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空间群（Space Group）与点群（Point Group）之间的对应关系

已有 45600 次阅读 2018-1-19 17:33 |系统分类:科研笔记

http://pd.chem.ucl.ac.uk/pdnn/symm3/allsgp.htm

Crystal System	Laue Class	Crystal Class	Lattice Centring	230 3-Dimensional Space Groups
Triclinic	-1	1	P	P1
Triclinic	-1	-1	P	P-1
Monoclinic	2/m	2	P	P2, P2₁
		2	C	C2
		m	P	Pm, Pc
		m	C	Cm, Cc
		2/m	P	P2/m, P2₁m, P2/c, P2₁c
		2/m	C	C2/m, C2/c
Orthorhombic	mmm	222	P	P222, P222₁, P2₁2₁2, P2₁2₁2₁
			C	C222, C222₁
			F	F222
			I	I222, I2₁2₁2₁
		mm2	P	Pmm2, Pmc2₁, Pcc2, Pma2, Pca2₁, Pnc2, Pmn2₁, Pba2, Pna2₁, Pnn2
			C or A	Cmm2, Cmc2₁, Ccc2, Amm2, Abm2, Ama2, Aba2
			F	Fmm2, Fdd2
			I	Imm2, Iba2, Ima2
		mmm	P	Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma
			C	Cmmm, Cmcm, Cmca, Cccm, Cmma, Ccca
			F	Fmmm, Fddd
			I	Immm, Ibam, Ibcm, Imma
Tetragonal	4/m	4	P	P4, P4₁, P4₂, P4₃
		4	I	I4, I4₁
		-4	P	P-4
		-4	I	I-4
		4/m	P	P4/m, P4₂/m, P4/n, P4₂/n
		4/m	I	I4/m, I4₁/a
	4/mmm	422	P	P422, P42₁2, P4₁22, P4₁2₁2, P4₂22, P4₂2₁2, P4₃22, P4₃2₁2
		422	I	I422, I42₁2
		4mm	P	P4mm, P4bm, P4₂cm, P4₂nm, P4cc, P4nc, P4₂mc, P4₂bc
		4mm	I	I4mm, I4cm, I4₁md, I4₁cd
		-42m	P	P-42m, P-42c, P-42₁m, P-42₁c
		-42m	I	I-42m, I-42d
		-4m2	P	P-4m2, P-4c2, P-4b2, P-4n2
		-4m2	I	I-4m2, I-4c2
		4/mmm	P	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/mcm, P4₂/nmc, P4₂/ncm
		4/mmm	I	I4/mmm, I4/mcm, I4₁/amd, I4₁/acd
Trigonal	-3	3	P	P3, P3₁, P3₂
		3	R	R3
		-3	P	P-3
		-3	R	R-3
	-3m	312	P	P312, P3₁12, P3₂12
		321	P	P321, P3₁21, P3₂21
		321	R	R32
		31m	P	P31m, P31c
		3m1	P	P3m1, P3c1
		3m1	R	R3m, R3c
		-31m	P	P-31m, P-31c
		-3m1	P	P-3m1, P-3c1
		-3m1	R	R-3m, R-3c
Hexagonal	6/m	6	P	P6, P6₁, P6₂, P6₃, P6₄, P6₅
		-6		P-6
		6/m		P6/m, P6₃/m
	6/mmm	622		P622, P6₁22, P6₂22, P6₃22, P6₄22, P6₅22
		6mm		P6mm, P6cc, P6₃cm, P6₃mc
		-6m2		P-6m2, P-6c2
		-62m		P-62m, P62c
		6/mmm		P6/mmm, P6/mcc, P6₃/mcm, P6₃/mmc
Cubic	m-3	23	P	P23, P2₁3
			F	F23
			I	I23, I2₁3
		m-3	P	Pm-3, Pn-3, Pa-3
			F	Fm-3, Fd-3
			I	Im-3, Ia-3
	m-3m	432	P	P432, P4₂32, P4₃32, P4₁32
			F	F432, F4₁32
			I	I432, I4₁32
		-43m	P	P-43m, P-43n
			F	F-43m, F-43c
			I	I-43m, I-43d
		m-3m	P	Pm-3m, Pn-3n, Pm-3n, Pn-3m
			F	Fm-3m, Fm-3m, Fd-3m, Fd-3c
			I	Im-3m, Ia-3d

点群（Point Group）的Hermann-Mauguin（International ) notation 标记符号与Schoenflies（spectroscopy）标记符号之间的对应关系：

https://en.wikipedia.org/wiki/Crystallographic_point_group

The correspondence between different notations[edit]

Crystal system	Hermann-Mauguin		Shubnikov^[1]	Schoenflies	Orbifold	Coxeter	Order
Crystal system	(full)	(short)	Shubnikov^[1]	Schoenflies	Orbifold	Coxeter	Order
Triclinic	1	1	$1\$	C₁	11	[ ]⁺	1
Triclinic	1	1	${\tilde {2}}$	C_i = S₂	×	[2⁺,2⁺]	2
Monoclinic	2	2	$2\$	C₂	22	[2]⁺	2
	m	m	$m\$	C_s = C_1h	*	[ ]	2
	${\tfrac {2}{m}}$	2/m	$2:m\$	C_2h	2*	[2,2⁺]	4
Orthorhombic	222	222	$2:2\$	D₂ = V	222	[2,2]⁺	4
	mm2	mm2	$2\cdot m\$	C_2v	*22	[2]	4
	${\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	mmm	$m\cdot 2:m\$	D_2h = V_h	*222	[2,2]	8
Tetragonal	4	4	$4\$	C₄	44	[4]⁺	4
	4	4	${\tilde {4}}$	S₄	2×	[2⁺,4⁺]	4
	${\tfrac {4}{m}}$	4/m	$4:m\$	C_4h	4*	[2,4⁺]	8
	422	422	$4:2\$	D₄	422	[4,2]⁺	8
	4mm	4mm	$4\cdot m\$	C_4v	*44	[4]	8
	42m	42m	${\tilde {4}}\cdot m$	D_2d = V_d	2*2	[2⁺,4]	8
	${\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	4/mmm	$m\cdot 4:m\$	D_4h	*422	[4,2]	16
Trigonal	3	3	$3\$	C₃	33	[3]⁺	3
	3	3	${\tilde {6}}$	S₆ = C_3i	3×	[2⁺,6⁺]	6
	32	32	$3:2\$	D₃	322	[3,2]⁺	6
	3m	3m	$3\cdot m\$	C_3v	*33	[3]	6
	3 ${\tfrac {2}{m}}$	3m	${\tilde {6}}\cdot m$	D_3d	2*3	[2⁺,6]	12
Hexagonal	6	6	$6\$	C₆	66	[6]⁺	6
	6	6	$3:m\$	C_3h	3*	[2,3⁺]	6
	${\tfrac {6}{m}}$	6/m	$6:m\$	C_6h	6*	[2,6⁺]	12
	622	622	$6:2\$	D₆	622	[6,2]⁺	12
	6mm	6mm	$6\cdot m\$	C_6v	*66	[6]	12
	6m2	6m2	$m\cdot 3:m\$	D_3h	*322	[3,2]	12
	${\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	6/mmm	$m\cdot 6:m\$	D_6h	*622	[6,2]	24
Isometric	23	23	$3/2\$	T	332	[3,3]⁺	12
	${\tfrac {2}{m}}$ 3	m3	${\tilde {6}}/2$	T_h	3*2	[3⁺,4]	24
	432	432	$3/4\$	O	432	[4,3]⁺	24
	43m	43m	$3/{\tilde {4}}$	T_d	*332	[3,3]	24
	${\tfrac {4}{m}}$ 3 ${\tfrac {2}{m}}$	m3m	${\tilde {6}}/4$	O_h	*432	[4,3]	48

Character table for point groups

https://www.webqc.org/symmetrypointgroup-c3v.html

在该网站可以查不同Point group的character table。

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The correspondence between different notations[edit]

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The correspondence between different notations[edit]

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