Electron Crystallography: Electron Microscopy and Electron Diffraction

Xiaodong Zou, Sven Hovmöller, and Peter Oleynikov

Print publication date: 2011

Print ISBN-13: 9780199580200

Published to Oxford Scholarship Online: January 2012

DOI: 10.1093/acprof:oso/9780199580200.001.0001

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(p. 271 ) Appendix 2: Tables for Space Group Determination

Source:: Electron Crystallography
Publisher:: Oxford University Press

Adopted from: International Tables for Crystallography, (2005), Vol. A. Ed. Th. Hahn, 5th edn, Springer, Dordrecht, with kind permission.

In X-ray crystallography the space group of a crystal is determined mainly from the systematic absences in diffraction. First, the crystal class is found (cubic, tetragonal, orthorhombic etc), then the systematically absent (forbidden) reflections. They come in three categories;

1. absences for all reflections hkl. These are due to centred lattices (A, B, C, F or I);
2. absences in planes hk0, h0l and 0kl. These are due to glide planes (a, b, c, d, e, n);
3. absences along lines h00, 0k0 and 00l. These are due to screw axes (2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄ or 6₅).

Note that in these tables the ‘conditions limiting possible reflections’ are given, rather than the absent reflections. Thus, they list the reflections that are present (in most cases with even indices n = 2) rather than the forbidden ones (which in most cases have odd indices n = 2 + 1).

If you know these absences, you can determine the space group uniquely in most cases, using these tables.

In a few cases there may be two possible space groups with the same absences (for example P1 and P̄1, Pnma and Pna2₁). They can only be distinguished based on statistics (see Chapter 9.9).

For electron crystallography the same rules apply, but we have to be extra careful because forbidden reflections often appear in ED patterns, due to multiple diffraction. This is in strong contrast to X-ray diffraction, where the forbidden reflections are really absent. In electron diffraction we may look for systematically weak/strong/weak/strong reflections along an axis – that is a strong indication of a 2₁ screw axis rather than a 2-fold rotation axis.

Note also that HRTEM images contain crystallographic structure-factor phase information, but this is not mentioned in these tables, because it is not available in diffraction (neither X-ray nor electron diffraction). (p. 272 )

TRICLINIC. Laue class 1̄

		Point group
Reflection conditions	Extinction symbol	1	1̄
None	P−	Pl(l)	P1̄ (2)

MONOCLINIC, Laue class 2/m

Unique axis b				Laue class 1 2/m 1
Reflection conditions				Point group
hkl0kl hk0	h0l h00 00l	0k0	Extinction symbol	2	m	2/m
			Pl−1	P121 (3)	P1m1 (6)	P1 2/m 1(10)
		k	P12₁l	P12 ₁ l (4)		P121/ml(ll)
	h		Plal		Plal (7)	P1 2/a 1 (13)
	h	k	P12₁/al			P1 2₁,/a 1 (14)
	l		Plcl		P 1 c1 (7)	P 12/c1 (13)
	l	k	P12₁,/cl			P 1 2 ₁,/c1 (14)
	h / l		Plnl		Plnl (7)	P1 2/n 1 (13)
	h + l	k	P1 2₁/n 1			P1 2₁/n 1 (14)
h+k	h	k	Cl‐1	c 121 (5)	C 1 _m 1 (8)	C 1 2/m 1(12)
h+k	h,l	k	Clcl		C 1 c 1 (9)	C 12/c l(15)
k+l		k	A1‐1	A121 (5)	A 1ml (8)	Al 2/m 1 (12)
k+l	h,l	k	Alnl		Alnl (9)	Al 2/n 1 (15)
h+k+l	h + l	k	I1‐1	I121 (5)	I1ml (8)	I1 2/m 1 (12)
h+k+l	h,l	k	I1a1		I1al (9)	I1 2/a 1 (15)

Unique axis c				Laue class 1 1 2/m
Reflection conditions				Point group
hkl0kl h0l	hkO h00 0k0	00l	Extinction symbol	2	m	2/m
			Pll‐	P112 (3)	Pllm(6)	Pll 2/m (10)
		l	P112₁,	P112₁, (4)		Pll 2₁,/m(ll)
	h		Plla		Plla (7)	Pll 2/a (13)
	h	l	Pll 2₁/a			Pll 2₁/a(14)
	k		P11b		Pllb(7)	Pll 2/6 (13)
	k	l	Pll 2₁/b			Pll 2/b (14)
	h + k		P11n		P1 1n (7)	Pll 2/n (13)
	h + k	l	Pll 2₁/n			Pll 2₁/n(14)
h + l	h	l	Bll‐	B112(5)	B11m (8)	B11 2/m (12)
h + l	h,k	l	Blln		Blln(9)	Bll 2/n (15)
k + l	k	l	All−	A112(5)	A 11m (8)	All 2/m (12)
k + l	h,k	l	Alla		Al1a (9)	All 2/a (15)
h + k + l	h + k	l	I11−	I112(5)	I11m, (8)	I11 2/m (12)
h + k + l	h,k	l	I11b		I11b(9)	I11 2/b(15)

Unique axis a				Laue class 2/m 1 1
Reflection conditions				Point group
hklh0l hk0	0kl0k0 00l	h00	Extinction symbol	2	m	2/m
			P−ll	P211 (3)	Pm 11 (6)	P2/mll (10)
		h	P2₁,ll	P2₁,ll (4)		P2₁/m 11 (11)
	k		Pb11		Pb11 (7)	P2/b11 (13)
	k	h	P2₁/b 11			P2₁/b 11 (14)
	l		Pcll		Pcll(7)	P2/c 11 (13)
	l	h	P2₁/c 11			P2₁/c 11 (14)
	k + l		Pnll		Pnll (7)	P2/nll (13)
	k + l	h	P2₁/n 11			P2₁/n 11 (14)
h + k	k	h	C−ll	C211 (5)	Cm 11 (8)	C2/mll (12)
h + k	k,l	h	Cnll		Cnll (9)	C2/n 11 (15)
h + l	l	h	B−ll	B211 (5)	Bmll (8)	B2/m 11 (12)
h + l	k,l	h	Bb1		Bb11 (9)	B2/b11 (15)
h + k + l	k + l	h	I−11	I211(5)	Im 11 (8)	I2/m 11 (12)
h + k + l	k,l	h	Icll		Ic 11 (9)	I2/c 11 (15)

(p. 273 )

ORTHORHOMBIC, Laue class mmm (2/m 2/m 2/m)

In this table, the symbol e in the space‐group symbol represents the two glide planes given between parentheses in the corresponding extinction symbol. Only for one of the two cases does a bold printed symbol correspond with the standard symbol.

Reflection conditions								Laue class mmm (2/m 2/m 2/m)
								Point group
hkl	0kl	h0l	hk0	h00	0k0	00l	Extinction symbol	222	mm2 m2m 2mm	mmm
							P−−−	P 222 (16)	*Pmm* 2 (25)	*Pmmm* (47)
									Pm2m (25)
									P2mm (25)
						l	P−−2₁,	P 222 ₁ (17)
					k		P−2₁−	P22₁2(17)
					k	l	P−2₁2₁	P22₁2₁ (18)
				h			P2₁−−	P2₁,22(17)
				h		l	P2₁−2₁,	P2₁,22₁, (18)
				h	k		P2₁2₁−	P 2 ₁ 2 ₁ 2 (18)
				h	k	l	P2₁2₁2₁	P 2 ₁ 2 ₁ 2 ₁(19)
			h	h			P−−a		Pm2a (28)
									P2₁,ma (26)	*Pmma* (51)
			k		k		P−−b		Pm2₁ b (26)
									P2mb (28)	Pmmb (51)
			h + k	h	k		P−−n		Pm2₁ n(31)
									P2₁ mn(31)	*Pmmn* (59)
		h		h			P−a−		*Pma* 2 (28)	Pmam (51)
									P2₁ am (26)
		h	h	h			P−aa		P2aa (27)	Pmaa (49)
		h	k	h	k		P−ab		P2₁ ab (29)	Pmab (57)
		h	h + k	h	k		P−an		P2an (30)	Pman (53)
		l				l	P−c−		*Pmc* 2 ₁ (26)
									P2cm (28)	Pmcm (51)
		l	h	h		l	P−ca		P2₁ ca(29)	Pmca (57)
		l	k		k	l	P−cb		P2cb (32)	Pmcb (55)
		l	h + k	h	k	l	P−cn		P2_l Cn(33)	Pmcn (62)
		h + l		h		l	P−n−		*Pmn* 2 ₁ (31)
									p2₁ (31)	Pmnm (59)
		h + l	h	h		l	P−na		P2na (30)	*Pmna* (53)
		h + l	k	h	k	l	P−nb		P2₁ nb(33)	Pmnb (62)
		h +l	h + k	h	k	l	P−nn		P2nn (34)	Pmnn (58)
	k				k		Pb−−		Pbm2 (28)
									Pb2₁ m (26)	Pbmm (51)
	k		h	h	k		Pb−a		Pb2₁ a (29)	Pbma (57)
	k		k		k		Pb−b		Pb2b (27)	Pbmb (49)
	k		h + k	h	k		Pb−n		Pb2n (30)	Pbmn (53)
	k	h		h	k		Pba−		*Pba* 2 (32)	*Pbam* (55)
	k	h	h	h	k		Pbaa			Pbaa (54)
	k	h	k	h	k		Pbab			Pbab (54)
	k	h	h + k	h	k		Pban			*Pban* (50)
	k	l			k		Pbc−		Pbc2₁ (29)	*Pbcm* (57)
	k	l	h	h	k		Pbca			*Pbca* (61)
	k	l	k		k		Pbcb			Pbcb (54)
	k	l	h +k	h	k		Pbcn			Pbcn (60)
	k	h + l		h	k		Pbn−		Pbn2₁ (33)	Pbnm (62)
	k	h + l	h	h	k		Pbna			Pbna (60)
	k	h + l	k	h	k		Pbnb			Pbnb (56)
	k	h + l	h +k	h	k		Plmn			Pbnn (52)
	l						Pc−−		Pcm2₁ (26)
									Pc2m (28)	Pcmm (51)
			h	h			Pc−a		Pc2a (32)	Pcma (55)
			k		k		Pc−b		Pc2₁ b (29)	Pcmb (57)
			h + k	h	k		Pc−n		Pc2₁ n (33)	Pcmn (62)
		h		h			Pca−		*Pca* 2 ₁ (29)	Pcam (57)
		h	h	h			Pcaa			Pcaa (54)
		h	k	h	k		Pcab			Pcab (61)
		h	h + k	h	k		Pcan			Pcan (60)
		l					Pcc−		*Pcc* 2 (27)	*Pccm* (49)
		l	h	h			Pcca			*Pcca* (54)
		l	k		k		Pccb			Pccb (54)
		l	h + k	h	k		Pccn			*Pccn* (56)
		h + l		h			Pcn −		Pcn2 (30)	Pcnm (53)
		h + l	h	h			Pcna			Pcna (50)
		h + l	k	h	k		Pcnb			Pcnb (60)
		h +l	h + k	h	k		Pcnn			Pcnn (52)
	k + l				k		Pn−−		Pnm2₁ (31)	Pnmm (59)
									pn2₁m(31)
	k + l		h	h	k	l	Pn−a		Pn2₁ a (33)	*Pnma* (62)
	k + l		k		k		Pn−b		Pn2b (30)	Pnmb (53)
	k + l		h + k	h	k		Pn−n		Pn2n (34)	Pnmn (58)
	k + l	h		h	k		Pna−		*Pna* 2₁ (33)	Pnam (62)
	k + l	h	h	h	k		Pnaa			Pnaa (56)
	k + l	h	k	h	k		Pnab			Pnab (60)
	k + l	h	h + k	h	k		Pnan			Pnan (52)
	k + l	l			k		Pnc−		*Pnc* 2 (30)	Pncm (53)
	k + l	l	h	h	k		Pnca			Pnca (60)
	k + l	l	k		k		Pncb			Pncb (50)
	k + l	l	h + k	h	k		Pncn			Pncn (52)
	k + l	h + l		h	k		Pnn −		*Pnn* 2 (34)	*Pnnm* (58)
	k + l	h + l	h	h	k		Pnna			*Pnna* (52)
	k + l	h + l	k	h	k		Pnnb			Pnnb (52)
	k + l	h + l	h + k	h	k		Pnnn			*Pnnn* (48)
h + k	k	h	h + k	h	k		C−−−	C 222(21)	*Cmm* 2 (35)	*Cmmm* (65)
									Cm2m (38)
									C2mm (38)
h − k	k	h	h + k	h	k		C−−2₁,	C 222 ₁ (20)
h + k	k	h	h,k	h	k		C−−(ab)		Cm2e (39)	*Cmme* (67)
									C2me (39)
h + k	k	h,l	h + k	h	k		C−c−		*Cmc* 2₁ (36)	*Cmcm* (63)
									C2cm (40)
h + k	k	h,l	hk	h	k		C−c(ab)		C2ce (41)	*Cmce* (64)
h + k	k,l	h	h + k	h	k		Cc−−		Ccm2₁, (36)	Ccmm (63)
									Cc2m (40)
h + k	k,l	h	hk	h	k		Cc −(ab)		Cc2e (41)	Ccme (64)
h + k	k,l	h,l	h + k	h	k		Ccc−		*Ccc* 2 (37)	*Cccm* (66)
h + k	k,l	h,l	h,k	h	k		Ccc(ab)			*Ccce* (68)
h + l	l	h +l	h	h		l	B−−−	B222 (21)	Bmm2 (38)	Bmmm (65)
									Bm2m (35)
									B2mm (38)
h + l	l	h + l	h	h	k		B−2₁−	B22₁2(20)
h + l	l	h +l	hk	h	k		B−−b		Bm2₁ b (36)	Bmmb (63)
									B2mb (40)
h + l	l	h,l	h	h		l	B −(ac)−		Bme2 (39)	Bmem (67)
									B2em (39)
h + l	l	h,l	h,k	h	k		B−(ac)b		B2eb (41)	Bmeb (64)
h +l	k,l	h + l	h	h	k		Bb−−		Bbm2 (40)	Bbmm (63)
									Bb2₁ m (36)
h + l	k,l	h +l	h,k	h	k		Bb−b		Bb2b (37)	Bbmb (66)
h + l	k,l	h,l	h	h	k		Bb(ac)−		Bbe2 (41)	Bbem (64)
h + l	k,l	h,l	h,k	h	k		Bb(ac)b			Bbeb (68)
k + l	k + l	l	k		k	l	A−−−	A222 (21)	*Amm* 2 (38)	Ammm (65)
									Am2m (38)
									A2mm (35)
k + l	k + l	l	k	h	k	l	A2₁−−−	A2₁22(20)
k + l	k + l	l	h k	h	k		A−−a		Am 2 a (40)	Amma (63)
									A2₁ ma (36)
k + l	k + l	h,l	k	h	k		A−a−		*Ama* 2 (40)	Amam (63)
									A2₁ am (36)
k + l	k + l	h,l	h.k	h	k		A−aa		A2aa (37)	Amaa (66)
k + l	k,l	l	k		k		A(bc)−−		*Aem* 2 (39)	Aemm (67)
									Ae2m (39)
k + l	k,l	l	h,k	h	k		A(bc)−a		Ae2a (41)	Aema (64)
k + l	k,l	h,i	k	h	k		A(bc)a−		*Aea* 2(41)	Aeam (64)
k + l	k,l	h,i	h,k	h	k		A(bc)aa			Aeaa (68)
h + k + l	k + l	h + l	h + k	h	k		I−−−	$[\begin{array}{l} I 222 (23) \\ I 2_{1} 2_{1} 2_{1} (24) \end{array}]$ *	*Imm* 2 (44) Im2m (44)	*Immm* (71)
h+k+l	k + l	h +l	h,k	h	k	l	l−−(ab)		Im2a (46)	*Imma* (74)
									I2mb (46)	Immb (74)
h + k + l	k + l	h,l	h + k	h	k	l	I−(ac)−		*Ima* 2(46)	Imam (74)
									I2cm (46)	lmcm (74)
h + k +l	k + l	h,l	h,k	h	k	l	I−cb		I2cb (45)	Imcb (72)
h + h +l	k,l	h + l	h + k	h	k	l	I(bc)−−		Item2 (46)	Iemm (74)
									Ie2m (46)
h + k +l	k,l	h + l	h,k	h	k	l	Ic−a		Ic2a (45)	Icma (72)
h + k +l	k,l	h,l	h + k	h	k	l	Iba−		*Iba* 2(45)	*Ibam* (72)
h + k + l	k,l	h,l	h,k	h	k	l	Ibca			*Ibca* (73)
										Icab (73)
h + k,h + l,k + l	k,l	h,l	h,k	h	k	l	F−−−	F 222(22)	*Fmm* 2(42)	*Fmmm* (69)
									Fm2m (42)
									F2mm (42)
h + k,h + l,k + l	k,l	h + l =4n; h, l	h + k = 4n;h,k	h = 4n	k = 4n	l = 4n	F−dd		F2dd(43)
h +k,h + l,k + l	k + l =4n;k, l	h,l	h + k = 4n;h,k	h = 4n	k = 4n	l = 4n	Fd−d		Fd2d(43)
h + k,h + l,k + l	k + l =4n; k, l	h + l = 4n;h,l	h,k	h = 4n	k = 4n	l = 4n	Fdd−		*Fdd* 2(43)
h + k,h + l,k + l	k + l =4n;k, l	h +l = 4n;h,l	h + k = 4n;h,k	h = 4n	k = 4n	l = 4n	Fddd			*Fddd* (70)

(*) Pair of space groups with common point group and symmetry elements hut differing in the relative location of these elements.

(p. 274 ) (p. 275 ) (p. 276 )

TETRAGONAL, Laue classes 4/m and 4/mmm

								Laue class
								4/m			4/mmm (4/m 2/m 2/m)
Reflection conditions								Point group
hkl	hk0	0kl	hhl	00l	0k0	hh0	Extinction symbol	4	4	4/m	422	4mm	4̄2m 4̄m2	4/mmm
							P−−−	P4 (75)	P4̄(81)	P4/m (83)	P422 (89)	P4mm (99)	P4̄2m(lll)	P4/mmm (123)
													P4̄m2(115)
					k		P−2₁−				P42₁2(90)		P4̄2₁ m(113)
				l			P4₂−−	P4₂ (77)		P4₂/m (84)	P4₂22 (93)
				l	k		p4₂2₁−				P4₂2₁2 (94)
				l = 4n			P4₁−−	${\begin{matrix} p 4_{1} (76) \\ p 4_{3} (78) \end{matrix}}$ †			${\begin{matrix} p 4_{1} 22 (91) \\ p 4_{3} 22 (95) \end{matrix}}$ †
				l = 4n	k		P4₁,2₁−				${\begin{matrix} p 4_{1} 2_{1} 2 (92) \\ p 4_{3} 2_{1} 2 (96) \end{matrix}}$ †
			l	l			P−−C					P4₂ mc (105)	P4̄2c(112)	P4₂/mmc (131)
			l	l	k		P−2₁ c						P4̄2₁ c(114)
		k			k		P−b−					P4bm (100)	P4̄b2(117)	P4/mbm (127)
		k	l	l	k	P−bc						P4₂ bc (106)		P4₂/mbc (135)
		l		l			P−c−					P4₂ cm (101)	P4̄c2(116)	P4₂/mcm (132)
		l	l	l			P −cc					P4cc (103)		P4/mcc (124)
		k + l		l	k		P−n−					P4₂ nm (102)	P4̄n2(118)	P4₂/mnm (136)
		k +l	l	l	k		P−nc					P4nc (104)		P4/mnc (128)
	h + k				k		Pn−−			P4/n (85)				P4/nmm (129)
	h + k			l	k		P4₂/n− −			P4₂/n (86)
	h + k		l	l	k		Pn −c							P4₂/nmc (137)
	h+k	k			k		Pnb−							P4/nbm(125)
	h+k	k	l	l	k		Pubc							P4₂/nbc (133)
	h+k	l		l	k		Pnc−							P4₂/ncm (138)
	h+k	l	l	l	k		Pncc							P4/ncc (130)
	h+k	k + l		l	k		Pnn−							P4₂/nnm (134)
	h+k	k + l	l	l	k		Pnnc							P4/nnc (126)
h + k + l	h+k	k + l	l	l	k		l−−−	l4 (79)	l4̄ (82)	l4/m (87)	l422 (97)	l4mm (107)	l4̄2m (121)	l4/mmm (139)
													l4̄m2(119)
h + k + l	h+k	k + l	l	l= 4n	k		l4₁−−	l4₁ (80)			l4₁22(98)
h + k +l	h+k	k + l	‡	l = 4n	k	h	l−−d					l4₁ md (109)	l4̄2d (122)
h +k +l	h+k	k,l	l	l	k		l−c−					l4cm (108)	l4̄c2 (120)	l4/mcm(140)
h +k +l	h+k	k,l	‡	l = 4n	k	h	l−cd					l4₁ cd(110)
h + k + l	h,k	k + l	l	l = 4n	k		14₁/a−−			l4₁/a (88)
h +k + l	h,k	k + l	‡	l = 4n	k	h	la−d							14₁/amd (141)
h + k + l	h,k	k,l	‡	l = 4n	k	h	lacd							l4₁/acd(142)

(†) Pair of enantioinorphic space groups, cf. Section 3.1.5.

(‡) Condition: 2h +l = 4n; l.

(p. 277 )

TRIGONAL, Laue classes 3̄ and 3̄m

					Lane class
Reflection conditions					3̄		3̄ml (3̄ 2/m 1) 3̄m			3̄1m (3̄ 1 2/m)
Hexagonal axes					Point.group
hkil	hh̄0l	h,h,2̄h̄,1	000l	Extinction symbol	3	3̄	321 32	3ml 3m	3̄ml 3̄m	312	31m	31m
				P−−−	P3 (143)	P3̄ (147)	P3̄21 (150)	P3ml (156)	P3̄ml (164)	P312 (149)	P31m (157)	P3̄m (162)
			l = 3n	P3₁−−	${\begin{matrix} p 3_{1} (144) \\ p 3_{2} (145) \end{matrix}} §$		${\begin{matrix} p 3_{1} 21 (152) \\ p 3_{2} 21 (154) \end{matrix}} §$			${\begin{matrix} p 3_{1} 12 (151) \\ p 3_{2} 12 (153) \end{matrix}} §$
		l	l	P−−C							P31c (159)	P3̄c (163)
	l		l	P−c−				P3cl(158)	P3̄cl (165)
−h+k + l = 3n	h + l = 3n	l = 3n	l = 3n	R(obv)− −	R3 (146)	R3̄ (148)	R32 (155)	R3m (160)	R3̄m (166)
−h+k + l = 3n	h + l = 3n; l	l = 3n	l = 6i	R(obv)− c				R3c(161)	R3̄c (167)
h − k + l = 3n	−h + l = 3n	l = 3n	l = 3n	R(rev)− −	R3 (146)	R3̄ (148)	R32 (155)	R3m (160)	R3̄m (166)
h−k + l = 3n	−h + l = 3n; l	l = 3n	l = 6n	R(rev)− c				R3c(161)	R3̄c (167)
Rhombohedral axes				Point group
hkl		hhl	hhh	Extinction symbol	3	3̄	32	3m	3̄m
				R −−	R3(146)	R3̄ (148)	R32 (155)	R3m (160)	R3m (166)
		l	h	R−c				R3c (161)	R3̄c(167)

§ Pair of enantioinorphic space groups: cf. Section 3.1.5.

¶ For obverse and reverse sellings cf Seclion 1.2.1. The obverse selling is slandard in these tables.

The transformation reverse → obverse is given by a(obv.) = −a(rev.), b(obv.) = −b(rev.), c(obv.) = c(rev.).

HEXAGONAL, Laue classes 6/m and 6/mmm

				Laue class
				6/m			6/mmm(6/m 2/m 2/m)
Reflection conditions				Point group
hh̄0l	hh2̄h̄l	000l	Extinction symbol	6	6̄	6/m	622	6mm	6̄2m 6̄m2	6/mmm
			P−−−	P6 (168)	P6̄ (174)	P6/m (175)	P622 (177)	P6mm(183)	P6̄2m (189)	P6/mmm(191)
									P6̄m2 (187)
		l	P6₃− −	P6₃ (173)		P6₃/m(176)	P6₃22 (182)
		l = 3n	P6₂−−	${\begin{matrix} p 6_{2} (171) \\ p 6_{4} (172) \end{matrix}} * *$			${\begin{matrix} p 6_{2} 22 (180) \\ p 6_{4} 22 (181) \end{matrix}} * *$
		l = 6n	P6₁−−	${\begin{matrix} p 6_{1} (169) \\ p 6_{3} (170) \end{matrix}} * *$			${\begin{matrix} p 6_{1} 22 (178) \\ p 6_{5} 22 (179) \end{matrix}} * *$
	l	l	P−−c					P6₃ mc(186)	P6̄2c(190)	P6₃/mmc(194)
l		1	P−c−					P6₃ cm(185)	P6̄c2 (188)	P6₃/mcm(193)
l	l	1	P− cc					P6cc(184)		P6/mcc(192)

**Pair of enantiomorphic space groups, cf. Section 3.1.5

(p. 278 )

CUBIC, Laue classes m3̄ and m3̄m

					Laue class
Reflection conditions (Indices are permutable, apart from space group No. 205) ††					m3̄ (2/m 3̄)		m3̄m (4/m 3̄ 2/m)
					Point group
hkl	0kl	hhl	00l	Extinction symbol	23	m3̄	432	4̄3m	m3̄m
				P −−−	P23 (195)	Pm3̄ (200)	P432 (207)	P4̄3m (215)	Pm3̄m (221)
			l	${\begin{matrix} P 2_{1} - - \\ P 4_{2} - - \end{matrix}$	P2₁3(198)		P4₂32 (208)
			l = 4n	P4₁−−			${\begin{matrix} P 4_{1} 32 (213) \\ P 4_{3} 32 (212) \end{matrix}}$ ‡‡
		l	l	P−−n				P4̄3n (218)	Pm3̄n (223)
	k ††		l	Pa−−		Pa3̄ (205)
	k+l		l	Pn−−		Pn3̄ (201)			Pn3̄m (224)
	k + i	l	l	Pn−n					Pn3̄n (222)
h + k + l	k + l	l	l	I−−−	$[\begin{matrix} I 23 (197) \\ I 2_{1} 3 (199) \end{matrix}]$ §§	Im3̄ (204)	I432(211)	I4̄3m (217)	Im3̄m (229)
h + k + l	k+l	I	l = 4n	14₁−−			I4₁32 (214)
h + k + l	k+l	2h + l = 4n,l	l = 4n	I−−d				l4̄3d (220)
h + k + l	k,l	l	l	Ia−−		la3̄ (206)
h + k + l	k,i	2h + l = 4n,l	l = 4n	Ia−d					Ia3̄d (230)
h + k,h + l,k + l	k,i	h + l	l	F−−−	F23 (196)	Fm3̄ (202)	F432 (209)	F4̄3m (216)	Fm3̄m (225)
h + k,h + l,k + l	k,i	h + l	l = 4n	F4₁−−			F4₁32(210)
h + k,h + l,k + l	k,i	h,l	l	F−−c				F4̄3c (219)	Fm3̄c (226)
h + k,h + l,k + l	k +l = 4n,k,l	h + l	l = 4n	Fd−−		Fd3̄ (203)			Fd3̄m (227)
h + k,h + l,k + l	k + l = 4n,k,l	h,l	l = 4n	Fd−c					Fd3̄c (228)

(††) For No. 205, only cyclic permutations are permitted. Conditions are 0kl: k = 2n; h0l: l = 2n; hk0: h = 2n.

(‡‡) Pair of enantiomorphic space groups, cf. Section 3.1.5.

(§§) Pair of space groups with common point group and symmetry elements but differing in the relative location of these elements.

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(p. 271 ) Appendix 2: Tables for Space Group Determination