Widths of Spectral Lines (continued)
Pressure Broadening
This cause of line broadening may be regarded as being due to random collisions
between atoms, which disturb internal interactions and shorten the lifetime.
e.g., Lorentz model (see Corney, p.241)
This model assumes that an atom excited at time t = 0 radiates classically until a collision
occurs. The decaying wavetrain is then terminated – i.e., the radiation is cut off early,
hence the lifetime is apparently shortened.
As with the natural linewidth, the lineshape is a Lorentzian.
L( ) 
 L

2
1
(   0 )    L 
2

2
2
with FWHM L  1 and c  mean time between collisions.
c
For example, consider a mercury lamp running at 150C – this gives a
vapour pressure. p, of 2.8 mmHg. For the collisions, a hard sphere
approximation and atoms of diameter d can be assumed.
Atoms collide only if the impact parameter  d
To obtain the mean time between collisions, the number of collisions per
second, c is required.
Consider an atom travelling with average speed vrms. Then, the “volume”, V, swept in 1
second by the atom, within which a collision can occur is:
V  d 2v rms
Atom “sweeping” a
cylinder of diameter d.
Atom undergoing
collision
Atom within
collision
region
Atom outside
collision region.
If there are n atoms per unit volume, the collision frequency is simply this density
multiplied by the volume swept by an atom per second:
c  nd 2v rms
Hence, the mean time between collisions is: (substituting for vrms from Maxwell-Boltzmann
statistics)
1  m 
c 



c nd 2  3kT 
1
1
2
For mercury, m = 200 a.u.
p
In the example of the mercury lamp, n  kT  6.31  10 22 m3 and v rms  229 ms1
If d  4.5 Å, then  c  109 ns or, l  2.93 MHz.
This is about 25% of the natural linewidth, but for higher pressures this type of broadening
is usually larger.
Sometimes, pressure broadening is referred to as homogeneous broadening, since if n is
reduced by removing a selection of atoms, the whole line narrows. Also, l is
independent of the frequency of the spectral line.
Doppler Broadening
Unlike the other forms of broadening described above, Doppler broadening does not
involve intra- or inter-atomic interactions. Rather, it is due to the motions of the atoms with
respect to an observer.
Atom moving towards
observer – light is
blue shifted.
Atom moving away
from observer – light is
red shifted.
To find the form of the distribution, it is first necessary to consider the velocity distribution
along the line of sight of the observer, vx. What is the probability, P(vx) of the velocity
being in the range vxvx + dvx? From Maxwell-Boltzmann statistics,
P(v x )dv x  e
 mv x2


2 kT 

dv x
The Doppler shift is given by

0
Doppler broadened line profile is:

vx
where 0 is the line centre frequency. Hence, the
c
 m c 2 (   0 ) 2 
D( )d  D 0 exp

  d where
 02
 2kT

c  m 
D0 


 0  2kt 
1
2
, and the FWHM,  D  2
D0
is
a
 0  2kT ln(2) 

c 
m


normalising
constant,
1
2
.
For a gas of atomic/molecular mass M in a.u. at a temperature T K, the above reduces to:
 D
0
 7.16  10 7
T
M
The form of this profile is
Gaussian
–
somewhat
different from the Lorentzian
profiles of the previous
cases. This reflects its
different origin – an effect
related to the motions of
atoms rather than decays of
radiation.
As an example, in the
mercury lamp discussed earlier; M = 200 a.u., T = 150ºC and 0 = 546 nm. Then, D =
570 MHz – ca. 50 the natural line width. This is usually the dominant source of
broadening in gas and vapour sources. Notice also that the amount of broadening
depends on the centre frequency of the spectral line.
Doppler broadening is sometimes called inhomogeneous broadening since each atom
contributed its own frequency to the profile. Hence, removal of a (non-random) sample of
atoms (e.g., by staturating a transition at a particular frquency) will distort the line shape. A
consequence is Doppler-free, or Saturated absorption spectroscopy.
Absorption of Radiation and Absorption Coefficients
On passing through a medium, radiation may be lost via a number of mechanisms. For
example:

Scattering

Absorption
x
Absorption
The final mechanism leads to
an increase in the internal
energy of the medium, through,
for
example,
electronic
excitation of constituent atoms.
I0
Scattering
I
To quantify these losses, consider a monochromatic, collimated beam incident on small
segment of the medium of thickness x, over which the intensity changes by an amount,
I = I(x+x) - I(x) . Then, if the medium is homogeneous, we can write:
I( x  x )  I( x )  I( x )x
(i.e., the change in intensity is proportional both to the
incident intensity and the amount of absorber)
I(x)
Hence, i  I( x )x
Writing as a differential equation,
dI(x )
 I(x )
dx
I(x+x)
x
This is clearly the form of an exponential, hence I( x )  I 0 e x where  is the absorption
coefficient.
In the context of a laser, we can ask:

How is the absorption coefficient related to the Einstein coefficients?

What is it's wavelength dependence?

How is it affected by line shapes?

Can a situation be arrived at where the incident radiation gains rather than loses
intensity - i.e., amplification?