
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. 






