International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 152-154

## Section 2.1.3.5. Patterson symmetry

Th. Hahna and A. Looijenga-Vosb
H. D. Flackd

#### 2.1.3.5. Patterson symmetry

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The entry Patterson symmetry in the headline gives the symmetry of the vector set' generated by the operation of the space group on an arbitrary set of general positions. More prosaically, it may be described as the symmetry of the set of the interatomic vectors of a crystal structure with the selected space group. The Patterson symmetry is a crystallographic space group denoted by its Hermann–Mauguin symbol. It is in fact one of the 24 centrosymmetric symmorphic space groups (see Section 1.3.3.3 ) in three dimensions and one of 7 in two dimensions. For each of the 230 space groups, the Patterson symmetry has the same Bravais-lattice type as the space group itself and its point group is the lowest-index centrosymmetric supergroup of the point group of the space group. The point-group part' of the symbol of the Patterson symmetry represents the Laue class to which the plane group or space group belongs (cf. Table 2.1.2.1). By way of examples: space group No. 100, P4bm, has a Bravais lattice of type tP and point group 4mm. The centrosymmetric supergroup of 4mm (see Fig. 3.2.1.3 ) is 4/mmm, so the Patterson symmetry is P4/mmm; space group No. 66, Cccm, has a Bravais lattice of type oC and point group mmm. This point group is centrosymmetric, so the Patterson symmetry is Cmmm.

Note: For the four space groups Amm2 (38), Aem2 (39), Ama2 (40) and Aea2 (41), the standard symbol for their Patterson symmetry, Cmmm, is added (between parentheses) after the actual symbol Ammm in the space-group tables.

The Patterson symmetry is intimately related to the symmetry of the Patterson function (see Flack, 2015). The latter, P|F|2(uvw), is the inverse Fourier transform of the squared structure-factor amplitudes. Patterson functions possess the crystallographic symmetry of the symmorphic space-group representative of the arithmetic crystal class (see Section 1.3.4.4.1 ) to which the space group belongs. Table 2.1.3.3 lists these crystallographic symmetries of the Patterson function and the Patterson symmetries for the space groups and plane groups. However, further symmetry is also present, as desribed below for the three common forms of the Patterson function:

• (a) P|F|2(uvw): The most general form of the Patterson function, the complex P|F|2(uvw), is the complex Fourier transform of |F(hkl)|2. The full symmetry of P|F|2(uvw) can be described in terms of the 1651 two-colour (Shubnikov) space groups (Fischer & Knof, 1987; Wilson, 1993; Shubnikov & Belov, 1964; cf. also Chapter 3.6 ). The real and imaginary parts of P|F|2(uvw) have the same symmetry as PA(uvw) and PD(uvw), respectively, described below. P|F|2(uvw) is real for centrosymmetric space groups and noncentrosymmetric ones in the absence of any resonant-scattering contribution.

 Table 2.1.3.3| top | pdf | Patterson symmetries and symmetries of Patterson functions for space groups and plane groups
 The space-group types of each row form an arithmetic crystal class. (In three instances the row is typeset on two lines.) The arithmetic crystal class is identified by its representative symmorphic space group for which both the Hermann–Mauguin symbol and the space-group-type number are shown in bold. A set of space groups with sequential numbers is indicated by the symbols of the first and last space group of the sequence separated by a dash. The column Patterson symmetry' indicates the symmetry of the set of interatomic vectors of crystal structures described in the space groups given in the column Space-group types'. The Patterson symmetry is also the symmetry of PA(uvw) and the real part of the complex P|F|2(uvw). The Patterson symmetry is given in the headline of each space-group table in Chapter 2.3. The crystallographic symmetry of both PD(uvw) and the imaginary part of the complex P|F|2(uvw) is that of the symmorphic space group of crystal structures described in the space groups given in the column `Space-group types'. To this crystallographic symmetry, the noncrystallographic operation of a centre of antisymmetry needs to be added to give . The full symmetry of PD(uvw) is not shown in this volume. The setting and origin choice of the chosen space group should also be used for the space group of the Patterson symmetry and the symmorphic space group. Similar remarks apply to the plane groups listed in part (b) of the table. (a) Space groups.
Space-group typesPatterson symmetry
Hermann–Mauguin symbolsNos.
Crystal family triclinic (anorthic), Bravais-lattice type aP
P1 1
2
Crystal family monoclinic, Bravais-lattice type mP
P2P21 3–4 P2/m
PmPc 6–7 P2/m
P2/mP21/m, 10–11 P2/m
P2/cP21/c 13–14 P2/m
Crystal family monoclinic, Bravais-lattice type mS
C2 5 C2/m
CmCc 8–9 C2/m
C2/m, C2/c 12, 15 C2/m
Crystal family orthorhombic, Bravais-lattice type oP
P222P212121 16–19 Pmmm
Pmm2Pnn2 25–34 Pmmm
PmmmPnma 47–62 Pmmm
Crystal family orthorhombic, Bravais-lattice type oS
C2221, C222 20, 21 Cmmm
Cmm2Ccc2 35–37 Cmmm
Amm2Aea2 38–41 Ammm
CmcmCmce, Cmmm, 63–64, 65 Cmmm
CccmCcce 66–68 Cmmm
Crystal family orthorhombic, Bravais-lattice type oF
F222 22 Fmmm
Fmm2Fdd2 42–43 Fmmm
FmmmFddd 69–70 Fmmm
Crystal family orthorhombic, Bravais-lattice type oI
I222I212121 23–24 Immm
Imm2Ima2 44–46 Immm
ImmmImma 71–74 Immm
Crystal family tetragonal, Bravais-lattice type tP
P4P43 75–78 P4/m
81 P4/m
P4/mP42/n 83–86 P4/m
P422P43212 89–96 P4/mmm
P4mmP42bc 99–106 P4/mmm
111–114 P4/mmm
115–118 P4/mmm
P4/mmmP42/ncm 123–138 P4/mmm
Crystal family tetragonal, Bravais-lattice type tI
I4, I41 79–80 I4/m
82 I4/m
I4/mI41/a 87–88 I4/m
I422I4122 97–98 I4/mmm
I4mmI41cd 107–110 I4/mmm
119–120 I4/mmm
121–122 I4/mmm
I4/mmmI41/acd 139–142 I4/mmm
Crystal family hexagonal, Bravais-lattice type hP
P3P32 143–145
147
P312, P3112, P3212 149, 151, 153
P321, P3121, P3221 150, 152, 154
P3m1, P3c1 156, 158
P31m, P31c 157, 159
162–163
164–165
P6P63 168–173 P6/m
174 P6/m
P6/mP63/m 175–176 P6/m
P622P6322 177–182 P6/mmm
P6mmP63mc 183–186 P6/mmm
187–188 P6/mmm
189–190 P6/mmm
P6/mmmP63/mmc 191–194 P6/mmm
Crystal family hexagonal, Bravais-lattice type hR
R3 146
148
R32 155
R3mR3c 160–161
166–167
Crystal family cubic, Bravais-lattice type cP
P23, P213 195, 198
, 200–201, 205
P432P4232, 207–208,
P4332–P4132 212–213
, 215, 218
221–224
Crystal family cubic, Bravais-lattice type cF
F23 196
202–203
F432F4132 209–210
216, 219
225–228
Crystal family cubic, Bravais-lattice type cI
I23, I213 197, 199
, 204, 206
I432, I4132 211, 214
, 217, 220
229–230
 (b) Plane groups.
Plane-group typesPatterson symmetry
Hermann–Mauguin symbolsNos.
Crystal family oblique (monoclinic), Bravais-lattice type mp
p1 1 p2
p2 2 p2
Crystal family rectangular (orthorhombic), Bravais-lattice type op
pmpg 3–4 p2mm
p2mmp2gg 6–8 p2mm
Crystal family rectangular (orthorhombic), Bravais-lattice type oc
cm 5 c2mm
c2mm 9 c2mm
Crystal family square (tetragonal), Bravais-lattice type tp
p4 10 p4
p4mmp4gm 11–12 p4mm
Crystal family hexagonal, Bravais-lattice type hp
p3 13 p6
p3m1 14 p6mm
p31m 15 p6mm
p6 16 p6
p6mm 17 p6mm
• (b) PA(uvw): , the average of the squared structure-factor amplitudes of the pair of Friedel opposites hkl and . Real PA(uvw) is the real cosine Fourier transform of A(hkl). The symmetry of PA(uvw) is generated by the symmorphic space-group representative of the arithmetic crystal class to which the space group belongs, combined with a centre of symmetry (inversion centre), i.e. . PA(uvw) thus possesses the Patterson symmetry of the space group whether the latter is centrosymmetric or not, and whether there is any resonant-scattering contribution or not (see Table 2.1.3.3).

• (c) PD(uvw): , the difference of the squared structure-factor amplitudes of the pair of Friedel opposites hkl and . Real PD(uvw) is the real sine Fourier transform of D(hkl). The crystallographic symmetry of PD(uvw) is that of the symmorphic space-group representative of the arithmetic crystal class to which the space group belongs (see Table 2.1.3.3). The full symmetry of PD(uvw) is given by the type-III black-and-white space group generated by the appropriate symmorphic space group and the centre of antisymmetry, (Fischer & Knof, 1987). Thus PD(uvw) does not possess the Patterson symmetry.

### References

Fischer, K. F. & Knof, W. E. (1987). Space groups for imaginary Patterson and for difference Patterson functions used in the lambda technique. Z. Kristallogr. 180, 237–242.
Flack, H. D. (2015). Patterson functions. Z. Kristallogr. 230, 743–748.
Shubnikov, A. V. & Belov, N. V. (1964). Coloured Symmetry, pp. 198–210. Oxford: Pergamon Press.
Wilson, A. J. C. (1993). Laue and Patterson symmetry in the complex case. Z. Kristallogr. 208, 199–206.