Crystallization models, derivation

The predicted solids based on thermodynamic measurements and models derived from them depend strongly on chemical composition. However, small aqueous aerosols remain meta-stable with decreasing RH until reaching a crystallization relative humidity(CRH). In contrast, solid aerosols t e up water at the thermodynamically favored deliquescence relative humidity(DRH). This hysteresis causes a dependence of aerosol phase on RH history. 

For proteins that are members of gene families, such as cytochrome P450, available structural models derived from X-ray crystallography and NMR data allow inference of the structure of newly discovered family members. The inference is based on the sequence variations derived from comparing the reference and new DNA sequences. While a high degree of computational power is needed, predictions based on this approach provide preliminary estimates of protein structure without a huge investment in expression, purification, and crystallization. A full structural elucidation requires several years for each protein, even if it can be crystallized. 

Gervais, C., Coquerel, G., Simple model designed to generate new crystal structures derived from a mother phase, application to molecular compounds. Acta Crystallogr. Sect. B-Struct. Commun. 2002, 58, 662-672. 

In 1995, an elaborated method was developed for accurate structure analysis using X-ray powder diffraction data, that is, the MEM/Rietveld method [1,9]. The method enables us to construct the fine structural model up to charge density level, and is a self-consistent analysis with MEM charge density reconstruction of powder diffraction data. It also includes the Rietveld powder pattern fitting based on the model derived from the MEM charge density. To start the methods, it is necessary to have a primitive (or preliminary) structural model. The Rietveld method using this primitive structural model is called the pre-Rietveld analysis. It is well known that the MEM can provide useful information purely from observed structure factor data beyond a presumed crystal structure model used in the pre-Rietveld analysis. The flow chart of the method is shown in Fig. 2. 

As established in the previous chapter (section 6.17), the anhydrous iron phosphate has a relatively simple crystal structure, however, poor crystallinity of the powder results in broad peaks where full widths at half maximum vary from 0.2 to 0.5° when Mo Ka radiation is employed. Therefore, the resolution of the diffraction data is quite low, as was illustrated in Figure 6.33. There are only 6 atoms in the asymmetric unit but Rietveld refinement of the model is complicated by the inadequate quality of the diffraction data. The model, derived from a suspected analogy with the hydrated FeP04 2H20, cannot be completed based solely on the powder diffraction data due to problems with the experiment. Thus, Rietveld refinement considered in this section starts from the model improved by... 

If the model derived from the diffraction data were correct in every way, and the measured data were perfect, then the agreement would be exact Fhkt-obs would always equal Fhki-caic and R = 0. But this is never the case. The data, of course, contain measurement errors, the atomic positions may be accurate and precise, but still not perfect, temperature factors and the ellipsoids of vibration may be only approximate, and so on. In general, even for a very well and correctly determined structure, R will commonly be in the range of 0.05 to 0.10 for a conventional crystal having an asymmetric unit of 50 or so atoms. For macro-molecular structure determinations, R is normally in the range of 0.15 to 0.25. The R factor is in most cases the ultimate criterion of model quality at the resolution it was determined. 

The structure gap concept derives from the difficulty of knowing to what extent idealized catalysts are representative of the results obtained with real-life catalysts. The most idealized catalysts expose only one well-defined single crystal plane with surface areas of the order of 1 cm and are most often studied under UHV conditions. In contrast to such simple single crystal model systems, real-life catalysts normally consist of small-supported nanoparticles buried in a porous support material. For example, in emission cleaning... 

On the basis of the combination of MAS NMR spectrometry a crystallization model capable of explaining the morphology change is derived. ... 

Equations (5.18) and (5.20) are special cases of the more general formulas derived in Ch. 3 they demonstrate for the given crystal model how the mixing of molecular configurations due to intermolecular interaction in the crystal affects the oscillator strengths of the dipole transitions. We shall only stress here the... 

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