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

International Tables for Crystallography (2016). Vol. A, ch. 1.6, p. 114

Section 1.6.4. Tables of reflection conditions and possible space groups

H. D. Flackb and U. Shmuelia

1.6.4. Tables of reflection conditions and possible space groups

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1.6.4.1. Introduction

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The primary order of presentation of these tables of reflection conditions of space groups is the Bravais lattice. This order has been chosen because cell reduction on unit-cell dimensions leads to the Bravais lattice as described as stage 1 in Section 1.6.2.1[link]. Within the space groups of a given Bravais lattice, the entries are arranged by Laue class, which may be obtained as described as stage 2 in Section 1.6.2.1[link]. As a consequence of these decisions about the way the tables are structured, in the hexagonal family one finds for the Bravais lattice hP that the Laue classes [\overline{3}], [\overline{3}m1], [\overline{3}1m], 6/m and 6/mmm are grouped together.

As an aid in the study of naturally occurring macromolecules and compounds made by enantioselective synthesis, the space groups of enantiomerically pure compounds (Sohncke space groups) are typeset in bold.

The tables show, on the left, sets of reflection conditions and, on the right, those space groups that are compatible with the given set of reflection conditions. The reflection conditions, e.g. h or k + l, are to be understood as [h=2n] or [k+l=2n], respectively. All of the space groups in each table correspond to the same Patterson symmetry, which is indicated in the table header. This makes for easy comparison with the entries for the individual space groups in Chapter 2.3[link] of this volume, in which the Patterson symmetry is also very clearly shown. All space groups with a conventional choice of unit cell are included in Tables 1.6.4.2–1.6.4.30. All alternative settings displayed in Chapter 2.3[link] are thus included. The following further alternative settings, not displayed in Chapter 2.3[link] , are also included: space group [Pb\overline{3}] (205) and all the space groups with an hR Bravais lattice in the reverse setting with hexagonal axes.

Table 1.6.2.1[link] gives some relevant statistics drawn from Tables 1.6.4.2–1.6.4.30. The total number of space-group settings mentioned in these tables is 416. This number is considerably larger than the 230 space-group types described in Part 2 of this volume. The following example shows why the tables include data for several descriptions of the space-group types. At the stage of space-group determination for a crystal in the crystal class mm2, it is not yet known whether the twofold rotation axis lies along a, b or c. Consequently, space groups based on the three point groups 2mm, m2m and mm2 need to be considered.

In some texts dealing with space-group determination, a `diffraction symbol' (sometimes also called an `extinction symbol') in the form of a Hermann–Mauguin space-group symbol is used as a shorthand code for the reflection conditions and Laue class. These symbols were introduced by Buerger (1935[link], 1942[link], 1969[link]) and a concise description is to be found in Looijenga-Vos & Buerger (2002)[link]. Nespolo et al. (2014)[link] use them.

1.6.4.2. Examples of the use of the tables

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  • (1) If the Bravais lattice is oI and the Laue class is mmm, Table 1.6.4.1[link] directs us to Table 1.6.4.11[link]. Given the observed reflection conditions[\eqalign{&hkl{:}\ h+k+l=2n,\quad\!\! 0kl{:}\ k=2n, l=2n,\quad\!\! h0l{:}\ h+l=2n,\cr &hk0{:}\ h+k=2n,\quad\!\! h00{:}\ h=2n,\quad\!\! 0k0{:}\ k=2n,\hfill \quad\!\! 00l{:}\ l=2n,}]it is seen from Table 1.6.4.11[link] that the possible settings of the space groups are: Ibm2 (46), Ic2m (46), Ibmm (74) and Icmm (74).

    Table 1.6.4.1| top | pdf |
    Summary of Tables 1.6.4.2–1.6.4.30

    Table No.Bravais latticeLaue classPatterson symmetryComment
    1.6.4.2[link] aP [\overline{1}] [P\overline{1}]  
    1.6.4.3[link] mP 2/m [P12/m1] Unique b
    1.6.4.4[link] mS (mC, mA, mI) 2/m [C12/m1], [A12/m1], [I12/m1] Unique b
    1.6.4.5[link] mP 2/m [P112/m] Unique c
    1.6.4.6[link] mS (mA, mB, mI) 2/m [A112/m], [B112/m], [I112/m] Unique c
    1.6.4.7[link] oP mmm Pmmm  
    1.6.4.8[link] oS (oC) mmm Cmmm  
    1.6.4.9[link] oS (oB) mmm Bmmm  
    1.6.4.10[link] oS (oA) mmm Ammm  
    1.6.4.11[link] oI mmm Immm  
    1.6.4.12[link] oF mmm Fmmm  
    1.6.4.13[link] tP [4/m] [P4/m]  
    1.6.4.14[link] tP [4/mmm] [P4/mmm]  
    1.6.4.15[link] tI [4/m] [I4/m]  
    1.6.4.16[link] tI [4/mmm] [I4/mmm]  
    1.6.4.17[link] hP [\overline{3}] [P\overline{3}]  
    1.6.4.18[link] hP [\overline{3}1m] and [\overline{3}m1] [P\overline{3}1m] and [P\overline{3}m1]  
    1.6.4.19[link] hP [6/m] [P6/m]  
    1.6.4.20[link] hP [6/mmm] [P6/mmm]  
    1.6.4.21[link] hR [\overline{3}] [R\overline{3}] Hexagonal axes
    1.6.4.22[link] hR [\overline{3}m] [R\overline{3}m] Hexagonal axes
    1.6.4.23[link] hR [\overline{3}] [R\overline{3}] Rhombohedral axes
    1.6.4.24[link] hR [\overline{3}m] [R\overline{3}m] Rhombohedral axes
    1.6.4.25[link] cP [m\overline{3}] [Pm\overline{3}]  
    1.6.4.26[link] cP [m\overline{3}m] [Pm\overline{3}m]  
    1.6.4.27[link] cI [m\overline{3}] [Im\overline{3}]  
    1.6.4.28[link] cI [m\overline{3}m] [Im\overline{3}m]  
    1.6.4.29[link] cF [m\overline{3}] [Fm\overline{3}]  
    1.6.4.30[link] cF [m\overline{3}m] [Fm\overline{3}m]  
  • (2) If the Bravais lattice is oP and the Laue class is mmm, Table 1.6.4.1[link] directs us to Table 1.6.4.7[link]. If there are no conditions on 0kl, the space groups P222 to Pmnn should be searched. If the condition is [0kl{:}\ k=2n] or [l=2n], the space groups Pbm2 to Pcnn should be searched. If the condition is [0kl{:}\ k+l= 2n], the space groups [Pnm2_{1}] to Pnnn should be searched.

  • (3) If the Bravais lattice is cP and the Laue class is [m\overline{3}], Table 1.6.4.1[link] directs us to Table 1.6.4.25[link]. If the conditions are [0kl{:}\ k=2n] and [h00{:}\ h=2n], it is readily seen that the space group is [Pa\overline{3}].

  • (4) If only the Bravais lattice is known or assumed, which is the case in powder-diffraction work (see Section 1.6.5.3[link]), all tables of this section corresponding to this Bravais lattice need to be consulted. For example, if it is known that the Bravais lattice is of type cP, Table 1.6.4.1[link] tells us that the possible Laue classes are [m\overline{3}] and [m\overline{3}m], and the possible space groups can be found in Tables 1.6.4.25[link] and 1.6.4.26[link], respectively. The appropriate reflection conditions are of course given in these tables. All relevant tables can thus be located with the aid of Table 1.6.4.1[link] if the Bravais lattice is known.[link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link][link]

    Table 1.6.4.2| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice aP and Laue class [\overline{1}]; Patterson symmetry [P\overline{1}]

    Reflection conditionsSpace groupNo.Space groupNo.
      P1 1 [P\overline{1}] 2

    Table 1.6.4.3| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice [mP] and Laue class 2/m; (monoclinic, unique axis b); Patterson symmetry [P12/m1]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    h0l0klhk00k0h0000l
                P2 3 Pm 6 [P2/m] 10
                           
          k     P21 4 [P2_{1}/m] 11    
                           
    h       h   Pa 7 [P2/a] 13    
                           
    h     k h   [P2_{1}/a] 14        
                           
    l         l Pc 7 [P2/c] 13    
                           
    l     k   l [P2_{1}/c] 14        
                           
    h + l       h l Pn 7 [P2/n] 13    
                           
    h + l     k h l [P2_{1}/n] 14        

    Table 1.6.4.4| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice mS (mC, mA, mI) and Laue class 2/m (monoclinic, unique axis b); Patterson symmetry [C12/m1], [A12/m1], [I12/m1]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklh0l0klhk00k0h0000l
    h + k h k h + k k h   C2 5 Cm 8 [C2/m] 12
                             
    h + k h, l k h + k k h l Cc 9 [C2/c] 15    
                             
    k + l l k + l k k   l A2 5 Am 8 [A2/m] 12
                             
    k + l h, l k + l k k h l An 9 [A2/n] 15    
                             
    h + k + l h + l k + l h + k k h l I2 5 Im 8 [I2/m] 12
                             
    h + k + l h, l k + l h + k k h l Ia 9 [I2/a] 15    

    Table 1.6.4.5| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice mP and Laue class 2/m (monoclinic, unique axis c); Patterson symmetry [P112/m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    h0l0klhk00k0h0000l
                P2 3 Pm 6 [P2/m] 10
                           
              l P21 4 [P2_{1}/m] 11    
                           
        h   h   Pa 7 [P2/a] 13    
                           
        h   h l [P2_{1}/a] 14        
                           
        k k     Pb 7 [P2/b] 13    
                           
        k k   l [P2_{1}/b] 14        
                           
        h + k k h   Pn 7 [P2/n] 13    
                           
        h + k k h l [P2_{1}/n] 14        

    Table 1.6.4.6| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice mS (mA, mB, mI) and Laue class 2/m (monoclinic, unique axis c); Patterson symmetry [A112/m], [B112/m1], [I112/m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklh0l0klhk00k0h0000l
    k + l l k + l k k   l A2 5 Am 8 [A2/m] 12
                             
    k + l l k + l h, k k h l Aa 9 [A2/a] 15    
                             
    h + l h + l l h   h l B2 5 Bm 8 [B2/m] 12
                             
    h + l h + l l h, k k h l Bn 9 [B2/n] 15    
                             
    h + k + l h + l k + l h + k k h l I2 5 Im 8 [I2/m] 12
                             
    h + k + l h + l k + l h, k k h l Ib 9 [I2/b] 15    

    Table 1.6.4.7| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oP and Laue class mmm; Patterson symmetry [Pmmm]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    0klh0lhk0h000k000l
                P222 16 [Pmm2] 25 [Pm2m] 25
                [P2mm] 25 Pmmm 47    
                           
              l P2221 17        
                           
            k   P2212 17        
                           
            k l P22121 18        
                           
          h     P2122 17        
                           
          h   l P21221 18        
                           
          h k   P21212 18        
                           
          h k l P212121 19        
                           
        h h     [P2_{1}ma] 26 [Pm2a] 28 Pmma 51
                           
        k   k   [Pm2_{1}b] 26 [P2mb] 28 Pmmb 51
                           
        h + k h k   [Pm2_{1}n] 31 [P2_{1}mn] 31 Pmmn 59
                           
      h   h     [P2_{1}am] 26 [Pma2] 28 Pmam 51
                           
      h h h     [P2aa] 27 Pmaa 49    
                           
      h k h k   [P2_{1}ab] 29 Pmab 57    
                           
      h h + k h k   [P2an] 30 Pman 53    
                           
      l       l [Pmc2_{1}] 26 [P2cm] 28 Pmcm 51
                           
      l h h   l [P2_{1}ca] 29 Pmca 57    
                           
      l k   k l [P2cb] 32 Pmcb 55    
                           
      l h + k h k l [P2_{1}cn] 33 Pmcn 62    
                           
      h + l   h   l [Pmn2_{1}] 31 [P2_{1}nm] 31 Pmnm 59
                           
      h + l h h   l [P2na] 30 Pmna 53    
                           
      h + l k h k l [P2_{1}nb] 33 Pmnb 62    
                           
      h + l h + k h k l [P2nn] 34 Pmnn 58    
                           
    k       k   [Pb2_{1}m] 26 [Pbm2] 28 Pbmm 51
                           
    k   h h k   [Pb2_{1}a] 29 Pbma 57    
                           
    k   k   k   [Pb2b] 27 Pbmb 49    
                           
    k   h + k h k   [Pb2n] 30 Pbmn 53    
                           
    k h   h k   [Pba2] 32 Pbam 55    
                           
    k h h h k   Pbaa 54        
                           
    k h k h k   Pbab 54        
                           
    k h h + k h k   Pban 50        
                           
    k l     k l [Pbc2_{1}] 29 Pbcm 57    
                           
    k l h h k l Pbca 61        
                           
    k l k   k l Pbcb 54        
                           
    k l h + k h k l Pbcn 60        
                           
    k h + l   h k l [Pbn2_{1}] 33 Pbnm 62    
                           
    k h + l h h k l Pbna 60        
                           
    k h + l k h k l Pbnb 56        
                           
    k h + l h + k h k l Pbnn 52        
                           
    l         l [Pcm2_{1}] 26 [Pc2m] 28 Pcmm 51
                           
    l   h h   l [Pc2a] 32 Pcma 55    
                           
    l   k   k l [Pc2_{1}b] 29 Pcmb 57    
                           
    l   h + k h k l [Pc2_{1}n] 33 Pcmn 62    
                           
    l h   h   l [Pca2_{1}] 29 Pcam 57    
                           
    l h h h   l Pcaa 54        
                           
    l h k h k l Pcab 61        
                           
    l h h + k h k l Pcan 60        
                           
    l l       l [Pcc2] 27 Pccm 49    
                           
    l l h h   l Pcca 54        
                           
    l l k   k l Pccb 54        
                           
    l l h + k h k l Pccn 56        
                           
    l h + l   h   l [Pcn2] 30 Pcnm 53    
                           
    l h + l h h   l Pcna 50        
                           
    l h + l k h k l [Pcnb] 60        
                           
    l h + l h + k h k l Pcnn 52        
                           
    k + l       k l [Pnm2_{1}] 31 [Pn2_{1}m] 31 Pnmm 59
                           
    k + l   h h k l [Pn2_{1}a] 33 Pnma 62    
                           
    k + l   k   k l [Pn2b] 30 Pnmb 53    
                           
    k + l   h + k h k l [Pn2n] 34 Pnmn 58    
                           
    k + l h   h k l [Pna2_{1}] 33 Pnam 62    
                           
    k + l h h h k l Pnaa 56        
                           
    k + l h k h k l Pnab 60        
                           
    k + l h h + k h k l Pnan 52        
                           
    k + l l     k l [Pnc2] 30 Pncm 53    
                           
    k + l l h h k l Pnca 60        
                           
    k + l l k   k l Pncb 50        
                           
    k + l l h + k h k l Pncn 52        
                           
    k + l h + l   h k l [Pnn2] 34 Pnnm 58    
                           
    k + l h + l h h k l Pnna 52        
                           
    k + l h + l k h k l Pnnb 52        
                           
    k + l h + l h + k h k l Pnnn 48        

    Table 1.6.4.8| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oS (oC setting) and Laue class mmm; Patterson symmetry Cmmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    h + k k h h + k h k   C222 21 [Cmm2] 35 [Cm2m] 38
                  [C2mm] 38 Cmmm 65    
                             
    h + k k h h + k h k l C2221 20        
                             
    h + k k h h, k h k   [Cm2e] 39 [C2me] 39 Cmme 67
                             
    h + k k h, l h + k h k l [Cmc2_{1}] 36 [C2cm] 40 Cmcm 63
                             
    h + k k h, l h, k h k l [C2ce] 41 Cmce 64    
                             
    h + k k, l h h + k h k l [Ccm2_{1}] 36 [Cc2m] 40 Ccmm 63
                             
    h + k k, l h h, k h k l [Cc2e] 41 Ccme 64    
                             
    h + k k, l h, l h + k h k l [Ccc2] 37 Cccm 66    
                             
    h + k k, l h, l h, k h k l Ccce 68        

    Table 1.6.4.9| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oS (oB setting) and Laue class mmm; Patterson symmetry Bmmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    h + l l h + l h h   l B222 21 [Bm2m] 35 [Bmm2] 38
                  [B2mm] 38 Bmmm 65    
                             
    h + l l h + l h h k l B2212 20        
                             
    h + l l h + l h, k h k l [Bm2_{1}b] 36 [B2mb] 40 Bmmb 63
                             
    h + l l h, l h h   l [Bme2] 39 [B2em] 39 Bmem 67
                             
    h + l l h, l h, k h k l [B2eb] 41 Bmeb 64    
                             
    h + l k, l h + l h h k l [Bb2_{1}m] 36 [Bbm2] 40 Bbmm 63
                             
    h + l k, l h + l h, k h k l [Bb2b] 37 Bbmb 66    
                             
    h + l k, l h, l h h k l [Bbe2] 41 Bbem 64    
                             
    h + l k, l h, l h, k h k l Bbeb 68        

    Table 1.6.4.10| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oS (oA setting) and Laue class mmm; Patterson symmetry Ammm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    k + l k + l l k   k l A222 21 [A2mm] 35 [Am2m] 38
                  [Amm2] 38 Ammm 65    
                             
    k + l k + l l k h k l A2122 20        
                             
    k + l k + l l h, k h k l [A2_{1}ma] 36 [Am2a] 40 Amma 63
                             
    k + l k + l h, l k h k l [A2_{1}am] 36 [Ama2] 40 Amam 63
                             
    k + l k + l h, l h, k h k l [A2aa] 37 Amaa 66    
                             
    k + l k, l l k   k l [Aem2] 39 [Ae2m] 39 Aemm 67
                             
    k + l k, l l h, k h k l [Ae2a] 41 Aema 64    
                             
    k + l k, l h, l k h k l [Aea2] 41 Aeam 64    
                             
    k + l k, l h, l h, k h k l Aeaa 68        

    Table 1.6.4.11| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oI and Laue class mmm; Patterson symmetry Immm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0klh0lhk0h000k000l
    h + k + l k + l h + l h + k h k l I222 23 I212121 24 [Imm2] 44
                  [Im2m] 44 [I2mm] 44 Immm 71
                             
    h + k + l k + l h + l h, k h k l [Im2a] 46 [I2mb] 46 Imma 74
                  [Immb] 74        
                             
    h + k + l k + l h, l h + k h k l [Ima2] 46 [I2cm] 46 Imam 74
                  Imcm 74        
                             
    h + k + l k + l h, l h, k h k l [I2cb] 45 [Imcb] 72    
                             
    h + k + l k, l h + l h + k h k l [Ibm2] 46 [Ic2m] 46 Ibmm 74
                  Icmm 74        
                             
    h + k + l k, l h + l h, k h k l [Ic2a] 45 Icma 72    
                             
    h + k + l k, l h, l h + k h k l [Iba2] 45 Ibam 72    
                             
    h + k + l k, l h, l h, k h k l Ibca 73 Icab 73    

    Table 1.6.4.12| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice oF and Laue class mmm; Patterson symmetry Fmmm

    Reflection conditionsSpace groupNo.
    hkl0klh0lhk0h000k000l
    h + k, h + l, k + l k, l h, l h, k h k l F222 22
                  [Fmm2] 42
                  [Fm2m] 42
                  [F2mm] 42
                  Fmmm 69
                     
    h + k, h + l, k + l k, l [h+l=4n;h,l] [h+k=4n;h,k] [h=4n] [k=4n] [l=4n] [F2dd] 43
                     
    h + k, h + l, k + l [k+l=4n;k,l] h, l [h+k=4n;h,k] [h=4n] [k=4n] [l=4n] [Fd2d] 43
                     
    h + k, h + l, k + l [k+l=4n;k,l] [h+l=4n;h,l] h, k [h=4n] [k=4n] [l=4n] [Fdd2] 43
                     
    h + k, h + l, k + l [k+l=4n;k,l] [h+l=4n;h,l] [h+k=4n;h,k] [h=4n] [k=4n] [l=4n] Fddd 70

    Table 1.6.4.13| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tP and Laue class 4/m; hk are permutable; Patterson symmetry P4/m

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hk00kl[h\,{\pm} h\, l]00lh00
              P4 75 [P\overline{4}] 81 [P4/m] 83
                         
          l   P42 77 [P4_{2}/m] 84    
                         
          [l=4n]   P41 76 P43 78    
                         
    h + k       h [P4/n] 85        
                         
    h + k     l h [P4_{2}/n] 86        

    Table 1.6.4.14| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tP and Laue class 4/mmm; hk are permutable; Patterson symmetry P4/mmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hk00kl[h\,{\pm} h\, l]00lh00
              P422 89 [P4mm] 99 [P\overline{4}2m] 111
              [P\overline{4}m2] 115 [P4/mmm] 123    
                         
            h P4212 90 [P\overline{4}2_{1}m] 113    
                         
          l   P4222 93        
                         
          l h P42212 94        
                         
          [l=4n]   P4122 91 P4322 95    
                         
          [l=4n] h P41212 92 P43212 96    
                         
        l l   [P4_{2}mc] 105 [P\overline{4}2c] 112 [P4_{2}/mmc] 131
                         
        l l h [P\overline{4}2_{1}c] 114        
                         
      k     h [P4bm] 100 [P\overline{4}b2] 117 [P4/mbm] 127
                         
      k l l h [P4_{2}bc] 106 [P4_{2}/mbc] 135    
                         
      l   l   [P4_{2}cm] 101 [P\overline{4}c2] 116 [P4_{2}/mcm] 132
                         
      l l l   [P4cc] 103 [P4/mcc] 124    
                         
      k + l   l h [P4_{2}nm] 102 [P\overline{4}n2] 118 [P4_{2}/mnm] 136
                         
      k + l l l h [P4nc] 104 [P4/mnc] 128    
                         
    h + k       h [P4/nmm] 129        
                         
    h + k   l l h [P4_{2}/nmc] 137        
                         
    h + k k     h [P4/nbm] 125        
                         
    h + k k l l h [P4_{2}/nbc] 133        
                         
    h + k l   l h [P4_{2}/ncm] 138        
                         
    h + k l l l h [P4/ncc] 130        
                         
    h + k k + l   l h [P4_{2}/nnm] 134        
                         
    h + k k + l l l h [P4/nnc] 126        

    Table 1.6.4.15| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tI and Laue class 4/m; hk are permutable; Patterson symmetry I4/m

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklhk00kl[h\,{\pm} h\, l]00lh00[h\,{\pm} h\, 0]
    h + k + l h + k k + l l l h   I4 79 [I\overline{4}] 82 [I4/m] 87
                             
    h + k + l h + k k + l l [l=4n] h   I41 80        
                             
    h + k + l h, k k + l l [l=4n] h h [I4_{1}/a] 88        

    Table 1.6.4.16| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice tI and Laue class 4/mmm; hk are permutable; Patterson symmetry I4/mmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hklhk00kl[h\,{\pm} h\, l]00lh00[h\,{\pm} h\, 0]
    h + k + l h + k k + l l l h   I422 97 [I4mm] 107 [I\overline{4}m2] 119
                  [I\overline{4}2m] 121 [I4/mmm] 139    
                             
    h + k + l h + k k + l l [l=4n] h   I4122 98        
                             
    h + k + l h + k k + l [2h+l=4n] [l=4n] h h [I4_{1}md] 109 [I\overline{4}2d] 122    
                             
    h + k + l h + k k, l l l h   [I4cm] 108 [I\overline{4}c2] 120 [I4/mcm] 140
                             
    h + k + l h + k k, l [2h+l=4n] [l=4n] h h [I4_{1}cd] 110        
                             
    h + k + l h, k k + l [2h+l=4n] [l=4n] h h [I4_{1}/amd] 141        
                             
    h + k + l h, k k, l [2h+l=4n] [l=4n] h h [I4_{1}/acd] 142        

    Table 1.6.4.17| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue class [\overline{3}]; hki are permutable; Patterson symmetry [P\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    [hh\overline{2h}l][h\overline{h}0l][000l]
          P3 143 [P\overline{3}] 147
                 
        [l=3n] P31 144 P32 145

    Table 1.6.4.18| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue classes [\overline{3}1m] and [\overline{3}m1]; hki are permutable; Patterson symmetry [P\overline{3}1m] and [P\overline{3}m1]

    Reflection conditionsClass [\overline{3}1m]Class [\overline{3}m1]
    [hh\overline{2h}l][h\overline{h}0l][000l]Space groupNo.Space groupNo.
          P312 149 P321 150
          [P31m] 157 [P3m1] 156
          [P\overline{3}1m] 162 [P\overline{3}m1] 164
                 
        [l=3n] P3112 151 P3121 152
          P3212 153 P3221 154
                 
    l   l [P31c] 159    
          [P\overline{3}1c] 163    
                 
      l l     [P3c1] 158
              [P\overline{3}c1] 165

    Table 1.6.4.19| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue class 6/m; hki are permutable; Patterson symmetry P6/m

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    [hh\overline{2h}l][h\overline{h}0l][000l]
          P6 168 [P\overline{6}] 174 [P6/m] 175
                     
        l P63 173 [P6_{3}/m] 176    
                     
        [l=3n] P62 171 P64 172    
                     
        [l=6n] P61 169 P65 170    

    Table 1.6.4.20| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hP and Laue class 6/mmm; hki are permutable; Patterson symmetry P6/mmm

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    [hh\overline{2h}l][h\overline{h}0l][000l]
          P622 177 [P6mm] 183 [P\overline{6}m2] 187
          [P\overline{6}2m] 189 [P6/mmm] 191    
                     
        l P6322 182        
                     
        [l=3n] P6222 180 P6422 181    
                     
        [l=6n] P6122 178 P6522 179    
                     
    l   l [P6_{3}mc] 186 [P\overline{6}2c] 190 [P6_{3}/mmc] 194
                     
      l l [P6_{3}cm] 185 [P\overline{6}c2] 188 [P6_{3}/mcm] 193
                     
    l l l [P6cc] 184 [P6 /mcc] 192    

    Table 1.6.4.21| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}] (hexagonal axes); hki are permutable; Patterson symmetry [R\overline{3}]; Ov = obverse setting; Rv = reverse setting

    Reflection conditionsSpace groupNo.Space groupNo. 
    [hkil][hki0][hh\overline{2h}l][h\overline{h}0l][000l][h\overline{h}00]
    [-h+k+l=3n] [-h+k=3n] [l=3n] [h+l=3n] [l=3n] [h=3n] R3 146 [R\overline{3}] 148 Ov
                         
    [h-k+l=3n] [h-k=3n] [l=3n] [-h+l=3n] [l=3n] [h=3n] R3 146 [R\overline{3}] 148 Rv

    Table 1.6.4.22| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}m] (hexagonal axes); hki are permutable; Patterson symmetry [R\overline{3}m]; Ov = obverse setting; Rv = reverse setting

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo. 
    [hkil][hki0][hh\overline{2h}l][h\overline{h}0l][000l][h\overline{h}00]
    [-h+k+l=3n] [-h+k=3n] [l=3n] [h+l=3n] [l=3n] [h=3n] R32 155 [R3m] 160 [R\overline{3}m] 166 Ov
                             
    [-h+k+l=3n] [-h+k=3n] [l=3n] [h+l=3n], [l=2m] [l=6n] [h=3n] [R3c] 161 [R\overline{3}c] 167     Ov
                             
    [h-k+l=3n] [h-k=3n] [l=3n] [-h+l=3n] [l=3n] [h=3n] R32 155 [R3m] 160 [R\overline{3}m] 166 Rv
                             
    [h-k+l=3n] [h-k=3n] [l=3n] [-h+l=3n,l=2m] [l=6n] [h=3n] [R3c] 161 [R\overline{3}c] 167     Rv

    Table 1.6.4.23| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}] (rhombohedral axes); hkl are permutable; Patterson symmetry [R\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    [hhl][hhh]
        R3 146 [R\overline{3}] 148

    Table 1.6.4.24| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice hR and Laue class [\overline{3}m] (rhombohedral axes); hkl are permutable; Patterson symmetry [R\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    [hhl][hhh]
        R32 155 [R3m] 160 [R\overline{3}m] 166
                   
    l h [R3c] 161 [R\overline{3}c] 167    

    Table 1.6.4.25| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cP and Laue class [m\overline{3}]; hkl are cyclically permutable; Patterson symmetry [Pm\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    0kl[h\,{\pm} h\, l]h00
          P23 195 [Pm\overline{3}] 200
                 
        h P213 198    
                 
    k   h [Pa\overline{3}] 205    
                 
    l   h [Pb\overline{3}] 205    
                 
    k + l   h [Pn\overline{3}] 201    

    Table 1.6.4.26| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cP and Laue class [m\overline{3}m]; hkl are permutable; Patterson symmetry [Pm\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    0kl[h\,{\pm} h\, l]h00
          P432 207 [P\overline{4}3m] 215 [Pm\overline{3}m] 221
                     
        h P4232 208        
                     
        [h=4n] P4332 212 P4132 213    
                     
      l h [P\overline{4}3n] 218 [Pm\overline{3}n] 223    
                     
    k + l   h [Pn\overline{3}m] 224        
                     
    k + l l h [Pn\overline{3}n] 222        

    Table 1.6.4.27| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cI and Laue class [m\overline{3}]; hkl are cyclically permutable; Patterson symmetry [Im\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0kl[h\,{\pm} h\, l]h00
    h + k + l k + l l h I23 197 I213 199 [Im\overline{3}] 204
                       
    h + k + l k, l l h [Ia\overline{3}] 206        

    Table 1.6.4.28| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cI and Laue class [m\overline{3}m]; hkl are permutable; Patterson symmetry [Im\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0kl[h\,{\pm} h\, l]h00
    h + k + l k + l l h I432 211 [I\overline{4}3m] 217 [Im\overline{3}m] 229
                       
    h + k + l k + l l [h=4n] I4132 214        
                       
    h + k + l k + l [2h+l=4n] [h=4n] [I\overline{4}3d] 220        
                       
    h + k + l k, l [2h+l=4n] [h=4n] [Ia\overline{3}d] 230        

    Table 1.6.4.29| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cF and Laue class [m\overline{3}]; hkl are cyclically permutable; Patterson symmetry [Fm\overline{3}]

    Reflection conditionsSpace groupNo.Space groupNo.
    hkl0kl[h\,{\pm} h\, l]h00
    [h+k,h+l,k+l] k, l h + l h F23 196 [Fm\overline{3}] 202
                   
    [h+k,h+l,k+l] [k+l=4n;k,l] h + l [h=4n] [Fd\overline{3}] 203    

    Table 1.6.4.30| top | pdf |
    Reflection conditions and possible space groups with Bravais lattice cF and Laue class [m\overline{3}m]; hkl are permutable; Patterson symmetry [Fm\overline{3}m]

    Reflection conditionsSpace groupNo.Space groupNo.Space groupNo.
    hkl0kl[h\pm h l]h00
    [h+k,h+l,k+l] k, l h + l h F432 [209] [F\overline{4}3m] 216 [Fm\overline{3}m] 225
                       
    [h+k,h+l,k+l] k, l h + l [h=4n] F4132 210        
                       
    [h+k,h+l,k+l] k, l h, l h [F\overline{4}3c] 219 [Fm\overline{3}c] 226    
                       
    [h+k,h+l,k+l] [k+l=4n;k,l] h + l [h=4n] [Fd\overline{3}m] 227        
                       
    [h+k,h+l,k+l] [k+l=4n;k,l] h, l [h=4n] [Fd\overline{3}c] 228        

References

Buerger, M. J. (1935). The application of plane groups to the interpretation of Weissenberg photographs. Z. Kristallogr. 91, 255–289.
Buerger, M. J. (1942). X-ray Crystallography, ch. 22. New York: Wiley.
Buerger, M. J. (1969). Diffraction symbols. In Physics of the Solid State, edited by S. Balakrishna, ch. 3, pp. 27–42. London: Academic Press.
Looijenga-Vos, A. & Buerger, M. J. (2002). Space-group determination and diffraction symbols. In International Tables for Crystallography, Volume A, Space-Group Symmetry, 5th ed., edited by Th. Hahn, Section 3.1.3. Dordrecht, Boston, London: Kluwer Academic Publishers.
Nespolo, M., Ferraris, G. & Souvignier, B. (2014). Effects of merohedric twinning on the diffraction pattern. Acta Cryst. A70, 106–125.








































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