Diffraction by crystals

A crystal consists of a large number of unit cells arranged regularly in three-dimensional space, with each unit cell having the identical atomic content. The shape and size of the unit cell are defined by the three unit cell vectors a, b, c. The origin of each unit cell is on a lattice point, whose position is specified as 

The content of the unit cell is defined by specifying the positions of all the atoms it contains. It is more convenient, however, to specify the unit cell content by the distribution pu(r) of the appropriate scattering length density, so that the same expression can be used in discussing both x-ray and neutron diffraction. The convolution product 

The amplitude A(s) of scattered x-rays or neutrons is equal to the Fourier transform of p(r), and therefore, by taking the Fourier transform of (1.95) and using the convolution theorem (see Appendix B), we obtain 

While F(s) is a continuously varying function of s, its magnitude is experimentally observable and therefore is meaningful only at discrete values of s corresponding to 

If Fhid is known for a large number of hkl reflections, (1.98) can be inverted to obtain pu(r) and hence the positions of all the atoms in the unit cell. Such an endeavor is called the crystal structure analysis and is explained in more detail in Section 3.3. The intensity of reflection, observed at s = r%kl is equal to F/ / 2. The absolute value of Fhki can therefore be obtained as the square root of the observed intensity of the hkl reflection, but the intensity data do not provide any direct information about the phase angle of the complex Fw A major task in crystal structure analysis is solving the phase problem to determine the phases of the structure factors. 

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Peng, L.-M. (1997) Anisotropic thermal vibrations and dynamical electron diffraction by crystals, Acta Cryst. A, 53, 663-672. 

The techniques of X-ray diffraction analyses of crystals of compounds of interest can be used to determine, with high precision, the three-dimensional arrangement of atoms, ions and molecules in such crystals (14) in each case the result is referred to as the "crystal structure." X-ray diffraction by crystals was discovered by von Laue, Friedrich and Knipping (15) and the technique was applied by the Braggs to the determination of the structures of... 

Max von Laue, 1879-1960. German physicist who in 1912 discovered the interference of X-rays diffracted by crystals, measured the wave lengths of X rays, and studied the structure of crystals. In 1914 he was awarded the Nobel Prize for physics. 

Hitherto only the positions of the X-ray beams diffracted by crystals have been considered unit cell dimensions are determined from the positions of diffracted beams without reference to their intensities. To discover the arrangement and positions of the atoms in the unit cell it is necessary to consider the intensities of the diffracted beams. 

Neutron diffraction by crystals has been found valuable for locating hydrogen atoms (especially deuterium atoms, which scatter neutrons strongly), for studying the arrangement of magnetic moments, and for other special purposes. A summary is given by G. E. Bacon, Neutron Diffraction, Clarendon Press, Oxford, 1955. 

The electrons, however, besides properties which suggest a conception of them as charged particles, have as much the character of a wave phenomenon. Their diffraction by crystal lattices is entirely analogous to that of X-rays (p. 106). 

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. 

Several types of diffraction by crystals are now studied. Neutron diffraction can be used with great effectiveness to give information on molecular structure. These results complement those from X-ray diffraction studies, because there are different mechanisms for the scattering of X rays and of neutrons by the various atoms. X rays are scattered by electrons, while neutrons are scattered by atomic nuclei. Neutron diffraction is important for the determination of the locations of hydrogen atoms which, because of their low electron count, are poor X-ray scatterers. Electron diffraction, while requiring much smaller crystals and therefore being potentially useful for the study of macromolecules, produces diffraction patterns that are more complicated. Their interpretation is hampered by the fact that the diffracted electron beams are rediffracted within the crystal much more than are X-ray beams. This has limited the practical use of electron diffraction in the determination of atomic arrangements in crystals to studies of surface structure. 

Once it had been shown that crystals diffract X rays, the relationship between the observed effect and the experimental conditions was put on a sound mathematical basis by Max von Laue, Paul P. Ewald and many others.X-ray diffraction by crystals represents the interference between X rays scattered by the electrons in the various atoms at various locations within the unit cell. It must, however, be stressed again that any molecule or ion can diffract X rays or neutrons. It is only when this diffraction is reinforced by the repetition of the unit cell in the crystal that diffraction by atoms is a conveniently observable effect, for example as spots of differing intensity on photographic film. Of particular interest to chemists and biochemists is the work by W. L. Bragg,who demonstrated that measurement of the diffraction patterns gives information on the distribution of electron density in the unit cell, (i.e., the arrangement of atoms within this unit cell). 

Methods for the experimental measurement of the intensities of the diffracted beams will be described in Chapter 7, and methods for deriving relative phases of these beams for recombination will be discussed in Chapter 8. The result of these mathematical calculations, which simulate the action of a lens, will be all the information that is needed for the calculation of a three-dimensional electron-density map. This is a map in which peaks are situated at or near atomic positions. In this way, measurements of the diffraction pattern lead to an image of the molecules or ions and their arrangement in the crystal under study. Details of the calculation of the electron density maps that reveal the atomic arrangement will be described in Chapter 9. For more information on each aspect of diffraction by crystals, the reader is referred to the many texts on the subject listed in the Preface to this book. 

An example is provided by a comparison of the diffraction patterns of the isostructural chlorides of sodium and potassium (see Figure 6.20). It is noted that alternate rows of diffraction spots are very faint in the potassium chloride diffraction pattern, unlike the situation for sodium chloride. This alternating pattern of intensity is due to the fact that potassium and chloride ions are isoelectronic (with 18 electrons), and therefore have approximately identical powers to scatter X rays. On the other hand, the difference in scattering power between a sodium ion (10 electrons) and a chloride ion (18 electrons) is appreciable. Therefore those diffraction spots in which scattering from the metal ion interferes with scattering from the chloride ion will have a measurable intensity for diffraction by crystals of sodium chloride but almost no intensity for diffraction by crystals of potassium chloride. 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation. ... 

About 50 years ago (1912), Max von Laue discovered X-ray diffraction by crystals with the application of the X-ray method to the determination of crystal structures, especially by Wilham Henry Bragg and Sir Lawrence Bragg (since 1913), the development of crystallography took a new direction, thereby making an enormous impact on science and technology. 

Explain how x-rays and nentrons are diffracted by crystals, and nse information from such experiments to calculate lattice spacings (Section 21.1, Problems 5-10). 

Figure 2.6 Bragg diffraction by crystal planes. The path difference between beams 1 and 2 is SQ + QT = 2 PQ sin0. (Reproduced with permission from W.J. Callister Jr., Materials Science and Engineering An Introduction, 7th ed., John Wiley Sons Inc., New York. 2006 John Wiley Sons Inc.)...

Figure 13-19 (a) X-ray diffraction by crystals (schematic), (b) A photograph of the X-ray diffraction pattern from a crystal of the enzyme histidine decarboxylase (MW 37,000 amu). The crystal was rotated so that many different lattice planes with different spacings were moved in succession into diffracting position (see Figure 13-20). 

Radiography was thus initiated without any precise understanding of the radiation used, because it was not until 1912 that the exact nature of x-rays was established. In that year the phenomenon of x-ray diffraction by crystals was discovered, and this discovery simultaneously proved the wave nature of x-rays and provided a new method for investigating the fine structure of matter. Although radiography is a very important tool in itself and has a wide field of applicability, it is ordinarily limited in the internal detail it can resolve, or disclose, to sizes of the order of 10 cm. Diffraction, on the other hand, can indirectly reveal details of internal structure of the order of 10 cm in size, and it is with this phenomenon, and its applications to metallurgical problems, that this book is concerned. The properties of x-rays and the internal structure of crystals are here described in the first two chapters as necessary preliminaries to the discussion of the diffraction of x-rays by crystals which follows. 

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