Ray Diffraction by Crystals

Virtually all we know about crystal structure has been learned from X-ray diffraction studies. X-ray diffraction refers to the scattering ofX rays by the units of a crystalline solid. The scattering, or diffraction, patterns produced are used to deduce the arrangement of particles in the soUd lattice. 

William H. Bragg and Sir William L. Bragg. The reinforced waves produce a dark spot on a photographic film for each value of 6 that satisfies the Bragg equation. Example 11.4 illustrates the use of Equation (11.1). 

Reinforced waves aie waves that have interacted constructively (see Fi9ure 10.22). 

X rays of wavelength 0.154 nm strike an aluminum crystal the rays are reflected at an angle of 19.3°. Assuming that n = 1, calculate the spacing between the planes of aluminum atoms (in pm) that is responsible for this angle of reflection. The conversion 

Solution Converting the wavelength to picometers and using the angle of reflection 

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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. 

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). 

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. 

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. 

In 1912, when M. Lane suggested to W. Friedrich and P. Knipping the irradiation of a crystal with an X-ray beam in order to see if the interaction between this beam and the internal atomic arrangement of the crystal could lead to interferences, it was mainly meant to prove the undulatoiy character of this X-ray discovered by W.C. Rontgen 17 years earher. The experiment was a success, and in 1914 M. Laue received the Nobel Prize for Physics for the discovery of X-ray diffraction by crystals. In 1916, this phenomenon was used for the first time to study the structure of polycrystalhne samples. Throughout the 20 century. X-ray diffraction was, on the one hand, studied as a physical phenomenon arrd explained in its kinematic approximation or in the more general context of the dynamic theory, and on the other, implemented to study material that is mainly solid. 

The introduction recounts the history of the emphasis on X-ray diffraction by crystals since the discovery of X-rays. The book is then divided irrto two parts. The first part focuses on the description of the basic theoretical concepts, the irrstrumerttation arrd the presentation of traditional methods for data processing and the irrterpretation of the results. The second part is devoted to a more specific domain which is the quantitative study of the rrricrostmcture by X-ray diffraction. 

At this point, one may wonder why the studies on modulated structures only recently acquired some momentum. One obvious answer lies in the availability of the tools required to perform these studies. The discovery of the 3D space groups that are fundamental for the symmetry description of crystalline material preceded by a quarter of a century the discovery of x-ray diffraction by crystals. Shortly after, a selected number of crystal structures were described for the first time by W.H. and W.L. Bragg. For aperiodic crystals and, in particular, modulated ones, all the tools and methodologies necessary to study chis new type of materials had to be developed before further progress could be made. The development of the mathematical theory of superspace was instrumental for further success. After more than three decades of development, the availability of good performing tools is just starting to appear and explains the relatively belated development of this speciality. 

The method of x-ray diffraction by crystals has been used to determine the detailed structures of a number of globular proteins. The alpha helix and the two pleated sheets have been found to be the main types of secondary structure in these proteins. The location of the catalytically active region of an enzyme can be discovered by the x-ray study of crystals of the enzyme combined with an inhibitor. 

A subject emphasized in this article concerns the role of non-negativity in structural research in terms of its immediate applications, its special mathematical features and the use of additional probabilistic insights. The experimental techniques considered are electron diffraction by gases and X-ray diffraction by crystals. 

In this historical outline of some of the main contributions of mathematics to structural research in the areas of electron diffraction by gases and X-ray diffraction by crystals, it has been seen that these areas have benefited considerably by the application of mathematics. The same is true of other areas of structural research. It is also so that improvements in the mathematical techniques bode well for the future. In view of the broad implications of structural research to many fields of science, it may be said that structural research is an example of a discipline in which those who enjoy the application of mathematics to scientific problems can also enjoy the broad fruitfulness of the results. 

A sample of nickel [in Davisson-Germer experiments] was accidentally converted into crystalline form and, when subjected to the electron beam, produced totally unexpected diffraction patterns. .. similar to those observed in X-ray diffraction by crystals. .. Such behavior indicated that electrons, like electromagnetic radiation, possess wave characteristics (O Connor 1974, p. 50). 

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