Spectra, Fourier transform

Infrared Spectroscopy can be used to gain important information about functional groups on surfaces of minerals, but quantitative determinations have been difficult. For complex materials, like coal, the spectra are still not resolved fully for example, there is great deal of uncertainty about the 1600cm-1 band which is the dominant feature of all coal spectra. Fourier-transform infrared spectroscopy, which is a considerable improvement in this technique, has recently been used to investigate low-temperature oxidation of coal (13). 

Keywords, bomb phenomenology, detonation fireball spectra, Fourier-transform spectroscopy, classification, feature extraction. 

In an interesting study, Chang et al. (146), have proposed the use of power spectra (Fourier transforms of r(t)) as a diagnostic tool for nonstatistical behavior in unimolecular decomposition. 

The nature of intermolecular interactions can also be studied by analyzing the frequency and intensity of the external vibrations, or lattice modes, of the imit cell (49,54,55). The coupling of chains can be characterized by the whole chain segment movements which generally occur in the very low frequency region, ie, typically <100 cm. These modes are in addition to the internal modes of each chain. The translatory modes can be seen in far-infrared absorption spectra. Fourier transform instruments have improved the signal-to-noise ratio, making... 

NMR spectrum. Fourier transform nuclear magnetic resonance (FTNMR) instruments, which are similar in principle to Fourier transform infrared spectrometry (FTIR) instruments, are popular today. We will briefly describe these instruments later in this section. 

In the last years the technique of Fourier transform NMR spectroscopy has become widely appreciated for its speed and increased sensitivity it has especially become the method of choice for studies of the less sensitive nuclei, such as 13C or 15 N. In principle, the same information is contained in the slow passage and Fourier-transformed, pulse-excited spectra, and each may be optimized for resolution or for studies of particular features of the spectrum. Fourier transformation is nor-... 

Technique 26, Section 26.1, describes the technique for preparing samples for proton NMR. Much of what is described there also applies to carbon NMR. There are some differences, however, in determining a carbon spectrum. Fourier transform instruments require a deuterium signal to stabilize (lock) the field. Therefore, the solvents must contain deuterium. Deuterated chloroform, CDClg, is used most commonly for this purpose because of its relatively low cost. Other deuterated solvents may also be used. 

The fundamental components of a Michelson interferometer are depicted schematically in Fig. 1. One possible misconception regarding the use of a Michelson interferometer at visible wavelengths is that it is difficult to obtain a sufficiently precise movement for the scanning mirror,, to provide useful resolution in the absorption spectrum (Fourier transformed inter-ferogram). This is definitely not the case. The actual reason that FT techniques are not commonplace in visible spectroscopy is that the usual visible spectrophotometer with a photomultiplier tube for a detector is source-noise-limited rather than detector-noise-limited so that FT methodology does not improve (and actually degrades) the quality of the measured spectrum. 

By incrementing the interval of the second time dimension, a series of information-rich spectra is produced that contains the chemical shift frequency in one dimension and either the interaction frequency or interaction correlation in the second dimension. The projection of the frequency of these interactions, in the form of time-dependent intensity modulation or magnetization transfers along a second axis creates the two-dimensional spectrum. Fourier Transformation along this second dimension results in a tremendous resolution of spectral information. Several reviews of applications of two-dimensional NMR have appeared (37, 48, 105, 259, 277, 377). Only those experiments in routine use or of high potential value for natural products chemists are presented. 

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

As mentioned, we also carried out IR studies (a fast vibrational spectroscopy) early in our work on carbocations. In our studies of the norbornyl cation we obtained Raman spectra as well, although at the time it was not possible to theoretically calculate the spectra. Comparison with model compounds (the 2-norbornyl system and nortri-cyclane, respectively) indicated the symmetrical, bridged nature of the ion. In recent years, Sunko and Schleyer were able, using the since-developed Fourier transform-infrared (FT-IR) method, to obtain the spectrum of the norbornyl cation and to compare it with the theoretically calculated one. Again, it was rewarding that their data were in excellent accord with our earlier work. 

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra-... 

Mathematical manipulation (Fourier transform) of the data to plot a spectrum... 

Other techniques for mass measurement are available, but they are not as popular as those outlined above. These other methods include mass measurements on a standard substance to calibrate the instrument. The standard is then withdrawn, and the unknown is let into the instrument to obtain a new spectrum that is compared with that of the standard. It is assumed that there are no instrumental variations during this changeover. Generally, this technique is less reliable than when the standard and unknown are in the instrument together. Fourier-transform techniques are used with ion cyclotron mass spectrometers and give excellent mass accuracy at lower mass but not at higher. 

The process of going from the time domain spectrum f t) to the frequency domain spectrum F v) is known as Fourier transformation. In this case the frequency of the line, say too MFtz, in Figure 3.7(b) is simply the value of v which appears in the equation... 

Conceptually, the problem of going from the time domain spectra in Figures 3.7(a)-3.9(a) to the frequency domain spectra in Figures 3.7(b)-3.9(b) is straightforward, at least in these cases because we knew the result before we started. Nevertheless, we can still visualize the breaking down of any time domain spectrum, however complex and irregular in appearance, into its component waves, each with its characteristic frequency and amplitude. Although we can visualize it, the process of Fourier transformation which actually carries it out is a mathematically complex operation. The mathematical principles will be discussed only briefly here. 

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). 

FID Free induction decay, decay of the induction (transverse magnetisation) back to equilibrium (transverse magnetisation zero) due to spin-spin relaxation, following excitation of a nuclear spin by a radio frequency pulse, in a way which is free from the influence of the radiofrequency field this signal (time-domain) is Fourier-transformed to the FT NMR spectrum (frequency domain)... 

One of the major advantages of SEXAFS over other surface structutal techniques is that, provided that single scattering applies (see below), one can go direcdy from the experimental spectrum, via Fourier transformation, to a value for bond length. The Fourier transform gives a real space distribudon with peaks in at dis-... 

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