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. 344352
Section 2.5.3.4. Symmetry determination of incommensurate crystals
M. Tanaka^{f}

Incommensurately modulated crystals do not have threedimensional lattice periodicity. The crystals, however, recover lattice periodicity in a space higher than three dimensions. de Wolff (1974, 1977) showed that onedimensional displacive and substitutionally modulated crystals can be described as a threedimensional section of a (3 + 1)dimensional periodic crystal. Janner & Janssen (1980a,b) developed a more general approach for describing a modulated crystal with n modulations as (3 + n)dimensional periodic crystals (n = 1, 2, …). Yamamoto (1982) derived a general structurefactor formula for ndimensionally modulated crystals (n = 1, 2, …), which holds for both displacive and substitutionally modulated crystals. Tables of the (3 + 1)dimensional space groups for onedimensional incommensurately modulated crystals were given by de Wolff et al. (1981), where the wavevector of the modulation was assumed to lie in the c direction. Later, some corrections to the tables were made by Yamamoto et al. (1985). The analysis of incommensurately modulated crystals using (3 + 1)dimensional space groups has become familiar in the field of Xray structure analysis.
Fung et al. (1980) applied the CBED method to the study of incommensurately modulated transitionmetal dichalcogenides. Steeds et al. (1985) applied the LACBED method (Tanaka et al., 1980) to the study of incommensurately modulated crystals of NiGe_{1−x}P_{x}. Tanaka et al. (1988, pp. 74–81) examined the symmetries of the incommensurate and fundamental reflections appearing in the CBED patterns obtained from the incommensurately modulated crystals of Sr_{2}Nb_{2}O_{7} and Mo_{8}O_{23}. Terauchi & Tanaka (1993) clarified theoretically the interrelation between the symmetries of CBED patterns and the (3 + 1)dimensional pointgroup symbols for incommensurately modulated crystals and verified experimentally the theoretical results for Sr_{2}Nb_{2}O_{7} and Mo_{8}O_{23}. Terauchi et al. (1994) investigated dynamical extinction for the (3 + 1)dimensional space groups. They clarified that approximate dynamical extinction lines appear in CBED discs of the reflections caused by incommensurate modulations when the amplitudes of the incommensurate modulation waves are small. They tabulated the dynamical extinction lines appearing in the CBED discs for all the (3 + 1)dimensional space groups of the incommensurately modulated crystals. The tables were stored in the British Library Document Supply Centre as Supplementary Publication No. SUP 71810 (65 pp.). They showed an example of the dynamical extinction lines obtained from Sr_{2}Nb_{2}O_{7}. The point and spacegroup determinations of the (3 + 1)dimensional crystals are described compactly in the book by Tanaka et al. (1994, pp. 156–205).
Fig. 2.5.3.17 illustrates (3 + 1)dimensional descriptions of a crystal structure without modulation (a), a onedimensional displacive modulated structure (b) and a onedimensional substitutionally modulated structure (c). The arrows labelled a_{1}–a_{3} (a, b and c) and a_{4} indicate the (3 + 1)dimensional crystal axes. The horizontal line labelled R_{3} represents the threedimensional space (external space). In the (3 + 1)dimensional description, an atom is not located at a point as in the threedimensional space, but extends as a string along the fourth direction a_{4} perpendicular to the threedimensional space R_{3}. The shaded parallelogram is a unit cell in the (3 + 1)dimensional space. The unit cell contains two atom strings in this case. In the case of no modulations, the atoms are shown as straight strings, as shown in Fig. 2.5.3.17(a). For a displacive modulation, atoms are expressed by wavy strings periodic along the fourth direction a_{4} as shown in Fig. 2.5.3.17(b). The width of the atom strings indicates the spread of the atoms in R_{3}. The atom positions of the modulated structure in R_{3} are given as a threedimensional (R_{3}) section of the atom strings in the (3 + 1)dimensional space. A substitutional modulation, which is described by a modulation of the atom form factor, is expressed by atom strings with a density modulation along the direction a_{4} as shown in Fig. 2.5.3.17(c).
The diffraction vector G is written aswhere a set of h_{1}h_{2}h_{3}h_{4} is a (3 + 1)dimensional reflection index, and a*, b* and c* are the reciprocallattice vectors of the reallattice vectors a, b and c of the average structure. The modulation vector k is written aswhere one coefficient k_{i} (i = 1–3) is an irrational number and the others are rational. Fig. 2.5.3.18(a) shows a diffraction pattern of a crystal with an incommensurate modulation wavevector k_{1}a* (k_{2} and k_{3} = 0). Large and small black spots show the fundamental reflections and incommensurate reflections, respectively, only the firstorder incommensurate reflections being shown. It should be noted that the diffraction pattern of a modulated crystal is obtained by a projection of the Fourier transform of the (3 + 1)dimensional periodic structure. Fig. 2.5.3.18(b) is assumed to be the Fourier transform of Fig. 2.5.3.17(b). Incommensurate reflections are obtained by a projection of the reciprocallattice points onto .
The displacive modulation is expressed by the atom displacement with x_{4}. The structure factor for the (3 + 1)dimensional crystal with a displacive modulation is given by de Wolff (1974, 1977) as follows:where and (i = 1–3) are, respectively, the atom form factor and the ith component of the position of the μth atom in the unit cell of the average structure. The symbol is the ith component of the displacement of the μth atom. Since the atom in the (3 + 1)dimensional space is continuous along a_{4} and discrete along R_{3}, the structure factor is expressed by summation in R_{3} and integration along a_{4} as seen in equation (2.5.3.6). The integration implies that the sum for the atoms with displacements is taken over the infinite number of unit cells of the average structure. That is, equation (2.5.3.6) is the structure factor for a unit cell with the lattice parameter of an infinite length in R_{3} along the direction of the modulation wavevector k.
CBED patterns are obtained from a finite area of a specimen crystal. For the symmetry analysis of CBED patterns obtained from modulated structures, the effect of the finite size was considered by Terauchi & Tanaka (1993). The integration over a unitcell length along a_{4} in equation (2.5.3.6) is rewritten in the following way with the summation over a finite number of threedimensional sections of the atom strings:where , and , being the number of unit cells of the average structure included in a specimen volume from which CBED patterns are taken.
The substitutional modulation arises from a periodic variation of the siteoccupation probability of the atoms. This modulation is expressed by a modulation of the atom form factor with x_{4}. The structure factor for a finitesize crystal is written aswhere .
The symmetries of the CBED patterns can be determined by examination of the symmetries of the structure factor in equations (2.5.3.7) or (2.5.3.8). We consider a displacive modulated structure, which has a modulation wavevector k = k_{3}c* and belongs to (3 + 1)dimensional space group . This spacegroup symbol implies the following.
For the twofold rotation axis (symbol 2) of this space group, the structure factor is written aswhere . It is found from equation (2.5.3.9) that two structure factors and are the same when reflections and are equivalent with respect to the twofold rotation axis of the average structure. Thus, not only fundamental reflections (h_{4} = 0) from the average structure but also the satellite reflections (h_{4} ≠ 0) from the incommensurate structure show twofold rotation symmetry about the c* axis.
For the mirror plane (symbol m), the structure factor is written in a similar manner to the case of the twofold rotation axis. It is found that is not equal to for the incommensurate reflections h_{4} ≠ 0. Hence, the incommensurate reflections do not show mirror symmetry with respect to the mirror plane of the average structure. For the fundamental reflections (h_{4} = 0), is equal to , indicating the existence of mirror symmetry. It should be noted that the mirror symmetry can be destroyed by the dynamical diffraction effect between the fundamental and incommensurate reflections. In most modulated structures, however, the amplitude of the modulation wave is not so large as to destroy the symmetry of the fundamental reflections. Thus, the fundamental reflections ought to show the symmetry of the average structure, while the incommensurate reflections lose the symmetry.
The problem of the finite size of the illuminated area is discussed using equations (2.5.3.7) and (2.5.3.8) in a paper by Terauchi & Tanaka (1993) and in the book by Tanaka et al. (1994, pp. 156–205). The results are as follows: Even if the size and position of an illuminated specimen area are changed, the intensity distribution in a CBED pattern changes but the symmetry of the pattern does not. To obtain the symmetries of incommensurate crystals, it is not necessary to take CBED patterns from an area whose diameter is larger than the period of the modulated structure. The symmetries of the modulated structure can appear when more than one unit cell of the average structure is illuminated for displacive modulations. For substitutional modulations, a specimen volume that produces the average atom form factor is needed, namely a volume of about 1 nm diameter area and 50 nm thick.
Table 2.5.3.13 shows the pointgroup symmetries (third column) of the incommensurate reflections for the two pointgroup subsymbols. For symmetry subsymbol 1, both the fundamental and incommensurate reflections show the symmetries of the average structure. For symmetry subsymbol , the fundamental reflections show the symmetries of the average structure but the incommensurate reflections do not have any symmetry. These facts imply that the symmetries of the incommensurate reflections are determined by the point group of the average structure and the modulation wavevector k. In other words, observation of the symmetries of the incommensurate reflections is not necessary for the determination of the point groups, although it can ascertain the point groups of the modulated crystals.

An example of pointgroup determination is shown for the incommensurate phase of Sr_{2}Nb_{2}O_{7}. Many materials of the A_{2}B_{2}O_{7} family undergo phase transformations from space group Cmcm to Cmc2_{1} and further to P2_{1} with decreasing temperature. An incommensurate phase appears between the Cmc2_{1} phase and the P2_{1} phase. Sr_{2}Nb_{2}O_{7} transforms at 488 K from the Cmc2_{1} phase into the incommensurate phase with a modulation wavevector k = (½ − δ)a* (δ = 0.009–0.023) but does not transform into the P2_{1} phase. The space group of Sr_{2}Nb_{2}O_{7} was reported as (Yamamoto, 1988). (Since the spacegroup notation Cmc2_{1} is broadly accepted, the direction of the modulation is taken as the a axis.) The point group of the phase is . The modulation wavevector k is transformed to −k by the mirror symmetry operation perpendicular to the a axis () and by the twofold rotation symmetry operation about the c axis (). The wavevector is transformed into itself by the mirror symmetry operation perpendicular to the b axis ().
Fig. 2.5.3.19(a) shows a CBED pattern of the incommensurate phase of Sr_{2}Nb_{2}O_{7} taken with the [010] incidence at an accelerating voltage of 60 kV. The reflections indicated by arrowheads are the incommensurate reflections. Other reflections are the fundamental reflections. Since the pattern is produced by the interaction of the reflections in the zerothorder Laue zone, symmetry operations () and () act the same. These symmetries are confirmed by the fact that the fundamental reflections show mirror symmetry perpendicular to the a* axis (twofold rotation symmetry about the c* axis) but the incommensurate reflections do not. Fig. 2.5.3.19(b) shows a CBED pattern of the incommensurate phase of Sr_{2}Nb_{2}O_{7} taken with the [201] incidence at 60 kV. The reflections in the two rows indicated by arrowheads are the incommensurate reflections and the others are the fundamental reflections. Symmetry symbol () implies that both the fundamental and incommensurate reflections display mirror symmetry perpendicular to the b* axis. Fig. 2.5.3.19(b) exactly exhibits the symmetry.
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.
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