Environmental Engineering Reference
In-Depth Information
order in
σ
p
to be
f
2
(
σ
p
)=
4
J
2
p
{
σ
p
}
,
Q
p
)
2
σ
p
+
2
A
ξ
−J
(
Q
p
)
}
K
(
Q
p
)
{
2(
σ
p
·
−
(2
.
1
.
42)
where
Q
p
(
q
)is
found at
q
=
Q
along the
η
-axis, or the other equivalent
b
-axes, with
Q
being about one fourth of the distance to the Brillouin-zone boundary,
and
=
Q
p
/Q
p
. In Pr and Nd, the maximum of
J
(
q
)
±K
(
Q
) is negative. The transition between the paramagnetic phase
and a phase described by (2.1.40), with
J
p
lying in the hexagonal plane,
then occurs when the coecient 2
A
ξ
−J
K
(
Q
) vanishes, at which
temperature the corresponding factor for the
c
-component of the mo-
ments, 2
A
ζ
−J
(
Q
)+
K
(
Q
), is still positive in Pr and Nd. Besides confining the
moments to the hexagonal planes,
K
(
Q
) also removes the degeneracy
between the two states in which
J
p
is parallel or perpendicular to
Q
p
.
With a negative
(
Q
), the anisotropic coupling favours a longitudinal
ordering of the moments at
T
N
,with
J
p
parallel to
Q
p
.Justbelow
T
N
,
the magnitude of the ordered moments is determined by
f
2
(
σ
p
), together
with the fourth-order contributions. When the moments lie in the basal
plane (
B
=
B
ξξ
=
B
ηη
=
B
ξη
), we obtain, from eqn (2.1.22),
f
4
(
σ
p
)=
B
1
K
N
i
J
i
·
J
i
2
=
BJ
4
8
σ
p
·
σ
p
)
2
.
(2
.
1
.
43)
σ
p
σ
p
+2(
σ
p
+
4
p
p
=
p
Introducing the effective order parameter
σ
, defined by
σ
2
=
p
σ
p
,we
obtain further:
f
J
2
2
A
ξ
−J
(
Q
)
σ
2
+
8
f
2
(
σ
p
)+
f
4
(
σ
p
)=
4
J
4
Bσ
4
,
(2
.
1
.
44)
assuming
J
p
parallel to
Q
p
along the three
b
-axes making an angle of
120
◦
with each other (
Q
p
·
K
(
Q
)+
Q
p
=
p
). Hence the free
energy, in this approximation, is independent of whether the ordering is
single-, double- or triple-
Q
. Instead of utilizing (2.1.22), we may appeal
to symmetry arguments, by which the fourth-order term may readily be
seen to have the general form
f
4
(
σ
p
)=
u
p
=
−
1
/
2when
p
v
p
=
p
σ
p
+
2
σ
p
σ
p
,
(2
.
1
.
45
a
)
σ
p
vectors remain at 120
◦
(Bak and Lebech 1978). Introducing the parameter
w
as long as the angles between the different
≡
v
−
2
u
,wemay
write this:
f
4
(
σ
p
)=
u
p
w
p
=
p
σ
p
2
+
2
σ
p
σ
p
=(
u
+
γw
)
σ
4
,
(2
.
1
.
45
b
)
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