Environmental Engineering Reference
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where the parameter Λ
γ
is given by (5.4.10). The magnetic susceptibil-
ities can be expressed in terms of the Green functions calculated above,
using (5.2.39) and (5.2.40), and we finally arrive at
χ
xx
(
q
,ω
)=
Jσ
{
Λ
γ
(
hω
q
)
2
/
(
hω
q
)
2
(
hω
)
2
A
q
(
T
)
−
B
q
(
T
)
}{
−
}−
(
q
,ω
)
(5
.
4
.
36
a
)
D
and
(
hω
q
)
2
(
hω
)
2
χ
yy
(
q
,ω
)=
Jσ
{
A
q
(
T
)+
B
q
(
T
)
}{
−
}
/
D
(
q
,ω
)
.
(5
.
4
.
36
b
)
Because
ω
q
∝
q
and
E
0
(
T
)
>
0, it is possible to satisfy the inequality
E
q
(
T
)
hω
q
by choosing a suciently small
q
. As mentioned ear-
lier,
E
0
(
T
) is always greater than zero, if the magnetoelastic coupling is
non-zero, on account of the constant-strain term Λ
γ
. Under these cir-
cumstances the elementary-excitation energies, determined by the poles
of the susceptibilities or by
D
(
q
,ω
) = 0, are found to be
(
hω
)
2
=
E
q
(
T
)+4
W
q
hω
q
/E
q
(
T
)
(
hω
q
)
2
(5
.
4
.
37)
4
W
q
hω
q
/E
q
(
T
)
,
−
to leading order in
hω
q
/E
q
(
T
). The different excitations have become
mixed magnetoelastic modes, which mutually repel due to the magneto-
elastic coupling, and their squared energies are shifted up or down by
an equal amount. When
E
q
(
T
)
hω
q
, the change in energy of the
upper, predominantly magnon-like branch can be neglected, whereas
the frequency of the lower phonon-like mode, as obtained from (5.4.37),
using the relation (5.4.35),
1
+
O
{
4
,
Λ
γ
ω
2
=
ω
q
−
hω
q
/E
q
(
T
)
}
(5
.
4
.
38
a
)
A
0
(
T
)
−
B
0
(
T
)
may be modified appreciably relative to the unperturbed phonon fre-
quency. This relation implies that the elastic constant, relative to the
unperturbed value, as determined by the velocity of these magneto-
acoustic sound waves, is
c
66
c
66
Λ
γ
=1
−
;
q
or
⊥
J
.
(5
.
4
.
38
b
)
A
0
(
T
)
−
B
0
(
T
)
At
q
=
0
, the dynamic coupling vanishes identically and the spin-wave
energy gap is still found at
hω
=
E
0
(
T
)=
A
0
(
T
)
B
0
(
T
)
1
/
2
,withthe
{
−
}
static-strain contributions included in
A
0
(
T
)
B
0
(
T
). Due to the van-
ishing of the eigenfrequencies of the elastic waves at zero wave-vector, the
lattice cannot respond to a uniform precession of the magnetic moments
±
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