Imaginary frequency from absorption spectrum

In general, and for the nonlinear hyperpolarizabilities to be derived below, one introduces r, for the transition between states w) and ). In effect the imaginary term iT o/2 takes the place of ie in Eq. (42). The linear absorption spectrum, which corresponds to the imaginary part of Eq. (45), will be built from smeared out Dirac delta functions of Lorentzian shape, i.e. the frequency-integrated absorption will remain constant regardless of the value of the lifetime broadening. The real part of the polarizability is related to the refractive index n of the sample... 

In the last step, the exponential exp —Afs/72 has been approximated by 1 because Ais 2 applies. It is seen from (2.2.28) that the phase shift of the recorded signal is linear in frequency, and two variables 0o and Atg are required for its determination. To obtain pure absorption mode real parts of experimental spectra, their phase is adjusted by multiplication with the exp —i(0o -t- X oAts) to cancel the phase shift. This process of mixing real and imaginary parts of the spectrum is called phase correction. It is a routine operation in obtaining phase-sensitive NMR spectra. In liquid-state NMR the time delay Ats is sometimes called 0i to indicate its function as a parameter for frequency-linear phase correction. 

For the interpretation of experiments it is important to know the corrections to the real and imaginary parts of the refractive index, n and k, respectively, where k is measured as a function of frequency and gives the absorption spectrum. These corrections can be found easily from the relations e = (n + in)2. Thus, the corrections to n and k, which we denote as 5n and Sk, read... 

Typically, both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye , and the first-order correction is then used to adjust the phase of a signal far away from the first in a similar manner. Ideally, the two chosen resonances should be as far apart in the spectrum as possible to maximise the frequency-dependent effect. Experimentally, this process of phase correction involves mixing of the real and imaginary parts of the spectra produced by the FT process such that the final displayed real spectrum is in pure absorption mode whereas the usually unseen imaginary spectrum is pure dispersion. 

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