Polder tensor explained

The Polder tensor is a tensor introduced by Dirk Polder^[1] for the description of magnetic permeability of ferrites.^[2] 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:^[3]

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+ (\omega _{0} \omega _{m})/(\omega ^{2} _{0} - \omega ^{2}) \right) \kappa = \mu _{0} (\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 x 10 ^{5} ⋅ 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 x 10 ^{4} ⋅ g Hz / (A / m) = 1 . 40 ⋅ g MHz / Oe. If CGS units are used, computations can be further simplified because the \mu _{0} factor can be dropped.

Notes and References

1. http://www.tandfonline.com/doi/abs/10.1080/14786444908561215 D. Polder, On the theory of ferromagnetic resonance, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 40, 1949
2. http://www.nature.com/nature/journal/v182/n4642/abs/1821080a0.html G. G. Robbrecht, J. L. Verhaeghe, Measurements of the Permeability Tensor for Ferroxcube, Letters to Nature, Nature 182, 1080 (18 October 1958)
3. Book: Marqués. Ricardo. Martin. Ferran. Sorolla. Mario. Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications. 2008. Wiley. 978-0-470-19172-9. 93.