8.5 Paramagnetism in Spin Systems
451
which better illustrates that the summation in Eq. (8.5.37) is a finite geometric
progression. By starting from this expression for z mag (T , B), we may, with a little
manipulation, obtain a closed-form expression for it, namely
z mag (T , B) = e
βg J μ B BJ (e −βg J μ B B ) 2J +1 − 1
e −βg J μ B B − 1
.
(8.5.44)
We may further rearrange this expression into the more symmetric form
z mag (T , B) =
(e βg J μ B B )
J +
1
2 − (e βg J μ B B )
−J −
1
2
(e βg J μ B B )
1
2 − (e βg J μ B B )
−
1
2
,
(8.5.45)
which specifically suggests that we may employ the definition of the hyperbolic sine
function to rewrite it as
z mag (T , B) =
sinh[
1
2 β(2J + 1)g J μ B B]
sinh(
1
2 βg J μ B B)
.
Magnetization in a bulk sample is simply the net magnetic moment associated
with the sample, i.e., it is given by the average magnetic moment of a single
paramagnetic atom or ion multiplied by the number of such ions in the bulk sample.
Thus, we may write the electronic magnetization M for a bulk sample as
M(T , B) = Nμ J (T , B) = −Nk B T
∂ ln z mag (T , B)
∂B
T
.
(8.5.46)
If we are working with a solid sample, the atoms or ions will be localized (at lattice
sites in the solid), and hence distinguishable, so that the appropriate form for the
canonical partition function for a solid sample containing N atoms or ions is
Z N (T , B) = [z mag (T , B)]
N
=
sinh
N
[
1
2 β(2J + 1)g J μ B B]
sinh
N (
1
2 βg J μ B B)
.
(8.5.47)
We can now make use of this result to rewrite our expression for the electronic bulk
magnetization M(N, T , B) as 7
M(N, T , B) = −k B T
∂ ln Z N (T , B)
∂B
T ,N
.
(8.5.48)
7 Note that the negative sign here arises specifically because we have explicitly taken the sign of
the electronic magnetic dipole moment into account in Eqs. (8.5.36) and (8.5.43). For nuclear spin
magnetization the overall sign in Eq. (8.5.48) would be positive.