Polder tensor

{{Use dmy dates|date=October 2019}}

The Polder tensor is a tensor introduced by Dirk Polder[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] {{doi|10.1080/14786444908561215}} for the description of magnetic permeability of ferrites.[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)], {{doi|10.1038/1821080a0}} 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:{{cite book|last1=Marqués|first1=Ricardo|last2= Martin|first2=Ferran|last3=Sorolla|first3=Mario|title=Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications|url=https://books.google.com/books?id=lqHsnZoa7wAC&pg=PA93|year=2008|publisher=Wiley|isbn=978-0-470-19172-9|page=93}}

::

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

:$\backslash mu\; =\; \backslash mu\_0\; \backslash left(\; 1+\; \backslash frac\{\backslash omega\_0\; \backslash omega\_m\}\{\backslash omega\_0^2\; -\; \backslash omega^2\}\; \backslash right)$

:$\backslash kappa\; =\; \backslash mu\_0\; \backslash frac\{\backslash omega\; \backslash omega\_m\}\{\{\backslash omega\_0\}^2\; -\; \backslash omega^2\}$

where

:$\backslash omega\_0\; =\; \backslash gamma\; \backslash mu\_0\; H\_0\; \backslash$

:$\backslash omega\_m\; =\; \backslash gamma\; \backslash mu\_0\; M\; \backslash$

:$\backslash omega\; =\; 2\; \backslash pi\; f$

$\backslash gamma\; =\; 1.11\; \backslash times\; 10^5\; \backslash cdot\; g\; \backslash ,\backslash ,$ (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 $\backslash mu\_0$ is the magnetic permeability of free space.

To simplify computations, the radian frequencies of $\backslash omega\_0,\; \backslash ,\; \backslash omega\_m,\; \backslash ,$ and $\backslash omega$ can be replaced with frequencies (Hz) in the equations for $\backslash mu$ and $\backslash kappa$ because the $2\; \backslash pi$ factor cancels. In this case, $\backslash gamma\; =\; 1.76\; \backslash times\; 10^4\; \backslash cdot\; g\; \backslash ,\backslash ,$ Hz / (A / m) $=\; 1.40\; \backslash cdot\; g\; \backslash ,\backslash ,$ MHz / Oe. If CGS units are used, computations can be further simplified because the $\backslash mu\_0$ factor can be dropped.