The Radiation Damping Absorption Coefficient Profile
- For a Classical Oscillator –
The absorption coefficient (cross section) in cm2 per classical oscillator is
2e 2
 
me c
 2

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.
Special Case I: Electron (“Thompson”) Scattering
In the limit of free electrons, k, the “spring constant” of the classical oscillator
goes to zero,
i.e., k → 0. But ν0 = 2πω0 = 2π√(k/me), so as k → 0, ν0 → 0, or, equivalently, ν >>
ν0. The
limiting value of the above expression for αν as ν0 → 0 is
 
8e 4
3me2 c 4
1
8e 2 2
1

3me2 c 4 3c 2
4
.
But
8e 4
 6.652455  10 25 cm 2 ,
2 4
3me c
and
8e 4 2
 1.55  10 45 s 2 .
2 4
2
3me c 3c
Therefore, for optical frequencies,
 
8e 4
  e  0.6652455  10 24 cm 2 .
2 4
3me c
This expression is valid for ν  1020 Hz. Note that over the frequency range of validity
for this expression, electron scattering is independent of frequency and is therefore a truly
“gray” phenomenon. . (Electron scattering is the dominant opacity source in the interiors
of very hot stars.)
Special Case II: Rayleigh Scattering
When the scattered radiation has a much lower frequency than the natural frequencies
of the scattering particle, i.e., ν << ν0 we say that Rayleigh scattering occurs. In this
case the limiting value of the above expression for αν as ν0 →  is
 R
8e 4

3me2 c 4


 0
4


   e 

 0
4


  0.6652455  10  24 cm 2 

 0
4

 .

Now consider a transition between upper energy level En and lower energy level En’.
E n  E n'  h 0  h nn' .
There is a probability of a scattering event occurring which is symbolized by fnn’ and is
called the “oscillator strength” or the “f-value” which we can interpret roughly as the
number of oscillators of natural frequency ν0 per absorbing particle. Hence the total
number density of scattering particles, N, can be expressed as N = Nn’ fnn’ . Now, since
we can write the relationship between the mass absorption coefficient, κ, and the
absorption coefficient per classical oscillator, αν, as Nαν = κνρ, we can write, if there
are multiple possible upper states, n, above a particular lower state, n’,
 Rayleigh   N n' f nn'
n
8e 4  4
.
4
3me2 e 4  nn
'
If absorption involves more than one lower state n’, we need sum over n’ also, i.e.,
 Rayleigh
8e 4  4
  N n ' f nn'
.
3me2 e 4  nn4 '
n' n
Finally, if more than one species of ion, atom or molecule is present, we must also sum
over all additional species, so we get,
 Rayleigh    N n' f nn' 82e 4  4
3m e 
4
all species n '
n
e
4
.
nn'
Rayleigh scattering is important in the earth’s atmosphere. The 
4
4
 nn
'
dependence
produces the blue color of the sky. In the earth’s atmosphere the n’ sum in the above
expression includes only one term per species, the ground state, since the temperature
is too low for significant populations in any other states, i.e., Nn’ is vanishingly small
for all excited states.