Lorentzian Line Profile of the Emitted Radiation

Because the amplitude x t) of the oscillation decreases gradually, the frequency of the emitted radiation is no longer monochromatic as it would be for an oscillation with constant amplitude. Instead, it shows a frequency distribution related to the function x t) in (3.5) by a Fourier transformation (Fig. 3.2). 

The damped oscillation x t) can be described as a superposition of monochromatic oscillations exp(icwt) with slightly different frequencies co and amplitudes A co) 

The lower integration limit is taken to be zero because x(t) = 0 for t 0. Equation (3.7) can readily be integrated to give the complex amplitudes 

The real intensity I(co) oc A(co)A (a)) contains terms with (co-coo) and cjo- -coo) in the denominator. In the vicinity of the central frequency coq of an atomic transition where (co — coq) coq, the terms with (co-j-coo) can be neglected and the intensity profile of the spectral line becomes 

For comparison of different line profiles it is useful to define a normalized intensity profile L a) — a)o) = I co — a)o)/Io with h = J I co)A(j) such that 

Any intensity distribution with a Lorentzian profile is then 

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More detailed consideration shows that a Doppler-broadened spectral line cannot be strictly represented by a pure Gaussian profile as has been assumed in the foregoing discussion, since not all molecules with a definite velocity component Vz emit or absorb radiation at the same frequency o) = o o(l - -Vz/c). Because of the finite lifetimes of the molecular energy levels, the frequency response of these molecules is represented by a Lorentzian profile, see (3.10)... 

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