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

International Tables for Crystallography (2006). Vol. A, ch. 9.1, p. 748

Table 9.1.8.1 

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

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]