|
Axes |
Coordinates |
Wyckoff positions |
|
|
|
2a |
6b |
8c |
12d |
12e |
24f |
| I Maximal translationengleiche subgroups |
| [4] R3 (146) |
(1/2)(-a + b + c), |
y + z, x + z, x + y |
1a |
3b |
1a; 3b |
2 × 3b |
2 × 3b |
4 × 3b |
|
(1/2)(a - b + c), |
(rhombohedral |
|
|
|
|
|
|
|
(1/2)(a + b - c) |
axes) |
|
|
|
|
|
|
|
-a + b, -b + c, |
(1/3)(-2x + y + z), (1/3)(-x - y + 2z), |
3a |
9b |
3a; 9b |
2 × 9b |
2 × 9b |
4 × 9b |
|
(1/2)(a + b + c) |
(2/3)(x + y + z) |
|
|
|
|
|
|
|
(hexagonal axes) |
|
|
|
|
|
|
| conjugate: |
a + b, -b - c, |
(1/3)(2x + y - z), (1/3)(x - y - 2z), |
|
|
|
|
|
|
|
(1/2)(-a + b - c) |
(2/3)(-x + y - z) |
|
|
|
|
|
|
|
(hexagonal axes) |
|
|
|
|
|
|
| conjugate: |
-a - b, b - c, |
(1/3)(-2x - y - z), (1/3)(-x + y - 2z), |
|
|
|
|
|
|
|
(1/2)(a - b - c) |
(2/3)(x - y - z) |
|
|
|
|
|
|
|
(hexagonal axes) |
|
|
|
|
|
|
| conjugate: |
a - b, b + c, |
(1/3)(2x - y + z), (1/3)(x + y + 2z), |
|
|
|
|
|
|
|
(1/2)(-a - b + c) |
(2/3)(-x - y + z) |
|
|
|
|
|
|
|
(hexagonal axes) |
|
|
|
|
|
|
| [3] I222 (23) |
|
|
2a |
2b; 2c; 2d |
8k |
4e; 4g; 4i |
4f; 4h; 4j |
3 × 8k |
| II Maximal klassengleiche subgroups |
| Loss of centring translations |
| [2] P23 (195) |
|
|
1a; 1b |
3c; 3d |
2 × 4e |
6f; 6i |
6g; 6h |
2 × 12j |
| Enlarged unit cell, isomorphic |
| [27] I23 |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z; ±((1/3), 0, 0); |
2a; 2 × 8c; 12d; |
6b; 12d; 2 × 12e; |
3 × 8c; |
3 × 12d; |
3 × 12e; |
27 × 24f |
|
|
±(0, (1/3), 0); ±(0, 0, (1/3)); |
24f |
5 × 24f |
8 × 24f |
12 × 24f |
12 × 24f |
|
|
|
±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
|
|
|
|
|
|
|
|
±((1/3), (1/3), 0); ±(0, (1/3), (2/3)); |
|
|
|
|
|
|
|
|
±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); ±((1/3), (1/3), (1/3)); |
|
|
|
|
|
|
|
|
±((1/3), (1/3), (2/3)); ±((1/3), (2/3), (1/3)); ±((2/3), (1/3), (1/3)) |
|
|
|
|
|
|
| [p3] I23 |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; +((u/p), (v/p), (w/p)) |
2a; (p - 1) × 8c; |
6b; ((p - 1)/2) × 12d; |
p × 8c; |
p × 12d; |
p × 12e; |
p3 × 24f |
|
p = prime > 2; |
((p - 1)/2) × 12d; |
(p - 1) × 12e; |
((p(p2 - 1))/3) × |
((p(p2 - 1))/2) × |
((p(p2 - 1))/2) × |
|
|
u, v, w = 1, . . ., p - 1 |
(((p - 1)(p - 2)(p + 3))/12) × |
(((p - 1)2(p + 2))/4) × |
24f |
24f |
24f |
|
|
|
24f |
24f |
|
|
|
|
