Crystal Structure Transformations Part 3. Factors

The most usual way is effectively the same as is used for completion of crystal structures during structure solution (Section 10.7.6 andEq. 10.12). When a reasonable part of the structure has been described and some atoms positions have been defined, along with correct scattering factors and thermal parameters, a difference map can be calculated between this (incomplete) model and the Fourier transform of the experimental structure amplitudes. This map is a representation of the difference between the electron density of the model and that determined by the experiment. The biggest accumulations of electron density are assigned as new atoms until the structural model is complete. But even then there is residual electron density, which describes the difference between the true electron density and the model of a sum of independent spherical atom contributions, taking some simplified model of displacement into account There is also some random noise. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Temperature is the main factor that affects sintering and solid-state transformation experimental observations, however, have shown that the nature of the atmosphere in which the catalyst is heated may also play a part to an extent in the sintering process. For example, the presence of water vapor accelerates crystallization and structure modifications in oxide supports (Forzatti and Lietti, 1999). 

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