# Nanoscale iron spheres

High-frequency permeability, long treated as a single bulk parameter, is now computed as a field that varies with position and frequency. The approach discretizes a sample into micromagnetic cells, extracts a permeability spectrum per cell through time-to-frequency conversion, and suppresses Fourier error with a verified pulse window. A planar square model shows localized negative imaginary permeability at 14.10 GHz, where magnetization leads the field by 9π/10; on nanoscale iron spheres, machine-learning predictions of defect-to-spectrum trends agree with broadband measurements on four etched samples.

High-frequency permeability, long treated as a single bulk parameter, is now computed as a field that varies with position and frequency. The approach discretizes a sample into micromagnetic cells, extracts a permeability spectrum per cell through time-to-frequency conversion, and suppresses Fourier error with a verified pulse window. A planar square model shows localized negative imaginary permeability at 14.10 GHz, where magnetization leads the field by 9π/10; on nanoscale iron spheres, machine-learning predictions of defect-to-spectrum trends agree with broadband measurements on four etched samples.