空间群

在数学和物理学中，空间群（space group）是空间中（通常是三维空间）一种形态的空间对称群。在三维空间中有219种不同的类型，或230种不同的手性类型。对超过三维的空间中的空间群也有研究，它们有时被称作比贝尔巴赫群，并且是离散的紧群，具有欧氏空间的等距同构。

在晶体学中，空间群也被称为费奥多罗夫群，是对晶体对称型的一种描述。三维空间群的权威参考文献是《国际晶体学表》。

三维的空间群

#	晶系 (count) Bravais lattice	点群					空间群 (国际短符号)
		国际符号	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	单斜晶系 (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 面是双滑移面，是在两个不同方向的滑移，存在于七个正交群，五个四方群和五个立方群中，都具有含有中心的晶格，官方的符号为e

参考资料

• 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, ISSN 0138-4821, MR1865535 
• 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 , 编, International Tables for Crystallography, Volume A: Space Group Symmetry A 5th, Berlin, New York: Springer-Verlag, 2002, ISBN 978-0-7923-6590-7, doi:10.1107/97809553602060000100 
• 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 
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• 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 (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4 
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外部链接

• International Union of Crystallography （页面存档备份，存于互联网档案馆）
• 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
• Huson, Daniel H., The Fibrifold Notation and Classification for 3D Space Groups (PDF), 1999 
• The Geometry Center: 2.1 Formulas for Symmetries in Cartesian Coordinates (two dimensions)
• The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)