Lesson 4 Extinction & Scattering
Transcript of Lesson 4 Extinction & Scattering
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Radiation Extinction & Scattering
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Basic radiative processes There are three radiation-matter interactions,absorption, emission and scattering
We can consider the radiation field in twoways, classical and quantum.
Classical the electromagnetic field is a
continuous function of space and time, with awe e ne e ec r c an magne c e a
every location and instant of time
Quantum the radiation field is a
, .
More for absorption
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The extinction law
Consider a small element of an absorbing medium, ds, within the total
medium s.
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The extinction law can be written as
dsIkdI )(
The constant of proportionality is defined as the. .
(1) by the length of the absorbing path with the gas
at one atmos here ressure
)()( 1 mdI
k s
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B mass
KgmdIdIkm 12.)(
dsdM
or y concen ra on
dIdI 2
ndsdN
mdNIndsIn
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extinction over a finite distance (path length)
nms nkdskdskds
0 0 0
)(')(')(')(
Where S() is the extinction optical depth
The integrated form of the extinction equation
becomes
)(exp),0(),( s
IsI
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= +
processes, scattering and absorption, hence
ascs
)',()( sds
s
ii
sc 0
s
i
,0
ssi
a
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Scattering
Air molecules scatter light as dipoles
Dipole induced
We use di ole and molecule nearl
synonymously molecules can be
approximated as dipoles. If sufficiently
small, any particle can be approximated by
a po e osc a or
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Scattering of radiation fields
Radiation fields scattered from the points P and P are90 degreed different in phase and therefore interfere
.
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Scattering without aborption (e.g. Rayleigh)
Scattering with aborption Coherent scattering
No/little change to the frequency
Inelastic scattering (Raman lidar)
Scatterin with an exchan e of internal
energy of the medium with that of radiation
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Lorentz theory of radiation-matter
n erac ons
(negative charges) and nucleus (positive.
Bound together by elastic forces .
Combined with the Maxwell theory of the
e ec romagne c e Classical theory
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Scattering from Damped Simple
armon c sc a or
Assume that a molecule is a simple harmonicoscillator with a single harmonic oscillation
0 When irradiated by monochromatic
,electron undergoes an acceleration, while thenucleus, being massive, is assumed not to
move. An accelerating charge gives rise to
.
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Damped Harmonic Oscillator
Without energy loss the oscillator would keep its motion
indefinitely forward beam would be unchanged. In
rea y we see a sorp on an energy oss.
Can only occur if there is some damping force acting on
the oscillator. The classical dam in force is iven b :
F mev where e 0
6 0mec3
eis the electronic charge
0is the vacuum permitivity
meis the mass of the electron.
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In the classical theor the inte rated cross
section is a constant. Under the quantumtheory there is usually more than one resonant,
integrated cross section given by the above
term, but multi lied b a constant . f is called the oscillator strength
2e
4 0
LiL
e
ncm
ere i s ca e e ne s reng i
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The fre uenc de endent art of of the
equation is called the Lorentz profile
220 )4/()()(
L
Since the Lorentz profile is normalized we
eres2
cmen
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Comparison of line shapes
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Scattering is governed byre rac ve n ex an
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Complex Refractive Indices of
Liquid Water
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Size Distribution Matters to Scattering: Lognormal Function
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Size Distribution Matters to Scattering: Gamma Function
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e c ou rop e ec ve ra us
weighted mean of the size distribution of cloud droplets.[1]
The term was defined in 1974 b James E. Hansen and Larr Travis
as the ratio of the third to the second moment of a droplet size distribution
to aid in the inversion of remotely sensed data.[2]
Physically, it is an area weighted radius of the cparticles.
Mathematically, this can be expressed as
.
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So far we have ignored the directional
dependence of the scattered radiation - phasefunction
Let the direction of incidence be , and
.
between these directions is cos = . is.
If is < /2 - forward scattering
If is > /2 - backward scattering
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Phase diagrams for aerosols
the wavelength of the incident radiation (left hand column)
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In polar coordinates
cos = coscos + sinsincos(-) We e ne t e p ase unct on as o ows
)(cos 1 n n )(cos
4
dn n
''
isionnormalisatThe2
1
4
.,sin
4 004
dddw
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Asymmetry Factor
It is a measure of the preferred scattering direction ( forward or backward ) .
In radiative transfer studies, asymmetry factor 'g' is equal to the
mean value of (the cosine of the scattering angle),
weighted by the angular scattering phase function P().
Phase function is defined as the energy scattered per unit solid anglein a given direction to the average energy in all directions.
The asymmetry factor approaches
+1 for scattering strongly peaked in the forward direction and
-1 for scattering strongly peaked in the backward direction. n ca es sca er ng rec ons even y s r u e
i.e isotropic scattering (e.g scattering from small particle
g 90 de . often backscatterin is referred to scatterin at 180 de .
g>0 scattering in the forward direction(i.e scattering angle < 90, often forward-scattering is referred to scattering at 0 deg.
For larger size or Mie particles, g is close to +1.App lications
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Single Scattering Albedo and Refractive Index
The ratio of scattering and total extinction coefficient:
s
Refractive index: m = mr+ i midI Ida
If only absorption is considered:
= absorption coefficient; = density; s = path, k = absorption coefficient)
a s s
2 mi
The value of mi depends on how easy it is to bounceelectrons to higher energy levels so that they dont fall back.
Usually: small values + sharp peaks at a few wavelengths
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(though learned about widening of absorption spectra)
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Single scattering albedo
Upper bound: 1.0
Lower bound for large particles: 0.5
T ical values for dro lets at visible wavelen ths: ust below 1.0
Some aerosols contain mix of water and carbon -> lower values
Wavelength-dependence: decrease with size
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Scattering by clear air 2
s Q r, 0
n r r2 dr Q x 0
n r r2 drWe know that x2r
In clean air, rremains constant, but of interest may vary
Size Parameter
Which part of Q(x) curve applies?
Q x x4 14
0.2 in blue0.03 in red
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Rayleigh Scattering by Air Molecules
Lord Rayleigh
John William Strutt
(third Baron Rayleigh)
1842-1919
Essex, Cambridge
Nobel Prize in Physics in 1904
"
of the most important gases
and for his discovery of argon
in connection with these studies"
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wavelength of light, the scattering cross section
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1 RAY ee
2
00
4
0
42
0
4 66 een
mccm
The molecular polarizability is defined as2
e02
00
4
e
pm
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rans orm ng rom angu ar requency to
wavelength we get
nRAY ()
8
3
2
4
p2
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Rayleigh scattering
The polarizability can be expressed in terms
of the real refractive index, m
nmrp 2/)1( () n n 32 (mr1) (m
)
w e e s e sca e g coe c e e
atmosphere)
mrvaries with wavelength, so the actualcross section deviates somewhat from the -4
dependence
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22
3)cos1(sin
4)cos1(
40
2
0
2
4
ddd
)cos1(4
)( 2 rayp
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Phase diagram for Rayleigh scattering
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Rayleigh scattering
, nm , cm2 , surface Exp(-)
-. . .
400 1.90 E-26 0.38 0.684
600 3.80 E-27 0.075 0.928
1000 4.90 E-28 0.0097 0.990
10,000 4.85 E-32 9.70 E-7 0.999
Sky appears blue at noon, red at sunrise andsunse - w y
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Lorenz-Mie-Debye Theory
Mie theory, also called Lorenz-Mie theory or
Lorenz-Mie-Debye theory, is an analytical
solution of Maxwell's equations for the scattering
of electromagnetic radiation by spherical
par c es a so ca e e sca er ng n erms o
infinite series. The Mie solution is named after its
, .
However, others like Danish physicist Lorenz
receded him inde endentl develo ed the
theory of electromagnetic plane wave scatteringby a dielectric sphere. The term "Mie solution" is
sometimes used more generically for any
analytical solution in terms of infinite series,
S h ti f tt i f l ti l
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Schematic of scattering from a large particle
In the diagram above 1 and 2 are points within the particle. In the forward
rec on e n uce ra a on rom an are n p ase. owever n e
backward direction the two induced waves can be completely out of
phase.
Mie Theory
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Mie Theory
Mie theory: scattering by arbitrary homogeneous sphere
Mie scattering is a theory (one of many), not a physicalprocess
Scattering by a sphere can also be determined by
Fraunhofer theory, geometrical optics, anomalous
, - ,
No distinct boundary between Mie and Rayleigh scatterers;
Mie theor includes Ra lei h theor a licable as x 0
Mie scattering by cylinders, spheroids, coated spheres? Mienever considered it; not Mie theory.
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Mie-Debye Scattering by Particles Unlike absorption, scattering can be apportioned into directions:
differential scattering cross section is the contribution to the total
scatterin cross section Csca from scatterin into a unit solid an le
in each direction:
Scattering coefficient of a suspension of N identical particles
per unit volume:
sum of the absorption and scattering cross sections is called the
extinction cross section: Cext = Cabs + Csca;
sca er ng an ex nc on are some mes norma ze y e r
geometrical (projected) cross-sectional areas G to yield
dimensionless efficiencies or efficiency factors for scattering and
Qext= Qsca+Qabs
Expand the incident, scattered, and internal EM fields in
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Expand the incident, scattered, and internal EM fields in
a series of vector spherical harmonics (general solutions
Coefficients of the expansion functions are
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Project 1 Mie Scattering Due Feb 17
1. Read the papers that I will send you to help you understand the Mie
theory and codes.
2. Compute Mie scattering optical properties for liquid cloud droplets
with number concentration of 100 cm3
, effective radius of 10 m, andeffective variance of 0.1 (=7 for a gamma distribution) at wavelengths
-. , . , . , . .
properties for a cloud with N = 100 cm3 and ref f =5 m at 1.64 m.
Finally, compute the Mie scattering optical properties for a mineral
aerosol layer index of refraction m =1.56 0.01i and size distribution N= cm , re = .7 m, = at . 5 m.
3. Plots the phase function, the extinction (km1 ), single scattering
albedo as mmetr arameter for the 6 cases.
4. Discuss the results with regard to the variations of the phase function
and single scattering albedo with particle size and refractive index, and
wave eng .