Absorption spectra of gases

We consider absorption of light from a collimated beam with a smooth continuous spectrum. Imagine that the absorbing gas is monatomic (for instance, an alkali vapor), and the absorbing resonance is an electronic transition from the ground state to some other state connected by a dipole transition (such as the transition from the 3²S_1/2 ground state of sodium to the 4²P_3/2 excited state). The absorption process consists of excitation of an atom followed by spontaneous emission, which is in a random direction and thus effectively takes a photon out of the beam.

The incident intensity in frequency interval f to f+f is I₀f. We suppose that I₀ is nearly constant over a limited frequency range of interest. After passing through path length x in the absorbing vapor, the remaining intensity is
 

I(f) = I0 exp(-k(f)nx)

(equation 1)

where n is the number of atoms per unit volume and k(f) is called the absorption coefficient. k(f) is nearly zero except near a transition frequency (say f₀), where it has a sharp peak.

The absorbing atom can be modeled classically as a driven harmonic oscillator. (Imagine an electron on a "spring" driven by an external oscillating electric field.) This leads to the absorption coefficient of the form
 

k(f) = (S b / ) / [(f - f0)2 + b2]

(equation 2)

This shape is called a Lorentz or Lorentzian profile. b is the half-width of the profile at half height. 1/4b is the lifetime in the excited state. (This is an uncertainty principle: the time uncertainty of emission is related to the energy uncertainty of the resonance.) The coefficient in Eq.(2) has been chosen so that k(f)df = S. (Note that the integral is independent of b). Classically, S = e²/mc, where e and m are the charge and mass of the electron. The quantum mechanical result is similar, except this absorption strength is shared among all of the allowed transitions from the ground state. (This is the "f-sum rule").

The natural width b is normally very small. Thermal motion of the atoms produces Doppler broadening, which is of order 1 GHz in the case of interest (and has a different profile -- Gaussian). However, in a vapor of appreciable density, the most important effect is collisional broadening (or "pressure broadening"): Collisions with other atoms greatly reduce the lifetime of the excited state, thus increasing b (to b_c). Then profile is essentially Lorentzian and the half-width b_c is proportional to pressure.

So what do we expect to see in the experiment?

At very low vapor pressure (small n), k(f)nx << 1, with the result that Eq.(1) is approximately I(f) / I₀ = 1 - k(f)n x. I.e., the absorption is linearly proportional to n at every frequency. Now if the vapor pressure is increased, according to Eq.(1) there is almost total absorption at the center of the peak, but the absorption remains small at frequencies sufficiently far from f₀, due to the (f - f₀)² factor in the denominator of Eq.(2). The instrumental width of our monochromator is rather large (f/f 10^-3), so we always see some of this off-resonance light. At still higher pressures the pressure broadening will make the absorption line as wide as the monochromator resolution, and then the dip we see in the scan can approach zero.

References:

• Kuhn, Atomic Spectra (1969).

 
• M.A.H. Smith et al., in Molecular Spectroscopy: Modern Research, Vol. III, chap. 3 (1985).