Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 700-708

Section 3.1.2. Bravais types of lattices and other classifications

H. Burzlaffa and H. Zimmermannb

3.1.2. Bravais types of lattices and other classifications

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By means of the above-mentioned 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[link]) 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 point-group symmetry and the centring mode (Bravais, 1866[link]). 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 point-group 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[link]) or algebraic procedures (Burckhardt, 1966[link]; Neubüser et al., 1971[link]). It can be shown (Wondratschek et al., 1971[link]) 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[link] ) differ in their centring mode. Thus, the Bravais types may be described by a lower-case letter designating the crystal family and an upper-case letter designating the centring mode. The relations between the point groups of the lattices and the crystal families are shown in Table[link]. 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[link] reprint of International Tables for X-ray Crystallography (1952[link]) and for Structure Reports since 1975 (cf. Trotter, 1975[link]). Description of Bravais types of lattices

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In Fig.[link], conventional cells for the 14 three-dimensional Bravais types of lattices are illustrated.


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Conventional cells of the three-dimensional Bravais types of lattices (for symbols see Table[link]).

In Tables[link] and[link], the two- and three-dimensional 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.

Table| top | pdf |
Two-dimensional Bravais types of lattices

Bravais type of latticeLattice parametersMetric tensorProjections
ConventionalPrimitiveConventionalPrimitive/transformation to primitive cellRelations of the components
mp a, b a, b [\matrix{g_{11} &g_{12}\hfill\cr &g_{22}\hfill\cr}] [\matrix{g_{11} &g_{12}\hfill\cr &g_{22}\hfill\cr}]   [Scheme scheme130]
γ γ
op a, b a, b [\matrix{g_{11} &0\hfill\cr &g_{22}\hfill\cr}] [\matrix{g_{11} &0\hfill\cr &g_{22}\hfill\cr}]   [Scheme scheme131]
γ = 90° γ = 90°
oc a1 = a2, γ P(c) [g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})] [Scheme scheme132]
  [\matrix{g'_{11} &g'_{12}\hfill\cr &g'_{11}\hfill\cr}] [g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]
[g_{11} = 2(g'_{11} + g'_{12})]
[g_{12} = 2(g'_{11} - g'_{12})]
tp a1 = a2 a1 = a2 [\matrix{g_{11} &0\hfill\cr &g_{11}\hfill\cr}] [\matrix{g_{11} &0\hfill\cr &g_{11}\hfill\cr}]   [Scheme scheme133]
γ = 90° γ = 90°
hp a1 = a2 a1 = a2 [\matrix{g_{11} &-{\textstyle{1 \over 2}}\,g_{11}\hfill\cr &\phantom{-{\textstyle{1 \over 2}}\,}g_{11}\hfill\cr}] [\matrix{g_{11} &-{\textstyle{1 \over 2}}\,g_{11}\hfill\cr &\phantom{-{\textstyle{1 \over 2}}\,}g_{11}\hfill\cr}]   [Scheme scheme134]
γ = 120° γ = 120°
The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985[link]).
[{\bi P}(c) =\textstyle{1 \over 2}(11/\bar{1}1).]

Table| top | pdf |
Three-dimensional Bravais types of lattices

Bravais type of latticeLattice parametersMetric tensorProjections
ConventionalPrimitiveConventionalPrimitive/transf.Relations of the components
aP a, b, c a, b, c [\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &g_{12} &g_{13}\hfill\cr &g_{22} &g_{23}\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme116]
α, β, γ α, β, γ
mP a, b, c a, b, c [\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &0 &g_{13}\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme117]
β, α = γ = 90° β, α = γ = 90°
mC a1 = a2, c P(C) [g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})] [Scheme scheme118]
(mS) γ, α = β [\matrix{g'_{11} &g'_{12} &g'_{13}\hfill\cr &g'_{11} &g'_{13}\hfill\cr &&g_{33}\hfill\cr}] [g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]
[g'_{13} = {1 \over 2}g_{13}]
[g_{11} = 2(g'_{11} + g'_{12})]
[g_{22} = 2(g'_{11} - g'_{12})]
[g_{13} = 2g'_{13}]
oP a, b, c a, b, c [\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{22} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme119]
α = β = γ = 90° α = β = γ = 90°  
oC a1 = a2, c P(C) [g'_{11} = {\textstyle{1 \over 4}}(g_{11} + g_{22})] [Scheme scheme120]
(oS) γ, α = β = 90° [\matrix{g'_{11} &g'_{12} &0\hfill\cr &g'_{11} &0\hfill\cr & &g_{33}\hfill\cr}] [g'_{12} = {\textstyle{1 \over 4}}(g_{11} - g_{22})]
[g_{11} = 2(g'_{11} + g'_{12})]
[g_{22} = 2(g'_{11} - g'_{12})]
oI a1 = a2 = a3 P(I) [g'_{12} = {\textstyle{1 \over 4}}(-g_{11} - g_{22} + g_{33})] [Scheme scheme121]
α, β, γ [\matrix{-\tilde g &g'_{12} &g'_{13}\hfill\cr &-\tilde g &g'_{23}\hfill\cr & &-\tilde g\hfill\cr}] [g'_{13} = {\textstyle{1 \over 4}}(-g_{11} + g_{22} - g_{33})]
cos α + cos β + cos γ = −1 [g'_{23} = {\textstyle{1 \over 4}}(g_{11} - g_{22} - g_{33})]
[\tilde{g} = g'_{12} + g'_{13} + g'_{23}] [g_{11} = -2(g'_{12} + g'_{13})]
[g_{22} = -2(g'_{12} + g'_{23})]
[g_{33} = -2(g'_{13} + g'_{23})]
oF a, b, c [{\bi P}(F)] [g'_{12} = {\textstyle{1 \over 4}}\;g_{33}] [Scheme scheme122]
α, β, γ [\matrix{\tilde{g}_{1} &g'_{12} &g'_{13}\hfill\cr &\tilde{g}_{2} &g'_{23}\hfill\cr & &\tilde{g}_{3}\hfill\cr}] [g'_{13} = {\textstyle{1 \over 4}}\;g_{22}]
[\eqalign{&\cos \alpha =\cr&\quad {\displaystyle{-a^{2} + b^{2} + c^{2} \over 2bc}}\cr &\cos \beta =\cr&\quad {\displaystyle{a^{2} + b^{2} + c^{2} \over 2ac}}\cr &\cos \gamma =\cr&\quad {\displaystyle{a^{2} + b^{2} - c^{2} \over 2ab}}}] [g'_{23} = {\textstyle{1 \over 4}}\;g_{11}]
[\displaylines{\openup-4pt\hfill\cr \tilde{g}_{1} = g'_{12} + g'_{13}\hfill\cr\tilde{g}_{2} = g'_{12} + g'_{23}\hfill\cr \tilde{g}_{3} = g'_{13} + g'_{23}\hfill\cr}] [g_{11} = 4g'_{23}]
[g_{22} = 4g'_{13}]
[g_{33} = 4g'_{12}]
tP a1 = a2, c a1 = a2, c [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme123]
α = β = γ = 90° α = β = γ = 90°
tI a1 = a2 = a3 [{\bi P}(I)] [g'_{12} = {\textstyle{1 \over 4}}(-2g_{11} + g_{33})] [Scheme scheme124]
γ, α = β [\matrix{\bar{g} &g'_{12} &g'_{13}\hfill\cr &\bar{g} &g'_{13}\hfill\cr & &\bar{g}\hfill\cr}] [g'_{13} = -{\textstyle{1 \over 4}}g_{33}]
2 cos α + cos γ = −1  
  [\bar{g} = -(g'_{12} + 2g'_{13})] [g_{11} = 2(g'_{12} + g'_{13})]
  [g_{33} = -4g'_{13}]
hR a1 = a2, c a1 = a2 = a3 [\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &\phantom{\textstyle{1 \over 2}}g_{11} &0\hfill\cr & &g_{33}\hfill\cr}] [{\bi P}(R)] [g'_{11} = {\textstyle{1 \over 9}}(3g_{11} + g_{33})] [Scheme scheme125]
α = β = 90° α = β = γ [\matrix{g'_{11} &g'_{12} &g'_{12}\hfill\cr &g'_{11} &g'_{12}\hfill\cr & &g'_{11}\hfill\cr}] [g'_{12} = {\textstyle{1 \over 9}}(-{3 \over 2}g_{11} + g_{33})]
γ = 120°  
[g_{11} = 2(g'_{11} - g'_{12})]
[g_{33} = 3(g'_{11} + 2g'_{12})]
hP a1 = a2, c [\matrix{g_{11} &-{\textstyle{1 \over 2}}g_{11} &0\hfill\cr &\phantom{ab\ }g_{11} &0\hfill\cr & &g_{33}\hfill\cr}]   [Scheme scheme126]
α = β = 90°
γ = 120°
cP a1 = a2 = a3 a1 = a2 = a3 [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}] [\matrix{g_{11} &0 &0\hfill\cr &g_{11} &0\hfill\cr & &g_{11}\hfill\cr}]   [Scheme scheme127]
α = β = γ = 90° α = β = γ = 90°
cI a1 = a2 = a3 P(I) [g'_{11} = {\textstyle{3 \over 4}}g_{11}] [Scheme scheme128]
α = β = γ = 109.5° [\matrix{g'_{11} &-{\textstyle{1 \over 3}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr &\phantom{{\textstyle{1 \over 3}}}g'_{11} &-{\textstyle{1 \over 3}}g'_{11}\hfill\cr& &\phantom{-{\textstyle{1 \over 3}}}g'_{11}\hfill\cr}] [g_{11} = {\textstyle{4 \over 3}}g'_{11}]
[\cos \alpha = -{\textstyle{1 \over 3}}]
cF a1 = a2 = a3 P(F) [g'_{11} = {\textstyle{1 \over 2}}g_{11}] [Scheme scheme129]
α = β = γ = 60° [\matrix{g'_{11} &{\textstyle{1 \over 2}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr &\phantom{{\textstyle{1 \over 3}}}g'_{11} &{\textstyle{1 \over 2}}g'_{11}\hfill\cr & &\phantom{{\textstyle{1 \over 3}}}g'_{11}\hfill\cr}] [g_{11} = 2g'_{11}]
The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985[link]). Symbols in parentheses are standard symbols, see Table[link] .
[{\bi P}(C) = \textstyle{1 \over 2}(110/\bar{1}10/002), {\bi P}(I) = \textstyle{1 \over 2}(\bar{1}11/1\bar{1}1/11\bar{1}), {\bi P}(F) = \textstyle{1 \over 2}(011/101/110), {\bi P}(R)=\textstyle{1 \over 3}(\bar{1}2\bar{1}/\bar{2}11/111).]

The last columns of Tables[link] and[link] 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[link], the special primitive basis mentioned above is obtained according to the following rules:

  • (i) For each type of centring, only one transformation matrix [\bi P] is used to obtain the primitive cell as given in Tables[link] and[link]. The transformation obeys equation ([link].

  • (ii) Among the different possible transformations, those are preferred which result in a metric tensor with simple relations among its components, as defined in Tables[link] and[link].

If a primitive basis is chosen according to these rules, basis vectors of the conventional cell have parallel face-diagonal or body-diagonal 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 symmetry-equivalent 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. Delaunay reduction and standardization

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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

  • (i) the Selling–Delaunay reduction (Selling, 1874[link]),

  • (ii) the Eisenstein–Niggli reduction.

The investigations by Gruber (cf. Section 3.1.4[link]) have shown the common root of both crystallographic approaches. As the Niggli reduction will be discussed in detail in Sections 3.1.3[link] and 3.1.4[link], we shall discuss the Delaunay reduction here.

We start with a lattice basis [({\bf b}_{i})_{1\leq i\leq n}\ (n =2,3)]. This basis is extended by a vector [{\bf b}_{n+1} = - ({\bf b}_{1}+ \ldots +{\bf b}_{n}).]All scalar products [{\bf b}_{i}\cdot {\bf b}_{k}\ (1 \leq i \lt k \leq n+1)]are considered. The reduction is performed minimizing the sum [\textstyle\sum = {{\bf b}_{1}}^{2}+ \ldots +{{\bf b}_{n+1}}^{2}.]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 [{\bf b}_{1}\cdot {\bf b}_{2}] is still positive, a transformation can be applied such that the sum [\textstyle\sum'] of the transformed [{\bf b}'^{2}_{i}] is smaller than [\textstyle\sum]: [{\bf b}'_{1} = -{\bf b}_{1},\ {\bf b}'_{2} = {\bf b}_{2},\ {\bf b}'_{3} = {\bf b}_{1} + {\bf b}_{3} \hbox{ and } {\bf b}'_{4} = {\bf b}_{1} + {\bf b}_{4}.]In the two-dimensional case, [{\bf b}'_{3} = 2{\bf b}_{1}+{\bf b}_{3}] 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 [\{{\bf b}_{1}, {\bf b}_{2}, {\bf b}_{3}, {\bf b}_{4}, {\bf b}_{1}+{\bf b}_{2}, {\bf b}_{2}+{\bf b}_{3}, {\bf b}_{3}+{\bf b}_{1}\}], 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 b1, b2, b3, b4, 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[link]).

Table| top | pdf |
Delaunay types of lattices (`Symmetrische Sorten')

Delaunay–Voronoi typeMetric conditionsSelling tetrahedronProjections along symmetry directionsDirichlet domain in the unit cellTransformation to the conventional cell
K1 V1 [Scheme scheme1] [Scheme scheme2] [Scheme scheme3] [Scheme scheme4] [Scheme scheme5] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
[{4\over m}\overline 3 {2\over m}]
v s s2
K2 V3 [Scheme scheme6] [Scheme scheme7] [Scheme scheme8] [Scheme scheme9] [Scheme scheme10] [\pmatrix{1&-1&1\cr 1&1&1\cr0&0&2}]
[{4 \over m}\overline 3{2\over m}]
v4 v3 v
K3 V5 [Scheme scheme11] [Scheme scheme12] [Scheme scheme13] [Scheme scheme14] [Scheme scheme15] [\pmatrix{1&0&0\cr0&0&1\cr0&1&1}]
[{4 \over m}\overline 3{2\over m}]
v v3 v2
  [Scheme scheme16] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
H V4 [Scheme scheme17] [Scheme scheme18] [Scheme scheme19] [Scheme scheme20] [Scheme scheme21] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
[{6\over m}{2\over m}{2 \over m}]
s v v2
R1 V1 [2c^2\,\lt\,3a^2] [Scheme scheme22] [Scheme scheme23] [Scheme scheme24] [Scheme scheme25] [\pmatrix{1&0&1\cr-1&1&1\cr0&-1&1}]
[\overline 3 {2\over m}]
s s2
R2 V3 [2c^2\,\gt\,3a^2] [Scheme scheme26] [Scheme scheme27] [Scheme scheme28] [Scheme scheme29] [\pmatrix{1&0&1\cr0&0&3\cr0&1&2}]
[\overline 3 {2\over m}]
v3 v
Q1 V1 [c^2\,\lt\,2a^2] [Scheme scheme30] [Scheme scheme31] [Scheme scheme32] [Scheme scheme33] [Scheme scheme34] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
[{4\over m}{2\over m}{2\over m}]
v v s2
Q2 V2 [c^2\,\gt\,2a^2] [Scheme scheme35] [Scheme scheme36] [Scheme scheme37] [Scheme scheme38] [Scheme scheme39] [\pmatrix{1&0&1\cr0&1&1\cr0&0&2}]
[{4\over m}{2\over m}{2\over m}]
v4 s s2
Q3 V5 [Scheme scheme40] [Scheme scheme41] [Scheme scheme42] [Scheme scheme43] [Scheme scheme44] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
[{4\over m}{2\over m}{2\over m}]
v v v2
  [Scheme scheme45] [\pmatrix{1&0&0\cr0&0&1\cr0&1&1}]
  [Scheme scheme46] [\pmatrix{0&0&1\cr1&1&0\cr0&1&0}]
O1 V1 [Scheme scheme47] [Scheme scheme48] [Scheme scheme49] [Scheme scheme50] [Scheme scheme51] [\pmatrix{1&-1&1\cr1&1&1\cr0&0&2}]
[{2\over m}{2\over m}{2\over m}]
s2 v s2
O2 V1 [a^2+b^2\,\gt\,c^2] [Scheme scheme52] [Scheme scheme53] [Scheme scheme54] [Scheme scheme55] [Scheme scheme56] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
[{2\over m}{2\over m}{2\over m}]
v v v
O3 V2 [a^2+b^2\,\lt\,c^2] [Scheme scheme57] [Scheme scheme58] [Scheme scheme59] [Scheme scheme60] [Scheme scheme61] [\pmatrix{1&0&1\cr0&1&1\cr0&0&2}]
[{2\over m}{2\over m}{2\over m}]
s s v4
O4 V3 [a^2+b^2=c^2] [Scheme scheme62] [Scheme scheme63] [Scheme scheme64] [Scheme scheme65] [Scheme scheme66] [\pmatrix{0&1&1\cr1&0&1\cr1&1&0}]
[{2\over m}{2\over m}{2\over m}]
v v v4
  [Scheme scheme67] [\pmatrix{1&0&1\cr0&1&1\cr0&0&2}]
O5 V4 [Scheme scheme68] [Scheme scheme69] [Scheme scheme70] [Scheme scheme71] [Scheme scheme72] [\pmatrix{2&0&0\cr1&1&0\cr0&0&1}]
[{2\over m}{2\over m}{2\over m}]
s v2 v
  [Scheme scheme73] [\pmatrix{1&1&0\cr-1&1&0\cr0&0&1}]
O6 V5 [Scheme scheme74] [Scheme scheme75] [Scheme scheme76] [Scheme scheme77] [Scheme scheme78] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
[{2\over m}{2\over m}{2\over m}]
v v v
  [Scheme scheme79] [\pmatrix{1&0&0\cr0&0&1\cr0&1&1}]

Delaunay–Voronoi typeMetric conditionsSelling tetrahedronProjections along symmetry directionsDirichlet domain in the unit cellTransformation to the conventional cell
M1 V1 [b^2\,\gt\,p^2] [Scheme scheme80] [Scheme scheme81] [Scheme scheme82] [Scheme scheme83] [Scheme scheme84] [\pmatrix{-1&1&0\cr-1&-1&0\cr-1&0&1}]
[{2\over m}]
  [A\colon\ b^2\,\gt\,c^2] [C\colon\ b^2\,\gt\,a^2] [I\colon\ b^2\,\gt\,f^2]
M2 V1 [p^2\,\gt\,b^2], [b^2\,\gt\,r^2-q^2] [Scheme scheme85] [Scheme scheme86] [Scheme scheme87] [Scheme scheme88] [Scheme scheme89] [\pmatrix{0&1&-1\cr1&1&0\cr1&0&-1}]
[{2\over m}]
  [A\colon c^2\,\gt\,b^2\,\gt\,f^2-a^2] [C\colon a^2\,\gt\,b^2\,\gt\,f^2-c^2] [I\colon f^2\,\gt\,b^2\,\gt\,c^2-a^2]
M3 V2 [r^2-q^2\,\gt\,b^2] [Scheme scheme90] [Scheme scheme91] [Scheme scheme92] [Scheme scheme93] [Scheme scheme94] [\pmatrix{-1&0&1\cr-1&1&0\cr-2&0&0}]
[{2\over m}]
  [A\colon\ f^2-a^2\,\gt\,b^2] [C\colon\ f^2-c^2\,\gt\,b^2] [I\colon\ c^2-a^2\,\gt\,b^2]
M4 V4 [b^2=p^2] [Scheme scheme95] [Scheme scheme96] [Scheme scheme97] [Scheme scheme98] [Scheme scheme99] [\pmatrix{0&1&-1\cr1&1&0\cr1&0&-1}]
[{2\over m}]
  [Scheme scheme100] [A\colon\ b^2=c^2] [C\colon\ b^2=a^2] [I\colon\ b^2=f^2] [\pmatrix{-1&1&0\cr-1&-1&0\cr-1&0&1}]
M5 V3 [b^2=r^2-q^2] [Scheme scheme101] [Scheme scheme102] [Scheme scheme103] [Scheme scheme104] [Scheme scheme105] [\pmatrix{-1&0&1\cr-1&1&0\cr-2&0&0}]
[{2\over m}]
  [Scheme scheme106] [A\colon\ b^2=f^2-a^2] [C\colon\ b^2=f^2-c^2] [I\colon\ b^2=c^2-a^2] [\pmatrix{1&0&-1\cr1&-1&0\cr0&-1&-1}]
M6 V4 [Scheme scheme107] [Scheme scheme108] [Scheme scheme109] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
[{2\over m}]
T1 V1 [Scheme scheme110]   [Scheme scheme111] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
T2 V2 [{\bf a}\cdot{\bf b}=0] [Scheme scheme112]   [Scheme scheme113] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
T3 V3 [{\bf a}\cdot{\bf b}=0] [Scheme scheme114]   [Scheme scheme115] [\pmatrix{1&0&0\cr0&1&0\cr0&0&1}]
aP [({\bf a}+{\bf b}+{\bf c})\cdot{\bf c}]
1 = 0

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, s2 indicates an edge between two hexagons and v2 an edge between two quadrangles, v4 is a vertex where four quadrangles meet and v3 is a vertex where three quadrangles meet. The five types are topologically characterized by: V1 (8s, 6v, 12s2), V2 (4s, 8v, 4s2), V3 (12v, 24v2, 8v3, 6v4), V4 (2s, 6v, 6v2) and V5 (6v, 12v2, 8v3). The numbers give the multiplicities of each feature.

Delaunay combined the topological description with the rotation groups of the crystallographic holohedries. He used upper-case 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[link]. 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 (p2, q2, r2). 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[link]).

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[link]) that fix the Voronoi types and the holohedries, and simultaneously lead directly to the conventional crystallographic cells by fixed transformations (cf. Patterson & Love, 1957[link]; Burzlaff & Zimmermann, 1993[link]).

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[link]) 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[link]). Example of Delaunay reduction and standardization of the basis

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Let the basis B = (b1, b2, b3) given by the scalar products[\pmatrix{g_{11}&g_{22}&g_{33}\cr g_{23}&g_{31}&g_{12}}=\pmatrix{6&8&8\cr4&2&3}]or by b1 = 2.449 ([\sqrt6]), b2 = b3 = 2.828 ([\sqrt8]) (in arbitrary units), β23 = 60° (cos β23 = ½), β13 = 73.22° ([\cos \beta_{13} = \sqrt3/6]), β12 = 64.34° ([\cos\beta_{12} =\sqrt3/4]).

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[link]), 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 = −abc. The edges are marked by the scalar products among these vectors. If positive scalar products can be found, choose the largest: [{\bf a}\cdot{\bf b}] (indicated as ab in Fig.[link]). The reduction transformation is: aD = a, bD = −b, cD = c + b, dD = d + b (see Fig.[link]a). In this example, this results in the Selling four flat shown in Fig.[link](b). The next step, shown in Fig.[link](c), uses the (maximal) positive scalar product for further reduction. Finally, using b2 + b3 + b4 = −b1 we get the result shown in Fig.[link](d).


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Delaunay reduction of Gruber's example (cf. Section[link]). 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[link]. 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 is[{\bf a}_{\rm D} = -{\bf b}_3, \quad {\bf b}_{\rm D} = {\bf b}_1,\quad {\bf c}_{\rm D} = {\bf b}_2-{\bf b}_1,\quad {\bf d}_{\rm D} = {\bf b}_3-{\bf b}_1.]The 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.

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Delaunay reduction for Gruber's example

abcdab, aDbDac, aDcDad, aDdDbc, bDcDbd, bDdDcd, cDdDaDbDcDdD
b2 b3 b1 b4 +4 +3 −15 +2 14 −11        
        −4 +7 −11 −10 +6 −15 b2 b3 b1 + b3 b4 + b3
b1 + b3 b2 b3 b3 + b4 +7 −10 −15 −4 −11 +6        
        −7 −3 −8 −4 +3 −1 b1 + b3 b2 b2b3 b2 + b3 + b4
b1 b2 b2b3 b1 + b3 +3 −1 −8 −4 −7 −3        
        −3 +2 −5 −4 −1 −6 b1 b2 b3 b1b2 + b3
b3 b1 b2 b1b2 + b3 +2 −4 −6 −3 −5 −1        
        −2 −2 −4 −3 −1 −3 b3 b1 b2b1 b3b1

For further standardization we consider the Delaunay set[\eqalign{&\{\pm {\bf a}_{\rm D},\pm {\bf b}_{\rm D}, \pm {\bf c}_{\rm D}, \pm {\bf d}_{\rm D}\cr &\quad=-({\bf a}_{\rm D}+{\bf b}_{\rm D}+{\bf c}_{\rm D}), \pm({\bf b}_{\rm D}+{\bf c}_{\rm D}), \pm({\bf a}_{\rm D}+{\bf c}_{\rm D}), \pm({\bf a}_{\rm D}+{\bf b}_{\rm D})\}.}]All bases of shortest lattice vectors ([{\bf b}_1^{\rm s}, {\bf b}_2^{\rm s}, {\bf b}_3^{\rm s}]) can be found:[\displaylines{|{\bf a}_{\rm D}|^2=8,\quad |{\bf b}_{\rm D}|^2=6,\quad |{\bf c}_{\rm D}|^2=8,\quad |{\bf d}_{\rm D}|^2=8,\cr |{\bf b}_{\rm D}+{\bf c}_{\rm D}|^2=8, \quad|{\bf a}_{\rm D}+{\bf c}_{\rm D}|^2=12,\quad |{\bf a}_{\rm D}+{\bf b}_{\rm D}|^2=10.}]Any basis of shortest lattice vectors contains [{\bf b}_1^{\rm s}={\bf b}_{\rm D}={\bf b}_1]. For [{\bf b}_2^{\rm s}] the vectors [{\bf a}_{\rm D}=-{\bf b}_3], [{\bf c}_{\rm D}={\bf b}_2-{\bf b}_1], [{\bf d}_{\rm D}={\bf b}_3-{\bf b}_1] and [({\bf b}_{\rm D}+{\bf c}_{\rm D})={\bf b}_2] are possible. [{\bf b}_3^{\rm s}] can only be chosen from these vectors such that a linear independent triplet results.

The resulting five choices are given in Table[link]. 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 non-acute 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.

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Discussion of Gruber's example using the cell surface

No.[({\bf b}_1^{\rm s}, {\bf b}_2^{\rm s}, {\bf b}_3^{\rm s})]Homogenous cornerSurface (surface units)
1 (+bD, +aD, +cD) Non-acute 41.25
2 (+bD, +aD, +dD) Non-acute 40.83
3 (+bD, −aD, bD + cD) Acute 39.61
4 (+bD, +cD, +dD) Non-acute 41.03
5 (+bD, −dD, bD + cD) Acute 40.06

The maximal surface has cell No. 1 with the metrical parameters[a = 2.449, b = c = 2.828\,{\rm \AA}, \alpha = 104.47, \beta = 115.66, \gamma = 106.78^\circ.]A 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:[a = 2.449, b = c = 2.828\,{\rm \AA}, \alpha = 104.47, \beta= 106.78, \gamma= 115.66^\circ.]


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