Mathematical Techniques in Crystallography and Materials Science

1 Matrices: Definitions and Fundamental Operations.- Fundamental Matrix Operations.- Linear Algebra.- Eigenvalues.- Linear Transformations.- Rotation of Axes.- The Metric Tensor.- 2 Symmetry of Finite Objects.- Groups.- Basis Functions.- 3 Symmetry of Infinitely Repeated Patterns.- Bravais Lattices.- Space Groups.- 4 Vectors.- Scalar and Vector Products.- The Reciprocal Lattice.- The Orientation Matrix.- Zones and Forms.- Sublattices and Superlattices.- 5 Tensors.- Covariance and Contravariance.- The Multivariate Normal Distribution.- Anisotropic Temperature Factors.- The Equivalent Isotropic Temperature Factor.- Effect of Symmetry.- Tensors of Higher Ranks.- Moments and Cumulants.- Rigid-Body Motion.- 6 Data Fitting.- Fitting Functions.- Finding the Minimum.- False Minima.- 7 Estimation of Uncertainty.- Estimates.- The Precision of Estimates of Precision.- Models with More than One Parameter.- Estimates of Uncertainty When the Algorithm Is Not Least Squares.- 8 Significance and Accuracy.- The F Distribution.- Student’s t Distribution.- Correlation.- The Relation between Precision and Accuracy.- Uncertainties of Derived Functions: Propagation of Errors.- 9 Constrained Crystal Structure Refinement.- The Observed Data.- The Model.- The General Form for a Constrained Model.- Shape Constraints.- Rigid-Body Thermal Motion Constraints.- Chemical Constraints.- Representing non-Gaussian Distributions.