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[link] ) 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[link]). 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[link] ) 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[link]). 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[link] ) to which the space group belongs. Table 2.1.3.3[link] 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[link]; Wilson, 1993[link]; Shubnikov & Belov, 1964[link]; cf. also Chapter 3.6[link] ). 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 [P_D(u v w) = -P_D(\bar u\bar v\bar w)]. 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 [P\bar 1]
    [{\bi P}\bar{\bf 1}] 2 [P\bar 1]
    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
    [{\bi P}\bar{\bf 4}] 81 P4/m
    P4/mP42/n 83–86 P4/m
    P422P43212 89–96 P4/mmm
    P4mmP42bc 99–106 P4/mmm
    [{\bi P}\bar{\bf 4}{\bf 2}{\bi m}][P\bar 42_1c] 111–114 P4/mmm
    [{\bi P}\bar{\bf 4}{\bi m}{\bf 2}][P\bar 4n2] 115–118 P4/mmm
    P4/mmmP42/ncm 123–138 P4/mmm
    Crystal family tetragonal, Bravais-lattice type tI
    I4, I41 79–80 I4/m
    [{\bi I}\bar{\bf 4}] 82 I4/m
    I4/mI41/a 87–88 I4/m
    I422I4122 97–98 I4/mmm
    I4mmI41cd 107–110 I4/mmm
    [{\bi I}\bar{\bf 4}{\bi m}{\bf 2}][I\bar 4c2] 119–120 I4/mmm
    [{\bi I}\bar{\bf 4}{\bf 2}{\bi m}][I\bar 42d] 121–122 I4/mmm
    I4/mmmI41/acd 139–142 I4/mmm
    Crystal family hexagonal, Bravais-lattice type hP
    P3P32 143–145 [P\bar 3]
    [{\bi P}\bar{\bf 3}] 147 [P\bar 3]
    P312, P3112, P3212 149, 151, 153 [P\bar 31m]
    P321, P3121, P3221 150, 152, 154 [P\bar 3m1]
    P3m1, P3c1 156, 158 [P\bar 3m1]
    P31m, P31c 157, 159 [P\bar 31m]
    [{\bi P}\bar{\bf 3}{\bf 1}{\bi m}][P\bar 31c] 162–163 [P\bar 31m]
    [{\bi P}\bar{\bf 3}{\bi m}{\bf 1}][P\bar 3c1] 164–165 [P\bar 3m1]
    P6P63 168–173 P6/m
    [{\bi P}\bar{\bf 6}] 174 P6/m
    P6/mP63/m 175–176 P6/m
    P622P6322 177–182 P6/mmm
    P6mmP63mc 183–186 P6/mmm
    [{\bi P}\bar{\bf 6}{\bi m}{\bf 2}][P\bar 6c2] 187–188 P6/mmm
    [{\bi P}\bar{\bf 6}{\bf 2}{\bi m}][P\bar 62c] 189–190 P6/mmm
    P6/mmmP63/mmc 191–194 P6/mmm
    Crystal family hexagonal, Bravais-lattice type hR
    R3 146 [R\bar 3]
    [{\bi R}\bar{\bf 3}] 148 [R\bar 3]
    R32 155 [R\bar 3m]
    R3mR3c 160–161 [R\bar 3m]
    [{\bi R}\bar{\bf 3}{\bi m}][R\bar 3c] 166–167 [R\bar 3m]
    Crystal family cubic, Bravais-lattice type cP
    P23, P213 195, 198 [Pm\bar 3]
    [{\bi Pm}\bar{\bf 3}][Pn\bar 3], [Pa\bar 3] 200–201, 205 [Pm\bar 3]
    P432P4232, 207–208, [Pm\bar 3m]
     P4332–P4132 212–213 [Pm\bar 3m]
    [{\bi P}\bar{\bf 4}{\bf 3}{\bi m}], [P\bar 43n] 215, 218 [Pm\bar 3m]
    [{\bi Pm}\bar{\bf 3}{\bi m}][Pn\bar 3 m] 221–224 [Pm\bar 3m]
    Crystal family cubic, Bravais-lattice type cF
    F23 196 [Fm\bar 3]
    [{\bi Fm}\bar{\bf 3}][Fd\bar 3] 202–203 [Fm\bar 3]
    F432F4132 209–210 [Fm\bar 3m]
    [{\bi F}\bar{\bf 4}{\bf 3}{\bi m}][F\bar 43c] 216, 219 [Fm\bar 3m]
    [{\bi Fm}\bar{\bf 3}{\bi m}][Fd\bar 3c] 225–228 [Fm\bar 3m]
    Crystal family cubic, Bravais-lattice type cI
    I23, I213 197, 199 [Im\bar 3]
    [{\bi Im}\bar{\bf 3}], [Ia\bar 3] 204, 206 [Im\bar 3]
    I432, I4132 211, 214 [Im\bar 3m]
    [{\bi I}\bar{\bf 4}{\bf 3}{\bi m}], [I\bar 43d] 217, 220 [Im\bar 3m]
    [{\bi Im}\bar{\bf 3}{\bi m}][Ia\bar 3d] 229–230 [Im\bar 3m]

    (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): [A(hkl) = {\textstyle{1\over 2}}[|F(hkl)|^2 + |F(\bar h\bar k\bar l)|^2]], the average of the squared structure-factor amplitudes of the pair of Friedel opposites hkl and [\bar h \bar k \bar l]. 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. [P_A(uvw)=P_A(\bar u\bar v\bar w)]. 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[link]).

  • (c) PD(uvw): [D(hkl) = |F(hkl)|^2 - |F(\bar h \bar k \bar l)|^2], the difference of the squared structure-factor amplitudes of the pair of Friedel opposites hkl and [\bar h \bar k \bar l]. 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[link]). 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, [P_D(uvw) = -P_D(\bar u \bar v \bar w)] (Fischer & Knof, 1987[link]). 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.








































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