Environmental Engineering Reference
In-Depth Information
of neighbouring ions, instead of the local strains. Evenson and Liu (1969)
have devised a simple procedure for replacing the local-strain variables in
the magnetoelastic Hamiltonian with the relative displacements of the
neighbouring ions. Using their procedure, and assuming the nearest-
neighbour interactions to be dominant, we find that eqn (5
.
4
.
41
b
)is
replaced by
A
k
(
T
)
2
H
ε
,
(5
.
4
.
43)
c
ε
√
N
2
c
sin (
kc/
2)
F
k
,
B
k
(
T
)
2
JσE
k
(
T
)
−
−
4
W
k
(
ε
)=
when
k
is along the
c
-axis.
c
is the lattice constant and
F
k
,
is the com-
ponent of
F
k
parallel to the magnetization vector, which is only non-zero
for one of the transverse-phonon modes. This interaction does not dis-
tinguish between the two sublattices in the hcp crystal. This means
that
W
k
(
ε
) only couples the magnons with the phonons at a certain
k
if
the modes are either both acoustic or both optical, consistent with the
double-zone representation in the
c
-direction. Except for the replace-
ment of (5
.
4
.
41
b
) by (5.4.43), the interaction Hamiltonian (5
.
4
.
41
a
)is
unchanged. From the equations of motion of the Green functions, we
may derive the susceptibilities, when
k
is along the
c
-direction, in the
same way as before, eqns (5.4.29-36), and the results are found to be:
(
hω
t
k
)
2
(
hω
)
2
χ
xx
(
k
,ω
)=
Jσ
{
A
k
(
T
)
−
B
k
(
T
)
}{
−
}
/
D
ε
(
k
,ω
)
χ
yy
(
k
,ω
)=
Jσ
{
A
k
(
T
)+
B
k
(
T
)
}
×
(
hω
t
k
)
2
4
W
k
(
ε
)
hω
t
k
/E
k
(
T
)
/
(
hω
)
2
−
−
D
ε
(
k
,ω
)
,
(5
.
4
.
44)
with
4
W
k
(
ε
)
hω
t
k
E
k
(
T
)
,
(5
.
4
.
45)
where
ω
t
k
is the angular frequency of the transverse phonon mode at
k
.
Introducing the parameter
E
k
(
T
)
(
hω
)
2
(
hω
t
k
)
2
(
hω
)
2
D
ε
(
k
,ω
)=
{
−
}{
−
}−
Υ
k
=
1+
16
hω
t
k
E
k
(
T
)
W
k
(
ε
)
2
,
(5
.
4
.
46)
{
E
k
(
T
)
−
(
hω
t
k
)
2
}
2
we find the poles in the susceptibilities at
=
±
2
E
k
(
T
)+(
hω
t
k
)
2
±
2
E
k
(
T
)
−
(
hω
t
k
)
2
Υ
k
2
,
(5
.
4
.
47
a
)
hω
=
±E
k
corresponding to
(
E
k
)
2
(
hω
)
2
(
E
k
)
2
(
hω
)
2
D
ε
(
k
,ω
)=
{
−
}{
−
}
.
(5
.
4
.
47
b
)
Search WWH ::
Custom Search