Environmental Engineering Reference
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neglecting a term proportional to
σ
2
Q
in the denominator.
The 3
Q
-
component is thus proportional to
σ
3
Q
, and hence to (
T
N
−
T
)
3
/
2
.De-
noting the wave-vector at which
(
q
) has its maximum by
Q
0
,wecon-
clude that the appearance of the third harmonic implies that
f
has its
minimum at a value of
Q
slightly different from
Q
0
. Minimizing the free
energy with respect to
Q
along the
c
-axis, by requiring
∂f/∂Q
=0,we
obtain to leading order
J
σ
3
Q
σ
Q
2
3
J
(3
Q
0
)
J
(
Q
0
)
Q
=
Q
0
−
.
(2
.
1
.
35
b
)
J
(
Q
0
) is negative, so the shift
Q
−
Q
0
has the same sign as
J
(3
Q
0
)
T
)
2
. The other special case, 3
q
=
Q
,re-
and is proportional to (
T
N
−
flects the possibility that, if
(
Q
0
), the system may
reduce its energy by making a
first order
transition to a state where
Q
Q
0
/
3 is the fundamental wave-vector, with the third harmonic be-
ing close to
Q
0
. The presence of a term in the free energy cubic in the
order parameter,
σ
Q
/
3
in this case, implies that the transition becomes
of first order, so that the order parameter changes discontinuously from
zero to a finite value. The
Q
0
/
3-transition appears to be of no impor-
tance in real systems, so we shall return to the discussion of the other
case. If the free energy is expanded to higher (even) powers in the rel-
ative magnetization, it is clear that the (2
n
+ 2)-power term leads to a
contribution proportional to
σ
(2
n
+1)
Q
σ
2
n
+
Q
which, in combination with
the term quadratic in
σ
(2
n
+1)
Q
, implies that the ordering at the fun-
damental wave-vector
Q
induces a (2
n
+ 1)-harmonic proportional to
σ
2
n
+1
Q
J
(
Q
0
/
3) is close to
J
T
)
(2
n
+1)
/
2
. Starting as a pure sinusoidally modulated
wave at
T
N
, the moments approach the
square wave
∝
(
T
N
−
π
cos
x
x
=
Q
·
R
i
+
ϕ
,
(2
.
1
.
36
a
)
4
J
−
3
cos 3
x
+
5
−
7
J
iζ
cos 5
x
cos 7
x
+
···
=
in the limit of zero temperature where
J
, neglecting strong
anisotropy effects. Although the behaviour of the angular momentum
is simple, the dependence of the free energy on the wave-vector is com-
plicated. It is only when the ordering is incommensurable, i.e.
mQ
is
different from any multiple of the length 4
π/c
of the reciprocal-lattice
vector along the
c
-axis, that the energy of the square-wave structure at
T
=0is
J
iζ
=
±
4
J
2
π
2
J
···
.
(
Q
)+
1
(3
Q
)+
1
f
(0) =
H
cf
−
9
J
25
J
(5
Q
)+
(2
.
1
.
36
b
)
An infinitesimal change of the ordering wave-vector from
Q
, which min-
imizes
f
(0), to
Q
c
may make it commensurable with the lattice, so that
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