Fourier double

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

Because of the double sound path involved in PE measurements of the back wall echo, we approximate the corresponding attenuation at a certain frequency to be twice as large as the attenuation that would be obtained by an ordinary TT measurement. We propose to use the logarithm of the absolute value of the Fourier transform of the back wall echo as input data, i.e... 

Due to the rather stringent requirements placed on the monochromator, a double or triple monocln-omator is typically employed. Because the vibrational frequencies are only several hundred to several thousand cm and the linewidths are only tens of cm it is necessary to use a monochromator with reasonably high resolution. In addition to linewidth issues, it is necessary to suppress the very intense Rayleigh scattering. If a high resolution spectrum is not needed, however, then it is possible to use narrow-band interference filters to block the excitation line, and a low resolution monocln-omator to collect the spectrum. In fact, this is the approach taken with Fourier transfonn Raman spectrometers. 

For quadnipolar nuclei, the dependence of the pulse response on Vq/v has led to the development of quadnipolar nutation, which is a two-dimensional (2D) NMR experiment. The principle of 2D experiments is that a series of FIDs are acquired as a fimction of a second time parameter (e.g. here the pulse lengdi applied). A double Fourier transfomiation can then be carried out to give a 2D data set (FI, F2). For quadnipolar nuclei while the pulse is on the experiment is effectively being carried out at low field with the spin states detemiined by the quadnipolar interaction. In the limits Vq v the pulse response lies at v and... 

Dowrey A E and Maroott C 1982 A double-modulation Fourier transform infrared approaoh to studying adsorbates on metal surfaoes Appl. Spectrosc. 36 414-16... 

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

A simple mass spectrometer of low resolution (many quadrupoles, magnetic sectors, time-of-flight) cannot easily be used for accurate mass measurement and, usually, a double-focusing magnetic/electric-sector or Fourier-transform ion cyclotron resonance instrument is needed. 

Consider the general class of laminated rectangular plates that are simply supported along edges x = 0, x = a, y = 0, and y = b and subjected to a distributed transverse load, p(x,y). In Figure 5-8. The transverse load can be expanded in a double Fourier sine series ... 

Molecules that are composed of atoms having a maximum valency of 4 (as essentially all organic molecules) are with a few exceptions found to have rotational profiles showing at most three minima. The first three terms in the Fourier series eq. (2.9) are sufficient for qualitatively reproducing such profiles. Force fields which are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos (3u ) and double bonds only cos (2u))). 

Identification of unknowns using GC/MS is greatly simplified if accurate mass measurements are made of all the ions in a spectrum so that reasonable elemental compositions of each ion are available. Unfortunately, obtaining a mass measurement that is accurate enough to significantly limit the number of possible elemental compositions requires expensive instrumentation such as a double-focusing magnetic sector or fourier transform ICR MS. 

In situ Fourier transform infrared and in situ infrared reflection spectroscopies have been used to study the electrical double layer structure and adsorption of various species at low-index single-crystal faces of Au, Pt, and other electrodes.206"210 It has been shown that if the ions in the solution have vibrational bands, it is possible to relate their excess density to the experimentally observed surface. 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

Surface forces measurement is a unique tool for surface characterization. It can directly monitor the distance (D) dependence of surface properties, which is difficult to obtain by other techniques. One of the simplest examples is the case of the electric double-layer force. The repulsion observed between charged surfaces describes the counterion distribution in the vicinity of surfaces and is known as the electric double-layer force (repulsion). In a similar manner, we should be able to study various, more complex surface phenomena and obtain new insight into them. Indeed, based on observation by surface forces measurement and Fourier transform infrared (FTIR) spectroscopy, we have found the formation of a novel molecular architecture, an alcohol macrocluster, at the solid-liquid interface. 

Recent work in our laboratory has shown that Fourier Transform Infrared Reflection Absorption Spectroscopy (FT-IRRAS) can be used routinely to measure vibrational spectra of a monolayer on a low area metal surface. To achieve sensitivity and resolution, a pseudo-double beam, polarization modulation technique was integrated into the FT-IR experiment. We have shown applicability of FT-IRRAS to spectral measurements of surface adsorbates in the presence of a surrounding infrared absorbing gas or liquid as well as measurements in the UHV. We now show progress toward situ measurement of thermal and hydration induced conformational changes of adsorbate structure. The design of the cell and some preliminary measurements will be discussed. 

Radiofrequency spectroscopy (NMR) was introduced in 1946 [158,159]. The development of the NMR method over the last 30 years has been characterised by evolution in magnet design and cryotechnology, the introduction of computer-based operating systems and pulsed Fourier transform methods, which permit the performance of new types of experiment that control production, acquisition and processing of the experimental data. New pulse sequences, double-resonance techniques and gradient spectroscopy allow different experiments and have opened up the area of multidimensional NMR and NMRI. 

Resolution does not affect the accuracy of the individual accurate mass measurements when no separation problem exists. When performing accurate mass measurements on a given component in a mixture, it may be necessary to raise the resolution of the mass spectrometer wherever possible. Atomic composition mass spectrometry (AC-MS) is a powerful technique for chemical structure identification or confirmation, which requires double-focusing magnetic, Fourier-transform ion-cyclotron resonance (FTICR) or else ToF-MS spectrometers, and use of a suitable reference material. The most common reference materials for accurate mass measurements are perfluorokerosene (PFK), perfluorotetrabutylamine (PFTBA) and decafluorotriph-enylphosphine (DFTPP). One of the difficulties of high-mass MS is the lack of suitable calibration standards. Reference inlets to the ion source facilitate exact mass measurement. When appropriately calibrated, ToF mass... 

Although the idea of generating 2D correlation spectra was introduced several decades ago in the field of NMR [1008], extension to other areas of spectroscopy has been slow. This is essentially on account of the time-scale. Characteristic times associated with typical molecular vibrations probed by IR are of the order of picoseconds, which is many orders of magnitude shorter than the relaxation times in NMR. Consequently, the standard approach used successfully in 2D NMR, i.e. multiple-pulse excitations of a system, followed by detection and subsequent double Fourier transformation of a series of free-induction decay signals [1009], is not readily applicable to conventional IR experiments. A very different experimental approach is therefore required. The approach for generation of 2D IR spectra defined by two independent wavenumbers is based on the detection of various relaxation processes, which are much slower than vibrational relaxations but are closely associated with molecular-scale phenomena. These slower relaxation processes can be studied with a conventional... 

Stump, M. J. Jones, J. J. Fleming, R. C. Lay, J. O., Jr. Wilkins, C. L. Use of double-depleted 13C and 15N culture media for analysis of whole cell bacteria by MALDI time-of-flight and Fourier transform mass spectrometry. J. Am. Soc. Mass Spectrom. 2003,14,1306-1314. 

Wunschel, D. S. Fox, K. F. Bruce, J. E. Muddiman, D. C. Smith, R. D. Analysis of double-stranded polymerase chain reaction products from the Bacillus cereus group by electrospray ionization Fourier transform ion cyclotron resonance mass spectrometry. Rapid Comm. Mass Spectrom 1996,10, 29-35. 

Figure 3.4 shows (i) a line spectrum (one-dimensional dispersive spec-trographic record), (ii) a spectrometric record, (iii) an interferogram obtained by a Fourier transform spectrometer, and (iv, v) two- and three-dimensional double dispersive spectra recorded e.g. by Echelle spectrometers. In principle, all forms may be obtained by OES. 

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