Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 16. Statistical thermodynamics models 151
here, but such complications are not needed for the description we need here. The corresponding
models are appropriate; although one might have somewhat more complicated
expressions for spin couplings. But what is important here is that when one takes the next step,
one should consider how electron spins may influence each other, when quantum mechanics
tells that the spins and the spatial distribution of the wave functions couple to each other
to provide a certain symmetry. This means that spins couple to each other, not because of
any direct, possibly magnetic fields but because the electron distributions, manifested in their
wave functions interact. And this can lead to strong effects, which can mean the magnetic
states of iron and some other magnetic materials.
Here we do not need to go further with the fundamentals of that—it forms rather complicated
parts of quantum mechanics, but we will discuss the information that there is a
coupling between electron spins and then also between magnetic moments that enhance
magnetic effects.
Our next step is to include such interactions in the formalism, and we shall go to two
rather different types of models. In the first, which is the simplest and maybe the one that best
tells what can happen, one starts with a view that one particular magnetic moment will feel
an influence of all other magnetic moments, which is a kind of mean field, that is an addition
to the external magnetic field, which is proportional to the total magnetisation, i.e. to the
difference of magnetic moments in the two possible directions according to our previous view.
We don’t change the basis of that.
Thus, for our model, let there be a term cM added to the magnetic field, B, of the previous
relations. We simply put this in our expression and get new expressions as:
BcM
M Nm
1
2
kT ⎡
B ( 1
MNm / ) ⎤
ln
m ⎣
⎢ ( 1
MNm / ) ⎦
⎥
exp( 2( B
cM)
m/
kBT
) 1
( B cM)
m
Nm tanh
exp( ( BcM) m/
k T)
2 1
kT
B
or
B
(16.7)
These are implicit relations, and we cannot get simple analytic expressions for general
situations. We may, however, start with the same linear approximation we had above, which
for this generalised expression becomes:
⎛ Nm2
⎞
M
B cM which can be rewritten as
⎝⎜
kT B ⎠⎟ ( ), :
Nm2/
kB
M B
T cNm2/
k
B
(16.8)
It has a similar form as eq. (16.6) but the temperature factor is now different. Now it grows more
strongly when the temperature decreases and it diverges at a temperature equal to cNm 2 /k B .