International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 289-291   | 1 | 2 |

Section 2.5.3.4. Auxiliary tables

P. Goodmanb

2.5.3.4. Auxiliary tables

| top | pdf |

Space groups may very well be identified using CBED patterns from an understanding of the diffraction properties of real-space symmetry elements, displayed for example in Table 2.5.3.2[link]. It is, however, of great assistance to have the symmetries tabulated in reciprocal space, to allow direct comparison with the pattern symmetries.

There are three generally useful ways in which this can be done, and these are set out in Tables 2.5.3.3[link] [link] to 2.5.3.5[link]. The simplest of these is by means of point group, following the procedures of Buxton et al. (1976)[link]. Next, the CBED pattern symmetries can be listed as diperiodic groups which are space groups in two dimensions, allowing identification with a restricted set of three-dimensional space groups (Goodman, 1984b[link]). Finally, the dynamic extinctions (GS bands and zero-layer absences) can be listed for each non-symmorphic space group, together with the diffraction conditions for their observation (Tanaka et al., 1983[link]; Tanaka & Terauchi, 1985[link]). Descriptions for these tables are given below.

  • Table 2.5.3.3[link]. BESR symbols (Buxton et al., 1976[link]) incorporate the subscript R to describe reciprocity-related symmetry elements, R being the operator that rotates the disc pattern by 180° about its centre. The symbols formed in this way are [1_{R}], [2_{R}], [4_{R}], [6_{R}], where [X_{R}] represents [2\pi / X] rotation about the zone axis, followed by R. Of these, [2_{R}] represents the ±H symmetry (two twofold rotations) described earlier [equation (2.5.3.2)[link]] as a transformation of crystalline centrosymmetry; [6_{R}] may be thought of as decomposing into [3 \cdot 2_{R}] for purposes of measurement. The mirror line [m_{R}] (Fig. 2.5.3.2[link]) is similarly generated by [m \cdot 1_{R}].

    Table 2.5.3.3| top | pdf |
    Diffraction point-group tables, giving whole-pattern and central-beam pattern symmetries in terms of BESR diffraction-group symbols and diperiodic group symbols

    IIIIIIIVV
    Whole patternBright field (central beam)BESR groupDiperiodic group (point group)Cubic point groups
    [100][110]
    1 1 1 1    
    1 2 [\underline{1}_{R}] m    
    2 2 2 2    
    1 1 [^{*}2_{R}] [{\bar 1}^{\prime}]    
    2 2 [^{*}\underline{21}_{R}] [2/m']    
    1 m [m_{R}] 2′   23
    m m m m    
    m 2mm [\underline{m1}_{R}] 2′mm   [\bar{4}3m]
    2 2mm [2m_{R}m_{R}] 2′2′2 23 432
    2mm 2mm 2mm mm2    
    m m [^{*}2_{R}mm_{R}] [2'/m]   m [\bar{3}]
    2mm 2mm [^{*}\underline{2mm1}_{R}] mmm m [\bar{3}] m [\bar{3}] m
    4 4 4 4    
    2 4 [4_{R}] [\bar{4}']    
    4 4 [^{*}\underline{41}_{R}] [4/m']    
    4 4mm [4m_{R}m_{R}] [42'2'] [\bar{4}32]  
    4mm 4mm 4mm 4mm    
    2mm 4mm [4_{R}mm_{R}] [\bar{4}'m2'] [\bar{4}3m]  
    4mm 4mm [^{*}\underline{4mm1}_{R}] [4/m'mm] m [\bar{3}] m  
    3 3 3 3    
    3 6 [\underline{31}_{R}] [\bar{6}']    
    3 3m [3m_{R}] 32′    
    3m 3m 3m 3m    
    3m 6mm [\underline{3m1}_{R}] [\bar{6}'m2']    
    6 6 6 6    
    3 3 [^{*}6_{R}] [\bar{3}']    
    6 6 [^{*}\underline{61}_{R}] [6/m']    
    6 6mm [6m_{R}m_{R}] [62'2']    
    6mm 6mm 6mm 6mm    
    3m 3m [^{*}6_{R}mm_{R}] [\bar{3}'m]    
    6mm 6mm [^{*}\underline{6mm1}_{R}] [6/m'mm]    

    Table 2.5.3.3[link] gives the BESR interrelation of pattern symmetries with point group (Buxton et al., 1976[link]; Steeds, 1983[link]). Columns I and II of the table list the point symmetries of the whole pattern and bright-field pattern, respectively; column III gives the BESR diffraction groups. [Note: following the Pond & Vlachavas (1983)[link] usage, `[^{*}]' has been appended to the centrosymmetric groups.]

    Inspection of columns I and II shows that 11 of the 31 diffraction groups can be determined from a knowledge of the whole pattern and bright-field (central-beam disc) point symmetries alone. The remaining 10 pairs of groups need additional observation of the dark-field pattern for their resolution. Disc symmetries [1_{R}], [m_{R}] (Fig. 2.5.3.2[link]; Table 2.5.3.2[link]) are sought (a) in general zero-layer discs and (b) in discs having an [m_{R}] line perpendicular to a proposed twofold axis, respectively; the ±H test is applied for centrosymmetry, to complete the classification.

    Column IV gives the equivalent diperiodic point-group symbol, which, unprimed, gives the corresponding three-dimensional symbol. This will always refer to a non-cubic point group. Column V gives the additional cubic point-group information indicating, where appropriate, how to translate the diffraction symmetry into [100] or [110] cubic settings, respectively.

    Of the groups listed in column III, those representing the projection group of their class are underlined. These groups all contain [1_{R}], the BESR symbol for m′. When the projection approximation is applicable, only those groups underlined will apply. The effect of this approximation is to add a horizontal mirror plane to the symmetry group.

  • Table 2.5.3.4[link]. This lists possible space groups for each of the classified zero-layer CBED symmetries. Since the latter constitute the 80 diperiodic groups, it is first necessary to index the pattern in diperiodic nomenclature; the set of possible space groups is then given by the table.

    Table 2.5.3.4| top | pdf |
    Tabulation of principal-axis CBED pattern symmetries against relevant space groups given as IT A numbers

    Three columns of diperiodic groups (central section) correspond to (i) symmorphic groups, (ii) non-symmorphic groups (GS bands) and (iii) non-symmorphic groups (zero-layer absences arising from horizontal glide planes). Cubic space groups are given underlined in the right-hand section with the code: underlining = [001](cyclic) setting; italics + underlining = [110](cyclic) setting. Separators `;' and `:' indicate change of Bravais lattice type and change of crystal system, respectively.

    DGIIISGIII
    Point groupsDiperiodic groupsSpace groups
    H–MBESR(i)(ii)(iii)Subgroups IIb (Subgroups 1)
    Oblique Triclinic
    1 1 1 p1     1  
    2* [\bar{1}] [2_{R}] [p\bar{1}']     2  
      Monoclinic (Oblique)
    3 12 2 p2     3 4, 5
    4 1m [1_{R}] pm     6 8
    5 1m       pb 7 9
    6* [2/m] [21_{R}] [p2/m']     10 11, 12
    7* [2/m] [21_{R}]     [p2/b'] 13 14, 15
    Rectangular (Rectangular)
    8 21 [m_{R}] p2′     3 52: 195; 197, 199
    9 21 [m_{R}]   [p2'_{1}]   4 198
    10 21 [m_{R}] c2′     5 196
    11 m1 m pm     62 7, 82
    12 m1 m   pa   72 92
    13 m1 m cm     8 9
    14* [12/m] [2_{R}mm_{R}] [p2'/m]     10 13, 122: 200, 201; 204
    15* [12/m] [2_{R}mm_{R}]   [p2'_{1}/m]   11 14
    16* [12/m] [2_{R}mm_{R}]   [p2'/a]   132 152: 206
    17* [12/m] [2_{R}mm_{R}]   [p2'_{1}/m]   142 205
    18* [12/m] [2_{R}mm_{R}] [c2'/m]     12 15: 202, 203
      Orthorhombic
    19 222 [2m_{R}m_{R}] p2′2′2     16 17; 212; 22: 195; 196, 207, 206; 211, 214
    20 222 [2m_{R}m_{R}]   [p2'_{1}2'2]   172 182; 202: 212, 213
    21 222 [2m_{R}m_{R}]   [p2'_{1}2'_{1}2]   18 19: 198
    22 222 [2m_{R}m_{R}] c2′2′2     21 20; 23, 24: 197, 199, 209, 210
    23 mm2 2mm pmm2     25 26, 27; 38, 39; 42
    24 mm2 2mm   pbm2   28 29, 30, 312; 40, 41
    25 mm2 2mm   pba2   32 33, 34; 43
    26 mm2 2mm cmm2     35 36, 37; 44, 45, 46
    27 mm2 [m1_{R}] p2′mm     252 281; 352, 422; 382, 392: 215; 217
    28 mm2 [m1_{R}]   [p2'_{1}m'a]   261 311; 361
    29 mm2 [m1_{R}]   [p2'_{1}ab'] [(p2'_{1}ab')] 292 332
    30 mm2 [m1_{R}]     [p2'_{1}ma'] 262 291; 362
    31 mm2 [m1_{R}]     [p2'_{1}mn'] 312 332
    32 mm2 [m1_{R}]     p2′mb 282 322, 402, 412
    33 mm2 [m1_{R}]     p2′aa 272 302; 372
    34 mm2 [m1_{R}]     [pb2'n'] 301 342; 432: 218; 219
    35 mm2 [m1_{R}] c2′mm     381 401; 442, 461: 216; 220
    36 mm2 [m1_{R}]     c2′mb 391 411; 452, 462
    37* mmm [2mm1_{R}] pmmm     47 49, 511; 652, 672; 69:
                  200; 202, 221, 224, 226, 228, 229
    38* mmm [2mm1_{R}]   [pbmm'\; (2'_{1})]   512 531, 57, 592; 631, 641
    39* mmm [2mm1_{R}]   [pbam'\; (2'_{1}2'_{1})]   55 58, 622
    40* mmm [2mm1_{R}]   [pmab'\; (2'_{1}2'_{1})] [(pmab')] 571 602, 61, 62: 205
    41* mmm [2mm1_{R}]   [pbaa'\; (2'_{1})] [(pbaa')] 542 52, 562, 601
    42* mmm [2mm1_{R}]     [pmma'\; (2'_{1})] 51 54, 552, 572; 632, 642
    43* mmm [2mm1_{R}]     pmmn[(2'_{1}2'_{1})] 59 56, 621
    44* mmm [2mm1_{R}]     [pbmn'\; (2'_{1})] 532 521, 581, 60
    45* mmm [2mm1_{R}]     pmaa 492 502, 53, 541; 662, 681: 222, 223
    46* mmm [2mm1_{R}]     pban 50 522, 48; 70: 201; 203, 230
    47* mmm [2mm1_{R}] cmmm     65 63, 66; 72, 742, 71: 204, 225, 227
    48* mmm [2mm1_{R}]     cmma 67 64, 68; 721, 74, 73: 206
    Square Tetragonal
    49 4 4 p4     75 77, 76, 78; 79, 80
    50 [4/m] [41_{R}] [p4/m']     83 84; 87
    51 [4/m] [41_{R}]     [p4/n'] 85 86, 88
    52 422 [4m_{R}m_{R}] [p42'2']     89 93, 91, 95; 97, 98:
                  207, 208; 209, 210; 211, 214
    53 422 [4m_{R}m_{R}]   [p42'_{1}2']   90 94, 92, 96: 212, 213
    54 4mm 4mm [p4mm]     99 101, 103, 105; 107, 108
    55 4mm 4mm   [p4bm]   100 102, 104, 106; 109, 110
    56* [4/mmm] [4mm1_{R}] [p4/m'mm]     123 124, 131, 132; 139, 140;
                  221, 223; 225, 226; 229
    57* [4/mmm] [4mm1_{R}]   [p4/m'bm\; (2'_{1})]   127 128, 135, 136
    58* [4/mmm] [4mm1_{R}]     [p4/n'bm] 125 126, 133, 134; 141, 142:
                  222, 224; 227, 228; 230
    59* [4/mmm] [4mm1_{R}]     [p4/n'mm\; (2'_{1})] 129 130, 137, 138
    60 [\bar{4}] [4_{R}] [p\bar{4}']     81 82
    61 [\bar{4}2m] [4_{R}mm_{R}] [p\bar{4}'m\bar{2}']     115 116; 119, 120
    62 [\bar{4}2m] [4_{R}mm_{R}]   [p\bar{4}b2']   117 118; 122: 220
    63 [\bar{4}2m] [4_{R}mm_{R}] [p\bar{4}'2'm]     111 112; 121: 215; 216; 217; 218; 219
    64 [\bar{4}2m] [4_{R}mm_{R}]   [p4'2'_{1}m]   113 114
    Hexagonal Trigonal
    65 3 3 p3     143 144, 145; 146
    66 [\bar{3}] [6_{R}] [p\bar{3}']     147 148
    67 32 [3m_{R}] p312′     149 151, 153
    68 32 [3m_{R}] [p32'1]     150 152, 154; 155
    69 3m 3m p31m     157 159
    70 3m 3m p3m1     156 158; 160, 161
    71* [\bar{3}m] [6_{R}mm_{R}] [p\bar{3}'1m]     162 163
    72* [\bar{3}m] [6_{R}mm_{R}] [p\bar{3}'m1]     164 165; 166, 167
      Hexagonal
    73 6 6 p6     168 171, 172, 173, 169, 170
    74 [\bar{6}] [31_{R}] [p3/m'\; (p\bar{6}')]     174  
    75 622 [6m_{R}m_{R}] [p62'2']     177 180, 181, 182, 178, 179
    76 6mm 6mm [p6mm]     183 184, 185, 186
    77* [6/m] [61_{R}] [p6/m']     175 176
    78* [6/mmm] [6mm1_{R}] [p6/m'mm]     191 192, 193, 194
    79 [\bar{6}m2] [3m1_{R}] [p3/m'2'm] [(p\bar{6}'m2')]     189 190
    80 [\bar{6}m2] [3m1_{R}] [p3/m'm2'] [(p\bar{6}'2'm)]     187 188

    A basic requirement for diperiodic group nomenclature has been that of compatibility with IT A and I. This has been met by the recent Pond & Vlachavas (1983) tabulation. For example, DG: [(^{*})pban'], where [^{*}] indicates centrosymmetry, becomes space group Pban when, in Seitz matrix description, the former group matrix is multiplied by the third primitive translation, [a_{3}]. Furthermore, in textual reference the prime can be optionally omitted, since the lower-case lattice symbol is sufficient indication of a two-dimensional periodicity (as pban).

    The three sections of Table 2.5.3.4[link] are:

    • I. Point-group entries, given in H–M and BESR symbols.

    • II. Pattern symmetries, in diperiodic nomenclature, have three subdivisions: (i) symmorphic groups: patterns without zero-layer absences or extinctions. Non-symmorphic groups are then given in two categories: (ii) patterns with zero-layer GS bands, and (iii) patterns with zero-layer absences resulting from a horizontal glide plane; where the pattern also contains dynamic extinctions (GS bands) and so is listed in column (ii), the column (iii) listing is given in parentheses.

      The `short' (Pond & Vlachavas) symbol has proved an adequate description for all but nine groups for which the screw-axis content was needed: here [(2'_{1})], or [(2'_{1}2'_{1})], have been added to the symbol.

    • III. Space-group entries are given in terms of IT A numbers. The first column of each row gives the same-name space group as illustrated by the example pban′ → Pban above. The groups following in the same row (which have the same zero-layer symmetry) complete an exhaustive listing of the IIb subgroups, given in IT A . Cubic space groups are underlined for the sake of clarity; hence, those giving rise to the zero-layer symmetry of the diffraction group in the [100] (cyclic) setting have a single underline: these are type I minimal supergroups in IT A nomenclature. The cubic groups are also given in the [110] setting, in underlined italics, since this is a commonly encountered high-symmetry setting. (Note: these then are no longer minimal supergroups and the relationship has to be found through a series of IT A listings.)

    The table relates to maximal-symmetry settings. For monoclinic and orthorhombic systems there are three equally valid settings. For monoclinic groups, the oblique and rectangular settings appear separately; where rectangular C-centred groups appear in a second setting this is indicated by superscript `2'. For orthorhombic groups, superscripts correspond to the `incident-beam' system adopted in Table 2.5.3.5[link], as follows: no superscript: [001] beam direction; superscript 1: [100] beam direction; superscript 2: [010] beam direction. The cubic system is treated specially as described above.

    Table 2.5.3.5| top | pdf |
    Conditions for observation of GS bands for the 137 space groups exhibiting these extinctions

    Point groups 2, m, [{\bf 2/{\bi m}}] (2nd setting unique axis [\|b])

    Space groupIncident-beam direction
    [u0w]
    4 [P2_{1}] 0k0 S
    [2_{1}]
    7 Pc h0l G
    c
    9 Cc h0l G
    c
    11 [P2_{1}/m] 0k0 S
    [2_{1}]
    13 [P2/c] h0l G
    c
    14 [P2_{1}/c] 0k0 S
    [2_{1}]
    h0l G
    c
    15 [C2/c] h0l G
    c

    Point groups 222, mm2

    Space groupIncident-beam direction
    [100][010][001][uv0][0vw][u0w]
    17 [P222_{1}] 00l S 00l S     00l S        
    [2_{1}] [2_{1}]     [2_{1}]        
    18 [P2_{1}2_{1}2] 0k0 S h00 S h00, 0k0 S     h00 S 0k0 S
    [2_{1}] [2_{1}] [2_{1}]     [2_{1}] [2_{1}]
    19 [P2_{1}2_{1}2_{1}] 0k0, 00l S h00, 00l S h00, 0k0 S 00l S h00 S 0k0 S
    [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}]
    20 [C222_{1}] 00l S 00l S     00l S        
    [2_{1}] [2_{1}]     [2_{1}]        
    26 [Pmc2_{1}] 00l GS 00l c′−     00l S     h0l G
    [c + 2_{1}] [2_{1}]     [2_{1}]     c
    27 [Pcc2] 00l c′− 00l c′−         0kl G h0l G
    c c         c c
    28 [Pma2]         h00 G         h0l G
            a         a
    29 [Pca2_{1}] 00l c′− 00l GS h00 G 00l S 0kl G h0l G
    [2_{1}] [c + 2_{1}] a [2_{1}] c a
    30 [Pnc2] 00l n′− 00l c′− 0k0 G     0kl G h0l G
    c n n     n c
    31 [Pmn2_{1}] 00l GS 00l n′− h00 G 00l S     h0l G
    [n + 2_{1}] [2_{1}] n [2_{1}]     n
    32 [Pba2]         h00, 0k0 G     0kl G h0l G
            a b     b a
    33 [Pna2_{1}] 00l n′− 00l GS h00, 0k0 G 00l S 0kl G h0l G
    [2_{1}] [n + 2_{1}] a b [2_{1}] n a
    34 [Pnn2] 00l n′− 00l n′− h00, 0k0 G     0kl G h0l G
    n n n     n n
    36 [Cmc2_{1}] 00l GS 00l c′−     00l S     h0l G
    [c + 2_{1}] [2_{1}]     [2_{1}]     c
    37 [Ccc2] 00l c′− 00l c′−         0kl G h0l G
    c c         c c
    39 [Abm2]                 0kl G    
                    b    
    40 [Ama2]         h00 G         h0l G
            a         a
    41 [Aba2]         h00 G     0kl G h0l G
            a     b a
    43 [Fdd2] 00l d′− 00l d′− h00, 0k0 G     0kl G h0l G
      d d     d d
    45 [Iba2]   b′−   a′−         0kl G h0l G
                b a
    46 [Ima2]       a′−             h0l G
                      a

    Point group mmm

    Space groupIncident-beam direction
    [100][010][001][uv0][0vw][u0w]
    48 P 2/n 2/n 2/n 00l, 0k0 n′− 00l, h00 n′− 0k0, h00 n′− hk0 G 0kl G h0l G
    n n n n n n
    49 P 2/c 2/c 2/m 00l c′− 00l c′−         0kl G h0l G
    c c         c c
    50 P 2/b 2/a 2/n 0k0 b′− h00 a′− 0k0, h00 n′− hk0 G 0kl G h0l G
    n n b a n b a
    51 [P\;2_{1}/m\;2/m\;2/a]     h00 GS h00 a′− hk0 G h00 S    
        [a + 2_{1}] [2_{1}] a [2_{1}]    
    52 [P\;2/n\;2_{1}/n\;2/a] 00l, 0k0 a′− 00l, h00 n′− 0k0 GS hk0 G 0kl G h0l G
    [n + 2_{1}] n
    [2_{1}] n a h00 n′− a n 0k0 S
    n [2_{1}]
    53 [P\;2/m\;2/n\;2_{1}/a] 00l GS h00, 00l a′− h00 a′− hk0 G     h0l G
    a
    [n + 2_{1}] [2_{1}] n 00l S n
    [2_{1}]
    54 [P\;2_{1}/c\;2/c\;2/a] 00l c′− h00 GS h00 a′− hk0 G 0kl G h0l G
    [a + 2_{1}] c
    c 00l c′− [2_{1}] a h00 S c
    c [2_{1}]
    55 [P\;2_{1}/b\;2_{1}/a\;2/m] 0k0 b′− h00 a′− 0k0, h00 GS     0kl G h0l G
    b a
    [2_{1}] [2_{1}] [b + 2_{1}, a + 2_{1}] h00 S 0k0 S
    [2_{1}] [2_{1}]
    56 [P\;2_{1}/c\;2_{1}/c\;2/n] 0k0 GS h00 GS 0k0, h00 n′− hk0 G 0kl G h0l G
    [n + 2_{1}] [n + 2_{1}] c c
    00l c′− 00l c′− [2_{1}] n h00 S 0k0 S
    c c [2_{1}] [2_{1}]
    57 [P\;2/b\;2_{1}/c\;2_{1}/m] 00l GS 00l c′− 0k0 GS 00l S 0kl G h0l G
    [c + 2_{1}] c
    0k0 b′− [2_{1}] [b + 2_{1}] [2_{1}] b 0k0 S
    [2_{1}] [2_{1}]
    58 [P\;2_{1}/n\;2_{1}/n\;2/m] 00l, 0k0 n′− 00l, h00 n′− 0k0, h00 GS     0kl G h0l G
    n n
    n [2_{1}] n [2_{1}] [n + 2_{1}] h00 S 0k0 S
    [2_{1}] [2_{1}]
    59 [P\;2_{1}/m\;2_{1}/m\;2/n] 0k0 GS h00 GS 0k0, h00 n′− hk0 G h00 S 0k0 S
    [n + 2_{1}] [n + 2_{1}] [2_{1}] n [2_{1}] [2_{1}]
    60 [P\;2_{1}/b\;2/c\;2_{1}/n] 00l GS h00 GS 0k0, h00 n′− hk0 G 0kl G h0l G
    [c + 2_{1}] [n + 2_{1}] n b
    0k0 b′− 00l c′− b [2_{1}] 00l S h00 S c
    n [2_{1}] [2_{1}] [2_{1}]
    61 [P\;2_{1}/b\;2_{1}/c\;2_{1}/a] 00l GS h00 GS 0k0 GS hk0 G 0kl G h0l G
    [c + 2_{1}] [a + 2_{1}] [b + 2_{1}] a b c
    0k0 b′− 00l c′− h00 a′− 00l S h00 S 0k0 S
    [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}] [2_{1}]
    62 [P\;2_{1}/n\;2_{1}/m\;2_{1}/a] 0k0, 00l n′− 00l GS 0k0 GS hk0 G 0kl G 0k0 S
    [n + 2_{1}] [n + 2_{1}] a n
    [2_{1}] h00 h00 a′− 00l S h00 S [2_{1}]
    [a + 2_{1}] [2_{1}] [2_{1}] [2_{1}]
    63 [C\;2/m\;2/c\;2_{1}/m] 00l GS 00l c′−     00l S     h0l G
    [c + 2_{1}] [2_{1}] [2_{1}] c
    64 [C\;2/m\;2/c\;2_{1}/a] 00l GS 00l c′−     hk0 G     h0l G
    a
    [c + 2_{1}] [2_{1}] 00l S c
    [2_{1}]
    66 [C\;2/c\;2/c\;2/m] 00l c′− 00l c′−         0kl G h0l G
    c c c c
    67 [C\;2/m\;2/m\;2/a]             hk0 G        
    a
    68 [C\;2/c\;2/c\;2/a] 00l c′− 00l c′−     hk0 G 0kl G h0l G
    c c a c c
    70 [F\;2/d\;2/d\;2/d] 00l, 0k0 d′− h00, 00l d′− 0k0, h00 d′− hk0 G 0kl G h0l G
    d d d d d d
    72 [I\;2/b\;2/a\;2/m]   b′−   a′−         0kl G h0l G
    b a
    73 [I\;2/b\;2/c\;2/a]   b′−   c′−   a′− hk0 G 0kl G h0l G
    a b c
    74 [I\;2/m\;2/m\;2/a]             hk0 G        
    a

    Point groups 4, [\bar{\bf{4}}], [\bf{4/{\bi m}}]

    Space groupIncident-beam direction
    [uv0]
    76 [P4_{1}] 00l S
    [4_{1}]
    78 [P4_{3}] 00l S
    [4_{3}]
    85 [P4/n] hk0 G
    n
    86 [P4_{2}/n] hk0 G
    n
    88 [I4_{1}/a] hk0 G
    a

    Point group 422

    Space groupIncident-beam direction
    [uv0][0vw]
    90 [P42_{1}2]     h00 S
    [2_{1}]
    91 [P4_{1}22] 00l S    
    [4_{1}]
    92 [P4_{1}2_{1}2] 00l S h00 S
    [4_{1}] [2_{1}]
    94 [P4_{2}2_{1}2]     h00 S
    [2_{1}]
    95 [P4_{3}22] 00l S    
    [4_{3}]
    96 [P4_{3}2_{1}2] 00l S h00 S
    [4_{3}] [2_{1}]

    Point group 4mm

    Space groupIncident-beam direction
    [100][001][110][u0w] and [0vw] [[u\bar{u}w]]
    100 [P4bm]     h00, 0k0 G     h0l, 0kl G    
    a b a b
    101 [P4_{2}cm] 00l c′−         h0l, 0kl G    
    c c
    102 [P4_{2}nm] 00l n′− h00, 0k0 G     h0l, 0kl G    
    n n n
    103 [P4cc] 00l c′−     00l c′− h0l, 0kl G hhl G
    c c c c
    104 [P4nc] 00l n′− h00, 0k0 G 00l c′− h0l, 0kl G hhl G
    n n c n c
    105 [P4_{2}mc]         00l c′−     hhl G
    c c
    106 [P4_{2}bc]     h00, 0k0 G 00l c′− h0l, 0kl G hhl G
    a b c a b c
    108 [I4cm]             h0l, 0kl G    
    c
    109 [I3_{1}md]     hh0 G 00l d′−     hhl G
    [{\bar h}h0]    
    d d d
    110 [I4_{1}cd]     hh0 G 00l d′− h0l, 0kl G hhl G
    [{\bar h}h0]      
    d d c d

    Point groups [\bar{\bf {4}}]2m, 4/mmm

    Space groupIncident-beam direction
    [100][001][110][u0w] and [0vw][uuw][uv0]
    112 [P{\bar 4}2c]         00l c′−     hhl G    
    c c
    113 [P{\bar 4}2_{1}m] 0k0 S h00, 0k0 S     0k0, h00 S        
    [2_{1}] [2_{1}] [2_{1}]
    114 [P{\bar 4}2_{1}c] 0k0 S h00, 0k0 S 00l c′− 0k0, h00 S hhl G    
    [2_{1}] [2_{1}] c [2_{1}] c
    116 [P{\bar 4}c2] 00l c′−         h0l, 0kl G        
    c c
    117 [P{\bar 4}b2]     h00, 0k0 G     h0l, 0kl G        
    a b a b
    118 [P{\bar 4}n2] 00l n′− h00, 0k0 G     h0l, 0kl G        
    n n n
    120 [I{\bar 4}c2]             h0l, 0kl G        
    c
    122 [I{\bar 4}2d]     hh0 G 00l d′−     hhl G    
    [{\bar h}h0]    
    d d d
    124 [P\;4/m\;2/c\;2/c] 00l c′−     00l c′− h0l, 0kl G hhl G    
    c c c c
    125 [P\;4/n\;2/b\;2/m] 0k0 b′− h00, 0k0 n′−     h0l, 0kl G     hk0 G
    n a b a b n
    126 [P\;4/n\;2/n\;2/c] 0k0 n′− h00, 0k0 n′− 00l c′− h0l, 0kl G hhl G hk0 G
    n
    00l n c n c n
    n
    127 [P\;4/m\;2_{1}/b\;2/m] 0k0 b′− h00 GS     h0l, 0kl G        
    [a + 2_{1}] a b
    [2_{1}] 0k0 0k0, h00 S
    [b + 2_{1}] [2_{1}]
    128 [P\;4/m\;2_{1}/n\;2/c] 00l, 0k0 n′− h00, 0k0 GS 00l c′− h0l, 0kl G hhl G    
    n
    [n\quad 2_{1}] [n + 2_{1}] c 0k0, h00 S c
    [2_{1}]
    129 [P\;4/n\;2_{1}/m\;2/m] 0k0 GS h00, 0k0 n′−     0k0, h00 S     hk0 G
    [n + 2_{1}] [2_{1}] [2_{1}] n
    130 [P\;4/n\;2_{1}/c\;2/c] 0k0 GS h00, 0k0 n′− 00l c′− h0l, 0kl G hhl G hk0 G
    [n + 2_{1}] c
    00l c′− [2_{1}] c 0k0, h00 S c n
    c [2_{1}]
    131 [P\;4_{2}/m\;2/m\;2/c]         00l c     hhl G    
    c c
    132 [P\;4_{2}/m\;2/c\;2/m] 00l c′−         h0l, 0kl G        
    c c
    133 [P\;4_{2}/n\;2/b\;2/c] 0k0 b′− h00, 0k0 n′− 00l c′− h0l, 0kl G hhl G hk0 G
    n a b c a b c n
    134 [P\;4_{2}/n\;2/n\;2/m] 0k0, 00l n′− h00, 0k0 n′−     h0l, 0kl G     hk0 G
    n n n n
    135 [P\;4_{2}/m\;2_{1}/b\;2/c] 0k0 b′− h00, 0k0 GS 00l c′− h0l, 0kl G hhl G    
    a b
    [2_{1}] [a + 2_{1}\;b + 2_{1}] c 0k0, h00 S c
    [2_{1}]
    136 [P\;4_{2}/m\;2_{1}/n\;2/m] 00l, 0k0 n′− h00, 0k0 GS     h0l, 0kl G        
    n
    n 2 [n + 2_{1}] 0k0, h00 S
    [2_{1}]
    137 [P\;4_{2}/n\;2_{1}/m\;2/c] 0k0 GS h00, 0k0 n′− 00l c′− 0k0, h00 S hhl G hk0 G
    [n + 2_{1}] [2_{1}] c [2_{1}] c n
    138 [P\;4_{2}/n\;2_{1}/c\;2/m] 0k0 GS h00, 0k0 n′−     h0l, 0kl G     hk0 G
    [n + 2_{1}] c
    00l c′− [2_{1}] 0k0, h00 S n
    c [2_{1}]
    140 [I\;4/m\;2/c\;2/m]             h0l, 0kl G        
    c
    141 [I\;4_{1}/a\;2/m\;2/d]     hh0 a′− 00l, [{\bar h}h0] d′−     hhl G hk0 G
    [{\bar h}h0]      
    d d a d a
    142 [I\;4_{1}/a\;2/c\;2/d]     hh0 a′− 00l, [{\bar h}h0] d′− h0l, 0kl G hhl G hk0 G
    [{\bar h}h0]        
    d d a c d a

    Point groups 3m, [{\bar{\bf 3}}] m, 6, 6/m, 622, 6mm, [{\bar{\bf 6}}] m2, 6/mmm

    Space groupIncident-beam direction
    [100][210][2u u w][v0w]
    158 P3c1     000l G [hh2{\bar h}l] G    
    c c
    159 P31c 000l G         [h{\bar h}0l] G
    c c
    161 R3c     000l G [hh2{\bar h}l] G    
    [l = 6n + 3]  
    c c
    163 [P{\bar 3}1c] 000l G         [h{\bar h}0l] G
    c c
    165 [P{\bar 3}c1]     000l G [hh2{\bar h}l] G    
    c c
    167 [R{\bar 3}c]     000l G [hh2{\bar h}l] G    
    [l = 6n + 3]  
    c c
    169 [P6_{1}] 000l S 000l S        
    [6_{1}] [6_{1}]
    170 [P6_{5}] 00l S 00l S        
    [6_{5}] [6_{5}]
    173 [P6_{3}] 000l S 000l S        
    [6_{3}] [6_{3}]
    176 [P6_{3}/m] 000l S 000l S        
    [6_{3}] [6_{3}]
    178 [P6_{1}22] 000l S 000l S        
    [6_{1}] [6_{1}]
    179 [P6_{5}22] 000l S 000l S        
    [6_{5}] [6_{5}]
    182 [P6_{3}22] 000l S 000l S        
    [6_{3}] [6_{3}]
    184 [P6cc] 000l c′− 000l c′− [hh2{\bar h}l] G [h{\bar h}0l] G
    c c c c
    185 [P6_{3}cm] 000l c′− 000l GS [hh2{\bar h}l] G    
    [6_{3}] [c + 6_{3}] c
    186 [P6_{3}mc] 000l GS 000l c′−     [h{\bar h}0l] G
    [c + 6_{3}] [6_{3}] c
    188 [P{\bar 6}c2]     000l G [hh2{\bar h}l] G    
    c c
    190 [P{\bar 6}c2] 000l G         [h{\bar h}0l] G
    c c
    192 [P6/mcc] 000l c′− 000l c′− [hh2{\bar h}l] G [h{\bar h}0l] G
    c c c c
    193 [P6_{3}/mcm] 00l c′− 000l GS [hh2{\bar h}l] G    
    [6_{3}] [c + 6_{3}] c
    194 [P6_{3}/mmc] 000l GS 000l c′−     [h{\bar h}0l] G
    [c + 6_{3}] [6_{3}] c

    Point groups 23, m3, 432, m3m

    Space groupIncident-beam direction
    [100][110][uv0][uuw]
    (cyclic)(cyclic)(cyclic)(cyclic)
    198 P [2_{1}]3 00l, 0k0 S 00l S 00l S    
    [2_{1}] [2_{1}] [2_{1}]
    201 [Pn\bar{3}] 00l, 0k0 n′−     [{\bar k}h0] G    
    [P2/n{\bar 3}] n n
    203 Pd [{\bar 3}] 00l, 0k0 d′−     [{\bar k}h0] G    
    [F2/d{\bar 3}] d d
    205 Pa [{\bar 3}] 00l GS 00l S 00l S    
    [c + 2_{1}] [2_{1}] [2_{1}]
    [P2_{1}/a{\bar 3}] 0k0 b′− [{\bar h}h0] G [{\bar k}h0] G
    [2_{1}] a a
    206 Ia [{\bar 3}]     [{\bar h}h0] G [{\bar k}h0] G    
    [I2_{1}/a{\bar 3}] a a
    212 [P4_{3}32]         00l S    
    [4_{3}]
    213 [P4_{1}32]         00l S    
    [4_{1}]
    218 [P{\bar 4}3n]     00l n hhl G    
    c c
    219 [F{\bar 4}3c]         hhl G    
    c
    220 [I{\bar 4}3d] 0kk G 00l d hhl G    
    [0{\bar k}k]    
    d d d
    222 [Pn{\bar 3}n] 00l, 0k0 n′− 00l n hk0 G hhl G
    n c n c
    223 [Pm{\bar 3}n]     00l n′−     hhl G
    c c
    224 [Pn{\bar 3}m] 00l, 0k0 n′−     hk0 G    
    n n
    226 [Fm{\bar 3}c]             hhl G
    c
    227 [Fd{\bar 3}m] 00l, 0k0 d′−     hk0 G    
    d d
    228 [Fd{\bar 3}c] 00l, 0k0 d′−     hk0 G hhl G
    d d c
    230 [Ia{\bar 3}d] 0kk b′− 00l, [{\bar h}h0] d′− hk0 G hhl G
    [0{\bar k}k]      
    d d a a d
    Conditions in this column are cyclic on h and k.
  • Table 2.5.3.5[link]. This lists conditions for observation of GS bands for the 137 space groups exhibiting these extinctions. These are entered as `G', `S', or `GS', indicating whether a glide plane, screw axis, or both is responsible for the GS band. All three possibilities will lead to a glide line (and hence to both extinction bands) in projection, and one of the procedures (a), (b), or (c) of Section 2.5.3.3[link](iv)[link] above is needed to complete the three-dimensional interpretation. In addition, the presence of horizontal glide planes, which result in systematic absences in these particular cases in the zero-layer pattern, is indicated by the symbol `−'. Where these occur at the site of prospective `G' or `S' bands from other glide or screw elements the symbol of that element is given and the `G' or `S' symbol is omitted.

    The following paragraphs give information on the real-space interpretation of GS band formation, and their specific extinction rules, considered useful in structural interpretation.

    Real-space interpretation of extinction conditions . Dynamic extinctions (GS bands) are essentially a property of symmetry in reciprocal space. However, since diagrams from IT I and A are used there is a need to give an equivalent real-space description. These bands are associated with the half-unit-cell-translational glide planes and screw axes represented in these diagrams. Inconsistencies between `conventional' and `physical' real-space descriptions, however, become more apparent in dynamical electron diffraction, which is dependent upon three-dimensional scattering physics, than in X-ray diffraction. Also, the distinction between general (symmorphic) and specific (non-symmorphic) extinctions is more basic (in the former case). This is clarified by the following points:

    • (i) Bravais lattice centring restricts the conditions for observation of GS bands. For example, in space group Abm2 (No. 39), `A' centring prevents observation of the GS bands associated with the `b' glide at the [001] zone-axis orientation; this observation, and hence verification of the b glide, must be made at the lower-symmetry zone axes [0vw] (see Table 2.5.3.5)[link]. In the exceptional cases of space groups [I2_{1}2_{1}2_{1}] and [I2_{1}3] (Nos. 24 and 199), conditions for the observation of the relevant GS bands are completely prevented by body centring; here the screw axes of the symmorphic groups I222 and I23 are parallel to the screw axes of their non-symmorphic derivatives. However, electron crystallographic methods also include direct structure imaging by HREM, and it is important to note here that while the indistinguishability encountered in data sets acquired in Fourier space applies to both X-ray diffraction and CBED (notwithstanding possible differences in HOLZ symmetries), this limitation does not apply to the HREM images (produced by dynamic scattering) yielding an approximate structure image for the (zone-axis) projection. This technique then becomes a powerful tool in space-group research by supplying phase information in a different form.

    • (ii) A different complication, relating to nomenclature, occurs in the space groups [P\bar{4}3n], Pn3n and Pm3n (Nos. 218, 222 and 223) where `c' glides parallel to a diagonal plane of the unit cell occur as primary non-symmorphic elements (responsible for reciprocal-space extinctions) but are not used in the Hermann–Maugin symbol; instead the derivative `n' glide planes are used as characters, resulting in an apparent lack of correspondence between the conventionally given real-space symbols and the reciprocal-space extinctions.

      (Note: In IT I non-symmorphic reflection rules which duplicate rules given by lattice centring, or those which are a consequence of more general rules, are given in parentheses; in IT A this clarification by parenthesizing, helpful for electron-diffraction analysis, has been removed.)

    • (iii) Finally, diamond glides (symbol `d') require special consideration since they are associated with translations [{1 \over 4}], [{1 \over 4}], [{1 \over 4}], and so would appear not to qualify for GS bands; however, this translation is a result of the conventional cell being defined in real rather than reciprocal space where the extinction symmetry is formed. Hence `d' glides occur only in F-centred lattices (most obviously Nos. 43, 70, 203, 277 and 228). These have correspondingly an I-centred reciprocal lattice for which the zero-layer two-dimensional unit cell has an edge of [a^{*\prime\prime} = 2a^{*}]. Consequently, the first-order row reflection along the diamond glide retains the reciprocal-space anti-symmetry on the basis of this physical unit cell (halved in real space), and leads to the labelling of odd-order reflections as [4n + 2] (instead of [2n + 1] when the cell is not halved). Additionally, although seven space groups are I-centred in real space with the conventional unit cell (Nos. 109, 110, 122, 141, 142, 220 and 230), these space groups are F-centred with the transformation [a'' = [110]], [b'' = [1\bar{1}0]], and correspondingly I-centred in the reciprocal-space cell as before, but the directions [100], [010] and reflection rows h00, 0k0 become replaced by directions [110] (or [[1\bar{1}0]]) and rows hh0, [h\bar{h}0], in the entries of Table 2.5.3.5[link].

    Extinction rules for symmetry elements appearing in Table 2.5.3.5[link]. Reflection indices permitting observation of G and S bands follow [here `zero-layer' and `out-of-zone' (i.e. HOLZ or alternative zone) serve to emphasize that these are zone-axis observations].

    • (i) Vertical glide planes lead to `G' bands in reflections as listed (`a', `b', `c' and `n' glides):

      h0l, hk0, 0kl out-of-zone reflections (for glide planes having normals [010], [001] and [100]) having [h + l], [h + k], [k + l = 2n + 1], respectively, in the case of `n' glides, or h, k, l odd in the case of `a', `b' or `c' glides, respectively;

      h00, 0k0, 00l zero-layer reflections with h, k or l odd.

      Correspondingly for `d' glides:

      • (a) In F-centred cells:

        h0l, hk0, 0kl out-of-zone reflections (for glide planes having normals [010], [100] and [001], having [h + l], [k + l], or [h + k = 4n + 2], respectively, with h, k and l even; and (space group No. 43 only) zero-layer reflections h00, 0k0 with h, k even and [= 4n + 2].

      • (b) In I-centred cells:

        hhl (cyclic on h, k, l for cubic groups) out-of-zone reflections having [2h + l = 4n + 2], with l even; and zero-layer reflections hh0, [h\bar{h}0] (cyclic on h, k, l for cubic groups) having h odd.

    • (ii) Horizontal screw axes, namely [2_{1}] or the [2_{1}] component of screw axes [4_{1}], [4_{3}], [6_{1}], [6_{3}], [6_{5}], lead to `S' bands in reflection rows parallel to the screw axis, i.e. either h00, 0k0 or 00l, with conventional indexing, for h, k or l odd.

    • (iii) Horizontal glide planes lead to zero-layer absences rather than GS bands. When these prevent observation of a specific GS band (by removing the two-dimensional conditions), the symbol `−' indicates a situation where, in general, there will simply be absences for the odd-order reflections. However, Ishizuka & Taftø (1982)[link] were the first to observe finite-intensity narrow bands under these conditions, and it is now appreciated that with a sufficient crystal thickness and a certain minimum for the z-axis repeat distance, GS bands can be recorded by violating the condition for horizontal-mirror-plane (m′) extinction while satisfying the condition for G or S, achieved by appropriate tilts away from the exact zone-axis orientation [see Section 2.5.3.3[link](iv)[link]].

References

First citation Buxton, B., Eades, J. A., Steeds, J. W. & Rackham, G. M. (1976). The symmetry of electron diffraction zone axis patterns. Philos. Trans. R. Soc. London Ser. A, 181, 171–193.Google Scholar
First citation Goodman, P. (1984b). A retabulation of the 80 layer groups for electron diffraction usage. Acta Cryst. A40, 633–642.Google Scholar
First citation Ishizuka, K. & Taftø, J. (1982). Kinematically allowed reflections caused by scattering via HOLZ. Proc. Electron Microsc. Soc. Am. pp. 688–689.Google Scholar
First citation Pond, R. C. & Vlachavas, D. S. (1983). Bicrystallography. Proc. R. Soc. London Ser. A, 386, 95–143.Google Scholar
First citation Steeds, J. W. (1983). Developments in convergent beam electron diffraction. Report to the Commission on Electron Diffraction of the International Union of Crystallography.Google Scholar
First citation Tanaka, M., Sekii, H. & Nagasawa, T. (1983). Space group determination by dynamic extinction in convergent beam electron diffraction. Acta Cryst. A39, 825–837.Google Scholar
First citation Tanaka, M. & Terauchi, M. (1985). Convergent-beam electron diffraction. Tokyo: JEOL Ltd.Google Scholar








































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