 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. 
 
Questionaire  Multiple Choice questions to 4.3.1 
