Environmental Engineering Reference
In-Depth Information
included in
E
k
(
T
) through (5.4.11). In general,
W
k
couples all three
phonon modes with the magnons. A simplification occurs when
k
is
along the 1- or 2-axis, i.e. when
k
is either parallel or perpendicular to
the magnetization vector. In this case,
W
k
is only different from zero
when
ν
specifies the mode as a transverse phonon with its polarization
vector parallel to the basal plane. In order to analyse this situation, we
introduce the four Green functions:
α
k
;
α
k
−
α
+
−
k
;
α
k
−
G
1
(
k
,ω
)=
α
−
k
G
2
(
k
,ω
)=
α
−
k
β
k
;
α
k
−
β
+
−
k
;
α
k
−
,
(5
.
4
.
29)
where the phonon mode is as specified above (the index
ν
is suppressed).
H
mp
then leads to the following coupled equations of motion for these
Green functions:
G
3
(
k
,ω
)=
α
−
k
G
4
(
k
,ω
)=
α
−
k
{
hω
−
E
k
(
T
)
}
G
1
(
k
,ω
)
−
W
k
{
G
3
(
k
,ω
)+
G
4
(
k
,ω
)
}
=1
{
hω
+
E
k
(
T
)
}
G
2
(
k
,ω
)
−
W
k
{
G
3
(
k
,ω
)+
G
4
(
k
,ω
)
}
=1
(5
.
4
.
30)
{
hω
−
hω
k
}
G
3
(
k
,ω
)+
W
−
k
{
G
1
(
k
,ω
)
−
G
2
(
k
,ω
)
}
=0
{
hω
+
hω
k
}
G
4
(
k
,ω
)
−
W
−
k
{
G
1
(
k
,ω
)
−
G
2
(
k
,ω
)
}
=0
.
These four equations may be solved straightforwardly and, using
W
−
k
=
−
W
k
, we obtain, for instance,
α
+
−
k
;
α
k
−
α
k
−
α
−
k
=
G
1
(
k
,ω
)
−
G
2
(
k
,ω
)
(
hω
)
2
(
hω
k
)
2
=2
E
k
(
T
)
{
−
}
/
D
(
k
,ω
)
,
(5
.
4
.
31)
where the denominator is
(
hω
)
2
E
k
(
T
)
(
hω
)
2
(
hω
k
)
2
4
W
k
hω
k
E
k
(
T
)
.
(5
.
4
.
32)
D
(
k
,ω
)=
{
−
}{
−
}−
In a similar way, introducing the appropriate Green functions, we find
=
2
E
k
(
T
)
+8
W
k
hω
k
/
α
k
+
α
+
−
k
;
α
k
+
α
−
k
(
hω
)
2
(
hω
k
)
2
{
−
}
D
(
k
,ω
)
.
(5
.
4
.
33)
In this situation, the polarization factor is (
k
1
f
k
,
2
+
k
2
f
k
,
1
)=
±
k
,with
k
=
. At long wavelengths, the velocity
v
=
ω
k
/k
of the transverse
sound waves is related to the elastic constant
c
66
=
ρv
2
, and hence
|
k
|
c
γ
=4
c
66
V/N
=4
Mω
k
/k
2
,
(5
.
4
.
34)
D
(
k
,ω
) can be written
and the coupling term in
4
W
k
hω
k
E
k
(
T
)=
(
hω
k
)
2
Λ
γ
,
{
A
k
(
T
)+
B
k
(
T
)
}
(5
.
4
.
35)
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