Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 233
and the isothermal susceptibility χ, which is defined as
∂M
χ = . (5.10)
∂B T
The susceptibility χ is a measure of the change of the magnetization due to the change in the
external magnetic field and is another example of a response function.
We will frequently omit the factor of µ in (5.9) so that M becomes the number of spins
pointing in a given direction minus the number pointing in the opposite direction. Often it is more
convenient to workwith the mean magnetization per spin m, an intensive variable, which is defined
as
m = 1
M. (5.11)
N
As for the discussion of the heat capacity and the specific heat, the meaning of M and m will be
clear from the context.
We can express M and χ in terms of derivatives of lnZ by noting that the total energy can
be expressed as
E = E0 −MB, (5.12)
where E0 is the energy of interaction of the spins with each other (the energy of the system when
B = 0) and −MB is the energy of interaction of the spins with the external magnetic field. (For
noninteracting spins E0 = 0.) The form of E in (5.12) implies that we can write Z in the form
Z =
e −β(E0,s−MsB) , (5.13)
where Ms and E0,s are the values of M and E0 in microstate s. From (5.13) we have
∂Z
∂B
and hence the mean magnetization is given by
M = 1
Mse
Z
−β(E0,s−MsB)
s
= βMse −β(E0,s−MsB) , (5.14)
s
= 1
βZ
s
∂Z
∂B
If we substitute the relation F = −kT lnZ, we obtain
(5.15a)
∂lnZN
= kT . (5.15b)
∂B
M = − ∂F
. (5.16)
∂B
Problem 5.2. Relation of the susceptibility to the magnetization fluctuations
Use considerations similar to that used to derive (5.15b) to show that the isothermal susceptibility
can be written as
χ = 1
kT [M2 −M 2 ] . (5.17)
Note the similarity of the form (5.17) with the form (4.88) for the heat capacity CV.