Doppler broadening of optical spectral lines

A moving particle (an atom or molecule) emits or absorbs radiation that is not exactly at the quantum transition frequency wq = 21 between two energy levels Ei and 2, which is determined by the Bohr quantization condition 

In a gas, particles move in all possible directions. For this reason, the Doppler shift differs between individual particles. At thermal equilibrium, all directions are equiprob-able, that is, the velocity distribution of the particles is isotropic. Therefore, the projection of the particle velocity on any direction (w = n v) is given by the Maxwell 

In the simple case where the broadening due to the Doppler effect and that caused by particle collisions are statistically independent, the shape of the spectral line S(u ) of the whole ensemble of particles is defined by the convolution of the line shape of an individual particle and the Doppler-shift distribution, that is, by the distribution of the projections of the atomic velocities onto the observation direction. The full width at half maximum (FWHM) of a spectral line due to the Doppler effect, Aljd, is given by 

Spectral line of a group of molecules moving with a given velocity 

a monochromatic traveling laser wave of frequency w interacts only with a group of particles that have a certain velocity projection v on the direction of the wave vector k within the limits of the homogeneous width  

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