Chemistry Reference
In-Depth Information
Substituting eq. (7.18) in (7.20) then gives
H
0
M
C
T
1
2
λ
M
=
−
(7.21)
which can be re-arranged as
C
=
M
2
H
0
(7.22)
T
+
λ
C
/
with the susceptibility
χ
now equal to
C
χ
=
(7.23)
T
+
T
N
where
T
N
2 is referred to as the Néel temperature.
The onset of spontaneous magnetisation occurs when we can get finite
sublattice magnetisation in zero applied field. Setting
H
0
=
λ
C
/
=
0 in eqs (7.18)
and (7.19), this occurs when
1
2
C
T
λ
1
2
C
T
λ
M
A
=−
M
B
M
B
=−
M
A
(7.24)
Both parts of eq. (7.24) can be satisfied for finite
M
A
and
M
B
if
T
=
T
N
, with mean field theory then predicting spontaneous ordering at the
Néel temperature, and for
T
=
λ
C
/
2
T
N
.
We can calculate the spontaneous magnetisation of the two sublattices
below
T
N
using a similar method to the ferromagnetic case. We apply, for
instance, the Langevin (classical) equation for paramagnetism to one of the
two sublattices. We write
<
2
m
0
coth
x
A
N
1
x
A
M
A
=
−
(7.25)
with the second equation linking
x
A
and
M
A
derived using the fact that
M
B
=−
M
A
:
=
µ
0
m
0
F
A
kT
=
µ
0
m
0
(
−
λ
M
B
)
=
µ
0
m
0
λ
M
A
x
A
(7.26)
kT
kT
Equations (7.25) and (7.26) are equivalent to eqs (7.15) and (7.17) in the
ferromagnetic case, implying that the zero-field ordering on each sublattice
in an antiferromagnet has a similar temperature dependence to that of a
ferromagnet.
Although an antiferromagnet has no net magnetisation (
M
=
0in
zero applied field), antiferromagnetic ordering has been widely studied
experimentally, particularly using neutron diffraction. Each neutron has
