  The Einstein-de Haas (EdH) effect is a fundamental, mechanical consequence of
any temporal change of magnetism in an object. EdH torque results from
conserving the object's total angular momentum: the angular momenta of all the
specimen's magnetic moments, together with its mechanical angular momentum.
Although the EdH effect is usually small and difficult to observe, it increases
in magnitude with detection frequency. We explore the frequency-dependence of
EdH torque for a thin film permalloy microstructure by employing a ladder of
flexural beam modes (with five distinct resonance frequencies spanning from 3
to 208 MHz) within a nanocavity optomechanical torque sensor via magnetic
hysteresis curves measured at mechanical resonances. At low DC fields the
gyrotropic resonance of a magnetic vortex spin texture overlaps the 208 MHz
mechanical mode. The massive EdH mechanical torques arising from this
co-resonance yield a fingerprint of vortex core pinning and depinning in the
sample. The experimental results are discussed in relation to mechanical
torques predicted from both macrospin (at high DC magnetic field) and
finite-difference solutions to the Landau-Lifshitz-Gilbert (LLG) equation. A
global fit of the LLG solutions to the frequency-dependent data reveals a
statistically significant discrepancy between the experimentally observed and
simulated torque phase behaviours at spin texture transitions that can be
reduced through the addition of a time constant to the conversion between
magnetic cross-product torque and mechanical torque, constrained by experiment
to be in the range of 0.5 - 4 ns.
