Fourier transform phase-coherent excitation

In a previous section, it was found that time evolutions of WPs taking into account nonadiabatic couplings strongly depend on the initial in-phase or out-of-phase coherent excitation of two quasi-degenerate states. This suggests that the initial direction of K-electron rotation can be identified by analyzing vibrational spectra. It is well known that Fourier transform of the autocorrelation function of WPs gives its frequency spectrum [37]. The frequency spectrum of (Q. 0 after the nonadiabatic transition from H) to L) is defined as... 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

Another light pulse of frequency comes at a time delay ta and interacts with the vibrationally excited molecules. The intensity of the probe light transmitted through the interface is modulated as a function of the delay. The modulation is Fourier-transformed to provide the frequency and phase of the vibrational coherence. 

Ifourth(fd, 2 Q) was multiplied with a window function and then converted to a frequency-domain spectrum via Fourier transformation. The window function determined the wavenumber resolution of the transformed spectrum. Figure 6.3c presents the spectrum transformed with a resolution of 6cm as the fwhm. Negative, symmetrically shaped bands are present at 534, 558, 594, 620, and 683 cm in the real part, together with dispersive shaped bands in the imaginary part at the corresponding wavenumbers. The band shapes indicate the phase of the fourth-order field c() to be n. Cosine-like coherence was generated in the five vibrational modes by an impulsive stimulated Raman transition resonant to an electronic excitation. 

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]. 

The multiple-quantum (MQ)/MAS NMR is one of the 2D NMR methods, which is capable of averaging out the second-order quadrupolar interaction in nuclei with spin > 1/2 such as H, "B, O, etc. The "B MQ/ MAS NMR measurements on boron as contained in silyl-carborane hybrid Si-based polymer networks considered here. The molded samples are cut into small pieces to insert them into a 4-mm NMR rotor and spun at 12 kHz in a MAS probe. The observation frequency of the "B nucleus (spin number I = 3/2 and isotope natural abundance = 80.42%) is 96.3 MHz. Excitation of both the echo (—3Q) and anti echo (+3Q) coherences is achieved by using a three-pulse sequence with a zero quantum filter (z-filter). The widths of the first, second, and third pulses are 3.0 4.1 ps, 1.1-1.6 ps, and 19-28 ps, respectively. The z-filter is 20 ps. The recycle delay time is 6-15 s and the data point of FI (vertical) axis is 64 and for each the number of scans is 144. Then, the total measurement time is 15-38 h. The phase cycling used in this experiment consists of 12 phases. Boron phosphate (BPO4 3 = 0 ppm) is used as an external standard for "B. The chemical shift value of BPO4 is —3.60 ppm from BF3 O(C2H5)2 which is used as a standard reference in " B NMR in the liquid state. The transmitter frequency of " B is set on peak of BPO4 for a trustworthy chemical shift after Fourier transform." " ... 

Syvitski et al43 reported a 3D sequence which uses TPPI to separate the spectra of different coherence orders. The pulse sequence employs a non-selective 2D excitation sequence (as in Fig. 2) but also systematically increments the phase of the first two pulses in the excitation sequence by A

) in the experiment is arbitrary but needs to be at least 2TV+ 1, where TV is the number of spins in the spin system. The 3D interferogram is then a function of t, Fourier transformation over these dimensions6,30,44 affords MQ spectra of various orders cleanly separated in the pseudo-frequency ... 

To obtain a mass spectrum over the desired m/z interval, all ions within this interval are excited simultaneously by a rapid frequency sweep of the voltage on the transmitter plates. The excitation pulse increases the orbital radii of all ions and puts ions of the same m/z ratio in phase. The orbiting ions create a complex wave signal in the circuit connecting the receiver plates, which is monitored over time as the coherent motion of the ions is destroyed by collisions (Figure 1.25). Fourier-transformation of this time-domain signal furnishes the individual cyclotron frequencies and, hence, the m/z values (Eq. 1.17) of the ions (Figure 1.25). 

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