{{Short description|Tensor describing the magnetic permeability of ferrites}} {{Use dmy dates|date=October 2019}} The '''Polder tensor''' is a tensor introduced by Dirk Polder in 1949<ref>[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}}</ref> for the description of magnetic permeability of ferrites.<ref>[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}}</ref> 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:<ref>{{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}}</ref>
::<math>B = \begin{bmatrix} \mu & j \kappa & 0 \\ -j \kappa & \mu & 0 \\ 0 & 0 & \mu_0 \end{bmatrix} H</math>
Neglecting the effects of damping, the components of the tensor are given by
:<math>\mu = \mu_0 \left( 1+ \frac{\omega_0 \omega_m}{\omega_0^2 - \omega^2} \right) </math> :<math>\kappa = \mu_0 \frac{\omega \omega_m}{{\omega_0}^2 - \omega^2}</math>
where
:<math>\omega_0 = \gamma \mu_0 H_0 \ </math> :<math>\omega_m = \gamma \mu_0 M \ </math> :<math>\omega = 2 \pi f</math>
<math>\gamma = 1.11 \times 10^5 \cdot g \,\, </math> (rad /s) /(A/m) is the effective gyromagnetic ratio and <math>g</math>, the so-called effective ''g''-factor, is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. <math>f</math> is the frequency of the RF/microwave signal propagating through the ferrite, <math>H_0</math> is the internal magnetic bias field, <math>M</math> is the magnetization of the ferrite material and <math>\mu_0</math> is the magnetic permeability of free space.
To simplify computations, the ''radian'' frequencies of <math>\omega_0, \, \omega_m, \,</math> and <math> \omega</math> can be replaced with frequencies (Hz) in the equations for <math> \mu </math> and <math> \kappa </math> because the <math> 2 \pi </math> factor cancels. In this case, <math>\gamma = 1.76 \times 10^4 \cdot g \,\, </math> Hz/ (A/m) <math> = 1.40 \cdot g \,\, </math> MHz/Oe. If CGS units are used, computations can be further simplified because the <math> \mu_0 </math> factor can be dropped.
==References== {{Reflist}}
{{DEFAULTSORT:Polder Tensor}} Category:Ferrites Category:Tensor physical quantities Category:Ferromagnetic materials Category:Magnetic ordering