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. 308318
Section 2.5.3.2. Pointgroup determination
M. Tanaka^{f}

When an electron beam traverses a thin slab of crystal parallel to a zone axis, one can easily imagine that symmetries parallel to the zone axis should appear in the resulting CBED pattern. It is, however, more difficult to imagine what symmetries appear due to symmetries perpendicular to the incident beam. Goodman (1975) pioneered the clarification of CBED symmetries for the twofold rotation axis and mirror plane perpendicular to the incident beam, and the symmetry of an inversion centre, with the help of the reciprocity theorem of scattering theory. Tinnappel (1975) solved many CBED symmetries at various crystal settings with respect to the incident beam using a grouptheoretical treatment. Buxton et al. (1976) also derived these results from first principles, and generalized them to produce a systematic method for the determination of the crystal point group. Tanaka, Saito & Sekii (1983) developed a method to determine the point group using simultaneously excited manybeam patterns. The pointgroupdetermination method given by Buxton et al. (1976) is described with the aid of the description by Tanaka, Saito & Sekii (1983) in the following.
Since CBED uses the Laue geometry, Buxton et al. (1976) assumed a perfectly crystalline specimen in the form of a parallelsided slab which is infinite in two dimensions. The symmetry elements of the specimen (as distinct from those of an infinite crystal) form `diffraction groups', which are isomorphic to the point groups of the diperiodic plane figures and Shubnikov groups of coloured plane figures. The diffraction groups of a specimen are determined from the symmetries of CBED patterns taken at various orientations of the specimen. The crystal pointgroup of the specimen is identified by referring to Fig. 2.5.3.4, which gives the relation between diffraction groups and crystal point groups.
A specimen that is parallelsided and is infinitely extended in the x and y directions has ten symmetry elements. The symmetry elements consist of six twodimensional symmetry elements and four threedimensional ones. The operation of the former elements transforms an arbitrary coordinate (x, y, z) into (x′, y′, z), with z remaining the same. The operation of the latter transforms a coordinate (x, y, z) into (x′, y′, z′), where . A vertical mirror plane m and one, two, three, four and sixfold rotation axes that are parallel to the surface normal z are the twodimensional symmetry elements. A horizontal mirror plane m′, an inversion centre i, a horizontal twofold rotation axis 2′ and a fourfold rotary inversion are the threedimensional symmetry elements, and are shown in Fig. 2.5.3.1. The fourfold rotary inversion was not recognized as a symmetry element until the point groups of the diperiodic plane figures were considered (Buxton et al., 1976). Table 2.5.3.1 lists these symmetry elements, where the symbols in parentheses express symmetries of CBED patterns expected from threedimensional symmetry elements.

The diffraction groups are constructed by combining these symmetry elements (Table 2.5.3.2). Twodimensional symmetry elements and their combinations are given in the top row of the table. The third symmetry m in parentheses is introduced automatically when the first two symmetry elements are combined. Threedimensional symmetry elements are given in the first column. The equations given below the table indicate that no additional threedimensional symmetry elements can appear by combination of two symmetry elements in the first column. As a result, 31 diffraction groups are produced by combining the elements in the first column with those in the top row. Diffraction groups in square brackets have already appeared ealier in the table. In the fourth row, three columns have two diffraction groups, which are produced when symmetry elements are combined at different orientations. In the last row, five columns are empty because a fourfold rotary inversion cannot coexist with threefold and sixfold rotation axes. In the last column, the number of independent diffraction groups in each row is given, the sum of the numbers being 31.

It is difficult to imagine the symmetries in CBED patterns generated by the threedimensional symmetry elements of the sample. The reason is that if a threedimensional symmetry element is applied to a specimen, it turns it upside down, which is impractical in most experiments. The reciprocity theorem of scattering theory (Pogany & Turner, 1968) enables us to clarify the symmetries of CBED patterns expected from these threedimensional symmetry elements. A graphical method for obtaining CBED symmetries due to sample symmetry elements is described in the papers of Goodman (1975), Buxton et al. (1976) and Tanaka (1989). The CBED symmetries of the threedimensional symmetries do not appear in the zoneaxis patterns, but do in a diffraction disc set at the Bragg condition, each of which we call a darkfield pattern (DP). The CBED symmetries obtained are illustrated in Fig. 2.5.3.2. A horizontal twofold rotation axis 2′, a horizontal mirror plane m′, an inversion centre i and a fourfold rotary inversion produce symmetries m_{R} (m_{2}), 1_{R}, 2_{R} and 4_{R} in DPs, respectively.

Illustration of symmetries appearing in darkfield patterns (DPs). (a) m_{R} and m_{2}; (b) 1_{R}; (c) 2_{R}; (d) 4_{R}, originating from 2′, m′, i and , respectively. 
Next we explain the symbols of the CBED symmetries. (1) Operation m_{R} is shown in the lefthand part of Fig. 2.5.3.2(a), which implies successive operations of (a) a mirror m with respect to a twofold rotation axis, transforming an open circle beam (○) in reflection G into a beam (+) in reflection G′ and (b) rotation R of this beam by π about the centre point of disc G′ (or the exact Bragg position of reflection G′), resulting in position ○ in reflection G′. The combination of the two operations is written as m_{R}. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the lefthand part of the figure become one reflection G, and a mirror symmetry, whose mirror line is perpendicular to vector G and passes through the centre of disc G, appears between the two beams (the righthand side figure of Fig. 2.5.3.2a). The mirror symmetry is labelled m_{2} after the twofold rotation axis. (2) Operation 1_{R} (Fig. 2.5.3.2b) for a horizontal mirror plane is a combination of a rotation by 2π of a beam (○) about a zone axis O (symbol 1), which is equivalent to no rotation, and a rotation by π of the beam about the exact Bragg position or the centre of disc G. (3) Operation 2_{R} is a rotation by π of a beam (○) in reflection G about a zone axis (symbol 2), which transforms the beam into a beam (+) in reflection −G, followed by a rotation by π of the beam (+) about the centre of disc −G, resulting in the beam (○) in disc −G (Fig. 2.5.3.2c). The symmetry is called translational symmetry after Goodman (1975) because the pattern of disc +G coincides with that of disc −G by a translation. It is emphasized that an inversion centre is identified by the test of translational symmetry about a pair of ±G darkfield patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4_{R} (Fig. 2.5.3.2d) can be understood in a similar manner. It is noted that regular letters are symmetries about a zone axis, while subscripts R represent symmetries about the exact Bragg position. We call a pattern that contains an exact Bragg position (if possible at the disc centre) a darkfield pattern. As far as CBED symmetries are concerned, we do not use the term darkfield pattern if a disc does not contain the exact Bragg position.
The four threedimensional symmetry elements are found to produce different symmetries in the DPs. These facts imply that these symmetry elements can be identified unambiguously from the symmetries of CBED patterns.
Twodimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zoneaxis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a brightfield pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the brightfield disc [the central or `direct' (000) beam]. The WP is composed of the BP and the pattern formed by the surrounding diffraction discs, which are not exactly excited. The twodimensional symmetry elements m, 1, 2, 3, 4 and 6 yield symmetry m_{v} and one, two, three, four and sixfold rotation symmetries, respectively, in WPs, where the suffix v for m_{v} is assigned to distinguish it from mirror symmetry m_{2} caused by a horizontal twofold rotation axis.
It should be noted that a BP shows not only the zoneaxis symmetry but also threedimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to threedimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the threedimensional symmetry elements m′, i, 2′ and produce, respectively, symmetries 1_{R}, 1, m_{2} and 4 in the BP, instead of 1_{R}, 2_{R}, m_{2} and 4_{R} in the DPs (Fig. 2.5.3.2). We mention that the BP cannot distinguish whether a specimen crystal has an inversion centre or not, because an inversion centre forms the lowest symmetry 1 in the BP.
In conclusion, all the twodimensional symmetry elements can be identified from the WP symmetries.
All the symmetry elements of the diffraction groups can be identified from the symmetries of a WP and DPs. But it is practical and convenient to use just the four patterns WP, BP, DP and ±DP to determine the diffraction group. The symmetries appearing in these four patterns are given for the 31 diffraction groups in Table 2.5.3.3 (Tanaka, Saito & Sekii, 1983), which is a detailed version of Table 2 of Buxton et al. (1976). All the possible symmetries of the DP and ±DP appearing at different crystal orientations are given in the present table. When a BP has a higher symmetry than the corresponding WP, the symmetry elements that produce the BP are given in parentheses in column II except only for the case of 4_{R}. When two types of vertical mirror planes exist, these are distinguished by symbols m_{v} and m_{v′}. Each of the two or three symmetries given in columns IV and V for many diffraction groups appears in a DP or ±DP in different directions.

It is emphasized again that no two diffraction groups exhibit the same combination of BP, WP, DP and ±DP, which implies that the diffraction groups are uniquely determined from an inspection of these pattern symmetries. Fig. 2.5.3.3 illustrates the symmetries of the DP and ±DP appearing in Table 2.5.3.3, which greatly eases the cumbersome task of determining the symmetries. The first four patterns illustrate the symmetries appearing in a single DP and the others treat those in ±DPs. The pattern symmetries are written beneath the figures. The other symbols are the symmetries of a specimen. The crosses outside the diffraction discs designate the zone axis. The crosses inside the diffraction discs indicate the exact Bragg position.

Illustration of symmetries appearing in darkfield patterns (DPs) and a pair of darkfield patterns (±DP) for the combinations of symmetry elements. 
When the four patterns appearing in three photographs are taken and examined using Table 2.5.3.3 with the aid of Fig. 2.5.3.3, one diffraction group can be selected unambiguously. It is, however, noted that many diffraction groups are determined from a WP and BP pair without using a DP or ±DP (or from one photograph) or from a set of a WP, a BP and a DP without using a ±DP (or from two photographs).
Fig. 2.5.3.4 provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 highsymmetry diffraction groups corresponds to only one crystal point group. In this case, the point group is uniquely determined from the diffraction group. When more than one point group falls under a diffraction group, a different diffraction group has to be obtained for another zone axis. A point group is identified by finding a common point group among the point groups obtained for different zone axes. It is clear from the figure that highsymmetry zones should be used for quick determination of point groups because lowsymmetry zone axes exhibit only a small portion of crystal symmetries in the CBED patterns. Furthermore, it should be noted that CBED cannot observe crystal symmetries oblique to an incident beam or horizontal three, four or sixfold rotation axes. The diffraction groups to be expected for different zone axes are given for all the point groups in Table 2.5.3.4 (Buxton et al., 1976). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance.

We shall explain the pointgroup determination procedure using an Si crystal. Fig. 2.5.3.5(a) shows a [111] ZAP of the Si specimen. The BP has threefold rotation symmetry and mirror symmetry or symmetry 3m_{v}, which are caused by the threefold rotation axis along the [111] direction and a vertical mirror plane. The WP has the same symmetry. Figs. 2.5.3.5(b) and (c) are and DPs, respectively. Both show symmetry m_{2} perpendicular to the reflection vector. This symmetry is caused by a twofold rotation axis parallel to the specimen surface. One DP coincides with the other upon translation. This translational or 2_{R} symmetry indicates the existence of an inversion centre. By consulting Table 2.5.3.3, the diffraction group giving rise to these pattern symmetries is found to be 6_{R}mm_{R}. Fig. 2.5.3.4 shows that there are two point groups and causing diffraction group 6_{R}mm_{R}. Fig. 2.5.3.6 shows another ZAP, which shows symmetry 4mm in the BP and the WP. The point group which has fourfold rotation symmetry is not but . The point group of Si is thus determined to be .

CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3m_{v}. (b) and (c) DPs show symmetry m_{2} and DP symmetry 2_{R}m_{v′}. 
HOLZ reflections appear as excess HOLZ rings far outside the ZOLZ reflection discs and as deficit lines in the ZOLZ discs. By ignoring these weak diffraction effects with components along the beam direction, we may obtain information about the symmetry of the sample as projected along the beam direction. Thus when HOLZ reflections are weak and no deficit HOLZ lines are seen in the ZOLZ discs, the symmetry elements found from the CBED patterns are only those of the specimen projected along the zone axis. The projection of the specimen along the zone axis causes horizontal mirror symmetry m′, the corresponding CBED symmetry being 1_{R}. When symmetry 1_{R} is added to the 31 diffraction groups, ten projection diffraction groups having symmetry symbol 1_{R} are derived as shown in column VI of Table 2.5.3.3. If only ZOLZ reflections are observed in CBED patterns, a projection diffraction group instead of a diffraction group is obtained, where only the pattern symmetries given in the rows of the diffraction groups having symmetry symbol 1_{R} in Table 2.5.3.3 should be consulted. Two projection diffraction groups obtained from two different zone axes are the minimum needed to determine a crystal point group, because it is constructed by the threedimensional combination of symmetry elements. It should be noted that if a diffraction group is determined carelessly from CBED patterns with no HOLZ lines, the wrong crystal point group is obtained.
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
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Howe, J. M., Sarikaya, M. & Gronsky, R. (1986). Spacegroup analyses of thin precipitates by different convergentbeam electron diffraction procedures. Acta Cryst. A42, 368–380.
Pogany, A. P. & Turner, P. S. (1968). Reciprocity in electron diffraction and microscopy. Acta Cryst. A24, 103–109.
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Tinnappel, A. (1975). PhD Thesis, Technische Universität Berlin, Germany.