engleski

Linear Transformations in Crystallography

Linear transformations are a basic mathematical tool in crystallography. They are mostly used and studied in the forms of coordinate transformations and of symmetry operations. The course will cover both applications. Symmetry operations are used to classify crystals. The point group elements are linear transformations and, provided a basis is chosen, can be represented by matrices. The translations are not linear transformations, but all the symmetry operators can be represented by one matrix for the linear transformation part and one column-matrix representing the translation vector. The first part of the course will cover symmetry operators and their usage. An often used coordinate transformation is the rotation of the coordinate system, but this is by far not the only coordinate transformation useful in crystallography. We shall explore various examples of the coordinate transformations occurring in crystallographic contexts, including the change of basis for the usual three-dimensional space when shifting from descriptions of crystals in direct space to the descriptions in reciprocal space.

linear operator; coordinate transformations; crystallographic bases; symmetry operators; reciprocal space

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