International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 9.1, pp. 745-749

Section 9.1.8. Delaunay reduction

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@knot.uni-erlangen.de

9.1.8. Delaunay reduction

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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. Chapter 9.3[link] ) have shown the common root of both crystallographic approaches. As in Chapters 9.2[link] and 9.3[link] the Niggli reduction will be discussed in detail, 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}. \eqno(9.1.8.1)] It can be shown that this sum can be reduced 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 performed such that the sum [\sum'] of the transformed [{\bf b}^{'2}_{i}] is smaller than [\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 forming the reduced basis are contained in the set [V = \{{\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}\},] which corresponds to the maximal set of faces of the Dirichlet domain (14 faces).

For practical application, it is useful to classify the patterns of the resulting [n(n-1)/2] scalar products regarding their equivalence or zero values. These classes of patterns correspond to the reduced bases and result in `Symmetrische Sorten' (Delaunay, 1933[link]) that lead directly to the conventional crystallographic cells by fixed transformations (cf. Patterson & Love, 1957[link]; Burzlaff & Zimmermann, 1993[link]). Table 9.1.8.1[link] gives the list of the 24 `symmetrische Sorten'. Column 1 contains Delaunay's symbols, column 2 the symbol of the Bravais type. For monoclinic centred lattices, the matrix [\bi P] of the last column transforms the primitive reduced cell into an I-centred cell, which has to be transformed to A or C according to the monoclinic standardization rules, if necessary. [\bi P] operates on the basis in the following form: [({\bf a}_{r},{\bf b}_{r},{\bf c}_{r})\cdot {\bi P} = ({\bf a}_{c},{\bf b}_{c},{\bf c}_{c}),] where [({\bf a}_{r},{\bf b}_{r},{\bf c}_{r})] denotes the [(1 \times 3)] matrix of the basis vectors of the reduced cell and [({\bf a}_{c},{\bf b}_{c},{\bf c}_{c})] the [(1 \times 3)] matrix of the conventional cell.

Table 9.1.8.1| top | pdf |
The 24 `Symmetrische Sorten'

In the centred monoclinic lattices, the set [\{{\bf a},{\bf c},{\bf a}+{\bf c}\}=\{{\bf p},{\bf q},{\bf r}\}] of the three shortest vectors in the ac plane is used to describe the metrical conditions. These vectors are renamed according to their relation to the projection of the centring point in the ac plane: p designates the vector that crosses the projection of the centring point, q is the shorter one of the two others and r labels the third one.

Delaunay symbolBravais typeMetrical conditions (parameters of conventional cells)Voronoi typeNotation of the scalar products according to equation (9.1.8.1)[link]Transformation matrix [\bi P]
121314232434
K1 cI I 12 12 12 12 12 12 [011/101/110]
K2 cF III 0 13 13 13 13 0 [1 \bar{1}1/111/002]
K3 cP V 0 0 14 14 14 0 [100/001/011]
0 0 14 0 14 14 [100/010/001]
H hP IV 12 0 12 0 12 34 [100/010/001]
R1 hR [2c^{2} \;\lt\; 3a^{2}] I 12 12 14 12 14 14 [101/\bar{1}11/0\bar{1}1]
R2 hR [2c^{2} \;\gt\; 3a^{2}] III 0 13 13 13 24 0 [101/003/012]
Q1 tI [c^{2} \;\lt\; 2a^{2}] I 12 13 13 13 13 12 [011/101/110]
Q2 tI [c^{2} \;\gt\; 2a^{2}] II 0 13 13 13 13 34 [101/011/002]
Q3 tP V 0 0 14 0 14 34 [100/010/001]
0 0 14 14 24 0 [100/001/011]
0 0 14 23 0 23 [001/110/010]
O1 oF I 12 13 13 13 13 34 [1\bar{1}1/111/002]
O2 oI [a^{2}+b^{2} \;\gt\; c^{2}] I 12 13 14 14 13 12 [011/101/110]
O3 oI [a^{2}+b^{2} \;\lt\; c^{2}] II 0 13 13 23 23 34 [101/011/002]
O4 oI [a^{2}+b^{2} = c^{2}] III 0 13 14 14 13 0 [011/101/110]
0 13 13 23 23 0 [101/011/002]
O5 o(AB)C IV 12 0 14 0 12 34 [200/110/001]
12 0 14 0 14 34 [110/\bar{1}10/001]
O6 oP V 0 0 14 0 24 34 [100/010/001]
0 0 14 23 24 0 [100/001/011]
M1 m(AC)I [b^{2} \;\gt\; p^{2}] I 12 13 14 13 14 34 [\bar{1}10/\bar{1}\bar{1}0/\bar{1}01]
M2 m(AC)I [p^{2} \;\gt\; b^{2} \;\gt\; r^{2}-q^{2}] I 12 13 14 14 13 34 [01\bar{1}/110/10\bar{1}]
M3 m(AC)I [r^{2}-q^{2} \;\gt\; b^{2}] II 0 13 14 23 23 34 [\bar{1}01/\bar{1}10/\bar{2}00]
M4 m(AC)I [b^{2} = p^{2}] II 0 13 14 14 13 34 [01\bar{1}/110/10\bar{1}]
0 13 14 13 14 34 [\bar{1}10/\bar{1}\bar{1}0/\bar{1}01]
M5 m(AC)I [b^{2} = r^{2}-q^{2}] III 0 13 14 23 23 0 [\bar{1}01/\bar{1}10/\bar{2}00]
0 13 14 23 13 0 [10\bar{1}/1\bar{1}0/0\bar{1}\bar{1}]
M6 mP IV 0 13 14 0 24 34 [100/010/001]
T1 aP I 12 13 14 23 24 34 [100/010/001]
T2 aP II 0 13 14 23 24 34 [100/010/001]
T3 aP III 0 13 14 23 24 0 [100/010/001]

Column 3 gives metrical conditions for the occurrence of certain Voronoi types. Column 5 indicates the relations among the scalar products of the reduced vector set. 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]).

References

Burzlaff, H. & Zimmermann, H. (1993). Kristallsymmetrie–Kristallstruktur, pp. 90–105. Erlangen: R. Merkel.
Delaunay, B. N. (1933). Neuere Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149.
Patterson, A. L. & Love, W. E. (1957). Remarks on the Delaunay reduction. Acta Cryst. 10, 111–116.
Schwarzenberger, R. L. E. (1980). N-dimensional crystallography. San Francisco: Pitman.
Selling, E. (1874). Über binäre und ternäre quadratische Formen. Crelles J. Reine Angew. Math. 77, 143ff.
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.








































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