Dependence of Brownian and Neel relaxation times on magnetic field strength

Purpose: In magnetic particle imaging (MPI) and magnetic particle spectroscopy (MPS) the relaxation time of the magnetization in response to externally applied magnetic fields is determined by the Brownian and Neel relaxation mechanisms. Here the authors investigate the dependence of the relaxation times on the magnetic field strength and the implications for MPI and MPS. Methods: The Fokker-Planck equation with Brownian relaxation and the Fokker-Planck equation with Neel relaxation are solved numerically for a time-varying externally applied magnetic field, including a step-function, a sinusoidally varying, and a linearly ramped magnetic field. For magnetic fields that are applied as a step function, an eigenvalue approach is used to directly calculate both the Brownian and Neel relaxation times for a range of magnetic field strengths. For Neel relaxation, the eigenvalue calculations are compared to Brown's high-barrier approximation formula. Results: The relaxation times due to the Brownian or Neel mechanisms depend on the magnitude of the applied magnetic field. In particular, the Neel relaxation time is sensitive to the magnetic field strength, and varies by many orders of magnitude for nanoparticle properties and magnetic field strengths relevant for MPI and MPS. Therefore, the well-known zero-field relaxation times underestimate the actual relaxation times and, in particular, can underestimate the Neel relaxation time by many orders of magnitude. When only Neel relaxation is present-if the particles are embedded in a solid for instance-the authors found that there can be a strong magnetization response to a sinusoidal driving field, even if the period is much less than the zero-field relaxation time. For a ferrofluid in which both Brownian and Neel relaxation are present, only one relaxation mechanism may dominate depending on the magnetic field strength, the driving frequency (or ramp time), and the phase of the magnetization relative to the applied magnetic field. Conclusions: A simple treatment of Neel relaxation using the common zero-field relaxation time overestimates the relaxation time of the magnetization in situations relevant for MPI and MPS. For sinusoidally driven (or ramped) systems, whether or not a particular relaxation mechanism dominates or is even relevant depends on the magnetic field strength, the frequency (or ramp time), and the phase of the magnetization relative to the applied magnetic field. c 2014 American Association of Physicists in Medicine.