Analytic solution for the nonlinear response of magnetic nanoparticles to large-amplitude oscillatory fields

Nonlinear responses of physical systems to strong perturbations are notoriously difficult to tackle analytically. Here, we present analytic results for the nonlinear response of magnetic nanoparticles to large-amplitude oscillatory magnetic fields based on a particular model for the magnetization dynamics. A number of characteristic features of the in-phase and out-of-phase higher harmonic response are found and analyzed. In particular, we find that the magnitude of higher harmonic contributions R_{n} depends on the field amplitude and frequency only via a single scaling variable that combines the two quantities. The decrease of |R_{n}| with increasing order n of harmonics is a key quantity monitored in biomedical applications such as magnetic particle spectroscopy and magnetic particle imaging. Except for the first few harmonics, we find that this decrease is exponential with a rate that depends on the scaling variable only. For not too high frequencies and not too large amplitudes, we find that these exact results for one particular model of magnetization dynamics hold approximately also for other, more frequently used models. Our results therefore offer not only deeper insight into strongly nonlinear responses of magnetic nanoparticles, especially for higher harmonics that are very difficult to determine numerically, but also suggest analyzing data in terms of a scaling variable.