Line broadening

We have now discussed a number of different absorption mechanisms of molecules and have also outlined that the strength of an absorption line depends both on the transition probability and on the population of the lower energy level. However, these absorption lines are not infinitely thin, but instead have a finite width as we have alluded to previously and which arise from a number of possible mechanisms that we shall now describe.

Natural broadening

In any transition, there is a natural linewidth that arises due to the finite time over which a photon is absorbed or emitted by an atom or molecule. From Heisenberg's uncertainty principal we know that AE At « «, or Av At « 1 since E = «v. This may be further refined to give the width of an absorption line in terms of wavenumbers (cm-1)

In practice natural broadening is usually negligible compared with the following mechanisms. As an order of magnitude, the natural-broadened linewidth is of the order of 1Q—7 cm-1 (Hanel et a/., 2QQ3).

Collision broadening or pressure broadening

In a gas, molecules suffer repeated collisions with other molecules and there is a certain probability that such a collision may occur while molecules are absorbing or emitting photons. This effectively shortens the length of the absorbed or emitted wavetrains, and thus from the uncertainty principle increases the spread of wavelengths. This process is known as co//ision broadening- or sometimes ^resswre broadening since this effect becomes dominant at high pressures. The absorption coefficient kv (usually defined in units of molecules per square meter) at wavenumber v due to a collision-broadened line centered at vQ is given by the Lorentz lineshape k, = V7* (6.27)

where s = Jo° kv dv is the line strength and 7£ = (2^tc)-1 is the linewidth (cm-1). The parameter t is the mean time between collisions, which depends on density and thus for atmospheres mostly on pressure. We may thus rewrite 7 as

where 7^ is the linewidth at a reference pressure of The temperature coefficient n is by simple theory equal to Q.5, although in reality it varies slightly from molecule to molecule. At a pressure of 1 bar, and room temperature, a typical collision-broadened linewidth is of the order of Q.1 cm-1.

Doppler broadening

This arises due the line-of-sight motion of the emitting/absorbing molecules in the gas, which is due to the molecules moving with a Maxwell-Boltzmann distribution of speeds. This distribution depends upon the temperature of the gas and the molecular weight of the molecules. Molecules approaching the observer will absorb at slightly higher frequencies than receding molecules due to the Doppler effect and the absorption spectrum of a Doppler-broadened line is given by where

is the Doppler linewidth; Mr is the molecular weight; and is the Doppler linewidth at a reference temperature. For example, for methane, for which Mr — 16g, the Doppler width at room temperature (293 K) and a wavenumber of 1,300 cm-1 is 0.002 cm-1.

Voigt broadening

Considering the previous two mechanisms it is found that pressure broadening dominates at high pressures in an atmosphere, whereas Doppler broadening dominates at low pressures since pressure broadening is directly proportional to pressure that falls exponentially with height in an atmosphere, while Doppler broadening is proportional to vT which decreases much less rapidly. At intermediate temperatures and pressures both mechanisms are significant and thus the lineshape of an observed absorption line is due to a combination of both pressure broadening and Doppler broadening giving rise to the Voigt lineshape sy id* 3/2

(x — t)2 + y where x = ( — v0)/^d and y = 7L/to. Unfortunately this equation does not have an analytical solution and so must be integrated numerically.

6.3.8 Giant planet gas transmission spectra

The absorption of gases in planetary atmospheres has been seen to be due to both vibration-rotation bands and collision-induced absorptions, and to demonstrate the absorptions of different gases Figure 6.5 shows the transmission of a typical path in an atmosphere of approximately solar composition between 0cm—1 and 2,500 cm—l. Here we can see the basic properties outlined above. The various vibration-rotation bands, and pure rotation bands if allowed, can be seen for the main gases of interest in the tropospheres of the giant planets. Together with the line spectra, the importance of H2-H2 and H2-He collision-induced absorption at long wavelengths is clearly seen. Figure 6.5 also shows similar transmission spectra for various hydrocarbons observed in the giant planet stratospheres.

At shorter wavelengths, Figure 6.6 shows the calculated transmission between 0.4 ^m and 5.5 ^m for the same solar composition path used in Figure 6.5, where further vibration-rotation bands can be seen. The absorption of methane, ammonia, and again H2-H2 and H2-He CIA are most important in this region. Although water vapor also has strong absorptions, the abundance of water vapor at the cloud tops of

Figure 6.5. Mid-IR to far-IR transmission of tropospheric and stratospheric gases for a solar composition path with p = 0.3 bar, T = 127 K, and pathlength = 10 km. Calculated transmissions are between 0 and 1 unless indicated otherwise. [NB: The stratospheric hydrocarbon abundances have been increased to give clear absorption spectra.]

Figure 6.5. Mid-IR to far-IR transmission of tropospheric and stratospheric gases for a solar composition path with p = 0.3 bar, T = 127 K, and pathlength = 10 km. Calculated transmissions are between 0 and 1 unless indicated otherwise. [NB: The stratospheric hydrocarbon abundances have been increased to give clear absorption spectra.]

the giant planets is so low that water vapor is not detectable at these wavelengths, except for Jupiter.

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