Internal and Brownian mode-coupling effects in the theory of magnetic relaxation and ferromagnetic resonance of ferrofluids

It is shown how the Langevin equation for the motion of the magnetization of a ferrofluid particle with uniaxial anisotropy in a strong uniform applied field reduces to those governing the Neel (i.e. the solid-state or internal) mechanism of reorientation of the magnetic moment in the non-axially symmetric potential created when a field is applied at an angle to the easy axis and a Lart-nor-like equation for the transverse motion. The field angle, unlike in the solid-state problem, is a function of the time due to the torques imposed by the fluid carrier. The Langevin equation for the Brownian rotational motion of the particle itself reduces to that describing Debye relaxation in the applied field but is coupled to the magnetic motion via the external field. The results indicate that the dissipation parameter of the internal solid-state mechanism is augmented by the external stochastic torques imposed by the carrier. However, the effect appears to be negligible because of the ratio of the Brownian (Debye) time to the free Neel diffusion time. Furthermore, just as in the pure solid-state process, pronounced precession-aided longitudinal relaxation and ferromagnetic resonance effects, having, their origin in the breaking of the axial symmetry due to the strong field, will occur. The precession-aided relaxation disappears, for weak fields since the potential becomes axially symmetric. Moreover, the equations of motion of the magnetic moment and the particle completely decouple and the overall decay function is simply the product of the decay functions of the internal (Neel) and Debye processes. It appears that the ferromagnetic resonance in this instance is accurately described by the known solid-state results, since the Brownian relaxation time greatly exceeds the effective relaxation times of the internal dipole and quadrupole modes associated with the ferromagnetic resonance. This conclusion is reinforced by the favourable agreement of the weak-field result with experimental observations of the complex susceptibility of four ferrofluid samples.