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- Maximum Entropy Method in Superspace Crystallography (2003)
- This thesis discusses several aspects of the combination of the Maximum Entropy Method (MEM) for the reconstructions of the electron density with the superspace approach to the description of structures of aperiodic crystals. It is shown that the MEM in superspace provides a parameter-free reconstruction of the modulation functions with sufficient accuracy. The MEM in superspace has been applied to diffraction data of several compounds. The computer program BayMEM was developed for this purpose. BayMEM allows electron densities of the ordinary 3D structures and the superspace electron densities of the aperiodic structures to be reconstructed using the same general principles. The program has been extended by adding features improving its versatility and accuracy of the results. The improvements include attaching of the set of subroutines MemSys5 to BayMEM, implementation of the method of the Generalized F-constraints and the static weighting, implementation of the G-constraints, of the Prior-derived F-constraints and of the two-channel entropy. The second major computer program EDMA is a software tool for analysis of the electron densities in arbitrary dimension. The program analyzes the MEM electron density and extracts quantitative information about the atoms according to Bader's formalism “Atoms in molecules“. Two new variants of the constraints in the MEM have been developed in order to solve the problems with artifacts in the MEM reconstructions. The two methods are the Generalized F-constraints and the Prior-derived F-constraints. The concept of the Generalized F-constraints is based in the observation, that the standard F-constraint is not sufficiently strong to constrain the histogram of the normalized residuals of the structure factors to the expected Gaussian shape. Higher moments of the distribution of the normalized residuals were therefore used as the constraint in the MEM calculations. With these constraints significantly improved histograms were obtained. The source of some artifacts in the MEM electron densities was identified to be the tendency of the MEM to estimate incorrectly those structure factors, that are not included in the experimental dataset. It is shown that the missing structure factors can successfully be replaced by the structure factors derived from the procrystal electron density. If the structure factors derived from the procrystal prior electron density (the Prior-derived F-constraints) are used as additional constraints in the MEM calculation, the result is free of sharp artifacts and the quality of the reconstruction of the electron density is comparable with the results of multipole refinements. To test the accuracy of the MEM in superspace, the method was applied to the dataset of the misfit-layer composite structure of (LaS)1.14NbS2. It has been shown, that the MEM on the model structure factors reproduces the model modulation functions with accuracy better that 10% of the pixel size of the grid, on which the electron density was sampled. The structure of the high-pressure phase III of Bi provided a prominent example illustrating the advantages of the MEM in superspace over the standard structure refinements. The MEM in superspace was applied to the diffraction data of Bi-III to extract more information about the modulation than obtained from the standard structure refinement. The modulation functions extracted from the MEM electron density revealed a block-wave-like shape of the modulation function of the Bi atom of the host structure that indicates shifts of the atom between two stable environments rather than smooth harmonic variation of the position indicated by the modulation function from the standard refinement. Secondly, the MEM modulation function of the Bi atoms in channels allowed to better understand the nature of the most prominent feature of the modulated structure - the occurrence of the pairs of Bi atoms along the channels. The incommensurately modulated structure of ammonium tetrafluoroberyllate (NH4)2BeF4, stable between 175K and 182K, was solved and refined in superspace. The known two-fold low-temperature superstructure of (NH4)2BeF4, that is stable below 175K has been described in superspace as a commensurately modulated structure. With aid of this description the close relationship between the two structures has been found. The MEM was applied to the incommensurate structure to test the appropriateness of the refined harmonic structure model. The MEM has shown that the harmonic model is very accurate. The MEM in superspace was established in this thesis as a reliable tool for the structure solutions of the modulated structure. The individual chapters together form a framework that allows to use the MEM in superspace to extract novel information from the diffraction data of both the periodic and aperiodic structures, that cannot be obtained from the structure refinements.