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and, when
h
y
=0,
F
φφ
E
0
(
T
)
χ
xx
(
0
,ω
)=
1
(
hω
)
2
,
(5
.
3
.
12
b
)
N
−
where the uniform-mode energy is
F
θθ
F
φφ
1
/
2
.
1
E
0
(
T
)=
(5
.
3
.
13)
N
J
z
This result for the uniform mode in an anisotropic ferromagnet was
derived by Smit and Beljers (1955). It may be generalized to an arbitrary
magnetization direction by defining (
θ, φ
) to be in a coordinate system
in which the polar axis is
perpendicular
to the
z
-axis (as is the case here),
and by replacing
F
θθ
F
φφ
by
F
θθ
F
φφ
−
F
θφ
if
F
θφ
=0.
The introduction of the averaged effective-field in (5.3.8) corre-
sponds to the procedure adopted in the RPA, and a comparison of the
results (5.3.12-13) with the RPA result (5.2.40), at
q
=
0
and
ω
=0,
shows that the relations
1
A
0
(
T
)
−
B
0
(
T
)=
F
φφ
N
J
z
(5
.
3
.
14)
1
A
0
(
T
)+
B
0
(
T
)=
F
θθ
N
J
z
must be valid to second order in 1
/J
. In this approximation,
A
0
(
T
)
±
B
0
(
T
) are directly determined by that part of the time-averaged two-
dimensional potential, experienced by the single moments, which is
quadratic in the components of the moments perpendicular to the mag-
netization axis. The excitation energy of the uniform mode is thus pro-
portional to the geometric mean of the two force constants characterizing
the parabolic part of this potential. Since
A
0
(
T
)
±
B
0
(
T
) are parameters
H
in (5.3.6), which are
not known, appear only in order 1
/J
3
in (5.3.14), when the magnetiz-
ation is along a high-symmetry direction.
B
2
of order 1
/J
, the second-order contributions of
B
0
(
T
), and this is in accordance
with eqn (5.3.14), as
Q
2
is independent of
φ
. Considering instead the
θ
-dependence, we find that the contribution to
F
θθ
is determined by
∂
2
Q
2
does not appear in
A
0
(
T
)
−
∂θ
2
=
−
6(
J
z
− J
x
)cos2
θ −
6(
J
z
J
x
+
J
x
J
z
)sin2
θ
θ
=
π/
2
=3
O
2
−
O
2
.
(5
.
3
.
15)
From (5.2.10) and (5.2.11), the thermal average is found to be
=2
J
(2)
1
3
J
a
+
a
+
3
2
J
2
O
2
−
O
2
a
+
a
+
aa
−
4
J
2
(
a
+
aaa
+
a
+
a
+
a
+
a
)
,
1
2
J
(1 +
1
1
4
J
)(
aa
+
a
+
a
+
)+
−
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