internal magnetic field:…”