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

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. Wilsona

aSt 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.

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[link]), Section 8.2.3[link] . They have, however, at least four applications in practical crystallography:

  • (1) in the classification of space groups (Section 1.4.2[link]);

  • (2) in forming symbols for certain space groups in higher dimensions (see Chapter 9.8[link] and the references cited therein);

  • (3) in modelling the frequency of occurrence of space groups (see Chapter 9.7[link] and the references cited therein); and

  • (4) in establishing `equivalent origins' (Wondratschek, 2005[link], Section 8.2.3[link] ).

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[link]) 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[link]). 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[link].

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[link].

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

Crystal systemCrystal classSpace group
GeometricArithmetic
NumberSymbolNumberSymbol
Oblique 1 1 1p 1 p1
2 2 2p 2 p2
Rectangular m 3 mp 3 pm
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
12 p4gm
Hexagonal 3 9 3p 13 p3
3m 10 3m1p 14 p3m1
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[link]), 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[link] and 1.4.2.1[link], and some statistics for the three-dimensional classes are given in Table 1.4.3.1[link]. 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[link], 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[link], Table 8.3.4.1[link]

Crystal systemCrystal classSpace group
GeometricArithmetic
NumberSymbolNumberSymbol
Triclinic 1 1 1P 1 P1
[\bar1] 2 [\bar1P] 2 [P\bar1]
Monoclinic 2 3 2P 3 [P2]
4 [P2_1]
4 2C 5 C2
m 5 mP 6 Pm
7 Pc
6 mC 8 Cm
9 Cc
2/m 7 2/mP 10 [P2/m]
11 [P2_1/m]
13 [P2/c]
14 [P2_1/c]
8 2/mC 12 [C2/m]
15 [C2/c]
Orthorhombic 222 9 222P 16 P222
17 [P222_1]
18 [P2_12_12]
19 [P2_12_12_1]
10 222C 20 [C222_1]
21 C222
11 222F 22 F222
12 222I 23 I222
24 [I2_12_12_1]
mm 13 mm2P 25 Pmm2
26 [Pmc2_1]
27 Pcc2
28 Pma2
29 [Pca2_1]
30 Pnc2
31 [Pmn2_1]
32 Pba2
33 [Pna2_1]
34 Pnn2
14 mm2C 35 Cmm2
36 [Cmc2_1]
37 Ccc2
15 2mmC
(Amm2)
38 C2mm
(Amm2)
39 [C2me]
(Aem2)
40 [C2cm]
[(Ama2)]
41 [C2ce]
(Aea2)
16 mm2F 42 Fmm2
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
70 Fddd
21 mmmI 71 Immm
72 Ibam
73 Ibca
74 Imma
Tetragonal 4 22 4P 75 P4
76 [P4_1]
77 [P4_2]
78 [P4_3]
23 4I 79 I4
80 [I4_1]
[\bar4] 24 [\bar4P] 81 [P\bar4]
25 [\bar4I] 82 [I\bar4]
4/m 26 4/mP 83 [P4/m]
84 [P4_2/m]
85 [P4/n]
86 [P4_2/n]
27 4/mI 87 [I4/m]
88 [I4_1/a]
422 28 422P 89 P422
90 [P42_12]
91 [P4_122]
92 [P4_12_12]
93 [P4_222]
94 [P4_22_12]
95 [P4_322]
96 [P4_32_12]
29 422I 97 [I422]
98 [I4_122]
4mm 30 4mmP 99 [P4mm]
100 [P4bm]
101 [P4_2cm]
102 [P4_2nm]
103 [P4cc]
104 [P4nc]
105 [P4_2mc]
106 [P4_2bc]
31 4mmI 107 [I4mm]
108 [I4cm]
109 [I4_1md]
110 [I4_1cd]
[\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]
120 [I\bar4c2]
35 [\bar42mI] 121 [I\bar42m]
122 [I\bar42d]
4/mmm 36 4/mmmP 123 [P4/mmm]
124 [P4/mcc]
125 [P4/nbm]
126 [P4/nnc]
127 [P4/mbm]
128 [P4/mnc]
129 [P4/nmm]
130 [P4/ncc]
131 [P4_2/mmc]
132 [P4_2/mcm]
133 [P4_2/nbc]
134 [P4_2/nnm]
135 [P4_2/mbc]
136 [P4_2/mnm]
137 [P4_2/nmc]
138 [P4_2/ncm]
37 4/mmmI 139 I4/mmm
140 I4/mcm
141 [I4_1/amd]
142 [I4_1/acd]
Trigonal 3 38 3P 143 P3
144 [P3_1]
145 [P3_2]
39 3R 146 R3
[\bar3] 40 [\bar3P] 147 [P\bar3]
41 [\bar3R] 148 [R\bar3]
32 42 312P 149 P312
151 [P3_112]
153 [P3_212]
43 321P 150 P321
152 [P3_121]
154 [P3_221]
44 32R 155 R32
3m 45 3m1P 156 [P3m1]
158 [P3c1]
46 31mP 157 [P31m]
159 [P31c]
47 3mR 160 [R3m]
161 [R3c]
[\bar3m] 48 [\bar31mP] 162 [P\bar31m]
163 [P\bar31c]
49 [\bar3m1P] 164 [P\bar3m1]
165 [P\bar3c1]
50 [\bar3mR] 166 [R\bar3m]
167 [R\bar3c]
Hexagonal 6 51 6P 168 P6
169 [P6_1]
170 [P6_5]
171 [P6_2]
172 [P6_4]
173 [P6_3]
[\bar6] 52 [\bar6P] 174 [P\bar6]
6/m 53 6/mP 175 [P6/m]
176 [P6_3/m]
622 54 622P 177 [P622]
178 [P6_122]
179 [P6_522]
180 [P6_222]
181 [P6_422]
182 [P6_322]
6mm 55 6mmP 183 P6mm
184 [P6cc]
185 [P6_3cm]
186 [P6_3mc]
[\bar6m] 56 [\bar6m2P] 187 [P\bar6m2]
188 [P\bar6c2]
57 [\bar62mP] 189 [P\bar62m]
190 [P\bar62c]
6/mmm 58 6/mmmP 191 [P6/mmm]
192 [P6/mmc]
193 [P6_3/mcm]
194 [P6_3/mmc]
Cubic 23 59 23P 195 P23
198 [P2_13]
60 23F 196 F23
61 23I 197 I23
199 [I2_13]
[m\bar3] 62 [m\bar3P] 200 [Pm\bar3]
201 [Pn\bar3]
205 [Pa\bar3]
63 [m\bar3F] 202 [Fm\bar3]
203 [Fd\bar3]
64 [m\bar3I] 204 [Im\bar3]
206 [Ia\bar3]
432 65 432P 207 P432
208 [P4_232]
213 [P4_132]
212 [P4_332]
66 432F 209 [F432]
210 [F4_132]
67 432I 211 I432
214 [I4_132]
[\bar43m] 68 [\bar43mP] 215 [P\bar43m]
218 [P\bar43n]
69 [\bar43mF] 216 [F\bar43m]
219 [F\bar43c]
70 [\bar43mI] 217 [I\bar43m]
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]
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 classSymbols of the arithmetic crystal classes
1 1P [\bar1P]        
2C          
222F          
[\bar4P] [\bar4I]        
3R [\bar3P] [\bar3R] 32R    
[\bar6P]          
23F          
2 2P mP mC [2/mC]    
222C 222I mm2F mmmF    
4I [4/mI] 422I [\bar4m2I] [\bar42mI]  
3P 312P 321P [3m1P] 31mP 3mR
[\bar31mP] [\bar3m1P] [\bar3mR]      
[6/mP] [\bar6m2P] [\bar62mP]      
23P 23I [m\bar3F] [m\bar3I] 432F 432I
[\bar43mP] [\bar43mF] [\bar43mI] [m\bar3mI]    
3 [mm2C] [mm2I]        
3P 312P 321P      
4P          
[m\bar3P] [432P]        
4 2/mP          
222P 2mmC [(=mm2A)] mmmI    
4P [4/mP] 4mmI [\bar42mP] [\bar4m2P] [4/mmmI]
6P 622P 6mmP [6/mmmP]    
432P [m\bar3mP] [m\bar3mF]      
6 mmmC          
422P          
6P 622P        
8 422P 4mmP        
10 mm2P          
16 mmmP          
4/mmmP          
Enantiomorphs combined.
Enantiomorphs distinguished.

The space groups in Table 1.4.2.1[link] 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[link], Section 8.3.4[link] . 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[link]) 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[link]); 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[link]), 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[link] .

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[link]). 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[link]); see also Wilson (1993[link]).

1.4.3.2. `Laue' symmetry

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Similarly, the eleven conventional `Laue' symmetries [International Tables for Crystallography (2005[link]), Volume A, Section 3.1.2[link] 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.
Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic groups of four-dimensional space. New York: Wiley.
Engel, P. (1986). Geometric crystallography. Dordrecht: Reidel. (Present distributor Kluwer Academic Publishers, Dordrecht.)
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.
International Tables for Crystallography (2005). Vol. A. Space-group symmetry, fifth ed., edited by Th. Hahn. Heidelberg: Springer.
Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North Holland.
Wilson, A. J. C. (1993). Laue and Patterson symmetry in the complex case. Z. Kristallogr. 208, 199–206.
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.
Wondratschek, H. (2005). Introduction to space-group symmetry. International tables for crystallography, Vol. A, edited by Th. Hahn, Part 8. Heidelberg: Springer.








































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