Arithmetic crystal classes and symmorphic space groups

Wilson, A. J. C.

doi:10.1107/97809553602060000575

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 1.4, pp. 15-22
https://doi.org/10.1107/97809553602060000575

Chapter 1.4. Arithmetic crystal classes and symmorphic space groups

A. J. C. Wilson^a ^‡

^a St John's College, Cambridge CB2 1TP, England

Arithmetic crystal classes have four main applications in practical crystallography: in the classification of space groups; in forming symbols for certain space groups in higher dimensions; in modelling the frequency of occurrence of space groups; and in establishing `equivalent origins'. Simple descriptions and complete enumerations of the arithmetic crystal classes in one, two and three dimensions are given in this chapter. Symmorphic space groups and the effect of dispersion on diffraction symmetry are also discussed.

Keywords: anomalous dispersion; arithmetic crystal classes; classification; crystal classes; dispersion; one-dimensional crystal classes; space groups; symmetry; symmorphism; three-dimensional crystal classes.

1.4.1. Arithmetic crystal classes

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Arithmetic crystal classes are of great importance in theoretical crystallography, and are treated from that point of view in Volume A of International Tables for Crystallography (2005 ), Section 8.2.3 . They have, however, at least four applications in practical crystallography:

(1) in the classification of space groups (Section 1.4.2 );
(2) in forming symbols for certain space groups in higher dimensions (see Chapter 9.8 and the references cited therein);
(3) in modelling the frequency of occurrence of space groups (see Chapter 9.7 and the references cited therein); and
(4) in establishing `equivalent origins ' (Wondratschek, 2005 , Section 8.2.3 ).

The tabulation of arithmetic crystal classes in Volume A is incomplete, and the relation of the notation used in complete tabulations found elsewhere (for example, in Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, 1978 ) to that of International Tables is not immediately obvious. Simple descriptions and complete enumerations of the arithmetic crystal classes in one , two and three dimensions are therefore given here.

1.4.1.1. Arithmetic crystal classes in three dimensions

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The 32 geometric crystal classes and the 14 Bravais lattices are familiar in three-dimensional crystallography. The three-dimensional arithmetic crystal classes are easily derived in an elementary fashion by enumerating the compatible combinations of geometric crystal class and Bravais lattice ; the symbol adopted by the International Union of Crystallography for an arithmetic crystal class is simply the juxtaposition of the symbol for the geometric crystal class and the symbol for the Bravais lattice (de Wolff et al., 1985 ). For example, in the monoclinic system the geometric crystal classes are 2, m, and 2/m, and the Bravais lattices are monoclinic P and monoclinic C. The six arithmetic crystal classes in the monoclinic system are thus 2P, 2C, mP, mC, 2/mP, and 2/mC. In certain cases (loosely, when the geometric crystal class and the Bravais lattice have unique directions that are not necessarily parallel), the crystal class and the lattice can be combined in two different orientations. The simplest example is the combination of the orthorhombic crystal class¹ mm with the end-centred lattice C. The intersection of the mirror planes of the crystal class defines one unique direction, the C centring of the lattice another. If these directions are placed parallel to one another, the arithmetic class mm2C is obtained; if they are placed perpendicular to one another, a different arithmetic class² 2mmC is obtained. The other combinations exhibiting this phenomenon are lattice P with geometric classes 32, 3m, $[\bar {3}m]$ , $[\bar {4}m]$ , and $[\bar {6}m]$ . By consideration of all possible combinations of geometric class and lattice, one obtains the 73 arithmetic classes listed in Table 1.4.2.1 .

1.4.1.2. Arithmetic crystal classes in one , two and higher dimensions

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In one dimension , there are two geometric crystal classes , 1 and m, and a single Bravais lattice , $[{\scr p}]$ . Two arithmetic crystal classes result, $[{\scr p}]$ and $[m{\scr p}]$ . In two dimensions, there are ten geometric crystal classes, and two Bravais lattices, p and c; 13 arithmetic crystal classes result. The two-dimensional geometric and arithmetic crystal classes are listed in Table 1.4.1.1 .

Table 1.4.1.1| top | pdf |
The two-dimensional arithmetic crystal classes

Crystal system	Crystal class			Space group
	Geometric	Arithmetic		Space group
	Geometric	Number	Symbol	Number	Symbol
Oblique	1	1	1p	1	p1
Oblique	2	2	2p	2	p2
Rectangular	m	3	mp	3	pm
		3	mp	4	pg
		4	mc	5	cm
	2mm	5	2mmp	6	p2mm
				7	p2mg
				8	p2gg
		6	2mmc	9	c2mm
Square	4	7	4p	10	p4
	4mm	8	4mmp	11	p4mm
	4mm	8	4mmp	12	p4gm
Hexagonal	3	9	3p	13	p3
	3m	10	3m1p	14	p3m1
	3m	11	31mp	15	p31m
	6	12	6p	16	p6
	6mm	13	6mmp	17	p6mm

The number of arithmetic crystal classes increases rapidly with increasing dimensionality; there are 710 (plus 70 enantiomorphs) in four dimensions (Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, 1978 ), but those in dimensions higher than three are not needed in this volume.

1.4.2. Classification of space groups

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Arithmetic crystal classes may be used to classify space groups on a scale somewhat finer than that given by the geometric crystal classes. Space groups are members of the same arithmetic crystal class if they belong to the same geometric crystal class, have the same Bravais lattice , and (when relevant) have the same orientation of the lattice relative to the point group . Each one-dimensional arithmetic crystal class contains a single space group, symbolized by $[{\scr p}1]$ and $[{\scr p}m]$ , respectively. Most two-dimensional arithmetic crystal classes contain only a single space group; only 2mmp has as many as three.

The space groups belonging to each geometric and arithmetic crystal class in two and three dimensions are indicated in Tables 1.4.1.1 and 1.4.2.1 , and some statistics for the three-dimensional classes are given in Table 1.4.3.1 . 12 three-dimensional classes contain only a single space group, whereas two contain 16 each. Certain arithmetic crystal classes (3P, 312P, 321P, 422P, 6P, 622P, 432P) contain enantiomorphous pairs of space groups , so that the number of members of these classes depends on whether the enantiomorphs are combined or distinguished. Such classes occur twice in Table 1.4.3.1 , as indicated by the footnotes.

Table 1.4.2.1| top | pdf |
The three-dimensional space groups, arranged by arithmetic crystal class ; in a few geometric crystal classes this differs somewhat from the conventional numerical order; see International Tables Volume A , Table 8.3.4.1

Crystal system	Crystal class			Space group
	Geometric	Arithmetic		Space group
	Geometric	Number	Symbol	Number	Symbol
Triclinic	1	1	1P	1	P1
Triclinic	$[\bar1]$	2	$[\bar1P]$	2	$[P\bar1]$
Monoclinic	2	3	2P	3
		3	2P	4
		4	2C	5	C2
	m	5	mP	6	Pm
		5	mP	7	Pc
		6	mC	8	Cm
		6	mC	9	Cc
	2/m	7	2/mP	10
				11
				13
				14
		8	2/mC	12
		8	2/mC	15
Orthorhombic	222	9	222P	16	P222
				17
				18
				19
		10	222C	20
		10	222C	21	C222
		11	222F	22	F222
		12	222I	23	I222
		12	222I	24
	mm	13	mm2P	25	Pmm2
				26
				27	Pcc2
				28	Pma2
				29
				30	Pnc2
				31
				32	Pba2
				33
				34	Pnn2
		14	mm2C	35	Cmm2
				36
				37	Ccc2
		15	2mmC (Amm2)	38	C2mm (Amm2)
				39	(Aem2)
				40
				41	(Aea2)
		16	mm2F	42	Fmm2
		16	mm2F	43	Fdd2
		17	mm2I	44	Imm2
				45	Iba2
				46	Ima2
	mmm	18	mmmP	47	Pmmm
				48	Pnnn
				49	Pccm
				50	Pban
				51	Pmma
				52	Pnna
				53	Pmna
				54	Pcca
				55	Pbam
				56	Pccn
				57	Pbcm
				58	Pnnm
				59	Pmmn
				60	Pbcn
				61	Pbca
				62	Pnma
		19	mmmC	63	Cmcm
				64	Cmce
				65	Cmmm
				66	Cccm
				67	Cmme
				68	Ccce
		20	mmmF	69	Fmmm
		20	mmmF	70	Fddd
		21	mmmI	71	Immm
				72	Ibam
				73	Ibca
				74	Imma
Tetragonal	4	22	4P	75	P4
				76
				77
				78
		23	4I	79	I4
		23	4I	80
	$[\bar4]$	24	$[\bar4P]$	81	$[P\bar4]$
	$[\bar4]$	25	$[\bar4I]$	82	$[I\bar4]$
	4/m	26	4/mP	83
				84
				85
				86
		27	4/mI	87
		27	4/mI	88
	422	28	422P	89	P422
				90
				91
				92
				93
				94
				95
				96
		29	422I	97
		29	422I	98
	4mm	30	4mmP	99
				100
				101
				102
				103
				104
				105
				106
		31	4mmI	107
				108
				109
				110
	$[\bar4m]$	32	$[\bar42mP]$	111	$[P\bar42m]$
				112	$[P\bar42c]$
				113	$[P\bar42_1m]$
				114	$[P\bar42_1c]$
		33	$[\bar4m2P]$	115	$[P\bar4m2]$
				116	$[P\bar4c2]$
				117	$[P\bar4b2]$
				118	$[P\bar4n2]$
		34	$[\bar4m2I]$	119	$[I\bar4m2]$
		34	$[\bar4m2I]$	120	$[I\bar4c2]$
		35	$[\bar42mI]$	121	$[I\bar42m]$
		35	$[\bar42mI]$	122	$[I\bar42d]$
	4/mmm	36	4/mmmP	123
				124
				125
				126
				127
				128
				129
				130
				131
				132
				133
				134
				135
				136
				137
				138
37		4/mmmI	139	I4/mmm
			140	I4/mcm
			141
			142
Trigonal	3	38	3P	143	P3
				144
				145
		39	3R	146	R3
	$[\bar3]$	40	$[\bar3P]$	147	$[P\bar3]$
	$[\bar3]$	41	$[\bar3R]$	148	$[R\bar3]$
	32	42	312P	149	P312
				151
				153
		43	321P	150	P321
				152
				154
		44	32R	155	R32
	3m	45	3m1P	156
		45	3m1P	158
		46	31mP	157
		46	31mP	159
		47	3mR	160
		47	3mR	161
	$[\bar3m]$	48	$[\bar31mP]$	162	$[P\bar31m]$
		48	$[\bar31mP]$	163	$[P\bar31c]$
		49	$[\bar3m1P]$	164	$[P\bar3m1]$
		49	$[\bar3m1P]$	165	$[P\bar3c1]$
		50	$[\bar3mR]$	166	$[R\bar3m]$
		50	$[\bar3mR]$	167	$[R\bar3c]$
Hexagonal	6	51	6P	168	P6
				169
				170
				171
				172
				173
	$[\bar6]$	52	$[\bar6P]$	174	$[P\bar6]$
	6/m	53	6/mP	175
	6/m	53	6/mP	176
	622	54	622P	177
				178
				179
				180
				181
				182
	6mm	55	6mmP	183	P6mm
				184
				185
				186
	$[\bar6m]$	56	$[\bar6m2P]$	187	$[P\bar6m2]$
		56	$[\bar6m2P]$	188	$[P\bar6c2]$
		57	$[\bar62mP]$	189	$[P\bar62m]$
		57	$[\bar62mP]$	190	$[P\bar62c]$
	6/mmm	58	6/mmmP	191
				192
				193
				194
Cubic	23	59	23P	195	P23
		59	23P	198
		60	23F	196	F23
		61	23I	197	I23
		61	23I	199
	$[m\bar3]$	62	$[m\bar3P]$	200	$[Pm\bar3]$
				201	$[Pn\bar3]$
				205	$[Pa\bar3]$
		63	$[m\bar3F]$	202	$[Fm\bar3]$
		63	$[m\bar3F]$	203	$[Fd\bar3]$
		64	$[m\bar3I]$	204	$[Im\bar3]$
		64	$[m\bar3I]$	206	$[Ia\bar3]$
	432	65	432P	207	P432
				208
				213
				212
		66	432F	209
		66	432F	210
		67	432I	211	I432
		67	432I	214
	$[\bar43m]$	68	$[\bar43mP]$	215	$[P\bar43m]$
		68	$[\bar43mP]$	218	$[P\bar43n]$
		69	$[\bar43mF]$	216	$[F\bar43m]$
		69	$[\bar43mF]$	219	$[F\bar43c]$
		70	$[\bar43mI]$	217	$[I\bar43m]$
		70	$[\bar43mI]$	220	$[I\bar43d]$
	$[m\bar3m]$	71	$[m\bar3mP]$	221	$[Pm\bar3m]$
				222	$[Pn\bar3n]$
				223	$[Pm\bar3n]$
				224	$[Pn\bar3m]$
		72	$[m\bar3mF]$	225	$[Fm\bar3m]$
				226	$[Fm\bar3c]$
				227	$[Fd\bar3m]$
				228	$[Fd\bar3c]$
		73	$[m\bar3mI]$	229	$[Im\bar3m]$
		73	$[m\bar3mI]$	230	$[Ia\bar3d]$

Table 1.4.3.1| top | pdf |
Arithmetic crystal classes classified by the number of space groups that they contain

Number of space groups in the class	Symbols of the arithmetic crystal classes
1	1P	$[\bar1P]$
	2C
	222F
	$[\bar4P]$	$[\bar4I]$
	3R	$[\bar3P]$	$[\bar3R]$	32R
	$[\bar6P]$
	23F
2	2P	mP	mC
	222C	222I	mm2F	mmmF
	4I		422I	$[\bar4m2I]$	$[\bar42mI]$
	3P^†	312P^†	321P^†		31mP	3mR
	$[\bar31mP]$	$[\bar3m1P]$	$[\bar3mR]$
		$[\bar6m2P]$	$[\bar62mP]$
	23P	23I	$[m\bar3F]$	$[m\bar3I]$	432F	432I
	$[\bar43mP]$	$[\bar43mF]$	$[\bar43mI]$	$[m\bar3mI]$
3
	3P^‡	312P^‡	321P^‡
	4P^†
	$[m\bar3P]$	^†
4	2/mP
	222P	2mmC		mmmI
	4P^‡		4mmI	$[\bar42mP]$	$[\bar4m2P]$
	6P^†	622P^†	6mmP
	432P^‡	$[m\bar3mP]$	$[m\bar3mF]$
6	mmmC
	422P^†
	6P^‡	622P^‡
8	422P^‡	4mmP
10	mm2P
16	mmmP
16	4/mmmP

^†Enantiomorphs combined.
^‡Enantiomorphs distinguished.

The space groups in Table 1.4.2.1 are listed in the order of the arithmetic crystal class to which they belong. It will be noticed that arrangement according to the conventional space-group numbering would separate members of the same arithmetic crystal class in the geometric classes 2/m, 3m, 23, $[m{\bar 3}]$ , 432, and $[{\bar 4}3m]$ . This point is discussed in detail in Volume A of International Tables, Section 8.3.4 . The symbols of five space groups [ [C2me] (Aem2), C2ce (Aea2), Cmce, Cmme, Ccce] have been conformed to those recommended in the fourth, revised edition of Volume A of International Tables.

1.4.2.1. Symmorphic space groups

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The 73 space groups known as `symmorphic' are in one-to-one correspondence with the arithmetic crystal classes, and their standard `short' symbols (Bertaut, 2005 ) are obtained by interchanging the order of the geometric crystal class and the Bravais cell in the symbol for the arithmetic space group. In fact, conventional crystallographic symbolism did not distinguish between arithmetic crystal classes and symmorphic space groups until recently (de Wolff et al., 1985 ); the symbol of the symmorphic group was used also for the arithmetic class.

This relationship between the symbols, and the equivalent rule-of-thumb symmorphic space groups are those whose standard (short) symbols do not contain glide planes or screw axes, reveal nothing fundamental about the nature of symmorphism; they are simply a consequence of the conventions governing the construction of symbols in International Tables for Crystallography.³

Although the standard symbols of the symmorphic space groups do not contain screw axes or glide planes, this is a result of the manner in which the space-group symbols have been devised. Most symmorphic space groups do in fact contain screw axes and/or glide planes. This is immediately obvious for the symmorphic space groups based on centred cells; C2 contains equal numbers of diad rotation axes and diad screw axes, and Cm contains equal numbers of reflection planes and glide planes. This is recognized in the `extended' space-group symbols (Bertaut, 2005 ), but these are clumsy and not commonly used; those for C2 and Cm are $[C1^{ 2}_{ 2_{1}}\!1]$ and $[C1^{m}_{ a}1]$ , respectively. In the more symmetric crystal systems, even symmorphic space groups with primitive cells contain screw axes and/or glide planes; [P422] ( $[P42^{ 2}_{2_{1}}]$ ) contains many diad screw axes and P4/mmm ( $[P4/m2/m^{ 2/m}_{ 2_{1}/g}]$ ) contains both screw axes and glide planes.

The balance of symmetry elements within the symmorphic space groups is discussed in more detail in Subsection 9.7.1.2 .

1.4.3. Effect of dispersion on diffraction symmetry

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In the absence of dispersion (`anomalous scattering'), the intensities of the reflections hkl and $[\bar {h}\bar {k}{\bar l}]$ are equal (Friedel's law), and statements about the symmetry of the weighted reciprocal lattice and quantities derived from it often rest on the tacit or explicit assumption of this law – the condition underlying it being forgotten. In particular, if dispersion is appreciable, the symmetry of the Patterson synthesis and the `Laue' symmetry are altered.

1.4.3.1. Symmetry of the Patterson function

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In Volume A of International Tables, the symmetry of the Patterson synthesis is derived in two stages. First, any glide planes and screw axes are replaced by mirror planes and the corresponding rotation axes, giving a symmorphic space group (Subsection 1.4.2.1 ). Second, a centre of symmetry is added. This second step involves the tacit assumption of Friedel's law, and should not be taken if any atomic scattering factors have appreciable imaginary components. In such cases, the symmetry of the Patterson synthesis will not be that of one of the 24 centrosymmetric symmorphic space groups, as given in Volume A, but will be that of the symmorphic space group belonging to the arithmetic crystal class to which the space group of the structure belongs. There are thus 73 possible Patterson symmetries.

An equivalent description of such symmetries, in terms of 73 of the 1651 dichromatic colour groups, has been given by Fischer & Knop (1987 ); see also Wilson (1993 ).

1.4.3.2. `Laue' symmetry

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Similarly, the eleven conventional `Laue' symmetries [International Tables for Crystallography (2005 ), Volume A, Section 3.1.2 and elsewhere] involve the explicit assumption of Friedel's law. If dispersion is appreciable, the `Laue' symmetry may be that of any of the 32 point groups. The point group, in correct orientation, is obtained by dropping the Bravais-lattice symbol from the symbol of the arithmetic crystal class or of the Patterson symmetry.

References

Bertaut, E. F. (2005). Synoptic tables of space-group symbols. International tables for crystallography, Vol. A, edited by Th. Hahn, Part 4. Heidelberg: Springer.Google Scholar

Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of four-dimensional space. New York: Wiley.Google Scholar

Engel, P. (1986). Geometric crystallography. Dordrecht: Reidel. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar

Fischer, K. F. & Knop, W. E. (1987). Space groups for imaginary Patterson and for difference Patterson functions in the lambda technique. Z. Kristallogr. 180, 237–242.Google Scholar

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, fifth ed., edited by Th. Hahn. Heidelberg: Springer.Google Scholar

Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North Holland.Google Scholar

Wilson, A. J. C. (1993). Laue and Patterson symmetry in the complex case. Z. Kristallogr. 208, 199–206.Google Scholar

Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Acta Cryst. A41, 278–280.Google Scholar

Wondratschek, H. (2005). Introduction to space-group symmetry. International tables for crystallography, Vol. A, edited by Th. Hahn, Part 8. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 1.4, pp. 15-22
https://doi.org/10.1107/97809553602060000575

Chapter 1.4. Arithmetic crystal classes and symmorphic space groups

1.4.1. Arithmetic crystal classes

1.4.1.1. Arithmetic crystal classes in three dimensions

1.4.1.2. Arithmetic crystal classes in one, two and higher dimensions

1.4.2. Classification of space groups

1.4.2.1. Symmorphic space groups

1.4.3. Effect of dispersion on diffraction symmetry

1.4.3.1. Symmetry of the Patterson function

1.4.3.2. `Laue' symmetry

References

1.4.1.2. Arithmetic crystal classes in one , two and higher dimensions