Layer group

In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane.

Table of the 80 layer groups, organized by crystal system or lattice type, and by their point groups[1][2]

Triclinic
1 p1 2 p1
Monoclinic/inclined
3 p112 4 p11m 5 p11a 6 p112/m 7 p112/a
Monoclinic/orthogonal
8 p211 9 p2111 10 c211 11 pm11 12 pb11
13 cm11 14 p2/m11 15 p21/m11 16 p2/b11 17 p21/b11
18 c2/m11
Orthorhombic
19 p222 20 p2122 21 p21212 22 c222 23 pmm2
24 pma2 25 pba2 26 cmm2 27 pm2m 28 pm21b
29 pb21m 30 pb2b 31 pm2a 32 pm21n 33 pb21a
34 pb2n 35 cm2m 36 cm2e 37 pmmm 38 pmaa
39 pban 40 pmam 41 pmma 42 pman 43 pbaa
44 pbam 45 pbma 46 pmmn 47 cmmm 48 cmme
Tetragonal
49 p4 50 p4 51 p4/m 52 p4/n 53 p422
54 p4212 55 p4mm 56 p4bm 57 p42m 58 p421m
59 p4m2 60 p4b2 61 p4/mmm 62 p4/nbm 63 p4/mbm
64 p4/nmm
Trigonal
65 p3 66 p3 67 p312 68 p321 69 p3m1
70 p31m 71 p31m 72 p3m1
Hexagonal
73 p6 74 p6 75 p6/m 76 p622 77 p6mm
78 p6m2 79 p62m 80 p6/mmm

Correspondence Between Layer Groups and Plane Groups

The surjective mapping from a layer group to a wallpaper group (plane group) can be obtained by disregarding symmetry elements along the stacking direction, typically denoted as the z-axis, and aligning the remaining elements with those of the plane groups.[3] The resulting surjective mapping provides a direct correspondence between layer groups and plane groups (wallpaper groups).

Surjective mapping from Layer Groups to Plane Groups
# Layer Group # Plane Group
1 p1 1 p1
2 p1 2 p2
3 p112 2 p2
4 p11m 1 p1
5 p11a 1 p1
6 p112/m 2 p2
7 p112/a 2 p2
8 p211 3 pm
9 p2111 4 pg
10 c211 5 cm
11 pm11 3 pm
12 pb11 4 pg
13 cm11 5 cm
14 p2/m11 6 p2mm
15 p21/m11 7 p2mg
16 p2/b11 7 p2mg
17 p21/b11 8 p2gg
18 c2/m11 9 c2mm
19 p222 6 p2mm
20 p2122 7 p2mg
21 p21212 8 p2gg
22 c222 9 c2mm
23 pmm2 6 p2mm
24 pma2 7 p2mg
25 pba2 8 p2gg
26 cmm2 9 c2mm
27 pm2m 3 pm
28 pm21b 3 pm
29 pb21m 4 pg
30 pb2b 3 pm
31 pm2a 3 pm
32 pm21n 4 pg
33 pb21a 4 pg
34 pb2n 5 cm
35 cm2m 5 cm
36 cm2e 3 pm
37 pmmm 6 p2mm
38 pmaa 6 p2mm
39 pban 10 p4
40 pmam 7 p2mg
41 pmma 6 p2mm
42 pman 9 c2mm
43 pbaa 7 p2mg
44 pbam 8 p2gg
45 pbma 7 p2mg
46 pmmn 10 p4
47 cmmm 9 c2mm
48 cmme 6 p2mm
49 p4 10 p4
50 p4 10 p4
51 p4/m 10 p4
52 p4/n 12 p4gm
53 p422 11 p4mm
54 p4212 12 p4gm
55 p4mm 11 p4mm
56 p4bm 12 p4gm
57 p42m 11 p4mm
58 p421m 12 p4gm
59 p4m2 11 p4mm
60 p4b2 12 p4gm
61 p4/mmm 11 p4mm
62 p4/nbm 11 p4mm
63 p4/mbm 12 p4gm
64 p4/nmm 11 p4mm
65 p3 13 p3
66 p3 16 p6
67 p312 14 p3m1
68 p321 15 p31m
69 p3m1 14 p3m1
70 p31m 15 p31m
71 p31m 17 p6mm
72 p3m1 17 p6mm
73 p6 16 p6
74 p6 13 p3
75 p6/m 16 p6
76 p622 17 p6mm
77 p6mm 17 p6mm
78 p6m2 14 p3m1
79 p62m 15 p31m
80 p6/mmm 17 p6mm

See also

References

  1. ^ Kopsky, V.; Litvin, D.B., eds. (2002). International Tables for Crystallography, Volume E: Subperiodic Groups. Vol. E (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000105. ISBN 978-1-4020-0715-6.
  2. ^ Hitzer, E.S.M.; Ichikawa, D. (17–19 Aug 2008). "Representation of crystallographic subperiodic groups by geometric algebra". Electronic Proc. of AGACSE (3). Leipzig, Germany. arXiv:1306.1280. Bibcode:2013arXiv1306.1280H.
  3. ^ Sze, W.H.R.; Xi, B.; Zhu, J. (2025). "Key difference of input data organization to the predictions of symmetry information and layer number for quasi-2D films from band structure". Computational Condensed Matter. 42 e01009. doi:10.1016/j.cocom.2025.e01009. ISSN 2352-2143.

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