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. 312318
Section 2.5.3.2.7. Symmetrical manybeam method
M. Tanaka^{f}

In the sections above, the pointgroup determination method established by Buxton et al. (1976) was described, where two and threedimensional symmetry elements were determined, respectively, from ZAPs and DPs.
The Laue circle is defined as the intersection of the Ewald sphere with the ZOLZ, and all reflections on this circle are simultaneously at the Bragg condition. If many such DPs are recorded (all simultaneously at the Bragg condition), many threedimensional symmetry elements can be identified from one photograph. Using a grouptheoretical method, Tinnappel (1975) studied the symmetries appearing in simultaneously excited DPs for various combinations of crystal symmetry elements. Based upon his treatment, Tanaka, Saito & Sekii (1983) developed a method for determining diffraction groups using simultaneously excited symmetrical hexagonal sixbeam, square fourbeam and rectangular fourbeam CBED patterns. All the CBED symmetries appearing in the symmetrical manybeam (SMB) patterns were derived by the graphical method used in the paper of Buxton et al. (1976). From an experimental viewpoint, it is advantageous that symmetry elements can be identified from one photograph. It was found that twenty diffraction groups can be identified from one SMB pattern, whereas ten diffraction groups can be determined by Buxton et al.'s method. An experimental comparison between the two methods was performed by Howe et al. (1986).
SMB patterns are easily obtained by tilting a specimen crystal or the incident beam from a zone axis into an orientation to excite loworder reflections simultaneously. Fig. 2.5.3.7 illustrates the symmetries of the SMB patterns for all the diffraction groups except for the five groups 1, 1_{R}, 2, 2_{R} and 21_{R}. For these groups, the twobeam method for exciting one reflection is satisfactory because manybeam excitation gives no more information than the twobeam case. In the sixbeam and square fourbeam cases, the CBED symmetries for the two crystal (or incidentbeam) settings which excite respectively the +G and −G reflections are drawn because the vertical rotation axes create the SMB patterns at different incidentbeam orientations. [This had already been experienced for the case of symmetry 2_{R} (Goodman, 1975; Buxton et al., 1976).] In the rectangular fourbeam case, the symmetries for four settings which excite the +G, +H, −G and −H reflections are shown. For the diffraction groups 3m, 3m_{R}, 3m1_{R} and 6_{R}mm_{R}, two different patterns are shown for the two crystal settings, which differ by π/6 rad from each other about the zone axis. Similarly, for the diffraction group 4_{R}mm_{R}, two different patterns are shown for the two crystal settings, which differ by π/4 rad. Illustrations of these different symmetries are given in Fig. 2.5.3.7. The combination of the vertical threefold axis and a horizontal mirror plane introduces a new CBED symmetry 3_{R}. Similarly, the combination of the vertical sixfold rotation axis and an inversion centre introduces a new CBED symmetry 6_{R}.
There is an empirical and conventional technique for reproducing the symmetries of the SMB patterns which uses three operations of twodimensional rotations, a vertical mirror at the centre of disc O and a rotation of π about the centre of a disc (1_{R}) without involving the reciprocal process. For example, we may consider 3_{R} between discs F and F′ in Table 2.5.3.5 in the case of diffraction group 31_{R}. Disc F′ is rotated anticlockwise not about the zone axis but about the centre of disc O by 2π/3 rad (symbol 3) to coincide with disc F, and followed by a rotation of π rad (symbol R) about the centre of disc F′, resulting in the correct symmetry seen in Fig. 2.5.3.7. When the symmetries appearing between different SMB patterns are considered, this technique assumes that the symmetry operations are conducted after discs O and are superposed. Another assumption is that the vertical mirror plane perpendicular to the line connecting discs O and acts at the centre of disc O when the symmetries between two SMB patterns are considered. As an example, symmetry 3_{R} between discs S and appearing in the two SMB patterns is reproduced by a threefold anticlockwise rotation of disc S about the centre of disc O (or ) and followed by a rotation of π rad (R) about the centre of disc .
Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7 express the symmetries illustrated in Fig. 2.5.3.7 with the symmetry symbols for the hexagonal sixbeam case, square fourbeam case and rectangular fourbeam case, respectively. In the fourth rows of the tables the symmetries of zoneaxis patterns (BP and WP) are listed because combined use of the zoneaxis pattern and the SMB pattern is efficient for symmetry determination. In the fifth row, the symmetries of the SMB pattern are listed. In the following rows, the symmetries appearing between the two SMB patterns are listed because the SMB symmetries appear not only in an SMB pattern but also in the pairs of SMB patterns. That is, for each diffraction group, all the possible SMB symmetries appearing in a pair of symmetric sixbeam patterns, two pairs AB and AC of the square fourbeam patterns and three pairs AB, AC and AD of the rectangular fourbeam patterns are listed, though such pairs are not always needed for the determination of the diffraction groups. It is noted that the symmetries in parentheses are the symmetries which add no new symmetries, even if they are present. In the last row, the point groups which cause the diffraction groups listed in the first row are given.



By referring to Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7, the characteristic features of the SMB method are seen to be as follows. CBED symmetry m_{2} due to a horizontal twofold rotation axis can appear in every disc of an SMB pattern. Symmetry 1_{R} due to a horizontal mirror plane, however, appears only in disc G or H of an SMB pattern. In the hexagonal sixbeam case, an inversion centre i produces CBED symmetry 6_{R} between discs S and S′ due to the combination of an inversion centre and a vertical threefold rotation axis (and/or of a horizontal mirror plane and a vertical sixfold rotation axis). This indicates that one hexagonal sixbeam pattern can reveal whether a specimen has an inversion centre or not, while the method of Buxton et al. (1976) requires two photographs for the inversion test. All the diffraction groups in Table 2.5.3.5 can be identified from one sixbeam pattern except groups 3 and 6. Diffraction groups 3 and 6 cannot be distinguished from the hexagonal sixbeam pattern because it is insensitive to the vertical axis. In the square fourbeam case, fourfold rotary inversion produces CBED symmetry 4_{R} between discs F and F′ in one SMB pattern, while Buxton et al.'s method requires four photographs to identify fourfold rotary inversion. Although an inversion centre itself does not exhibit any symmetry in the square fourbeam pattern, it causes symmetry 1_{R} due to the horizontal mirror plane produced by the combination of an inversion centre and the twofold rotation axis. Thus, symmetry 1_{R} is an indication of the existence of an inversion centre in the square fourbeam case. All of the seven diffraction groups in Table 2.5.3.6 can be identified from one square fourbeam pattern. One rectangular fourbeam pattern can distinguish all the diffraction groups in Table 2.5.3.7 except the groups m and 2mm. It is emphasized again that the inversion test can be carried out using one sixbeam pattern or one square fourbeam pattern.
Fig. 2.5.3.8 shows CBED patterns taken from a [111] pyrite (FeS_{2}) plate with an accelerating voltage of 100 kV. The space group of FeS_{2} is . The diffraction group of the plate is 6_{R} due to a threefold rotation axis and an inversion centre. The zoneaxis pattern of Fig. 2.5.3.8(a) shows threefold rotation symmetry in the BP and WP. The hexagonal sixbeam pattern of Fig. 2.5.3.8(b) shows no symmetry higher than 1 in discs O, G, F and S but shows symmetry 6_{R} between discs S and S′, which proves the existence of a threefold rotation axis and an inversion centre. The same symmetries are also seen in Fig. 2.5.3.8(c), where reflections , , , , and are excited. Table 2.5.3.5 indicates that diffraction group 6_{R} can be identified from only one hexagonal sixbeam pattern, because no other diffraction groups give rise to the same symmetries in the six discs. When Buxton et al.'s method is used, three photographs or four patterns are necessary to identify diffraction group 6_{R} (see Table 2.5.3.3). In addition, if the symmetries between Figs. 2.5.3.8(b) and (c) are examined, symmetry 2_{R} between discs G and and symmetry 6_{R} between discs F and are found. All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.5.
Fig. 2.5.3.9 shows CBED patterns taken from a [110] V_{3}Si plate with an accelerating voltage of 80 kV. The space group of V_{3}Si is Pm3n. The diffraction group of the plate is 2mm1_{R} due to two vertical mirror planes and a horizontal mirror plane, a twofold rotation axis being produced at the intersection line of two perpendicular mirror planes. The zoneaxis pattern of Fig. 2.5.3.9(a) shows symmetry 2mm in the BP and WP. The rectangular fourbeam pattern of Fig. 2.5.3.9(b) shows symmetry 1_{R} in disc H due to the horizontal mirror plane and symmetry m_{2} in both discs and F′ due to the twofold rotation axes in the [001] and [110] directions, respectively. The same symmetries are also seen in Fig. 2.5.3.9(c), where reflections , S′ and are excited. Table 2.5.3.7 implies that the diffraction group 2mm1_{R} can be identified from only one rectangular fourbeam pattern, because no other diffraction groups give rise to the same symmetries in the four discs. When Buxton et al.'s method is used, two photographs or three patterns are necessary to identify diffraction group 2mm1_{R} (see Table 2.5.3.3). One can confirm the theoretically predicted symmetries between Fig. 2.5.3.9(b) and Fig. 2.5.3.9(c). All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.7.
These experiments show that the SMB method is quite effective for determining the diffraction group of slabs. Buxton et al.'s method identifies twodimensional symmetry elements in the first place using a zoneaxis pattern, and threedimensional symmetry elements using DPs. On the other hand, the SMB method primarily finds many threedimensional symmetry elements in an SMB pattern, and twodimensional symmetry elements from a pair of SMB patterns, as shown in Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7. Therefore, the use of a ZAP and SMB patterns is the most efficient way to find as many crystal symmetry elements in a specimen as possible.
References
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.Goodman, P. (1975). A practical method of threedimensional spacegroup analysis using convergentbeam electron diffraction. Acta Cryst. A31, 804–810.
Howe, J. M., Sarikaya, M. & Gronsky, R. (1986). Spacegroup analyses of thin precipitates by different convergentbeam electron diffraction procedures. Acta Cryst. A42, 368–380.
Tanaka, M., Saito, R. & Sekii, H. (1983). Pointgroup determination by convergentbeam electron diffraction. Acta Cryst. A39, 357–368.
Tinnappel, A. (1975). PhD Thesis, Technische Universität Berlin, Germany.