Chemistry Reference
In-Depth Information
elementary excitation energies and, therefore, predicts that the spin align-
ment approaches perfect ordering more rapidly as
T
0 than is observed
experimentally. We leave it as an exercise to the end of this chapter to show
that in mean field theory the magnetisation
M
approaches the saturation
value,
M
s
as
→
e
−
β/
T
M
(
T
)
=
M
s
(
1
−
)
(7.31)
where
β
is a material constant. Experimentally,
it is found that
M
T
3
/
2
. The experi-
mental data can be understood by considering the energetics of magnon
excitation.
Using eqs (7.30) and (5.10) it can be shown that the number of allowed
magnon modes per unit volume with frequency between
approaches
M
s
much more slowly, as
M
(
T
)
=
M
s
−
α
ω
and
ω
+
d
ω
is
given by
2
Js
2
a
2
3
/
2
1
1
/
2
d
g
(ω)
d
ω
=
ω
ω
(7.32)
2
4
π
where
g
is the magnon density of states. The average number of
magnons,
n
at temperature
T
for a mode of frequency
(ω)
ω
is given by the
Bose-Einstein distribution function, as
1
e
ω/
kT
n
(ω)
=
(7.33)
−
1
The number of magnons excited per unit volume,
N
, is then given by
∞
N
=
n
(ω)
g
(ω)
d
ω
0
C
∞
0
1
/
2
ω
=
1
d
ω
(7.34)
e
ω/
kT
−
If we make the change of variable
x
=
ω/
kT
, the integral in eq. (7.34) can
be rewritten as
2
∞
0
x
1
/
2
e
x
3
/
3
/
2
N
=
C
(
kT
)
1
d
x
=
2.32
C
(
kT
)
(7.35)
−
so that the number of magnons excited per unit volume increases as
T
3
/
2
.
As each magnon reduces the overall magnetic moment by
g
µ
B
, the net
T
3
/
2
, as observed
magnetisation is indeed found to vary as
M
(
T
)
=
M
s
−
α
experimentally.
Mean field theory also breaks down near the Curie temperature,
T
c
.
We have seen that the susceptibility above
T
c
is predicted to vary as
