Diffraction methods unit cell

There has been much activity in the study of monolayer phases via the new optical, microscopic, and diffraction techniques described in the previous section. These experimental methods have elucidated the unit cell structure, bond orientational order and tilt in monolayer phases. Many of the condensed phases have been classified as mesophases having long-range correlational order and short-range translational order. A useful analogy between monolayer mesophases and die smectic mesophases in bulk liquid crystals aids in their characterization (see [182]). 

Several methods are also available for determination of the isothermal compressibility of materials. High pressures and temperatures can for example be obtained through the use of diamond anvil cells in combination with X-ray diffraction techniques [10]. kt is obtained by fitting the unit cell volumes measured as a function of pressure to an equation of state. Very high pressures in excess of 100 GPa can be obtained, but the disadvantage is that the compressed sample volume is small and that both temperature and pressure gradients may be present across the sample. 

The total surface areas determined by the N2 BET method for the calcined, supported catalysts are listed in Table II. The X-ray diffraction (XRD) results showed diffraction peaks from a cubic lattice with a unit cell distance of 6.1 A were present on all of the calcined catalysts. Both C03O4 and C0AI2O4 have structures consistent with that lattice spacing, making assignment of the type of crystalline cobalt species present on the alumina supports difficult. 

In amorphous solids there is a considerable disorder and it is impossible to give a description of their structure comparable to that applicable to crystals. In a crystal indeed the identification of all the atoms in the unit cell, at least in principle, is possible with a precise determination of their coordinates. For a glass, only a statistical description may be obtained to this end different experimental techniques are useful and often complementary to each other. Especially important are the methods based on diffraction experiments only these will be briefly mentioned here. The diffraction pattern of an amorphous alloy does not show sharp diffraction peaks as for crystalline materials but only a few broadened peaks. Much more limited information can thus be extracted and only a statistical description of the structure may be obtained. The so-called radial distribution function is defined as ... 

As an example for the specific case of vanadium alloys with palladium, the trend of the average atomic volume of the alloys is shown in Fig. 4.20 and compared with the phase diagram. These data were obtained by Ellner (2004) who studied the solid solutions of several metals (Ti, V, Cr, Mn, Fe, Co and Ni) in palladium. The alloys were heat treated at 800°C and water-quenched. From the unit cell parameters measured by X-ray diffraction methods, the average atomic volume was obtained Vat = c 14 (see Table 4.3). These data together with those of the literature were reported in a graph, and the partial molar (atomic) value of the vanadium volume in Pd solid solution (Fv)... 

From a comparison of various spot electron diffraction patterns of a given crystal, a three-dimensional system of axis in the reeiproeal lattice may be established. The reeiproeal unit cell may be eompletely determined, if all the photographs indexed. For this it is sufficient to have two electron diffraction patterns and to know the angle between the seetions of the reeiproeal lattice represented by them, or to have three patterns which do not all have a particular row of points in common (Fig.5). Crystals of any compound usually grow with a particular face parallel to the surface of the specimen support. Various sections of the reciprocal lattice may, in this case, be obtained by the rotation method (Fig.5). 

One big problem, which arises mainly in crystals with rather large unit cells, is the overlapping of reflections. This prevents an accurate measurement of the local integrated intensities of a large number of reflections. A solution to this problem can be the 2D pattern decomposition method, which is based on the same principles as in X-ray powder diffraction. This method takes into account the dependence of intensities on the particle orientation function and the size of microcrystals. It is therefore necessary to establish the mathematical formalism that describes the dififiaction pattern taking into account these parameters. 

The modeling of electron diffraction by the pattern decomposition method, for which no structural information is required, can be successfully applied for extraction of the diffraction information from the pattern. Several parameters can be refined during the procedure of decomposition, including the tilt angle of the specimen the unit cell parameters peak-shape parameters intensities. The procedure consists of fitting, usually with a least-squares refinement, a calculated model to the whole observed diffraction pattern. 

Electron dynamic scattering must be considered for the interpretation of experimental diffraction intensities because of the strong electron interaction with matter for a crystal of more than 10 nm thick. For a perfect crystal with a relatively small unit cell, the Bloch wave method is the preferred way to calculate dynamic electron diffraction intensities and exit-wave functions because of its flexibility and accuracy. The multi-slice method or other similar methods are best in case of diffraction from crystals containing defects. A recent description of the multislice method can be found in [8]. 

Information content in a powder diffraction pattern is reduced as compared to that in single crystal diffraction, due to the collapse of the three dimensional reciprocal space into a one dimensional space where the only independent variable is the scattering angle. The poorer the resolution of the diffraction method, the less the information content in the pattern (Altomare et al. 1995 David 1999). As a consequence, structure of less complex phases can be determined from power diffraction alone (fewer atoms in the asymmetric unit of the unit cell). However, refinement of the structure is not limited so seriously with resolution issues, so powder diffraction data are used in Rietveld refinement more frequently than in structure determination. Electron powder diffraction patterns can be processed and refined using public domain computer programs. The first successful applications of electron diffraction in this field were demonstrated on fairly simple structures. 

Model building remains a useful technique for situations where the data are not amenable to solution in any other way, and for which existing related crystal structures can be used as a starting point. This usually happens because of a combination of structural complexity and poor data quality. For recent examples of this in the structure solution of polymethylene chains see Dorset [21] and [22]. It is interesting to note that model building methods for which there is no prior information are usually unsuccessful because the data are too insensitive to the atomic coordinates. This means that the recent advances in structure solution from powder diffraction data (David et al. [23]) in which a model is translated and rotated in a unit cell and in which the torsional degrees of freedom are also sampled by rotating around bonds which are torsionally free will be difficult to apply to structure solution with electron data. 

One deduces the space group from the symmetry in the crystal s diffraction pattern and the systematic absence of specific reflections in that pattern. The crystal s cell dimensions are derived from the diffraction pattern for the crystal collected on X-ray film or measured with a diffractometer. An estimation of Z (the number of molecules per unit cell) can be carried out using a method called Vm proposed by Matthews. For most protein crystals the ratio of the unit cell volume and the molecular weight is a value around 2.15 AVOa. Calculation of Z by this method must yield a number of molecules per unit cell that is in agreement with the decided-upon space group. 

The isomorphous replacement method requires attachment of heavy atoms to protein molecules in the crystal. In this method, atoms of high atomic number are attached to the protein, and the coordinates of these heavy atoms in the unit cell are determined. The X-ray diffraction pattern of both the native protein and its heavy atom derivative(s) are determined. Application of the so-called Patterson function determines the heavy atom coordinates. Following the refinement of heavy atom parameters, the calculation of protein phase angles proceeds. In the final step the electron density of the protein is calculated. 

Effect of Steam Treatment. X-ray diffraction analyses indicated that ZSM-5 retained in excess of 90% of its crystallinity after the steam treatment described in the methods section. Unit cell constant of the REY zeolite in Super-D declined from 24.658. to 24.38a due to the steam treatment. Independent measurements... 

Typical X-ray diffraction patterns of the Fe oxides are shown in Figure 7.16. They provide the three parameters, namely line (angle) position, width and intensity from which the nature of the oxide, its quantity (in a mixture), its unit cell parameters and its crystallinity (crystal size and order) can be deduced. The crystal structure of unknown compounds may also be determined. XRD is still the most reliable way to identify a particular oxide because it is based on the long range order of the atoms, whereas most other methods (e.g. Mossbauer spectroscopy, EXAFS) characterize the atoms and their immediate (short range) environment. 

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