International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 3.1, pp. 45-46

Section 3.1.4. Deduction of possible space groups

A. Looijenga-Vosa and M. J. Buergerb§

aLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands, and bDepartment of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA

3.1.4. Deduction of possible space groups

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Reflection conditions, diffraction symbols, and possible space groups are listed in Table 3.1.4.1[link]. For each crystal system, a different table is provided. The monoclinic system contains different entries for the settings with b, c and a unique. For monoclinic and orthorhombic crystals, all possible settings and cell choices are treated. In contradistinction to Table 4.3.2.1[link] , which lists the space-group symbols for different settings and cell choices in a systematic way, the present table is designed with the aim to make space-group determination as easy as possible.

Table 3.1.4.1| top | pdf |
Reflection conditions, diffraction symbols and possible space groups

TRICLINIC. Laue class [\bar{1}]

Reflection conditionsExtinction symbolPoint group
1[\bar{1}]
None P [P1(1)] [P\bar{1}\;(2)]

MONOCLINIC, Laue class [2/m]

Unique axis bExtinction symbolLaue class [1\;2/m\;1]
Reflection conditionsPoint group
hkl
0kl hk0
h0l
h00 00l
0k02m[2/m]
      P1–1 P121 (3) P1m1 (6) [{\bi P} {\bf 1\;2/}{\bi m}{\hbox to 2pt{}}{\bf 1}\;(10)]
    k [P12_{1}1] [{\bi P}{\bf 12}_{\bf 1}{\bf 1}\;(4)]   [{\bi P}{\bf 1\;2}_{\bf 1}/{\bi m}{\hbox to 2pt{}}{\bf 1}\;(11)]
  h   P1a1   P1a1 (7) [P1\;2/a\;1\;(13)]
  h k [P1\;2_{1}/a\;1]     [P1\;2_{1}/a\;1\;(14)]
  l   P1c1   P1c1 (7) [{\bi P}{\bf 1\;2}/{\bi c}{\bf \;1}\;(13)]
  l k [P1\;2_{1}/c\;1]     [{\bi P}{\bf 1} {\bf \;2}_{\bf 1}/{\bi c}{\bf \;1}\;(14)]
  [h + l]   P1n1   P1n1 (7) [P1\;2/n\;1\;(13)]
  [h + l] k [P1\;2_{1}/n\;1]     [P1\;2_{1}/n\;1\;(14)]
[h + k] h k C1–1 C121 (5) C1m1 (8) [{\bi C}{\bf 1\;2}/{\bi m}{\bf \;1}\;(12)]
[h + k] h, l k C1c1   C1c1 (9) [{\bi C}{\bf 1\;2}/{\bi c}{\bf \;1}\;(15)]
[k + l] l k A1–1 A121 (5) A1m1 (8) [A1\; 2/m\; 1\;(12)]
[k + l] h, l k A1n1   A1n1 (9) [A1\; 2/n\; 1\;(15)]
[h + k + l] [h + l] k I1–1 I121 (5) I1m1 (8) [I1\; 2/m\; 1\;(12)]
[h + k + l] h, l k I1a1   I1a1 (9) [I1\; 2/a\; 1\;(15)]
Unique axis c Extinction symbol Laue class [1\; 1\; 2/m]
Reflection conditions Point group
hkl
0kl h0l
hk0
h00 0k0
00l 2 m [2/m]
      P11– P112 (3) P11m (6) [P11\; 2/m\;(10)]
    l [P112_{1}] [P112_{1}\;(4)]   [P11\; 2_{1}/m\;(11)]
  h   P11a   P11a (7) [P11\; 2/a\;(13)]
  h l [P11\; 2_{1}/a]     [P11\; 2_{1}/a\;(14)]
  k   P11b   P11b (7) [P11\; 2/b\;(13)]
  k l [P11\; 2_{1}/b]     [P11\; 2_{1}/b\;(14)]
  [h + k]   P11n   P11n (7) [P11\; 2/n\;(13)]
  [h + k] l [P11\; 2_{1}/n]     [P11\; 2_{1}/n\;(14)]
[h + l] h l B11– B112 (5) B11m (8) [B11\; 2/m\;(12)]
[h + l] h, k l B11n   B11n (9) [B11\; 2/n\;(15)]
[k + l] k l A11– A112 (5) A11m (8) [A11\; 2/m\;(12)]
[k + l] h, k l A11a   A11a (9) [A11\; 2/a\;(15)]
[h + k + l] [h + k] l I11– I112 (5) I11m (8) [I11\; 2/m\;(12)]
[h + k + l] h, k l I11b   I11b (9) [I11\; 2/b\;(15)]
Unique axis a Extinction symbol Laue class [2/m\; 1\; 1]
Reflection conditions Point group
hkl
h0l hk0
0kl
0k0 00l
h00 2 m [2/m]
      P–11 P211 (3) Pm11 (6) [P2/m\;11\;(10)]
    h [P2_{1}]11 [P2_{1}]11 (4)   [P2_{1}/m\;11\;(11)]
  k   Pb11   Pb11 (7) [P2/b\;11\;(13)]
  k h [P 2_{1}/b\ 11]     [P2_{1}/b\;11\;(14)]
  l   Pc11   Pc11 (7) [P2/c\;11\;(13)]
  l h [P 2_{1}/c\;11]     [P2_{1}/c\ 11\;(14)]
  [k + l]   Pn11   Pn11 (7) [P2/n\;11\;(13)]
  [k + l] h [P 2_{1}/n\;11]     [P2_{1}/n\;11\;(14)]
[h + k] k h C–11 C211 (5) Cm11 (8) [C2/m\;11\;(12)]
[h + k] k, l h Cn11   Cn11 (9) [C2/n\;11\;(15)]
[h + l] l h B–11 B211 (5) Bm11 (8) [B2/m\;11\;(12)]
[h + l] k, l h Bb11   Bb11 (9) [B2/b\;11\;(15)]
[h + k + l] [k + l] h I–11 I211 (5) Im11 (8) [I2/m\;11\;(12)]
[h + k + l] k, l h Ic11   Ic11 (9) [I2/c\;11\;(15)]

ORTHORHOMBIC, Laue class mmm ([2/m\; 2/m\; 2/m])

In this table, the symbol e in the space-group symbol represents the two glide planes given between parentheses in the corresponding extinction symbol. Only for one of the two cases does a bold printed symbol correspond with the standard symbol.

Reflection conditionsLaue class mmm ([2/m\; 2/m\; 2/m])
hkl0klh0lhk0h000k000lExtinction symbolPoint group
222[\!\matrix{mm2\cr m2m\cr 2mm\cr}]mmm
              P– – – P222 (16) Pmm2 (25) Pmmm (47)
                  Pm2m (25)  
                  P2mm (25)  
            l P– –[2_{1}] [{\bi P}{\bf 222}_{{\bf 1}}\;(17)]    
          k   P[2_{1}] [P22_{1}2\;(17)]    
          k l P[2_{1}2_{1}] [P22_{1}2_{1}\;(18)]    
        h     [P2_{1}]– – [P2_{1}22\;(17)]    
        h   l [P2_{1}][2_{1}] [P2_{1}22_{1}\;(18)]    
        h k   [P2_{1}2_{1}] [{\bi P}{\bf 2}_{{\bf 1}}{\bf 2}_{{\bf 1}}{\bf 2}\;(18)]    
        h k l [P2_{1}2_{1}2_{1}] [{\bi P}{\bf 2}_{{\bf 1}}{\bf 2}_{{\bf 1}}{\bf 2}_{{\bf 1}}\;(19)]    
      h h     P– –a   Pm2a (28)  
                  [P2_{1}ma\;(26)] Pmma (51)
      k   k   P– –b   [Pm2_{1}b\;(26)]  
                  P2mb (28) Pmmb (51)
      [h + k] h k   P– –n   [Pm2_{1}n\;(31)]  
                  [P2_{1}mn\;(31)] Pmmn (59)
    h   h     P–a–   Pma2 (28) Pmam (51)
                  [P2_{1}am\;(26)]  
    h h h     P–aa   P2aa (27) Pmaa (49)
    h k h k   P–ab   [P2_{1}ab\;(29)] Pmab (57)
    h [h + k] h k   P–an   P2an (30) Pman (53)
    l       l P–c–   [{\bi P}{\bi m}{\bi c}{\bf 2}_{{\bf 1}}\;(26)]  
                  P2cm (28) Pmcm (51)
    l h h   l P–ca   [P2_{1}ca\;(29)] Pmca (57)
    l k   k l P–cb   P2cb (32) Pmcb (55)
    l [h + k] h k l P–cn   [P2_{1}cn\;(33)] Pmcn (62)
    [h + l]   h   l P–n–   [{\bi P}{\bi m}{\bi n}{\bf 2}_{{\bf 1}}\;(31)]  
                  [P2_{1}nm\;(31)] Pmnm (59)
    [h + l] h h   l P–na   P2na (30) Pmna (53)
    [h + l] k h k l P–nb   [P2_{1}nb\;(33)] Pmnb (62)
    [h + l] [h + k] h k l P–nn   P2nn (34) Pmnn (58)
  k       k   Pb– –   Pbm2 (28)  
                  [Pb2_{1}m\;(26)] Pbmm (51)
  k   h h k   Pb–a   [Pb2_{1}a\;(29)] Pbma (57)
  k   k   k   Pb–b   Pb2b (27) Pbmb (49)
  k   [h + k] h k   Pb–n   Pb2n (30) Pbmn (53)
  k h   h k   Pba–   Pba2 (32) Pbam (55)
  k h h h k   Pbaa     Pbaa (54)
  k h k h k   Pbab     Pbab (54)
  k h [h + k] h k   Pban     Pban (50)
  k l     k l Pbc–   [Pbc2_{1}\;(29)] Pbcm (57)
  k l h h k l Pbca     Pbca (61)
  k l k   k l Pbcb     Pbcb (54)
  k l [h + k] h k l Pbcn     Pbcn (60)
  k [h + l]   h k l Pbn–   [Pbn2_{1}\;(33)] Pbnm (62)
  k [h + l] h h k l Pbna     Pbna (60)
  k [h + l] k h k l Pbnb     Pbnb (56)
  k [h + l] [h + k] h k l Pbnn     Pbnn (52)
  l         l Pc– –   [Pcm2_{1}\;(26)]  
                  Pc2m (28) Pcmm (51)
  l   h h   l Pc–a   Pc2a (32) Pcma (55)
  l   k   k l Pc–b   [Pc2_{1}b\;(29)] Pcmb (57)
  l   [h + k] h k l Pc–n   [Pc2_{1}n\;(33)] Pcmn (62)
  l h   h   l Pca–   [{\bi P}{\bi c}{\bi a}{\bf 2}_{{\bf 1}}\;(29)] Pcam (57)
  l h h h   l Pcaa     Pcaa (54)
  l h k h k l Pcab     Pcab (61)
  l h [h + k] h k l Pcan     Pcan (60)
  l l       l Pcc–   Pcc2 (27) Pccm (49)
  l l h h   l Pcca     Pcca (54)
  l l k   k l Pccb     Pccb (54)
  l l [h + k] h k l Pccn     Pccn (56)
  l [h + l]   h   l Pcn   Pcn2 (30) Pcnm (53)
  l [h + l] h h   l Pcna     Pcna (50)
  l [h + l] k h k l Pcnb     Pcnb (60)
  l [h + l] [h + k] h k l Pcnn     Pcnn (52)
  [k + l]       k l Pn – –   [Pnm2_{1}\;(31)] Pnmm (59)
                  [Pn2_{1}m\;(31)]  
  [k + l]   h h k l Pn–a   [Pn2_{1}a\;(33)] Pnma (62)
  [k + l]   k   k l Pn–b   Pn2b (30) Pnmb (53)
  [k + l]   [h + k] h k l Pn–n   Pn2n (34) Pnmn (58)
  [k + l] h   h k l Pna   [{\bi P}{\bi n}{\bi a}{\bf 2}_{{\bf 1}}\;(33)] Pnam (62)
  [k + l] h h h k l Pnaa     Pnaa (56)
  [k + l] h k h k l Pnab     Pnab (60)
  [k + l] h [h + k] h k l Pnan     Pnan (52)
  [k + l] l     k l Pnc   Pnc2 (30) Pncm (53)
  [k + l] l h h k l Pnca     Pnca (60)
  [k + l] l k   k l Pncb     Pncb (50)
  [k + l] l [h + k] h k l Pncn     Pncn (52)
  [k + l] [h + l]   h k l Pnn   Pnn2 (34) Pnnm (58)
  [k + l] [h + l] h h k l Pnna     Pnna (52)
  [k + l] [h + l] k h k l Pnnb     Pnnb (52)
  [k + l] [h + l] [h + k] h k l Pnnn     Pnnn (48)
[h + k] k h [h + k] h k   C – – – C222 (21) Cmm2 (35) Cmmm (65)
                  Cm2m (38)  
                  C2mm (38)  
[h + k] k h [h + k] h k l C– –[2_{1}] [{\bi C}{\bf 222}_{{\bf 1}}\;(20)]    
[h + k] k h h, k h k   C– –(ab)   Cm2e (39) Cmme (67)
                  C2me (39)  
[h + k] k h, l [h + k] h k l C–c   [{\bi C}{\bi m}{\bi c}{\bf 2}_{{\bf 1}}\;(36)] Cmcm (63)
                  C2cm (40)  
[h + k] k h, l h, k h k l C–c(ab)   C2ce (41) Cmce (64)
[h + k] k, l h [h + k] h k l Cc – –   [Ccm2_{1}\;(36)] Ccmm (63)
                  Cc2m (40)  
[h + k] k, l h h, k h k l Cc –(ab)   Cc2e (41) Ccme (64)
[h + k] k, l h, l [h + k] h k l Ccc   Ccc2 (37) Cccm (66)
[h + k] k, l h, l h, k h k l Ccc(ab)     Ccce (68)
[h + l] l [h + l] h h   l B – – – B222 (21) Bmm2 (38) Bmmm (65)
                  Bm2m (35)  
                  B2mm (38)  
[h + l] l [h + l] h h k l B[2_{1}] [B22_{1}2\;(20)]    
[h + l] l [h + l] h, k h k l B– –b   [Bm2_{1}b\;(36)] Bmmb (63)
                  B2mb (40)  
[h + l] l h, l h h   l B –(ac)–   Bme2 (39) Bmem (67)
                  B2em (39)  
[h + l] l h, l h, k h k l B –(ac)b   B2eb (41) Bmeb (64)
[h + l] k, l [h + l] h h k l Bb – –   Bbm2 (40) Bbmm (63)
                  Bb21m (36)  
[h + l] k, l [h + l] h, k h k l Bb–b   Bb2b (37) Bbmb (66)
[h + l] k, l h, l h h k l Bb(ac)–   Bbe2 (41) Bbem (64)
[h + l] k, l h, l h, k h k l Bb(ac)b     Bbeb (68)
[k + l] [k + l] l k   k l A – – – A222 (21) Amm2 (38) Ammm (65)
                  Am2m (38)  
                  A2mm (35)  
[k + l] [k + l] l k h k l [A2_{1}]– – [A2_{1}22\;(20)]    
[k + l] [k + l] l h, k h k l A– –a   Am2a (40) Amma (63)
                  [A2_{1}ma\;(36)]  
[k + l] [k + l] h, l k h k l A–a   Ama2 (40) Amam (63)
                  [A2_{1}am\;(36)]  
[k + l] [k + l] h, l h, k h k l A–aa   A2aa (37) Amaa (66)
[k + l] k, l l k   k l A(bc)– –   Aem2 (39) Aemm (67)
                  Ae2m (39)  
[k + l] k, l l h, k h k l A(bc)– a   Ae2a (41) Aema (64)
[k + l] k, l h, l k h k l A(bc)a   Aea2 (41) Aeam (64)
[k + l] k, l h, l h, k h k l A(bc)aa     Aeaa (68)
[h + k + l] [k + l] [h + l] [h + k] h k l I – – – [\left[\!\matrix{{\bi I}{\bf 222}\ (23)\hfill\cr{\bi I}{\bf 2}_{\bf 1}{\bf 2}_{\bf 1}{\bf 2}_{\bf 1}(24)}\!\!\right]\!] Imm2 (44) Immm (71)
                Im2m (44)  
                  I2mm (44)  
[h + k + l] [k + l] [h + l] h, k h k l I – –(ab)   Im2a (46) Imma (74)
                  I2mb (46) Immb (74)
[h + k + l] [k + l] h, l [h + k] h k l I –(ac)–   Ima2 (46) Imam (74)
                  I2cm (46) lmcm (74)
[h + k + l] [k + l] h, l h, k h k l Icb   I2cb (45) Imcb (72)
[h + k + l] k, l [h + l] [h + k] h k l I(bc)– –   Iem2 (46) Iemm (74)
                  Ie2m (46)  
[h + k + l] k, l [h + l] h, k h k l Ica   Ic2a (45) Icma (72)
[h + k + l] k, l h, l [h + k] h k l Iba   Iba2 (45) Ibam (72)
[h + k + l] k, l h, l h, k h k l Ibca     Ibca (73)
                    Icab (73)
[h + k,h + l,k + l] [k, l] [h, l] [h, k] h k l F – – – F222 (22) Fmm2 (42) Fmmm (69)
                  Fm2m (42)  
                  F2mm (42)  
[h + k,h + l,k + l] k, l [h + l = 4n]; h, l [h + k = 4n]; h, k [h = 4n] [k = 4n] [l = 4n] F–dd   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] Fd–d   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] Fdd–   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     Fddd (70)

TETRAGONAL, Laue classes [4/m] and [4/mmm]

Reflection conditionsExtinction symbolLaue class
[4/m][4/mmm\;(4/m\; 2/m\; 2/m)]
Point group
hklhk00klhhl00l0k0hh04[\bar{4}][4/m]4224mm[\bar{4}2m\ \ \bar{4}m2][4/mmm]
              P – – – P4 (75) [P\bar{4}\;(81)] [P4/m\;(83)] P422 (89) P4mm (99) [P\bar{4}]2m (111) [P4/mmm\;(123)]
                          [P\bar{4}m2\;(115)]  
          k   P[2_{1}]       [P42_{1}2\;(90)]   [P\bar{4}2_{1}m\;(113)]  
        l     [P4_{2}]– – [P4_{2}\;(77)]   [P4_{2}/m\;(84)] [P4_{2}22\;(93)]      
        l k   [P4_{2}2_{1}]       [P4_{2}2_{1}2\;(94)]      
        [l = 4n]     [P4_{1}]– – [\!\left\{\!\matrix{P4_{1}\; (76)\cr P4_{3}\; (78)\cr}\!\right\}]     [\!\left\{\!\matrix{P4_{1}22\; (91)\cr P4_{3}22\; (95)\cr}\!\right\}]      
        [l = 4n] k   [P4_{1}2_{1}]       [\!\left\{\!\matrix{P4_{1}2_{1}2\; (92)\cr P4_{3}2_{1}2\; (96)\cr}\!\right\}]      
      l l     P – – c         [P4_{2}mc\;(105)] [P\bar{4}2c\;(112)] [P4_{2}/mmc\;(131)]
      l l k   P[2_{1}c]           [P\bar{4}2_{1}c\;(114)]  
    k     k   Pb         P4bm (100) [P\bar{4}b2\;(117)] [P4/mbm\;(127)]
    k l l k   Pbc         [P4_{2}bc\;(106)]   [P4_{2}/mbc\;(135)]
    l   l     Pc         [P4_{2}cm\;(101)] [P\bar{4}c2\;(116)] [P4_{2}/mcm\;(132)]
    l l l     Pcc         P4cc (103)   [P4/mcc\;(124)]
    [k + l]   l k   Pn         [P4_{2}nm\;(102)] [P\bar{4}n2\;(118)] [P4_{2}/mnm\;(136)]
    [k + l] l l k   Pnc         P4nc (104)   [P4/mnc\;(128)]
  [h + k]       k   Pn – –     [P4/n\;(85)]       [P4/nmm\;(129)]
  [h + k]     l k   [P4_{2}/n]– –     [P4_{2}/n\;(86)]        
  [h + k]   l l k   Pnc             [P4_{2}/nmc\;(137)]
  [h + k] k     k   Pnb             [P4/nbm\;(125)]
  [h + k] k l l k   Pnbc             [P4_{2}/nbc\;(133)]
  [h + k] l   l k   Pnc             [P4_{2}/ncm\;(138)]
  [h + k] l l l k   Pncc             [P4/ncc\;(130)]
  [h + k] [k + l]   l k   Pnn             [P4_{2}/nnm\;(134)]
  [h + k] [k + l] l l k   Pnnc             [P4/nnc\;(126)]
[h + k + l] [h + k] [k + l] l l k   I – – – I4 (79) [I\bar{4}\;(82)] [I4/m\;(87)] I422 (97) I4mm (107) [I\bar{4}2m\;(121)] [I4/mmm\;(139)]
                          [I\bar{4}m2\;(119)]  
[h + k + l] [h + k] [k + l] l [l = 4n] k   [I4_{1}]– – [I4_{1}\;(80)]     [I4_{1}]22 (98)      
[h + k + l] [h + k] [k + l] § [l = 4n] k h I – – d         [I4_{1}md\;(109)] [I\bar{4}2d\;(122)]  
[h + k + l] [h + k] k, l l l k   Ic         I4cm (108) [I\bar{4}c2\;(120)] [I4/mcm\;(140)]
[h + k + l] [h + k] k, l § [l = 4n] k h Icd         [I4_{1}cd\;(110)]    
[h + k + l] h, k [k + l] l [l = 4n] k   [I4_{1}/a]– –     [I4_{1}/a\;(88)]        
[h + k + l] h, k [k + l] § [l = 4n] k h Iad             [I4_{1}/amd\;(141)]
[h + k + l] h, k k, l § [l = 4n] k h Iacd             [I4_{1}/acd\;(142)]

TRIGONAL, Laue classes [\bar{3}] and [\bar{3}m]

Reflection conditionsExtinction symbolLaue class
[\bar{3}][\matrix{\bar{3}m1\;(\bar{3}\;2/m\; 1)\hfill\cr \bar{3}m\hfill\cr}][\bar{3}1m\; (\bar{3}\; 1\; 2/m)]
Hexagonal axesPoint group
hkil[h\bar{h}0l][hh\overline{2h}l]000l3[\bar{3}]3213m1[\bar{3}m1]31231m[\bar{3}1m]
323m[\bar{3}m]
        P – – – P3 (143) [P\bar{3}\;(147)] P321 (150) P3m1 (156) [P\bar{3}m1\;(164)] P312 (149) P31m (157) [P\bar{3}1m\;(162)]
      [l = 3n] [P3_{1}]– – [\!\left\{\!\matrix{P3_{1} (144)\cr P3_{2} (145)\cr}\!\right\}]   [\!\left\{\!\matrix{P3_{1}21\;(152)\cr P3_{2}21\;(154)\cr}\!\right\}]     [\!\left\{\!\matrix{P3_{1}12\;(151)\cr P3_{2}12\;(153)\cr}\!\right\}]    
    l l P – – c             P31c (159) [P\bar{3}1c\;(163)]
  l   l Pc       P3cl (158) [P\bar{3}c1\;(165)]      
[- h + k + l = 3n] [h + l = 3n] [l = 3n] [l = 3n] R(obv)– – R3 (146) [R\bar{3}\;(148)] R32 (155) R3m (160) [R\bar{3}m\;(166)]      
[- h + k + l = 3n] [h + l = 3n;\;l] [l = 3n] [l = 6n] R(obv)– c       R3c (161) [R\bar{3}c\;(167)]      
[h - k + l = 3n] [- h + l = 3n] [l = 3n] [l = 3n] R(rev)– – R3 (146) [R\bar{3}\;(148)] R32 (155) R3m (160) [R\bar{3}m\;(166)]      
[h - k + l = 3n] [- h + l = 3n;\; l] [l = 3n] [l = 6n] R(rev)– c       R3c (161) [R\bar{3}c\;(167)]      
Rhombohedral axes Extinction symbol Point group  
hkl hhl hhh 3 [\bar{3}] 32 3m [\bar{3}m]
        R – – R3(146) [R\bar{3}\;(148)] R32 (155) R3m (160) [R\bar{3}m\;(166)]
    l h Rc       R3c (161) [R\bar{3}c\;(167)]

HEXAGONAL, Laue classes [6/m] and [6/mmm]

Reflection conditionsExtinction symbolLaue class
[6/m][6/mmm\;(6/m\; 2/m\; 2/m)]
Point group
[h\bar{h}0l][hh\overline{2h}l]000l6[\bar{6}][6/m]6226mm[\matrix{\bar{6}2m\cr \bar{6}m2\cr}][6/mmm]
      P – – – P6 (168) [P\bar{6}\;(174)] [P6/m\;(175)] P622 (177) P6mm (183) [P\bar{6}2m\;(189)] [P6/mmm\;(191)]
                  [P\bar{6}m2\;(187)]  
    l [P6_{3}]– – [P6_{3}\;(173)]   [P6_{3}/m\;(176)] [P6_{3}22\;(182)]      
    [l = 3n] [P6_{2}]– – [\!\left\{\!\matrix{P6_{2}\; (171)\cr P6_{4}\; (172)\cr}\!\right\}]     [\!\left\{\!\matrix{P6_{2}22\; (180)\cr P6_{4}22\; (181)\cr}\!\right\}]      
    [l = 6n] [P6_{1}]– – [\!\left\{\!\matrix{P6_{1}\; (169)\cr P6_{3}\; (170)\cr}\!\right\}]     [\!\left\{\!\matrix{P6_{1}22\; (178)\cr P6_{5}22\; (179)\cr}\!\right\}]      
  l l P– – c         [P6_{3}mc\;(186)] [P\bar{6}2c\;(190)] [P6_{3}/mmc\;(194)]
l   l Pc         [P6_{3}cm\;(185)] [P\bar{6}c2\;(188)] [P6_{3}/mcm\;(193)]
l l l Pcc         P6cc (184)   P[6/]mcc (192)

CUBIC, Laue classes [m\bar{3}] and [m\bar{3}m]

Reflection conditions (Indices are permutable, apart from space group No. 205)††Extinction symbolLaue class
[m\bar{3}\; (2/m\; \bar{3})][m\bar{3}m\; (4/m\; \bar{3}\; 2/m)]
Point group
hkl0klhhl00l23[m\bar{3}]432[\bar{4}3m][m\bar{3}m]
        P – – – P23 (195) [Pm\bar{3}\; (200)] P432 (207) [P\bar{4}3m\; (215)] [Pm\bar{3}m\; (221)]
      l [\!\left\{\!\matrix{P2_{1}\hbox{--}\;\hbox{--}\cr P4_{2}\hbox{--}\;\hbox{--}\cr}\right.] [P2_{1}3\; (198)]   [P4_{2}32\; (208)]    
      [l = 4n] [P4_{1}]– –     [\!\left\{\!\matrix{P4_{1}32\; (213)\cr P4_{3}32 \;(212)\cr}\!\right\}]    
    l l P– –n       [P\bar{4}3n\; (218)] [Pm\bar{3}n\; (223)]
  k††   l Pa – –   [Pa\bar{3}\; (205)]      
  [k + l]   l Pn – –   [Pn\bar{3}\; (201)]     [Pn\bar{3}m\; (224)]
  [k + l] l l Pn–n         [Pn\bar{3}n\; (222)]
[h + k + l] [k + l] l l I – – – [\left[\matrix{I23\ (197)\hfill\cr I2_{1}3\ (199)\cr}\right]] [Im\bar{3}\; (204)] I432 (211) [I\bar{4}3m\; (217)] [Im\bar{3}m\; (229)]
[h + k + l] [k + l] l [l = 4n] [I4_{1}]– –     [I4_{1}32\; (214)]    
[h + k + l] [k + l] [2h + l = 4n,l] [l = 4n] I– –d       [l\bar{4}3d\; (220)]  
[h + k + l] k, l l l Ia – –   [Ia\bar{3}\; (206)]      
[h + k + l] k, l [2h + l = 4n,l] [l = 4n] Ia–d         [Ia\bar{3}d\; (230)]
[h + k,h + l,k + l] k, l [h + l] l F – – – F23 (196) [Fm\bar{3}\; (202)] F432 (209) [F\bar{4}3m\; (216)] [Fm\bar{3}m\; (225)]
[h + k,h + l,k + l] k, l [h + l] [l = 4n] [F4_{1}]– –     [F4_{1}32\; (210)]    
[h + k,h + l,k + l] k, l h, l l F– –c       [F\bar{4}3c\; (219)] [Fm\bar{3}c\; (226)]
[h + k,h + l,k + l] [k + l = 4n,k,l] [h + l] [l = 4n] Fd – –   [Fd\bar{3}\;(203)]     [Fd\bar{3}m\; (227)]
[h + k,h + l,k + l] [k + l = 4n,k,l] h, l [l = 4n] Fd–c         [Fd\bar{3}c\;(228)]
Pair of space groups with common point group and symmetry elements but differing in the relative location of these elements.
Pair of enantiomorphic space groups, cf. Section 3.1.5[link].
§Condition: [2h + l = 4n{\rm ;}\;l].
For obverse and reverse settings cf. Section 1.2.1[link] . The obverse setting is standard in these tables. The transformation reverse [\rightarrow] obverse is given by [{\bf a}(\hbox{obv.}) = - {\bf a}(\hbox{rev.})], [{\bf b}(\hbox{obv.}) = - {\bf b}(\hbox{rev.})], [{\bf c}(\hbox{obv.}) = {\bf c}(\hbox{rev.})].
††For No. 205, only cyclic permutations are permitted. Conditions are 0kl: [k = 2n]; h0l: [l = 2n]; hk0: [h = 2n].

The left-hand side of the table contains the Reflection conditions. Conditions of the type [h = 2n] or [h + k = 2n] are abbreviated as h or [h + k]. Conditions like [h = 2n, k = 2n, h + k = 2n] are quoted as h, k; in this case, the condition [h + k = 2n] is not listed as it follows directly from [h = 2n, k = 2n]. Conditions with [l = 3n], [l = 4n], [{l = 6n}] or more complicated expressions are listed explicitly.

From left to right, the table contains the integral, zonal and serial conditions. From top to bottom, the entries are ordered such that left columns are kept empty as long as possible. The leftmost column that contains an entry is considered as the `leading column'. In this column, entries are listed according to increasing complexity. This also holds for the subsequent columns within the restrictions imposed by previous columns on the left. The make-up of the table is such that observed reflection conditions should be matched against the table by considering, within each crystal system, the columns from left to right.

The centre column contains the Extinction symbol. To obtain the complete diffraction symbol, the Laue-class symbol has to be added in front of it. Be sure that the correct Laue-class symbol is used if the crystal system contains two Laue classes. Particular care is needed for Laue class [\bar{3}m] in the trigonal system, because there are two possible orientations of this Laue symmetry with respect to the crystal lattice, [\bar{3}m1] and [\bar{3}1m]. The correct orientation can be obtained directly from the diffraction record.

The right-hand side of the table gives the Possible space groups which obey the reflection conditions. For crystal systems with two Laue classes, a subdivision is made according to the Laue symmetry. The entries in each Laue class are ordered according to their point groups. All space groups that match both the reflection conditions and the Laue symmetry, found in a diffraction experiment, are possible space groups of the crystal.

The space groups are given by their short Hermann–Mauguin symbols, followed by their number between parentheses, except for the monoclinic system, where full symbols are given (cf. Section 2.2.4[link] ). In the monoclinic and orthorhombic sections of Table 3.1.4.1[link], which contain entries for the different settings and cell choices, the `standard' space-group symbols (cf. Table 4.3.2.1[link] ) are printed in bold face. Only these standard representations are treated in full in the space-group tables.

Example

The diffraction pattern of a compound has Laue class mmm. The crystal system is thus orthorhombic. The diffraction spots are indexed such that the reflection conditions are [0kl: l = 2n]; [h0l: h + l = 2n]; [h00: h = 2n]; [00l:l = 2n]. Table 3.1.4.1[link] shows that the diffraction symbol is mmmPcn–. Possible space groups are Pcn2 (30) and Pcnm (53). For neither space group does the axial choice correspond to that of the standard setting. For No. 30, the standard symbol is Pnc2, for No. 53 it is Pmna. The transformation from the basis vectors [{\bf a}_{e}, {\bf b}_{e}, {\bf c}_{e}], used in the experiment, to the basis vectors [{\bf a}_{s}, {\bf b}_{s}, {\bf c}_{s}] of the standard setting is given by [{\bf a}_{s} = {\bf b}_{e}, {\bf b}_{s} = -{\bf a}_{e}] for No. 30 and by [{\bf a}_{s} = {\bf c}_{e}, {\bf c}_{s} = -{\bf a}_{e}] for No. 53.

Possible pitfalls

Errors in the space-group determination may occur because of several reasons.

  • (1) Twinning of the crystal

    Difficulties that may be encountered are shown by the following example. Say that a monoclinic crystal (b unique) with the angle β fortuitously equal to [\sim 90^{\circ}] is twinned according to (100). As this causes overlap of the reflections hkl and [\bar{h}kl], the observed Laue symmetry is mmm rather than [2/m]. The same effect may occur within one crystal system. If, for instance, a crystal with Laue class [4/m] is twinned according to (100) or (110), the Laue class [4/mmm] is simulated (twinning by merohedry, cf. Catti & Ferraris, 1976[link], and Koch, 2004[link]). Further examples are given by Buerger (1960)[link]. Errors due to twinning can often be detected from the fact that the observed reflection conditions do not match any of the diffraction symbols.

  • (2) Incorrect determination of reflection conditions

    Either too many or too few conditions may be found. For serial reflections, the first case may arise if the structure is such that its projection on, say, the b direction shows pseudo-periodicity. If the pseudo-axis is [b/p], with p an integer, the reflections 0k0 with [k \neq p] are very weak. If the exposure time is not long enough, they may be classified as unobserved which, incorrectly, would lead to the reflection condition [0k0: k = p]. A similar situation may arise for zonal conditions, although in this case there is less danger of errors. Many more reflections are involved and the occurrence of pseudo-periodicity is less likely for two-dimensional than for one-dimensional projections.

    For `structural' or non-space-group absences, see Section 2.2.13[link] .

    The second case, too many observed reflections, may be due to multiple diffraction or to radiation impurity. A textbook description of multiple diffraction has been given by Lipson & Cochran (1966)[link]. A well known case of radiation impurity in X-ray diffraction is the contamination of a copper target with iron. On a photograph taken with the radiation from such a target, the iron radiation with [\lambda \hbox{(Fe)} \sim 5/4\lambda \hbox{(Cu)}] gives a reflection spot [4h_{\prime} 4k_{\prime} 4l] at the position [5h_{\prime} 5k_{\prime} 5l] for copper [[\lambda (\hbox{Cu}\; K\bar{\alpha})\! =\! 1.5418\;\hbox{\AA}], [\lambda (\hbox{Fe}\; K\bar{\alpha})\! =\! 1.9373\;\hbox{\AA}]]. For reflections 0k0, for instance, this may give rise to reflected intensity at the copper 050 position so that, incorrectly, the condition [0k0: k = 2n] may be excluded.

  • (3) Incorrect assignment of the Laue symmetry

    This may be caused by pseudo-symmetry or by `diffraction enhancement'. A crystal with pseudo-symmetry shows small deviations from a certain symmetry, and careful inspection of the diffraction pattern is necessary to determine the correct Laue class. In the case of diffraction enhancement, the symmetry of the diffraction pattern is higher than the Laue symmetry of the crystal. Structure types showing this phenomenon are rare and have to fulfil specified conditions. For further discussions and references, see Perez-Mato & Iglesias (1977)[link].

References

Buerger, M. J. (1960). Crystal-structure analysis, Chap. 5. New York: Wiley.
Catti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray crystal structure determination. Acta Cryst. A32, 163–165.
Koch, E. (2004). Twinning. International Tables for Crystallography Vol. C, 3rd ed., edited by E. Prince, ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Lipson, H. & Cochran, W. (1966). The determination of crystal structures, Chaps. 3 and 4.4. London: Bell.
Perez-Mato, J. M. & Iglesias, J. E. (1977). On simple and double diffraction enhancement of symmetry. Acta Cryst. A33, 466–474.








































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