
What we really want is the magnetic polarization J as a function
of the external field H. Unfortunately we have a transcendental equation
for J which can not be written down directly without a "J" on the righthand side. 


What we also like to have is the value of the spontaneous magnetization J for
no external field, i.e. for H = 0. Again, there is no analytical solution for this case. 


There is an easy graphical solution, however: We actually have two
equations for which must hold at the same time: 


The argument b of the Langevin function is 
 
b  = 
m · µ_{0} · (H + w · J) kT_{ } 




Rewritten for J, we get our first equation: 
 
J  = 
kT_{ } · b
w · m · µ_{0}  – 
H w 




This is simply a straight line with a slope and intercept value determined by the interesting
variables H, w, and T. 

On the other hand we have the equation for J, and this is our second
independent equation . 


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





This is simply the Langevin function which we know for any numerical value for b 

All we have to do is to draw both functions
in a J  b diagram 


We can do that by simply putting in some number for b and
calculating the results. The intersection of the two curves gives the solutions of the equation for J. 


This looks like this 





Without knowing anything about b, we can draw a definite
conclusion: 


For H = 0 we have two solutions (or none
at all, if the straight line is too steep): One for J = 0 and one for a rather large J.



It can be shown that the solution for J = 0 is unstable (it disappears for an
arbitrarily small field H) so we are left with a spontaneous large magnetic polarization
without an external magnetic field as the first big result of the mean field theory. 

We can do much more with the mean field theory, however. 


First, we note that switching on an external
magnetic field does not have a large effect. J increases somewhat, but for realistic values of
H/w the change remains small. 


Second, we can look at the temperature
dependence of J by looking at the straight lines. For T
® 0, the intersection point moves all the way out to infinity. This means that all
dipoles are now lined up in the field and L(b) becomes 1. We obtain the saturation value J_{sat} 
 



Third, we look at the effect of increasing temperatures.
Raising T increases the slope of the straight line, and the two points of intersection move together. When
the slope is equal to the slope of the Langevin function (which, as we
know, is 1/3), the two points of solution merge at J = 0; if we increase the slope for the straight
line even more by increasing the temperature by an incremental amount, solutions do no longer exist and the spontaneous
magnetization disappears. 

This means, there is a critical temperature
above which ferromagnetism disappears. This is, of course, the Curie temperature
T_{C}. 


At the Curie temperature T_{C}, the slope of the straight line and the
slope of the Langevin function for b = 0 must be identical. In formulas we obtain: 
 

dJ db 
=  kT_{C}
w · m · µ_{0}  = 
slope of the straight line 
dJ
db 
÷
÷ 
b = 0 
= 
N · m · µ_{0} · 
dL(b) db 
=  N · m · µ_{0}
3  




We made use of our old insight
that the slope of the Langevin function for b
® 0 is 1/3. 

Equating
both slopes yields for T_{C} 
 
T_{C}  = 
N · m
^{2} · µ_{0}^{2} · w 3k 



This is pretty cool. We did not solve an transcendental equation nor go into deep
quantum physical calculations, but still could produce rather simple equations for prime material parameters like the Curie
temerature. 


If we only would know w, the Weiss factor! Well, we do not
know w, but now we can turn the equation around: If we know T_{C}, we can calculate
the Weiss factor w and thus the fictive magnetic field that we need to
keep the spins in line. 


In Fe, for example, we have T_{C} = 1043 K, m = 2,2
· m_{Bohr}. It follows that 


H_{Weiss}  = 
w · J = 1.7 · 10^{9} A/m 




This is a truly gigantic field strength telling us that
quantum mechanical spin interactions, if existent, are not to be laughed at. 


If you do not have a feeling of what this number means, consider the unit of H:
A field of 1,7 · 10^{9} A/m is produced if a current of 1,7 · 10^{9} A flows through
a loop (= coil) with 1 m^{2} area. Even if you make the loop to cover only 1 cm^{2}, you still
need 1,7 · 10^{5} A. 

We can go one step further and approximate
the Langevin function again for temperatures >T_{C}, i.e. for b
< 1 by 
 



This yields 


J(T > T_{C}) 
» 
N · m^{2} · µ_{0}^{2} 3kT 
·  (H + w · J) 




From the equation for T_{C} we can extract w and insert it, arriving at

 
J(T > T_{C}) 
» 
N · m^{2} · µ_{0}^{2}
3k(T – T_{C}) 
·  H 




Dividing by H gives the susceptibility c for
T > T_{C} and the final formula 
 
c  = 
J H  = 
N · m^{2} · µ_{0}^{2}
3k · (T – T_{C}) 
=  const._{ }
T – T_{C} 


 



This is the famous Curie law for the paramagnetic
regime at high temperatures which was a phenomenological thing so far. Now we derived it with a theory and will therefore
call it Curie  Weiss law. 

In summary, the mean field approach ain´t that bad! It can be used for attacking
many more problems of ferromagnetism, but you have to keep in mind that it is only a description, and not based on sound
principles. 

