Niggli lattice characters: definition and graphical representation

An exact definition of the 44 lattice characters listed by Niggli is thoroughly discussed and is elucidated by examples. In order to represent the characters graphically, use is made of the projection of the Niggli-reduced basis vector c on the a, b plane. Not only is the projected end point of c restricted to certain domains in the plane by the reduction rules - cf. International Tables for Crystallography (1987), ch. 9.3 (Dordrecht: Kluwer) - but for given constants A, B and F in the Niggli-reduced form this polygonal domain contains the locus of each of the characters as a vertex or an edge or the area of the polygon. For each of the cases a = b = c, a = b < c, a < b = c and a < b < c, nine figures fully cover all alternatives determined by five special values F = A/2, 0, -A/4, -A/3 and -A/2 and the four open intervals between them. Also, all normalized Buerger bases which are not Niggli-reduced bases are shown in the same figures.