Physics:Polder tensor
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]
- [math]\displaystyle{ 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]\displaystyle{ \mu = \mu_0 \left( 1+ \frac{\omega_0 \omega_m}{\omega_0^2 - \omega^2} \right) }[/math]
- [math]\displaystyle{ \kappa = \mu_0 \frac{\omega \omega_m}{{\omega_0}^2 - \omega^2} }[/math]
where
- [math]\displaystyle{ \omega_0 = \gamma \mu_0 H_0 \ }[/math]
- [math]\displaystyle{ \omega_m = \gamma \mu_0 M \ }[/math]
- [math]\displaystyle{ \omega = 2 \pi f }[/math]
[math]\displaystyle{ \gamma = 1.11 \times 10^5 \cdot g \,\, }[/math] (rad / s) / (A / m) is the effective gyromagnetic ratio and [math]\displaystyle{ g }[/math], 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. [math]\displaystyle{ f }[/math] is the frequency of the RF/microwave signal propagating through the ferrite, [math]\displaystyle{ H_0 }[/math] is the internal magnetic bias field, [math]\displaystyle{ M }[/math] is the magnetization of the ferrite material and [math]\displaystyle{ \mu_0 }[/math] is the magnetic permeability of free space.
To simplify computations, the radian frequencies of [math]\displaystyle{ \omega_0, \, \omega_m, \, }[/math] and [math]\displaystyle{ \omega }[/math] can be replaced with frequencies (Hz) in the equations for [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \kappa }[/math] because the [math]\displaystyle{ 2 \pi }[/math] factor cancels. In this case, [math]\displaystyle{ \gamma = 1.76 \times 10^4 \cdot g \,\, }[/math] Hz / (A / m) [math]\displaystyle{ = 1.40 \cdot g \,\, }[/math] MHz / Oe. If CGS units are used, computations can be further simplified because the [math]\displaystyle{ \mu_0 }[/math] factor can be dropped.
References
- ↑ 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
- ↑ 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
- ↑ Marqués, Ricardo; Martin, Ferran; Sorolla, Mario (2008). Metamaterials with Negative Parameters: Theory, Design, and Microwave Applications. Wiley. p. 93. ISBN 978-0-470-19172-9. https://books.google.com/books?id=lqHsnZoa7wAC&pg=PA93.
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