International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 312-318   | 1 | 2 |

Section 2.5.3.2.7. Symmetrical many-beam method

M. Tanakaf

2.5.3.2.7. Symmetrical many-beam method

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In the sections above, the point-group determination method established by Buxton et al. (1976[link]) was described, where two- and three-dimensional 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 three-dimensional symmetry elements can be identified from one photograph. Using a group-theoretical method, Tinnappel (1975[link]) studied the symmetries appearing in simultaneously excited DPs for various combinations of crystal symmetry elements. Based upon his treatment, Tanaka, Saito & Sekii (1983[link]) developed a method for determining diffraction groups using simultaneously excited symmetrical hexagonal six-beam, square four-beam and rectangular four-beam CBED patterns. All the CBED symmetries appearing in the symmetrical many-beam (SMB) patterns were derived by the graphical method used in the paper of Buxton et al. (1976[link]). 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[link]).

SMB patterns are easily obtained by tilting a specimen crystal or the incident beam from a zone axis into an orientation to excite low-order reflections simultaneously. Fig. 2.5.3.7[link] illustrates the symmetries of the SMB patterns for all the diffraction groups except for the five groups 1, 1R, 2, 2R and 21R. For these groups, the two-beam method for exciting one reflection is satisfactory because many-beam excitation gives no more information than the two-beam case. In the six-beam and square four-beam cases, the CBED symmetries for the two crystal (or incident-beam) settings which excite respectively the +G and −G reflections are drawn because the vertical rotation axes create the SMB patterns at different incident-beam orientations. [This had already been experienced for the case of symmetry 2R (Goodman, 1975[link]; Buxton et al., 1976[link]).] In the rectangular four-beam case, the symmetries for four settings which excite the +G, +H, −G and −H reflections are shown. For the diffraction groups 3m, 3mR, 3m1R and 6RmmR, 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 4RmmR, 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[link]. The combination of the vertical threefold axis and a horizontal mirror plane introduces a new CBED symmetry 3R. Similarly, the combination of the vertical sixfold rotation axis and an inversion centre introduces a new CBED symmetry 6R.

[Figure 2.5.3.7]
[Figure 2.5.3.7]
[Figure 2.5.3.7]

Figure 2.5.3.7 | top | pdf |

Illustration of symmetries appearing in hexagonal six-beam, square four-beam and rectangular four-beam dark-field patterns expected for all the diffraction groups except for 1, 1R, 2, 2R and 21R.

There is an empirical and conventional technique for reproducing the symmetries of the SMB patterns which uses three operations of two-dimensional rotations, a vertical mirror at the centre of disc O and a rotation of π about the centre of a disc (1R) without involving the reciprocal process. For example, we may consider 3R between discs F and F′ in Table 2.5.3.5[link] in the case of diffraction group 31R. 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[link]. When the symmetries appearing between different SMB patterns are considered, this technique assumes that the symmetry operations are conducted after discs O and [\bar{O}] are superposed. Another assumption is that the vertical mirror plane perpendicular to the line connecting discs O and [\bar{O}] acts at the centre of disc O when the symmetries between two SMB patterns are considered. As an example, symmetry 3R between discs S and [\bar{S}] appearing in the two SMB patterns is reproduced by a threefold anticlockwise rotation of disc S about the centre of disc O (or [\bar{O}]) and followed by a rotation of π rad (R) about the centre of disc [\bar{S}].

Tables 2.5.3.5[link], 2.5.3.6[link] and 2.5.3.7[link] express the symmetries illustrated in Fig. 2.5.3.7[link] with the symmetry symbols for the hexagonal six-beam case, square four-beam case and rectangular four-beam case, respectively. In the fourth rows of the tables the symmetries of zone-axis patterns (BP and WP) are listed because combined use of the zone-axis 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 six-beam patterns, two pairs AB and AC of the square four-beam patterns and three pairs AB, AC and AD of the rectangular four-beam 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.

Table 2.5.3.5| top | pdf |
Symmetries of hexagonal six-beam CBED patterns for diffraction groups

 Projection diffraction group
31R3m1R61R
Diffraction group 3 31R 3mR 3m 3m1R 6 6R 61R
Two-dimensional symmetry 3 3 3 3m 3m 6 3 6
Three-dimensional symmetry   m 2′   m′, (2′)   i m′, (i)
Zone-axis pattern Bright-field pattern 3 6 3m 3m 6mm 6 3 6
Whole-field pattern 3 3 3 3m 3m 6 3 6
Hexagonal six-beam pattern O 1 1 1 m2 1 mv m2 mv 1 1 1
G 1 1R m2 1 1 mv 1R 1Rmv(m2) 1 1 1R
F 1 1 m2 1 1 1 1 m2 1 1 1
S 1 1 1 m2 1 1 m2 1 1 1 1
FF 1 3R 1 1 1 mv 3R 3Rmv 1 1 3R
SS 1 1 1 1 1 mv 1 mv 1 6R 6R
A pair of symmetrical six-beam patterns [Scheme scheme13] ±O 1 1R m2 1 mv 1 mv1R 1Rm2 2 1 2(1R)
±G 1 1 1 mR mv 1 mvmR 1 2 2R 21R
±F 1 1 1 1 mv 1 mv 1 1 6R 6R
±S 1 3R 1 1 mv 1 3Rmv 3R 1 1 3R
[F'\bar F] 1 1 1 mR 1 1 mR 1 2 1 2
[S'\bar S] 1 1 mR 1 1 1 1 mR 2 1 2
Point group 23, 3 [\bar 6] 432, 32 [\bar 43m], 3m [\bar 6m2] 6 m3, 3 6/m

 Projection diffraction group
6mm1R
Diffraction group 6mRmR 6mm 6RmmR 6mm1R
Two-dimensional symmetry 6 6mm 3m 6mm
Three-dimensional symmetry 2′   i, (2′) m′, (i, 2′)
Zone-axis pattern Bright-field pattern 6mm 6mm 3m 6mm
Whole-field pattern 6 6mm 3m 6mm
Hexagonal six-beam pattern O m2 mv 1 mv(m2) mv(m2)
G m2 mv m2 mv 1Rmv(m2)
F m2 1 m2 1 m2
S m2 1 1 m2 m2
FF 1 mv 1 mv 3Rmv
SS 1 mv 6R 6Rmv 6Rmv
A pair of symmetrical six-beam patterns [Scheme scheme113] ±O 2m2 2mv′ mv(m2) 1 2(1R)mv(m2)
±G 2mR 2mv′ 2Rmv 2RmR 21Rmv′(mR)
±F 1 mv 6Rmv 6R 6Rmv′
±S 1 mv′ mv 1 3Rmv′
[F'\bar F] 2mR 2 1 mR 2mR
[S'\bar S] 2mR 2 mR 1 2mR
Point group 622 6mm m3m, [\bar 3 m] 6/mmm

Table 2.5.3.6| top | pdf |
Symmetries of square four-beam CBED patterns for diffraction groups

 Projection diffraction group
41R4mm1R
Diffraction group 4 4R 41R 4mRmR 4mm 4RmmR 4mm1R
Two-dimensional symmetry 4 (2) 4 4 4mm (2mm) 4mm
Three-dimensional symmetry   [\bar 4] m′, (i, [\bar 4]) 2′   [\bar 4], 2′ m′, (i, 2′, [\bar 4])
Zone-axis pattern Bright-field pattern 4 4 4 4mm 4mm 4mm 4mm
Whole-field pattern 4 2 4 4 4mm 2mm 4mm
Square four-beam pattern   O 1 1 1 m2 mv m2 mv mv(m2)
G 1 1 1R m2 mv m2 mv 1Rmv(m2)
F 1 1 1 m2 1 1 m2 m2
FF 1 4R 4R 1 mv 4R 4Rmv 4Rmv
Two pairs of square four-beam patterns [Scheme scheme14] AB ±O 2 2 2(1R) 2m2 2mv′ 2m2 2mv′ 2(1R)mv(m2)
±G 2 2 21R 2mR 2mv′ 2mR 2mv′ 21Rmv(mR)
FF 2 2 2 2mR 2 2 2mR 2mR
±F 1 4R 4R 1 mv′ 4R 4Rmv′ 4Rmv
AC OO 4 4 4 4m2 4mv′′ 4mv 4m2 4mv′′(m2)
GG 4 4R 41R 4mR 4mv′′ 4Rmv 4RmR 41Rmv′′(mR)
FS 4 1 4 4mR 4 mR 1 41Rmv′′(mR)
FS 1 1 1R 1 mv′′ mv 1 1Rmv′′
Point group 4 [\bar 4] 4/m 432, 422 4mm [\bar 4 3 m], [\bar 4 2 m] m3m, 4/mmm

Table 2.5.3.7| top | pdf |
Symmetries of rectangular four-beam CBED patterns for diffraction groups

 Projection diffraction group
m1R2mm1R
Diffraction group mR m m1R 2mRmR 2mm 2RmmR 2mm1R
Two-dimensional symmetry   m m 2 2mm m 2mm
Three-dimensional symmetry 2′   m′, 2′ 2′   2′, i m′, 2′, i
Zone-axis pattern Bright-field pattern m m 2mm 2mm 2mm m 2mm
Whole-field pattern 1 m m 2 2mm m 2mm
Rectangular four-beam pattern   O 1 1 1 1 1 1 1
G 1 1 1R 1 1 1 1R
F m2 1 m2 m2 1 m2 m2
S 1 1 1 m2 1 1 m2
Three pairs of rectangular four-beam patterns [Scheme scheme15] AB [O_GO_{\bar H}] m2 1 m2 m2 mv mv(m2) mv(m2)
[G\bar H] 1 1 1 mR mv mv mvmR
[F \bar F] 1 1 1 1 mv 2Rmv 2Rmv
SS 1 1 1R 1 mv mv mv1R
AC OGOH 1 mv mv m2 mv′ 1 mv(m2)
GH mR mv mvmR mR mv′ mR mv′mR
FF 1 mv mv1R 1 mv′ 1 mv′1R
[S\bar S] 1 mv mv 1 mv′ 2R 2Rmv′
AD [O_GO_{\bar G}] 1 1 1R 2 2 1 2(1R)
GG 1 1 1 2 2 2R 21R
[F\bar F'] 1 1 1 2mR 2 1 2mR
[S\bar S'] mR 1 mR 2mR 2 mR 2mR
Point group 2, 222, mm2, 4, [\bar 4], 422, 4mm, [\bar 4 2 m], 32, 6, 622, 6mm, [\bar 6m2], 23, 432, [\bar 43m] m, mm2, 4mm, [\bar 42m], 3m, [\bar 6], 6mm, [\bar 6 m 2], [\bar 4 3 m] mm2, 4mm, 42m, 6mm, [\bar 6 m 2], [\bar 4 3 m] 222, 422, [\bar 42m], 622, 23, 432 mm2, [\bar 6 m2] 2/m, mmm, 4/m, 4/mmm, [\bar 3 m], [\bar 6/m], 6/mmm, m3, m3m mmm, 4/mmm, m3, m3m, 6/mmm

By referring to Tables 2.5.3.5[link], 2.5.3.6[link] and 2.5.3.7[link], the characteristic features of the SMB method are seen to be as follows. CBED symmetry m2 due to a horizontal twofold rotation axis can appear in every disc of an SMB pattern. Symmetry 1R due to a horizontal mirror plane, however, appears only in disc G or H of an SMB pattern. In the hexagonal six-beam case, an inversion centre i produces CBED symmetry 6R 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 six-beam pattern can reveal whether a specimen has an inversion centre or not, while the method of Buxton et al. (1976[link]) requires two photographs for the inversion test. All the diffraction groups in Table 2.5.3.5[link] can be identified from one six-beam pattern except groups 3 and 6. Diffraction groups 3 and 6 cannot be distinguished from the hexagonal six-beam pattern because it is insensitive to the vertical axis. In the square four-beam case, fourfold rotary inversion [\bar{4}] produces CBED symmetry 4R 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 four-beam pattern, it causes symmetry 1R due to the horizontal mirror plane produced by the combination of an inversion centre and the twofold rotation axis. Thus, symmetry 1R is an indication of the existence of an inversion centre in the square four-beam case. All of the seven diffraction groups in Table 2.5.3.6[link] can be identified from one square four-beam pattern. One rectangular four-beam pattern can distinguish all the diffraction groups in Table 2.5.3.7[link] except the groups m and 2mm. It is emphasized again that the inversion test can be carried out using one six-beam pattern or one square four-beam pattern.

Fig. 2.5.3.8[link] shows CBED patterns taken from a [111] pyrite (FeS2) plate with an accelerating voltage of 100 kV. The space group of FeS2 is [P2_1/a\bar{3}]. The diffraction group of the plate is 6R due to a threefold rotation axis and an inversion centre. The zone-axis pattern of Fig. 2.5.3.8[link](a) shows threefold rotation symmetry in the BP and WP. The hexagonal six-beam pattern of Fig. 2.5.3.8[link](b) shows no symmetry higher than 1 in discs O, G, F and S but shows symmetry 6R 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[link](c), where reflections [\bar{O}], [\bar{G}], [\bar{F}], [\bar{S}], [\bar{F}'] and [\bar{S}'] are excited. Table 2.5.3.5[link] indicates that diffraction group 6R can be identified from only one hexagonal six-beam 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 6R (see Table 2.5.3.3[link]). In addition, if the symmetries between Figs. 2.5.3.8[link](b) and (c) are examined, symmetry 2R between discs G and [\bar{G}] and symmetry 6R between discs F and [\bar{F}] are found. All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7[link] and Table 2.5.3.5[link].

[Figure 2.5.3.8]

Figure 2.5.3.8 | top | pdf |

CBED patterns of FeS2 taken with the [111] incidence. (a) Zone-axis pattern, (b) hexagonal six-beam pattern with excitation of reflection +G, (c) hexagonal six-beam pattern with excitation of reflection −G. Symmetry 6R is noted between discs S and S′ and discs [\bar F] and [\bar F'].

Fig. 2.5.3.9[link] shows CBED patterns taken from a [110] V3Si plate with an accelerating voltage of 80 kV. The space group of V3Si is Pm3n. The diffraction group of the plate is 2mm1R 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 zone-axis pattern of Fig. 2.5.3.9[link](a) shows symmetry 2mm in the BP and WP. The rectangular four-beam pattern of Fig. 2.5.3.9[link](b) shows symmetry 1R in disc H due to the horizontal mirror plane and symmetry m2 in both discs [\bar{S}] 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[link](c), where reflections [\bar{H}], S′ and [\bar{F}] are excited. Table 2.5.3.7[link] implies that the diffraction group 2mm1R can be identified from only one rectangular four-beam 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 2mm1R (see Table 2.5.3.3[link]). One can confirm the theoretically predicted symmetries between Fig. 2.5.3.9[link](b) and Fig. 2.5.3.9[link](c). All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7[link] and Table 2.5.3.7[link].

[Figure 2.5.3.9]

Figure 2.5.3.9 | top | pdf |

CBED patterns of V3Si taken with the [110] incidence. (a) Zone-axis pattern, (b) rectangular four-beam pattern with excitation of reflections H, [\bar S] and F, (c) rectangular four-beam pattern with excitation of reflections [\bar H], S and [\bar F].

These experiments show that the SMB method is quite effective for determining the diffraction group of slabs. Buxton et al.'s method identifies two-dimensional symmetry elements in the first place using a zone-axis pattern, and three-dimensional symmetry elements using DPs. On the other hand, the SMB method primarily finds many three-dimensional symmetry elements in an SMB pattern, and two-dimensional symmetry elements from a pair of SMB patterns, as shown in Tables 2.5.3.5[link], 2.5.3.6[link] and 2.5.3.7[link]. 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 three-dimensional space-group analysis using convergent-beam electron diffraction. Acta Cryst. A31, 804–810.
Howe, J. M., Sarikaya, M. & Gronsky, R. (1986). Space-group analyses of thin precipitates by different convergent-beam electron diffraction procedures. Acta Cryst. A42, 368–380.
Tanaka, M., Saito, R. & Sekii, H. (1983). Point-group determination by convergent-beam electron diffraction. Acta Cryst. A39, 357–368.
Tinnappel, A. (1975). PhD Thesis, Technische Universität Berlin, Germany.








































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