Structure Invariants, Seminvariants and Origin Definition

Abstract

In a crystal a system of basic vectors a, b, c is chosen that complies with the crystal symmetry. These basic vectors represent the translational symmetry of the crystal; this means that in the three-dimensional lattice, generated by the basic vectors, the lattice points represent positions that are translational equivalent, i.e. points that have identical surroundings in the same orientation. In order to define a crystal structure completely it is sufficient to specify the atomic positions within one translational unit, the unit cell, of the lattice. However, the lattice, and thus the origin of the unit cell, can be moved freely, in a parallel fashion, within the crystal; in order to specify the atomic coordinates unambiguously, the origin of the unit cell must be defined in a unique way. Though the position of the origin can be chosen anywhere, some positions are preferred, dependent on the symmetry of the crystal. An example is space group P1, in which one of the eight different positions of the centres of symmetry is usually chosen as the origin position (Fig. 1.1). The choice of origin is not trivial, as the eight positions are not equivalent (do not have the same surroundings).