Interference among waves

It is worth noting that the phase of g Da ESk ) is not equal to that of (g Db ESk ) even in the above limit, a point that was the source of confusion in the previous literature [40]. It is only upon integration over scattering angles that interference among different partial waves is eliminated (see Section IVA), and an observed phase implies interference within a partial wave. Hence angle-resolved measurements may observe a nonzero phase regardless of the nature of the continuum. 

As indicated in Section 3.4, after detection the FID from a system giving a single NMR line consists of an exponentially decaying cosine wave of frequency (to — Wrf), as illustrated in Fig. 3.4a. If there are several nuclei that differ in Larmor frequency because of chemical shifts and/or spin—spin coupling, each line corresponds to a different frequency in the FID, and the interference among all these signals generates a response of the sort depicted in Fig. 3.46. To extract those component frequencies we usually turn to Fourier transform procedures, which result in a clear display of the component frequencies present in the FID as an NMR spectrum. 

When more than one point is affected by the same incident wave, the overall scattered amplitude will be a result of interference among multiple spherical waves. As established above Figure 2.21) the amplitude will vary depending on the difference in the phases of multiple waves with parallel propagation vectors but originating from different points. 

So far we have discussed only independent scattering from particles, that is, the particles are so far apart from each other that the effect of interference among the waves scattered by different particles can be ignored. We now consider what happens to the observed intensity as the concentration of particles in the system is increased and the interference effect becomes no longer negligible. 

X-ray diffractometry is widely used for the characterization of electrode materials for the battery field, electrocatalysis, etc. Both areas require the development of new materials whose three-dimensional stracture is critical for their electrochemical activity. In brief, XRD is based on a monochromatic X-ray beam that hits the sample and is reflected from it at a variety of scattering angles. Since the X-rays are reflected by the atoms in the sample s lattice, and since the wave length is of the same order of magnitude as interatom distances in the solid state, interference among the reflected X-rays occurs, leading to typical, imique diffraction patterns for each specific material. A completed analysis of lattice stractures can be obtained fiom judicious treatment of the data in the XRD patterns. 

In die following sections we desc e the interaction of monochromatic, coherent radiation witti scattering centers, whidi results in spherical scattered waves. Interference among these waves creates the intensity pattern sensed by a detector. The connection between the observed scattered wave intensity and the structure of matter is ultimately sought. We are especially interested to find die spatial periodicities within our material that lead to interference. To this end, the mathematical techniques of Fourier transformation and convolution are presented. We end the chapter with sections on the small angle scattering from lamellar systems, and neutron scattering. 

In a collection of atoms containing many electrons, the scattered radiation collected at a distant point P shown in Figure 3 (drawn after Ref. 6) will depend upon the phase relationship among the waves scattered from each atom. A large intensity will be measured at P provided that there is constructive interference among the scattraed waves emitted by the atoms. On the other hand, if crests from some scattered waves overlap... 

Diffraction is based on wave interference, whether the wave is an electromagnetic wave (optical, x-ray, etc), or a quantum mechanical wave associated with a particle (electron, neutron, atom, etc), or any other kind of wave. To obtain infonnation about atomic positions, one exploits the interference between different scattering trajectories among atoms in a solid or at a surface, since this interference is very sensitive to differences in patii lengths and hence to relative atomic positions (see chapter B1.9). 

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