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. 348349
Section 2.5.3.4.2. Pointgroup determination
M. Tanaka^{f}

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.
References
Tanaka, M., Terauchi, M. & Tsuda, K. (1994). ConvergentBeam Electron Diffraction III. Tokyo: JEOL–Maruzen.Terauchi, M. & Tanaka, M. (1993). Convergentbeam electron diffraction study of incommensurately modulated crystals. I. (3+1)dimensional point groups. Acta Cryst. A49, 722–729.
Yamamoto, N. (1988). Electron microscope study of incommensurate phases. Kotaibutsui, 23, 547.