The single-crystal X-ray diffraction pattern from the -phase of the industrially important Pigment Red 170 (-P.R.170) consists of a difficult-to-disentangle mixture of Bragg diffraction superimposed by rods of diffuse scattering and satellite peaks. This extremely complicated diffraction pattern illustrates the complexity of real world crystals, whose underlying structure is far from the concept of a crystal being a regular periodic arrangement of unit cells usually presented in introductory

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... stallography textbooks. Such complex structures still present a big challenge to practitioners of X-ray crystallography. Understanding of the photochemical properties of this pigment would benefit from knowledge of the specific local arrangement of molecules in the crystal structure, but such information was not available due to the disordered nature of this material. The focus of this thesis was to model the crystal structure of this material by an analysis of the total diffraction pattern. The disorder in this material manifests itself as rods of strong diffuse scattering in the diffraction pattern. According to the mathematical description presented in the first part of the thesis, the type of disorder present in this material is stacking faults. These faults can occur during the stacking of the two dimensionally ordered molecular layers when the crystal grows. A detailed analysis of the diffraction pattern revealed that the rods of diffuse scattering pass through the Bragg reflections. Furthermore, it showed that a considerable percentage of the Bragg reflections is completely immersed in the strong diffuse streaks. As a result, the unit cell indexation and the accurate Bragg intensity estimation were extremely difficult. An analysis of only the Bragg reflections resulted in two plausible average structures. Both structures have the same unit cell dimensions, but occur in different space groups, namely B21/g and P21/a. The model developed in B21/g has only one symmetry-independent disordered molecular layer in which there are two symmetry-independent molecules, both of which are disordered over two positions with an occupancy ratio 0.91:0.09 related by the vector [0, -0.158b, 0]. In contrast, the model developed in P21/a has two symmetry independent molecular layers of which only one is disordered. The disordered layer is similar to the unique layer of the other model, but this time the occupancy ratio is 0.65:0.35. In addition, the two models differ in the number of molecules in the asymmetric unit, relative placement of molecular layers in the unit cell and the number of crystallographic and non-crystallographic symmetry elements in the average unit cell. The agreement R-factors calculated from both models implied that the B21/g model is the better description of the average structure. The basic structural unit in both models is the same. It possesses the layer group symmetry p 1 21/c 1. The geometries of all adjacent layer pairs in both models are equivalent. According to Order-Disorder theory, this implies that the two models belong to the same polytypic family, but they differ in their layer stacking sequences. The last part of the work presents the initial attempts taken to estimate the layer stacking sequence in the real crystal using model crystals. Two model crystals were constructed in the computer with the aid of a random number generator using the atomic coordinates and site occupancies obtained from the two average structures. The correlations between the interacting layers were introduced and the total interaction energy of each crystal was minimized according to the Monte Carlo (MC) method. The MC minimized crystals were then used to calculate total scattering intensities. Both disordered model crystals constructed and tested in this work produced broad diffuse scattering features superimposed with some fine structure. So far, the match with the experimental data is poor. It is not yet known whether the observed fine structure in each calculated pattern is due to some underlying periodicity of the molecular layers in the model crystal, or is just a consequence of the statistical noise in the MC simulations. Resolution of this problem will require future additional time-consuming calculations. Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der ii iii Acknowledgements I wish to express my sincere gratitude to following people and organizations for their support during this work. Prof. Dr. Anthony Linden for giving me the opportunity to undertake my studies in his lab and offering guidance and support throughout the work. Prof. Dr. Hans-Beat Bürgi for supervision, motivation, advice, discussions, openness and placing confidence in me. Prof. Dr. Karl Heinz Ernst for examining the thesis and providing constructive criticism about the work. Dr. Thomas Weber for the XCAVATE source code and helpful discussions. Dr. Arkadiy Simonov for MATLAB scripts and continuous help during diffuse scattering data processing. Dr. Lukas Ahrenberg for computational support and helpful discussions. Dr. Michal Chodkiewicz for guidance with the ZODS program and tutorials. The Swiss National Science Foundation and the University of Zürich for financial support. Present and former colleagues for warm discussions, coffee and lunches. Timm Reumann for the nice company, chats and dinners. My family for support and sympathy. My wife Sandamali Warshamanage for her patience, awesome foods and filling me with love. iv v Abstract The single-crystal X-ray diffraction pattern from the β-phase of the industrially important Pigment Red 170 (β-P.R.170) consists of a difficult-to-disentangle mixture of Bragg diffraction superimposed by rods of diffuse scattering and satellite peaks. This extremely complicated diffraction pattern illustrates the complexity of real world crystals, whose underlying structure is far from the concept of a crystal being a regular periodic arrangement of unit cells usually presented in introductory crystallography textbooks. Such complex structures still present a big challenge to practitioners of X-ray crystallography. Understanding of the photochemical properties of this pigment would benefit from knowledge of the specific local arrangement of molecules in the crystal structure, but such information was not available due to the disordered nature of this material. The focus of this thesis was to model the crystal structure of this material by an analysis of the total diffraction pattern. The disorder in this material manifests itself as rods of strong diffuse scattering in the diffraction pattern. According to the mathematical description presented in the first part of the thesis, the type of disorder present in this material is stacking faults. These faults can occur during the stacking of the two dimensionally ordered molecular layers when the crystal grows. A detailed analysis of the diffraction pattern revealed that the rods of diffuse scattering pass through the Bragg reflections. Furthermore, it showed that a considerable percentage of the Bragg reflections is completely immersed in the strong diffuse streaks. As a result, the unit cell indexation and the accurate Bragg intensity estimation were extremely difficult. An analysis of only the Bragg reflections resulted in two plausible average structures. Both structures have the same unit cell dimensions, but occur in different space groups, namely B2 1 /g and P2 1 /a. The model developed in B2 1 /g has only one symmetry-independent disordered molecular layer in which there are two symmetry-independent molecules, both of which are disordered over two positions with an occupancy ratio 0.91:0.09 related by the vector [0, -0.158b, 0]. In contrast, the model developed in P2 1 /a has two symmetry independent molecular layers of which only one is disordered. The disordered layer is similar to the unique layer of the other model, but this time the occupancy ratio is 0.65:0.35. In addition, the two models differ in the number of molecules in the asymmetric unit, relative placement of molecular layers in the unit cell and the number of crystallographic and noncrystallographic symmetry elements in the average unit cell. The agreement R-factors 5.2.2 Preparation of diffuse data for modeling 91 5.3 Construction of the model crystal 94 5.3.