High-order refinement

Difference density maps from the current data and the SC [11] data after high order refinements are shown in Figure 1(a) and (b). The main features of the maps agree... 

Figure 1. Difference map after high order refinement - data cutoff at sin 0/k = 0.9A1 for reflections with F2 > (a) this work (b) SC. Model map after multipole refinement - data cutoff at... 

An example of a standard deformation density, for oxalic acid dihydrate, obtained at limited resolution using parameters from a high-order refinement, is shown in Fig. 5.2. Oxalic acid dihydrate has been the subject of an extensive study aimed at calibrating the techniques used in different laboratories. The map shows... 

The reliable experimental information on the absolute scale and thermal vibrations of beryllium metal made it possible to analyze the effect of the model on the least-squares scale factor, and test for a possible expansion of the 1 s core electron shell. The 0.03 A y-ray structure factors were found to be 0.7% lower than the LH data, when the scale factor from a high-order refinement (sin 6/X) > 0.65 A l) is applied. Larsen and Hansen (1984) conclude that because of the delocalization of the valence electrons, it is doubtful that diffraction data from a metallic substance can be determined reliably by high-order refinement, even with very high sin 0/X cut-off values. This conclusion, while valid for the lighter main-group metals, may not fully apply to metals of the transition elements, which have much heavier cores and show more directional bonding. 

Biological fibers, such as can be formed by DNA and fibrous proteins, may contain crystallites of highly ordered molecules whose structure can in principle be solved to atomic resolution by x-ray crystallography. In practice, however, these crystallites are rarely as ordered as true crystals, and in order to locate individual atoms it is necessary to introduce stereochemical constraints in the x-ray analysis so that the structure can be refined by molecular modeling. 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

Dunitz and Seiler (1973) have used the equivalence to modify least-squares weighting, such as to emphasize the fit near the density peak positions, in order to obtain parameters less biased by bonding effects. The resulting weights emphasize high-order reflections, similar to the higher-order refinement method, but with a smoothly varying cut-off rather than a sharp sin 6//. limit. 

As discussed in the previous section, a residual density calculated after least-squares refinement will have minimal features. This is confirmed by experience (Dawson 1964, O Connell et al. 1966, Ruysink and Vos 1974). Least-biased structural parameters are needed if the adequacy of a charge density model is to be investigated. Such parameters can be obtained by neutron diffraction, from high-order X-ray data, or by using the modified scattering models discussed in chapter 3. 

As discussed in chapter 3, the valence electrons scatter mainly in the low-order region. Consequently, refinement of high-order data, first proposed by Jeffrey and Cruickshank (1953), yields parameters less biased by bonding effects. The X-X deformation density is calculated with the high-order X-ray parameters, and is defined as... 

Up to now we have seen how lattice distortions are detected and characterized. This does not provide a direct observation of the molecular translations, rotations, and deformations associated with the distortion. However, for a few compounds it has been possible to measure a large enough number of satellite or superstructure reflections so that the distorted structure can be parametrized and refined (rigid-body or full structural study). We consider below four examples, taken from materials selected in Section IV. A, which show that such studies are not easy and that the data collection requires special attention. Indeed, it is generally difficult to measure enough satellite reflections, especially if several kinds of the latter coexist (e.g., 2kp and 4kF satellites, high-order satellites, etc.). 

Thus, the level of sophistication which one may consider for the application of rubber-like elasticity theory to epoxy networks may depend on the application. For highly crosslinked systems (M < 1,000), a quantitative dependence of the rubbery modulus on network chain length has recently been demonstrated , but the relevance of higher order refinements in elasticity theory is questionable. Less densely crosslinked epoxies, however, are potentially suitable for testing modern elasticity theories because they form via near quantitative stepwise reactions. Detailed investigations of such networks have been reported by Dusek and coworkers in recent studies ... 

An excellent example in which the information obtained from X-X/,0 studies of deformation density maps has been compared with that from theoretical calculations is given by tetrafluoroterephthalonitrile. This compound is ideal for these studies because it contains no hydrogen atoms, and all the atoms have similar scattering power (6 to 9 electrons for C, N, and F). The molecule has high symmetry which simplifies the number of parameters to be refined. A very precise set of X-ray Bragg reflections were measured at a low temperature (98 K) and to sin0/A = 1.15 Least-squares refinement based on the high-order data... 

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