Basic theorems of vector symmetry in crystallography

Abstract

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 z or Gv/G,,, 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, is the inversion group, and when the vector symmetry of X is C n, for example, X is specified as C,-vector-symmetric. When X is homometric with itself by a rotation of 2z#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 spacegroupoid 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.

Groupoid of orientational variants

“…The OD theory is actually a crystallographic (and algebraic) description of polytypism, but order-disorder transitions are far more universal and complex than order-disorder in polytypes. Dornberger-Schiff points out that the set of POs does not form a group but a groupoid since, for example, the operation that transforms the layer L p in L q and the operation that transforms the layer L r in L s cannot be combined, unless q = r. Such considerations explain some diffraction enhancements of symmetry not due to Friedel's law 10 in polytypes (Sadanaga, 1978;Sadanaga & Ohsumi, 1979) and in quasicrystals (Yamamoto & Ishihara, 1988). Those works introduce the concept of space groupoid in order to define the local symmetry operations on a space group, assuming that this one is constituted by a repeated substructure.…”

Does mathematical crystallography still have a role in the XXI century?

“…The symmetry operations of a space group are isometries operating on the whole crystal space and are also called 'total operations ' (Dornberger-Schiff, 1964a) or 'global operations' (Sadanaga & Ohsumi, 1979;Sadanaga et al, 1980). These are the operations that normally come to mind when one thinks of the symmetry of a crystal.…”

“…when the operation is È(S i ) ! S i and brings a component to coincide with itself, the partial operation is of special type and is called local (Sadanaga & Ohsumi, 1979;Sadanaga et al, 1980). A local operation is in fact a symmetry operation, which is defined only on a part of the crystal space: local operations may constitute a subperiodic group (Kopský & Litvin, 2002), and in particular a diperiodic group (Holser, 1958) when S i corresponds to a layer.…”

Penrose patterns and related structures. II. Decagonal quasicrystals

“…The operators hi (i = 1, 2,..., 6) operate only on the first substructure and superimposes it on the ith substructure, while hi I operate on the ith substructure and superimpose it on the first one. Then the set of operators which transforms a substructure into another substructure or into itself forms a groupoid (Sadanaga & Ohsumi, 1979). The groupoid T is given by {hiGhj-~[ hi, hj ~ H}, where G is the space group of the substructure and H is the set of hi.…”

“…Such an extinction rule is often observed in polytypes (Verma & Krishna, 1966). We point out that the two models can be regarded as polytypes in the quasicrystal and we discuss their interrelation and symmetry based on groupoid theory (Sadanaga & Ohsumi, 1979). In order to consider realistic models the density of the structure should also be taken into account.…”