Deduction of possible space groups

Looijenga-Vos, A.; Buerger, M. J.

doi:10.1107/97809553602060000506

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 3.1, pp. 45-46

Section 3.1.4. Deduction of possible space groups

A. Looijenga-Vos^a ^‡ and M. J. Buerger^b ^§

^a Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands, and ^bDepartment of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

3.1.4. Deduction of possible space groups

| top | pdf |

Reflection conditions, diffraction symbols, and possible space groups are listed in Table 3.1.4.1. For each crystal system, a different table is provided. The monoclinic system contains different entries for the settings with b, c and a unique. For monoclinic and orthorhombic crystals, all possible settings and cell choices are treated. In contradistinction to Table 4.3.2.1 , which lists the space-group symbols for different settings and cell choices in a systematic way, the present table is designed with the aim to make space-group determination as easy as possible.

Table 3.1.4.1| top | pdf |
Reflection conditions, diffraction symbols and possible space groups

TRICLINIC. Laue class $[\bar{1}]$

Reflection conditions	Extinction symbol	Point group
Reflection conditions	Extinction symbol	1	$[\bar{1}]$
None	P–		$[P\bar{1}\;(2)]$

MONOCLINIC, Laue class [2/m]

Unique axis b			Extinction symbol	Laue class $[1\;2/m\;1]$
Reflection conditions				Point group
hkl 0kl hk0	h0l h00 00l	0k0		2	m
			P1–1	P121 (3)	P1m1 (6)	$[{\bi P} {\bf 1\;2/}{\bi m}{\hbox to 2pt{}}{\bf 1}\;(10)]$
		k	$[P12_{1}1]$	$[{\bi P}{\bf 12}_{\bf 1}{\bf 1}\;(4)]$		$[{\bi P}{\bf 1\;2}_{\bf 1}/{\bi m}{\hbox to 2pt{}}{\bf 1}\;(11)]$
	h		P1a1		P1a1 (7)	$[P1\;2/a\;1\;(13)]$
	h	k	$[P1\;2_{1}/a\;1]$			$[P1\;2_{1}/a\;1\;(14)]$
	l		P1c1		P1c1 (7)	$[{\bi P}{\bf 1\;2}/{\bi c}{\bf \;1}\;(13)]$
	l	k	$[P1\;2_{1}/c\;1]$			$[{\bi P}{\bf 1} {\bf \;2}_{\bf 1}/{\bi c}{\bf \;1}\;(14)]$
			P1n1		P1n1 (7)	$[P1\;2/n\;1\;(13)]$
		k	$[P1\;2_{1}/n\;1]$			$[P1\;2_{1}/n\;1\;(14)]$
	h	k	C1–1	C121 (5)	C1m1 (8)	$[{\bi C}{\bf 1\;2}/{\bi m}{\bf \;1}\;(12)]$
	h, l	k	C1c1		C1c1 (9)	$[{\bi C}{\bf 1\;2}/{\bi c}{\bf \;1}\;(15)]$
	l	k	A1–1	A121 (5)	A1m1 (8)	$[A1\; 2/m\; 1\;(12)]$
	h, l	k	A1n1		A1n1 (9)	$[A1\; 2/n\; 1\;(15)]$
		k	I1–1	I121 (5)	I1m1 (8)	$[I1\; 2/m\; 1\;(12)]$
	h, l	k	I1a1		I1a1 (9)	$[I1\; 2/a\; 1\;(15)]$
Unique axis c			Extinction symbol	Laue class $[1\; 1\; 2/m]$
Reflection conditions				Point group
hkl 0kl h0l	hk0 h00 0k0	00l		2	m
			P11–	P112 (3)	P11m (6)	$[P11\; 2/m\;(10)]$
		l	$[P112_{1}]$	$[P112_{1}\;(4)]$		$[P11\; 2_{1}/m\;(11)]$
	h		P11a		P11a (7)	$[P11\; 2/a\;(13)]$
	h	l	$[P11\; 2_{1}/a]$			$[P11\; 2_{1}/a\;(14)]$
	k		P11b		P11b (7)	$[P11\; 2/b\;(13)]$
	k	l	$[P11\; 2_{1}/b]$			$[P11\; 2_{1}/b\;(14)]$
			P11n		P11n (7)	$[P11\; 2/n\;(13)]$
		l	$[P11\; 2_{1}/n]$			$[P11\; 2_{1}/n\;(14)]$
	h	l	B11–	B112 (5)	B11m (8)	$[B11\; 2/m\;(12)]$
	h, k	l	B11n		B11n (9)	$[B11\; 2/n\;(15)]$
	k	l	A11–	A112 (5)	A11m (8)	$[A11\; 2/m\;(12)]$
	h, k	l	A11a		A11a (9)	$[A11\; 2/a\;(15)]$
		l	I11–	I112 (5)	I11m (8)	$[I11\; 2/m\;(12)]$
	h, k	l	I11b		I11b (9)	$[I11\; 2/b\;(15)]$
Unique axis a			Extinction symbol	Laue class $[2/m\; 1\; 1]$
Reflection conditions				Point group
hkl h0l hk0	0kl 0k0 00l	h00		2	m
			P–11	P211 (3)	Pm11 (6)	$[P2/m\;11\;(10)]$
		h	$[P2_{1}]$ 11	$[P2_{1}]$ 11 (4)		$[P2_{1}/m\;11\;(11)]$
	k		Pb11		Pb11 (7)	$[P2/b\;11\;(13)]$
	k	h	$[P 2_{1}/b\ 11]$			$[P2_{1}/b\;11\;(14)]$
	l		Pc11		Pc11 (7)	$[P2/c\;11\;(13)]$
	l	h	$[P 2_{1}/c\;11]$			$[P2_{1}/c\ 11\;(14)]$
			Pn11		Pn11 (7)	$[P2/n\;11\;(13)]$
		h	$[P 2_{1}/n\;11]$			$[P2_{1}/n\;11\;(14)]$
	k	h	C–11	C211 (5)	Cm11 (8)	$[C2/m\;11\;(12)]$
	k, l	h	Cn11		Cn11 (9)	$[C2/n\;11\;(15)]$
	l	h	B–11	B211 (5)	Bm11 (8)	$[B2/m\;11\;(12)]$
	k, l	h	Bb11		Bb11 (9)	$[B2/b\;11\;(15)]$
		h	I–11	I211 (5)	Im11 (8)	$[I2/m\;11\;(12)]$
	k, l	h	Ic11		Ic11 (9)	$[I2/c\;11\;(15)]$

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 ( $[2/m\; 2/m\; 2/m]$ )
hkl	0kl	h0l	hk0	h00	0k0	00l	Extinction symbol	Point group
hkl	0kl	h0l	hk0	h00	0k0	00l	Extinction symbol	222	$[\!\matrix{mm2\cr m2m\cr 2mm\cr}]$	mmm
							P– – –	P222 (16)	*Pmm*2 (25)	*Pmmm* (47)
									Pm2m (25)
									P2mm (25)
						l	P– – $[2_{1}]$	$[{\bi P}{\bf 222}_{{\bf 1}}\;(17)]$
					k		P– $[2_{1}]$ –	$[P22_{1}2\;(17)]$
					k	l	P– $[2_{1}2_{1}]$	$[P22_{1}2_{1}\;(18)]$
				h			$[P2_{1}]$ – –	$[P2_{1}22\;(17)]$
				h		l	$[P2_{1}]$ – $[2_{1}]$	$[P2_{1}22_{1}\;(18)]$
				h	k		$[P2_{1}2_{1}]$ –	$[{\bi P}{\bf 2}_{{\bf 1}}{\bf 2}_{{\bf 1}}{\bf 2}\;(18)]$
				h	k	l	$[P2_{1}2_{1}2_{1}]$	$[{\bi P}{\bf 2}_{{\bf 1}}{\bf 2}_{{\bf 1}}{\bf 2}_{{\bf 1}}\;(19)]$
			h	h			P– –a		Pm2a (28)
									$[P2_{1}ma\;(26)]$	*Pmma* (51)
			k		k		P– –b		$[Pm2_{1}b\;(26)]$
									P2mb (28)	Pmmb (51)
				h	k		P– –n		$[Pm2_{1}n\;(31)]$
									$[P2_{1}mn\;(31)]$	*Pmmn* (59)
		h		h			P–a–		*Pma*2 (28)	Pmam (51)
									$[P2_{1}am\;(26)]$
		h	h	h			P–aa		P2aa (27)	Pmaa (49)
		h	k	h	k		P–ab		$[P2_{1}ab\;(29)]$	Pmab (57)
		h		h	k		P–an		P2an (30)	Pman (53)
		l				l	P–c–		$[{\bi P}{\bi m}{\bi c}{\bf 2}_{{\bf 1}}\;(26)]$
									P2cm (28)	Pmcm (51)
		l	h	h		l	P–ca		$[P2_{1}ca\;(29)]$	Pmca (57)
		l	k		k	l	P–cb		P2cb (32)	Pmcb (55)
		l		h	k	l	P–cn		$[P2_{1}cn\;(33)]$	Pmcn (62)
				h		l	P–n–		$[{\bi P}{\bi m}{\bi n}{\bf 2}_{{\bf 1}}\;(31)]$
									$[P2_{1}nm\;(31)]$	Pmnm (59)
			h	h		l	P–na		P2na (30)	*Pmna* (53)
			k	h	k	l	P–nb		$[P2_{1}nb\;(33)]$	Pmnb (62)
				h	k	l	P–nn		P2nn (34)	Pmnn (58)
	k				k		Pb– –		Pbm2 (28)
									$[Pb2_{1}m\;(26)]$	Pbmm (51)
	k		h	h	k		Pb–a		$[Pb2_{1}a\;(29)]$	Pbma (57)
	k		k		k		Pb–b		Pb2b (27)	Pbmb (49)
	k			h	k		Pb–n		Pb2n (30)	Pbmn (53)
	k	h		h	k		Pba–		Pba2 (32)	*Pbam* (55)
	k	h	h	h	k		Pbaa			Pbaa (54)
	k	h	k	h	k		Pbab			Pbab (54)
	k	h		h	k		Pban			*Pban* (50)
	k	l			k	l	Pbc–		$[Pbc2_{1}\;(29)]$	*Pbcm* (57)
	k	l	h	h	k	l	Pbca			*Pbca* (61)
	k	l	k		k	l	Pbcb			Pbcb (54)
	k	l		h	k	l	Pbcn			*Pbcn* (60)
	k			h	k	l	Pbn–		$[Pbn2_{1}\;(33)]$	Pbnm (62)
	k		h	h	k	l	Pbna			Pbna (60)
	k		k	h	k	l	Pbnb			Pbnb (56)
	k			h	k	l	Pbnn			Pbnn (52)
	l					l	Pc– –		$[Pcm2_{1}\;(26)]$
									Pc2m (28)	Pcmm (51)
	l		h	h		l	Pc–a		Pc2a (32)	Pcma (55)
	l		k		k	l	Pc–b		$[Pc2_{1}b\;(29)]$	Pcmb (57)
	l			h	k	l	Pc–n		$[Pc2_{1}n\;(33)]$	Pcmn (62)
	l	h		h		l	Pca–		$[{\bi P}{\bi c}{\bi a}{\bf 2}_{{\bf 1}}\;(29)]$	Pcam (57)
	l	h	h	h		l	Pcaa			Pcaa (54)
	l	h	k	h	k	l	Pcab			Pcab (61)
	l	h		h	k	l	Pcan			Pcan (60)
	l	l				l	Pcc–		*Pcc*2 (27)	*Pccm* (49)
	l	l	h	h		l	Pcca			*Pcca* (54)
	l	l	k		k	l	Pccb			Pccb (54)
	l	l		h	k	l	Pccn			*Pccn* (56)
	l			h		l	Pcn –		Pcn2 (30)	Pcnm (53)
	l		h	h		l	Pcna			Pcna (50)
	l		k	h	k	l	Pcnb			Pcnb (60)
	l			h	k	l	Pcnn			Pcnn (52)
					k	l	Pn – –		$[Pnm2_{1}\;(31)]$	Pnmm (59)
									$[Pn2_{1}m\;(31)]$
			h	h	k	l	Pn–a		$[Pn2_{1}a\;(33)]$	*Pnma* (62)
			k		k	l	Pn–b		Pn2b (30)	Pnmb (53)
				h	k	l	Pn–n		Pn2n (34)	Pnmn (58)
		h		h	k	l	Pna –		$[{\bi P}{\bi n}{\bi a}{\bf 2}_{{\bf 1}}\;(33)]$	Pnam (62)
		h	h	h	k	l	Pnaa			Pnaa (56)
		h	k	h	k	l	Pnab			Pnab (60)
		h		h	k	l	Pnan			Pnan (52)
		l			k	l	Pnc –		*Pnc*2 (30)	Pncm (53)
		l	h	h	k	l	Pnca			Pnca (60)
		l	k		k	l	Pncb			Pncb (50)
		l		h	k	l	Pncn			Pncn (52)
				h	k	l	Pnn –		*Pnn*2 (34)	*Pnnm* (58)
			h	h	k	l	Pnna			*Pnna* (52)
			k	h	k	l	Pnnb			Pnnb (52)
				h	k	l	Pnnn			*Pnnn* (48)
	k	h		h	k		C – – –	C222 (21)	*Cmm*2 (35)	*Cmmm* (65)
									Cm2m (38)
									C2mm (38)
	k	h		h	k	l	C– – $[2_{1}]$	$[{\bi C}{\bf 222}_{{\bf 1}}\;(20)]$
	k	h	h, k	h	k		C– –(ab)		Cm2e (39)	*Cmme* (67)
									C2me (39)
	k	h, l		h	k	l	C–c –		$[{\bi C}{\bi m}{\bi c}{\bf 2}_{{\bf 1}}\;(36)]$	*Cmcm* (63)
									C2cm (40)
	k	h, l	h, k	h	k	l	C–c(ab)		C2ce (41)	*Cmce* (64)
	k, l	h		h	k	l	Cc – –		$[Ccm2_{1}\;(36)]$	Ccmm (63)
									Cc2m (40)
	k, l	h	h, k	h	k	l	Cc –(ab)		Cc2e (41)	Ccme (64)
	k, l	h, l		h	k	l	Ccc –		*Ccc*2 (37)	*Cccm* (66)
	k, l	h, l	h, k	h	k	l	Ccc(ab)			*Ccce* (68)
	l		h	h		l	B – – –	B222 (21)	Bmm2 (38)	Bmmm (65)
									Bm2m (35)
									B2mm (38)
	l		h	h	k	l	B– $[2_{1}]$ –	$[B22_{1}2\;(20)]$
	l		h, k	h	k	l	B– –b		$[Bm2_{1}b\;(36)]$	Bmmb (63)
									B2mb (40)
	l	h, l	h	h		l	B –(ac)–		Bme2 (39)	Bmem (67)
									B2em (39)
	l	h, l	h, k	h	k	l	B –(ac)b		B2eb (41)	Bmeb (64)
	k, l		h	h	k	l	Bb – –		Bbm2 (40)	Bbmm (63)
									Bb2₁m (36)
	k, l		h, k	h	k	l	Bb–b		Bb2b (37)	Bbmb (66)
	k, l	h, l	h	h	k	l	Bb(ac)–		Bbe2 (41)	Bbem (64)
	k, l	h, l	h, k	h	k	l	Bb(ac)b			Bbeb (68)
		l	k		k	l	A – – –	A222 (21)	*Amm*2 (38)	Ammm (65)
									Am2m (38)
									A2mm (35)
		l	k	h	k	l	$[A2_{1}]$ – –	$[A2_{1}22\;(20)]$
		l	h, k	h	k	l	A– –a		Am2a (40)	Amma (63)
									$[A2_{1}ma\;(36)]$
		h, l	k	h	k	l	A–a –		Ama2 (40)	Amam (63)
									$[A2_{1}am\;(36)]$
		h, l	h, k	h	k	l	A–aa		A2aa (37)	Amaa (66)
	k, l	l	k		k	l	A(bc)– –		*Aem*2 (39)	Aemm (67)
									Ae2m (39)
	k, l	l	h, k	h	k	l	A(bc)– a		Ae2a (41)	Aema (64)
	k, l	h, l	k	h	k	l	A(bc)a –		*Aea*2 (41)	Aeam (64)
	k, l	h, l	h, k	h	k	l	A(bc)aa			Aeaa (68)
				h	k	l	I – – –	$[\left[\!\matrix{{\bi I}{\bf 222}\ (23)\hfill\cr{\bi I}{\bf 2}_{\bf 1}{\bf 2}_{\bf 1}{\bf 2}_{\bf 1}(24)}\!\!\right]\!]$ ^†	Imm2 (44)	*Immm* (71)
									Im2m (44)
									I2mm (44)
			h, k	h	k	l	I – –(ab)		Im2a (46)	*Imma* (74)
									I2mb (46)	Immb (74)
		h, l		h	k	l	I –(ac)–		Ima2 (46)	Imam (74)
									I2cm (46)	lmcm (74)
		h, l	h, k	h	k	l	I – cb		I2cb (45)	Imcb (72)
	k, l			h	k	l	I(bc)– –		Iem2 (46)	Iemm (74)
									Ie2m (46)
	k, l		h, k	h	k	l	Ic – a		Ic2a (45)	Icma (72)
	k, l	h, l		h	k	l	Iba –		Iba2 (45)	*Ibam* (72)
	k, l	h, l	h, k	h	k	l	Ibca			*Ibca* (73)
										Icab (73)
				h	k	l	F – – –	F222 (22)	Fmm2 (42)	*Fmmm* (69)
									Fm2m (42)
									F2mm (42)
	k, l	; h, l	; h, k				F–dd		F2dd (43)
	; k, l	h, l	; h, k				Fd–d		Fd2d (43)
	; k, l	; h, l	h, k				Fdd–		Fdd2 (43)
	; k, l	; h, l	; h, k				Fddd			*Fddd* (70)

TETRAGONAL, Laue classes [4/m] and [4/mmm]

Reflection conditions							Extinction symbol	Laue class
											$[4/mmm\;(4/m\; 2/m\; 2/m)]$
								Point group
hkl	hk0	0kl	hhl	00l	0k0	hh0		4	$[\bar{4}]$		422	4mm	$[\bar{4}2m\ \ \bar{4}m2]$
							P – – –	P4 (75)	$[P\bar{4}\;(81)]$	$[P4/m\;(83)]$	P422 (89)	P4mm (99)	$[P\bar{4}]$ 2m (111)	$[P4/mmm\;(123)]$
													$[P\bar{4}m2\;(115)]$
					k		P– $[2_{1}]$ –				$[P42_{1}2\;(90)]$		$[P\bar{4}2_{1}m\;(113)]$
				l			$[P4_{2}]$ – –	$[P4_{2}\;(77)]$		$[P4_{2}/m\;(84)]$	$[P4_{2}22\;(93)]$
				l	k		$[P4_{2}2_{1}]$ –				$[P4_{2}2_{1}2\;(94)]$
							$[P4_{1}]$ – –	$[\!\left\{\!\matrix{P4_{1}\; (76)\cr P4_{3}\; (78)\cr}\!\right\}]$ ^‡			$[\!\left\{\!\matrix{P4_{1}22\; (91)\cr P4_{3}22\; (95)\cr}\!\right\}]$ ^‡
					k		$[P4_{1}2_{1}]$ –				$[\!\left\{\!\matrix{P4_{1}2_{1}2\; (92)\cr P4_{3}2_{1}2\; (96)\cr}\!\right\}]$ ^‡
			l	l			P – – c					$[P4_{2}mc\;(105)]$	$[P\bar{4}2c\;(112)]$	$[P4_{2}/mmc\;(131)]$
			l	l	k		P– $[2_{1}c]$						$[P\bar{4}2_{1}c\;(114)]$
		k			k		P – b –					P4bm (100)	$[P\bar{4}b2\;(117)]$	$[P4/mbm\;(127)]$
		k	l	l	k		P – bc					$[P4_{2}bc\;(106)]$		$[P4_{2}/mbc\;(135)]$
		l		l			P – c –					$[P4_{2}cm\;(101)]$	$[P\bar{4}c2\;(116)]$	$[P4_{2}/mcm\;(132)]$
		l	l	l			P – cc					P4cc (103)		$[P4/mcc\;(124)]$
				l	k		P – n –					$[P4_{2}nm\;(102)]$	$[P\bar{4}n2\;(118)]$	$[P4_{2}/mnm\;(136)]$
			l	l	k		P – nc					P4nc (104)		$[P4/mnc\;(128)]$
					k		Pn – –			$[P4/n\;(85)]$				$[P4/nmm\;(129)]$
				l	k		$[P4_{2}/n]$ – –			$[P4_{2}/n\;(86)]$
			l	l	k		Pn – c							$[P4_{2}/nmc\;(137)]$
		k			k		Pnb –							$[P4/nbm\;(125)]$
		k	l	l	k		Pnbc							$[P4_{2}/nbc\;(133)]$
		l		l	k		Pnc –							$[P4_{2}/ncm\;(138)]$
		l	l	l	k		Pncc							$[P4/ncc\;(130)]$
				l	k		Pnn –							$[P4_{2}/nnm\;(134)]$
			l	l	k		Pnnc							$[P4/nnc\;(126)]$
			l	l	k		I – – –	I4 (79)	$[I\bar{4}\;(82)]$	$[I4/m\;(87)]$	I422 (97)	I4mm (107)	$[I\bar{4}2m\;(121)]$	$[I4/mmm\;(139)]$
													$[I\bar{4}m2\;(119)]$
			l		k		$[I4_{1}]$ – –	$[I4_{1}\;(80)]$			$[I4_{1}]$ 22 (98)
			^§		k	h	I – – d					$[I4_{1}md\;(109)]$	$[I\bar{4}2d\;(122)]$
		k, l	l	l	k		I – c –					I4cm (108)	$[I\bar{4}c2\;(120)]$	$[I4/mcm\;(140)]$
		k, l	^§		k	h	I – cd					$[I4_{1}cd\;(110)]$
	h, k		l		k		$[I4_{1}/a]$ – –			$[I4_{1}/a\;(88)]$
	h, k		^§		k	h	Ia – d							$[I4_{1}/amd\;(141)]$
	h, k	k, l	^§		k	h	Iacd							$[I4_{1}/acd\;(142)]$

TRIGONAL, Laue classes $[\bar{3}]$ and $[\bar{3}m]$

Reflection conditions				Extinction symbol	Laue class
Reflection conditions					$[\bar{3}]$		$[\matrix{\bar{3}m1\;(\bar{3}\;2/m\; 1)\hfill\cr \bar{3}m\hfill\cr}]$			$[\bar{3}1m\; (\bar{3}\; 1\; 2/m)]$
Hexagonal axes					Point group
hkil	$[h\bar{h}0l]$	$[hh\overline{2h}l]$	000l		3	$[\bar{3}]$	321	3m1	$[\bar{3}m1]$	312	31m	$[\bar{3}1m]$
hkil	$[h\bar{h}0l]$	$[hh\overline{2h}l]$	000l		3	$[\bar{3}]$	32	3m	$[\bar{3}m]$	312	31m	$[\bar{3}1m]$
				P – – –	P3 (143)	$[P\bar{3}\;(147)]$	P321 (150)	P3m1 (156)	$[P\bar{3}m1\;(164)]$	P312 (149)	P31m (157)	$[P\bar{3}1m\;(162)]$
				$[P3_{1}]$ – –	$[\!\left\{\!\matrix{P3_{1} (144)\cr P3_{2} (145)\cr}\!\right\}]$ ^‡		$[\!\left\{\!\matrix{P3_{1}21\;(152)\cr P3_{2}21\;(154)\cr}\!\right\}]$ ^‡			$[\!\left\{\!\matrix{P3_{1}12\;(151)\cr P3_{2}12\;(153)\cr}\!\right\}]$ ^‡
		l	l	P – – c							P31c (159)	$[P\bar{3}1c\;(163)]$
	l		l	P – c –				P3cl (158)	$[P\bar{3}c1\;(165)]$
				R(obv)– – ^¶	R3 (146)	$[R\bar{3}\;(148)]$	R32 (155)	R3m (160)	$[R\bar{3}m\;(166)]$
	$[h + l = 3n;\;l]$			R(obv)– c				R3c (161)	$[R\bar{3}c\;(167)]$
				R(rev)– –	R3 (146)	$[R\bar{3}\;(148)]$	R32 (155)	R3m (160)	$[R\bar{3}m\;(166)]$
	$[- h + l = 3n;\; l]$			R(rev)– c				R3c (161)	$[R\bar{3}c\;(167)]$
Rhombohedral axes				Extinction symbol	Point group
hkl		hhl	hhh	Extinction symbol	3	$[\bar{3}]$	32	3m	$[\bar{3}m]$
				R – –	R3(146)	$[R\bar{3}\;(148)]$	R32 (155)	R3m (160)	$[R\bar{3}m\;(166)]$
		l	h	R – c				R3c (161)	$[R\bar{3}c\;(167)]$

HEXAGONAL, Laue classes [6/m] and [6/mmm]

Reflection conditions			Extinction symbol	Laue class
							$[6/mmm\;(6/m\; 2/m\; 2/m)]$
				Point group
$[h\bar{h}0l]$	$[hh\overline{2h}l]$	000l		6	$[\bar{6}]$		622	6mm	$[\matrix{\bar{6}2m\cr \bar{6}m2\cr}]$
			P – – –	P6 (168)	$[P\bar{6}\;(174)]$	$[P6/m\;(175)]$	P622 (177)	P6mm (183)	$[P\bar{6}2m\;(189)]$	$[P6/mmm\;(191)]$
									$[P\bar{6}m2\;(187)]$
		l	$[P6_{3}]$ – –	$[P6_{3}\;(173)]$		$[P6_{3}/m\;(176)]$	$[P6_{3}22\;(182)]$
			$[P6_{2}]$ – –	$[\!\left\{\!\matrix{P6_{2}\; (171)\cr P6_{4}\; (172)\cr}\!\right\}]$ ^‡			$[\!\left\{\!\matrix{P6_{2}22\; (180)\cr P6_{4}22\; (181)\cr}\!\right\}]$ ^‡
			$[P6_{1}]$ – –	$[\!\left\{\!\matrix{P6_{1}\; (169)\cr P6_{3}\; (170)\cr}\!\right\}]$ ^‡			$[\!\left\{\!\matrix{P6_{1}22\; (178)\cr P6_{5}22\; (179)\cr}\!\right\}]$ ^‡
	l	l	P– – c					$[P6_{3}mc\;(186)]$	$[P\bar{6}2c\;(190)]$	$[P6_{3}/mmc\;(194)]$
l		l	P– c –					$[P6_{3}cm\;(185)]$	$[P\bar{6}c2\;(188)]$	$[P6_{3}/mcm\;(193)]$
l	l	l	P– cc					P6cc (184)		Pmcc (192)

CUBIC, Laue classes $[m\bar{3}]$ and $[m\bar{3}m]$

Reflection conditions (Indices are permutable, apart from space group No. 205)^††				Extinction symbol	Laue class
					$[m\bar{3}\; (2/m\; \bar{3})]$		$[m\bar{3}m\; (4/m\; \bar{3}\; 2/m)]$
					Point group
hkl	0kl	hhl	00l		23	$[m\bar{3}]$	432	$[\bar{4}3m]$	$[m\bar{3}m]$
				P – – –	P23 (195)	$[Pm\bar{3}\; (200)]$	P432 (207)	$[P\bar{4}3m\; (215)]$	$[Pm\bar{3}m\; (221)]$
			l	$[\!\left\{\!\matrix{P2_{1}\hbox{--}\;\hbox{--}\cr P4_{2}\hbox{--}\;\hbox{--}\cr}\right.]$	$[P2_{1}3\; (198)]$		$[P4_{2}32\; (208)]$
				$[P4_{1}]$ – –			$[\!\left\{\!\matrix{P4_{1}32\; (213)\cr P4_{3}32 \;(212)\cr}\!\right\}]$ ^‡
		l	l	P– –n				$[P\bar{4}3n\; (218)]$	$[Pm\bar{3}n\; (223)]$
	k^††		l	Pa – –		$[Pa\bar{3}\; (205)]$
			l	Pn – –		$[Pn\bar{3}\; (201)]$			$[Pn\bar{3}m\; (224)]$
		l	l	Pn–n					$[Pn\bar{3}n\; (222)]$
		l	l	I – – –	$[\left[\matrix{I23\ (197)\hfill\cr I2_{1}3\ (199)\cr}\right]]$ ^†	$[Im\bar{3}\; (204)]$	I432 (211)	$[I\bar{4}3m\; (217)]$	$[Im\bar{3}m\; (229)]$
		l		$[I4_{1}]$ – –			$[I4_{1}32\; (214)]$
				I– –d				$[l\bar{4}3d\; (220)]$
	k, l	l	l	Ia – –		$[Ia\bar{3}\; (206)]$
	k, l			Ia–d					$[Ia\bar{3}d\; (230)]$
	k, l		l	F – – –	F23 (196)	$[Fm\bar{3}\; (202)]$	F432 (209)	$[F\bar{4}3m\; (216)]$	$[Fm\bar{3}m\; (225)]$
	k, l			$[F4_{1}]$ – –			$[F4_{1}32\; (210)]$
	k, l	h, l	l	F– –c				$[F\bar{4}3c\; (219)]$	$[Fm\bar{3}c\; (226)]$
				Fd – –		$[Fd\bar{3}\;(203)]$			$[Fd\bar{3}m\; (227)]$
		h, l		Fd–c					$[Fd\bar{3}c\;(228)]$

^†Pair of space groups with common point group and symmetry elements but differing in the relative location of these elements.
^‡Pair of enantiomorphic space groups, cf. Section 3.1.5

.
^§Condition: $[2h + l = 4n{\rm ;}\;l]$ .
^¶For obverse and reverse settings cf. Section 1.2.1

. The obverse setting is standard in these tables. The transformation reverse $[\rightarrow]$ obverse is given by $[{\bf a}(\hbox{obv.}) = - {\bf a}(\hbox{rev.})]$ , $[{\bf b}(\hbox{obv.}) = - {\bf b}(\hbox{rev.})]$ , $[{\bf c}(\hbox{obv.}) = {\bf c}(\hbox{rev.})]$ .
^††For No. 205, only cyclic permutations are permitted. Conditions are 0kl: [k = 2n]

; h0l:

; hk0:

The left-hand side of the table contains the Reflection conditions. Conditions of the type [h = 2n] or [h + k = 2n] are abbreviated as h or [h + k] . Conditions like [h = 2n, k = 2n, h + k = 2n] are quoted as h, k; in this case, the condition is not listed as it follows directly from [h = 2n, k = 2n] . Conditions with [l = 3n] , [l = 4n] , [{l = 6n}] or more complicated expressions are listed explicitly.

From left to right, the table contains the integral, zonal and serial conditions. From top to bottom, the entries are ordered such that left columns are kept empty as long as possible. The leftmost column that contains an entry is considered as the `leading column'. In this column, entries are listed according to increasing complexity. This also holds for the subsequent columns within the restrictions imposed by previous columns on the left. The make-up of the table is such that observed reflection conditions should be matched against the table by considering, within each crystal system, the columns from left to right.

The centre column contains the Extinction symbol. To obtain the complete diffraction symbol, the Laue-class symbol has to be added in front of it. Be sure that the correct Laue-class symbol is used if the crystal system contains two Laue classes. Particular care is needed for Laue class $[\bar{3}m]$ in the trigonal system, because there are two possible orientations of this Laue symmetry with respect to the crystal lattice, $[\bar{3}m1]$ and $[\bar{3}1m]$ . The correct orientation can be obtained directly from the diffraction record.

The right-hand side of the table gives the Possible space groups which obey the reflection conditions. For crystal systems with two Laue classes, a subdivision is made according to the Laue symmetry. The entries in each Laue class are ordered according to their point groups. All space groups that match both the reflection conditions and the Laue symmetry, found in a diffraction experiment, are possible space groups of the crystal.

The space groups are given by their short Hermann–Mauguin symbols, followed by their number between parentheses, except for the monoclinic system, where full symbols are given (cf. Section 2.2.4 ). In the monoclinic and orthorhombic sections of Table 3.1.4.1, which contain entries for the different settings and cell choices, the `standard' space-group symbols (cf. Table 4.3.2.1 ) are printed in bold face. Only these standard representations are treated in full in the space-group tables.

Example

The diffraction pattern of a compound has Laue class mmm. The crystal system is thus orthorhombic. The diffraction spots are indexed such that the reflection conditions are [0kl: l = 2n] ; [h0l: h + l = 2n] ; [h00: h = 2n] ; [00l:l = 2n] . Table 3.1.4.1 shows that the diffraction symbol is mmmPcn–. Possible space groups are Pcn2 (30) and Pcnm (53). For neither space group does the axial choice correspond to that of the standard setting. For No. 30, the standard symbol is Pnc2, for No. 53 it is Pmna. The transformation from the basis vectors $[{\bf a}_{e}, {\bf b}_{e}, {\bf c}_{e}]$ , used in the experiment, to the basis vectors $[{\bf a}_{s}, {\bf b}_{s}, {\bf c}_{s}]$ of the standard setting is given by $[{\bf a}_{s} = {\bf b}_{e}, {\bf b}_{s} = -{\bf a}_{e}]$ for No. 30 and by $[{\bf a}_{s} = {\bf c}_{e}, {\bf c}_{s} = -{\bf a}_{e}]$ for No. 53.

Possible pitfalls

Errors in the space-group determination may occur because of several reasons.

(1) Twinning of the crystal

Difficulties that may be encountered are shown by the following example. Say that a monoclinic crystal (b unique) with the angle β fortuitously equal to $[\sim 90^{\circ}]$ is twinned according to (100). As this causes overlap of the reflections hkl and $[\bar{h}kl]$ , the observed Laue symmetry is mmm rather than . The same effect may occur within one crystal system. If, for instance, a crystal with Laue class is twinned according to (100) or (110), the Laue class is simulated (twinning by merohedry, cf. Catti & Ferraris, 1976, and Koch, 2004). Further examples are given by Buerger (1960). Errors due to twinning can often be detected from the fact that the observed reflection conditions do not match any of the diffraction symbols.
(2) Incorrect determination of reflection conditions

Either too many or too few conditions may be found. For serial reflections, the first case may arise if the structure is such that its projection on, say, the b direction shows pseudo-periodicity. If the pseudo-axis is , with p an integer, the reflections 0k0 with $[k \neq p]$ are very weak. If the exposure time is not long enough, they may be classified as unobserved which, incorrectly, would lead to the reflection condition . A similar situation may arise for zonal conditions, although in this case there is less danger of errors. Many more reflections are involved and the occurrence of pseudo-periodicity is less likely for two-dimensional than for one-dimensional projections.

For `structural' or non-space-group absences, see Section 2.2.13 .

The second case, too many observed reflections, may be due to multiple diffraction or to radiation impurity. A textbook description of multiple diffraction has been given by Lipson & Cochran (1966). A well known case of radiation impurity in X-ray diffraction is the contamination of a copper target with iron. On a photograph taken with the radiation from such a target, the iron radiation with $[\lambda \hbox{(Fe)} \sim 5/4\lambda \hbox{(Cu)}]$ gives a reflection spot $[4h_{\prime} 4k_{\prime} 4l]$ at the position $[5h_{\prime} 5k_{\prime} 5l]$ for copper $[[\lambda (\hbox{Cu}\; K\bar{\alpha})\! =\! 1.5418\;\hbox{\AA}]$ , $[\lambda (\hbox{Fe}\; K\bar{\alpha})\! =\! 1.9373\;\hbox{\AA}]]$ . For reflections 0k0, for instance, this may give rise to reflected intensity at the copper 050 position so that, incorrectly, the condition may be excluded.
(3) Incorrect assignment of the Laue symmetry

This may be caused by pseudo-symmetry or by `diffraction enhancement'. A crystal with pseudo-symmetry shows small deviations from a certain symmetry, and careful inspection of the diffraction pattern is necessary to determine the correct Laue class. In the case of diffraction enhancement, the symmetry of the diffraction pattern is higher than the Laue symmetry of the crystal. Structure types showing this phenomenon are rare and have to fulfil specified conditions. For further discussions and references, see Perez-Mato & Iglesias (1977).

References

Buerger, M. J. (1960). Crystal-structure analysis, Chap. 5. New York: Wiley.Google Scholar

Catti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray crystal structure determination. Acta Cryst. A32, 163–165.Google Scholar

Koch, E. (2004). Twinning. International Tables for Crystallography Vol. C, 3rd ed., edited by E. Prince, ch. 1.3. Dordrecht: Kluwer Academic Publishers.Google Scholar

Lipson, H. & Cochran, W. (1966). The determination of crystal structures, Chaps. 3 and 4.4. London: Bell.Google Scholar

Perez-Mato, J. M. & Iglesias, J. E. (1977). On simple and double diffraction enhancement of symmetry. Acta Cryst. A33, 466–474.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 3.1, pp. 45-46