METO 621

Lesson 5

Natural broadening• The line width (full width at half maximum) of the

Lorentz profile is the damping parameter, . • For an isolated molecule the damping parameter

can be interpreted as the inverse of the lifetime of the excited quantum state.

• This is consistent with the Heisenberg Uncertainty Principle

Et h

2t

h

2 .h

1

2• If absorption line is dampened solely by the If absorption line is dampened solely by the natural lifetime of the state this is natural natural lifetime of the state this is natural broadeningbroadening

Pressure broadening

• For an isolated molecule the typical natural lifetime is about 10-8 s, 5x10-4 cm-1 line width

• However as the pressure increases the distance between molecules becomes shorter. We can view the outcome in two ways

• (1) Collisions between molecules can shorten the lifetime, and hence the line width becomes larger.

• (2) As the molecules get closer their potential fields overlap and this can change the ‘natural line width’.

• The resultant line shape leads to a Lorentz line shape.• Except at very high pressures when the fields overlap

strongly - assymetric line shapes - Holtzmach broadening.

Pressure broadening

• Clearly the line width will depend on the number of collisions per second,i.e. on the number density of the molecules (Pressure) and the relative speed of the molecules (the square root of the temperature)

0

)()(v

v)(

Tn

TnSTP

STPn

nSTP

L

LrelL

relLL

Doppler broadening

• Second major source of line broadening• Molecules are in motion when they absorb. This

causes a change in the frequency of the incoming radiation as seen in the molecules frame of reference

• Let the velocity be v, and the incoming frequency be , then

)cosv

1(cos vcosv'

cc

v

Doppler broadening

• In the atmosphere the molecules are moving with velocities determined by the Maxwell Boltzmann distribution

mTk

dvTk

mdf

B

XXB

XX

/2 vwhere

)v/vexp(2

v)v(

0

20

2

2/1

Doppler broadening

•The cross section at a frequency is the sum of all line of sight components

20

20

20

2

2/1

20

2

2/1

v/)(exp2

)/v()v/vexp(2

)/v1()v(v)(

vcTk

mS

cdvTk

m

cfd

B

xnxxB

xnxxn

Doppler broadening

• We now define the Doppler width as

220

00

/)(exp)( )(

/v

D

D

Dn

D

SS

cv

Voigt profile

• In general the overall broadening is a mixture of Lorentz and Doppler. This is known as the Voigt profile

D

DL

Dn

ratiodampinga

ay

ydyaS

/)(v

/

)v(

)exp()(

0

22

2

2/3

Voigt profile

• For small damping ratios, a 0, we retrieve the Doppler result. For a > 1 we retrieve the Lorentz result

• In general the Voigt profile shows a Doppler-like behavior in the line core, and a Lorentz-like behavior in the line wings.

• The Voigt profile must be evaluated by numerical integration

Comparison of the line shapes

Rayleigh scattering

200

24

40

420

4

44

6

1

6

ee

RAYn m

e

ccm

e

•If the driving frequency is much less than the natural frequency then the scattering cross section for a damped simple oscillator becomes

•The molecular polarizability is defined as

0200

2

for 4

e

p m

e

Rayleigh scattering

• Transforming from angular frequency to wavelength we get

nRAY ()

83

2

4

p2

Rayleigh scattering

• The polarizability can be expressed in terms of the real refractive index, mr

RAY () nRAY n 32 3(mr 1)2 (m 1)

where RAY() is the scattering coefficient (per atmosphere)

• mr varies with wavelength, so the actual cross section deviates somewhat from the -4 dependence

nmrp 2/)1(

Relation between Cartesian and spherical coordinates

Scattering in the planes of polarization

Scattering phase function• So far we have ignored the directional dependence of the scattered radiation - phase function

• Let the direction of incidence be ’, and direction of observation be . The angle between these directions is cos = ’ . is the scattering angle.

•If is < /2 - forward scattering

•If is > /2 - backward scattering

Scattering phase function• In polar coordinates

cos = cos’cos + sin’sincos(’- )

• We define the phase function as follows

14

).;','(sin

4

)(cos

is ionnormalisat The

)( )(cos

)(cos)(cos

0

2

04

1

4

pdd

pdw

srdn

np

n

n

Rayleigh phase function• The radiation pattern for the far field of a classical dipole is proportional to sin2 , where is the polar angle measured from the axis, and is the induced dipole moment.

• We can take the incoming radiation and break it up into two linearly polarized incident waves, one with the electric vector parallel to the scattering plane, the other perpendicular to the scattering plane.

• These waves give rise to induced dipoles

Rayleigh scattering phase function• If the incident electric field lies in the

scattering plane then the scattering angle is (/2+), if perpendicular to the scattering plane the angle is /2.

• Hence)2/(sin)2/(sin)( 2

||2

|| IIIRAY

• given that the parallel and perpendicular given that the parallel and perpendicular intensities are equal intensities are equal

)cos1( )( 2 IIRAY

Rayleigh scattering phase function

• If we normalize the equation

)cos1(4

3)(

3

4)cos1(sin

4

1)cos1(

4

1

2

2

0

22

0

2

4

rayp

ddd

Phase diagram for Rayleigh scattering

Rayleigh scattering

• Sky appears blue at noon, red at sunrise and sunset - why?

, nm , cm2 , surface Exp(-)

300 6.00 E-26 1.2 0.301

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

Schematic of scattering from a large particle

In the diagram above 1 and 2 are points within the particle. In the forward direction the induced radiation from 1 and 2 are in phase. However in the backward direction the two induced waves can be completely out of phase.

Mie-Debye scattering• For particles which are not small compared with

the wavelength one has to deal with multiple waves from different molecules/atoms within the particle

• Forward moving waves tend to be in phase and this gives a large resultant amplitude.

• Backward waves tend to be out of phase and this results in a small resultant amplitude

• Hence the scattering phase function for a particle has a much larger forward component (forward peak) than the backward component

Phase diagrams for aerosols

Phase diagrams for different values of the ratio of the aerosol radius to the wavelength of the incident radiation (left hand column)