Full Description
Crystallographers have to apply many mathematical methods in their daily work. Mathematical Techniques in Crystallography and Materials Science brings together common and less familiar mathematical procedures used in studies of the structures and physical properties of solids. This practical guide and reference serves as a unified source book for students and professionals, and it provides a solid basis for further studies in more specialized literature. Based Prince's decades of practical experience, it can be recommended as an introduction for beginners in crystallography, as a refresher and handy guide for crystallographers working on specific problems, and as a reference for others seeking a dictionary of basic mathematical and crystallographic terms. The third edition further clarifies key points, as well as offers new sections on two topics: the projection matrix and the fast Fourier transform.
Contents
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.- Representations.- Point 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 Disribution.- Anisotropic Atomic Displacement 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 Precision When the Model Is Not Least Squares.- 8 Significance and Accuracy.- The F Distribution.- Student's t Distribution.- Correlation.- Relationship Between Precision and Accuracy.- Uncertainties of Derived Functions.- The Projection Matrix.- 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.- 10 The Fast Fourier Transform.- The Discrete Fourier Transform.- The Good-Thomas Algorithm.- The Cooley-Tukey Algorithm.- Prime Numbers.- FFTs for Real Sequences.- Space Group Symmetry.- Appendix A Stereographic Projection.- Appendix B Eigenvalues and Eigenvectors of 3 x 3 Symmetric Matrices.- Appendix C Sublattices and Superlattices.-Appendix D The Probability Integral, the Gamma Function, and Related Topics.- Appendix E The Harmonic Oscillator in Quantum Mechanics: Bloch's Theorem.- Appendix F Symmetry Restrictions on Second, Third, and Fourth Rank Tensors.- Appendix G Some Useful Computer Programs.- ndex.
