International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 700708

By means of the abovementioned lattice properties, it is possible to classify lattices according to various criteria. Lattices can be subdivided with respect to their topological types of domains, resulting in two classes in two dimensions and five classes in three dimensions. They are called Voronoi types. If the classification involves topological and symmetry properties of the domains, 24 Symmetrische Sorten (Delaunay, 1933) are obtained in three dimensions and 5 in two dimensions. Other classifications consider either the centring type or the point group of the lattice.
The most important classification takes into account both the lattice pointgroup symmetry and the centring mode (Bravais, 1866). The resulting classes are called Bravais types of lattices or, for short, Bravais lattices. Two lattices belong to the same Bravais type if and only if they coincide both in their pointgroup symmetry and in the centring mode of their conventional cells. The Bravais lattice characterizes the translational subgroup of a space group. The number of Bravais lattices is 1 in one dimension, 5 in two dimensions, 14 in three dimensions and 64 in four dimensions. The Bravais lattices may be derived by topological (Delaunay, 1933) or algebraic procedures (Burckhardt, 1966; Neubüser et al., 1971). It can be shown (Wondratschek et al., 1971) that `all Bravais types of the same [crystal] family can be obtained from each other by the process of centring'. As a consequence, different Bravais types of the same crystal family (cf. Section 1.3.4 ) differ in their centring mode. Thus, the Bravais types may be described by a lowercase letter designating the crystal family and an uppercase letter designating the centring mode. The relations between the point groups of the lattices and the crystal families are shown in Table 3.1.1.1. Since the hexagonal and rhombohedral Bravais types belong to the same crystal family, the rhombohedral lattice is described by hR, h indicating the family and R the centring type. This nomenclature was adopted for the 1969 reprint of International Tables for Xray Crystallography (1952) and for Structure Reports since 1975 (cf. Trotter, 1975).
In Fig. 3.1.2.1, conventional cells for the 14 threedimensional Bravais types of lattices are illustrated.

Conventional cells of the threedimensional Bravais types of lattices (for symbols see Table 3.1.2.2). 
In Tables 3.1.2.1 and 3.1.2.2, the two and threedimensional Bravais types of lattices are described in detail. For each entry, the tables contain conditions that must be fulfilled by the lattice parameters and the metric tensor. These conditions are given with respect to two different basis systems, first the conventional basis related to symmetry, second a special primitive basis (see below). In columns 2 and 3, basis vectors not required by symmetry to be of the same length are designated by different letters. Columns 4 and 5 contain the metric tensors for the two related bases. Column 6 shows the relations between the components of the two tensors.


The last columns of Tables 3.1.2.1 and 3.1.2.2 show parallel projections of the appropriate conventional unit cells. Among the different possible choices of the primitive basis, as discussed in Section 3.1.1, the special primitive basis mentioned above is obtained according to the following rules:
If a primitive basis is chosen according to these rules, basis vectors of the conventional cell have parallel facediagonal or bodydiagonal orientation with respect to the basis vectors of the primitive cell. For cubic and rhombohedral lattices, the primitive basis vectors are selected such that they are symmetryequivalent with respect to a threefold axis. In all cases, a face of the `domain of influence' is perpendicular to each basis vector of these primitive cells.
Further classifications use reduction theory. There are different approaches to the reduction of quadratic forms in mathematics. The two most important in our context are
The investigations by Gruber (cf. Section 3.1.4) have shown the common root of both crystallographic approaches. As the Niggli reduction will be discussed in detail in Sections 3.1.3 and 3.1.4, we shall discuss the Delaunay reduction here.
We start with a lattice basis . This basis is extended by a vector All scalar products are considered. The reduction is performed minimizing the sum It can be shown that this sum can be reduced by a sequence of transformations as long as one of the scalar products is still positive. If e.g. the scalar product is still positive, a transformation can be applied such that the sum of the transformed is smaller than : In the twodimensional case, holds.
If all the scalar products are less than or equal to zero, the three shortest vectors of the reduced basis are contained in the set , called the Delaunay set, which corresponds to the maximal set of faces of the Dirichlet domain (at most 14 faces).
The result of a reduction can be presented by a graphical symbol, the Selling tetrahedron. The four corners of the tetrahedron correspond to the vectors b_{1}, b_{2}, b_{3}, b_{4}, the mutual scalar products are attached to the edges. A scalar product that is zero is indicated by `0'; equal scalar products are designated by the same graphical symbol (cf. Table 3.1.2.3).

Delaunay's classification is based on Voronoi types. Voronoi distinguishes five classes of Dirichlet domains. To describe these, the following symbols are used to represent particular topological features: s is used for a hexagon and for v for a quadrangle, s^{2} indicates an edge between two hexagons and v^{2} an edge between two quadrangles, v^{4} is a vertex where four quadrangles meet and v^{3} is a vertex where three quadrangles meet. The five types are topologically characterized by: V1 (8s, 6v, 12s^{2}), V2 (4s, 8v, 4s^{2}), V3 (12v, 24v^{2}, 8v^{3}, 6v^{4}), V4 (2s, 6v, 6v^{2}) and V5 (6v, 12v^{2}, 8v^{3}). The numbers give the multiplicities of each feature.
Delaunay combined the topological description with the rotation groups of the crystallographic holohedries. He used uppercase letters for these groups (K – cubic, H – hexagonal, R – rhombohedral, Q – tetragonal, O – orthorhombic, M – monoclinic, T – triclinic) followed by a incremental number if more than one Voronoi type with the same symmetry exists. The results are presented in Table 3.1.2.3. In each row a `Symmetrische Sorte' is described.
Column 1 contains the Delaunay description followed by the Voronoi type. Beneath these, the Bravais lattice and the symbol of its holohedry are given. Next the topological features that are compatible with the symmetry axes referred to the `blickrichtungen' of the holohedry are listed. Column 2 gives the metric conditions for the occurrence of certain Voronoi types. For the monoclinic cases with centred cells (M1–M5) it is useful to introduce in addition to the vectors a, c, f = a + c special parameters (p^{2}, q^{2}, r^{2}). p designates the vector below the centring point in the projection in the net perpendicular to b. q is the shorter one of the other two vectors and r labels the remaining one (cf. Burzlaff & Zimmermann, 1985).
For practical applications, it is useful to classify the patterns of the resulting six scalar products regarding their equivalence or zero values in the form of a symbolic (Selling) tetrahedron (column 3). These classes of patterns correspond to the reduced bases. They result in 24 `Symmetrische Sorten' (Delaunay, 1933) that fix the Voronoi types and the holohedries, and simultaneously lead directly to the conventional crystallographic cells by fixed transformations (cf. Patterson & Love, 1957; Burzlaff & Zimmermann, 1993).
Column 4 contains projections of the Dirichlet domain along the symmetry directions indicated by the topological/symmetry symbol in column 1. Column 5 shows the relation between the Dirichlet domain and the conventional cell. Column 6 contains the transformation matrix from the reduced basis to the conventional basis. (Note: In the monoclinic centred case it leads to the I centring.)
In some cases, different Selling patterns are given for one `Symmetrische Sorte'. This procedure avoids a final reduction step (cf. Patterson & Love, 1957) and simplifies the computational treatment significantly. The number of `Symmetrische Sorten', and thus the number of transformations which have to be applied, is smaller than the number of lattice characters according to Niggli. Note that the introduction of reduced bases using shortest lattice vectors causes complications in more than three dimensions (cf. Schwarzenberger, 1980).
Let the basis B = (b_{1}, b_{2}, b_{3}) given by the scalar productsor by b_{1} = 2.449 (), b_{2} = b_{3} = 2.828 () (in arbitrary units), β_{23} = 60° (cos β_{23} = ½), β_{13} = 73.22° (), β_{12} = 64.34° ().
The aim is to find a standardized basis of shortest lattice vectors using Delaunay reduction. This example, given by B. Gruber (cf. Burzlaff & Zimmermann, 1985), shows the standardization problems remaining after the reduction.
The general reduction step can be described using Selling four flats. The corners are designated by the vectors a, b, c, d = −a − b − c. The edges are marked by the scalar products among these vectors. If positive scalar products can be found, choose the largest: (indicated as ab in Fig. 3.1.2.2a). The reduction transformation is: a_{D} = a, b_{D} = −b, c_{D} = c + b, d_{D} = d + b (see Fig. 3.1.2.2a). In this example, this results in the Selling four flat shown in Fig. 3.1.2.2(b). The next step, shown in Fig. 3.1.2.2(c), uses the (maximal) positive scalar product for further reduction. Finally, using b_{2} + b_{3} + b_{4} = −b_{1} we get the result shown in Fig. 3.1.2.2(d).

Delaunay reduction of Gruber's example (cf. Section 3.1.2.4). The edges of Selling tetrahedra are labelled by the scalar products of the vectors which designate the corners of the tetrahedra. 
The complete procedure can be expressed in a table, as shown in Table 3.1.2.4. Each pair of lines contains the starting basis and its scalar products before transformation as the first line, and then the transformed scalar products and the Delaunay basis after transformation below. In our case, four transformation steps are necessary. The result isThe final Selling tetrahedron shows that the Dirichlet domain belongs to Voronoi type 1. It fulfils no symmetry condition and thus corresponds to an anorthic (triclinic) lattice.

For further standardization we consider the Delaunay setAll bases of shortest lattice vectors () can be found:Any basis of shortest lattice vectors contains . For the vectors , , and are possible. can only be chosen from these vectors such that a linear independent triplet results.
The resulting five choices are given in Table 3.1.2.5. Any case corresponds to eight combinations of signs for the three basis vectors. The principle of the `homogenous corner' (i.e., there is always a pair of opposite corners of the corresponding cell where all angles are either nonacute or all three are acute) selects one of the bases in each case, thus five different bases remain. For the final choice the surfaces of the corresponding cells are given.

The maximal surface has cell No. 1 with the metrical parametersA last possibility for the standardization is the interchange of b and c with inversion of all basis vectors. In this way the sequence of β and γ can be interchanged:
References
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