Derivative Crystal Structures

There exists a class of crystal structures which are derived from others by generalization. These are here designated derivative structures. A special case of derivative structure is the popular ``superstructure.'' The symmetry of a derivative structure is a subgroup of the symmetry of the basic structure. Here the term ``subgroup'' is used in a wider sense than common in mathematical crystallography, and includes subgroups with multiple cells. The methods of finding the derivative symmetries are discussed. There are two important kinds of derivative structures, here designated substitution structures and distortion structures. Substitution structures result when a set of different atoms is substituted for a set of like atoms in the basic structure. For a given kind of substitution, it is possible to predict the cell dimensions of the crystal resulting from the substitution, as well as all the possible symmetries it could have. Examples are cited of the applications of derivative structure theory to several branches of crystallography.