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

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 308-318   | 1 | 2 |

Section 2.5.3.2. Point-group determination

M. Tanakaf

2.5.3.2. Point-group determination

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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[link]) 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[link]) solved many CBED symmetries at various crystal settings with respect to the incident beam using a group-theoretical treatment. Buxton et al. (1976[link]) 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[link]) developed a method to determine the point group using simultaneously excited many-beam patterns. The point-group-determination method given by Buxton et al. (1976[link]) is described with the aid of the description by Tanaka, Saito & Sekii (1983[link]) in the following.

2.5.3.2.1. Symmetry elements of a specimen and diffraction groups

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Since CBED uses the Laue geometry, Buxton et al. (1976[link]) assumed a perfectly crystalline specimen in the form of a parallel-sided 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 point-group of the specimen is identified by referring to Fig. 2.5.3.4[link], which gives the relation between diffraction groups and crystal point groups.

A specimen that is parallel-sided and is infinitely extended in the x and y directions has ten symmetry elements. The symmetry elements consist of six two-dimensional symmetry elements and four three-dimensional 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 [z'\ne z]. A vertical mirror plane m and one-, two-, three-, four- and sixfold rotation axes that are parallel to the surface normal z are the two-dimensional symmetry elements. A horizontal mirror plane m′, an inversion centre i, a horizontal twofold rotation axis 2′ and a fourfold rotary inversion [\bar 4] are the three-dimensional symmetry elements, and are shown in Fig. 2.5.3.1[link]. 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[link]). Table 2.5.3.1[link] lists these symmetry elements, where the symbols in parentheses express symmetries of CBED patterns expected from three-dimensional symmetry elements.

Table 2.5.3.1| top | pdf |
Two- and three-dimensional symmetry elements of an infinitely extended parallel-sided specimen

Symbols in parentheses show CBED symmetries appearing in dark-field patterns.

Two-dimensional symmetry elementsThree-dimensional symmetry elements
1 m′ (1R)
2 i (2R)
3 2′ (m2, mR)
4 [\bar 4] (4R)
5  
6  
m  
[Figure 2.5.3.1]

Figure 2.5.3.1 | top | pdf |

Four symmetry elements m′, i, 2′ and [\bar 4] of an infinitely extended parallel-sided specimen.

The diffraction groups are constructed by combining these symmetry elements (Table 2.5.3.2[link]). Two-dimensional 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. Three-dimensional symmetry elements are given in the first column. The equations given below the table indicate that no additional three-dimensional 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.

Table 2.5.3.2| top | pdf |
Symmetry elements of an infinitely extended parallel-sided specimen and diffraction groups

 12346m2m(m)3m4m(m)6m(m) 
1 1 2 3 4 6 m 2m(m) 3m 4m(m) 6m(m) 10
(m′) 1R 1R 21R 31R 41R 61R m1R 2m(m)1R 3m1R 4m(m)1R 6m(m)1R 10
(i) 2R 2R [21R] 6R [41R] [61R] 2Rm(mR) [2m(m)1R] 6Rm(mR) [4m(m)1R] [6m(m)1R] 4
            [2Rm(mR)] [2m(m)1R] [3m1R]      
(2′) mR mR 2mR(mR) 3mR 4mR(mR) 6mR(mR) [m1R] [4R(m)mR] [6Rm(mR)] [4m(m)1R] [6Rm(mR)] 5
[(\bar 4)] 4R   4R   [41R]   4Rm(mR) [4Rm(mR)]   [4m(m)1R]   2

1R × 2R = 2, 2R × 2R = 1, mR × 2R = m, 4R × 2R = 4, 1R × mR = m × mR, 1R × 4R = 4 × 1R, mR × 4R = m × 4R.

2.5.3.2.2. Identification of three-dimensional symmetry elements

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It is difficult to imagine the symmetries in CBED patterns generated by the three-dimensional symmetry elements of the sample. The reason is that if a three-dimensional 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[link]) enables us to clarify the symmetries of CBED patterns expected from these three-dimensional symmetry elements. A graphical method for obtaining CBED symmetries due to sample symmetry elements is described in the papers of Goodman (1975[link]), Buxton et al. (1976[link]) and Tanaka (1989[link]). The CBED symmetries of the three-dimensional symmetries do not appear in the zone-axis patterns, but do in a diffraction disc set at the Bragg condition, each of which we call a dark-field pattern (DP). The CBED symmetries obtained are illustrated in Fig. 2.5.3.2[link]. A horizontal twofold rotation axis 2′, a horizontal mirror plane m′, an inversion centre i and a fourfold rotary inversion [\bar 4] produce symmetries mR (m2), 1R, 2R and 4R in DPs, respectively.

[Figure 2.5.3.2]

Figure 2.5.3.2 | top | pdf |

Illustration of symmetries appearing in dark-field patterns (DPs). (a) mR and m2; (b) 1R; (c) 2R; (d) 4R, originating from 2′, m′, i and [\bar 4], respectively.

Next we explain the symbols of the CBED symmetries. (1) Operation mR is shown in the left-hand part of Fig. 2.5.3.2[link](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 mR. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the left-hand 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 right-hand side figure of Fig. 2.5.3.2[link]a). The mirror symmetry is labelled m2 after the twofold rotation axis. (2) Operation 1R (Fig. 2.5.3.2[link]b) 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 2R 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. 2.5.3.2[link]c). The symmetry is called translational symmetry after Goodman (1975[link]) 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 dark-field patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4R (Fig. 2.5.3.2[link]d) 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 dark-field pattern. As far as CBED symmetries are concerned, we do not use the term dark-field pattern if a disc does not contain the exact Bragg position.

The four three-dimensional 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.

2.5.3.2.3. Identification of two-dimensional symmetry elements

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Two-dimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zone-axis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a bright-field pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the bright-field 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 two-dimensional symmetry elements m, 1, 2, 3, 4 and 6 yield symmetry mv and one-, two-, three-, four- and sixfold rotation symmetries, respectively, in WPs, where the suffix v for mv is assigned to distinguish it from mirror symmetry m2 caused by a horizontal twofold rotation axis.

It should be noted that a BP shows not only the zone-axis symmetry but also three-dimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to three-dimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the three-dimensional symmetry elements m′, i, 2′ and [\bar 4] produce, respectively, symmetries 1R, 1, m2 and 4 in the BP, instead of 1R, 2R, m2 and 4R in the DPs (Fig. 2.5.3.2[link]). 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 two-dimensional symmetry elements can be identified from the WP symmetries.

2.5.3.2.4. Diffraction-group determination

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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[link] (Tanaka, Saito & Sekii, 1983[link]), which is a detailed version of Table 2 of Buxton et al. (1976[link]). 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 4R. When two types of vertical mirror planes exist, these are distinguished by symbols mv and mv. 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.

Table 2.5.3.3| top | pdf |
Symmetries of different patterns for diffraction and projection diffraction groups

(II) Bright-field patterns (BPs); (III) whole patterns (WPs); (IV) dark-field patterns (DPs); and (V) ±dark-field patterns (±DPs) for diffraction groups (I) and projection diffraction groups (VI).

IIIIIIIVVVI
1 1 1 1 1 1R
1R 2 1 2 = 1R 1
(1R)
2 2 2 1 2 21R
2R 1 1 1 2R
21R 2 2 2 21R
mR m 1 1 1 m1R
(m2) mR
m2 1
m mv mv 1 1
mv
mv 1
m1R 2mm mv 2 1
[mv + m2 + (1R)] mv1R
2mvm2 1
2mRmR 2mm 2 1 2 2mm1R
(2 + m2) m2 2mR(m2)
2mm 2mvmv 2mvmv 1 2
mv 2mv(mv)
2RmmR mv mv 1 2R
m2 2Rmv(m2)
mv 2RmR(mv)
2mm1R 2mvmv 2mvmv 2 21R
2mvm2 21Rmv(mv)
4 4 4 1 2 41R
4R 4 2 1 2
41R 4 4 2 21R
4mRmR 4mm 4 1 2 4mm1R
(4 + m2) m2 2mR(m2)
4mm 4mvmv 4mvmv 1 2
mv 2mv(mv)
4RmmR 4mm 2mvmv 1 2
(2mvmv + m2) m2 2mR(m2)
mv 2mv(mv)
4mm1R 4mvmv 4mvmv 2 21R
2mvm2 21Rmv(mv)
3 3 3 1 1 31R
31R 6 3 2 1
(3 + 1R)
3mR 3m 3 1 1 3m1R
(3 + m2) mR
m2 1
3m 3mv 3mv 1 1
mv
mv 1
3m1R 6mm 3mv 2 1
[3mv + m2 + (1R)] mv1R
2mvm2 1
6 6 6 1 2 61R
6R 3 3 1 2R
61R 6 6 2 21R
6mRmR 6mm 6 1 2 6mm1R
(6 + m2) m2 2mR(m2)
6mm 6mvmv 6mvmv 1 2
mv 2mv(mv)
6RmmR 3mv 3mv 1 2R
m2 2Rmv(m2)
mv 2RmR(mv)
6mm1R 6mvmv 6mvmv 2 21R
2mvm2 21Rmv(mv)

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[link] illustrates the symmetries of the DP and ±DP appearing in Table 2.5.3.3[link], 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.

[Figure 2.5.3.3]

Figure 2.5.3.3 | top | pdf |

Illustration of symmetries appearing in dark-field patterns (DPs) and a pair of dark-field 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[link] with the aid of Fig. 2.5.3.3[link], 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).

2.5.3.2.5. Point-group determination

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Fig. 2.5.3.4[link] provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976[link]). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 high-symmetry 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 high-symmetry zones should be used for quick determination of point groups because low-symmetry 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[link] (Buxton et al., 1976[link]). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance.

Table 2.5.3.4| top | pdf |
Diffraction groups expected at various crystal orientations for 32 point groups

This table is adapted from Buxton et al. (1976[link]).

Point groupZone-axis symmetries
<111><100><110><uv0><uuw>[uvw]
m3m 6RmmR 4mm1R 2mm1R 2RmmR 2RmmR 2R
[\bar 4 3 m] 3m 4RmmR m1R mR m 1
432 3mR 4mRmR 2mRmR mR mR 1

Point groupZone-axis symmetries
<111><100><uv0>[uvw]
m3 6R 2mm1R 2RmmR 2R
23 3 2mRmR mR 1

Point groupZone-axis symmetries
[0001][\langle 11\bar 2 0\rangle][\langle 1\bar 1 00\rangle][uv.0][uu.w][[u\bar u.w]][uv.w]
6/mmm 6mm1R 2mm1R 2mm1R 2RmmR 2Rmm 2RmmR 2R
[\bar 6 m 2] 3m1R m1R 2mm m mR m 1
6mm 6mm m1R m1R mR m m 1
622 6mRmR 2mRmR 2mRmR mR mR mR 1

Point groupZone-axis symmetries
[0001][uv.0][uv.w]
6/m 61R 2RmmR 2R
[\bar 6] 31R m 1
6 6 mR 1

Point groupZone-axis symmetries
[0001][\langle 11\bar 2 0\rangle][[u\bar u.w]][uv.w]
[\bar 3 m] 6RmmR 21R 2RmmR 2R
3m 3m 1R m 1
32 3mR 2 mR 1

Point groupZone-axis symmetries
[0001][uv.w]
[\bar 3] 6R 2R
3 3 1

Point groupZone-axis symmetries
[001]<100><110>[u0w][uv0][uuw][uvw]
4/mmm 4mm1R 2mm1R 2mm1R 2RmmR 2RmmR 2RmmR 2R
[\bar 4 2 m] 4RmmR 2mRmR m1R mR mR m 1
4mm 4mm m1R m1R m mR m 1
422 4mRmR 2mRmR 2mRmR mR mR mR 1

Point groupZone-axis symmetries
[001][uv0][uvw]
4/m 41R 2RmmR 2R
[\bar 4] 4R mR 1
4 4 mR 1

Point groupZone-axis symmetries
[001]<100>[u0w][uv0][uvw]
mmm 2mm1R 2mm1R 2RmmR 2RmmR 2R
mm2 2mm m1R m mR 1
222 2mRmR 2mRmR mR mR 1

Point groupZone-axis symmetries
[010][u0w][uvw]
2/m 21R 2RmmR 2R
m 1R m 1
2 2 mR 1

Point groupZone-axis symmetry
[uvw]
[\bar 1] 2R
1 1
[Figure 2.5.3.4]

Figure 2.5.3.4 | top | pdf |

Relation between diffraction groups and crystal point groups (after Buxton et al., 1976[link]).

We shall explain the point-group determination procedure using an Si crystal. Fig. 2.5.3.5[link](a) shows a [111] ZAP of the Si specimen. The BP has threefold rotation symmetry and mirror symmetry or symmetry 3mv, 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[link](b) and (c) are [2\bar{2}0] and [\bar{2}20] DPs, respectively. Both show symmetry m2 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 2R symmetry indicates the existence of an inversion centre. By consulting Table 2.5.3.3[link], the diffraction group giving rise to these pattern symmetries is found to be 6RmmR. Fig. 2.5.3.4[link] shows that there are two point groups [\bar{3}m] and [m\bar{3}m] causing diffraction group 6RmmR. Fig. 2.5.3.6[link] shows another ZAP, which shows symmetry 4mm in the BP and the WP. The point group which has fourfold rotation symmetry is not [\bar{3}m] but [m\bar{3}m]. The point group of Si is thus determined to be [m\bar{3}m].

[Figure 2.5.3.5]

Figure 2.5.3.5 | top | pdf |

CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3mv. (b) and (c) DPs show symmetry m2 and DP symmetry 2Rmv.

[Figure 2.5.3.6]

Figure 2.5.3.6 | top | pdf |

CBED pattern of Si taken with the [100] incidence. The BP and WP show symmetry 4mm.

2.5.3.2.6. Projection diffraction groups

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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 1R. When symmetry 1R is added to the 31 diffraction groups, ten projection diffraction groups having symmetry symbol 1R are derived as shown in column VI of Table 2.5.3.3[link]. 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 1R in Table 2.5.3.3[link] 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 three-dimensional 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.

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.
Pogany, A. P. & Turner, P. S. (1968). Reciprocity in electron diffraction and microscopy. Acta Cryst. A24, 103–109.
Tanaka, M. (1989). Symmetry analysis. J. Electron Microsc. Tech. 13, 27–39.
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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