空間群
外觀
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在數學和物理學中,空間群(space group)是空間中(通常是三維空間)一種形態的空間對稱群。在三維空間中有219種不同的類型,或230種不同的手性類型。對超過三維的空間中的空間群也有研究,它們有時被稱作比貝爾巴赫群,並且是離散的緊群,具有歐氏空間的等距同構。它是由俄國結晶學家費多洛夫和德國結晶學家薛弗利斯(Artur Moritz Schoenflies,1853-1928)於1890至1891年間各自獨立地先後推導得出來的。
在晶體學中,空間群也被稱為費奧多羅夫群,是對晶體對稱型的一種描述。三維空間群的權威參考文獻是《國際晶體學表》。空間群可以分為兩類:一類稱為簡單空間群或稱點空間群;一類稱為複雜空間群或稱非點空間群。其中73種為簡單空間群,餘下的157種為複雜空間群。
三維的空間群
# | 晶系 (空間群數量) 布拉維晶格 |
點群 | 空間群 (國際短符號) | ||||
---|---|---|---|---|---|---|---|
國際符號 | Schön. | Orbifold | Cox. | Ord. | |||
1 | 三斜晶系 (2) |
1 | C1 | 11 | [ ]+ | 1 | P1 |
2 | 1 | Ci | 1× | [2+,2+] | 2 | P1 | |
3–5 | 單斜晶系 (13) |
2 | C2 | 22 | [2]+ | 2 | P2, P21 C2 |
6–9 | m | Cs | *11 | [ ] | 2 | Pm, Pc Cm, Cc | |
10–15 | 2/m | C2h | 2* | [2,2+] | 4 | P2/m, P21/m C2/m, P2/c, P21/c C2/c | |
16–24 | 正交晶系 (59) |
222 | D2 | 222 | [2,2]+ | 4 | P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121 |
25–46 | mm2 | C2v | *22 | [2] | 4 | Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2 Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2 | |
47–74 | mmm | D2h | *222 | [2,2] | 8 | Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma | |
75–80 | 四方晶系 (68) |
4 | C4 | 44 | [4]+ | 4 | P4, P41, P42, P43, I4, I41 |
81–82 | 4 | S4 | 2× | [2+,4+] | 4 | P4, I4 | |
83–88 | 4/m | C4h | 4* | [2,4+] | 8 | P4/m, P42/m, P4/n, P42/n I4/m, I41/a | |
89–98 | 422 | D4 | 224 | [2,4]+ | 8 | P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212 I422, I4122 | |
99–110 | 4mm | C4v | *44 | [4] | 8 | P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc I4mm, I4cm, I41md, I41cd | |
111–122 | 42m | D2d | 2*2 | [2+,4] | 8 | P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2 I4m2, I4c2, I42m, I42d | |
123–142 | 4/mmm | D4h | *224 | [2,4] | 16 | P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm I4/mmm, I4/mcm, I41/amd, I41/acd | |
143–146 | 三方晶系 (25) |
3 | C3 | 33 | [3]+ | 3 | P3, P31, P32 R3 |
147–148 | 3 | S6 | 3× | [2+,6+] | 6 | P3, R3 | |
149–155 | 32 | D3 | 223 | [2,3]+ | 6 | P312, P321, P3112, P3121, P3212, P3221 R32 | |
156–161 | 3m | C3v | *33 | [3] | 6 | P3m1, P31m, P3c1, P31c R3m, R3c | |
162–167 | 3m | D3d | 2*3 | [2+,6] | 12 | P31m, P31c, P3m1, P3c1 R3m, R3c | |
168–173 | 六方晶系 (27) |
6 | C6 | 66 | [6]+ | 6 | P6, P61, P65, P62, P64, P63 |
174 | 6 | C3h | 3* | [2,3+] | 6 | P6 | |
175–176 | 6/m | C6h | 6* | [2,6+] | 12 | P6/m, P63/m | |
177–182 | 622 | D6 | 226 | [2,6]+ | 12 | P622, P6122, P6522, P6222, P6422, P6322 | |
183–186 | 6mm | C6v | *66 | [6] | 12 | P6mm, P6cc, P63cm, P63mc | |
187–190 | 6m2 | D3h | *223 | [2,3] | 12 | P6m2, P6c2, P62m, P62c | |
191–194 | 6/mmm | D6h | *226 | [2,6] | 24 | P6/mmm, P6/mcc, P63/mcm, P63/mmc | |
195–199 | 立方晶系 (36) |
23 | T | 332 | [3,3]+ | 12 | P23, F23, I23 P213, I213 |
200–206 | m3 | Th | 3*2 | [3+,4] | 24 | Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3 | |
207–214 | 432 | O | 432 | [3,4]+ | 24 | P432, P4232 F432, F4132 I432 P4332, P4132, I4132 | |
215–220 | 43m | Td | *332 | [3,3] | 24 | P43m, F43m, I43m P43n, F43c, I43d | |
221–230 | m3m | Oh | *432 | [3,4] | 48 | Pm3m, Pn3n, Pm3n, Pn3m Fm3m, Fm3c, Fd3m, Fd3c Im3m, Ia3d |
注: e 面是雙滑移面,是在兩個不同方向的滑移,存在於七個正交群,五個四方群和五個立方群中,都具有含有中心的晶格,官方的符號為e
參考資料
- Barlow, W, Über die geometrischen Eigenschaften starrer Strukturen und ihre Anwendung auf Kristalle, Z. Kristallogr., 1894, 23: 1–63
- Bieberbach, Ludwig, Über die Bewegungsgruppen der Euklidischen Räume, Mathematische Annalen, 1911, 70 (3): 297–336, ISSN 0025-5831, doi:10.1007/BF01564500
- Bieberbach, Ludwig, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Mathematische Annalen, 1912, 72 (3): 400–412, ISSN 0025-5831, doi:10.1007/BF01456724
- Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans, Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], 1978, ISBN 978-0-471-03095-9, MR0484179
- Burckhardt, Johann Jakob, Die Bewegungsgruppen der Kristallographie, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften 13, Verlag Birkhäuser, Basel, 1947, MR0020553
- Burckhardt, Johann Jakob, Zur Geschichte der Entdeckung der 230 Raumgruppen, Archive for History of Exact Sciences, 1967, 4 (3): 235–246, ISSN 0003-9519, doi:10.1007/BF00412962, MR0220837
- Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P., On three-dimensional space groups, Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry, 2001, 42 (2): 475–507 [2013-03-04], ISSN 0138-4821, MR1865535, (原始內容存檔於2021-04-18)
- Fedorov, E. S., Symmetry of Regular Systems of Figures, Zap. Mineral. Obch., 1891, 28 (2): 1–146
- Fedorov, E. S., Symmetry of crystals, ACA Monograph 7, American Crystallographic Association, 1971
- Hahn, Th., Hahn, Theo , 編, International Tables for Crystallography, Volume A: Space Group Symmetry A 5th, Berlin, New York: Springer-Verlag, 2002 [2013-03-04], ISBN 978-0-7923-6590-7, doi:10.1107/97809553602060000100, (原始內容存檔於2021-04-28)
- Hall, S.R., Space-Group Notation with an Explicit Origin, Acta Cryst., 1981, A37: 517–525
- Janssen, T.; Birman, J.L.; Dénoyer, F.; Koptsik, V.A.; Verger-Gaugry, J.L.; Weigel, D.; Yamamoto, A.; Abrahams, S.C.; Kopsky, V., Report of a Subcommittee on the Nomenclature of n-Dimensional Crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups, Acta Cryst. A, 2002, 58 (Pt 6): 605–621, doi:10.1107/S010876730201379X
- Kim, Shoon K., Group theoretical methods and applications to molecules and crystals, Cambridge University Press, 1999, ISBN 978-0-521-64062-6, MR1713786
- Litvin, D.B., Tables of crystallographic properties of magnetic space groups, Acta Cryst. A, May 2008, 64 (Pt 3): 419–24, Bibcode:2008AcCrA..64..419L, PMID 18421131, doi:10.1107/S010876730800768X
- Litvin, D.B., Tables of properties of magnetic subperiodic groups, Acta Cryst. A, May 2005, 61 (Pt 3): 382–5, Bibcode:2005AcCrA..61..382L, PMID 15846043, doi:10.1107/S010876730500406X
- Neubüser, J.; Souvignier, B.; Wondratschek, H., Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) [New York: Wiley and Sons], Acta Cryst. A, 2002, 58 (Pt 3): 301, doi:10.1107/S0108767302001368
- Opgenorth, J; Plesken, W; Schulz, T, Crystallographic Algorithms and Tables, Acta Cryst. A, 1998, 54 (Pt 5): 517–531, doi:10.1107/S010876739701547X
- Plesken, Wilhelm; Hanrath, W, The lattices of six-dimensional space, Math. Comp., 1984, 43 (168): 573–587
- Plesken, Wilhelm; Schulz, Tilman, Counting crystallographic groups in low dimensions, Experimental Mathematics, 2000, 9 (3): 407–411 [2013-03-04], ISSN 1058-6458, MR1795312, (原始內容存檔於2021-04-18)
- Schönflies, Arthur Moritz, Theorie der Kristallstruktur, Gebr. Bornträger, Berlin., 1891
- Souvignier, Bernd, The four-dimensional magnetic point and space groups, Z. Kristallogr., 2006, 221: 77–82
- Vinberg, E., Crystallographic group, Hazewinkel, Michiel (編), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
- Zassenhaus, Hans, Über einen Algorithmus zur Bestimmung der Raumgruppen, Commentarii Mathematici Helvetici, 1948, 21: 117–141 [2013-03-04], ISSN 0010-2571, doi:10.1007/BF02568029, MR0024424, (原始內容存檔於2012-11-28)
外部連結
- International Union of Crystallography (頁面存檔備份,存於互聯網檔案館)
- Point Groups and Bravais Lattices
- [1] Bilbao Crystallographic Server
- Space Group Info (old)
- Space Group Info (new)
- Crystal Lattice Structures: Index by Space Group
- Full list of 230 crystallographic space groups
- Interactive 3D visualization of all 230 crystallographic space groups (頁面存檔備份,存於互聯網檔案館)
- Huson, Daniel H., The Fibrifold Notation and Classification for 3D Space Groups (PDF), 1999[永久失效連結]
- The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions) (頁面存檔備份,存於互聯網檔案館)
- The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions) (頁面存檔備份,存於互聯網檔案館)