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Polder tensor

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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 = [ μ j κ 0 − j κ μ 0 0 0 μ 0 ] H {\displaystyle 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

μ = μ 0 ( 1 + ω 0 ω m ω 0 2 − ω 2 ) {\displaystyle \mu =\mu _{0}\left(1+{\frac {\omega _{0}\omega _{m}}{\omega _{0}^{2}-\omega ^{2}}}\right)}
κ = μ 0 ω ω m ω 0 2 − ω 2 {\displaystyle \kappa =\mu _{0}{\frac {\omega \omega _{m}}{{\omega _{0}}^{2}-\omega ^{2}}}}

where

ω 0 = γ μ 0 H 0   {\displaystyle \omega _{0}=\gamma \mu _{0}H_{0}\ }
ω m = γ μ 0 M   {\displaystyle \omega _{m}=\gamma \mu _{0}M\ }
ω = 2 π f {\displaystyle \omega =2\pi f}

γ = 1.11 × 10 5 ⋅ g {\displaystyle \gamma =1.11\times 10^{5}\cdot g\,\,} (rad / s) / (A / m) is the effective gyromagnetic ratio and g {\displaystyle 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 {\displaystyle f} is the frequency of the RF/microwave signal propagating through the ferrite, H 0 {\displaystyle H_{0}} is the internal magnetic bias field, M {\displaystyle M} is the magnetization of the ferrite material and μ 0 {\displaystyle \mu _{0}} is the magnetic permeability of free space.

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

References

  1. 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
  2. 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
  3. 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.
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