Basic theorems of vector symmetry in crystallography

An attempt has been made to deduce the condition necessary for diffraction enhancement of symmetry to occur in the diffraction pattern of a structure X, and because the symmetry of the diffraction pattern of X coincides with that of its vector set V, the symmetric feature of X derived from the symmetry of V was studied. The symmetry with the point group G_V or G_V/G_I according as X is inversion-symmetric or not, is defined as the vector symmetry of X, where G_V is the point group of V and G_I, is the inversion group, and when the vector symmetry of X is C_n, for example, X is specified as C_n-vector-symmetric. When X is homometric with itself by a rotation of 2π/n, it is specified as n-fold self-homometric. X being n-fold self-homometric is the necessary and sufficient condition for X to be n-fold vector-symmetric. Also, X exhibits an enhanced vector (diffraction) symmetry if it is a space-groupoid structure with the kernel whose point-group symmetry is, other than by addition of an inversion, higher than the point-group symmetry of X. Four examples of enhanced vector symmetry are examined.