Environmental Engineering Reference
In-Depth Information
D
αβ
(
q
)
−
D
αβ
(
0
).
In addition to the condition
q
2
π/a
,weshall
assume that
q
10
/L
(Keffer 1966),
where
L
is a length dimension of the crystal, in which case the effects
of the boundaries on
D
αβ
(
q
) are averaged out because of the relatively
rapid variation of the exponential factor on the surface. Using these two
conditions, we find
2
π/L
, or more specifically
q
≥
D
αβ
(
q
)=
D
αβ
(
0
)+
3(
α
·
r
)(
ˆ
δ
αβ
r
2
β
·
r
)
−
e
i
q
·
r
1
d
r
−
r
5
=
D
αβ
(
0
)
L
+
3(
α
·
r
)(
ˆ
∞
[4
π
(2
l
+1)]
1
/
2
i
l
j
l
(
qr
)
Y
l
0
(
θ, φ
)
r
2
drd
Ω
.
δ
αβ
r
2
β
·
r
)
−
r
5
l
=0
The
q
-independent term in the first integral leads to the same result as in
(5.5.6), but without the lattice-sum contribution, and adding
D
αβ
(
0
),
we are left with the term
D
αβ
(
0
)
L
.The
q
-dependent exponential
is expanded in terms of the spherical Bessel functions, as in (4.1.8),
with the polar axis chosen to be parallel to
q
. The dipole factor in the
resulting integral may be written as a linear combination of the spherical
harmonics of second rank
Y
2
m
(
θ, φ
), multiplied by
r
−
3
, ensuring that
only the term with
l
=2inthesumover
l
survives the integration over
solid angles. Further, if
α
and
ˆ
are either parallel or perpendicular to
q
, only the diagonal components may differ from zero. With
α
β
and
ˆ
β
both parallel to
q
, the longitudinal component is
−
D
(
0
)
L
=
[16
π/
5]
1
/
2
Y
20
(
θ, φ
)
r
−
3
[4
π
D
(
q
)
5]
1
/
2
(
1)
j
2
(
qr
)
Y
20
(
θ, φ
)
r
2
drd
Ω
·
−
8
π
∞
0
8
π
∞
1
ρ
j
2
(
ρ
)
dρ
=
j
1
(
ρ
)
ρ
8
π
3
=
−
−
−
=
−
,
0
1
recalling that
j
1
(
ρ
)
/ρ
→
3
or 0, for respectively
ρ
→
0or
∞
. This result
implies that the two transverse components are
−
D
⊥
(
0
)
L
=
D
(
q
)
−
D
(
0
)
L
=
4
π
−
2
D
⊥
(
q
)
;
(5
.
5
.
7)
3
when
2
π/L
q
2
π/a.
The dipole-coupling components change from the values given by (5.5.6)
to those above within a very narrow range of
q
, i.e. when
q
goes from
zero to about 10
/L
, as shown by the detailed analysis of Keffer (1966).
At larger wave-vectors, the variation of
D
αβ
(
q
) is smooth and gradual,
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