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This chapter describes general approaches to the ab initio solution of crystal structures from X-ray or neutron powder diffraction data. The steps in the process, unit cell determination and indexing, intensity extraction, space group determination, structure solution, and structure refinement are described. Indexing methods such as zone indexing, exhaustive methods or recently developed whole pattern methods, and the use of a figure of merit (M₂₀) are presented. Intensity extraction is shown to be reasonably straightforward but for the problem of peak overlap that occurs in powder patterns. The phase problem makes structure solution more difficult: Fourier and Patterson methods, direct methods, or global optimization methods (simulated annealing, genetic algorithms) are brought to bear. The chapter concludes with a section on advanced refinement techniques, including the interpretation of displacement and site occupancy parameters, and the use of constraints. The discussion is illustrated by frequent reference to structure solution for the Ruddlesden-Popper compound Ca₃Ti₂O₇.

Global optimisation methods that involve the assessment of multiple trial crystal structures against measured diffraction data offer a powerful alternative to Direct methods of structure solution. This chapter describes the problem of searching an N-dimensional hypersurface, populated with a great many function minima, for the global minimum that corresponds to the true crystal structure. Key factors discussed include model construction, agreement factor calculation and estimating the likelihood of success. Various global optimisation methods including simulated annealing and genetic algorithms are covered, and the chapter is illustrated with numerous examples of solved molecular crystal structures.

This chapter lists many of the common physical properties that can be easily calculated using the OLCAO method discussed in Chapter 3. The most fundamental quantities are the band structures, density of states, interatomic bonding, effective charges, and optical excitations. Simple examples obtained by using the OLCAO method are presented here while modern practical applications of such calculations of more complex systems are described in later chapters. The details of how OLCAO is used for core-level spectroscopy are discussed separately in Chapter 11.

This chapter discusses how the critical problems faced by dislocation dynamics simulations are solved. Available references to the technical bases of current nodal and lattice-based simulation codes are presented in a specific section. Curved dislocation lines are most often discretized into a succession of connected straight segments. This simplification allows implementing in tractable form the elastic properties of dislocations. The main issues are the treatment of the self-stress and the optimization of accuracy and computing efficiency by parallelization or the fast-multipole method. Elastic properties are complemented by local rules that allow incorporating dislocation velocities, cross-slip and other mechanisms at the mesoscale. Periodic boundary conditions are used mainly for large-scale simulations of single crystals. Two types of boundary condition are also available for materials with finite sizes.