Section 9.3.3. A finer division of lattices

The 44 lattice characters form a subdivision of the 14 Bravais types. There is another commonly known subdivision of the Bravais types, namely the 24 Delaunay sorts (symmetrische Sorten) (Delaunay, 1933; International Tables for X-ray Crystallography, 1952, Vol. I; cf. Section 9.1.8 ). However, both divisions, being based on quite different principles, are incompatible: the 44 lattice characters do not form a subdivision of the 24 Delaunay sorts.

A natural problem arises to construct a division of lattices which would be a subdivision of both the lattice characters and the Delaunay sorts. However, we do not admit a purely mechanical intersection of both these divisions; we insist that their common subdivision be crystallographically meaningful.

Such a division was proposed recently (Gruber, 1997a). It uses the fact that the Niggli points of all lattices lie in two five-dimensional polyhedra, say and . The underlying idea, originating from H. Wondratschek, is based on the distribution of Niggli points among the vertices, edges, faces, three- and four-dimensional hyperfaces, and the interior of and . This leads to a natural division of Niggli points and further to a division of lattices. This division has 67 classes, but is not suitable for crystallography because it does not constitute a subdivision of the Bravais types.

A modification of the idea is necessary. It consists of representing a lattice L by several points (instead of by one Niggli point) and the addition of two minor conditions. One of them concerns the diagonals of the Niggli cell and the other the bases of L which describe the Niggli cell.

Though these conditions are of little importance in themselves, they lead to a very useful notion, viz the division of all lattices into 127 classes which is a subdivision of both the lattice characters and the Delaunay sorts. The equivalence classes of this division are called genera. They form, in a certain sense, building blocks of both lattice characters and Delaunay sorts and show their mutual relationship.

The distribution of genera along the Bravais types is the following (the number of genera is given in parentheses): cP(1), cI(1), cF(1), tP(2), tI(5), oP(1), oC(8), oI(7), oF(3), hP(3), hR(4), mP(5), mC(43), aP(43). Thus, genera seem to be especially suitable for a finer classification of lattices of low symmetry.

The genus of a given lattice L can be determined – provided that the Niggli point of L is known – by means of a table containing explicit descriptions of all genera. These descriptions are formed by open linear systems of inequalities. Consequently, the ranges of conventional parameters of genera are open unlike those concerning the lattice characters.

Genera are denoted by symbols derived from the geometrical shape of and . They can be visualized in the three-dimensional cross sections of these bodies. This gives a fairly good illustration of the relationships between genera.

However, the most important feature of genera seems to be the fact that lattices of the same genus agree in a surprisingly great number of crystallographically significant properties, such as the number of Buerger cells, the densest directions and planes, the symmetry of these planes etc. Even the formulae for the conventional cells are the same. The genus appears to be a remarkably strong bond between lattices.