Methods of Determining Crystal Structure

Many semicrystalline polymers are between 40% and 75% crystalline. (See Section 6.5.4 for further information.) 

Before beginning the study of the structure of crystalline polymers, the subject of crystallography and molecular order in crystalline substances is reviewed. Long before X-ray analysis was available, scientists had already deduced a great deal about the atomic order within crystals. 

The science of geometric crystallography was concerned with the outward spatial arrangement of crystal planes and the geometric shape of crystals. Workers of that day arrived at three fundamental laws (a) the law of constancy of interfacial angles, (b) the law of rationality of indexes, and (c) the law of symmetry (14). 

Briefly, the law of constancy of interfadal angles states that for a given substance, corresponding faces or planes that form the external surface of a crystal always intersect at a definite angle. This angle remains constant, independent 

The third law of crystallography states that all crystals of the same compound possess the same elements of symmetry. There are three types of symmetry a plane of symmetry, a line of symmetry, and a center of symmetry (14). A plane of symmetry passes through the center of the crystal and divides it into two equal portions, each of which is the mirror image of the other. If it is possible to draw an imaginary line through the center of the crystal and then revolve the crystal about this line in such a way as to cause the crystal to appear unchanged two, three, four, or six times in 360° of revolution, then the crystal 

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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. 

There are four principal methods of determining crystal structures at high pressures employing powders or single-crystals, using X-rays or neutrons. Here 1 will give a brief review of the relevant diffraction techniques and analysis methods used with each technique, focusing on recent developments. 

The standard method of determining crystal structure is by X-ray diffraction and this method was applied to ice by Bragg and others some fifty years ago. The early results were not, however, in agreement and the situation was finally clarified by Barnes (1929). X-rays are scattered primarily by the electron distribution in the crystal and, from the discussion of chapter i, this is concentrated around the oxygen nucleus in each water molecule, though there is an appreciable electron density around the two protons. The X-ray diffraction results thus basically determine the positions of the oxygen atoms, the proton positions being much more difficult to fix. 

I n the method of trial, crystal structures are determined by considering what atomic positions will account for the intensities of the diffracted X ray beams. This method is not only very laborious (except for very simple structures) but also has all the disadvantages of an indirect method so much depends on the chances of postulating an approximately correct structure. The opposite method is to record and measure the diffraction pattern, and then combine the results by suitable mathe matical or experimental operations to give a picture of the crystal structure. 

During the last five years, a powerful new method of getting crystal structural information from powder diffraction patterns has become widely used. Known variously as the Rietveld method, profile refinement1, or, more descriptively, whole-pattern-fitting structure refinement, the method was first introduced by Rietveld (X, 2) for use with neutron powder diffraction patterns. It has now been successfully used with neutron data to determine crystal structural details of more than 200 different materials in polycrystalline powder form. Later modified to work with x-ray powder patterns (3, X) the method has now been used for the refinement of more than 30 crystal structures, in 15 space groups, from x-ray powder data. Neutron applications have been reviewed by Cheetham and Taylor (5) and those for x-ray by Young (6). 

Thus, melting points (even in the absence of a full melting point phase diagram) and X-ray powder diffraction patterns are useful criteria for assessing the results of such crystallizations if suitable single crystals are formed, the more elaborate (but far more informative in terms of structural detail) method of single-crystal structure determination provides more definitive confirmation and structural descriptions of phase identities. 

Also, the structures of many substances have been determined by the methods of diffraction of electrons and diffraction of x-rays. In the following pages we shall describe many atomic structures that have been determined by these methods. The x-ray diffraction method of determining the structure of crystals is discussed in Appendix IV. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

X-ray, neutron, and electron diffraction techniques are used to determine crystal structures and can thus be used for molecular structure determinations. Because of its high resolution and applicability to small and often weakly diffracting samples, x-ray crystallography and powder diffraction are by far the methods of choice for most structure determinations on crystalline compounds,... 

As most of the nitroxyl spin-labelled synthetic derivatives of conjugated polyenes are light yellow crystals, the bond lengths were determined by X-ray crystallography38. The spectroscopic method used to measure the conformation is electron nuclear double resonance (ENDOR). It is beyond the scope of the present review to explain the method38 but the authors of the pertinent paper conclude that ENDOR is an accurate non-crystallographic method to determine polyene structures in solution. 

Some of the major areas of activity in this field have been the application of the method to more complex materials, molecular dynamics, [28] and the treatment of excited states. [29] We will deal with some of the new materials in the next section. Two major goals of the molecular dynamics calculations are to determine crystal structures from first principles and to include finite temperature effects. By combining molecular dynamics techniques and ah initio pseudopotentials within the local density approximation, it becomes possible to consider complex, large, and disordered solids. 

Many essential strnctural and functional features of hydrogenases have been derived from a wealth of various biochemical and spectroscopic methods. However, the knowledge of their atomic architectures have been obtained only very recently with the determination of the crystal structures of several hydrogenases belonging to both [NiFe] and [Fe] families (Table 6.1). These results have given a firm and nniqne strnctural basis to understand how these enzymes are actually working. 

A complete structure determination contains the two distinct steps solving and refining the structure. The refinement can only be started after the structure has been solved. By solving a structure we mean that most of the most strongly scattering atoms are found to within an accuracy of 0.2 to 0.3 A. All methods for solving crystal structures from X-ray diffraction data in most cases give just a fraction of the complete structure. Patterson... 

Diffraction analysis—whether employing x-rays, electrons, or neutrons—is the method of choice for obtaining structural information on crystalline substances. The application of the well understood principles and methods of diffraction analysis to single crystals of sufficient size and perfection can lead to a detailed determination of the crystal structure, without recourse to any auxiliary methodology. Hundreds of mono- and oligosaccharide molecules have been characterized by these means (1), yielding not only an increased understanding of their structures in the solid state, but also a data base useful for extrapolation to other states and molecular interactions. 

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