The observed frequency (ω) of radiation from an atom that moves with the velocity v at an angle θ to the line of sight is given by
where ω0 is the frequency that the atom radiates in its own frame of referenece. The Doppler shift is then
As the radiating atoms are subject to random thermal motion, a variety of Doppler shifts will be displayed. In equilibrium the Maxwellian distribution gives the fraction dN/N of atoms with x-component of velocity lying between vx and vx + dvx
where u/√2 is the root-mean-square velocity for particles of mass M at temperature T . Now
Further from (2) d(Δω) = dω. The relative distribution of Doppler shift is
Thus a Gaussian distribution is produced in the Doppler shift due to the random thermal motion of the source (Fig. 2.4). The intensity of radiation is
centered around the unshifted frequency ω0. The width of the distribution at the frequencies where I(ω) falls to half the central intensity I(ω0) is known as the half width
Thermal broadening is most pronounced for light atoms such as hydrogen and high temperatures, for example the Hα line (6,563) has a Doppler width of 0.6 at 400 K.
