Geoscience Reference
In-Depth Information
In addition to the above-mentioned, the cross-sections are defined as mono-
chromatic ones at wavelength
λ
(for the non-stationary case - at time
t
as
well).
Consider the process of the light scattering along direction
r
(Fig. 1.4). Here
the value
dE
d
(
r
) is the energy of scattered radiation (per intervals [
λ
λ
λ
,
+
d
],
Ω
[
t
,
t
+
dt
]) per solid angle
d
encircledarounddirection
r
.Define
the directional
scattering cross-section
analogously to the scattering cross-section expressed
by (1.12).
dE
d
(
r
)
F
0
d
=
C
d
(
r
)
.
(1.13)
λ
Ω
dtd
λ
Wave l e ng t h
and time
t
are corresponding to the cross-section
C
d
(
r
).
Total scattering energy is equal to the integral from
dE
d
(
r
)overalldirections
dE
s
=
4
π
Ω
. Obtain the link between the cross-sections of scattering and
directed scattering after substituting of
dE
d
(
r
)tothisintegral:
dE
d
d
=
Ω
C
s
C
d
d
.
(1.14)
π
4
Passing to a spherical coordinate system as in Sect. 1.1, introduce two pa-
rameters:
the scattering angle
γ
defined as an angle between directions of the
incident and scattered radiation (
ϕ
counted off an angle between the projection of vector
r
to the plane perpen-
dicular to
r
0
andanarbitrarydirectiononthisplane.Thenrewrite(1.14)as
follows:
2
γ
=
(
r
0
,
r
)) and
the scattering azimuth
π
π
2
=
ϕ
γ
ϕ
γ
γ
C
s
d
C
d
(
,
) sin
d
.
(1.15)
0
0
γ
ϕ
The directional scattering cross-section
C
d
(
,
) according to its definition
γ
ϕ
could be treated as follows: as the value
C
d
(
,
) is higher, then light scatters
γ
ϕ
stronger to the very direction (
) comparing to other directions. It is neces-
sary to pass to a dimensionless value for comparison of the different particles
using the directional scattering cross-section. For that the value
C
d
(
,
γ
ϕ
)has
to be normalized to the integral
C
s
expressed by (1.15) and the result has to
be multiplied by a solid angle. The resulting characteristic is called
aphase
function
and specified with the following relation:
,
γ
ϕ
C
d
(
,
)
γ
ϕ
=
π
x
(
,
)
4
.
(1.16)
C
s
2
It is called also “differential scattering cross-section” in another terminology and the scattering
cross-section is called “integral scattering cross-section”. The sense of these names is evident from
(1.12)-(1.15).
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