Polder tensor

Last updated September 25, 2023

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+{\frac {\omega _{0}\omega _{m}}{\omega _{0}^{2}-\omega ^{2}}}\right)

\kappa =\mu _{0}{\frac {\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

$\gamma =1.11\times 10^{5}\cdot g\,\,$ (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, $\gamma =1.76\times 10^{4}\cdot g\,\,$ Hz / (A / m) $=1.40\cdot g\,\,$ MHz / Oe. If CGS units are used, computations can be further simplified because the $\mu _{0}$ factor can be dropped.

Related Research Articles

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.

A ferrimagnetic material is a material that has populations of atoms with opposing magnetic moments, as in antiferromagnetism, but these moments are unequal in magnitude so a spontaneous magnetization remains. This can for example occur when the populations consist of different atoms or ions (such as Fe²⁺ and Fe³⁺).

<span class="mw-page-title-main">Hooke's law</span> Physical law: force needed to deform a spring scales linearly with distance

In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, $F s = kx$ , where $k$ is a constant factor characteristic of the spring, and $x$ is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s.

In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter μ. It is the ratio of the magnetic induction $to the magnetizing field as a function of the field in a material. The term was coined by William Thomson, 1st Baron Kelvin in 1872, and used alongside permittivity by Oliver Heaviside in 1885. The reciprocal of permeability is magnetic reluctivity.$

In physics, the gyromagnetic ratio of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol $γ$ , gamma. Its SI unit is the radian per second per tesla (rad⋅s⁻¹⋅T⁻¹) or, equivalently, the coulomb per kilogram (C⋅kg⁻¹).

In differential geometry, the four-gradient $is the four-vector analogue of the gradient from vector calculus.$

In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another, except that it is additionally multiplied or weighted by a power W of the Jacobian determinant of the coordinate transition function or its absolute value. A tensor density with a single index is called a vector density. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight W are called tensor capacity. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.

In physics, Larmor precession is the precession of the magnetic moment of an object about an external magnetic field. The phenomenon is conceptually similar to the precession of a tilted classical gyroscope in an external torque-exerting gravitational field. Objects with a magnetic moment also have angular momentum and effective internal electric current proportional to their angular momentum; these include electrons, protons, other fermions, many atomic and nuclear systems, as well as classical macroscopic systems. The external magnetic field exerts a torque on the magnetic moment,

Scalar–tensor–vector gravity (STVG) is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario. The theory is also often referred to by the acronym MOG.

In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux.

In continuum mechanics, a compatible deformation tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

In mathematical physics, the Belinfante–Rosenfeld tensor is a modification of the energy–momentum tensor that is constructed from the canonical energy–momentum tensor and the spin current so as to be symmetric yet still conserved.

The Maxwell–Bloch equations, also called the optical Bloch equations describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. They are analogous to the Bloch equations which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made.

Gauge theory gravity (GTG) is a theory of gravitation cast in the mathematical language of geometric algebra. To those familiar with general relativity, it is highly reminiscent of the tetrad formalism although there are significant conceptual differences. Most notably, the background in GTG is flat, Minkowski spacetime. The equivalence principle is not assumed, but instead follows from the fact that the gauge covariant derivative is minimally coupled. As in general relativity, equations structurally identical to the Einstein field equations are derivable from a variational principle. A spin tensor can also be supported in a manner similar to Einstein–Cartan–Sciama–Kibble theory. GTG was first proposed by Lasenby, Doran, and Gull in 1998 as a fulfillment of partial results presented in 1993. The theory has not been widely adopted by the rest of the physics community, who have mostly opted for differential geometry approaches like that of the related gauge gravitation theory.

The optical metric was defined by German theoretical physicist Walter Gordon in 1923 to study the geometrical optics in curved space-time filled with moving dielectric materials.

In quantum mechanics, magnetic resonance is a resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum (azimuthal) quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.

In mathematical physics, the Gordon decomposition of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

Adiabatic radio frequency (RF) pulses are used in magnetic resonance imaging (MRI) to achieve excitation that is insensitive to spatial inhomogeneities in the excitation field or off-resonances in the sampled object.

In mathematics, Rathjen's $psi function$ is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals $to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below is closed under . Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.$

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.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[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.

[1]

[2]

[3]