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

International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 165-166

Table 2.1.3.7 

Th. Hahna and A. Looijenga-Vosb

Table 2.1.3.7| top | pdf |
Zonal and serial reflection conditions for glide planes and screw axes (cf. Table 2.1.2.1[link])

(a) Glide planes

Type of reflectionsReflection conditionGlide planeCrystallographic coordinate system to which condition applies
Orientation of planeGlide vectorSymbol
0kl [k = 2n] (100) [{\bf b}/2] b [\!\left.\matrix{\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (a\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[l = 2n] [{\bf c}/2] c
[k + l = 2n] [{\bf b}/2 + {\bf c}/2] n
[\!\matrix{k + l = 4n\hfill\cr \quad (k,l = 2n)\hfill\cr}] [{\bf b}/4 \pm {\bf c}/4] d
h0l [l = 2n] (010) [{\bf c}/2] c [\!\left.\matrix{\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (b\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[h = 2n] [{\bf a}/2] a
[l + h = 2n] [{\bf c}/2 + {\bf a}/2] n
[\!\matrix{l + h = 4n\hfill\cr \quad (l,h = 2n)\hfill\cr}] [{\bf c}/4 \pm {\bf a}/4] d
hk0 [h = 2n] (001) [{\bf a}/2] a [\!\left.\matrix{\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic (}c\ \hbox{unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[k = 2n] [{\bf b}/2] b
[h + k = 2n] [{\bf a}/2 + {\bf b}/2] n
[\!\matrix{h + k = 4n\hfill\cr \quad (h,k = 2n)\hfill\cr}] [{\bf a}/4 \pm {\bf b}/4] d
[\matrix{h\bar{h}0l\hfill\cr 0k\bar{k}l\hfill\cr \bar{h}0hl\hfill\cr}] [l = 2n] [\left.\!\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}\right\} \{11\bar{2}0\}] [{\bf c}/2] c [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]
[\matrix{hh.\overline{2h}.l\hfill\cr \overline{2h}.hhl\hfill\cr h.\overline{2h}.hl\hfill\cr}] [l = 2n] [\left.\!\matrix{(1\bar{1}00)\hfill\cr (01\bar{1}0)\hfill\cr (\bar{1}010)\hfill\cr}\right\} \{1\bar{1}00\}] [{\bf c}/2] c [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]
[\matrix{hhl\hfill\cr hkk\hfill\cr hkh\hfill\cr}] [\matrix{l = 2n\hfill\cr h = 2n\hfill\cr k = 2n\hfill\cr}] [\left.\!\matrix{(1\bar{1}0)\hfill\cr (01\bar{1})\hfill\cr (\bar{1}01)\hfill\cr}\right\} \{1\bar{1}0\}] [\matrix{{\bf c}/2\hfill\cr {\bf a}/2\hfill\cr {\bf b}/2\hfill\cr}] [\matrix{c,n\hfill\cr a,n\hfill\cr b,n\hfill\cr}] [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Rhombohedral}]
[hhl, h\bar{h}l] [l = 2n] [(1\bar{1}0), (110)] [{\bf c}/2] c, n [\!\left.\let\normalbaselines\relax\openup4pt\matrix{\cr\cr}\right\}\hbox{Tetragonal}]§ [\!\left.\let\normalbaselines\relax\openup1pt\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\hbox{Cubic}]
[2h + l = 4n] [{\bf a}/4 \pm {\bf b}/4 \pm {\bf c}/4] d
[hkk, hk\bar{k}] [h = 2n] [(01\bar{1}), (011)] [{\bf a}/2] a, n
[2k + h = 4n] [\pm {\bf a}/4 + {\bf b}/4 \pm {\bf c}/4] d
[hkh, \bar{h}kh] [k = 2n] [(\bar{1}01), (101)] [{\bf b}/2] b, n
[2h + k = 4n] [\pm {\bf a}/4 \pm {\bf b}/4 + {\bf c}/4] d

(b) Screw axes

Type of reflectionsReflection conditionsScrew axisCrystallographic coordinate system to which condition applies
Direction of axisScrew vectorSymbol
h00 [h = 2n] [100] [{\bf a}/2] [2_{1}] [\left\{\matrix{\hbox{Monoclinic }(a\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[h = 4n] [{\bf a}/4] [4_{1},4_{3}]
0k0 [k = 2n] [010] [{\bf b}/2] [2_{1}] [\left\{\matrix{\hbox{Monoclinic }(b\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[k = 4n] [{\bf b}/4] [4_{1},4_{3}]
00l [l = 2n] [001] [{\bf c}/2] [2_{1}] [\matrix{\left\{\matrix{\hbox{Monoclinic } (c\hbox{ unique}),\hfill\cr\quad\hbox{ Orthorhombic}\hfill\cr}\right.\cr\left.\matrix{\noalign{\vskip31pt}\cr}\right\}\hbox{Tetragonal}\hfill}] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[l = 4n] [{\bf c}/4] [4_{1},4_{3}]
000l [l = 2n] [001] [{\bf c}/2] [6_{3}] [\left.{\hbox to 2pt{}}\matrix{\noalign{\vskip44pt}}\right\}\hbox{Hexagonal}]
[l = 3n] [{\bf c}/3] [3_{1},3_{2},6_{2},6_{4}]
[l = 6n] [{\bf c}/6] [6_{1},6_{5}]
Glide planes d with orientations (100), (010) and (001) occur only in orthorhombic and cubic F space groups. Combination of the integral reflection condition (hkl: all odd or all even) with the zonal conditions for the d glide planes leads to the further conditions given between parentheses.
For rhombohedral space groups described with `rhombohedral axes', the three reflection conditions [(l = 2n, h = 2n, k = 2n)] imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin space-group symbols, c is always used, as in R3c (161) and [R\bar{3}c\ (167)], because c glides also occur in the hexagonal description of these space groups.
§For tetragonal P space groups, the two reflection conditions (hhl and [h\bar{h}l] with [l = 2n]) imply interleaving of c and n glides. In the Hermann–Mauguin space-group symbols, c is always used, irrespective of which glide planes contain the origin: cf. P4cc (103), [P\bar{4}2c\ (112)] and [P4/nnc\ (126)].
For cubic space groups, the three reflection conditions [(l = 2n, h = 2n, k = 2n)] imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin space-group symbols, either c or n is used, depending upon which glide plane contains the origin, cf. [P\bar{4}3n\ (218)], [Pn\bar{3}n\ (222)], [Pm\bar{3}n\ (223)] versus [F\bar{4}3c\ (219)], [Fm\bar{3}c\ (226)], [Fd\bar{3}c\ (228)].