Scattering of X-rays

Guinier A and Fournet G 1955 Smaii Angie Scattering of X-rays (New York Wiley)... 

X-ray fluorescence spectrometry consists of the measurement of the incoherent scattering of x-rays (phenomenon 3 above). It is used primarily to determine the elemental composition of a sample. 

X-ray diffraction consists of the measurement of the coherent scattering of x-rays (phenomenon 4 above). X-ray diffraction is used to determine the identity of crystalline phases in a multiphase powder sample and the atomic and molecular stmctures of single crystals. It can also be used to determine stmctural details of polymers, fibers, thin films, and amorphous soflds and to study stress, texture, and particle size. 

The spectroscopic techniques that have been most frequently used to investigate biomolecular dynamics are those that are commonly available in laboratories, such as nuclear magnetic resonance (NMR), fluorescence, and Mossbauer spectroscopy. In a later chapter the use of NMR, a powerful probe of local motions in macromolecules, is described. Here we examine scattering of X-ray and neutron radiation. Neutrons and X-rays share the property of being found in expensive sources not commonly available in the laboratory. Neutrons are produced by a nuclear reactor or spallation source. X-ray experiments are routinely performed using intense synclirotron radiation, although in favorable cases laboratory sources may also be used. 

A. Guinier, G. Foumet, Small-Angle Scattering of X-Rays (John Wiley Sons,N, Y., 1955),... 

Narten, A. H., Levy, H. A. Liquid Water Scattering of X-Rays, in Water — a Comprehensive Treatise, (ed. Franks, F.), Vol. 1, chapter 8, New York, Plenum Press 1972... 

The reflection 7 of x-rays by a crystal (x-ray diffraction) results fundamentally from the unmodified scattering of x-rays by the atoms or ions in the different planes of the crystal. For many purposes, it suffices to regard the atoms or ions as the scattering centers, but a complete description of the process requires that the electrons be considered. 

As one may infer from the quotation, W. L. Bragg realized that a crystal can act as an x-ray grating made up of equidistant parallel planes (Bragg planes) of atoms or ions from which unmodified scattering of x-rays can occur in such fashion that the waves from different planes are in phase and reinforce each other. When this happens, the x-rays are said to undergo Bragg reflection by the crystal and a diffraction pattern results. 

Guinier A, Fournet G (1955) Small angle scattering of X-rays. Wiley, New York... 

X-ray diffraction occurs in the elastic scattering of X-ray photons by atoms in a periodic lattice. The scattered monochromatic X-rays that are in phase give constructive interference. Figure 4.4 illustrates how diffraction of X-rays by crystal planes allows one to derive lattice spacings by using the Bragg relation ... 

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

Projectors often arise in attempts to describe experiments within the structure of Quantum Mechanics. For example, in the case of the coherent scattering of X-rays by crystals the ideal measured intensities are given by the square of the structure factors... 

Although single-crystal x-ray diffraction undoubtedly represents the most powerful method for the characterization of crystalline materials, it does suffer from the drawback of requiring the existence of a suitable single crystal. Very early in the history of x-ray diffraction studies, it was recognized that the scattering of x-ray radiation by powdered crystalline solids could be used to obtain structural information, leading to the practice of x-ray powder diffraction (XRPD). 

A broadening in Mark s intellect is shown clearly in his publications of this period. The topics in 1926 and 1927 alone ranged from atomic structure and quantum theory (1), and Compton radiation (2) to the scattering of x-rays by an ideal gas (3) and mineral structure (4). The shear diversity of his contacts and interests made him qualified for expanded responsibilities. 

Work on the structure of crystals and fibers was not the only way in which Mark made use of x-rays. With several collaborators, he reported the results of a number of significant investigations of the physics of x-rays in 1926 and 1927. With Ehrenberg he reported studies of the index of refraction of x-rays, and with Leo Szilard studies verifying the linear polarization of x-rays scattered from electrons at 90. An investigation of the width of x-ray lines was carried out by Mark and Ehrenberg, and Mark and Kallmann reported work on the properties of Compton-scattered x-radiation and on the theory of the dispersion and scattering of x-rays. 

This discussion brings out an important principle that will reappear time and again in this chapter. If we can assign the absolute direction of the hand vis-a-vis the polar axis b (Figure 2), that is, establish whether the wrist or the fingers of the hand point up, then the absolute chirality of the hand is fixed as well. This determination of absolute polarity can be achieved in several ways other than anomalous scattering of X-rays or neutrons, for example, by an external means such as a solvent, a chemical reagent, or a physical measurement (e.g.,... 

The structure factor S(q as defined in Eq. (54) in terms of the Ising pseudospins Si, in the framework of the first Bom approximation describes elastic scattering of X-rays, neutrons, or electrons, from the adsorbed layer. SCq) is particularly interesting, since in the thermodynamic limit it allows to estimate both the order parameter amplitude tj/, the order parameter susceptibility X4, and correlati length since for q near the superstructure Bragg reflection q we have (k = q— q%)... 

The one major difficulty with use of a flow cell is that the combination of pipe cleaner fibers and a large volume of mother liquor around the crystid greatly increases the background scatter of X-rays, particularly in the 3 A resolution region. For diffractometric data collection—and, probably, for area detector work as well—this problem is not serious, but... 

We will not treat in ary detail in this chapter the more advanced topic of scattering of X-rays by distorted crystals. Appropriate theory will be introduced in later chapters as required. In this section we will merely give qualitative descriptions of the effects of various imperfections and qnote some results. 

Z.G.PINSKER Dynamical scattering of X-rays in crystals (Springer, Berlin, 1977). 

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