Environmental Engineering Reference
In-Depth Information
These parameters determine the (
xy
)-components of
χ
o
(
ω
) in (6.1.9),
and are valid, at least, to first order in 1
/J
. From (6.1.8), we finally
obtain
A
q
−
B
q
χ
xx
(
q
,ω
)=
J
z
A
q
−
B
q
−
(
hω
−
C
q
)
2
A
q
+
B
q
χ
yy
(
q
,ω
)=
J
z
(6
.
1
.
18)
A
q
−
B
q
−
(
hω
−
C
q
)
2
i
(
hω
−
C
q
)
χ
xy
(
q
,ω
)=
J
z
C
q
)
2
,
A
q
−
B
q
−
(
hω
−
where the parameters are now
A
q
−
−
2
J
−
2
J
}
A
q
+
B
q
=(
A
q
− B
q
)cos
2
θ
0
+
L
+
J
z
{J
(
0
)
−J
(
q
)
}
sin
2
θ
0
C
q
=
2
B
q
=
J
z
{J
(
Q
)
(
q
+
Q
)
(
q
−
Q
)
J
z
{J
(
q
−
Q
)
−J
(
q
+
Q
)
}
cos
θ
0
,
(6
.
1
.
19)
and the axial anisotropy constant is
L
=
+
f
(
u
0
)
/
J
z
{J
(
Q
)
−J
(
0
)
}
J
z
,
with
f
(
u
0
)=3
κ
2
(
T
)+
15
κ
4
(
T
)(7 cos
2
θ
0
−
1)
(6
.
1
.
20)
2
+
10
8
κ
6
(
T
)(33 cos
4
θ
0
−
18 cos
2
θ
0
+1)
.
This constant, to order 1
/J
, is that determined by the
c
-axis bulk sus-
ceptibility:
χ
ζζ
(
0
,
0) =
/L
. The dispersion relation, derived from
the pole at positive energies, is
E
q
=
C
q
+
A
q
−
J
z
B
q
1
/
2
,
(6
.
1
.
21)
which is no longer even with respect to
q
, because the parameter
C
q
changes sign, whereas
A
q
and
B
q
are unaffected, if
q
is replaced by
−
q
.
The other pole, with a minus before the square root, lies at negative
energies. If the two energies for
q
were both positive, the two poles at
−
q
would both lie at negative energies, indicating an instability of the
magnetic system. Hence in a stable cone
C
q
<A
q
−
B
q
(Cooper
et al.
1962).
The scattering cross-section of the spin waves is still determined by
(6
.
1
.
12
a
), but (6
.
1
.
12
b
) is replaced by
χ
ζζ
(
,ω
)sin
2
θ
0
κ
,ω
)=
χ
xx
(
κ
4
χ
xx
(
+
Q
,ω
)
cos
2
θ
0
,ω
)=
1
χ
ξξ
(
κ
,ω
)=
χ
ηη
(
κ
κ
−
Q
,ω
)+
χ
xx
(
κ
4
χ
yy
(
+
Q
,ω
)
+
1
κ
−
Q
,ω
)+
χ
yy
(
κ
2
χ
xy
(
+
Q
,ω
)
cos
θ
0
.
(6
.
1
.
22)
i
−
κ
−
Q
,ω
)
−
χ
xy
(
κ
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