Fourier diffraction

Until the advent of superresolution microscopes the only way to observe the minute world was based on the common Fourier-type microscopes. The maximum theoretical resolution limit for these apparatuses was established by Abbe, applying the Rayleigh-Fourier diffraction resolution rule [28]. The basic principle underlying the operational working of these ordinary microscopes is, in the eyes of Niels Bohr, a textbook example of Heisenberg uncertainty relations. 

Fig. 24.11 TEM image of ordered nanoporous carbon (ONC) and its Fourier diffraction (a), schematic of ONC (b), mass activities of Pt on ONC and carbon black depending on the Pt loading (c), and ORR polarization curves measured in 02-saturated 0.1 M HCIO4 at 10,000 RPM (d) [70]...

Patterson function The Fourier transform of observed intensities after diffraction (e.g. of X-rays). From the Patterson map it is possible to determine the positions of scattering centres (atoms, electrons). 

We have seen that the intensities of diffraction of x-rays or neutrons are proportional to the squared moduli of the Fourier transfomi of the scattering density of the diffracting object. This corresponds to the Fourier transfomi of a convolution, P(s), of the fomi... 

We have thus far discussed the diffraction patterns produced by x-rays, neutrons and electrons incident on materials of various kinds. The experimentally interesting problem is, of course, the inverse one given an observed diffraction pattern, what can we infer about the stmctirre of the object that produced it Diffraction patterns depend on the Fourier transfonn of a density distribution, but computing the inverse Fourier transfomi in order to detemiine the density distribution is difficult for two reasons. First, as can be seen from equation (B 1.8.1), the Fourier transfonn is... 

We have seen that the intensities of diffraction are proportional to the Fourier transfomi of the Patterson fimction, a self-convolution of the scattering matter and that, for a crystal, the Patterson fimction is periodic in tln-ee dimensions. Because the intensity is a positive, real number, the Patterson fimction is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

Flarker D and Kasper J S 1948 Phases of Fourier coefficients directly from crystal diffraction data Aota Crystallogr. 70-5... 

In a diffraction experiment a quantity F(S) can be measured which follows from equation (B 1.17.8) and equation (B 1.17.9) in Fourier space as... 

As in all Fourier transform methods in spectroscopy, the FTIR spectrometer benefits greatly from the multiplex, or Fellgett, advantage of detecting a broad band of radiation (a wide wavenumber range) all the time. By comparison, a spectrometer that disperses the radiation with a prism or diffraction grating detects, at any instant, only that narrow band of radiation that the orientation of the prism or grating allows to fall on the detector, as in the type of infrared spectrometer described in Section 3.6. 

This expression is the main tool used in describing diffraction effects associated with Fourier optics. Holographic techniques and effects can, likewise, be approached similarly by describing first the plane wave case which can then be generalized to address more complex distribution problems by using the same superposition principle. 

How do we find phase differences between diffracted spots from intensity changes following heavy-metal substitution We first use the intensity differences to deduce the positions of the heavy atoms in the crystal unit cell. Fourier summations of these intensity differences give maps of the vectors between the heavy atoms, the so-called Patterson maps (Figure 18.9). From these vector maps it is relatively easy to deduce the atomic arrangement of the heavy atoms, so long as there are not too many of them. From the positions of the heavy metals in the unit cell, one can calculate the amplitudes and phases of their contribution to the diffracted beams of the protein crystals containing heavy metals. 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

It is shown by empirical tests that the radial distribution function given by a sum of Fourier terms corresponding to the rings observed on an electron diffraction photograph of gas molecules... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

By deriving or computing the Maxwell equation in the frame of a cylindrical geometry, it is possible to determine the modal structure for any refractive index shape. In this paragraph we are going to give a more intuitive model to determine the number of modes to be propagated. The refractive index profile allows to determine w and the numerical aperture NA = sin (3), as dehned in equation 2. The near held (hber output) and far field (diffracted beam) are related by a Fourier transform relationship Far field = TF(Near field). 

To check the identity and purity of the products obtained in the above reactions it is not sufficient to analyze for the sulfur content since a mixture may incidentally have the same S content. Either X-ray diffraction on single crystals or Raman spectra of powder-like or crystalline samples will help to identify the anion(s) present in the product. However, the most convincing information comes from laser desorption Fourier transform ion cyclotron resonance (FTICR) mass spectra in the negative ion mode (LD mass spectra). It has been demonstrated that pure samples of K2S3 and K2S5 show peaks originating from S radical anions which are of the same size as the dianions in the particular sample no fragment ions of this type were observed [28]. 

Spectroscopic techniques as 13C-NMR [28], ESR [29], pyrolysis-GC/MS, and pyrolysis-Fourier transform infrared (FTIR) [30], x-ray diffraction [31], and SEM [32] techniques are also used to study mbber oxidation. 

When applied to the XRD patterns of Fig. 4.5, average diameters of 4.2 and 2.5 nm are found for the catalysts with 2.4 and 1.1 wt% Pd, respectively. X-ray line broadening provides a quick but not always reliable estimate of the particle size. Better procedures to determine particle sizes from X-ray diffraction are based on line-profile analysis with Fourier transform methods. 

---