Scattering in X-ray diffraction

Crystals also diffract neutrons, and this fact is useful to the crystal-lographer because it is the nuclei of the atoms that scatter neutrons rather than the electrons (which are the scatterers in X-ray diffraction). Among the difficulties encountered if one wishes to use neutron diffraction, however, are a need for bigger crystals and the uncertainty of availability of time at a nuclear reactor where neutron flux is available. The results of such neutron diffraction studies are particularly valuable for precise location of hydrogen atoms, for the differentiation of atoms of nearly the. same atomic number, and for distinguishing isotopes. 

The RMC method is more general than this simple algorithm in that any set(s) of data that can be directly related to the structure can be modelled. It can be applied to isotopic substitution in neutron diffraction, or equivalently to anomalous scattering in X-ray diffraction, to EXAFS (multiple edges) and possibly to NMR data. All data sets can be modelled simultaneously simply by adding the respective y1 values. 

Interference of Waves. The coherent scattering property of x-rays is used in x-ray diffraction appHcations. Two waves traveling in the same direction with identical wavelengths, X, and equal ampHtudes (the intensity of a wave is equal to the square of its ampHtude) can interfere with each other so that the resultant wave can have anywhere from zero ampHtude to two times the ampHtude of one of the initial waves. This principle is illustrated in Figure 1. The resultant ampHtude is a function of the phase difference between the two initial waves. 

When Davisson and Germer reported in 1927 that the elastic scattering of low-energy electrons from well ordered surfaces leads to diffraction spots similar to those observed in X-ray diffraction [2.238-2.240], this was the first experimental proof of the wave nature of electrons. A few years before, in 1923, De Broglie had postulated that electrons have a wavelength, given in A, of ... 

P. Canton, C. Meneghini, P. Riello, A. Benedetti, in B. M. Weekhuysen (ed.) In-Situ Spectroscopy of Catalysts in X-ray Diffraction and Scattering, American Scientific Publishers, 2004, 281. 

This chapter deals with the study of structural properties of catalysts and catalytic model surfaces by means of interference effects in scattered radiation. X-ray diffraction is one of the oldest and most frequently applied techniques in catalyst characterization. It is used to identify crystalline phases inside catalysts by means of lattice structural parameters, and to obtain an indication of particle size. Low energy electron diffraction is the surface sensitive analog of XRD, which, however, is only applicable to single crystal surfaces. LEED reveals the structure of surfaces and of ordered adsorbate layers. Both XRD and LEED depend on the constructive interference of radiation that is scattered by relatively large parts of the sample. As a consequence, these techniques require long-range order. 

Finally, in x-ray diffraction measurements the knowledge of the equation of state (EOS) of specific materials is used to calculate the pressure by the measurement of selected reflections. For this purpose, Au [266], NaCl [267], Re [268], Pt [269], and MgO [231] have been used in x-ray scattering measurements when no optical access was available. 

In X-ray diffraction one is interested in exploring the intensity of X-rays diffracted from the crystal planes. Note that the Bragg equation does not contain information about the scattered intensity from a given plane. It only provides the... 

Extinction, which is the failure of the kinematic scattering theory (Ihki hki) is only a minor problem in X-ray diffraction. In neutron diffraction, extinction is serious and pervasive throughout the whole data, as shown by the examples in Thble 3.2. The best methods available for extinction correction require careful measurement of crystal dimensions. Although somewhat empirical, it has proved to be very effective [184, 185]. At least one, and sometimes six, additional extinction parameters, gis0 or gij, have to be added to the variable parameters. Uncertainty in the validity of these extinction parameters appears to have very little effect on atomic positional coordinates, but may influence the absolute values of the atomic temperature factors. This is important in charge density or electrostatic potential... 

Scattering angle The angle at which a scattered wave deviates from the direct beam. Conventionally in X-ray diffraction, as a result of Bragg s Law, it is designated dhkl-... 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

It is worth noting that it is the radial distribution of core electrons in an atom, which is responsible for the reduction of the intensity when the diffraction angle increases. Thus, it is a specific feature observed in x-ray diffraction from ordered arrangements of atoms. If, for example, the diffraction of neutrons is of concern, they are scattered by nuclei, which may be considered as points. Hence, neutron scattering functions (factors) are independent of the diffraction angle and they remain constant for a given type of nuclei (also see Table 2.2). 

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