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Existuje 230 prostorových grup ve třech rozměrech, které jsou označeny číselným indexem, plným jménem v Hermann-Mauguinově značení a krátkým názvem (mezinárodní symbol). Dlouhé názvy jsou uvedeny s mezerami pro čitelnost. Každá grupa má bodovou grupu elementární buňka.
V Hermann-Mauguinově značení, jsou prostorové grupy pojmenovány po symbolu složeného z identifikátoru bodové grupy s velkým písmenem popisujícím typ mřížky. Translace v mřížce ve formě skluzové roviny a šroubové osy jsou také zaznamenány, čímž vzniká úplná krystalografická prostorová grupa.
Existují následující Bravaisovy buňky ve 3D:
A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.
A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector. The possible screw axes are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.
Wherever there is both a rotation or screw axis n and a mirror or glide plane m along the same crystallographic direction, they are represented as a fraction or n/m. For example, 41/a means that the crystallographic axis in question contains both a 41 screw axis as well as a glide plane along a.
In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form which specifies the Bravais lattice. Here is the lattice system, and is the centering type.[2]
In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups.
The 73 symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups, for example, the space groups P4/mmm (, 36s) and I4/mmm (, 37s).
The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Hemisymmorphic space groups contain the axial combination 422, which are P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc (, 37h), and I4/mcm (, 38h).
The remaining 103 space groups are asymmorphic, for example, those derived from the point group 4/mmm ().
Číslo | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Fibrifoldovo značení |
---|---|---|---|---|---|---|---|---|
1 | 1 | P1 | P 1 | 1s | ||||
2 | 1 | P1 | P 1 | 2s |
Number | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Fibrifoldovo značení (primární) | Fibrifoldovo značení (sekundární) | |
---|---|---|---|---|---|---|---|---|---|---|
3 | 2 | P2 | P 1 2 1 | P 1 1 2 | 3s | |||||
4 | P21 | P 1 21 1 | P 1 1 21 | 1a | ||||||
5 | C2 | C 1 2 1 | B 1 1 2 | 4s | , | |||||
6 | m | Pm | P 1 m 1 | P 1 1 m | 5s | |||||
7 | Pc | P 1 c 1 | P 1 1 b | 1h | , | |||||
8 | Cm | C 1 m 1 | B 1 1 m | 6s | , | |||||
9 | Cc | C 1 c 1 | B 1 1 b | 2h | , | |||||
10 | 2/m | P2/m | P 1 2/m 1 | P 1 1 2/m | 7s | |||||
11 | P21/m | P 1 21/m 1 | P 1 1 21/m | 2a | ||||||
12 | C2/m | C 1 2/m 1 | B 1 1 2/m | 8s | , | |||||
13 | P2/c | P 1 2/c 1 | P 1 1 2/b | 3h | , | |||||
14 | P21/c | P 1 21/c 1 | P 1 1 21/b | 3a | , | |||||
15 | C2/c | C 1 2/c 1 | B 1 1 2/b | 4h | , |
Number | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Fibrifoldovo značení (primární) | Fibrifoldovo značení (sekundární) |
---|---|---|---|---|---|---|---|---|---|
16 | 222 | P222 | P 2 2 2 | 9s | |||||
17 | P2221 | P 2 2 21 | 4a | ||||||
18 | P21212 | P 21 21 2 | 7a | ||||||
19 | P212121 | P 21 21 21 | 8a | ||||||
20 | C2221 | C 2 2 21 | 5a | ||||||
21 | C222 | C 2 2 2 | 10s | ||||||
22 | F222 | F 2 2 2 | 12s | ||||||
23 | I222 | I 2 2 2 | 11s | ||||||
24 | I212121 | I 21 21 21 | 6a | ||||||
25 | mm2 | Pmm2 | P m m 2 | 13s | |||||
26 | Pmc21 | P m c 21 | 9a | , | |||||
27 | Pcc2 | P c c 2 | 5h | ||||||
28 | Pma2 | P m a 2 | 6h | , | |||||
29 | Pca21 | P c a 21 | 11a | ||||||
30 | Pnc2 | P n c 2 | 7h | , | |||||
31 | Pmn21 | P m n 21 | 10a | , | |||||
32 | Pba2 | P b a 2 | 9h | ||||||
33 | Pna21 | P n a 21 | 12a | , | |||||
34 | Pnn2 | P n n 2 | 8h | ||||||
35 | Cmm2 | C m m 2 | 14s | ||||||
36 | Cmc21 | C m c 21 | 13a | , | |||||
37 | Ccc2 | C c c 2 | 10h | ||||||
38 | Amm2 | A m m 2 | 15s | , | |||||
39 | Aem2 | A b m 2 | 11h | , | |||||
40 | Ama2 | A m a 2 | 12h | , | |||||
41 | Aea2 | A b a 2 | 13h | , | |||||
42 | Fmm2 | F m m 2 | 17s | ||||||
43 | Fdd2 | F dd2 | 16h | ||||||
44 | Imm2 | I m m 2 | 16s | ||||||
45 | Iba2 | I b a 2 | 15h | ||||||
46 | Ima2 | I m a 2 | 14h | , | |||||
47 | Pmmm | P 2/m 2/m 2/m | 18s | ||||||
48 | Pnnn | P 2/n 2/n 2/n | 19h | ||||||
49 | Pccm | P 2/c 2/c 2/m | 17h | ||||||
50 | Pban | P 2/b 2/a 2/n | 18h | ||||||
51 | Pmma | P 21/m 2/m 2/a | 14a | , | |||||
52 | Pnna | P 2/n 21/n 2/a | 17a | , | |||||
53 | Pmna | P 2/m 2/n 21/a | 15a | , | |||||
54 | Pcca | P 21/c 2/c 2/a | 16a | , | |||||
55 | Pbam | P 21/b 21/a 2/m | 22a | ||||||
56 | Pccn | P 21/c 21/c 2/n | 27a | ||||||
57 | Pbcm | P 2/b 21/c 21/m | 23a | , | |||||
58 | Pnnm | P 21/n 21/n 2/m | 25a | ||||||
59 | Pmmn | P 21/m 21/m 2/n | 24a | ||||||
60 | Pbcn | P 21/b 2/c 21/n | 26a | , | |||||
61 | Pbca | P 21/b 21/c 21/a | 29a | ||||||
62 | Pnma | P 21/n 21/m 21/a | 28a | , | |||||
63 | Cmcm | C 2/m 2/c 21/m | 18a | , | |||||
64 | Cmce | C 2/m 2/c 21/a | 19a | , | |||||
65 | Cmmm | C 2/m 2/m 2/m | 19s | ||||||
66 | Cccm | C 2/c 2/c 2/m | 20h | ||||||
67 | Cmme | C 2/m 2/m 2/e | 21h | ||||||
68 | Ccce | C 2/c 2/c 2/e | 22h | ||||||
69 | Fmmm | F 2/m 2/m 2/m | 21s | ||||||
70 | Fddd | F 2/d 2/d 2/d | 24h | ||||||
71 | Immm | I 2/m 2/m 2/m | 20s | ||||||
72 | Ibam | I 2/b 2/a 2/m | 23h | ||||||
73 | Ibca | I 2/b 2/c 2/a | 21a | ||||||
74 | Imma | I 2/m 2/m 2/a | 20a |
Number | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Fibrifoldovo značení |
---|---|---|---|---|---|---|---|---|
75 | 4 | P4 | P 4 | 22s | ||||
76 | P41 | P 41 | 30a | |||||
77 | P42 | P 42 | 33a | |||||
78 | P43 | P 43 | 31a | |||||
79 | I4 | I 4 | 23s | |||||
80 | I41 | I 41 | 32a | |||||
81 | 4 | P4 | P 4 | 26s | ||||
82 | I4 | I 4 | 27s | |||||
83 | 4/m | P4/m | P 4/m | 28s | ||||
84 | P42/m | P 42/m | 41a | |||||
85 | P4/n | P 4/n | 29h | |||||
86 | P42/n | P 42/n | 42a | |||||
87 | I4/m | I 4/m | 29s | |||||
88 | I41/a | I 41/a | 40a | |||||
89 | 422 | P422 | P 4 2 2 | 30s | ||||
90 | P4212 | P4212 | 43a | |||||
91 | P4122 | P 41 2 2 | 44a | |||||
92 | P41212 | P 41 21 2 | 48a | |||||
93 | P4222 | P 42 2 2 | 47a | |||||
94 | P42212 | P 42 21 2 | 50a | |||||
95 | P4322 | P 43 2 2 | 45a | |||||
96 | P43212 | P 43 21 2 | 49a | |||||
97 | I422 | I 4 2 2 | 31s | |||||
98 | I4122 | I 41 2 2 | 46a | |||||
99 | 4mm | P4mm | P 4 m m | 24s | ||||
100 | P4bm | P 4 b m | 26h | |||||
101 | P42cm | P 42 c m | 37a | |||||
102 | P42nm | P 42 n m | 38a | |||||
103 | P4cc | P 4 c c | 25h | |||||
104 | P4nc | P 4 n c | 27h | |||||
105 | P42mc | P 42 m c | 36a | |||||
106 | P42bc | P 42 b c | 39a | |||||
107 | I4mm | I 4 m m | 25s | |||||
108 | I4cm | I 4 c m | 28h | |||||
109 | I41md | I 41 m d | 34a | |||||
110 | I41cd | I 41 c d | 35a | |||||
111 | 42m | P42m | P 4 2 m | 32s | ||||
112 | P42c | P 4 2 c | 30h | |||||
113 | P421m | P 4 21 m | 52a | |||||
114 | P421c | P 4 21 c | 53a | |||||
115 | P4m2 | P 4 m 2 | 33s | |||||
116 | P4c2 | P 4 c 2 | 31h | |||||
117 | P4b2 | P 4 b 2 | 32h | |||||
118 | P4n2 | P 4 n 2 | 33h | |||||
119 | I4m2 | I 4 m 2 | 35s | |||||
120 | I4c2 | I 4 c 2 | 34h | |||||
121 | I42m | I 4 2 m | 34s | |||||
122 | I42d | I 4 2 d | 51a | |||||
123 | 4/m 2/m 2/m | P4/mmm | P 4/m 2/m 2/m | 36s | ||||
124 | P4/mcc | P 4/m 2/c 2/c | 35h | |||||
125 | P4/nbm | P 4/n 2/b 2/m | 36h | |||||
126 | P4/nnc | P 4/n 2/n 2/c | 37h | |||||
127 | P4/mbm | P 4/m 21/b 2/m | 54a | |||||
128 | P4/mnc | P 4/m 21/n 2/c | 56a | |||||
129 | P4/nmm | P 4/n 21/m 2/m | 55a | |||||
130 | P4/ncc | P 4/n 21/c 2/c | 57a | |||||
131 | P42/mmc | P 42/m 2/m 2/c | 60a | |||||
132 | P42/mcm | P 42/m 2/c 2/m | 61a | |||||
133 | P42/nbc | P 42/n 2/b 2/c | 63a | |||||
134 | P42/nnm | P 42/n 2/n 2/m | 62a | |||||
135 | P42/mbc | P 42/m 21/b 2/c | 66a | |||||
136 | P42/mnm | P 42/m 21/n 2/m | 65a | |||||
137 | P42/nmc | P 42/n 21/m 2/c | 67a | |||||
138 | P42/ncm | P 42/n 21/c 2/m | 65a | |||||
139 | I4/mmm | I 4/m 2/m 2/m | 37s | |||||
140 | I4/mcm | I 4/m 2/c 2/m | 38h | |||||
141 | I41/amd | I 41/a 2/m 2/d | 59a | |||||
142 | I41/acd | I 41/a 2/c 2/d | 58a |
Number | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Fibrifoldovo značení |
---|---|---|---|---|---|---|---|---|
143 | 3 | P3 | P 3 | 38s | ||||
144 | P31 | P 31 | 68a | |||||
145 | P32 | P 32 | 69a | |||||
146 | R3 | R 3 | 39s | |||||
147 | 3 | P3 | P 3 | 51s | ||||
148 | R3 | R 3 | 52s | |||||
149 | 32 | P312 | P 3 1 2 | 45s | ||||
150 | P321 | P 3 2 1 | 44s | |||||
151 | P3112 | P 31 1 2 | 72a | |||||
152 | P3121 | P 31 2 1 | 70a | |||||
153 | P3212 | P 32 1 2 | 73a | |||||
154 | P3221 | P 32 2 1 | 71a | |||||
155 | R32 | R 3 2 | 46s | |||||
156 | 3m | P3m1 | P 3 m 1 | 40s | ||||
157 | P31m | P 3 1 m | 41s | |||||
158 | P3c1 | P 3 c 1 | 39h | |||||
159 | P31c | P 3 1 c | 40h | |||||
160 | R3m | R 3 m | 42s | |||||
161 | R3c | R 3 c | 41h | |||||
162 | 3 2/m | P31m | P 3 1 2/m | 56s | ||||
163 | P31c | P 3 1 2/c | 46h | |||||
164 | P3m1 | P 3 2/m 1 | 55s | |||||
165 | P3c1 | P 3 2/c 1 | 45h | |||||
166 | R3m | R 3 2/m | 57s | |||||
167 | R3c | R 3 2/c | 47h |
Number | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Fibrifoldovo značení |
---|---|---|---|---|---|---|---|---|
168 | 6 | P6 | P 6 | 49s | ||||
169 | P61 | P 61 | 74a | |||||
170 | P65 | P 65 | 75a | |||||
171 | P62 | P 62 | 76a | |||||
172 | P64 | P 64 | 77a | |||||
173 | P63 | P 63 | 78a | |||||
174 | 6 | P6 | P 6 | 43s | ||||
175 | 6/m | P6/m | P 6/m | 53s | ||||
176 | P63/m | P 63/m | 81a | |||||
177 | 622 | P622 | P 6 2 2 | 54s | ||||
178 | P6122 | P 61 2 2 | 82a | |||||
179 | P6522 | P 65 2 2 | 83a | |||||
180 | P6222 | P 62 2 2 | 84a | |||||
181 | P6422 | P 64 2 2 | 85a | |||||
182 | P6322 | P 63 2 2 | 86a | |||||
183 | 6mm | P6mm | P 6 m m | 50s | ||||
184 | P6cc | P 6 c c | 44h | |||||
185 | P63cm | P 63 c m | 80a | |||||
186 | P63mc | P 63 m c | 79a | |||||
187 | 6m2 | P6m2 | P 6 m 2 | 48s | ||||
188 | P6c2 | P 6 c 2 | 43h | |||||
189 | P62m | P 6 2 m | 47s | |||||
190 | P62c | P 6 2 c | 42h | |||||
191 | 6/m 2/m 2/m | P6/mmm | P 6/m 2/m 2/m | 58s | ||||
192 | P6/mcc | P 6/m 2/c 2/c | 48h | |||||
193 | P63/mcm | P 63/m 2/c 2/m | 87a | |||||
194 | P63/mmc | P 63/m 2/m 2/c | 88a |
Number | Bodová grupa | Orbifoldovo značení | Krátké jméno | Celé jméno | Schoenfliesovo značení | Fedorovovo značení | Shubnikovo značení | Conway | Fibrifold (zachování ) | Fibrifold (zachování , , ) |
---|---|---|---|---|---|---|---|---|---|---|
195 | 23 | P23 | P 2 3 | 59s | ||||||
196 | F23 | F 2 3 | 61s | |||||||
197 | I23 | I 2 3 | 60s | |||||||
198 | P213 | P 21 3 | 89a | |||||||
199 | I213 | I 21 3 | 90a | |||||||
200 | 2/m 3 | Pm3 | P 2/m 3 | 62s | ||||||
201 | Pn3 | P 2/n 3 | 49h | |||||||
202 | Fm3 | F 2/m 3 | 64s | |||||||
203 | Fd3 | F 2/d 3 | 50h | |||||||
204 | Im3 | I 2/m 3 | 63s | |||||||
205 | Pa3 | P 21/a 3 | 91a | |||||||
206 | Ia3 | I 21/a 3 | 92a | |||||||
207 | 432 | P432 | P 4 3 2 | 68s | ||||||
208 | P4232 | P 42 3 2 | 98a | |||||||
209 | F432 | F 4 3 2 | 70s | |||||||
210 | F4132 | F 41 3 2 | 97a | |||||||
211 | I432 | I 4 3 2 | 69s | |||||||
212 | P4332 | P 43 3 2 | 94a | |||||||
213 | P4132 | P 41 3 2 | 95a | |||||||
214 | I4132 | I 41 3 2 | 96a | |||||||
215 | 43m | P43m | P 4 3 m | 65s | ||||||
216 | F43m | F 4 3 m | 67s | |||||||
217 | I43m | I 4 3 m | 66s | |||||||
218 | P43n | P 4 3 n | 51h | |||||||
219 | F43c | F 4 3 c | 52h | |||||||
220 | I43d | I 4 3 d | 93a | |||||||
221 | 4/m 3 2/m | Pm3m | P 4/m 3 2/m | 71s | ||||||
222 | Pn3n | P 4/n 3 2/n | 53h | |||||||
223 | Pm3n | P 42/m 3 2/n | 102a | |||||||
224 | Pn3m | P 42/n 3 2/m | 103a | |||||||
225 | Fm3m | F 4/m 3 2/m | 73s | |||||||
226 | Fm3c | F 4/m 3 2/c | 54h | |||||||
227 | Fd3m | F 41/d 3 2/m | 100a | |||||||
228 | Fd3c | F 41/d 3 2/c | 101a | |||||||
229 | Im3m | I 4/m 3 2/m | 72s | |||||||
230 | Ia3d | I 41/a 3 2/d | 99a |
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