Theory of the Dispersion of Magnetic Permeability in Ferromagnetic Materials at Microwave Frequencies

TLDR

The initial magnetic permeability of iron and nickel drops from about 100 in the d.c. regime to roughly 1 in the infrared, a transition that occurs mainly in the microwave frequency range, and the applicability of Becker’s eddy‑current damping theory is also examined. The study proposes an explanation of the observed permeability transition by modeling domain‑boundary motion in high‑frequency fields and suggests further experiments to test the theory. The authors derive an analytic solution of Maxwell’s equations for a single‑domain layer, linking the complex permeability μ to resistive and reactive measurements (μ_R and μ_L) and establishing the conditions under which the skin depth is less than the domain thickness.

Abstract

The transition in the value of the initial magnetic permeability of iron and nickel from the d.c. permeability \ensuremath{\sim}100 to the infra-red permeability \ensuremath{\sim}1 is known to occur principally in the microwave frequency range. An explanation of the experimental facts is proposed by considering the equations of motion of a domain boundary in an applied magnetic field for frequencies such that the skin depth of the magnetic field is smaller than the thickness of a domain. An analytic solution of Maxwell's equations is found for the magnetization of a layer one domain thick. The definition of the permeability at high frequencies is considered carefully, and it is shown that the natural definition leads to complex values for the permeability. In experiments two different types of permeabilities are found. The relationship of the complex $\ensuremath{\mu}$ to the ${\ensuremath{\mu}}_{R}$ determined from resistive losses in a circuit element and to the ${\ensuremath{\mu}}_{L}$ determined from reactance changes is developed. A criticism is given of theories of ferromagnetic resonance. The status of Becker's theory of eddy current damping is considered. Several suggestions are made for further experiments.