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

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 307-356   | 1 | 2 |

Section 2.5.3. Point-group and space-group determination by convergent-beam electron diffraction

M. Tanakaf

2.5.3. Point-group and space-group determination by convergent-beam electron diffraction

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2.5.3.1. Introduction

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Because the cross section for electron scattering is at least a thousand times greater than that for X-rays, and because multiple Bragg scattering preserves information on symmetry (such as the absence of inversion symmetry), electron diffraction is exquisitely sensitive to symmetry. The additional ability of modern electron-optical lenses to focus an electron probe down to nanometre dimensions, and so allow the study of nanocrystals too small for analysis by X-rays, has meant that the method of convergent-beam diffraction described here has now become the preferred method of symmetry determination for very small crystals, domains, twinned structures, quasicrystals, incommensurate structures and other imperfectly crystalline materials.

Convergent-beam electron diffraction (CBED) originated with the experiments of Kossel & Möllenstedt (1938[link]). However, modern crystallographic investigations by CBED began with the studies performed by Goodman & Lehmpfuhl (1965[link]) in a modified transmission electron microscope. They obtained CBED patterns by converging a conical electron beam with an angle of more than 10−3 rad on an ~30 nm diameter specimen area, which had uniform thickness and no bending. Instead of the usual diffraction spots, diffraction discs (in Laue or transmission geometry) were produced. The diffraction intensity within a disc shows a specific symmetry, which enables one to determine the point groups and space groups of microcrystals. Unlike X-ray diffraction, the method is extremely sensitive to the presence or absence of inversion symmetry.

The method corresponding to CBED in the field of light optics is the conoscope method. Using a conoscope, we can identify whether a crystal is isotropic, uniaxial or biaxial, and determine the optic axis and the sign of birefringence of a crystal. When CBED, a conoscope method using an electron beam, is utilized, more basic properties of a crystal – the crystal point group and space group – can be determined.

Point- and space-group determinations are routinely also carried out by X-ray diffraction. This method, to which kinematical diffraction is applicable, cannot determine whether a crystal is polar or nonpolar unless anomalous absorption is utilized. As a result, the X-ray diffraction method can only identify 11 Laue groups among 32 point groups. CBED, based fully upon dynamical diffraction, can distinguish polar crystals from nonpolar crystals using only a nanometre-sized crystal, thus allowing the unique identification of all the point groups by inspecting the symmetries appearing in CBED discs.

As pointed out above, an unambiguous experimental determination of crystal symmetry, in the case of X-ray diffraction, is usually not possible because of the apparent centrosymmetry of the diffraction pattern, even for noncentrosymmetric crystals. However, methods based on structure-factor and X-ray intensity statistics remain useful for the resolution of space-group ambiguities, and are routinely applied to structure determinations from X-ray data. These methods are described in Chapter 2.1[link] of this volume.

In the field of materials science, correct space-group determination by CBED is often requested prior to X-ray or neutron structure refinement, in particular in the case of Rietveld refinements based on powder diffraction data.

CBED can determine not only the point and space groups of crystals but also crystal structure parameters – lattice parameters, atom positions, Debye–Waller factors and low-order structure factors. The lattice parameters can be determined from sub-micron regions of thin crystals by using higher-order Laue zone (HOLZ) reflections with an accuracy of 1 × 10−4. Cherns et al. (1988[link]) were the first to perform strain analysis of artificial multilayer materials using the large-angle technique (LACBED) (Tanaka et al., 1980[link]). Since then, many strain measurements at interfaces of various multilayer materials have been successfully conducted. In recent years, strain analysis has been conducted using automatic analysis programs, which take account of dynamical diffraction effects (Krämer et al., 2000[link]). We refer to the book of Morniroli (2002[link]), which carries many helpful figures, clear photographs and a comprehensive list of papers on this topic.

Vincent et al. (1984a[link],b[link]) first applied the CBED method to the determination of the atom positions of AuGeAs. They analysed the intensities of HOLZ reflections by applying a quasi-kinematical approximation. Tanaka & Tsuda (1990[link], 1991[link]) and Tsuda & Tanaka (1995[link]) refined the structural parameters of SrTiO3 by applying the dynamical theory of electron diffraction. The method was extended to the refinements of CdS, LaCrO3 and hexagonal BaTiO3 (Tsuda & Tanaka, 1999[link]; Tsuda et al., 2002[link]; Ogata et al., 2004[link]). Rossouw et al. (1996[link]) measured the order parameters of TiAl through a Bloch-wave analysis of HOLZ reflections in a CBED pattern. Midgley et al. (1996[link]) refined two positional parameters of AuSn4 from the diffraction data obtained with a small convergence angle using multislice calculations.

Low-order structure factors were first determined by Goodman & Lehmpfuhl (1967[link]) for MgO. After much work on low-order structure-factor determination, Zuo & Spence determined the 200 and 400 structure factors of MgO in a very modern way, by fitting energy-filtered patterns and many-beam dynamical calculations using a least-squares procedure. For the low-order structure-factor determinations, the excellent com­pre­hensive review of Spence (1993[link]) should be referred to. Saunders et al. (1995[link]) succeeded in obtaining the deformation charge density of Si using the low-order crystal structure factors determined by CBED. For the reliable determination of the low-order X-ray crystal structure factors or the charge density of a crystal, accurate determination of the Debye–Waller factors is indispensable. Zuo et al. (1999[link]) determined the bond-charge distribution in cuprite. Simultaneous determination of the Debye–Waller factors and the low-order structure factors using HOLZ and zeroth-order Laue zone (ZOLZ) reflections was performed to determine the deformation charge density of LaCrO3 accurately (Tsuda et al., 2002[link]).

CBED can also be applied to the determination of lattice defects, dislocations (Cherns & Preston, 1986[link]), stacking faults (Tanaka, 1986[link]) and twins (Tanaka, 1986[link]). Since this topic is beyond the scope of the present chapter, readers are referred to pages 156 to 205 of the book by Tanaka et al. (1994[link]).

We also mention the book by Spence & Zuo (1992[link]), which deals with the whole topic of CBED, including the basic theory and a wealth of literature.

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.

2.5.3.3. Space-group determination

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2.5.3.3.1. Lattice-type determination

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When the point group of a specimen crystal is determined, the crystal axes may be found from a spot diffraction pattern recorded at a high-symmetry zone axis, using the orientations of the symmetry elements determined in the course of point-group determination. Integral-number indices are assigned to the spots of the diffraction patterns. The systematic absence of reflections indicates the lattice type of the crystal. It should be noted that reflections forbidden by the lattice type are always absent, even if dynamical diffraction takes place. (This is true for all sample thicknesses and accelerating voltages.) By comparing the experimentally obtained absences and the extinction rules known for the lattice types [P, C (A, B), I, F and R], a lattice type may be identified for the crystal concerned.

2.5.3.3.2. Identification of screw axes and glide planes

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There are three space-group symmetry elements of diperiodic plane figures: (1) a horizontal screw axis [2_1'], (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the point-group symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten point-group symmetry elements form 80 space groups.)

The ordinary extinction rules for screw axes and glide planes hold only in the approximation of kinematical diffraction. The kinematically forbidden reflections caused by these symmetry elements appear owing to Umweganregung of dynamical diffraction. Extinction of intensity, however, does take place in these reflections at certain crystal orientations with respect to the incident beam (i.e. in certain regions within a CBED disc). This dynamical extinction was first predicted by Cowley & Moodie (1959[link]) and was discussed by Miyake et al. (1960[link]) and Cowley et al. (1961[link]). Goodman & Lehmpfuhl (1964[link]) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965[link]) discussed the dynamical extinction in a more general way considering not only ZOLZ reflections but also HOLZ reflections. They completely clarified the dynamical extinction rules by considering the exact cancellation which may occur along certain symmetry-related multiple-scattering paths. Based on the results of Gjønnes & Moodie (1965[link]), Tanaka, Sekii & Nagasawa (1983[link]) tabulated the dynamical extinctions expected at all the possible crystal orientations for all the space groups. These were later tabulated in a better form on pages 162 to 172 of the book by Tanaka & Terauchi (1985[link]).

Fig. 2.5.3.10[link](a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a b-glide plane exists perpendicular to the a axis and/or a 21 screw axis exists in the b direction. Let us consider an Umweganregung path a in the zeroth-order Laue zone to the 010 forbidden reflection and path b which is symmetric to path a with respect to axis k. Owing to the translation of one half of the lattice parameter b caused by the glide plane and/or the 21 screw axis, the following relations hold between the crystal structure factors:[\eqalignno{F(h,k) &= F(\bar{h},k)\quad \hbox{for }k=2n,&\cr F(h,k) &= -F(\bar{h},k)\quad\hbox{for }k=2n+1. &(2.5.3.1)}]That is, the structure factors of reflections hk0 and [\bar{h}k0] have the same phase for even k but have opposite phases for odd k.

[Figure 2.5.3.10]

Figure 2.5.3.10 | top | pdf |

Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a b-glide plane and a 21 screw axis. (a) Umweganregung paths a, b and c. (b) Dynamical extinction lines A are formed in forbidden reflections 0k0 (k = odd). Extinction line B perpendicular to the lines A is formed in the exactly excited 010 reflection.

Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_1,k_1)F(\bar{h}_2,k_2)\ldots F(\bar{h}_n,k_n)\quad\hbox{for path }b,&\cr&&(2.5.3.2)}]where[\textstyle\sum\limits_{i=1}^n h_i=0,\quad \textstyle\sum\limits_{i=1}^n k_i = k \,\,(k={\rm odd})]and the functions including the excitation errors are omitted because only the cases in which the functions are the same for all the paths are considered. The excitation errors for paths a and b become the same when the projection of the Laue point along the zone axis concerned, L, lies on axis k. Since the two waves passing through paths a and b have the same amplitude but opposite signs, these waves are superposed on the 0k0 discs (k = odd) and cancel out, resulting in dark lines A in the forbidden discs, as shown in Fig. 2.5.3.10[link](b). The line A runs parallel to axis k passing through the projection point of the zone axis.

In path c, the reflections are arranged in the reverse order to those in path b. When the 010 reflection is exactly excited, two paths a and c are symmetric with respect to the bisector m′–m′ of the 010 vector having the same excitation errors. The following equation holds:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_n,k_n)F(\bar{h}_{n-1},k_{n-1})\ldots F(\bar{h}_1,k_1)\quad\hbox{for path }c.&\cr&&(2.5.3.3)}]Since the waves passing through these paths have the same amplitude but opposite signs, these waves are superposed on the 010 discs and cancel out, resulting in dark line B in this disc, as shown in Fig. 2.5.3.10[link](b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zeroth-order Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A2 and B2 lines, subscript 2 indicating that the Umweganregung paths lie in the zeroth-order Laue zone.

The dynamical extinction effect is analogous to interference phenomena in the Michelson interferometer. That is, the incident beam is split into two beams by Bragg reflections in a crystal. These beams take different paths, in which they suffer a relative phase shift of π and are finally superposed on a kinematically forbidden reflection to cancel out.

When the paths include higher-order Laue zones, the glide plane produces only extinction lines A but the screw axis causes only extinction lines B. These facts are attributed to the different relations between structure factors for a 21 screw axis and a glide plane,[\eqalignno{F(hkl)&=(-1)^kF(\bar{h}k\bar{l})\,\,\,\hbox{for a }2_1\hbox{ screw axis in the [010] direction,}&\cr &&(2.5.3.4)\cr F(hkl)&=(-1)^kF(\bar{h}kl)\,\,\,\hbox{for a }b\hbox{ glide in the (100) plane.}&\cr&&(2.5.3.5)}%fd2o5o3o5]In the case of the glide plane, extinction lines A are still formed because two waves passing through paths a and b have opposite signs to each other according to equation (2.5.3.5)[link], but extinction lines B are not produced because equation (2.5.3.4)[link] holds only for the 21 screw axis. In the case of the 21 screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4)[link], forming extinction lines B only. We call these lines A3 and B3 dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higher-order Laue zones.

It was predicted by Gjønnes & Moodie (1965[link]) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10[link]b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987[link]) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965[link]). Table 2.5.3.8[link] summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 21 screw axis. The three space-group symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

Table 2.5.3.8| top | pdf |
Dynamical extinction rules for an infinitely extended parallel-sided specimen

Symmetry elements of plane-parallel specimenOrientation to specimen surfaceDynamical extinction lines
Two-dimensional (zeroth Laue zone) interactionThree-dimensional (HOLZ) interaction
Glide planes perpendicular: g A2 and B2 A3
parallel: g intersection of A3 and B3
Twofold screw axis perpendicular: 21
parallel: [2_1^\prime] A2 and B2 B3

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A3 and B3. It is, however, not easy to observe the extinction lines A3 and B3 because broad extinction lines A2 and B2 appear at the same time. The presence of the extinction lines A3 and B3 can be revealed by inspecting the symmetries of fine defect HOLZ lines appearing in the forbidden reflections instead of by direct observation of the lines A3 and B3 (Tanaka, Sekii & Nagasawa, 1983[link]). That is, if HOLZ lines form lines A3 and B3, HOLZ lines are symmetric with respect to the extinction lines A2 and B2. If HOLZ lines do not form lines A3 and B3, HOLZ lines are asymmetric with respect to the extinction lines A2 and B2. When the HOLZ lines are symmetric about the extinction lines A2, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B2, a 21 screw axis exists. It should be noted that a relatively thick specimen area has to be selected to observe HOLZ lines in ZOLZ reflection discs.

Fig. 2.5.3.11[link] shows CBED patterns taken from (a) thin and (b) thick areas of FeS2, whose space group is [P2_1/a\bar{3}], at the 001 Bragg setting with the [100] electron-beam incidence. In the case of the thin specimen (Fig. 2.5.3.11[link]a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the odd-order discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11[link]b). The HOLZ lines are symmetric with respect to both A2 and B2 extinction lines. This fact proves the presence of the extinction lines A3 and B3, or both the c glide in the (010) plane and the 21 screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9[link]. Fig. 2.5.3.12[link] shows a [110] zone-axis CBED pattern of FeS2. A2 extinction lines are seen in both the 001 and [\bar{1}10] discs. Fine HOLZ lines are symmetric with respect to the A2 extinction lines in the [\bar{1}10] disc but asymmetric about the A2 extinction line in the 001 disc, indicating formation of the A3 extinction line only in the [\bar{1}10] disc. This proves the existence of a 21 screw axis in the [001] direction and an a glide in the (001) plane. The appearance of HOLZ lines is easily changed by a change of a few hundred volts in the accelerating voltage. Steeds & Evans (1980[link]) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV.

[Figure 2.5.3.11]

Figure 2.5.3.11 | top | pdf |

CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS2. (a) Dynamical extinction lines A2 and B2 are seen. (b) Extinction lines A3 and B3 as well as A2 and B2 are formed because HOLZ lines are symmetric about lines A2 and B2.

[Figure 2.5.3.12]

Figure 2.5.3.12 | top | pdf |

CBED pattern of FeS2 taken with the [110] electron-beam incidence. In the 001 and [00\bar 1] discs, HOLZ lines are asymmetric with respect to extinction lines A2, indicating the existence of a 21 screw axis parallel to the c axis. In the [\bar 1 10] and [1\bar10] discs, HOLZ lines are symmetric with respect to extinction lines A2, indicating existence of a glide plane perpendicular to the c axis.

Another practical method for distinguishing between glide planes and 21 screw axes is that reported by Steeds et al. (1978[link]). The method is based on the fact that the extinction lines are observable even when a crystal is rotated with a glide plane kept parallel and with a 21 screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10[link](a), extinction lines A3 produced by a glide plane remain even when the crystal is rotated with respect to axis h but the lines are destroyed by a rotation of the crystal about axis k. Extinction lines B3 originating from a 21 screw axis are not destroyed by a crystal rotation about axis k but the lines are destroyed by a rotation with respect to axis h.

2.5.3.3.3. Space-group determination

| top | pdf |

We now describe a space-group determination method which uses the dynamical extinction lines caused by the horizontal screw axis [2_1'] and the vertical glide plane g of an infinitely extended parallel-sided specimen. We do not use the extinction due to the glide plane g′ because observation of the extinction requires a laborious experiment. It should be noted that a vertical glide plane with a glide vector not parallel to the specimen surface cannot be a symmetry element of a specimen of finite thickness; however, the component of the glide vector perpendicular to the incident beam acts as a symmetry element g. (Which symmetry elements are observed by CBED is discussed in Section 2.5.3.3.5[link].) The 21, 41, 43, 61, 63 and 65 screw axes of crystal space groups that are set perpendicular to the incident beam act as a symmetry element [2_1'] because two or three successive operations of 41, 43, 61, 63 and 65 screw axes make them equivalent to a 21 screw axis: (41)2 = (43)2 = (61)3 = (63)3 = (65)3 = 21. The 42, 31, 32, 62 and 64 screw axes that are set perpendicular to the incident beam do not produce dynamical extinction lines because the 42 screw axis acts as a twofold rotation axis due to the relation (42)2 = 2, the 31 and 32 screw axes give no specific symmetry in CBED patterns, and the 62 and 64 screw axes are equivalent to 31 and 32 screw axes due to the relations (62)2 = 32 and (64)2 = 31. Modifications of the dynamical extinction rules were investigated by Tanaka, Sekii & Nagasawa (1983[link]) when more than one crystal symmetry element (that gives rise to dynamical extinction lines) coexists and when the symmetry elements are combined with various lattice types. Using these results, dynamical extinction lines A2, A3, B2 and B3 expected from all the possible crystal settings for all the space groups were tabulated.

Table 2.5.3.9[link] shows all the dynamical extinction lines appearing in the kinematically forbidden reflections for all the possible crystal settings of all the space groups. The first column gives space groups. In each of the following pairs of columns, the left-hand column of the pair gives the reflection indices and the symmetry elements causing the extinction lines and the right-hand column gives the types of the extinction lines. The (second) suffixes 1, 2 and 3 of a 21 screw axis in each column distinguish the first, the second and the third screw axis of the space group (as in the symbols 211 and 212 of space group P21212). The glide symbols in the [001] column for space group P4/nnc have two suffixes (n21 and n22). The first suffix 2 denotes the second glide plane of the space group. The second suffixes 1 and 2, which appear in the tetragonal and cubic systems, distinguish two equivalent glide planes which lie in the x and y planes. The equivalent planes are distinguished only for the cases of [100], [010] and [001] electron-beam incidences, for convenience. The c-glide planes of space group Pcc2 are distinguished with symbols c1 and c2 (the first suffix only), because the equivalent planes do not exist. The glide symbol in the [001] column for space group P4/mbm has only one suffix 1 or 2. The suffix distinguishes the equivalent glide planes lying in the x and y planes. The first suffix to distinguish the first and the second glide planes is not necessary because the space group has only one glide symbol b. When the index of the incident-beam direction is expressed with a symbol like [h0l] for point groups 2, m and 2/m, the index h or l can take a value of zero. That is, the extinction rules are applicable to the [100] and [001] electron-beam incidences. However, if columns for [100], [010] and [001] incidences are present, as in the case of point group mm2, [hk0], [0kl] and [h0l] incidences are only for nonzero h, k and l. The reflections in which the extinction lines appear are always perpendicular to the corresponding incident-beam directions [(0k'l'\perp [0kl],\, h'k'0 \perp [hk0],\ldots)]. The indices of the reflections in which extinction lines appear are odd if no remark is given. For c-glide planes of space groups R3c and [R\bar{3}c] and for d-glide planes, the reflections in which extinction lines appear are specified as 6n + 3 and 4n + 2 orders, respectively.

Table 2.5.3.9| top | pdf |
Dynamical extinction lines appearing in ZOLZ reflections for all crystal space groups except Nos. 1 and 2

Point groups 2, m, 2/m (second setting, unique axis b)

Space groupIncident-beam direction
[h0l]
3 P2      
4 P21 0k0 A2 B2
21   B3
5 C2      
6 Pm      
7 Pc h0lo A2 B2
c A3  
8 Cm      
9 Cc he0lo A2 B2
c A3  
10 P2/m      
11 P21/m 0k0 A2 B2
21   B3
12 C2/m      
13 P2/c h0lo A2 B2
c A3  
14 P21/c 0k0 A2 B2
21   B3
h0lo A2 B2
c A3  
15 C2/c he0lo A2 B2
c A3  

Point group 222

Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
16 P222                                    
17 P2221 00l A2 B2 00l A2 B2       00l A2 B2            
21   B3 21   B3       21   B3            
18 P21212 0k0 A2 B2 h00 A2 B2 h00 A2 B2       h00 A2 B2 0k0 A2 B2
212   B3 211   B3 211   B3       211   B3 212   B3
            0k0                      
            212                      
19 P212121 0k0 A2 B2 h00 A2 B2 h00 A2 B2 00l A2 B2 h00 A2 B2 0k0 A2 B2
212   B3 211   B3 211   B3 213   B3 211   B3 212   B3
00l     00l     0k0                      
213     213     212                      
20 C2221 00l A2 B2 00l A2 B2       00l A2 B2            
21   B3 21   B3       21   B3            
21 C222                                    
22 F222                                    
23 I222                                    
24 I212121                                    

Point group mm2

Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
25 Pmm2                                    
26 Pmc21 00l A2 B2 00l           00l A2 B2       h0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
27 Pcc2 00l     00l                 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
28 Pma2             h00 A2 B2             ho0l A2 B2
            a A3               a A3  
29 Pca21 00l     00l A2 B2 h00 A2 B2 00l A2 B2 0klo A2 B2 ho0l A2 B2
21   B3 c, 21 A3 B3 a A3   21   B3 c A3   a A3  
30 Pnc2 00l
c

A3
  00l
n

A3
  0k0
n
A2
A3
B2       0kl: k + l = 2n + 1
n
A2
A3
B2 h0lo
c
A2
A3
B2
31 Pmn21 00l
n, 21
A2
A3
B2
B3
00l
21
 
B3
h00
n
A2
A3
B2 00l
21
A2 B2
B3
      h0l: h + l = 2n + 1
n
A2
A3
B2
32 Pba2             h00 A2 B2       0kol A2 B2 ho0l A2 B2
            a A3         b A3   a A3  
            0k0                      
            b                      
33 Pna21 00l
21
 
B3
00l
n, 21
A2
A3
B2
B3
h00
a
0k0
n
A2
A3
B2 00l
21
A2 B2
B3
0kl: k + l = 2n + 1
n
A2
A3
B2 ho0l
a
A2
A3
B2
34 Pnn2 00l
n2

A3
  00l
n1

A3
  h00
n2
0k0
n1
A2
A3
B2       0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
35 [\matrix{\hfill Cmm2\cr \hfill ba2}]                                  
36 [\matrix{\hfill Cmc2_1\cr\hfill bn2_1}] 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
37 [\matrix{\hfill Ccc2\cr\hfill nn2}] 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
38 [\matrix{\hfill Amm2\phantom{_1}\cr\hfill nc2_1}]                                    
39 [\matrix{\hfill Abm2\phantom{_1}\cr\hfill cc2_1}]                         0kolo A2 B2      
                        b A3        
40 [\matrix{\hfill Ama2\phantom{_1}\cr\hfill nn2_1}]             h00 A2 B2             ho0le A2 B2
            a A3               a A3  
41 [\matrix{\hfill Aba2\phantom{_1}\cr\hfill cn2_1}]             h00 A2 B2       0kolo A2 B2 ho0le A2 B2
            a A3         b A3   a A3  
42 Fmm2                                    
43 [\matrix{\hfill Fdd2\phantom{_1}\cr\hfill dd2_1}] 00l: l = 4n + 2
d2

A3
  00l: l = 4n + 2
d1

A3
  h00: h = 4n + 2
d2
0k0: k = 4n + 2
d1
A2
A3
B2       0kele: ke + le = 4n + 2
d1
A2
A3
B2 he0le: he + le = 4n + 2
d2
A2
A3
B2
44 [\matrix{\hfill Imm2\phantom{_1}\cr\hfill nn2_1}]                                    
45 [\matrix{\hfill Iba2\phantom{_1}\cr\hfill cc2_1}]                         0kolo A2 B2 ho0lo A2 B2
                        b A3   a A3  
46 [\matrix{\hfill Ima2\phantom{_1}\cr\hfill nc2_1}]                               ho0lo A2 B2
                              a A3  

Point group mmm

Space groupIncident-beam direction
[100][010][001][hk0][0kl][h0l]
47 P2/m2/m2/m                                    
48 P2/n2/n2/n 00l
n2
0k0
n3

A3
  00l
n1
h00
n3

A3
  0k0
n1
h00
n2

A3
  hk0: h + k = 2n + 1
n3
A2
A3
B2 0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
49 P2/c2/c2/m 00l     00l                 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
50 P2/b2/a2/n 0k0
n

A3
  h00
n

A3
  0k0
b
h00
a

A3
  hk0: h + k = 2n + 1
n
A2
A3
B2 0kol
b
A2
A3
B2 ho0l
a
A2
A3
B2
51 P21/m2/m2/a       h00 A2 B2 h00     hok0 A2 B2 h00 A2 B2      
      21, a A3 B3 21   B3 a A3   21   B3      
52 P2/n21/n2/a 00l
n2

A3
  00l
n1
h00
a

A3
  0k0
n1, 21
A2
A3
B2
B3
hok0
a
A2
A3
B2 0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
0k0           h00                 0k0 A2 B2
21   B3       n2 A3               21   B3
53 P2/m2/n21/a 00l
n, 21
A2
A3
B2
B3
h00
a

A3
  h00
n

A3
  hok0
a
A2
A3
B2       h0l: h + l = 2n + 1
n
A2
A3
B2
      00l           00l A2 B2            
      21   B3       21   B3            
54 P21/c2/c2/a 00l     00l     h00     hok0 A2 B2 0klo A2 B2 h0lo A2 B2
c2 A3   c1 A3   21   B3 a A3   c1 A3   c2 A3  
      h00 A2 B2             h00 A2 B2      
      a, 21 A3 B3             21   B3      
55 P21/b21/a2/m 0k0     h00     0k0 A2 B2       0kol A2 B2 ho0l A2 B2
212   B3 211   B3 b, 212 A3 B3       b A3   a A3  
            h00           h00 A2 B2 0k0 A2 B2
            a, 211           211   B3 212   B3
56 P21/c21/c2/n 00l
c2

A3
  00l
c1

A3
  0k0
212
h00
211
 
B3
hk0: h + k = 2n + 1
n
A2
A3
B2 0klo
c1
A2
A3
B2 h0lo
c2
A2
A3
B2
0k0 A2 B2 h00 A2 B2             h00 A2 B2 0k0 A2 B2
212, n A3 B3 211, n A3 B3             211   B3 212   B3
57 P2/b21/c21/m 00l A2 B2 00l     0k0 A2 B2 00l A2 B2 0kol A2 B2 h0lo A2 B2
c, 212 A3 B3 212   B3 b, 211 A3 B3 212   B3 b A3   c A3  
0k0                             0k0 A2 B2
211   B3                         211   B3
58 P21/n21/n2/m 00l
n2

A3
  00l
n1

A3
  0k0
n1, 212
h00
n2, 211
A2
A3
B2
B3
      0kl: k + l = 2n + 1
n1
A2
A3
B2 h0l: h + l = 2n + 1
n2
A2
A3
B2
0k0     h00                 h00 A2 B2 0k0 A2 B2
212   B3 211   B3             211   B3 212   B3
59 P21/m21/m2/n 0k0
n, 212
A2
A3
B2
B3
h00
n, 211
A2
A3
B2
B3
0k0
212
h00
211
 
B3
hk0: h + k = 2n + 1
n
A2
A3
B2 h00
211
A2 B2
B3
0k0
212
A2 B2
A3
60 P21/b2/c21/n 00l
c, 212
A2
A3
B2
B3
h00
n, 211
A2
A3
B2
B3
0k0
b

A3
  hk0: h + k = 2n + 1
n
A2
A3
B2 0kol
b
A2
A3
B2 h0lo
c
A2
A3
B2
0k0     00l     h00     00l A2 B2 h00 A2 B2      
n A3   212   B3 211   B3 212   B3 211   B3      
61 P21/b21/c21/a 00l A2 B2 00l     0k0 A2 B2 hok0 A2 B2 0kol A2 B2 h0lo A2 B2
c, 213 A3 B3 213   B3 b, 212 A3 B3 a A3   b A3   c A3  
0k0     h00 A2 B2 h00     00l A2 B2 h00 A2 B2 0k0 A2 B2
212   B3 a, 211 A3 B3 211   B3 213   B3 211   B3 212   B3
62 P21/n21/m21/a 00l
213
0k0
212
 
B3
00l
n, 213
h00
a, 211
A2
A3
B2
B3
0k0
n, 212
A2
A3
B2
B3
hok0
a
A2
A3
B2 0kl: k + l = 2n + 1
n
A2
A3
B2 0k0
212
A2 B2
B3
            h00     00l A2 B2 h00 A2 B2      
            211   B3 213   B3 211   B3      
63 C2/m2/c21/m 00l A2 B2 00l           00l A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       21   B3       c A3  
64 C2/m2/c21/a 00l A2 B2 00l           hoko0 A2 B2       he0lo A2 B2
c, 21 A3 B3 21   B3       a A3         c A3  
                  00l A2 B2            
                  21   B3            
65 C2/m2/m2/m                                    
66 C2/c2/c2/m 00l     00l                 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3               c1 A3   c2 A3  
67 C2/m2/m2/a                   hoko0 A2 B2            
                  a A3              
68 C2/c2/c2/a 00l     00l           hoko0 A2 B2 0kelo A2 B2 he0lo A2 B2
c2 A3   c1 A3         a A3   c1 A3   c2 A3  
69 F2/m2/m2/m                                    
70 F2/d2/d2/d 00l: l =
4n + 2
d2
0k0: k =
4n + 2
d3

A3
  h00: h =
4n + 2
d3
00l: l =
4n + 2
d1

A3
  0k0: k =
4n + 2
d1
h00: h =
4n + 2
d2

A3
  heke0: he + ke = 4n + 2
d3
A2
A3
B2 0kele: ke + le = 4n + 2
d1
A2
A3
B2 he0le: he + le = 4n + 2
d2
A2
A3
B2
71 I2/m2/m2/m                                    
72 I2/b2/a2/m                         0kolo A2 B2 ho0lo A2 B2
                        b A3   a A3  
73 I21/b21/c21/a                   hoko0 A2 B2 0kolo A2 B2 ho0lo A2 B2
                  a A3   b A3   c A3  
74 I21/m21/m21/a                   hoko0 A2 B2            
                  a A3              

Point groups [4, \bar 4, 4/m]

Space groupIncident-beam direction
[hk0]
75 P4      
76 P41 00l A2 B2
41   B3
77 P42      
78 P43 00l A2 B2
43   B3
79 I4      
80 I41      
81 [P\bar 4]      
82 [I\bar 4]      
83 P4/m      
84 P42/m      
85 P4/n hk0: h + k = 2n + 1 A2 B2
n A3  
86 P42/n hk0: h + k = 2n + 1 A2 B2
n A3  
87 I4/m      
88 I41/a hoko0 A2 B2
a A3  

Point group 422

Space groupIncident-beam direction
[hk0][0kl]
89 P422            
90 P4212       h00 A2 B2
      21   B3
91 P4122 00l A2 B2      
41   B3      
92 P41212 00l A2 B2 h00 A2 B2
41   B3 21   B3
93 P4222            
94 P42212       h00 A2 B2
      21   B3
95 P4322 00l A2 B2      
43   B3      
96 P43212 00l A2 B2 h00 A2 B2
43   B3 21   B3
97 I422            
98 I4122            

Point group 4mm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][h0l][hhl]
99 P4mm                              
100 P4bm     h00 A2 B2     ho0l A2 B2      
      a2 A3         a A3        
      0k0                      
      b1                      
101 P42cm 00l                 h0lo A2 B2      
c2 A3               c A3        
102 P42nm 00l     h00 A2 B2       h0l: h + l = 2n + 1 A2 B2      
n2 A3   n2 A3         n A3        
      0k0                      
      n1                      
103 P4cc 00l           00l     h0lo A2 B2 hhlo A2 B2
c12 A3         c2 A3   c1 A3   c2 A3  
104 P4nc 00l     h00 A2 B2 00l     h0l: h + l = 2n + 1 A2 B2 hhlo A2 B2
n2 A3   n2 A3   c A3   n A3   c A3  
      0k0                      
      n1                      
105 P42mc             00l           hhlo A2 B2
            c A3         c A3  
106 P42bc       h00 A2 B2 00l     ho0l A2 B2 hhlo A2 B2
      a2 A3   c A3   a A3   c A3  
      0k0                      
      b1                      
107 I4mm                              
108 I4cm                   ho0lo A2 B2      
                  c A3        
109 I41md       hh0, [\bar h h 0] A2 B2 00l: l = 4n + 2           hhle: 2h + le = 4n + 2 A2 B2
      d A3   d A3         d A3  
110 I41cd       hh0, [\bar h h 0] A2 B2 00l: l = 4n + 2     ho0lo A2 B2 hhle: 2h + le = 4n + 2 A2 B2
      d A3   d A3   c A3   d A3  

Point group [\bar 4 2 m]. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][h0l][hhl]
111 [P\bar 4 2 m]                              
112 [P\bar 4 2 c]             00l         hhlo A2 B2
            c A3         c A3  
113 [P\bar 4 2_1 m] 0k0 A2 B2 h00 A2 B2       0k0 A2 B2      
212   B3 211   B3       21   B3      
      0k0                      
      212                      
114 [P\bar 4 2_1 c] 0k0 A2 B2 h00 A2 B2 00l     0k0 A2 B2 hhlo A2 B2
212   B3 211   B3 c A3   21   B3 c A3  
      0k0                      
      212                      
115 [P \bar 4 m 2]                              
116 [P\bar 4 c 2] 00l                 h0lo A2 B2      
c2 A3               c A3        
117 [P\bar 4 b 2]       h00 A2 B2       ho0l A2 B2      
      a2 A3         a A3        
      0k0                      
      b1                      
118 [P\bar 4 n 2] 00l     h00 A2 B2       h0l: h + l = 2n + 1 A2 B2      
n2 A3   n2 A3         n A3        
      0k0                      
      n1                      
119 [I\bar 4 m 2]                              
120 [I\bar 4 c 2]                   ho0lo A2 B2      
                  c A3        
121 [I\bar 4 2 m]                              
122 [I\bar 4 2 d]       hh0, [\bar h h 0] A2 B2 00l: l = 4n + 2           hhle: 2h + le = 4n + 2 A2 B2
      d A3   d A3         d A3  

Point group 4/mmm. The symbol a in the column [h0l] is equivalent to the symbol b in the space groups of the first column.

Space groupIncident-beam direction
[100][001][110][h0l][hhl][hk0]
123 P4/mmm                                    
P4/m2/m2/m                                    
124 P4/mcc 00l           00l     h0lo A2 B2 hhlo A2 B2      
P4/m2/c2/c c12 A3         c2 A3   c1 A3   c2 A3        
125 P4/nbm
P4/n2/b2/m
0k0
n

A3
  h00
a2
0k0
b1

A3
      ho0l
a
A2
A3
B2       hk0: h + k = 2n + 1
n
A2
A3
B2
126 P4/nnc
P4/