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 Néel relaxation mechanisms. Here the authors investigate the dependence of the relaxation times on the magnetic field strength and the implications for MPI and MPS.
The Fokker–Planck equation with Brownian relaxation and the Fokker–Planck equation with Néel 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 Néel relaxation times for a range of magnetic field strengths. For Néel relaxation, the eigenvalue calculations are compared to Brown's high-barrier approximation formula.
The relaxation times due to the Brownian or Néel mechanisms depend on the magnitude of the applied magnetic field. In particular, the Néel 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 Néel relaxation time by many orders of magnitude. When only Néel 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 Néel 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.
A simple treatment of Néel 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.
The authors thank Mark Griswold and Lisa Bauer for suggesting the study of nanoparticles fixed in a polymer, as well as other helpful discussions. The authors thank Bob Brown for his guidance, his careful reading of the manuscript, and many useful discussions. The authors are grateful for support from the Ohio Third Frontier and from the NSF (Grant No. 1318206).
II.A. The Fokker–Planck equation for Brownian relaxation and its numerical solution
II.B. The Fokker–Planck equation for Néel relaxation and its numerical solution
II.C. Eigenvalue calculation for response to a magnetic field step function
III. RESULTS AND DISCUSSION
III.A. Magnetization response to a step function magnetic field
III.B. Eigenvalue calculations
III.C. Sinusoidal and linearly ramped magnetic fields
- Relaxation times
- Magnetic fields
- Magnetic moments
- Magnetic nanoparticles
Data & Media loading...
Article metrics loading...
Full text loading...