### 4.1.4 Summary to: Magnetic Materials - Definitions and General Relations

The relative permeability µr of a material "somehow" describes the interaction of magnetic (i.e. more or less all) materials and magnetic fields H, e.g. vial the equations Þ
B   =  µo · µr · H

L  =  µ0 · µr · A · w2
l

n  = (er· µr)½
B is the magnetic flux density or magnetic induction, sort of replacing H in the Maxwell equations whenever materials are encountered.
L is the inductivity of a linear solenoid (also called coil or inductor) with length l, cross-sectional area A, and number of turns t, that is "filled" with a magnetic material with µr.
n is still the index of refraction; a quantity that "somehow" describes how electromagnetic fields with extremely high frequency interact with matter.
For all practical purposes, however, µr = 1 for optical frequencies

Magnetic fields inside magnetic materials polarize the material, meaning that the vector sum of magnetic dipoles inside the material is no longer zero.
The decisive quantities are the magnetic dipole moment m, a vector, and the magnetic Polarization J, a vector, too.
B  =  µo · H  +  J

J  =  µo · Sm
V

M  =  J
µo
Note: In contrast to dielectrics, we define an additional quantity, the magnetization M by simply including dividing J by µo.
The magnetic dipoles to be polarized are either already present in the material (e.g. in Fe, Ni or Co, or more generally, in all paramagnetic materials, or are induced by the magnetic fields (e.g. in diamagnetic materials).
The dimension of the magnetization M is [A/m]; i.e. the same as that of the magnetic field.

The magnetic polarization J or the magnetization M are not given by some magnetic surface charge, because Þ.
 There is no such thing as a magnetic monopole, the (conceivable) counterpart of a negative or positive electric charge

The equivalent of "Ohm's law", linking current density to field strength in conductors is the magnetic Polarization law:
M  =  r - 1) · H

M  :=  cmag · H

B  =  µo · (H + M)
The decisive material parameter is cmag = (µr – 1) = magnetic susceptibility.
The "classical" induction B and the magnetization are linked as shown. In essence, M only considers what happens in the material, while B looks at the total effect: material plus the field that induces the polarization.

Magnetic polarization mechanisms are formally similar to dielectric polarization mechanisms, but the physics can be entirely different.
 Atomic mechanisms of magnetization are not directly analogous to the dielectric case
Magnetic moments originate from:
The intrinsic magnetic dipole moments m of elementary particles with spin is measured in units of the Bohr magnetonmBohr.
mBohr  =   h · e
4p · m*e
=  9.27 · 10–24 Am2

me   =   2 · h · e · s
4p · m*e
=  2 · s · m Bohr   =  ± mBohr
The magnetic moment me of the electron is Þ
Electrons "orbiting" in an atom can be described as a current running in a circle thus causing a magnetic dipole moment; too
The total magnetic moment of an atom in a crystal (or just solid) is a (tricky to obtain) sum of all contributions from the electrons, and their orbits (including bonding orbitals etc.), it is either:
Zero - we then have a diamagnetic material.
 Magnetic field induces dipoles, somewhat analogous to elctronic polarization in dielectrics. Always very weak effect (except for superconductors) Unimportant for technical purposes
In the order of a few Bohr magnetons - we have a essentially a paramagnetic material.
 Magnetic field induces some order to dipoles; strictly analogous to "orientation polarization" of dielectrics. Always very weak effect Unimportant for technical purposes

In some ferromagnetic materials spontaneous ordering of magnetic moments occurs below the Curie (or Neél) temperature. The important families are
• Ferromagnetic materials ÝÝÝÝÝÝÝ
large µr, extremely important.
• Ferrimagnetic materials ÝßÝßÝßÝ
still large µr, very important.
• Antiferromagnetic materials ÝßÝßÝßÝ
µr » 1,
unimportant

 Ferromagnetic materials: Fe, Ni, Co, their alloys "AlNiCo", Co5Sm, Co17Sm2, "NdFeB"

There is characteristic temperature dependence of µr for all cases
 Questionnaire Multiple Choice questions to all of 4.1

© H. Föll (Electronic Materials - Script)