Diffraction methods data analysis

The most frequently used technique for the determination of crystal structures is single crystal analysis. However, if no single crystals of suitable size and quality are available, powder diffraction is the nearest alternative. Furthermore, single crystal analysis does not provide information on the bulk material and is not a routinely used technique for the determination of microstructural properties. Neither is it often used to characterize disorder in materials. Studies of macroscopic stresses in components, both residual from processing and in situ under load, are studied by powder diffraction, as is the texture of polycrystalline samples. Powder diffraction remains to this day a crucial tool in the characterization of materials, with increasing importance and breadth of application as instrumentation, methods, data analysis and modeling become more powerful and quantitative. 

The powder diffraction experiment (WAXS) remains a crucial tool in the characterization of materials, and it has been used for many decades with increasing importance and breadth of application as instrumentation, methods, data analysis, and modeling become more powerful and quantitative. Although powder data usually lack the 3-D of the diffraction image, the fundamental nature of the method is easily appreciated from the fact that each powder diffraction pattern represents 1-D snapshot of the 3-D reciprocal lattice of a crystal. The quality of the data is usually limited by the resolution of the powder diffractometer and by the physical and chemical conditions of the specimen (Figure 8.2). 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

Order and polydispersity are key parameters that characterize many self-assembled systems. However, accurate measurement of particle sizes in concentrated solution-phase systems, and determination of crystallinity for thin-film systems, remain problematic. While inverse methods such as scattering and diffraction provide measures of these properties, often the physical information derived from such data is ambiguous and model dependent. Hence development of improved theory and data analysis methods for extracting real-space information from inverse methods is a priority. 

There are many types of data in chemistry that are not specifically covered in this book. For example, we do not discuss NMR data. NMR spectra of solutions that do not include fast equilibria (fast on the NMR time scale) can be treated essentially in the same way as absorption spectra. If fast equilibria are involved, e.g. protonation equilibria, other methods need to be applied. We do not discuss the highly specialised data analysis problems arising from single crystal X-ray diffraction measurements. Further, we do not investigate any kind of molecular modelling or molecular dynamics methods. While these methods use a lot of computing time and power, they are more concerned with data generation than with data analysis. 

The basic modem data describing the atomic stmcture of matter have been obtained by the using of diffraction methods - X-ray, neutron and electron diffraction. All three radiations are used not only for the stmcture analysis of various natural and synthetic crystals - inorganic, metallic, organic, biological crystals but also for the analysis of other condensed states of matter - quasicrystals, incommensurate phases, and partly disordered system, namely, for high-molecular polymers, liquid crystals, amorphous substances and liquids, and isolated molecules in vapours or gases. This tremendous... 

Correlation times and activation energy parameters obtained from different techniques may or may not agree with one another. Comparison of these data enables one to check the applicability of the model employed and examine whether any particular basic molecular process is reflected by the measurement or whether the method of analysis employed is correct. In order to characterize rotational motion in plastic crystals properly it may indeed be necessary to compare correlation times obtained by several methods. Thus, values from NMR spectroscopy and Rayleigh scattering enable us to distinguish uncorrelated and correlated rotations. Molecular disorder is not reflected in NMR measurements to this end, diffraction studies would be essential. 

It is possible that the future may also see the use of digital calculators in qualitative spectrometric analyses. Various types of punched cards have been used as a method of recording spectral data on pure compounds. The purpose of these files is to facilitate the identification of spectral data on unknown substances. Their use in infrared analysis has been covered by Mecke and Schmid (M6), Keuntzel (K3), and Baker, Wright, and Opler (B2). The last named authors describe a file of 3150 spectra which was expected eventually to be expanded to include up to 10,000 spectra. Zemany (Zl) discussed the use of edge-notched cards in cataloging mass spectra and Matthews (M4) describes a similar application in connection with X-ray diffraction powder data. These two applications made use of only hand-sorting methods the files of Baker et al. were intended to be processed by machine. 

There are methods used Lo study enzymes other than those of chemical instrumental analysis, such as chromatography, that have already been mentioned. Many enzymes can be crystallized, and their structure investigated by x-ray or electron diffraction methods. Studies of the kinetics of enzyme-catalyzed reactions often yield useful data, much of this work being based on the Michaelis-Menten treatment. Basic to this approach is the concept (hat the action of enzymes depends upon the formation by the enzyme and substrate molecules of a complex, which has a definite, though transient, existence, and then decomposes into the products, of the reaction. Note that this point of view was the basis of the discussion of the specilicity of the active sites discussed abuve. 

The Protein Data Bank (PDB, http //www.rcsb.org/pdb/) is the de facto repository for macromolecular structures resolved by NMR or diffraction methods [42]. The structures of many protein-ligand complexes have been resolved and their atomic coordinates can be downloaded from the PDB web portal for further analysis. 

The internal structure of a polyatomic ligand is obtained as an additional result in these investigations. Its high concentration and the sharp intramolecular distances often result in dominant contributions to the diffraction curves and make possible a precise determination of its bonding distances. When careful analyses have been made, no significant differences from values found in crystals have been found, however. Therefore, the derived structure for the ligand can serve as an internal check on the quality of the data. Large deviations from values found in crystal structures may be an indication of errors in the data or in the method of analysis. 

In part, the widespread and routine use of single-crystal X-ray diffraction techniques in the modern day is a result not only of advances in instrumentation but also of the development of tremendously powerful methods for data analysis, such that, in the vast majority of cases, the crystal structure can be determined almost routinely even from experimental data of only modest quality. Thus, provided a single crystal of sufficient size and quality is available for the material of interest, successful structure determination by analysis of single-crystal X-ray diffraction data is nowadays a virtual formality. 

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