
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 

J  = 
N · m · µ_{0} · L(b) 
= 
N · m · µ_{0} · L 
æ è 
m · µ_{0} · (H + w · J)
kT_{ }  ö ø





The term w · J describes the Weiss field via H_{loc}
= H_{ext} + w · J; the Weiss factor w is the decisive (and unknown)
parameter of this approach.  


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.  Saturation field strength.
 Curie temperature T_{C}.
T_{C}  = 
N · m
^{2} · µ_{0}^{2} · w 3k 

 Paramagnetic behavior above the Curie temperature.
 Strength of spinspin interaction via determining w from T_{C}.
 


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. 

Easy directions: Fe (bcc) <100> Ni (fcc) <111>
Co (hcp) <001> (cdirection) 

 
 

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:
 Bloch walls, usually found in bulk materials.
 Neél walls, usually found in thin films.
 


 

 
 

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. 



