Chemistry Reference
In-Depth Information
so that
J
N
J
z
=−
J
e
−
g
µ
B
J
z
B
/
kT
e
−
g
µ
B
J
z
B
/
kT
M
=
J
(
−
g
µ
B
J
z
)
(6.35)
J
z
=−
We leave it as an exercise at the end of the chapter to show that eq. (6.35)
can be simplified to give
M
=
Ng
µ
B
JB
J
(
x
)
(6.36)
where
x
kT
is again a measure of the relative strength of the
magnetic and thermal energy.
B
J
=
g
µ
B
JB
/
(
x
)
is referred to as the Brillouin function
and takes the form
+
coth
(
+
)
2
J
1
2
J
1
x
2
J
coth
x
1
B
J
(
x
)
=
−
(6.37)
2
J
2
J
2
J
The behaviour of the Brillouin function is qualitatively very similar to the
Langevin function.
B
J
(
x
)
→
1 for large
x
, so that the magnetisation,
M
,
saturates at
M
=
Ng
µ
B
J
when the magnetic energy is large compared to
the thermal energy.
B
J
(
x
)
varies linearly with
x
for small
x
, where it can be
shown that
µ
0
M
=
χ
(
J
)
B
(6.38)
with the paramagnetic susceptibility,
χ
(
J
)
, given by
0
Ng
2
2
χ
(
J
)
=
µ
µ
B
J
(
J
+
1
)
(6.39)
3
kT
This is identical to the classical result for the susceptibility in eq. (6.26),
if we identify
m
=−
p
µ
B
(6.40)
2
can be thought of as the effective Bohr magneton
number. Figure 6.6 compares the experimentally determined variation of
the magnetisation,
M
, for several paramagnetic salts with the theoretically
predicted variation, using the Brillouin function. The measurements were
all carried out at low temperature (below 5 K), enabling large values of
x
1
/
=
[
(
+
)
]
where
p
g
J
J
1
kT
to be obtained. It can be seen that excellent agreement was
achieved in each case by assuming the net orbital angular momentum,
L
=
mB
/
=
0, so that the splitting factor,
g
=
2, with
J
=
S
=
3
/
2, 5/2, and 7/2
respectively.
