Chemistry Reference
In-Depth Information
6.7 Brillouin (quantum mechanical) theory of
paramagnetism
The Langevin theory we considered earlier provides a good description
of the behaviour of paramagnetic materials. It describes accurately the
response to a weak magnetic field, when
x
kT
is small, and also
reproduces the saturation behaviour for large
x
, where all the magnetic
moments tend to align with the applied field,
B
. However, it tends
to underestimate the net magnetisation
M
for intermediate values of
x
.
The behaviour in this region can be reproduced more accurately using
Brillouin theory, which takes account of the quantised nature of the angu-
lar momentum and magnetic moment. For an ion whose total angular
momentum quantum number is
J
, the allowed values for the compo-
nent of magnetic moment along the field direction are given by eq. (6.30),
m
z
=
mB
/
=−
g
µ
B
J
z
, where
J
z
is an integer or half-integer, which takes values
−
1,
J
. We canuse a technique similar to the one used before
to calculate the net magnetisation
M
in the field
B
. We again assume that
there are
N
independent paramagnetic ions per unit volume. The number
of ions,
J
,
−
J
+
1,
...
,
J
−
N
(
J
z
)
, with magnetic moment component
m
z
=−
g
µ
B
J
z
along
the field direction is given by
C
N
e
m
z
B
/
kT
N
(
J
z
)
=
(6.32)
C
N
e
−
g
µ
B
J
z
B
/
kT
=⇒
N
(
J
z
)
=
where
C
N
is again a constant of proportionality, and exp
is the
Boltzmann probability function. We find, summing over the allowed total
angular momentum values that
(
m
z
B
/
kT
)
J
e
−
g
µ
B
J
z
B
/
kT
N
=
C
N
(6.33)
J
z
=−
J
which can be re-arranged to give
N
C
N
=
(6.34)
J
J
z
=−
J
e
−
g
µ
B
J
z
B
/
kT
The net magnetisation,
M
, in the field direction can then be found by
summing over the allowed energy levels
J
M
=
N
m
=
J
−
g
µ
B
J
z
N
(
J
z
)
J
z
=−
