Spectrum Dirac delta function

So suppose that we apply this property to our relaxation integral (Equation 4.47) such that the relaxation spectrum is replaced by a Dirac delta function at time rm ... 

The Dirac delta function clearly provides one form of spectra which has an analytical transform to the viscoelastic experimental regimes discussed so far. An often overlooked function was developed by Tobolsky6 and Smith.7 They noted that particular forms of the relaxation or retardation spectra have exact analytical transforms. These functions give well defined spectra and provide good fits to experimental data. The relaxation spectrum is defined by the function ... 

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... 

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

In a number of cases, it can be advantageous to compute the absorption spectrum without any reference to the eigenstates of the system but rather using the time-dependent wavepacket computed through the solution of the time-dependent Schrodinger equation. Using the integral form of the Dirac delta function, Eq. (4.55) can be recast as... 

Figure 3Ab illustrates the Dirac delta comb as a function of wavenumber, v. The function in Eq. 3.2 is an inbnite series. We multiply the analog interferogram with the Dirac delta comb of Eq. 3.1, and consequently, the Eourier transform of the interferogram (i.e., the spectrum) is convolved with the transformed comb (see Section 2.3). The effect of this convolution is to repeat the spectrum ad infinitum. If the spectrum covers the bandwidth 0 to v ax, the transformed Dirac delta comb must have a period of at least 2Vniax otherwise, the spectra will overlap as a result of the convolution. In other words. 

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