Polder tensor

The Polder tensor is a tensor introduced by Dirk Polder for the description of magnetic permeability of ferrites. The tensor notation needs to be used because ferrimagnetic material becomes anisotropic in the presence of a magnetizing field.

The tensor is described mathematically as:

$B={\begin{bmatrix}\mu &j\kappa &0\\-j\kappa &\mu &0\\0&0&\mu _{0}\end{bmatrix}}H$

Neglecting the effects of damping, the components of the tensor are given by

$\mu =\mu _{0}\left(1+{\frac {\omega _{0}\omega _{m}}{\omega _{0}^{2}-\omega ^{2}}}\right)$

$\kappa =\mu _{0}{\frac {\omega \omega _{m}}{{\omega _{0}}^{2}-\omega ^{2}}}$

where

$\omega _{0}=\gamma \mu _{0}H_{0}\$

$\omega _{m}=\gamma \mu _{0}M\$

$\omega =2\pi f$

$\gamma =1.11\times 10^{5}\cdot g\,\,$ (rad / s) / (A / m) is the effective gyromagnetic ratio and $g$, the so-called effective g-factor (physics), is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. $f$ is the frequency of the RF/microwave signal propagating through the ferrite, $H_{0}$ is the internal magnetic bias field, $M$ is the magnetization of the ferrite material and $\mu _{0}$ is the magnetic permeability of free space.

To simplify computations, the radian frequencies of $\omega _{0},\,\omega _{m},\,$ and $\omega$ can be replaced with frequencies (Hz) in the equations for $\mu$ and $\kappa$ because the $2\pi$ factor cancels. In this case, $\gamma =1.76\times 10^{4}\cdot g\,\,$ Hz / (A / m) $=1.40\cdot g\,\,$ MHz / Oe. If CGS units are used, computations can be further simplified because the $\mu _{0}$ factor can be dropped.