Natural line broadening

Natural line broadening is the consequence of a finite lifetime of an atom in any excited state. The absorption process is very fast being about 10 s. The lifetime of the excited state is longer (about 10 s), but sufficiently short that the Heisenberg s Uncertainty Principle is appreciable. If the mean time the atom spends in an excited state, E, is Af, then there will be an uncertainty, in the value of Ei. 

This equation may be simplified for a resonance line. Ej will then be the ground state with an infinitely large Atj value. At is then proportional to the radiative lifetime of the excited state, which is defined for the resonance level as /Af 

At the wavelength of 300 nm it will correspond to about 0.00001 nm, which is negligible in comparison with other line broadening factors. 

t is the time taken for to fall to 1 /e of its initial value (where e is the base of natural logarithms) and is referred to as the lifetime of state n. If spontaneous emission is the only process by which M decays, comparison with Equation (2.9) shows that 

The dependence of Av, the frequency spread, on results in a much larger value for an excited electronic state, typically 30 MHz, than for an excited rotational state, typically 10 Hz, because of the much greater v for an excited electronic state. 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

Natural line broadening is usually very small compared with other causes of broadening. However, not only is it of considerable theoretical importance but also, in the ingenious technique of Lamb dip spectroscopy (see Section 2.3.5.2), observations may be made of spectra in which all other sources of broadening are removed. 

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Many of the processes which determine line widths can be removed by appropriately designed experiments, but it is almost impossible to avoid so-called natural line broadening. This arises from the spontaneous emission process (governed by the Einstein A coefficient) described in the previous section. Spontaneous emission terminates the lifetime of the upper state involved in a transition, and the Heisenberg uncertainty principle states that the lifetime of the state (At) and uncertainty in its energy (A E) are related by the expression... 

If the probability Tik(co) of absorption or emission of radiation with frequency a> causing a transition Ek is equal for all the molecules of a sample that are in the same level i ,, we call the spectral line profile of this transition homogeneously broadened. Natural line broadening is an example that yields a homogeneous line profile. In this case, the probability for emission of light with frequency u> on a transition with the normalized Lorentzian profile L (o—u>o) and central frequency coq is given by... 

Natural line broadening is usually much smaller than the broadening seen in planetary spectra, but it is of fundamental importance as the ultimate limit of the narrowness of a spectral line. Natural line broadening arises from the finite coherence length of the wave train associated with a transition. Fourier analysis of radiation composed of a wave train of finite time, Ar, shows a frequency spread, A/, that is related to this time by... 

The line width A/ is just the range of radiation frequencies that have a high probability of interacting with the molecule. The time At can be vmderstood as the spontaneous emission lifetime of the transition discussed in Section 3.5. The spontaneous decay has an exponential dependence, and therefore the line shape due to natural line broadening is Lorentzian. The natural line width. 

Eq. (3.6.1) from the natural lifetime of the transition to the mean time between collisions, leading in this case to a larger line width, Av = Wc- Under atmospheric conditions of interest, the time between collisions is usually much shorter than the natural transition lifetime. Collision broadening in the infrared can thus be orders of magnitudes larger than natural line broadening. The line profile generated by collisions is... 

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