Spectral density half-width Lorentzians

The next two temis (Lorentzians) arise from the mechanical part of the density fluctuations, the pressure fluctuations at constant entropy. These are the adiabatic sound modes (l/y)exp[-FA t ]cos[co(A) t ] with (D(k) = ck, and lead to the two spectral lines (Lorentzians) which are shifted in frequency by -ck (Stokes line) and +ck (anti-Stokes line). These are known as the Brillouin-Mandehtarn, doublet. The half-width at... 

As seen, the spectral density involving direct damping is the double sum over m and n of Lorentzians centered on 0) = o>° (n m)il — 2ao2Q and having the same half-width y°, but different intensities, given by c x" Am (a°) 2. 

Besides, examination of Eq. (201) shows that the spectral density is the sum of different components, each of them being a superposition of Lorentzians involving different half-widthes and intensities. Note that this result differs deeply from the situation without indirect damping given by Eq. (82), for which all the Lorentzians forming the line shape have the same half-width. This Ending may be verified by taking y = 0 in Eq. (201). 

Figure 10.6C shows the spectral density function obtained from the correlation function shown in Fig. 10.6B. If the correlation function M mjk decays exponentially with a single time constant t, Jmijkio)) is a Lorentzian peaking at w = 0 (Appendix A3 and Eq. (2.70)). The Lorentzian has a peak amplitude of VnmVjk and a width at half-maximal amplitude of 2/7. ...

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