|
Axes |
Coordinates |
Wyckoff positions |
|
|
origin 1 |
origin 2 |
2a |
4b |
4c |
6d |
8e |
12f |
|
|
|
|
12g |
24h |
24i |
24j |
24k |
48l |
| I Maximal translationengleiche subgroups |
| [2] P-43m |
|
|
x + (1/4), y + (1/4), z + (1/4) |
1a; 1b |
4e |
4e |
3c; 3d |
2 × 4e |
12h |
| (215) |
|
|
|
6f; 6g |
2 × 12h |
24j |
24j |
2 × 12i |
2 × 24j |
| [2] P4232 |
|
|
x + (1/4), y + (1/4), z + (1/4) |
2a |
4b |
4c |
6d |
8g |
6e; 6f |
| (208) |
|
|
|
12h |
12i; 12j |
2 × 12k |
2 × 12l |
24m |
2 × 24m |
| [2] Pn-3 |
|
|
|
2a |
4b |
4c |
6d |
8e |
12g |
| (201) |
|
|
|
12f |
2 × 12g |
24h |
24h |
24h |
2 × 24h |
| [4] R-3m |
|
x - (1/4), y - (1/4), z - (1/4) |
|
2c |
1a; 3e |
1b; 3d |
6h |
2c; 6h |
6f; 6g |
| (166) |
(rhombohedral axes) |
|
2 × 6h |
2 × 12i |
2 × 6f; 12i |
2 × 6g; 12i |
2 × 6h; 12i |
4 × 12i |
|
a - b, b - c, |
(1/3)(2x - y - z), |
(1/3)(2x - y - z), |
6c |
3a; 9e |
3b; 9d |
18h |
6c; 18h |
18f; 18g |
|
a + b + c |
(1/3)(x + y - 2z), |
(1/3)(x + y - 2z), |
2 × 18h |
2 × 36i |
2 × 18f; 36i |
2 × 18g; 36i |
2 × 18h; 36i |
4 × 36i |
|
(hexagonal axes) |
(1/3)(x + y + z) - (1/4) |
(1/3)(x + y + z) |
|
|
|
|
|
|
| conjugate: |
-a - b, b + c, |
(1/3)(-2x - y + z), |
(1/3)(-2x - y + z) + (1/2), |
|
|
|
|
|
|
|
-a + b - c |
(1/3)(-x + y + 2z), |
(1/3)(-x + y + 2z) + (1/2), |
|
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(-x + y - z) - (1/4) |
(1/3)(-x + y - z) |
|
|
|
|
|
|
| conjugate: |
a + b, -b + c, |
(1/3)(2x + y + z), |
(1/3)(2x + y + z), |
|
|
|
|
|
|
|
a - b - c |
(1/3)(x - y + 2z), |
(1/3)(x - y + 2z) + (1/2), |
|
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(x - y - z) - (1/4) |
(1/3)(x - y - z) |
|
|
|
|
|
|
| conjugate: |
-a + b, -b - c, |
(1/3)(-2x + y - z), |
(1/3)(-2x + y - z) + (1/2), |
|
|
|
|
|
|
|
-a - b + c |
(1/3)(-x - y - 2z), |
(1/3)(-x - y - 2z), |
|
|
|
|
|
|
|
(hexagonal axes) |
(1/3)(-x - y + z) - (1/4) |
(1/3)(-x - y + z) |
|
|
|
|
|
|
| [3] P42/nnm |
|
|
x, y + (1/2), z |
2a |
4e |
4f |
2b; 4c |
8m |
4d; 8j |
| (134) |
|
|
|
4g; 8i |
8h; 2 × 8j |
8l; 16n |
8k; 16n |
8m; 16n |
3 × 16n |
| conjugate: |
b, c, a |
y, z, x |
y + (1/2), z, x |
|
|
|
|
|
|
| conjugate: |
c, a, b |
z, x, y |
z, x, y + (1/2) |
|
|
|
|
|
|
| II Maximal klassengleiche subgroups |
| Enlarged unit cell, non-isomorphic |
| [2] Fd-3c |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0) |
16a |
32c |
32b |
48d |
64e |
96g |
| (228) |
|
(1/2)z + (1/4); |
|
96f |
192h |
192h |
2 × 96g |
192h |
2 × 192h |
|
|
+((1/2), 0, 0) |
|
|
|
|
|
|
|
| [2] Fd-3c |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; |
(1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); |
16a |
32b |
32c |
48d |
64e |
96g |
| (228) |
|
+((1/2), 0, 0) |
+((1/2), 0, 0) |
96f |
192h |
2 × 96g |
192h |
192h |
2 × 192h |
| [2] Fd-3m |
2a, 2b, 2c |
(1/2)x, (1/2)y, (1/2)z; |
(1/2)x, (1/2)y, (1/2)z; +((1/2), 0, 0) |
8a; 8b |
16c; 16d |
32e |
48f |
2 × 32e |
96h |
| (227) |
|
+((1/2), 0, 0) |
|
2 × 48f |
192i |
2 × 96h |
192i |
2 × 96g |
2 × 192i |
| [2] Fd-3m |
2a, 2b, 2c |
(1/2)x + (1/4), (1/2)y + (1/4), |
(1/2)x + (1/4), (1/2)y + (1/4), (1/2)z + (1/4); |
8a; 8b |
32e |
16c; 16d |
48f |
2 × 32e |
96h |
| (227) |
|
(1/2)z + (1/4); |
+((1/2), 0, 0) |
2 × 48f |
192i |
192i |
2 × 96h |
2 × 96g |
2 × 192i |
|
|
+((1/2), 0, 0) |
|
|
|
|
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|
|
|
Axes |
Coordinates |
Wyckoff positions |
|
|
origin 1 |
origin 2 |
2a |
4b |
4c |
6d |
8e |
|
|
|
|
|
12f |
12g |
24h |
24i |
|
|
|
|
|
|
24j |
24k |
48l |
|
| Enlarged unit cell, isomorphic |
| [27] Pn-3m |
3a, 3b, 3c |
(1/3)x, (1/3)y, (1/3)z |
(1/3)x, (1/3)y, (1/3)z; |
2a; 2 × 8e; 12g; |
4b(c*); 8e; |
4c(b*); 8e; |
6d; 12g; 24h; |
3 × 8e; 6 × 24k; 48l |
|
|
|
|
|
|
|
|
±((1/3), 0, 0); ±(0, (1/3), 0); |
24k |
24i(j*); 3 × 24k |
24j(i*); 3 × 24k |
3 × 24k; 48l |
|
|
|
±(0, 0, (1/3)); ±(0, (1/3), (2/3)); |
|
12f; 24h; 24i; |
3 × 12g; 6 × 24k; |
3 × 24h; 12 × 48l |
3 × 24i(j*); 12 × 48l |
|
|
±((1/3), 0, (2/3)); ±((1/3), (2/3), 0); |
|
24j; 5 × 48l |
3 × 48l |
|
|
|
|
±(0, (1/3), (1/3)); ±((1/3), 0, (1/3)); |
|
|
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|
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|
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±((1/3), (1/3), 0); ±((1/3), (1/3), (1/3)); |
|
|
3 × 24j(i*); |
9 × 24k; 9 × 48l |
27 × 48l |
|
|
±((2/3), (1/3), (1/3)); ±((1/3), (2/3), (1/3)); |
|
|
12 × 48l |
|
|
|
|
±((1/3), (1/3), (2/3)) |
|
|
|
|
|
| [p3] Pn-3m |
pa, pb, pc |
(1/p)x, (1/p)y, (1/p)z; |
(1/p)x, (1/p)y, (1/p)z; |
2a; (p - 1) × 8e; |
4b(c† ); ((p - 1)/2) × 8e; |
4c(b† ); ((p - 1)/2) × 8e; |
6d; ((p - 1)/2) × 12g; |
p × 8e; |
|
|
+((u/p), (v/p), (w/p)) |
+((u/p), (v/p), (w/p)) |
((p - 1)/2) × 12g; |
((p - 1)/2) × 24i(j† ); |
((p - 1)/2) × 24j(i† ); |
((p - 1)/2) × 24h; |
p(p - 1) × 24k; |
|
p = prime > 2; |
|
(((p - 1)(p - 2))/2) × 24k; |
((p(p - 1))/2) × 24k; |
((p(p - 1))/2) × 24k; |
((p(p - 1))/2) × 24k; |
((p(p - 1)(p - 2))/6) × 48l |
|
u, v, w = 1, . . ., p - 1 |
(((p - 1)(p - 2)(p - 3))/24) × |
(((p2 - 1)(p - 3))/12) × 48l |
(((p2 - 1)(p - 3))/12) × 48l |
(((p2 - 1)(p - 2))/8) × 48l |
|
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|
|
48l |
|
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|
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|
|
12f; ((p - 1)/2) × 24h; |
p × 12g; |
p × 24h; |
p × 24i(j† ); |
|
|
|
|
|
((p - 1)/2) × 24i; |
p(p - 1) × 24k; |
((p(p2 - 1))/2) × 48l |
((p(p2 - 1))/2) × 48l |
|
|
|
|
|
((p - 1)/2) × 24j; |
((p(p - 1)2)/4) × 48l |
|
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|
|
(((p - 1)2(p + 2))/4) × 48l |
|
|
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p × 24j(i† ); |
p2 × 24k; |
p3 × 48l |
|
|
|
|
|
|
((p(p2 - 1))/2) × 48l |
((p2(p - 1))/2) × 48l |
|
| * origin 1 |
| † origin 1 and p = 4n - 1 |
