空间群

三维的空间群

更多信息 #, 晶系（空间群数量）布拉维晶格 ...

#	晶系（空间群数量）布拉维晶格	点群					空间群 (国际短符号)
#	晶系（空间群数量）布拉维晶格	国际标记法	熊夫利标记法（英语：Schoenflies notation）^[1]	轨形	考克斯特符号（英语：Coxeter notation）	点群阶	空间群 (国际短符号)
1	三斜晶系 (2)	1	C₁	11	[ ]⁺	1	P1
2	三斜晶系 (2)	1	C_i	1×	[2⁺,2⁺]	2	P1
3–5	单斜晶系 (13)	2	C₂	22	[2]⁺	2	P2, P2₁ C2
6–9		m	C_s	*11	[ ]	2	Pm, Pc Cm, Cc
10–15		2/m	C_2h	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	C_2v	*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	D_2h	*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	C_4h	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	C_4v	*44	[4]	8	P4mm, P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc, P4₂bc I4mm, I4cm, I4₁md, I4₁cd
111–122		42m	D_2d	2*2	[2⁺,4]	8	P42m, P42c, P42₁m, P42₁c, P4m2, P4c2, P4b2, P4n2 I4m2, I4c2, I42m, I42d
123–142		4/mmm	D_4h	*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	C_3v	*33	[3]	6	P3m1, P31m, P3c1, P31c R3m, R3c
162–167		3m	D_3d	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	C_3h	3*	[2,3⁺]	6	P6
175–176		6/m	C_6h	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	C_6v	*66	[6]	12	P6mm, P6cc, P6₃cm, P6₃mc
187–190		6m2	D_3h	*223	[2,3]	12	P6m2, P6c2, P62m, P62c
191–194		6/mmm	D_6h	*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	T_h	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	T_d	*332	[3,3]	24	P43m, F43m, I43m P43n, F43c, I43d
221–230		m3m	O_h	*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

Remove ads

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, MR 0484179
Burckhardt, Johann Jakob, Die Bewegungsgruppen der Kristallographie, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften 13, Verlag Birkhäuser, Basel, 1947, MR 0020553
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, MR 0220837

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, MR 1865535, （原始内容存档于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 , 编, 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, （原始内容存档于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, MR 1713786
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, MR 1795312, （原始内容存档于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, 数学百科全书, EMS Press, 2001 （英语）
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, MR 0024424, （原始内容存档于2012-11-28）

Remove ads

三维的空间群

参考资料

外部链接

Wikiwand - on