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
h
Et 
2
h
1
t 

2 .h 2
• If absorption line is dampened solely by the natural
lifetime of the state this is natural broadening
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)
nv rel
n T
 L  L ( STP)
  L ( STP)
nL v rel ( STP)
nL T0
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
  
'
v cos 

v v cos 
v
 
  (1  cos  )
c
c
Doppler broadening
• In the atmosphere the molecules are moving with
velocities determined by the Maxwell Boltzmann
distribution
1/ 2
 m 

f ( v X )dv X  
 2k BT 
where v 0  2k BT / m
exp(  v 2X / v 02 )dv X
Doppler broadening
•The cross section at a frequency  is the sum of all
line of sight components

 n ( )   dv x f ( v x ) n  (1  v x / c)

1/ 2  
 m 

 
 2k BT 
2
2
v
/
v

exp(
dv
0 ) n (  v x / c )
x
 x

1/ 2
 m 
2 2
2
2


exp  c (  0 ) / v 0 v0
S

 2k BT 


Doppler broadening
•
We now define the Doppler width as
 D  v 0 v0 / c
 n ( )  S  D ( ) 
S
D

exp  (  0 ) 2 /  D2

Voigt profile
• In general the overall broadening is a
mixture of Lorentz and Doppler. This is
known as the Voigt profile

dy exp(  y )
 n ( )  S 3 / 2
2
2

  D  ( v  y )  a
a
a   L /  D  damping ratio
v  (  0 ) /  D
2
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
•If the driving frequency is much less than the
natural frequency then the scattering cross
section for a damped simple oscillator becomes

2


e
1  
e






6 me4 02 c 4 04 6  c   me 0 02 
4
RAY
n
4
4
•The molecular polarizability is defined as
2
e
p 
for    0
2
4 me 0 0
Rayleigh scattering
• Transforming from angular frequency to
wavelength we get
4


8 2
RAY
2
 n ( ) 
   p
3   

Rayleigh scattering
• The polarizability can be expressed in terms
of the real refractive index, mr
 p  (mr  1) / 2n

RAY
( )  
RAY
n
n  32 (mr 1)
3
2
1
(m )
where RAY() is the scattering coefficient (per
atmosphere)
• mr varies with wavelength, so the actual
cross section deviates somewhat from the -4
dependence
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’sincos(’- )
• We define the phase function as follows
n n (cos )
1
p(cos ) 
( sr )
n  d n (cos )
4
The normalisat ion is
2

p(cos )
p( ' ,  ' ; . )
1
4 dw 4  0 d 0 d sin 
4
Rayleigh phase function
• The radiation pattern for the far field of a
classical dipole is proportional to Psin2 ,
where  is the polar angle measured from the
axis, and P 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
2
I RAY  ( I   I|| )  P  sin ( / 2)  P|| sin ( / 2  )
• given that the parallel and perpendicular
intensities are equal
I RAY ()  I (1  cos 2 )
Rayleigh scattering phase function
• If we normalize the equation
1
4
1
4 d (1  cos  )  4
2
2
2
4
0 d 0 d sin  (1  cos  )  3
3
pray ()  (1  cos 2 )
4
2
Phase diagram for Rayleigh scattering
Rayleigh scattering
, 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
• Sky appears blue at noon, red at sunrise and
sunset - why?
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)