Fourier inverse

First, the amplitude cj,(0,Of, is calculated by Eq. (91). Then, taking the Fourier inversion of the amplitude, the space-variable component of the surface concentration fluctuation is obtained,... 

Figure 50. Conversion from the point (kx, ky) on the kx-ky plane to the point (x,y) on the x-y plane accompanied by Fourier inversion." (Reprinted from M. Asanuma and R Aogaki, Morphological pattern formation in pitting corrosion, J. Electroanal. Chem. 396, 241, 1995, Fig. 7. Copyright 1995, reproduced with permission from Elsevier Science.)...

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

In the field of Compton scattering the real space function it B(r) for the electron system is defined by the Fourier inversion of the distribution function of electron... 

Kobayasi, T. (1994) Fourier inversion formalism for the calculation of angular correlation of positron annihilation radiation of semiconductors, Bull. Coll. Med. Sci. Tohoku Univ., 3, 11-22. 

After a simple Fourier inversion of a set of magnetic structure factors MbU, one can retrieve the magnetisation density. A much better result, e.g. the most probable density map, can be obtained using the Maximum Entropy (MaxEnt) method. It takes into account the lack and the uncertainty of the information not all the Bragg reflections are accessible on the instrument, and all the values contained in the error bars are satisfactory and have to be considered. However, as this method extracts all the information contained in the data, it is important to keep in mind that it may show spurious small details associated to a low accuracy and/or a specific lack of information located in (/-space. 

The indirect method described here returns the weight-average crystal size [121], irrespective of the model shape chosen. On the other hand, the direct Fourier inversion according to Warren-Averbach returns the number average of the crystal size distribution. 

The function g(k) is called the Fourier transform of f(x) and (8) the Fourier inversion formula. 

Fig. 7f. Neutron correlation functions for amorphous solid and for liquid D2O obtained by Fourier inversion of curves shown in Fig. 7(e). (From Ref. 27>)...

Unfortunately, the small-angle scattering techniques used in the investigations of Nafion morphology generally probe but a small region of reciprocal space and Fourier inversion methods of analyzing the... 

As is evident from Eqs. (1.36) and (1.37), and from the classical treatment as well, the effect of resonance on the intensity of X-ray scattering is pronounced when E E0, that is, in the vicinity of the absorption edges. Even for data collected at other wavelengths, it is necessary to correct the structure factors for anomalous scattering before the electron density can be calculated by the Fourier inversions of Eqs. (1.22) and (1.26), as further discussed in chapter 5. The anomalous scattering factors needed for this purpose are available in the literature (International Tables for X-ray Crystallography 1974, Kissel and Pratt 1990). 

We have presented two deconvolution methods from an intuitive point of view. The approach that suits the reader s intuition best depends, of course, on the reader s background. For those versed in linear algebra, methods that stem from a basic matrix formulation of the problem may lend particular insight. In this section we demonstrate a matrix approach that can be related to Van Cittert s method. In Section IV.D, both approaches will be shown to be equivalent to Fourier inverse filtering. Similar connections can be made for all linear methods, and many limitations of a given linear method are common to all. 

We defer additional analysis of the relaxation method until we have properly introduced the concept of Fourier inverse filtering. 

As with the methods of analysis based on Fourier inversion, it is necessary to characterize the lens response. This is again performed using a specimen in which waves are not excited in the surface. Lead is again suitable, and the measured curve is denoted Vi. VLI approximates closely to VGI Exactly how the analysis proceeds depends on lens geometry. 

Usually the available Hv data are insufficient for a Fourier inversion of Eq. (IV-34), needed to obtain the orientation correlation function Thus, one is faced with the necessity of reverting to simple models... 

The dipole autocorrelation function, , defined previously. The full-time dependence of this function for liquid carbon monoxide has been successfully determined experimentally from Fourier inversion of infrared band shapes.2,15 In fact, this was one of the reasons this system was studied. This function has also been successfully evaluated in terms of models of the molecular reorientation process.58 s memory function, KD(t), is defined by... 

DRS has been applied both to the analysis of vibrational spectra of surface species in the fundamental, overtone, and combination band regions,and to the determination of time correlation motion of adsorbed molecules by Fourier inversion of the spectra onto... 

The information given by an X-ray diffraction curve for a solution is one dimensional only. By a Fourier inversion it can be transformed into... 

The reduced function intensity, i(s), can be transformed into a radial distribution function, D(r), by means of a Fourier inversion... 

The corresponding contribution to the radial distribution function can be obtained by means of a Fourier inversion analogous to the one for the experimental data using the same modification function and upper integration limit. 

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