Determination of the Crystal Structure

For crystals of other symmetries, the relationship between the direct and reciprocal lattice distances is more complex (see Section SI.7) 

Crystal stmctures are determined by using diffraction (see Section 14.7.3). The extent of diffraction is significant only when the wavelength of the radiation is very similar to the dimensions of the object that is irradiated. In the case of crystals, radiation with a wavelength similar to that of the spacing of the atoms in the crystal will be diffracted. X-ray diffi action is the most widespread technique used for structure determination, but diffraction of electrons and neutrons is also of great importance, as these reveal feamres that are not readily observed with X-rays. 

The physics of diffraction by crystals has been worked out in detail. It is found that the incident radiation is diffracted in a characteristic way, called a diffraction pattern. If the positions of the 

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H. A. Hauptman (Buffalo, NY) and J. Karle (Washington, DC) outstanding achievements in the development of direct methods for the determination of crystal structures. 

The preceding quotation serves admirably as a brief historical introduction to x-ray diffraction. This field of research has proved enormously fruitful in yielding information about crystal structure, and in providing a means of obtaining monochromatic x-rays and of measuring their wavelength. The determination of crystal structure, though important to analytical chemistry, is outside the scope of this book.31... 

The theoretically obtained electron densities of ions may be used for the calculation of the so-called F curves, which give the effective reflecting power of the ion as a function of the angle of reflection and the wave-length of X-rays, and which are of use in the determination of crystal structures. It may be mentioned that the high maximum value of the electron density at the nucleus given by our calculations provides considerable justification for the method of determining crystal structures with the aid of the relative intensities of Laue spots produced by crystal planes with complicated indices. 

HAUPTMAN, HERBERT A. (1917-). An American biophysicist who won ihe Nobel prize for chemistry in 1985 along with Jerome Karle for their outstanding achievements in the development nf direct methods for the determination of crystal structures. Hauptman s work involved developing equations that allow determination of phase information from X-ray crystallography intensity patterns. The use of computers permitted use of the equations to determine the conformation of thousands of chemicals. Haaptmao was director of research and v ice president of the Medical Foundation of Buffalo and a professor of biophysics in Buffalo at the Stale University of New York. 

The atoms that comprise a solid can be considered for many purposes to be hard balls which rest against each other in a regular repetitive pattern called the crystal structure. Most elements have relatively simple crystal structures of high symmetry, but many compounds have complex crystal structures of low symmetry. The determination of crystal structures, of atom location in the crystal, and of the dependence of many physical properties upon the inherent charactensdcs of the perfect solid is an absorbing study, one that has occupied the lives of numerous geologists, mineralogists, physicists, and other scientists for many years. 

The determination of crystal structure in synthetic polymers is often made difficult by the lack of resolution in the diffraction data. The diffuseness of the reflections observed in most x-ray fiber patterns results from the small size and imperfect lattice nature of the polymer crystallites. Resolution of individual reflections is also made difficult from misorientation of the crystallites about the fiber axis. This lack of resolution leads to poor accuracy in measurement of peak positions. In particular, this lack of accuracy makes determination of layer line heights difficult with a corresponding loss of significant figures in evaluation of the repeat distance for the molecular conformation. In the case of helical conformations, the repeat distance may be of considerable length or, as we shall show, indeterminate and, in effect, nonperiodic. This evaluation requires high accuracy in measurements of layer line heights. 

The determination of crystal structures by X-ray crystallography provides precise and unambiguous data on intermolecular interactions. Crystal engineering has been defined by Desiraju as the understanding of intermolecular interactions in the context of crystal packing and in the utilization of such knowledge in the design of new solids with desired physical and chemical properties. ... 

Lipson, H., and Cochran, W. The Determination of Crystal Structures, G. Bell and Sons, Ltd., London (1953). 

Hydrogen bonds are also formed in the solid state. It is true that in the determination of crystal structure by X-rays the position of the hydrogen atom can hardly be determined directly but an abnormally short distance between two oxygen atoms points to the presence of a hydrogen bond. 

Sheldrick G. M., SHELXS-86 A Program for the Determination of Crystal Structures from Diffraction Data, Univeersity of Gottingen, Germany (1990). 

A combination of powder diffraction and HRTEM thus becomes a very powerful tool in the determination of crystal structures. 

Structure is the fundamental property of polymorphs. In this section we will deal with the definition of that structure, how it may be best viewed and understood, and how polymorphic structures may be compared. The determination of crystal structure, by... 

The structure of 6-(tert-butylthio)-y0-CD was characterized by Tabushi et al. [21]. The compound of 6-O-(tert-butylthio)-y0-CD was prepared from the reaction of 6-O-(p-toluenesulfonyl)-y0-CD with tert-butylmercaptan and recrystallized in water. This is the first example of the determination of crystal structure of monosubstituted CD derivatives and the first evidence concerning the supramolecular polymer of an inclusion complex of a monosubstituted CD. The crystal structure of 6-O-(tert-butylthio)-y0-CD was arranged around the two-fold axis to yield a polymeric structure, in which the tert-butyl group is intermolecularly included in the cavity of CD (Fig. 3). 

The determination of crystal structure is then immediate, in principle, since any standard diffraction pattern will be related to, e.g., the product of an appropriate combination of three such delta functions (periodic in x,y,z directions), with atomic form factors. Inversion to get the real space atomic positions from the diffraction pattern is then possible via the convolution theorem for Fourier transforms, provided the purely technical problem of the undetermined phase can be solved. 

Bragg, W. H., von Laue, M., and Hermann, C. (Eds.) Internationale Tabellen zur Bestimmung von Kristallstrukturen. Ersier Band. Gruppentheoretische Tafeln. [International Tables for the Determination of Crystal Structures. First Volume. Tables on the Theory of Groups.] [In German, English and French.) Borntraeger Berlin (1935). 

Hill, R. J., and Madsen, I. C. The effect of profile step width on the determination of crystal structure parameters and estimated standard deviations by X-ray Rietveld analysis. J. Appl. Cryst. 19, 10-18 (1986). 

The Patterson function is a map that indicates all the possible relationships (vectors) between atoms in a crystal structure. It was introduced by A. Lindo Patterson " in 1934, inspired by earlier work on radial distribution functions in liquids and powders. In crystals the directionality as well as the lengths of vectors between atoms (atomic distances) can be deduced. By contrast, in liquids and powders the geometric information that can be obtained is limited to interatomic distances, because in these the molecules are randomly oriented. While the use of the Patterson function revolutionized the determination of crystal structures of small molecules in the 1930s to 1950s, direct methods are now the most widely used methods for obtaining structures of small organic molecules. The Patterson function, however, continues to play an essential part in the determination of crystal structures of inorganic compounds and macromolecules. It is also very useful when the structure of a small molecule proves difficult to solve by direct methods. 

Fitzgerald, P. M. D. MERLOT, an integrated package of computer programs for the determination of crystal structures by molecular replacement. J. Appl. Cryst. 21, 273-278 (1988). 

The determination of crystal structures of small molecules has now become routine provided that suitable crystals are available [362,363,364,365]. The advent of higher power X-ray sources, more synchronized light, and better recording devices has meant that smaller and smaller crystals can be solved by diffraction methods. At the same time, the analysis of patterns from larger and larger molecules has become easier so that the limit on the use of X-ray crystallography is the production of crystals. 

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. 

The back-reflection Laue pattern of an aluminum crystal rotated into the orientation described above is shown in Fig. 8-22. Note that the arrangement of spots has 2-fold rotational symmetry about the primary beam, corresponding to the 2-fold rotational symmetry of cubic crystals about their <110> axes. (Conversely, the observed symmetry of the Laue pattern of a crystal of unknown structure is an indication of the kind of symmetry possessed by that crystal. Thus the Laue method can be used as an aid in the determination of crystal structure.)... 

G.12 H. Lipson and W. Cochrane. The Crystalline State. Vol. Ill The Determination of Crystal Structures (London George Bell, 1953). Advanced structure analysis by means of space group theory and Fourier series. Experimental methods are not included i.e., the problem of structure analysis is covered from the point at which jFp values have been determined by experiment to the final solution. Contains many illustrative examples. 

X-ray diffraction is a tool for the investigation of the fine structure of matter. This technique had its beginnings in von Laue s discovery in 1912 that crystals diffract x-rays, the manner of the diffraction revealing the structure of the crystal. At first, x-ray diffraction was used only for the determination of crystal structure. Later on, however, other uses were developed, and today the method is applied not only to structure determination, but to such diverse problems as chemical analysis and stress measurement, to the study of phase equilibria and the measurement of particle size, to the determination of the orientation of one crystal or the ensemble of orientations in a polycrystalline aggregate. 

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