International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 349352
Section 2.5.3.4.3. Spacegroup determination
M. Tanaka^{f}

Table 2.5.3.13 shows the spacegroup symbols (fourth column) of the modulated crystals. When a glide (screw) component τ_{4} between the modulation waves of two atom rows is 0, 1/2, ±1/3, ±1/4 or ±1/6, symbol 1, s, t, q or h is given, respectively (de Wolff et al., 1981). Such glide components are allowed for pointgroup symmetry 1 but are not for pointgroup symmetry . Dynamical extinction occurs for glide components s, q and h but does not for glide component t. When the average structure does not have a glide component, dynamical extinction due to a glide component τ_{4} appears in oddorder incommensurate reflections. When the average structure has a glide component, dynamical extinction due to a glide component τ_{4} appears in incommensurate reflections with h_{i} + h_{4} = 2n + 1, where h_{i} and h_{4} are the reflection indices for the average structure and incommensurate structure, respectively. Details are given in the paper by Terauchi et al. (1994).
Fig. 2.5.3.20(a) illustrates mirror symmetry () between atom rows A and B, which is perpendicular to the b axis with no glide component (τ_{4} = 0). Here, the wave number vector of the modulation is assumed to be k = k_{3}c* following the treatment of de Wolff et al. (1981). Fig. 2.5.3.20(b) illustrates glide symmetry () with a glide component τ_{4} = ½. The structure factor is written for the glide plane () of an infinite incommensurate crystal asThus, the following phase relations are obtained between the two structure factors:These relations are analogous to the phase relations between the two structure factors for an ordinary threedimensional crystal with a glide plane. The relations imply that dynamical extinction occurs for the glide planes and screw axes of the (3 + 1)dimensional crystal with an infinite dimension along the direction of the incommensurate modulation wavevector k. Terauchi et al. (1994) showed that approximate dynamical extinction occurs for an incommensurate crystal of finite dimension.

(a) Mirror symmetry of modulation waves () τ_{4} = 0. (b) Glide symmetry of modulation waves () τ_{4} = ½. The wave number vector of modulation is k_{3}c*. 
Fig. 2.5.3.21(a) and (b) illustrate a spot diffraction pattern and a CBED pattern, respectively, expected from a modulated crystal with a (3 + 1)dimensional space group (k = k_{3}c*) at the [100] incidence. The large and small spots in Fig. 2.5.3.21(a) designate the fundamental (h_{4} = 0) and incommensurate reflections (h_{4} ≠ 0), respectively. The 00h_{3}h_{4} (h_{4} = odd) reflections shown by crosses are kinematically forbidden by the glide plane () perpendicular to the b axis. Umweganregung paths a, b and c in the ZOLZ to a kinematically forbidden reflection are drawn. The two paths a and b are geometrically equivalent with respect to the line m–m perpendicular to the b axis. Since every Umweganregung path to a kinematically forbidden reflection contains an odd number of F(0h_{2,i}h_{3,i}h_{4,i}) with odd h_{4,i}, the following equation is obtained.where , and (h_{4} = odd).
When reflection 00h_{3}h_{4} (h_{4} = odd) is exactly excited, the two paths a and c are symmetric with respect to the bisector m′–m′ of the diffraction vector of the reflection and have the same excitation error. The waves passing through these paths have the same amplitude but different signs. Thus the following relation is obtained.where , and (h_{4} = odd).
Therefore, dynamical extinction occurs in kinematically forbidden reflections of incommensurate crystals. Fig. 2.5.3.21(b) schematically shows the extinction lines in oddorder incommensurate reflections, where the 0011 reflection is exactly excited.
We consider the dynamical extinction from Sr_{2}Nb_{2}O_{7} whose space group is . The glide plane () is perpendicular to the b axis with a glide vector (c + a_{4})/2. The wave number vector of the modulation is k = (½ − δ)a*. (Since spacegroup notation Cmc2_{1} is broadly accepted, the direction of the modulation is taken as the a axis.) The reflections h_{1}0h_{3}h_{4} with h_{3} + h_{4} = 2n + 1 (n = integer) are kinematically forbidden. Fig. 2.5.3.22 shows a schematic diffraction pattern of Sr_{2}Nb_{2}O_{7} at the [001] incidence. The large and small spots indicate the fundamental (h_{4} = 0) and incommensurate (h_{4} ≠ 0) reflections, respectively. Umweganregung paths a and b to the kinematically forbidden 0001 reflection via a fundamental reflection in the ZOLZ are drawn.
Fig. 2.5.3.23(a) shows a spot diffraction pattern of the incommensurate phase of Sr_{2}Nb_{2}O_{7} taken with the [001] incidence at 60 kV. The incommensurate reflections in which dynamical extinction lines appear at this incidence are those with the indices h_{1,even}00h_{4,odd} because h_{3} = 0 and h_{1} + h_{2} = 2n due to the lattice type C of the average structure.
The reflections in the four columns indicated by black arrowheads are incommensurate reflections. The reflections 0001, , and designated by white arrowheads are kinematically forbidden but exhibit certain intensities, which are caused by multiple diffraction. Other reflections are fundamental reflections due to the average structure.
Fig. 2.5.3.23(b) shows a CBED pattern corresponding to Fig. 2.5.3.23(a), taken from a specimen area 3 nm in diameter. The excitation errors of two Umweganregung paths a and b are the same at this electron incidence. The reflections 0001, , and indicated by white arrowheads show no intensity. Dynamical extinction does not appear as a line in the present case because the width of the extinction line exceeds the disc size of the reflections. Fig. 2.5.3.23(c) shows a CBED pattern taken at an incidence slightly tilted toward the b* axis from that for Fig. 2.5.3.23(b) or the [001] zoneaxis incidence. The excitation errors are no longer the same for the two Umweganregung paths. Thus, it is seen that the kinematically forbidden reflections indicated by white arrowheads have intensities due to incomplete cancellation of waves coming through different paths, which is an additional proof of the dynamical extinction.
References
Terauchi, M., Takahashi, M. & Tanaka, M. (1994). Convergentbeam electron diffraction study of incommensurately modulated crystals. II. (3+1)dimensional space groups. Acta Cryst. A50, 566–574.Wolff, P. M. de, Janssen, T. & Janner, A. (1981). The superspace groups for incommensurate crystal structures with a onedimensional modulation. Acta Cryst. A37, 625–636.