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
[edit]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).
# | 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
[edit]References
[edit]- ^ 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.
- ^ 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.
{{cite journal}}
: CS1 maint: date and year (link) - ^ 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.
External links
[edit]- Bilbao Crystallographic Server, under "Subperiodic Groups: Layer, Rod and Frieze Groups"
- Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin
- CVM 1.1: Vibrating Wallpaper by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.