The relative permeability µ_{r} 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 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, or )coil or inductor) with length l, crosssectional area A, and number of turns t, that is "filled" with a magnetic material with µ_{r}.  
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, µ_{r} = 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. 
 
Note: In contrast to dielectrics, we define an additional quantity, the magnetization M by simply including dividing J by µ_{o}.  
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. 
 
The equivalent of "Ohm's law", linking current density to field strength in conductors is the magnetic Polarization law: 
 
The decisive material parameter is c_{mag} = (µ_{r} – 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. 
 
Magnetic moments originate from:  
The intrinsic magnetic dipole moments m of elementary particles with spin is measured in units of the Bohr magnetonm_{Bohr}. 
 
The magentic moment m^{e} 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 magentic 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 diamagmetic material. 
 
In the order of a few Bohr magnetons  we have a essentially a paramagnetic material. 
 
In some ferromagnetic materials spontaneous ordering of magenetic moments occurs below the Curie (or Neél) temperature. The important familiess are  

 
There is characteristic temperatuer dependence of µ_{r} for all cases  
Dia and Paramagentic propertis of materials are of no consequence whatsoever for products of electrical engineering (or anything else!) 
 
Only their common denominator of being essentially "nonmagnetic" is of interest (for a submarine, e.g., you want a nonmagnetic steel)  
For research tools, however, these forms of magnitc behavious can be highly interesting ("paramagentic resonance")  
Diamagnetism can be understood in a semiclassical (Bohr) model of the atoms as the response of the current ascribed to "circling" electrons to a changing magnetic field via classical induction (µ dH/dt). 
 
The net effect is a precession of the circling electron, i.e. the normal vector of its orbit plane circles around on the green cone. Þ  
The "Lenz rule" ascertains that inductive effects oppose their source; diamagnetism thus weakens the magnetic field, c_{dia} < 0 must apply.  
Running through the equations gives a result that predicts a very small effect.
Þ A proper quantum mechanical treatment does not change this very much. 
 
The formal treatment of paramagnetic materuials is mathematically completely identical to the case of orientation polarization 
 
The range of realistc b values (given by largest H technically possible) is even smaller than in the case of orientation polarization. This allows tp approximate L(b) by b/3; we obtain:  
 
Insertig numbers we find that c_{para} is indeed a number just slightly larger than 0.  
In ferromagnetic materials the magnetic moments of the atoms are "correlated" or linedup, i.e. they are all pointing in the same direction 
 
The physical reason for this is a quantummechanical spinspin interaction that has no simple classical analogue.  
However, exactly the same result  complete lineup  could be obtained, if the magnetic moments would feel a strong magnetic field.  
In the "mean field" approach or the "Weiss" approach to ferromagnetism, we simply assume such a magnetic field H_{Weiss} to be the cause for the lineup of the magnetic moments. This allows to treat ferromagnetism as a "special" case of paramagnetism, or more generally, "orientation polarization".  
For the magnetization we obtain Þ 
 
The term w · J describes the Weiss field via  
Unfortunately the resulting equation for J, the quantity we are after, cannot be analytically solved, i.e. written down in a closed way.  
Graphical solutions are easy, however Þ 
 
From this, and with the usual approximation for the Langevin function for small arguments,
we get all the major ferromagnetic properties, e.g.
 
As it turns out, the Weiss field would have to be far stronger than what is technically achievable  in other words, the spinspin interaction can be exceedingly strong!  
In single crystals it must be expected that the alignments of the magnetic moments of the atom has some preferred crystallographic direction, the "easy" direction. 
 
A single crystal of a ferromagnetic material with all magnetic moments aligned in its easy direction would carry a high energy because: 
 
It would have a large external magnetic field, carrying field energy.  
In order to reduce this field energy (and other energy terms not important here), magnetic domains are formed Þ. But the energy gained has to be "payed for" by:  
Energy of the domain walls = planar "defects" in the magnetization structure. It follows: Many small domains —> optimal field reduction —> large domain wall energy "price".  
In polycrystals the easy direction changes from grain to grain, the domain structure has to account for this.  
In all ferromagnetic materials the effect of magnetostriction (elastic deformation tied to direction of magnetization) induces elastic energy, which has to be minimized by producing a optimal domain structure.  
The domain structures observed thus follows simple principles but can be fantastically complicated in reality Þ.  
For ferromagnetic materials in an external magnetic field, energy can be gained by increasing the total volume of domains with magnetization as parallel as possible to the external field  at the expense of unfavorably oriented domains. 
 
Domain walls must move for this, but domain wall movement is hindered by defects because of the elastic interaction of magnetostriction with the strain field of defects.  
Magnetization curves and hystereses curves result Þ, the shape of which can be tailored by "defect engineering".  
Domain walls (mostly) come in two varieties:
 
 
Depending on the shape of the hystereses curve (and described by the values of the remanence M_{R} and the coercivity H_{C}, we distinguish hard and soft magnets Þ. 
 
Tailoring the properties of the hystereses curve is important because magnetic losses and the frequency behavior is also tied to the hystereses and the mechanisms behind it.  
Magnetic losses contain the (trivial) eddy current losses (proportional to the conductivity and the square of the frequency) and the (notsotrivial) losses proportional to the area contained in the hystereses loop times the frequency.  
The latter loss mechanism simply occurs because it needs work to move domain walls.  
It also needs time to move domain walls, the frequency response of ferromagnetic materials is therefore always rather bad  most materials will not respond anymore at frequencies far below GHz.  
Uses of ferromagnetic materials may be sorted according to:  
Soft magnets; e.g. Fe  alloys 
 
Hard magnets; e.g. metal oxides or "strange" compounds. 
 
Even so we have essentially only Fe, Ni and Co (+ Cr, O and Mn in compounds) to work with, innumerable magnetic materials with optimized properties have been developed. 
 
New complex materials (including "nano"materials) are needed and developed all the time.  
Data storage provides a large impetus to magnetic material development and to employing new effects like "GMR"; giant magneto resistance; a purely quantum mechanical effect.  

© H. Föll (Electronic Materials  Script)