The determination of crystal structures by X-ray diffraction

The determination of the crystal structure of a substance with small molecules is nowadays almost a mechanical procedure for a polymer the problem is more difficult. The reason for the difficulty lies in the fact that large (a few tenths of a millimetre) fairly perfect single crystals can readily be obtained for most small-molecule compounds. As discussed in more detail in the next chapter, the best that can be obtained for a polymer, apart from a few very special cases, is a piece of material in which a mass of crystallites is embedded in a matrix of amorphous material. 

In order to obtain the maximum amount of information about the crystal structure it is necessary to align the crystallites, which can be done by methods described in detail in chapter 10. It is sufficient to note here that suitable orientation is often produced by stretching a fibre of the polymer. In the simplest cases the chain axis of each crystallite, which is designated the c-axis, becomes aligned towards the fibre axis, but there is no preferred orientation of the other two axes around the c-axis. From such a sample a fibre pattern can be obtained, of the type shown in fig. 3.10. 

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With sufficiently large crystals of the bacterial reaction centers available, the determination of crystal structure by X-ray diffraction became possible and in 1983, Deisenhofer, Epp, Miki, Huber and Michel determined tbe crystal structure of the Rp. viridis reaction centers at 3 A resolution [later refined to 2.3 A ]. From the electron-density map, the spatial arrangement of the polypeptide subunits, the pigment molecules and the electron carriers in the reaction center was determined. For this work, Deisenhofer, Michel and Huber were awarded the Nobel Prize in 1988. 

The determination of crystal structure by x-ray diffraction is one of the most important ways of determining the structures of molecules. Because of its ordered structure, a crystal consists of repeating planes of the same kind of atom. These planes can act as reflecting surfaces for x rays. When x rays are reflected from these planes, they show a diffraction pattern, which can be recorded on a photographic plate as a series of spots (see Figure 11.47). By analyzing the diffraction pattern, you can determine the positions of the atoms in the unit cell of the crystal. Once you have determined the positions of each atom in the unit cell of a molecular solid, you have also found the positions of the atoms in the molecule. ... 

Isomorphorous replacement The replacement of one atom in a macromolecule with a heavy metal atom in such a way that the structure of the macromolecule does not change. It is used in the determination of molecular structure by X-ray crystal diffraction. 

Another way to represent the structure factor is shown in Eq. 18, where p(r) is the electron density of the atoms in the unit cell (r = the coordinates of each point in vector notation). As you may recall, this is in the form of a Fourier transform that is, the structure factor and electrOTi density are related to each other by Fourier verse Fourier transforms (Eq. 19). Accordingly, this relation is paramount for the determination of crystal stractures using X-ray diffraction analysis. That is, this equation enables one to prepare a 3-D electron density map for the entire unit cell, in which maxima represent the positions of individual atoms. ... 

Metallurgists originally, and now materials scientists (as well as solid-state chemists) have used erystallographic methods, certainly, for the determination of the structures of intermetallic compounds, but also for such subsidiary parepistemes as the study of the orientation relationships involved in phase transformations, and the study of preferred orientations, alias texture (statistically preferential alignment of the crystal axes of the individual grains in a polycrystalline assembly) however, those who pursue such concerns are not members of the aristocracy The study of texture both by X-ray diffraction and by computer simulation has become a huge sub-subsidiary field, very recently marked by the publication of a major book (Kocks el al. 1998). 

X-rays. This was followed by the mathematical solution of crystal structure from X-ray diffraction data in 1913 by Bragg. Since that, many applications of X-ray were foimd including structure determination of fine-grained materials, like soils and days, which had been previously thought to be amorphous. Since then, crystals structures of the day minerals were well studied (Ray and Okamoto, 2003). 

We saw in the historical introduction that the study of crystal structure, that is the determination of the nature and the positions of the atoms inside the crystal cell, quickly became an essential application of X-ray diffraction. W.H. and W.L. Bragg, for example, were awarded the Nobel Prize for their works on the determination of the crystal stracture of several simple phases. Throughout the 20 century, the determination of crystal structure was one of the major driving forces in developing the study of condensed matter by X-ray diffraction with regard to both the improvements made to the instraments and the implementation of more efficient methods of data processing. 

In Chapter 5, we described in detail the effect on the diffiracted intensity s distribntion of the discrepancies between a crystal s mean stmcture and its actual structure. The quantitative study of these effects makes it possible to accurately determine the density of punctual defects, of dislocations or of stacking faults, the size of the grains, as well as the microstrain rate, etc. This type of analysis was developed in the second half of the 20 century and, due to the tremendous progress made by computers, became widespread in the late 1980s. In 1999, the International Union of Crystallography published a book describing most of the aspects of microstractirral analysis by X-ray diffraction on polycrystalline samples [SYN 99],... 

A novel crystalline form of the boron-containing antibacterial drug (5 )-3-(aminomethyl)-7-(3-hydroxypropoxy)benzo[c][l,2]oxaborol-l(3H)-ol hydrochloride has been studied by solid-state NMR and single-crystal X-ray diffraction techniques. After determination of the crystal structure by X-ray diffraction, solid-state NMR spectroscopy of this form is performed to obtain structural information using experimental approaches based on dipolar correlation, chemical shift analysis, and quadrupolar interaction analysis. solid-state NMR experiments at 16.4 T using MAS and homonuclear dipolar decoupling, 2D solid-state NMR experiments based on and dipolar heteronuclear correlation, and DFT... 

The basic study was performed on copper complexes with N,N,N, N1-tetramethylethane-1,2-diamine (TMED), which were known to be very effective oxidative coupling catalysts (7,12). From our first kinetic studies it appeared that binuclear copper complexes are the active species as in some copper-containing enzymes. By applying the very strongly chelating TMED we were able to isolate crystals of the catalyst and to determine its structure by X-ray diffraction (13). Figure 1 shows this structure for the TMED complex of basic copper chloride Cu(0H)Cl prepared from CuCl by oxidation in moist pyridine. 

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... 

Later, Tieke reported the UV- and y-irradiation polymerization of butadiene derivatives crystallized in perovskite-type layer structures [21,22]. He reported the solid-state polymerization of butadienes containing aminomethyl groups as pendant substituents that form layered perovskite halide salts to yield erythro-diisotactic 1,4-trans polymers. Interestingly, Tieke and his coworker determined the crystal structure of the polymerized compounds of some derivatives by X-ray diffraction [23,24]. From comparative X-ray studies of monomeric and polymeric crystals, a contraction of the lattice constant parallel to the polymer chain direction by approximately 8% is evident. Both the carboxylic acid and aminomethyl substituent groups are in an isotactic arrangement, resulting in diisotactic polymer chains. He also referred to the y-radiation polymerization of molecular crystals of the sorbic acid derivatives with a long alkyl chain as the N-substituent [25]. More recently, Schlitter and Beck reported the solid-state polymerization of lithium sorbate [26]. However, the details of topochemical polymerization of 1,3-diene monomers were not revealed until very recently. 

In order to obtain detailed structure, a knowledge of diffraction intensities is essential, the intensities being related to the structure factor. Computer-controlled single-crystal X-ray diffractometers with structure (software) packages have made structure elucidation a routine matter. The availability of synchrotron X-radiation of continuously variable wavelength has made X-ray diffraction a still more powerful structural tool for the study of solids. A technique of great utility to solid state chemists is the Rietveld treatment of powder X-ray diffraction profiles (Rietveld, 1969 Manohar, 1983). Automated structure packages for the determination of unknown structures by this method are now commercially available (see section 2.2.3). In Fig. 2.1, we show a typical set of profile data. 

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. 

There are several indications that a crystalline solid is the most appropriate state to model the protein interior (Chothia, 1984). The very fact that protein structures can be determined to high resolution by X-ray diffraction is indicative of the crystalline nature of the protein. Additionally, the packing density and volume properties of amino acid residues in proteins are characteristic of amino acid crystals (Richards, 1974, 1977). In spite of the apparent crystallinity of the protein interior, most model compound studies have investigated either the transfer of compounds from an organic liquid into water (see, for example, Nozaki and Tanford, 1971 Gill et al., 1976 Fauch-ere and Pliska, 1983), or the association of solute molecules in aqueous solution (see, for example, Schellman, 1955 Klotz and Franzen, 1962 Susi et al., 1964 Gill and Noll, 1972). Both these approaches tacitly assume a liquidlike protein interior. 

Crystals of 1-6 prepared as above were employed for the determination of molecular structure by single crystal X-ray diffraction. The relevant details of the crystal structures of the co-crystals are listed in Table l.f The intensity data were collected on a SMART system, Siemens, equipped with a CCD area detector,13 using Mo-Koc radiation. The structures were solved and refined using SHELXTL14 software. The refinements were uncomplicated and converged to good R-factors as... 

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