Natural Linewidth and Pressure Broadening

The natural linewidth of a molecular spectral line in the MMW region is related inversely to the spontaneous emission coefficient of the upper state (Equation 1.5) and consequently imparts a Lorenteian shape to the line profile (Equation 1.32). As the upper state can radiate to more than one lower state, the actual natural linewidth is related to the sum of the squared dipole moment matrix elements of the states involved. In any case its contribution to the overall linewidth is negligible in comparison with the other broadening contributions at 100 GHz it would be 10 Hz. 

The origin of Equation 1.32 can be justified thus the collision frequency is proportional to the pressure, as is the number density of absorbing molecules. Hence Av will be proportional to p. This proportionality is conventionally introduced into the equation by reference to a standard state (ref. 4, p. 819), where Avi is the width at px — 1/760 atm (133 Pa) and Tx = 300 K. Using the relationship 

At the line peak, when the absorption coefficient is maximum, v = Vq and the peak absorption coefficient is independent of pressure. 

A more rigorous description of the pressure dependence of line shape can be obtained, that is useful at higher pressures. At the end of the collision that terminated the life of the excited state, the molecule was not in fact oriented at random, but was oriented in a direction at LTE with the MMW electric field existing immediately after each collision, in accordance with the Boltzmann distribution 

Making the substitutions that gave Equation 1.34 now gives for the modified peak absorption coefficient a ax 

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