The Radiation Damping Absorption Coefficient Profile
- Quantum Mechanically Correct Version The absorption coefficient (cross section) in cm2 per classical oscillator is
 
 2
2
2e
me c

2
0


2 2
2

   
 2 
2
,
2
where
 
8 2 2 e 2
,
3me c 3
is the classical damping constant and ν0 is the resonant or natural frequency. This can
also be written, in the vicinity of an absorption line
 
e
2
mc
 v , where  

4 2
, since  02   2   [( 0   )( 0   )] 2  4 2    0  .
2
 

 4 
   0 2  
2
2
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Two adjustments need be made to the above expression to bring it into agreement with
the quantum mechanically derived expression:
(1) The coefficient needs to be multiplied by f, the so-called “oscillator strength” or
“f-value.” This is a number between 0 and 1 which can be regarded as the number of
classical oscillators per absorbing particle, or as a probability that an absorbing
particle which is otherwise in the correct position and energy configuration, will
actually absorb a passing photon.
(2) The classical damping constant, γ, needs to be replaced by a quantum mechanically
defensible quantity. We note that the half width of the symmetrical classical αν
profile at one half its peak value is given by ν  ν0 =  γ/4π. Because of the
uncertainty principle, ΔEΔt  h, there is an uncertainty in the energy of each of the
two energy levels associated with a given transition of ΔEu,l = h/tu,l, where tu,l is the
lifetime of the atom, ion or molecule in that energy level. The probability that an
absorbed or emitted photon will have an energy in the range E to E + dE can be
shown to be
dP 
h
dE
4 2 t ul
E  E 0 2
 h
 
 4 t ul



2
,
where E0 = hν0 is the average energy of a photon associated with a transition between
the levels and tul is defined by the equation
1
1 1
  , where tu and tl are the lifetimes
t ul t u t l
of the upper and lower energy levels respectively. For purely radiative conditions
(negligible collisional transition rates) these quantities are determined by the local
radiative conditions and the Einstein coefficients of the involved particles:
1
  Ann '   Bnn ' J  nn '   Bnn '' J  nn '' .
tn
n'
n'
n ''
In this expression n can represent either u or l; in both cases n’ represents lower
energy states which are available for spontaneous and stimulated emissions and n’’
represents higher energy states which are available for absorptions. To take full
advantage of the similarity between the expressions for αν and dP we define
Гul  1/tul = 1/tu + 1/tl = Гu + Гl.
In terms of Гul, the probability an emitted or absorbed photon between levels u and l
will have a frequency between ν and ν + dν is given by the “line intensity
distribution” or “broadening function” φν, which is therefore
nn '
 d 
   0 2
4 2
d ,
2
nn ' 


4 

where we have replaced the subscripts u and l with the subscripts n (upper state) and
n’ (lower state). The absorption cross section per particle, is therefore
nn '
4 2
 
f nn ' v , where  
.
2
mc
 nn ' 
2
   0    
 4 
e 2
This is identical with the classical result except for the two adjustments,
(1)
 e2
mc

 e2
mc
f nn ' and (2)   nn ' .