Diffraction process

With XRD applied to bulk materials, a detailed structural analysis of atomic positions is rather straightforward and routine for structures that can be quite complex (see chapter B 1.9) direct methods in many cases give good results in a single step, while the resulting atomic positions may be refined by iterative fitting procedures based on simulation of the diffraction process. 

The value of k is not universal and depends on the nature of the reactive mixture (for example, fhe value of k may be around 26 for mixtures highly diluted with a mono-atomic gas [29], or around 20 for hydrogen/air mixtures [30]), as well as on the diffraction process at the tube... 

The pressures on the sides and roof of the structure build up to the incident overpressure as the blast wave traverses the structure. Traveling behind the blast wave front there is a short period of low pressure caused by a vortex formed at the front edge during the diffraction process (Figures A. 8c and A. 9c). After the vortex has passed, the pressure returns essentially to that in the incident blast wave. The air flow causes some reduction in the loading to the sides and roof, because the drag pressure has a negative value for these surfaces. 

Later chapters will deal with a more complete description of the diffraction process, but we now have enough to discuss the selection of radiations and techniques. If the structure factor and scattering strength of the radiation are high, the penetration is low and the rocking curve is broad. This is the case with electron radiation. For X-rays and even more for neutrons, the structure and absorption factors are small, penetration is high and rocking curves are narrow. These factors have three main consequences for X-rays and also for neutrons ... 

Cope rearrangement is known to take place in semibullvalene (S4)368 and bull-valene ( 5)369. This process is quite slow compared to the electron-diffraction process, and the bond distances of (84) and (85) are therefore found to be similar to normal single and double bonds. 

One advantage of LEED is that the diffraction process filters out effects due to local defects or deviations from long-range order. The contribution of defects to I-V curves is proportional to the first power of the number of defects, while the contribution of the part of the surface with long-range order is proportional to the square of the number of atoms involved, so the LEED beam integrated intensity reflects the equilibrium geometry of the ordered surface structure. 

Chapter 1 is concerned with the fundamental principles of image formation by a lens. These principles were first formulated by Ernst Abbe in 1873 and are basic to the chapters that follow. According to the Abbe theory, the image of an illuminated object is the result of a twofold diffraction process. First, the Fraunhofer diffraction pattern of the object is formed in the back focal plane of the lens. Second, the light waves travel... 

The detailed interpretation of electron microscope images produced using any of the operating modes discussed in this chapter requires as complete an understanding as possible of the diffraction process. The next two chapters develop and explain as simply as possible the current theories of electron diffraction by crystals in order to provide a basis for the interpretation of images of crystal defects (such as dislocations, stacking faults, and twins) and of lattice images. 

In the case of plan view images the information due to the particle and the support overlap in the space. From the point of view of the interaction of the electron beam with the sample, the most relevant aspect in this case is the occurrence of a double diffraction process. The electron beams, which aregenerated in a first diffraction process in the particles, are further diffiacted by the support crystallites. Doubly... 

How is it possible to derive phase information when only structure amplitudes have been measured An answer can be found in what are called direct methods of structure determination. By these methods the crys-tallographer estimates the relative phase angles directly from the values of F hkl) (the experimental data). An electron-density map is calculated with the phases so derived, and the atomic arrangement is searched for in the map that results. This is why the method is titled direct. Other methods of relative phase determination rely on the computation of phase angles after the atoms in a trial structure have been found, and therefore they may be considered indirect methods. Thus, the argument that phase information is lost in the diffraction process is not totally correct. The phase problem therefore lies in finding methods for extracting the correct phase information from the experimental data. 

In the mid-1970 s, with the availability of intense X-ray synchrotron sources, a powerful new technique. X-ray absorption spectroscopy (XAS), emerged. This is a local structural probe, the information content of which derives from electron diffraction. For a metalloprotein, the electron source and detector is the metal atom that is probed, because selective excitation is achieved by scanning a range of X-ray wavelengths particularly appropriate to the element of central interest. The selectivity and the local nature of the diffraction process give the technique its major strength. For example, metal-ligand distances can be determined to an accuracy of approximately 0.02 A. In addition, XAS does not require crystalline materials thus, aqueous protein samples are readily probed under a variety of conditions. 

The crystallographic plane is a geometrical concept (mathematical abstraction) introduced to illustrate the phenomenon of diffraction from ideal crystal lattices since algebraic equations that govern diffraction process are difficult to visualize. It is important to realize and remember that no real ... 

The use of a monochromator produces a change in the relative intensities of the beams diffracted by the specimen. Equation (4-19), for example, was derived for the completely unpolarized incident beam obtained from the x-ray tube. Any beam diffracted by a crystal, however, becomes partially polarized by the diffraction process itself, which means that the beam from a crystal monochromator is partially polarized before it reaches the specimen. Under these circumstances, the usual polarization factor (1 - - cos 26)12, which is included in Eqs. (4-19) through (4-21), must be replaced by the factor (1 + cos 2a cos 20)/(l -I- cos 2a), where 2a is the diffraction angle in the monochromator (Fig. 6-16). Since the denominator in this expression is independent of 6, it may be omitted the combined Lorentz-polarization factor for crystal-monochromated radiation is therefore (1 + cos 2a cos 20)/sin 6 cos 6. This factor may be substituted into Eqs. (4-19) and (4-20), although a monochromator is not often used with a Debye-Scherrer camera, or into Eq. (4-21), when a monochromator is used with a diffractometer (Sec. 7-13). But note that Eq. (4-20) does not apply to the focusing cameras of the next section. 

Elton and Salt have used both theoretical and experimental methods to estimate the number of crystallites diffracting (Adiff) in a sample. Fluctuations in line intensity between replicate samples arise largely from statistical variation in the number of particles contributing to the diffraction process. It has been shown that small changes to the instrumental and sample configurations can... 

Each diffraction spot is caused by reflection of X-rays by a particular set of planes in the crystal. If the crystal contains layers of atoms with the same spacing and orientation as a particular set of planes which would satisfy Bragg s law (if the set of planes is physically present), the corresponding diffraction spot will be strong. On the other hand, if only few atoms in a crystal correspond to a particular set of planes, the corresponding reflection will be weak. The complicated structure present in the crystal is transformed by the diffraction process into a set of diffraction spots which correspond to sets of planes (more precisely, sinusoidal density waves), just as our ear converts a complicated sound signal into a series of (sinusoidal) tones when we listen to music. This conversion of a complicated function into a series of simple sine and cosine functions is called a Fourier transformation. 

Since the mass of the neutron (w , kg) is known, as are the neutron flight distances d, m), for an elastic scattering (diffraction) process the flight time t, ps) determines the neutron velocity (v ) and hence its energy E, cm ) since (cf Eq. 2.21)... 

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