lesson 4 extinction & scattering

8/12/2019 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

10


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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

00

4


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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

28

(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

- -

30


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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"

31


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wavelength of light, the scattering cross section

2444

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 .

lesson 4 extinction & scattering

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