  We found resonantly excited precession motions of a three-dimensional vortex
core in soft magnetic nanospheres and controllable precession frequency with
the sphere diameter 2R, as studied by micromagnetic numerical and analytical
calculations. The precession angular frequency for an applied static field
$H_{DC}$ is given as $\omega_{MV}= \gamma_{eff} H_{DC}$, where $\gamma_{eff} =
\gamma <m_{\Gamma}>$ is the effective gyromagnetic ratio in collective vortex
dynamics, with the gyromagnetic ratio $\gamma$ and the average magnetization
component $<m_{\Gamma}>$ of the ground-state vortex in the core direction.
Fitting to the micromagnetic simulation data for $<m_{\Gamma}>$ yields a simple
explicit form of $<m_{\Gamma}> = (73.6 \pm 3.4)(l_{ex}/2R)^{2.20 \pm 0.14}$,
where $l_{ex}$ is the exchange length of a given material. This dynamic
behavior might serve as a foundation for potential bio-applications of
size-specific resonant excitation of magnetic vortex-state nanoparticles, for
example, magnetic particle resonance imaging.
