Diffraction of X-rays

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... 

Many scientifically and teclmologically important substances caimot be prepared as single-crystals large enough to be studied by single crystal diffraction of x-rays and, especially, neutrons. If a sample composed of... 

James R W 1965 The Optical Principles of the Diffraction of X-Rays (Cornell University Press) ch III... 

Bragg W L 1913 The structure of some crystals as indicated by their diffraction of X-rays Proo. R Soo. A 89 248-60... 

Diffraction of x-rays from many atoms with many electrons results in a two-dimensional continuous pattern of peaks and valleys. Diffraction from a hquid or an amorphous (noncrystalline) soHd results in a continuous pattern with few features, mainly broad peaks and valleys. 

Because of Bragg s explanation of diffraction of x-rays from a crystal as being like reflections from famihes of planes, the diffraction spots ate usually called "reflections." Each reflection is identified with three integer indices, h, k, and / For the set of planes shown in Figure 7, the indices of the corresponding reflection are /i = 1, = 0, and I = 2. 

Figure 18.6 Diffraction of x-rays by a crystal, (a) When a beam of x-rays (red) shines on a crystal all atoms (green) in the crystal scatter x-rays in all directions. Most of these scattered x-rays cancel out, but in certain directions (blue arrow) they reinforce each other and add up to a diffracted beam, (b) Different sets of parallel planes can be arranged through the crystal so that each corner of all unit cells is on one of the planes of the set. The diagram shows in two dimensions three simple sets of parallel lines red, blue, and green. A similar effect is seen when driving past a plantation of regularly spaced trees. One sees the trees arranged in different sets of parallel rows.

Wilson, H.R. Diffraction of X-rays by Proteins, Nucleic Acids and Viruses. London Edward Arnold, 1966. 

Wavelength Spectrometry (WDS) is based upon the phenomenon of Bragg diffraction of X rays incident on a crystal. The difiraction phenomenon is described by the expression ... 

The classical approach for determining the structures of crystalline materials is through diflfiaction methods, i.e.. X-ray, neutron-beam, and electron-beam techniques. Difiiaction data can be analyzed to yield the spatial arrangement of all the atoms in the crystal lattice. EXAFS provides a different approach to the analysis of atomic structure, based not on the diffraction of X rays by an array of atoms but rather upon the absorption of X rays by individual atoms in such an array. Herein lie the capabilities and limitations of EXAFS. 

Bijvoet, J.M., Burgers, W.G. and Hagg, G. (1972) Early Papers on Diffraction of X-rays by Crystals (Int. Union of Crystallography, Utrecht) pp. 5. 

M. von Laue (Frankfurt) discovery of the diffraction of X-rays by crystals. 

Bragg equation An equation relating the angle of diffraction of x-rays to the spacing of layers of atoms in a crystal (X = 2d sin 6). 

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 notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

Figure 2.68 The diffraction of X-rays from a crystal lattice.

The technique of using the diffraction of X-rays to study the structure of fibres, plastics and elastomers, to determine the crystalline nature of such polymers, and to measure the particle size of finely divided materials. 

Structure determination has greatly advanced with the invention of new ways to use x-ray crystallography, mainly new mathematical methods that permit the interpretation of the observed patterns of diffraction of x-rays by a crystal, translating it into the molecular structures in the crystal. A Nobel Prize in 1985 to Herbert Hauptmann and Jerome Karle recognized such an advance. 

The diffraction of x-rays by crystalline substances is of great analytical interest, since it is only by pure coincidence that two compounds would form crystals in... 

The mathematics necessary to understand the diffraction of X rays by a crystal will not be discussed in any detail here. Chapter 4 of reference 10 contains an excellent discussion. The arrangement of unit cells in a crystal in a periodic manner leads to the Laue diffraction conditions shown in equations 3.3 where vectors a, b, and c as well as lattice indices h, k, and l have been defined in Figure 3.5 and S is a vector quantity equal to the difference between the resultant vector s after diffraction and the incident X-ray beam wave vector So so that S = s - So-... 

Vainshtein, B. K. (1966). Diffraction of X-Rays by Chain Molecules. Elsevier, Amsterdam. Valery, C., Patemostre, M., Robert, B., Gulik-Krzywicki, T., Narayanan, T., Dedieu, J. C., Keller, G., Torres, M. L., Cherif-Cheikh, R., Calvo, P., and Artzner, F. (2003). Biomimetic organization Octapeptide self-assembly into nanotubes of viral capsid-like dimension. Proc. Natl. Acad. Sci. USA 100, 10258-10262. 

Unlike the case of diffraction of light by a ruled grating, the diffraction of x-rays by a crystalline solid leads to the observation that constructive interference (i.e., reflection) occurs only at the critical Bragg angles. When reflection does occur, it is stated that the plane in question is reflecting in the nth order, or that one observes nth order diffraction for that particular crystal plane. Therefore, one will observe an x-ray scattering response for every plane defined by a unique Miller index of (h k l). 

As noted earlier, the diffraction of X-rays, unlike the diffraction of neutrons, is primarily sensitive to the distribution of 00 separations. Although many of the early studies 9> of amorphous solid water included electron or X-ray diffraction measurements, the nature of the samples prepared and the restricted angular range of the measurements reported combine to prevent extraction of detailed structural information. The most complete of the early X-ray studies is by Bon-dot 26>. Only scanty description is given of the conditions of deposition but it appears likely his sample of amorphous solid water had little or no contamination with crystalline ice. He found a liquid-like distribution of 00 separations at 83 K, with the first neighbor peak centered at 2.77 A. If the pair correlation function is decomposed into a superposition of Gaussian peaks, the area of the near neighbor peak is found to correspond to 4.23 molecules, and to have a root mean square width of 0.50 A. 

S.W.WILKINS A.W.STEVENSON, Nucl. Inst. Meth., A269,321 (1988). R.W.JAMES, The optical principles of the diffraction of X-rays (Ox Bow Press, Connecticut 1982) p. 308. ... 

It is of interest to attempt to cast the absorption-edge spectrum in a form resembling a diffraction function. In conventional diffraction of X-rays or electrons by gases, liquids, or crystals, the general diffraction function expresses the ratio of scattered intensity, 7 to incident intensity, 7o, as a function of (sin 0)/, where d is half the scattering angle ... 

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