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

International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 29-32

Section 2.2.13. Reflection conditions

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.13. Reflection conditions

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The Reflection conditions5 are listed in the right-hand column of each Wyckoff position.

These conditions are formulated here, in accordance with general practice, as `conditions of occurrence' (structure factor not systematically zero) and not as `extinctions' or `systematic absences' (structure factor zero). Reflection conditions are listed for all those three-, two- and one-dimensional sets of reflections for which extinctions exist; hence, for those nets or rows that are not listed, no reflection conditions apply.

There are two types of systematic reflection conditions for diffraction of crystals by radiation:

  • (1) General conditions. They apply to all Wyckoff positions of a space group, i.e. they are always obeyed, irrespective of which Wyckoff positions are occupied by atoms in a particular crystal structure.

  • (2) Special conditions (`extra' conditions). They apply only to special Wyckoff positions and occur always in addition to the general conditions of the space group. Note that each extra condition is valid only for the scattering contribution of those atoms that are located in the relevant special Wyckoff position. If the special position is occupied by atoms whose scattering power is high, in comparison with the other atoms in the structure, reflections violating the extra condition will be weak.

2.2.13.1. General reflection conditions

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These are due to one of three effects:

  • (i) Centred cells. The resulting conditions apply to the whole three-dimensional set of reflections hkl. Accordingly, they are called integral reflection conditions. They are given in Table 2.2.13.1[link]. These conditions result from the centring vectors of centred cells. They disappear if a primitive cell is chosen instead of a centred cell. Note that the centring symbol and the corresponding integral reflection condition may change with a change of the basis vectors (e.g. monoclinic: [C \rightarrow A \rightarrow I]).

    Table 2.2.13.1| top | pdf |
    Integral reflection conditions for centred cells (lattices)

    Reflection conditionCentring type of cellCentring symbol
    None Primitive [\left\{\matrix{P\cr R\cr}\right.][{\lower6pt\hbox{(rhombohedral axes)}}]
    [h + k = 2n] C-face centred C
    [k + l = 2n] A-face centred A
    [h + l = 2n] B-face centred B
    [h + k + l = 2n] Body centred I
    [h + k, h + l] and All-face centred F
    [k + l = 2n] or:    
    [h, k, l] all odd or all even (`unmixed')    
    [{-h + k + l = 3n}] Rhombohedrally centred, obverse setting (standard) [\!\!\left.{\matrix{{}\cr{}\cr{}\cr{}\cr{}\cr{}}}\right\}{R}] (hexagonal axes)
    [h - k + l = 3n] Rhombohedrally centred, reverse setting
    [h - k = 3n] Hexagonally centred H
    For further explanations see Chapters 1.2[link] and 2.1[link] .
    For the use of the unconventional H cell, see Chapter 1.2[link] .
  • (ii) Glide planes. The resulting conditions apply only to two-dimensional sets of reflections, i.e. to reciprocal-lattice nets containing the origin (such as hk0, h0l, 0kl, hhl). For this reason, they are called zonal reflection conditions. The indices hkl of these `zonal reflections' obey the relation [hu + kv + lw = 0], where [uvw], the direction of the zone axis, is normal to the reciprocal-lattice net. Note that the symbol of a glide plane and the corresponding zonal reflection condition may change with a change of the basis vectors (e.g. monoclinic: [c \rightarrow n \rightarrow a]).

  • (iii) Screw axes. The resulting conditions apply only to one-dimensional sets of reflections, i.e. reciprocal-lattice rows containing the origin (such as h00, 0k0, 00l). They are called serial reflection conditions.

Reflection conditions of types (ii)[link] and (iii)[link] are listed in Table 2.2.13.2[link]. They can be understood as follows: Zonal and serial reflections form two- or one-dimensional sections through the origin of reciprocal space. In direct space, they correspond to projections of a crystal structure onto a plane or onto a line. Glide planes or screw axes may reduce the translation periods in these projections (cf. Section 2.2.14[link]) and thus decrease the size of the projected cell. As a consequence, the cells in the corresponding reciprocal-lattice sections are increased, which means that systematic absences of reflections occur.

Table 2.2.13.2| top | pdf |
Zonal and serial reflection conditions for glide planes and screw axes (cf. Chapter 1.3[link] )

(a) Glide planes

Type of reflectionsReflection conditionGlide planeCrystallographic coordinate system to which condition applies
Orientation of planeGlide vectorSymbol
0kl [k = 2n] (100) [{\bf b}/2] b [\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (a\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[l = 2n] [{\bf c}/2] c
[k + l = 2n] [{\bf b}/2 + {\bf c}/2] n
[\!\matrix{k + l = 4n\hfill\cr \quad (k,l = 2n)\hfill\cr}] [{\bf b}/4 \pm {\bf c}/4] d
h0l [l = 2n] (010) [{\bf c}/2] c [\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic } (b\hbox{ unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[h = 2n] [{\bf a}/2] a
[l + h = 2n] [{\bf c}/2 + {\bf a}/2] n
[\!\matrix{l + h = 4n\hfill\cr \quad (l,h = 2n)\hfill\cr}] [{\bf c}/4 \pm {\bf a}/4] d
hk0 [h = 2n] (001) [{\bf a}/2] a [\!\left.\matrix{\cr\cr\cr\cr\cr}\!\right\}\!\matrix{\hbox{Monoclinic (}c\ \hbox{unique)},\hfill\cr\quad\hbox{Tetragonal}\hfill\cr}] [\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr}\right\}\!\matrix{\hbox{Orthorhombic,}\hfill\cr\quad\hbox{Cubic}\hfill\cr}]
[k = 2n] [{\bf b}/2] b
[h + k = 2n] [{\bf a}/2 + {\bf b}/2] n
[\!\matrix{h + k = 4n\hfill\cr \quad (h,k = 2n)\hfill\cr}] [{\bf a}/4 \pm {\bf b}/4] d
[\matrix{h\bar{h}0l\hfill\cr 0k\bar{k}l\hfill\cr \bar{h}0hl\hfill\cr}] [l = 2n] [\left.\!\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}\right\} \{11\bar{2}0\}] [{\bf c}/2] c [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]
[\matrix{hh.\overline{2h}.l\hfill\cr \overline{2h}.hhl\hfill\cr h.\overline{2h}.hl\hfill\cr}] [l = 2n] [\left.\!\matrix{(1\bar{1}00)\hfill\cr (01\bar{1}0)\hfill\cr (\bar{1}010)\hfill\cr}\right\} \{1\bar{1}00\}] [{\bf c}/2] c [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Hexagonal}]
[\matrix{hhl\hfill\cr hkk\hfill\cr hkh\hfill\cr}] [\matrix{l = 2n\hfill\cr h = 2n\hfill\cr k = 2n\hfill\cr}] [\left.\!\matrix{(1\bar{1}0)\hfill\cr (01\bar{1})\hfill\cr (\bar{1}01)\hfill\cr}\right\} \{1\bar{1}0\}] [\matrix{{\bf c}/2\hfill\cr {\bf a}/2\hfill\cr {\bf b}/2\hfill\cr}] [\matrix{c,n\hfill\cr a,n\hfill\cr b,n\hfill\cr}] [\left.\vphantom{\matrix{(11\bar{2}0)\hfill\cr (\bar{2}110)\hfill\cr (1\bar{2}10)\hfill\cr}}\right\}\hbox{Rhombohedral}]
[hhl, h\bar{h}l] [l = 2n] [(1\bar{1}0), (110)] [{\bf c}/2] c, n [\!\left.\matrix{\cr\cr\cr}\right\}\hbox{Tetragonal}]§ [\!\left.\matrix{\cr\cr\cr\cr\cr\cr\cr\cr\cr\cr}\right\}\hbox{Cubic}]
[2h + l = 4n] [{\bf a}/4 \pm {\bf b}/4 \pm {\bf c}/4] d
[hkk, hk\bar{k}] [h = 2n] [(01\bar{1}), (011)] [{\bf a}/2] a, n
[2k + h = 4n] [\pm {\bf a}/4 + {\bf b}/4 \pm {\bf c}/4] d
[hkh, \bar{h}kh] [k = 2n] [(\bar{1}01), (101)] [{\bf b}/2] b, n
[2h + k = 4n] [\pm {\bf a}/4 \pm {\bf b}/4 + {\bf c}/4] d

(b) Screw axes

Type of reflectionsReflection conditionsScrew axisCrystallographic coordinate system to which condition applies
Direction of axisScrew vectorSymbol
h00 [h = 2n] [100] [{\bf a}/2] [2_{1}] [\left\{\matrix{\hbox{Monoclinic }(a\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[h = 4n] [{\bf a}/4] [4_{1},4_{3}]
0k0 [k = 2n] [010] [{\bf b}/2] [2_{1}] [\left\{\matrix{\hbox{Monoclinic }(b\hbox{ unique}),\hfill\cr\quad\hbox{Orthorhombic, Tetragonal}\cr}\right.] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[k = 4n] [{\bf b}/4] [4_{1},4_{3}]
00l [l = 2n] [001] [{\bf c}/2] [2_{1}] [\matrix{\left\{\matrix{\hbox{Monoclinic } (c\hbox{ unique}),\hfill\cr\quad\hbox{ Orthorhombic}\hfill\cr}\right.\cr\left.\matrix{\noalign{\vskip31pt}\cr}\right\}\hbox{Tetragonal}\hfill}] [\left.\matrix{\noalign{\vskip56pt}\cr}\right\}\hbox{Cubic}]
[4_{2}]
[l = 4n] [{\bf c}/4] [4_{1},4_{3}]
000l [l = 2n] [001] [{\bf c}/2] [6_{3}] [\left.{\hbox to 6pt{}}\matrix{\noalign{\vskip54pt}}\right\}\hbox{Hexagonal}]
[l = 3n] [{\bf c}/3] [3_{1},3_{2},6_{2},6_{4}]
[l = 6n] [{\bf c}/6] [6_{1},6_{5}]
Glide planes d with orientations (100), (010) and (001) occur only in orthorhombic and cubic F space groups. Combination of the integral reflection condition (hkl: all odd or all even) with the zonal conditions for the d glide planes leads to the further conditions given between parentheses.
For rhombohedral space groups described with `rhombohedral axes' the three reflection conditions [(l = 2n, h = 2n, k = 2n)] imply interleaving of c and n glides, a and n glides, b and n glides, respectively. In the Hermann–Mauguin space-group symbols, c is always used, as in R3c (161) and [R\bar{3}c\ (167)], because c glides occur also in the hexagonal description of these space groups.
§For tetragonal P space groups, the two reflection conditions (hhl and [h\bar{h}l] with [l = 2n]) imply interleaving of c and n glides. In the Hermann–Mauguin space-group symbols, c is always used, irrespective of which glide planes contain the origin: cf. P4cc (103), [P\bar{4}2c\ (112)] and [P4/nnc\ (126)].
For cubic space groups, the three reflection conditions [(l = 2n, h = 2n, k = 2n)] imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin space-group symbols, either c or n is used, depending upon which glide plane contains the origin, cf. [P\bar{4}3n\ (218)], [Pn\bar{3}n\ (222)], [Pm\bar{3}n\ (223)] vs [F\bar{4}3c\ (219)], [Fm\bar{3}c\ (226)], [Fd\bar{3}c\ (228)].

For the two-dimensional groups, the reasoning is analogous. The reflection conditions for the plane groups are assembled in Table 2.2.13.3[link].

Table 2.2.13.3| top | pdf |
Reflection conditions for the plane groups

Type of reflectionsReflection conditionCentring type of plane cell; or glide line with glide vectorCoordinate system to which condition applies
hk None Primitive p All systems
[h + k = 2n] Centred c Rectangular
[h - k = 3n] Hexagonally centred h Hexagonal
h0 [h = 2n] Glide line g normal to b axis; glide vector [{1 \over 2}{\bf a}] [\left.\matrix{\noalign{\vskip 54pt}}\right\}\matrix{\hbox{Rectangular, }\hfill\cr\quad\hbox{Square}\hfill}]
0k [k = 2n] Glide line g normal to a axis; glide vector [{1 \over 2}{\bf b}]
For the use of the unconventional h cell see Chapter 1.2[link] .

For the interpretation of observed reflections, the general reflection conditions must be studied in the order (i)[link][link] to (iii)[link], as conditions of type (ii)[link] may be included in those of type (i)[link], while conditions of type (iii)[link] may be included in those of types (i)[link] or (ii)[link]. This is shown in the example below.

In the space-group tables, the reflection conditions are given according to the following rules:

  • (i) for a given space group, all reflection conditions are listed; hence for those nets or rows that are not listed no conditions apply. No distinction is made between `independent' and `included' conditions, as was done in IT (1952)[link], where `included' conditions were placed in parentheses;

  • (ii) the integral condition, if present, is always listed first, followed by the zonal and serial conditions;

  • (iii) conditions that have to be satisified simultaneously are separated by a comma or by `AND'. Thus, if two indices must be even, say h and l, the condition is written [h, l = 2n] rather than [h = 2n] and [l = 2n]. The same applies to sums of indices. Thus, there are several different ways to express the integral conditions for an F-centred lattice: `[h + k, h + l, k + l = 2n]' or `[h + k, h + l = 2n] and [k + l = 2n]' or `[h + k = 2n] and [h + l, k + l = 2n]' (cf. Table 2.2.13.1[link]);

  • (iv) conditions separated by `OR' are alternative conditions. For example, `[hkl: h = 2n + 1] or [h + k + l = 4n]' means that hkl is `present' if either the condition [h = 2n + 1] or the alternative condition [h + k + l = 4n] is fulfilled. Obviously, hkl is a `present' reflection also if both conditions are satisfied. Note that `or' conditions occur only for the special conditions described in Section 2.2.13.2[link];

  • (v) in crystal systems with two or more symmetrically equivalent nets or rows (tetragonal and higher), only one representative set (the first one in Table 2.2.13.2[link]) is listed; e.g. tetragonal: only the first members of the equivalent sets 0kl and h0l or h00 and 0k0 are listed;

  • (vi) for cubic space groups, it is stated that the indices hkl are `cyclically permutable' or `permutable'. The cyclic permutability of h, k and l in all rhombohedral space groups, described with `rhombohedral axes', and of h and k in some tetragonal space groups are not stated;

  • (vii) in the `hexagonal-axes' descriptions of trigonal and hexagonal space groups, Bravais–Miller indices hkil are used. They obey two conditions:

    • (a) [h + k + i = 0,\ i.e.\ i = - (h + k)];

    • (b) the indices h, k, i are cyclically permutable; this is not stated. Further details can be found in textbooks of crystallography.

Note that the integral reflection conditions for a rhombohedral lattice, described with `hexagonal axes', permit the presence of only one member of the pair hkil and [\bar{h}\bar{k}\bar{i{\phantom l}}\!\!l] for [l \neq 3n] (cf. Table 2.2.13.1[link]). This applies also to the zonal reflections [h\bar{h}0l] and [\bar{h}h0l], which for the rhombohedral space groups must be considered separately.

Example

For a monoclinic crystal (b unique), the following reflection conditions have been observed:

  • (1) [hkl\!\! :h + k = 2n{\hbox{;}}]

  • (2) [0kl\!\! :k = 2n{\hbox{;}}\quad h0l\!\!:h,l = 2n{\hbox{;}}\quad hk0\!\!:h + k = 2n{\hbox{;}}]

  • (3) [h00\!\! :h = 2n{\hbox{;}}\quad 0k0\!\!:k = 2n{\hbox{;}}\quad 00l\!\!:l = 2n.]

Line (1) states that the cell used for the description of the space group is C centred. In line (2), the conditions 0kl with [k = 2n], h0l with [h = 2n] and hk0 with [h + k = 2n] are a consequence of the integral condition (1), leaving only h0l with [l = 2n] as a new condition. This indicates a glide plane c. Line (3) presents no new condition, since h00 with [h = 2n] and 0k0 with [k = 2n] follow from the integral condition (1), whereas 00l with [l = 2n] is a consequence of a zonal condition (2). Accordingly, there need not be a twofold screw axis along [010]. Space groups obeying the conditions are Cc (9, b unique, cell choice 1) and [C2/c] (15, b unique, cell choice 1). On the basis of diffraction symmetry and reflection conditions, no choice between the two space groups can be made (cf. Part 3[link] ).

For a different choice of the basis vectors, the reflection conditions would appear in a different form owing to the transformation of the reflection indices (cf. cell choices 2 and 3 for space groups Cc and [C2/c] in Part 7[link] ).

2.2.13.2. Special or `extra' reflection conditions

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These apply either to the integral reflections hkl or to particular sets of zonal or serial reflections. In the space-group tables, the minimal special conditions are listed that, on combination with the general conditions, are sufficient to generate the complete set of conditions. This will be apparent from the examples below.

Examples

  • (1) [P4_{2}22\ (93)]

    General position [8p\!: 00l\!\!:\;l = 2n], due to [4_{2}]; the projection on [001] of any crystal structure with this space group has periodicity [{1 \over 2}c].

    Special position [4i\!\!: hkl\!\!:\;h + k + l = 2n]; any set of symmetrically equivalent atoms in this position displays additional I centring.

    Special position [4n\!\!: 0kl\!\!:\;l = 2n]; any set of equivalent atoms in this position displays a glide plane [c \perp [100]]. Projection of this set along [100] results in a halving of the original c axis, whence the special condition. Analogously for [h0l]: [l=2n].

  • (2) [C12/c1] (15, unique axis b, cell choice 1)

    General position [8f\!\!: hkl\!\!:\;h + k = 2n], due to the C-centred cell.

    Special position [4d\!\!: hkl\!\!:\;k + l = 2n], due to additional A and B centring for atoms in this position. Combination with the general condition results in [hkl\!\!:\;h + k, h + l, k + l = 2n] or hkl all odd or all even; this corresponds to an F-centred arrangement of atoms in this position.

    Special position [4b\!\!:\;hkl\!\!: l = 2n], due to additional halving of the c axis for atoms in this position. Combination with the general condition results in [hkl\!\!:\;h + k,l = 2n]; this corresponds to a C-centred arrangement in a cell with half the original c axis. No further condition results from the combination.

  • (3) [I12/a1] (15, unique axis b, cell choice 3)

    For the description of space group No. 15 with cell choice 3 (see Section 2.2.16[link] and space-group tables), the reflection conditions appear as follows:

    General position [8f\!\!: hkl\!\!:\; h + k + l = 2n], due to the I-centred cell.

    Special position [4b\!\!:\;hkl\!\!:\;h = 2n], due to additional halving of the a axis. Combination gives [hkl\!\!: h,k + l = 2n], i.e. an A-centred arrangement of atoms in a cell with half the original a axis.

    An analogous result is obtained for position 4d.

  • (4) Fmm2 (42)

    General position [16e\!\!: hkl\!\!: h + k,h + l,k + l = 2n], due to the F-centred cell.

    Special position [8b\!\!: hkl\!\!:\;h = 2n], due to additional halving of the a axis. Combination results in [hkl\!\!: h,k,l = 2n], i.e. all indices even; the atoms in this position are arranged in a primitive lattice with axes [{1 \over 2}a,\; {1 \over 2}b] and [{1 \over 2}c].

For the cases where the special reflection conditions are described by means of combinations of `OR' and `AND' instructions, the `AND' condition always has to be evaluated with priority, as shown by the following example.

Example: [P\bar{4}3n] (218)

Special position [6d\!\!: hkl\!\!: h+k+l = 2n] or [h = 2n+1], [k = 4n] and [l = 4n+2].

This expression contains the following two conditions:

(a) [hkl\!\!: h+k+l=2n];

(b) [h = 2n+1] and [k = 4n] and [l = 4n+2].

A reflection is `present' (occurring) if either condition (a) is satisfied or if a permutation of the three conditions in (b) are simultaneously fulfilled.

2.2.13.3. Structural or non-space-group absences

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Note that in addition non-space-group absences may occur that are not due to the symmetry of the space group (i.e. centred cells, glide planes or screw axes). Atoms in general or special positions may cause additional systematic absences if their coordinates assume special values [e.g. `noncharacteristic orbits' (Engel et al., 1984[link])]. Non-space-group absences may also occur for special arrangements of atoms (`false symmetry') in a crystal structure (cf. Templeton, 1956[link]; Sadanaga et al., 1978[link]). Non-space-group absences may occur also for polytypic structures; this is briefly discussed by Durovič in Section 9.2.2.2.5[link] of International Tables for Crystallography (2004)[link], Vol. C. Even though all these `structural absences' are fortuitous and due to the special arrangements of atoms in a particular crystal structure, they have the appearance of space-group absences. Occurrence of structural absences thus may lead to an incorrect assignment of the space group. Accordingly, the reflection conditions in the space-group tables must be considered as a minimal set of conditions.

The use of reflection conditions and of the symmetry of reflection intensities for space-group determination is described in Part 3[link] .

References

International Tables for Crystallography (2004). Vol. C, 3rd ed., edited by E. Prince. Dordrecht: Kluwer Academic Publishers.
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H. (1984). The non-characteristic orbits of the space groups. Z. Kristallogr., Supplement Issue No. 1.
Sadanaga, R., Takeuchi, Y. & Morimoto, N. (1978). Complex structures of minerals. Recent Prog. Nat. Sci. Jpn, 3, 141–206, esp. pp. 149–151.
Templeton, D. H. (1956). Systematic absences corresponding to false symmetry. Acta Cryst. 9, 199–200.








































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