Mathematical Basis for X-Ray Crystallography and Analysis of Diffraction Patterns

Videos 3.1-3.9 are the most mathematical in the series, and require some knowledge of vectors and calculus. The most calculus-heavy videos are 3.1-3.5, and these can be skipped by those with less mathematical background.

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3.1 - Sines, Cosines, and Fourier Transforms

Paul A. Heiney

The use of trigonometric functions in describing electromagnetic waves. Fourier transforms and waves with multiple frequencies present. The Euler formulation of sines and cosines as exponentials. Time 7:35.
3.2 - Some Fourier Transforms in One Dimension

Paul A. Heiney

Examples of Fourier transforms (that are relevant to X-ray scattering): Fourier transforms of box function, Gaussian, sum of Gaussians. General features of Fourier transforms. Time 9:36.
3.3 - Fourier Transforms and Bragg's Law

Paul A. Heiney

Connection between Fourier transforms and X-ray scattering. Scalar product of vectors. Fourier transforms in three dimensions. The Patterson function. Time 8:13.
3.4 - Crystal Structure, the Lattice, and the Basis

Paul A. Heiney

Review of crystal structure, lattice, and basis. Calculating the Fourier transform of the electron structure. Atomic form factor and reciprocal lattice. Time 8:30
3.5 - The Reciprocal Lattice

Paul A. Heiney

Definition and use of reciprocal lattice. The cross product of two vectors. Recipe for reciprocal lattice vectors. Reformulation of Bragg's Law in terms of wave vectors and reciprocal lattice vectors. General features of scattering from crystals. Relationship between Bragg's Law and Fourier transform formulations. Time 8:42.
3.6 - Some Simple Lattices

Paul A. Heiney

Review of reciprocal lattice. Cubic lattices and hexagonal lattice and their reciprocal lattices. Time 11:09.
3.7 - Peak Width and Correlation Length

Paul A. Heiney

Relate Fourier transform of finite sum of Gaussians to peak widths in X-ray scattering measurements. Analogy with diffraction gratings for visible light. Peak width and crystal perfection. The Scherrer Equation. Strain broadening. Williamson-Hall plots. Instrumental resolution. Time 9:04.
3.8 - Limitations to X-ray Crystallography

Paul A. Heiney

Limitations of crystallography due to: powder samples, finite angular range, the phase problem. Common solutions to these problems. Time 7:02.
3.9 - Electron Density Reconstruction for Partially Ordered Materials

Paul A. Heiney

Features of soft self-assembled materials such as liquid crystals and block copolymers. Approaches to solving the phase problem. Example of how to reconstruct electron density of a simple model system (silver behenate). Time 11:44.