Environmental Engineering Reference
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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 )=
α k
G 2 ( k )=
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 )=
α
k
G 4 ( k )=
α
k
{
E k ( T )
}
G 1 ( k )
W k {
G 3 ( k )+ G 4 ( k )
}
=1
{
+ E k ( T )
}
G 2 ( k )
W k {
G 3 ( k )+ G 4 ( k )
}
=1
(5 . 4 . 30)
{
k }
G 3 ( k )+ W k {
G 1 ( k )
G 2 ( k )
}
=0
{
+ k }
G 4 ( k )
W k {
G 1 ( k )
G 2 ( k )
}
=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 )
G 2 ( k )
( ) 2
( k ) 2
=2 E k ( T )
{
}
/
D
( k ) ,
(5 . 4 . 31)
where the denominator is
( ) 2
E k ( T )
( ) 2
( k ) 2
4 W k k E k ( T ) . (5 . 4 . 32)
D
( k )=
{
}{
}−
In a similar way, introducing the appropriate Green functions, we find
= 2 E k ( T )
+8 W k k /
α k + α +
k ; α k + α k
( ) 2
( k ) 2
{
}
D
( k ) .
(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 k /k 2 ,
(5 . 4 . 34)
D
( k ) can be written
and the coupling term in
4 W k k E k ( T )=
( k ) 2 Λ γ ,
{
A k ( T )+ B k ( T )
}
(5 . 4 . 35)
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