Fourier transform time-dependent observables

The final application we discuss is one where the maximum entropy formalism is used not only to fit the spectrum but also to extract new results. Specifically we discuss the determination of the time cross-correlation function, Cf, t) (Eq. (43)), which is the Fourier transform of the Raman scattering amplitude a/((Tu) (Eq. (44)) when what is measured is the Raman scattering cross section afi(m) a/((Tii) 2. The problem is that the experiment does not appear to determine the phase of the amplitude. The application proceeds in two stages (i) Representing the Raman spectrum as one of maximal entropy, using as constraints the Fourier transform of the observed spectrum. At the end of this stage one has a parametrization of a/,( nr) 2 whose accuracy can be determined by how well it fits the observed frequency dependence, (ii) The fact that the Raman spectrum can be written as a square modulus as in Eq. (97) implies that it can be uniquely factorized into a minimum phase function... 

The quantity of interest, A(t), is a function of the variables of the system of interest, and the indicated time dependence in A(t) is that due to the normal unperturbed motion of the system. The observable properties are ordinarily related to the Fourier transform of the correlation function... 

To conclude, the second dimension is introduced if the switching time ti (Fig. 2.48) is incremented in a series of single experiments so as to reach all possible double quantum frequencies vDQ within a sample molecule by the reciprocals l/t1. Again, the acquired FID signals will depend on two variable times t1 and t2, respectively. A first Fourier transformation in the t2 domain generates 13C — 13C satellite spectra. The corresponding AB or AX type doublet pairs, however, are modulated by the individual double quantum frequencies which characterize each AB or AX pair. The second Fourier transformation in the tl domain liberates the double quantum frequency as the second dimension Maximum AB or AX 13C—13C subspectra are observed at the corresponding double quantum frequencies, so that each doublet appears with unique coordinates,... 

While in the frequency domain all the spectroscopic information regarding vibrational frequencies and relaxation processes is obtained from the positions and widths of the Raman resonances, in the time domain this information is obtained from coherent oscillations and the decay of the time-dependent CARS signal, respectively. In principle, time- and frequency-domain experiments are related to each other by Fourier transform and carry the same information. However, in contrast to the driven motion of molecular vibrations in frequency-multiplexed CARS detection, time-resolved CARS allows recording the Raman free induction decay (RFID) with the decay time T2, i.e., the free evolution of the molecular system is observed. While the non-resonant contribution dephases instantaneously, the resonant contribution of RFID decays within hundreds of femtoseconds in the condensed phase. Time-resolved CARS with femtosecond excitation, therefore, allows the separation of nonresonant and vibrationally resonant signals [151]. 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

Cvi is the number of vinyl groups per chain. The direct observation of the decrease in amplitude of the Fourier Transform-Infrared spectroscopy (FT-IR) spectrum of vinyl groups leads to a square root dependence of a(t) on time. The threshold of gelation occurs at a time t0 such that e(t0) is equal to zero. Again, the magnetic relaxation rate is a function of the variable, e (Figure 8.4). 

The force-force correlation function used here has a complicated form that can be determined by numerical evaluation. We examined the correlation function for ethane at 50° C as well as the critical density. With the Egelstaff quantum correction, the correlation function initially decays as 1-at2 for a very short time ( 15 fs). It then slows and becomes progressively slower at longer times. As mentioned above and as will be discussed in detail in connection with the experiments, the Fourier transform is taken at a relatively low frequency (150 cm-1), not the 2000 cm 1 oscillator frequency. For low frequencies, the very short time details of the correlation function are not of prime importance. Without the quantum correction, the strictly classical correlation function does not begin with zero slope at zero time, but, rather, it initially falls steeply. However, the quantum corrected function and the classical function have virtually identical shapes after 15 fs. As will be demonstrated below, the force-force correlation function contained in Equation (21) with Equations (22), (24), (25), (26), and (27) does a remarkable job of reproducing the density dependence observed experimentally. The treatment also works very well... 

In some cases cyclic events occur, dependent, for example, on time of day, season of tire year or temperature fluctuations. These can be modelled using sine functions, and are tire basis of time series analysis (Section 3.4). In addition, cyclicity is also observed in Fourier spectroscopy, and Fourier transform techniques (Section 3.5) may on occasions be combined with methods for time series analysis. 

When the environment is not stationary, response functions such as x,M, (t, t ) and Xvx(t. t1) depend separately on the two times t and t7 entering into play, and not only on the time difference or observation time z t f. However, the observation time continues to play an essential role in the description. Hence, it has been proposed to define time- and frequency-dependent response functions as Fourier transforms with respect to x of the corresponding two-time quantities [5,6,58]. The time f, which represents the waiting time or the age of the system, then plays the role of a parameter. 

This equation, demonstrated in [62], is indeed independent of the m/z ratio. Thus broadband excitation will bring all the ions onto the same radius, but at frequencies depending on their m/z ratio, provided that the voltage is the same at each frequency. This can be best performed by applying a waveform calculated by the inverse Fourier transform, namely SWIFT [63], As usual for a technique based on Fourier transform, the resolution depends on the observation time, which is linked with the disappearance of the detected signal (relaxation time). Here the disappearance of the signal mainly results from the ions being slowed by... 

Figure 2.4 shows the Fourier transform of oscillating components of the real-time spectra of AT/T t) (the time dependence of the normalized transmittance change). In both PC- and NC-pulse excitations, two strong vibrating components are clearly observed in the AT/T (t) trace. These peaks, at 1140 and 1410cm are attributed to the C-N and N=N stretching modes, respectively, in DMAAB molecules from the Raman data in literature. ... 

Now we can show the explicit relation with experiment. What is usually measured in spectroscopic or scattering experiments is the spectral density function /(to), which is the Fourier transform of some correlation function. For example, the absorption intensity in infrared spectroscopy is given by the Fourier transform of the time-dependent dipole-dipole correlation function <[/x(r), ju,(0)]>. If one expands the observables, i.e., the dipole operator in the case of infrared spectroscopy, as a Taylor series in the molecular displacement coordinates, the absorption or scattering intensity corresponding to the phonon branch r at wave vector q can be written as (Kobashi, 1978)... 

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