LESSON 4
METO 621
The extinction law
Consider a small element of an absorbing medium, ds, within the total
medium s.
Extinction Law
• The extinction law can be written as
dI   k ( ) I ds
• The constant of proportionality is defined as the
extinction coefficient. k can be defined in three ways.
(1) by the length of the absorbing path with the gas
at one atmosphere pressure
dI
k ( ) 
I ds
(m 1 )
Extinction coefficient
• By mass
dI
dI
km ( )  

I ds I dM

m .Kg 
2
1
or by concentration
dI
dI
kn ( )  

I nds
I dN
(m 2 )
Optical depth
• Normally we are interested in the total
extinction over a finite distance (path length)
s
s
s
0
0
0
 s ( )   ds' k ( )   ds' km ( )    ds' kn ( )n
Where S() is the extinction optical depth
• The integrated form of the extinction equation
becomes
I ( s,  )  I (0,  ) exp   s ( )
Extinction = scattering + absorption
• Extinction really consists of two distinct
processes, scattering and absorption, hence
 s ( )  sc ( )  a ( )
where
s
 sc ( )    ds  ( , s ' )
i
i
i
0
s
 a ( )    ds  ( , s' )
i
i
0
i
Differential equation of radiative
transfer
• We must now add the process called emission.
Consider a slab of thickness ds, filled with an optically
active material giving rise to radiative energy of
frequency  in time dt. This energy emerges from the
slab as an angular beam within the solid angle dω,
around a propagation vector Ω. The emission
coefficient is defined as the ratio

j (r , ) 
d 4E
dAdsdtdd
d 4E

(W .m 3 .Hz 1.sr 1 )
dVdtdd
Differential equation of radiative transfer
• Combining the extinction law with the definition
of the emission coefficient
dI  k ( ) I ds  j ds
noting that
k ( )ds  d s
dI
j
  I 
d s
k ( )
Differential equation of radiative
transfer
• The ratio j/k() is known as the source
function,
j
S 
k ( )
dI
  I  S
d s
This is the differential equation of radiative
transfer
Basic scattering processes
•
•
•
•
We have identified three radiation-matter
interactions, absorption, emission and
scattering
We can consider the radiation field in two
ways, classical and quantum.
Classical – the electromagnetic field is a
continuous function of space and time, with a
well defined electric and magnetic field at
every location and instant of time
Quantum – the radiation field is a
concentration of discrete values of energy, h.
Scattering of radiation fields
•
Radiation fields scattered from the points P’’ and P’ are
90 degreed different in phase and therefore interfere
destructively.
Lorentz theory of radiation-matter
interactions
• Neutral atoms consist of electrons
(negative charges) and nucleus (positive
charge.
• Bound together by elastic forces –
Hooke’s Law.
• Combined with the Maxwell theory of the
electromagnetic field
• Classical theory
Lorentz theory
• Could not explain the black-body
frequency distribution law. Planck in 1900.
• Quantized states
• Could not explain the photoelectric effect
• Einstein 1914
• Photons
Scattering from Damped Simple
Harmonic Oscillator
• Assume that a molecule is a simple harmonic
oscillator with a single harmonic oscillation
frequency ω0 (2p)
• When irradiated by linearly polarized
monochromatic electromagnetic wave of
frequency ω0 , the electron undergoes an
acceleration, while the nucleus, being massive,
is assumed not to move.
• An accelerating charge gives rise to
electromagnetic radiation.
Damped Harmonic Oscillator
• Without energy loss the oscillator would increase its
motion indefinitely – forward beam would be
unchanged. In reality we see absorption – an energy
loss.
• Can only occur if there is some damping force acting on
the oscillator. The classical damping force is given by:
e 2 02
F  me  v where  
6p 0 me c 3
e is the electronic charge
0 is the vacuum permitivity
me is the mass of the electron.
Damped Harmonic Oscillator
An accelerated charge radiates with an average
power of

e 4 E '2 
4
P( ) 
2 2
2
3 
2
2 2
12p me  0 c  (   0 )    
The ratio of P to the power carried in the incident
field is the scattering cross section
P( )
e4
 n ( ) 

2
2 2 4
 0 cE ' / 2 6p me  0 c


4
 2
2 2
2 2
 (   0 )    
Resonance scattering – Lorentz profile
• Let the frequency of the incident light be close to the
resonance, i.e. close to ω0 . We can write
    ( 0   ).( 0   )  2 ( 0   )
2
0
2
Substituting this relation, and the formula for 



e
( / 4)
( ) 

2
2
me c 0  (0   )  ( / 2) 
2
res
n
or because   2p



e
( / 4p )
( ) 

2
2
4pme c 0  ( 0  )  ( / 4p ) 
2
res
n
Lorentz profile
• The frequency dependent part of of the
equation is called the Lorentz profile
 / 4p
 L ( ) 
p ( 0  ) 2  ( / 4p ) 2 
Since the Lorentz profile is normalized we
find by integrating over all frequencies

 d
0
2
res
n
e
( ) 
4me 0c
Oscillator Strength
• In the classical theory the integrated cross
section is a constant. Under the quantum
theory there is usually more than one resonant
frequency, and each resonance has an
integrated cross section given by the above
term, but multiplied by a constant f.
• f is called the oscillator strength
2
e
fi
rea
 n ( ) 
 L ( )  Si  L ( )
4me 0 c
Where Si is called the line strengthi
Comparison of line shapes