FUNDAMENTALS AND PROPERTIES FOR RADIATION IN ABSORBING, EMITTING, AND SCATTERING MEDIA Attenuation...
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Transcript of FUNDAMENTALS AND PROPERTIES FOR RADIATION IN ABSORBING, EMITTING, AND SCATTERING MEDIA Attenuation...
FUNDAMENTALS AND PROPERTIES FOR RADIATION
IN ABSORBING, EMITTING, AND SCATTERING MEDIA • Attenuation by absorption and
scattering
• Absorption and scattering coefficients
• Augmentation of intensity by emission
• Augmentation of intensity by incoming scattering
Attenuation by Absorption and Scattering
i di idi
i dsds
s + dss
di i ds
(experimental observation)or
dii
ds
: extinction coefficient, units of reciprocal length [cm-1]
( , , , )iT P C
di i ds
a
a : absorption coefficient
: scattering coefficient
0 0
( )* *
( )( )
i s s
i
dis ds
i
,di
ids
00* *( )
ln ( )( )
si ss ds
i
00 * *( ) ( )exp ( )
si s i s ds Bouguer’s la
w
Radiation mean penetration distance energy absorbed and scattered
at s( ) ( )i s i s ds
fraction of energy absorbed and scattered at s
1
0( )i ds
i
0
* *exp ( )s
s ds ds
( ) ( )di di
i s i s ds dsds ds
1
0( )
dids
i ds
0
( )
( )s
sd
i
i
Mean penetration depth
0
* *( )exp ( )s
s s ds ds
when = constant
0exp( )ml s s ds
lm : average penetration distance
before absorption or scattering
s
0 0
* *( )exp ( )s
ml s s s ds ds
1
Optical thickness (Opacity)
0
* *( ) ( )s
s s ds
0( ) ( )exp ( )i s i s
for a uniform gas
s m
s
l
the number of mean penetration distances
00 * *( ) ( )exp ( )
si s i s ds
1 : optically thick (photon continuum)• A volume element within the
material is only influenced by surrounding neighbors.• diffusion approximation (Rosseland approximation)
1 : optically thin• interact directly with medium
boundary• Radiation emitted within the material is not reabsorbed by the material.• negligible self-absorption
Limiting cases
0 : transparent medium
: completely opaque
0rq
0rq
The absorption and Scattering Coefficients Absorption coefficient
if no scattering,
0( ), a
00 * *( ) ( )exp ( )
si s i a s ds
in the case of a uniform medium0( ) ( )exp( )i s i a s
in the electromagnetic theory of radiant energy propagation in conducting media4
0( ) ( )exp ,s
i s i
4a
True absorption coefficientordinary or spontaneous emissionstimulated or induced emission
The observed emerging intensity is the result of the actual absorption modified by the addition of induced emission along the path
(negative absorption): resulting from the presence of the radiation field. At the same frequency and in the same direction as the incident photon
actual absorbed energy : true absorption coefficient ( , , )a T P
larger than a calculated by using observed attenuation media
for a gas with reflective index n = 1
01 exphc
a ak T
except at a large value of T a a
1% deviation T < 3120 m
within 5% deviation T < 4800 m
Mie Solution1) Definitions
Cs : scattering cross-section
the ratio of the rate of scattered energy by a spherical particle to the incident energy rate per unit area
Ca : absorption cross-section
Ce : extinction cross-section
e a sC C C
Q : efficiency factor
the ratio of the cross-section to the geometric cross-section
e a sQ Q Q
r : radius of the sphere
: absorption efficiency factor
2a
a
CQ
r
2s
s
CQ
r : scattering efficiency
factor
2) Mie’s solution
Re : real part
x : size parameter,
an, bn: Mie coefficient (Riccati-Bessel function)
2 2
21
22 1 ,s n n
n
Q n a bx
21
22 1 Ree n n
n
Q n a bx
D
z = x or y, y = mx, m = n - i
( ) ( ) / ( ) ( )
( ) ( ) / ( ) ( )n n n n
n
n n n n
x y y m xa
x y y m x
( ) ( ) / ( ) ( )
( ) ( ) / ( ) ( )n n n n
n
n n n n
m x y y xb
m x y y x
1 2
1
22
/
( ) ( ),nn
zz J z
1 2
1 1
2 2
12
/
( ) ( ) ( )n
nn n
zz J z iJ z
Basic parameters
1)index of refraction m = n - i relative to the surrounding medium
( = 0 : pure scatter)
3) scattering angle : phase function For a medium containing N particles per unit volume of the same composition and uniform size
2) size parameter:
D
2a aa C N r Q N
2s sC N r Q N
For a cloud of particles of different sizes
2
0 0( ) ( )a aa C dN r r Q N r dr
2
0 0( ) ( )s sC dN r r Q N r dr
volume element dV
dss = ds
s = 0
dAs
dA
dib
Increase of Intensity by Emission
dAs: a projected area normal to i(0)
R
spherical blackenclosure at T
nonparticipatingmedium
R
dss = ds
s = 0
dAs
Volume element dV
dA
d
i b
spherical blackenclosure at T
Nonparticipatingmedium
▪ spectral intensity incident at a location of dAs on dV, from element dA on the surface of the black enclosure: 0( ) ( , )bi i T
▪ the change of this intensity in dV as a result of absorption
0( ) ( , )bdi a i ds a i T ds
▪ the energy absorbed by the differential subvolume dsdAs :
R
dss = ds
s = 0
dAs
Volume element dV
dA
d
i b
spherical blackenclosure at T
Nonparticipatingmedium
( , )b sa i T dsdA d d
2sdA
dR
▪ the energy emitted by dA and absorbed by all of dV, sdV dA ds
( , )b sdVa i T d d dA ds
( , ) ( , )b s ba i T d d dA ds a i T dVd d
▪ for all energy incident upon dV from the entire spherical enclosure
R
dss = ds
s = 0
dAs
Volume element dV
dA
d
i b
spherical blackenclosure at T
Nonparticipatingmedium
4 ( , )ba i T dVd
4 4
0 0( , ) ( , )b ba i T dVd d a i T dVd d
▪ in equilibrium the energy spontaneously emitted by an isothermal volume element
R
dss = ds
s = 0
dAs
Volume element dV
dA
d
i b
spherical blackenclosure at T
Nonparticipatingmedium
4 4( , ) ( , )b ba i T dVd a e T dVd
▪ the radiation intensity emitted by a volume element into any direction
44,e p bdi dA d d a e dVd
4 4,e p bdi dA d a e dVd ,e bdi a i ds
,eb
dia i
ds
or
dAp: the projected area of dV normal to the direction of emission
p
dVds
dA : the mean thickness of dV parallel
to the direction of emission
Increase of Intensity by Incoming Scattering
ds
Incoming scattering
( , )i r
d d
2, ( , , )sd i r
scattered intensity in the direction2
, ( , , )sd i r
( , ) ( , )i r di r
d
ds( , )i r
d
( , )i r ds
phase function: directional distribution of scattered radiation
2 1
4, ,( , , ) ( , ) ( , )s sd i r di r P
,( , ) ( , )si r di r
where 2
4, ,( , ) ( , , )s sdi r d i r d
4
11
4( , )P d
4
1
4 , ( , ) ( , )sdi r P d
2
4 , ( , , )sd i r d
isotropic scattering 4
11
4P d P
physical interpretation4
1
4( , )P d
: probability that the incident radiation at will be scattered into the element of solid angle d about the direction
Augmentation by incoming scattering
scattering of the incident radiation per unit time, per unit wavelength, per unit volume, into an element of solid angle d about
2 1
4, ( , , ) ( , ) ( , )sd i r i r ds P
ds( , )i r
d d
2, ( , , )sd i r
All scattered intensity in the direction
44, ( , ) ( , ) ( , )s
dsdi r i r P d
44, ( , ) ( , )sdi
i r P dds
or
ds( , )i r
d d
2, ( , , )sd i r
2 1
4, ( , , ) ( , ) ( , )sd i r i r ds P
( )ba i r
44( , ) ( , )i r P d
, ,a sdii
ds
,eb
dia i
ds
44, ( , ) ( , )isdi
i r P dds
Radiative Transfer Equation (RTE)
( , )di r
ds
( , )i r