Origin on 3 m
| Asymmetric unit | 0 ≤ x ≤ 2/3; 0 ≤ y ≤ 2/3; 0 ≤ z ≤ 1/3; x ≤ 2y; y ≤ min(1 - x, 2x) |
| Vertices | | 0, 0, 0 | 2/3, 1/3, 0 | 1/3, 2/3, 0 | | 0, 0, 1/3 | 2/3, 1/3, 1/3 | 1/3, 2/3, 1/3 |
|
For (0, 0, 0)+ set
| (1) 1 | (2) 3+ 0, 0, z | (3) 3- 0, 0, z |
| (4) m x, -x, z | (5) m x, 2x, z | (6) m 2x, x, z |
For (2/3, 1/3, 1/3)+ set
| (1) t(2/3, 1/3, 1/3) | (2) 3+(0, 0, 1/3) 1/3, 1/3, z | (3) 3-(0, 0, 1/3) 1/3, 0, z |
| (4) g(1/6, -1/6, 1/3) x + 1/2, -x, z | (5) g(1/6, 1/3, 1/3) x + 1/4, 2x, z | (6) g(2/3, 1/3, 1/3) 2x, x, z |
For (1/3, 2/3, 2/3)+ set
| (1) t(1/3, 2/3, 2/3) | (2) 3+(0, 0, 2/3) 0, 1/3, z | (3) 3-(0, 0, 2/3) 1/3, 1/3, z |
| (4) g(-1/6, 1/6, 2/3) x + 1/2, -x, z | (5) g(1/3, 2/3, 2/3) x, 2x, z | (6) g(1/3, 1/6, 2/3) 2x - 1/2, x, z |
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); t(2/3, 1/3, 1/3); (2); (4)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates | Reflection conditions |
| | (0, 0, 0)+ (2/3, 1/3, 1/3)+ (1/3, 2/3, 2/3)+ | General:
|
| | (1) x, y, z | (2) -y, x - y, z | (3) -x + y, -x, z | | (4) -y, -x, z | (5) -x + y, y, z | (6) x, x - y, z |
| hkil : -h + k + l = 3n
hki0 : -h + k = 3n
hh(-2h)l : l = 3n
h-h0l : h + l = 3n
000l : l = 3n
h-h00 : h = 3n
|
| | | Special: as above, plus
|
| | x, -x, z | x, 2x, z | (-2x), -x, z |
| no extra conditions |
| | no extra conditions |
Symmetry of special projections
Along [001] p31m
a' = 1/3(2a + b) b' = 1/3(-a + b)
Origin at 0, 0, z | Along [100] p1
a' = 1/2(a + 2b) b' = 1/3(-a - 2b + c)
Origin at x, 0, 0 | Along [210] p1m1
a' = 1/2b b' = 1/3c
Origin at x, 1/2x, 0 |
Maximal non-isomorphic subgroups
| I | | [2] R31 (R3, 146) | (1; 2; 3)+ |
| | ![[brace]](/graphics/sg/brace3.gif) | [3] R1m (Cm, 8) | (1; 4)+ | | | [3] R1m (Cm, 8) | (1; 5)+ | | | [3] R1m (Cm, 8) | (1; 6)+ |
|
| IIa | | [3] P3m1 (156) | 1; 2; 3; 4; 5; 6 |
| IIb | [2] R3c (a' = -a, b' = -b, c' = 2c) (161) |
Maximal isomorphic subgroups of lowest index
| IIc | [2] R3m (a' = -a, b' = -b, c' = 2c) (160); [4] R3m (a' = -2a, b' = -2b) (160) |
Minimal non-isomorphic supergroups
| I | [2] R-3m (166); [4] P-43m (215); [4] F-43m (216); [4] I-43m (217) |
| II | [3] P31m (a' = 1/3(2a + b), b' = 1/3(-a + b), c' = 1/3c) (157) |
Origin on 3 m
| Asymmetric unit | 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; y ≤ x; z ≤ y |
| Vertices | | 0, 0, 0 | 1, 0, 0 | 1, 1, 0 | 1, 1, 1 |
|
| (1) 1 | (2) 3+ x, x, x | (3) 3- x, x, x |
| (4) m x, y, x | (5) m x, x, z | (6) m x, y, y |
Generators selected (1); t(1, 0, 0); t(0, 1, 0); t(0, 0, 1); (2); (4)
Multiplicity, Wyckoff letter,
Site symmetry | Coordinates | Reflection conditions |
| | | General:
|
| | (1) x, y, z | (2) z, x, y | (3) y, z, x | | (4) z, y, x | (5) y, x, z | (6) x, z, y |
| no conditions |
| | | Special: as above, plus
|
| | no extra conditions |
| | no extra conditions |
Symmetry of special projections
Along [111] p31m
a' = 1/3(2a - b - c) b' = 1/3(-a + 2b - c)
Origin at x, x, x | Along [1-10] p1
a' = 1/2(a + b - 2c) b' = c
Origin at x, -x, 0 | Along [2-1-1] p1m1
a' = 1/2(b - c) b' = 1/3(a + b + c)
Origin at 2x, -x, -x |
Maximal non-isomorphic subgroups
| I | | [2] R31 (R3, 146) | 1; 2; 3 |
| | | [3] R1m (Cm, 8) | 1; 4 | | | [3] R1m (Cm, 8) | 1; 5 | | | [3] R1m (Cm, 8) | 1; 6 |
|
| IIb | [2] F3c (a' = 2a, b' = 2b, c' = 2c) (R3c, 161); [3] P3m1 (a' = a - b, b' = b - c, c' = a + b + c) (156) |
Maximal isomorphic subgroups of lowest index
| IIc | [2] R3m (a' = b + c, b' = a + c, c' = a + b) (160); [4] R3m (a' = -a + b + c, b' = a - b + c, c' = a + b - c) (160) |
Minimal non-isomorphic supergroups
| I | [2] R-3m (166); [4] P-43m (215); [4] F-43m (216); [4] I-43m (217) |
| II | [3] P31m (a' = 1/3(2a - b - c), b' = 1/3(-a + 2b - c), c' = 1/3(a + b + c)) (157) |
