Doppler broadening

The study of gaseous samples is very important, as it can inform our understanding of atmospheric chemistry. In some cases, meaningful spectroscopic data can be obtained only from gaseous samples. For example, they are essential for rotational spectroscopy, for only in gases can molecules rotate freely. One important broadening process in gaseous samples is the Doppler effect, in which radiation is shifted in frequency when the source is moving towards or away from the observer. When a source emitting electromagnetic radiation of frequency ν moves with a speed s relative to an observer, the observer detects radiation of frequency

where c is the speed of light (see Further reading for derivations). For nonrelativistic speeds (s << c), these expressions simplify to

Molecules reach high speeds in all directions in a gas, and a stationary observer detects the corresponding Doppler-shifted range of frequencies. Some molecules approach the observer, some move away; some move quickly, others slowly. The detected spectral ‘line’ is the absorption or emission profile arising from all the resulting Doppler shifts. As shown in the following Justification, the profile reflects the distribution of molecular velocities parallel to the line of sight, which is a bell-shaped Gaussian curve. The Doppler line shape is therefore also a Gaussian (Fig. 13.7), and we show in the Justification that, when the temperature is T and the mass of the molecule is m, then the observed width of the line at half-height (in terms of frequency or wavelength) is

For a molecule like N2 at room temperature (T ≈ 300 K), δν/ν ≈ 2.3 × 10⁻⁶. For a typical rotational transition wavenumber of 1 cm⁻¹ (corresponding to a frequency of 30 GHz), the linewidth is about 70 kHz. Doppler broadening increases with tempera ture because the molecules acquire a wider range of speeds. Therefore, to obtain spectra of maximum sharpness, it is best to work with cold samples.

Justification 13.3 Doppler broadening We know from the Boltzmann distribution (Molecular interpretation 3.1) that the probability that a gas molecule of mass m and speed s in a sample with temperature T has kinetic energy EK = 1–2ms2 is proportional to e^−ms2/2kT. The observed frequencies, ν_obs, emitted or absorbed by the molecule are related to its speed by eqn 13.16:

Where ν is the unshifted frequency. When s << c, the Doppler shift in the frequency is , ν_obs − ν ≈± νs/c ,which implies a symmetrical distribution of observed frequencies with respect to molecular speeds. More specifically, the intensity I of a transition at νobs is proportional to the probability of finding the molecule that emits or absorbs at νobs, so it follows from the Boltzmannm distribution and the expression for the Doppler shift that, I(ν_obs) ∝ e^{−mc2(νobs−ν)2/2ν2kT} , which has the form of a Gaussian function. The width at half-height can be calculated directly from the exponent (see Comment 13.3) to give eqn 13.17.

Fig. 13.7 The Gaussian shape of a Doppler broadened spectral line reflects the Maxwell distribution of speeds in the sample at the temperature of the experiment. Notice that the line broadens as the temperature is increased.

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