Two-dimensional Fourier methods

These molecules were studied by two-dimensional Fourier methods and, with the exception of pyrene, attention was confined to the most favourable projection only. By assuming that these molecules are planar and making allowance for their orientation in the unit cell, bond lengths were estimated which agree with those predicted by molecular... 

The crystal structure analysis (by two-dimensional Fourier methods) was facilitated by the fact that the crystal space group requires the molecule to have symmetry 222, the asymmetric crystal unit consisting of one-quarter of the chemical molecule. If there were no distortions from a regular planar model with a trigonal arrangement of bonds... 

The crystal structure of 2,3-dichloro-l,4-naphthoquinone (61) has been investigated (Metras, 1961) by two-dimensional Fourier methods. 

Preliminary X-ray investigations of crystals of 9,10-dihydro-anthracene (66) (Iball, 1938) showed that the most likely space group was P2X which, with two molecules in the unit cell, gave no indication of the molecular symmetry. A non-planar conformation for 9,10-dihydroanthracene has been established by Ferrier and Iball (1954). Their crystal structure analysis, using two-dimensional Fourier methods, shows clearly that the molecule is not planar but is bent about the line joining the carbon atoms 9 and 10. Each half of the molecule appears to be planar, the two halves being inclined to each other at approximately 145°. 

In contrast to 9,10-dihydroanthracene, 9,10-dihydro-l,2-5,6-dibenzanthracene (67) approximates closely to a planar conformation. Preliminary X-ray analysis (Iball, 1938) established that the molecule possesses a centre of symmetry and therefore that the naphthalenic portions of the molecule must be parallel. A more detailed crystal structure analysis, using two-dimensional Fourier methods, has been completed by Iball and Young (1958), who concluded that the molecule was essentially planar (the r.m.s. deviation of the carbon atoms from the mean molecular plane is 0-039 A, the maximum displacement, 0-081 A). This planar conformation has been explained by Herbstein (1959) as arising from the need to minimize the repulsion between the... 

Two-dimensional NMR spectroscopy may be defined as a spectral method in which the data are collected in two different time domains acquisition of the FID tz), and a successively incremented delay (tj). The resulting FID (data matrix) is accordingly subjected to two successive sets of Fourier transformations to furnish a two-dimensional NMR spectrum in the two frequency axes. The time sequence of a typical 2D NMR experiment is given in Fig. 3.1. The major difference between one- and two-dimensional NMR methods is therefore the insertion of an evolution time, t, that is systematically incremented within a sequence of pulse cycles. Many experiments are generally performed with variable /], which is incremented by a constant Atj. The resulting signals (FIDs) from this experiment depend... 

The applicability of the ESE envelope modulation technique has been extended by two recent publications115,1161. Merks and de Beer1151 introduced a two-dimensional Fourier transform technique which is able to circumvent blind spots in the one-dimensional Fourier transformed display of ESE envelope modulation spectra, whereas van Ormondt and Nederveen1161 could enhance the resolution of ESE spectroscopy by applying the maximum entropy method for the spectral analysis of the time domain data. 

About 1915 W.H. Bragg suggested to use Fourier series to describe the arrangement of the atoms in a crystal [1]. The proposed technique was somewhat later extended by W. Duane [2] and W.H. Zachariasen was the first who used a two-dimensional Fourier map in 1929 for structure determination [3], Since then Fourier synthesis became a standard method in almost in every structure determination from diffraction data. 

There has been no controversy about the structure of fluorene (31) but its true conformation was in doubt for a number of years. From an early X-ray analysis, Iball (1936a) concluded that the fluorene molecule had a folded conformation and, in a review, Cook and Iball (1936) discussed further evidence for a non-planar conformation, provided by optical activity studies of unsymmetrically substituted fluorene derivatives. Later stereochemical studies (Weisburger et al., 1950) suggested that fluorene had, in fact, a planar conformation. A reinvestigation of the crystal structure by Burns and Iball (1954, 1955) and, independently, by Brown and Bortner (1954) showed that the early X-ray work was in error and confirmed the planar conformation. The refinement of the crystal structure (Burns and Iball, 1954, 1955), by two-dimensional Fourier and least-squares methods, reveals that the maximum deviation of the carbon atoms from the mean molecular plane is 0-030 A, the r.m.s. deviation being 0-017 A. This deviation, 0-017 A, is taken by Burns and Iball to be a measure of the accuracy of their analysis, assuming now that the molecule is strictly planar. 

Hexamethylbenzene forms a 1 1 molecular complex with chloranil, formulated as (44). The crystal structure of this complex has been investigated by Harding and Wallwork (1955) by two-dimensional and partial three-dimensional Fourier methods. A crystal structure analysis of the chloranil molecule itself was later reported by Ueda (1961). Several interesting structural differences between the molecules of the complex and the parent molecules are evident. In the complex, neither... 

The computational labor associated with two-dimensional Fourier syntheses is not too formidable, and two-dimensional Fourier maps can be constructed without machine help. The labor associated with two-dimensional Patterson sysntheses is even less, and a two-dimensional vector map can often be obtained from measured intensities in a few hours. For Fourier and Patterson syntheses in three-dimensions, however, machine help is virtually indispensable. Before application of automatic computers to x-ray diffraction, the main obstacle standing in the way of a structure determination was generally the computational effort involved. In the 1950 8, the use of computers became commonplace, and the main obstacle became the conversion of measured intensities to amplitudes (the so-called phase problem ). There is still no general way of attacking this problem that is applicable in all situations, but enough methods have been developed so that by use of one, or a combination of them, all but very complicated structures may, with time and ingenuity, be determined. 

Figure 50.6. MRI signal, two dimensional Fourier transform and grey scale representation of the Fourier transformed data. Left grey scale representation of the single slice MRI of a spherical phantom filled with water. Center spectral representation of the Fourier transformed data. Right image represented in gray scale, the standard viewing method.

Lipson. H., and Beevers, C. A. An improved numerical method of two-dimensional Fourier synthesis for crystals. Proc. Phys. Soc. 48, 772-780 (1936). Patterson, A. L., and Tunell, G. A method for the summation of the Fourier series used in the X-ray analysis of crystal structures. Amer. Mineralogist 27. 655-679 (1942). 

In this type of experiment, the echo and antiecho are linearly combined with the same amplitude to yield an amplitude-modulated signal in Pure absorption lineshapes may then be obtained in the frequency domain spectrum after a two-dimensional Fourier transform is performed. The disadvantage of this method is that it is not possible to discriminate the sign of the MQ coher-... 

Radoslovich tried to answer these problems using a crystal structure refinement technique involving X-ray diffraction intensity data collected by the multiple-film Weissenberg method. His refinement involved a modest use of an electronic computer with two-dimensional Fourier syntheses using intensity data transferred to perforated tape... 

From a numerical point of view, equation (40) is integrated with a simple first-order integration method in parallel with the quantum propagation within the low-dimensional Hilbert space. Due to the periodicity of the surface potential, the quantum propagation was performed using a two-dimensional Fourier basis for the X and Y degrees of freedom. 

Abstract Multi-resonance involves ENDOR, TRIPLE and ELDOR in continuous-wave (CW) and pulsed modes. ENDOR is mainly used to increase the spectral resolution of weak hyperfine couplings (hfc). TRIPLE provides a method to determine the signs of the hfc. The ELDOR method uses two microwave (MW) frequencies to obtain distances between specific spin-labeled sites in pulsed experiments, PELDOR or DEER. The electron-spin-echo (ESE) technique involves radiation with two or more MW pulses. The electron-spin-echo-envelope-modulation (ESEEM) method is particularly used to resolve weak anisotropic hfc in disordered solids. HYSCORE (Hyperfine Sublevel Correlation Spectroscopy) is the most common two-dimensional ESEEM method to measure weak hfc after Fourier transformation of the echo decay signal. The ESEEM and HYSCORE methods are not applicable to liquid samples, in which case the FID (free induction decay) method finds some use. Pulsed ESR is also used to measure magnetic relaxation in a more direct way than with CW ESR. 

Two-dimensional Fourier Spectroscopy. Two-dimensional (2D) spectroscopy is a general concept that can be applied to different brimches of spectroscopy which make it possible to acquire more detailed information about the molecular system under investigation. Since the first proposal in the 1971 (35) and the first experimental realization in the 1974 (36-38) a large number of 2D NMR methods have been invented and applied (1) to solve structural and dynamical problems in physics, chemistry, biology, and medicine. 

Another useful QST method was developed using the two-dimensional Fourier transform technique [29]. In this method the diagonal elements of the density matrix are obtained in a ID experiment where a short pulse, similar to that discussed above, is applied to retrieve the populations. The quantum coherences, including those not directly observable, are codified in a 2D spectrum S co, oyi) so that the 2D intensities depend on these coherences. This allows to extract the quantum coherences by fitting the 2D spectrum. The main advantage of such method is its scalability, as the 2D acquisition provides an efficient way of extracting the coherences even for a large number of qubits (>5). 

See also Contrast Mechanisms in MRI Fourier Transformation and Sampling Theory Magnetic Field Gradients in High Resolution NMR MRI Applications, Biological MRI Applications, Clinical MRI Applications, Clinical Flow Studies MRI Instrumentation NMR Principles Two-Dimensional NMR Methods. 

See also Fourier Transformation and Sampling Theory Laboratory Information Management Systems (LIMS) NMR Principles NMR Pulse Sequences NMR Spectrometers Two-Dimensional NMR, Methods. 

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