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METO 621
Lesson 6
Absorption by gaseous species
• Particles in the atmosphere are absorbers of
radiation.
• Absorption is inherently a quantum process.
• A transition takes place from an initial quantum
state to a higher quantum state.
• When the photon energy is close to the energy
difference between the lower and the higher states,
the process is called resonance. And the absorption
is high.
• When the opposite is true the absorption is small.
Absorption by gaseous species
• This energy selectivity is an outstanding characteristic of
absorption. Scattering is generally much less selective and
occurs over a much larger region of the spectrum.
• As we shall see later, molecules have a myriad of discrete
energy levels and hence have complex absorption spectra.
• One reason for our interest in the spectra of molecules lies
in the fact that many molecules have absorption features in
the thermal infra-red and these are responsible for the
thermal equilibrium of the atmosphere.
• Another is that many molecules absorb strongly on the
ultraviolet, where the photons have enough energy to break
apart the molecules. The resulting radicals start the
chemical processes in both the troposphere and
stratosphere.
Blackbody
radiation
• Consider a tiny opening in a hollow sphere
• The chance of incoming radiation being
reflected back out of the hole is extremely
small
• Hence the opening is perfectly absorbing - it
is ‘black.
• Within the sphere the radiation will have
reached thermal equilibrium.
• The radiation emanating from the internal
surface is called ‘blackbody radiation’
Planck’s spectral distribution law
• Planck introduced in 1901 his hypothesis
of quantized oscillators in a radiating body.
• He derived an expression for the
hemispherical blackbody spectral radiative
flux
BB
F
2
r
2
m

c
2h
exp( h /kB T) 1
3
Where h is Planck’s constant, mr is the real
index of refraction, kB is Boltzmann’s constant
Planck’s spectral distribution law
Approximations for the spectral distribution
law are
Wien' s limit for high energies :
2
r
2
m
3
F 
2h exp( h / k BT )
c
Rayleigh - Jean' s limit for low energies
BB
FBB
2 2 mr2 k BT

c
Planck’s function
As the blackbody radiation is isotropic, the
intensity is related to the hemispherical flux
through
BB
F
  I  B
BB
B is known as the Planck function, and has the
same units as intensity

m
2h
B (T) 
c exp(h /kB T) 1
2
r
2
3
When dealing with gases mr is set equal to one
Planck’s function
(1) By differentiating the Planck function
and equating the result to zero we find the
wavelength corresponding to the maximum:
mT = 2,897.8
Wien’s displacement law
(2) By integrating the blackbody flux over
all frequencies we get :
FBB = BT4
Stefan-Boltzmann Law
Comparison of solar and earth’s
blackbody intensity
Absorption in molecular lines and bands
• Molecules have three types of energy levels electronic, vibrational, and rotational
• Transitions between electronic levels occur
mainly in the ultraviolet
• Transitions between vibrational levels visible/near IR
• Transitions between rotational levels - far IR/ mm
wave region
• O2 and N2 have essentially no absorption in the IR
• 4 most important IR absorbers H2O, CO2, O3, CH4
Vibrational levels
• Consider a diatomic molecule. The two
atoms are bound together by a force, and
can oscillate along the axis of the molecule.
• The force between the two atoms is given
by
2
dx
F  k.x  m 2
dt
• The solution of which is
x  x0 sin( 2 0t  )
Vibrational levels
• 0 is known as the vibrational frequency
1
0 
2
k
m
• theoretically 0 can assume all values
• However in quantum mechanics these
values
must be discrete
E  h 0 (v 1/2)
• v is the vibrational quantum number
Vibrational levels
• In general k depends on the separation of
the atoms and we have an ‘anharmonic
oscillator’
E  hc e (v 1/2)  hc e x e (v 1/2)  hc e y e (v 1/2)
2
3
Selection Rules
• Not all vibrational excitation in molecules
produces radiation.
• To produce radiation one needs to have an
oscillating dipole. When the molecule is set into a
vibration mode, the combined electric field at the
center of mass must also oscillate
• O2 and N2, homonuclear molecules , do not
possess an oscillating dipole at their center of
mass when in the ground state and so do not show
vibrational spectra in the IR.
• On the other hand O16O18 does.
Schematic of vibrational levels
Various forms of molecular vibration

Rotational levels
Consider a diatomic molecule with different
atoms of mass m1 and m2, whose distance from
the center of mass are r1 and r2 respectively
•

o
m1
r1
| r2 
o
m2
• The moment of inertia of the system about the
center of mass is:
Imr m r
2
11
2
2 2
Rotational levels
• The classical expression for energy of rotation is
I 2 L2
Er 

where L angular momentum
2
2I
2
 h 
L 
 J ( J  1)
 2 
2
• where J is the rotational quantum number
2
1  h 
E(J ) 
  J ( J  1)  h c B J ( J  1)
2 I  2 
h
B 
the rotational constant
2
8 c I
Vibrating Rotator
• If there were no interaction between the
rotation and vibration, then the total energy of a
quantum state would be the sum of the two
energies. But there is, and we get
E ( v, J )
2
  0 (v  1 / 2)   0 xe (v  1 / 2)
hc
2
2
 Bv J ( J  1)  Dv J ( J  1)
• The wavenumber of a spectral line is given
by the difference of the term values of the two
states