Environmental Engineering Reference
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where
ν
=
±
1 denotes the sign of
J
2
(
0
).
J
1
(
κ
)=
J
1
(
q
) is real, whereas
J
2
(
q
)
e
i
τ
·
ρ
=
ν
e
iϕ
,
J
2
(
κ
)=
|J
2
(
q
)
|
(5
.
1
.
6
a
)
defining the phase
ϕ
=
ϕ
(
=
d
2
−
d
1
is the vector joining the
two sublattices. In terms of these quantities, the susceptibility (5.1.4)
may be written
κ
), and
ρ
,ω
)=
2
(1 + cos
ϕ
)
χ
Ac
(
q
,ω
)+
2
χ
(
q
+
τ
(1
−
cos
ϕ
)
χ
Op
(
q
,ω
)
.
(5
.
1
.
6
b
)
The phase
ϕ
vanishes in the limit
q
→
0
if
=
0
, and the scattering
cross-section then only depends on the isolated pole in the acoustic re-
sponse function, in accordance with our definition above. Introducing
the following lattice vectors of the hexagonal lattice:
τ
2
,
√
3
a
a
2
=
−
,
0
a
a
1
=(
a,
0
,
0)
a
3
=(0
,
0
,c
)
,
(5
.
1
.
7
a
)
2
we find the corresponding reciprocal lattice vectors:
b
1
=
2
π
a
√
3
a
,
0
b
2
=
0
,
√
3
a
,
0
b
3
=
0
,
0
,
2
π
c
.
2
π
4
π
(5
.
1
.
7
b
)
,
=
a
2
,
2
√
3
,
c
a
Since
ρ
2
,
=
4
π
3
h
+
2
π
τ
·
ρ
k
+
πl
with
τ
=(
hkl
)=
h
b
1
+
k
b
2
+
l
b
3
.
(5
.
1
.
8)
3
If
q
is parallel to the
c
-axis,
J
2
(
q
) is real. The phase
ϕ
in (5.1.6) is then
τ
·
ρ
and, if the Miller indices
h
and
k
are both zero,
ϕ
=
τ
·
ρ
=
lπ
.Inthis
case, with
in the
c
-direction, the inelastic scattering detects only the
acoustic or the optical excitations, depending on whether
l
is respectively
even or odd, and no energy gap appears at the zone boundary, even
though
l
changes, because
κ
J
2
(
b
3
/
2) = 0 by symmetry. We may therefore
use a
double-zone representation
, in which the dispersion relation for the
excitations is considered as comprising a single branch extending twice
the distance to the Brillouin zone boundary, corresponding to an effective
unit cell of height
c/
2. We shall generally use this representation when
discussing excitations propagating in the
c
-direction.
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