Chemistry Reference
In-Depth Information
Eq. (7.17), T > T
c
Eq. (7.17), T< T
c
Eq. (7.15)
x
Figure 7.5
The graphical method to determine the spontaneous magnetisation of a fer-
romagnet, using eqs (7.15) (curve) and eq. (7.17) (straight line). The slope
of eq. (7.17) is proportional to the temperature. At high temperatures
(
T
>
T
c
) the straight line and curve only intersect at the origin. For
T
<
T
c
,
the two curves intersect for a finite value of the magnetisation
M
, and
spontaneous magnetisation occurs.
We rewrite eq. (7.14) as
kT
M
=
x
(7.17)
µ
0
m
0
λ
Equations (7.15) and (7.17) are both true so, for a chosen temperature,
T
,we
can plot
M
in two ways, one from each equation. Any intersections of
the two curves indicate values of
M
which are in fact solutions of both. This
procedure can then be repeated for a range of temperatures, resulting in a
graph representing the spontaneous magnetisation
M
as a function of tem-
perature. This is illustrated in fig. 7.5. At high temperatures
(
x
)
(
T
>
T
c
)
and
with zero external field, the two curves only intersect at
M
=
0. But, once
T
drops below
T
c
, the two lines cut both at
M
0 and at finite
M
, giving
a spontaneous net magnetisation. (The solution at the origin is unstable;
once any spontaneous magnetisation occurs,
M
will grow to the finite,
stable solution.) Figure 7.6 shows the calculated variation of the magneti-
sation
M
as a function of temperature below
T
c
, compared to experimental
data for Ni. The overall agreement between the two curves looks excellent,
confirming the usefulness of the mean field theory introduced here; there
are, however, minor but significant differences between the two curves
both near
T
=
=
0 and near
T
=
T
c
, to which we will return later.
7.5 Spontaneous magnetisation and susceptibility of
an antiferromagnet
The exchange interaction
J
ij
between neighbouring sites is negative in an
antiferromagnet, favouring an anti-parallel alignment of neighbouring
