Because the cross section for electron scattering is at least a thousand times greater than that for Xrays, 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 electronoptical lenses to focus an electron probe down to nanometre dimensions, and so allow the study of nanocrystals too small for analysis by Xrays, has meant that the method of convergentbeam 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.
Convergentbeam electron diffraction (CBED) originated with the experiments of Kossel & Möllenstedt (1938). However, modern crystallographic investigations by CBED began with the studies performed by Goodman & Lehmpfuhl (1965) 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 Xray 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 spacegroup determinations are routinely also carried out by Xray 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 Xray 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 nanometresized 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 Xray diffraction, is usually not possible because of the apparent centrosymmetry of the diffraction pattern, even for noncentrosymmetric crystals. However, methods based on structurefactor and Xray intensity statistics remain useful for the resolution of spacegroup ambiguities, and are routinely applied to structure determinations from Xray data. These methods are described in Chapter 2.1
of this volume.
In the field of materials science, correct spacegroup determination by CBED is often requested prior to Xray 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 loworder structure factors. The lattice parameters can be determined from submicron regions of thin crystals by using higherorder Laue zone (HOLZ) reflections with an accuracy of 1 × 10^{−4}. Cherns et al. (1988) were the first to perform strain analysis of artificial multilayer materials using the largeangle technique (LACBED) (Tanaka et al., 1980). 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). We refer to the book of Morniroli (2002), which carries many helpful figures, clear photographs and a comprehensive list of papers on this topic.
Vincent et al. (1984a,b) first applied the CBED method to the determination of the atom positions of AuGeAs. They analysed the intensities of HOLZ reflections by applying a quasikinematical approximation. Tanaka & Tsuda (1990, 1991) and Tsuda & Tanaka (1995) refined the structural parameters of SrTiO_{3} by applying the dynamical theory of electron diffraction. The method was extended to the refinements of CdS, LaCrO_{3} and hexagonal BaTiO_{3} (Tsuda & Tanaka, 1999; Tsuda et al., 2002; Ogata et al., 2004). Rossouw et al. (1996) measured the order parameters of TiAl through a Blochwave analysis of HOLZ reflections in a CBED pattern. Midgley et al. (1996) refined two positional parameters of AuSn_{4} from the diffraction data obtained with a small convergence angle using multislice calculations.
Loworder structure factors were first determined by Goodman & Lehmpfuhl (1967) for MgO. After much work on loworder structurefactor determination, Zuo & Spence determined the 200 and 400 structure factors of MgO in a very modern way, by fitting energyfiltered patterns and manybeam dynamical calculations using a leastsquares procedure. For the loworder structurefactor determinations, the excellent comprehensive review of Spence (1993) should be referred to. Saunders et al. (1995) succeeded in obtaining the deformation charge density of Si using the loworder crystal structure factors determined by CBED. For the reliable determination of the loworder Xray crystal structure factors or the charge density of a crystal, accurate determination of the Debye–Waller factors is indispensable. Zuo et al. (1999) determined the bondcharge distribution in cuprite. Simultaneous determination of the Debye–Waller factors and the loworder structure factors using HOLZ and zerothorder Laue zone (ZOLZ) reflections was performed to determine the deformation charge density of LaCrO_{3} accurately (Tsuda et al., 2002).
CBED can also be applied to the determination of lattice defects, dislocations (Cherns & Preston, 1986), stacking faults (Tanaka, 1986) and twins (Tanaka, 1986). 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).
We also mention the book by Spence & Zuo (1992), which deals with the whole topic of CBED, including the basic theory and a wealth of literature.
When an electron beam traverses a thin slab of crystal parallel to a zone axis, one can easily imagine that symmetries parallel to the zone axis should appear in the resulting CBED pattern. It is, however, more difficult to imagine what symmetries appear due to symmetries perpendicular to the incident beam. Goodman (1975) pioneered the clarification of CBED symmetries for the twofold rotation axis and mirror plane perpendicular to the incident beam, and the symmetry of an inversion centre, with the help of the reciprocity theorem of scattering theory. Tinnappel (1975) solved many CBED symmetries at various crystal settings with respect to the incident beam using a grouptheoretical treatment. Buxton et al. (1976) also derived these results from first principles, and generalized them to produce a systematic method for the determination of the crystal point group. Tanaka, Saito & Sekii (1983) developed a method to determine the point group using simultaneously excited manybeam patterns. The pointgroupdetermination method given by Buxton et al. (1976) is described with the aid of the description by Tanaka, Saito & Sekii (1983) in the following.
Since CBED uses the Laue geometry, Buxton et al. (1976) assumed a perfectly crystalline specimen in the form of a parallelsided slab which is infinite in two dimensions. The symmetry elements of the specimen (as distinct from those of an infinite crystal) form `diffraction groups', which are isomorphic to the point groups of the diperiodic plane figures and Shubnikov groups of coloured plane figures. The diffraction groups of a specimen are determined from the symmetries of CBED patterns taken at various orientations of the specimen. The crystal pointgroup of the specimen is identified by referring to Fig. 2.5.3.4, which gives the relation between diffraction groups and crystal point groups.
A specimen that is parallelsided and is infinitely extended in the x and y directions has ten symmetry elements. The symmetry elements consist of six twodimensional symmetry elements and four threedimensional ones. The operation of the former elements transforms an arbitrary coordinate (x, y, z) into (x′, y′, z), with z remaining the same. The operation of the latter transforms a coordinate (x, y, z) into (x′, y′, z′), where . A vertical mirror plane m and one, two, three, four and sixfold rotation axes that are parallel to the surface normal z are the twodimensional symmetry elements. A horizontal mirror plane m′, an inversion centre i, a horizontal twofold rotation axis 2′ and a fourfold rotary inversion are the threedimensional symmetry elements, and are shown in Fig. 2.5.3.1. The fourfold rotary inversion was not recognized as a symmetry element until the point groups of the diperiodic plane figures were considered (Buxton et al., 1976). Table 2.5.3.1 lists these symmetry elements, where the symbols in parentheses express symmetries of CBED patterns expected from threedimensional symmetry elements.
Twodimensional symmetry elements  Threedimensional symmetry elements 
1 
m′ (1_{R}) 
2 
i (2_{R}) 
3 
2′ (m_{2}, m_{R}) 
4 
(4_{R}) 
5 

6 

m 



Figure 2.5.3.1
 top  pdf  Four symmetry elements m′, i, 2′ and of an infinitely extended parallelsided specimen.

The diffraction groups are constructed by combining these symmetry elements (Table 2.5.3.2). Twodimensional symmetry elements and their combinations are given in the top row of the table. The third symmetry m in parentheses is introduced automatically when the first two symmetry elements are combined. Threedimensional symmetry elements are given in the first column. The equations given below the table indicate that no additional threedimensional symmetry elements can appear by combination of two symmetry elements in the first column. As a result, 31 diffraction groups are produced by combining the elements in the first column with those in the top row. Diffraction groups in square brackets have already appeared ealier in the table. In the fourth row, three columns have two diffraction groups, which are produced when symmetry elements are combined at different orientations. In the last row, five columns are empty because a fourfold rotary inversion cannot coexist with threefold and sixfold rotation axes. In the last column, the number of independent diffraction groups in each row is given, the sum of the numbers being 31.
 1  2  3  4  6  m  2m(m)  3m  4m(m)  6m(m)  
1 
1 
2 
3 
4 
6 
m 
2m(m) 
3m 
4m(m) 
6m(m) 
10 
(m′) 1_{R} 
1_{R} 
21_{R} 
31_{R} 
41_{R} 
61_{R} 
m1_{R} 
2m(m)1_{R} 
3m1_{R} 
4m(m)1_{R} 
6m(m)1_{R} 
10 
(i) 2_{R} 
2_{R} 
[21_{R}] 
6_{R} 
[41_{R}] 
[61_{R}] 
2_{R}m(m_{R}) 
[2m(m)1_{R}] 
6_{R}m(m_{R}) 
[4m(m)1_{R}] 
[6m(m)1_{R}] 
4 






[2_{R}m(m_{R})] 
[2m(m)1_{R}] 
[3m1_{R}] 



(2′) m_{R} 
m_{R} 
2m_{R}(m_{R}) 
3m_{R} 
4m_{R}(m_{R}) 
6m_{R}(m_{R}) 
[m1_{R}] 
[4_{R}(m)m_{R}] 
[6_{R}m(m_{R})] 
[4m(m)1_{R}] 
[6_{R}m(m_{R})] 
5 
4_{R} 

4_{R} 

[41_{R}] 

4_{R}m(m_{R}) 
[4_{R}m(m_{R})] 

[4m(m)1_{R}] 

2 
1 _{R} × 2 _{R} = 2, 2 _{R} × 2 _{R} = 1, m_{R} × 2 _{R} = m, 4 _{R} × 2 _{R} = 4, 1 _{R} × m_{R} = m × m_{R}, 1 _{R} × 4 _{R} = 4 × 1 _{R}, m_{R} × 4 _{R} = m × 4 _{R}.

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

Figure 2.5.3.2
 top  pdf  Illustration of symmetries appearing in darkfield patterns (DPs). (a) m_{R} and m_{2}; (b) 1_{R}; (c) 2_{R}; (d) 4_{R}, originating from 2′, m′, i and , respectively.

Next we explain the symbols of the CBED symmetries. (1) Operation m_{R} is shown in the lefthand part of Fig. 2.5.3.2(a), which implies successive operations of (a) a mirror m with respect to a twofold rotation axis, transforming an open circle beam (○) in reflection G into a beam (+) in reflection G′ and (b) rotation R of this beam by π about the centre point of disc G′ (or the exact Bragg position of reflection G′), resulting in position ○ in reflection G′. The combination of the two operations is written as m_{R}. When the twofold rotation axis is parallel to the diffraction vector G, two beams (○) in the lefthand part of the figure become one reflection G, and a mirror symmetry, whose mirror line is perpendicular to vector G and passes through the centre of disc G, appears between the two beams (the righthand side figure of Fig. 2.5.3.2a). The mirror symmetry is labelled m_{2} after the twofold rotation axis. (2) Operation 1_{R} (Fig. 2.5.3.2b) for a horizontal mirror plane is a combination of a rotation by 2π of a beam (○) about a zone axis O (symbol 1), which is equivalent to no rotation, and a rotation by π of the beam about the exact Bragg position or the centre of disc G. (3) Operation 2_{R} is a rotation by π of a beam (○) in reflection G about a zone axis (symbol 2), which transforms the beam into a beam (+) in reflection −G, followed by a rotation by π of the beam (+) about the centre of disc −G, resulting in the beam (○) in disc −G (Fig. 2.5.3.2c). The symmetry is called translational symmetry after Goodman (1975) because the pattern of disc +G coincides with that of disc −G by a translation. It is emphasized that an inversion centre is identified by the test of translational symmetry about a pair of ±G darkfield patterns – if one disc can be translated into coincidence with the other, an inversion centre exists. We call the pair ±DP. (4) Operation 4_{R} (Fig. 2.5.3.2d) can be understood in a similar manner. It is noted that regular letters are symmetries about a zone axis, while subscripts R represent symmetries about the exact Bragg position. We call a pattern that contains an exact Bragg position (if possible at the disc centre) a darkfield pattern. As far as CBED symmetries are concerned, we do not use the term darkfield pattern if a disc does not contain the exact Bragg position.
The four threedimensional symmetry elements are found to produce different symmetries in the DPs. These facts imply that these symmetry elements can be identified unambiguously from the symmetries of CBED patterns.
Twodimensional symmetry elements that belong to a zone axis exhibit their symmetries in CBED patterns or zoneaxis patterns (ZAPs) directly, even if dynamical diffraction takes place. A ZAP contains a brightfield pattern (BP) and a whole pattern (WP). The BP is the pattern appearing in the brightfield disc [the central or `direct' (000) beam]. The WP is composed of the BP and the pattern formed by the surrounding diffraction discs, which are not exactly excited. The twodimensional symmetry elements m, 1, 2, 3, 4 and 6 yield symmetry m_{v} and one, two, three, four and sixfold rotation symmetries, respectively, in WPs, where the suffix v for m_{v} is assigned to distinguish it from mirror symmetry m_{2} caused by a horizontal twofold rotation axis.
It should be noted that a BP shows not only the zoneaxis symmetry but also threedimensional symmetries, indicating that the BP can have a higher symmetry than the symmetry of the corresponding WP. The symmetries of the BP due to threedimensional symmetry elements are obtained by moving the DPs to the zone axis. As a result, the threedimensional symmetry elements m′, i, 2′ and produce, respectively, symmetries 1_{R}, 1, m_{2} and 4 in the BP, instead of 1_{R}, 2_{R}, m_{2} and 4_{R} in the DPs (Fig. 2.5.3.2). We mention that the BP cannot distinguish whether a specimen crystal has an inversion centre or not, because an inversion centre forms the lowest symmetry 1 in the BP.
In conclusion, all the twodimensional symmetry elements can be identified from the WP symmetries.
All the symmetry elements of the diffraction groups can be identified from the symmetries of a WP and DPs. But it is practical and convenient to use just the four patterns WP, BP, DP and ±DP to determine the diffraction group. The symmetries appearing in these four patterns are given for the 31 diffraction groups in Table 2.5.3.3 (Tanaka, Saito & Sekii, 1983), which is a detailed version of Table 2 of Buxton et al. (1976). All the possible symmetries of the DP and ±DP appearing at different crystal orientations are given in the present table. When a BP has a higher symmetry than the corresponding WP, the symmetry elements that produce the BP are given in parentheses in column II except only for the case of 4_{R}. When two types of vertical mirror planes exist, these are distinguished by symbols m_{v} and m_{v′}. Each of the two or three symmetries given in columns IV and V for many diffraction groups appears in a DP or ±DP in different directions.
I  II  III  IV  V  VI 
1 
1 
1 
1 
1 
1_{R} 
1_{R} 
2 
1 
2 = 1_{R} 
1 
(1_{R}) 
2 
2 
2 
1 
2 
21_{R} 
2_{R} 
1 
1 
1 
2_{R} 
21_{R} 
2 
2 
2 
21_{R} 
m_{R} 
m 
1 
1 
1 
m1_{R} 
(m_{2}) 
m_{R} 
m_{2} 
1 
m 
m_{v} 
m_{v} 
1 
1 
m_{v} 
m_{v} 
1 
m1_{R} 
2mm 
m_{v} 
2 
1 
[m_{v} + m_{2} + (1_{R})] 
m_{v}1_{R} 
2m_{v}m_{2} 
1 
2m_{R}m_{R} 
2mm 
2 
1 
2 
2mm1_{R} 
(2 + m_{2}) 
m_{2} 
2m_{R}(m_{2}) 
2mm 
2m_{v}m_{v′} 
2m_{v}m_{v′} 
1 
2 
m_{v} 
2m_{v′}(m_{v}) 
2_{R}mm_{R} 
m_{v} 
m_{v} 
1 
2_{R} 
m_{2} 
2_{R}m_{v′}(m_{2}) 
m_{v} 
2_{R}m_{R}(m_{v}) 
2mm1_{R} 
2m_{v}m_{v′} 
2m_{v}m_{v′} 
2 
21_{R} 
2m_{v}m_{2} 
21_{R}m_{v′}(m_{v}) 
4 
4 
4 
1 
2 
41_{R} 
4_{R} 
4 
2 
1 
2 
41_{R} 
4 
4 
2 
21_{R} 
4m_{R}m_{R} 
4mm 
4 
1 
2 
4mm1_{R} 
(4 + m_{2}) 
m_{2} 
2m_{R}(m_{2}) 
4mm 
4m_{v}m_{v′} 
4m_{v}m_{v′} 
1 
2 
m_{v} 
2m_{v′}(m_{v}) 
4_{R}mm_{R} 
4mm 
2m_{v}m_{v′} 
1 
2 
(2m_{v}m_{v′} + m_{2}) 
m_{2} 
2m_{R}(m_{2}) 
m_{v} 
2m_{v′}(m_{v}) 
4mm1_{R} 
4m_{v}m_{v′} 
4m_{v}m_{v′} 
2 
21_{R} 
2m_{v}m_{2} 
21_{R}m_{v′}(m_{v}) 
3 
3 
3 
1 
1 
31_{R} 
31_{R} 
6 
3 
2 
1 
(3 + 1_{R}) 
3m_{R} 
3m 
3 
1 
1 
3m1_{R} 
(3 + m_{2}) 
m_{R} 
m_{2} 
1 
3m 
3m_{v} 
3m_{v} 
1 
1 
m_{v} 
m_{v} 
1 
3m1_{R} 
6mm 
3m_{v} 
2 
1 
[3m_{v} + m_{2} + (1_{R})] 
m_{v}1_{R} 
2m_{v}m_{2} 
1 
6 
6 
6 
1 
2 
61_{R} 
6_{R} 
3 
3 
1 
2_{R} 
61_{R} 
6 
6 
2 
21_{R} 
6m_{R}m_{R} 
6mm 
6 
1 
2 
6mm1_{R} 
(6 + m_{2}) 
m_{2} 
2m_{R}(m_{2}) 
6mm 
6m_{v}m_{v′} 
6m_{v}m_{v′} 
1 
2 
m_{v} 
2m_{v′}(m_{v}) 
6_{R}mm_{R} 
3m_{v} 
3m_{v} 
1 
2_{R} 
m_{2} 
2_{R}m_{v′}(m_{2}) 
m_{v} 
2_{R}m_{R}(m_{v}) 
6mm1_{R} 
6m_{v}m_{v′} 
6m_{v}m_{v′} 
2 
21_{R} 
2m_{v}m_{2} 
21_{R}m_{v′}(m_{v}) 

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

Figure 2.5.3.3
 top  pdf  Illustration of symmetries appearing in darkfield patterns (DPs) and a pair of darkfield patterns (±DP) for the combinations of symmetry elements.

When the four patterns appearing in three photographs are taken and examined using Table 2.5.3.3 with the aid of Fig. 2.5.3.3, one diffraction group can be selected unambiguously. It is, however, noted that many diffraction groups are determined from a WP and BP pair without using a DP or ±DP (or from one photograph) or from a set of a WP, a BP and a DP without using a ±DP (or from two photographs).
Fig. 2.5.3.4 provides the relationship between the 31 diffraction groups for slabs and the 32 point groups for infinite crystals given by Buxton et al. (1976). When a diffraction group is determined, possible point groups are selected by consulting this figure. Each of the 11 highsymmetry diffraction groups corresponds to only one crystal point group. In this case, the point group is uniquely determined from the diffraction group. When more than one point group falls under a diffraction group, a different diffraction group has to be obtained for another zone axis. A point group is identified by finding a common point group among the point groups obtained for different zone axes. It is clear from the figure that highsymmetry zones should be used for quick determination of point groups because lowsymmetry zone axes exhibit only a small portion of crystal symmetries in the CBED patterns. Furthermore, it should be noted that CBED cannot observe crystal symmetries oblique to an incident beam or horizontal three, four or sixfold rotation axes. The diffraction groups to be expected for different zone axes are given for all the point groups in Table 2.5.3.4 (Buxton et al., 1976). The table is useful for finding a suitable zone axis to distinguish candidate point groups expected in advance.
Point group  Zoneaxis symmetries 
<111>  <100>  <110>  <uv0>  <uuw>  [uvw] 
m3m 
6_{R}mm_{R} 
4mm1_{R} 
2mm1_{R} 
2_{R}mm_{R} 
2_{R}mm_{R} 
2_{R} 

3m 
4_{R}mm_{R} 
m1_{R} 
m_{R} 
m 
1 
432 
3m_{R} 
4m_{R}m_{R} 
2m_{R}m_{R} 
m_{R} 
m_{R} 
1 
Point group  Zoneaxis symmetries 
<111>  <100>  <uv0>  [uvw] 
m3 
6_{R} 
2mm1_{R} 
2_{R}mm_{R} 
2_{R} 
23 
3 
2m_{R}m_{R} 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[0001]    [uv.0]  [uu.w]   [uv.w] 
6/mmm 
6mm1_{R} 
2mm1_{R} 
2mm1_{R} 
2_{R}mm_{R} 
2_{R}mm 
2_{R}mm_{R} 
2_{R} 

3m1_{R} 
m1_{R} 
2mm 
m 
m_{R} 
m 
1 
6mm 
6mm 
m1_{R} 
m1_{R} 
m_{R} 
m 
m 
1 
622 
6m_{R}m_{R} 
2m_{R}m_{R} 
2m_{R}m_{R} 
m_{R} 
m_{R} 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[0001]  [uv.0]  [uv.w] 
6/m 
61_{R} 
2_{R}mm_{R} 
2_{R} 

31_{R} 
m 
1 
6 
6 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[0001]    [uv.w] 

6_{R}mm_{R} 
21_{R} 
2_{R}mm_{R} 
2_{R} 
3m 
3m 
1_{R} 
m 
1 
32 
3m_{R} 
2 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[0001]  [uv.w] 

6_{R} 
2_{R} 
3 
3 
1 
Point group  Zoneaxis symmetries 
[001]  <100>  <110>  [u0w]  [uv0]  [uuw]  [uvw] 
4/mmm 
4mm1_{R} 
2mm1_{R} 
2mm1_{R} 
2_{R}mm_{R} 
2_{R}mm_{R} 
2_{R}mm_{R} 
2_{R} 

4_{R}mm_{R} 
2m_{R}m_{R} 
m1_{R} 
m_{R} 
m_{R} 
m 
1 
4mm 
4mm 
m1_{R} 
m1_{R} 
m 
m_{R} 
m 
1 
422 
4m_{R}m_{R} 
2m_{R}m_{R} 
2m_{R}m_{R} 
m_{R} 
m_{R} 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[001]  [uv0]  [uvw] 
4/m 
41_{R} 
2_{R}mm_{R} 
2_{R} 

4_{R} 
m_{R} 
1 
4 
4 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[001]  <100>  [u0w]  [uv0]  [uvw] 
mmm 
2mm1_{R} 
2mm1_{R} 
2_{R}mm_{R} 
2_{R}mm_{R} 
2_{R} 
mm2 
2mm 
m1_{R} 
m 
m_{R} 
1 
222 
2m_{R}m_{R} 
2m_{R}m_{R} 
m_{R} 
m_{R} 
1 
Point group  Zoneaxis symmetries 
[010]  [u0w]  [uvw] 
2/m 
21_{R} 
2_{R}mm_{R} 
2_{R} 
m 
1_{R} 
m 
1 
2 
2 
m_{R} 
1 
Point group  Zoneaxis symmetry 
[uvw] 

2_{R} 
1 
1 


Figure 2.5.3.4
 top  pdf  Relation between diffraction groups and crystal point groups (after Buxton et al., 1976).

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

Figure 2.5.3.5
 top  pdf  CBED patterns of Si taken with the [111] incidence. (a) BP and WP show symmetry 3m_{v}. (b) and (c) DPs show symmetry m_{2} and DP symmetry 2_{R}m_{v′}.


Figure 2.5.3.6
 top  pdf  CBED pattern of Si taken with the [100] incidence. The BP and WP show symmetry 4mm.

HOLZ reflections appear as excess HOLZ rings far outside the ZOLZ reflection discs and as deficit lines in the ZOLZ discs. By ignoring these weak diffraction effects with components along the beam direction, we may obtain information about the symmetry of the sample as projected along the beam direction. Thus when HOLZ reflections are weak and no deficit HOLZ lines are seen in the ZOLZ discs, the symmetry elements found from the CBED patterns are only those of the specimen projected along the zone axis. The projection of the specimen along the zone axis causes horizontal mirror symmetry m′, the corresponding CBED symmetry being 1_{R}. When symmetry 1_{R} is added to the 31 diffraction groups, ten projection diffraction groups having symmetry symbol 1_{R} are derived as shown in column VI of Table 2.5.3.3. If only ZOLZ reflections are observed in CBED patterns, a projection diffraction group instead of a diffraction group is obtained, where only the pattern symmetries given in the rows of the diffraction groups having symmetry symbol 1_{R} in Table 2.5.3.3 should be consulted. Two projection diffraction groups obtained from two different zone axes are the minimum needed to determine a crystal point group, because it is constructed by the threedimensional combination of symmetry elements. It should be noted that if a diffraction group is determined carelessly from CBED patterns with no HOLZ lines, the wrong crystal point group is obtained.
In the sections above, the pointgroup determination method established by Buxton et al. (1976) was described, where two and threedimensional symmetry elements were determined, respectively, from ZAPs and DPs.
The Laue circle is defined as the intersection of the Ewald sphere with the ZOLZ, and all reflections on this circle are simultaneously at the Bragg condition. If many such DPs are recorded (all simultaneously at the Bragg condition), many threedimensional symmetry elements can be identified from one photograph. Using a grouptheoretical method, Tinnappel (1975) studied the symmetries appearing in simultaneously excited DPs for various combinations of crystal symmetry elements. Based upon his treatment, Tanaka, Saito & Sekii (1983) developed a method for determining diffraction groups using simultaneously excited symmetrical hexagonal sixbeam, square fourbeam and rectangular fourbeam CBED patterns. All the CBED symmetries appearing in the symmetrical manybeam (SMB) patterns were derived by the graphical method used in the paper of Buxton et al. (1976). From an experimental viewpoint, it is advantageous that symmetry elements can be identified from one photograph. It was found that twenty diffraction groups can be identified from one SMB pattern, whereas ten diffraction groups can be determined by Buxton et al.'s method. An experimental comparison between the two methods was performed by Howe et al. (1986).
SMB patterns are easily obtained by tilting a specimen crystal or the incident beam from a zone axis into an orientation to excite loworder reflections simultaneously. Fig. 2.5.3.7 illustrates the symmetries of the SMB patterns for all the diffraction groups except for the five groups 1, 1_{R}, 2, 2_{R} and 21_{R}. For these groups, the twobeam method for exciting one reflection is satisfactory because manybeam excitation gives no more information than the twobeam case. In the sixbeam and square fourbeam cases, the CBED symmetries for the two crystal (or incidentbeam) settings which excite respectively the +G and −G reflections are drawn because the vertical rotation axes create the SMB patterns at different incidentbeam orientations. [This had already been experienced for the case of symmetry 2_{R} (Goodman, 1975; Buxton et al., 1976).] In the rectangular fourbeam case, the symmetries for four settings which excite the +G, +H, −G and −H reflections are shown. For the diffraction groups 3m, 3m_{R}, 3m1_{R} and 6_{R}mm_{R}, two different patterns are shown for the two crystal settings, which differ by π/6 rad from each other about the zone axis. Similarly, for the diffraction group 4_{R}mm_{R}, two different patterns are shown for the two crystal settings, which differ by π/4 rad. Illustrations of these different symmetries are given in Fig. 2.5.3.7. The combination of the vertical threefold axis and a horizontal mirror plane introduces a new CBED symmetry 3_{R}. Similarly, the combination of the vertical sixfold rotation axis and an inversion centre introduces a new CBED symmetry 6_{R}.

Figure 2.5.3.7
 top  pdf  Illustration of symmetries appearing in hexagonal sixbeam, square fourbeam and rectangular fourbeam darkfield patterns expected for all the diffraction groups except for 1, 1_{R}, 2, 2_{R} and 21_{R}.

There is an empirical and conventional technique for reproducing the symmetries of the SMB patterns which uses three operations of twodimensional rotations, a vertical mirror at the centre of disc O and a rotation of π about the centre of a disc (1_{R}) without involving the reciprocal process. For example, we may consider 3_{R} between discs F and F′ in Table 2.5.3.5 in the case of diffraction group 31_{R}. Disc F′ is rotated anticlockwise not about the zone axis but about the centre of disc O by 2π/3 rad (symbol 3) to coincide with disc F, and followed by a rotation of π rad (symbol R) about the centre of disc F′, resulting in the correct symmetry seen in Fig. 2.5.3.7. When the symmetries appearing between different SMB patterns are considered, this technique assumes that the symmetry operations are conducted after discs O and are superposed. Another assumption is that the vertical mirror plane perpendicular to the line connecting discs O and acts at the centre of disc O when the symmetries between two SMB patterns are considered. As an example, symmetry 3_{R} between discs S and appearing in the two SMB patterns is reproduced by a threefold anticlockwise rotation of disc S about the centre of disc O (or ) and followed by a rotation of π rad (R) about the centre of disc .
Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7 express the symmetries illustrated in Fig. 2.5.3.7 with the symmetry symbols for the hexagonal sixbeam case, square fourbeam case and rectangular fourbeam case, respectively. In the fourth rows of the tables the symmetries of zoneaxis patterns (BP and WP) are listed because combined use of the zoneaxis pattern and the SMB pattern is efficient for symmetry determination. In the fifth row, the symmetries of the SMB pattern are listed. In the following rows, the symmetries appearing between the two SMB patterns are listed because the SMB symmetries appear not only in an SMB pattern but also in the pairs of SMB patterns. That is, for each diffraction group, all the possible SMB symmetries appearing in a pair of symmetric sixbeam patterns, two pairs AB and AC of the square fourbeam patterns and three pairs AB, AC and AD of the rectangular fourbeam patterns are listed, though such pairs are not always needed for the determination of the diffraction groups. It is noted that the symmetries in parentheses are the symmetries which add no new symmetries, even if they are present. In the last row, the point groups which cause the diffraction groups listed in the first row are given.
 Projection diffraction group 
31_{R}  3m1_{R}  61_{R} 
Diffraction group 
3 
31_{R} 
3m_{R} 
3m 
3m1_{R} 
6 
6_{R} 
61_{R} 
Twodimensional symmetry 
3 
3 
3 
3m 
3m 
6 
3 
6 
Threedimensional symmetry 

m′ 
2′ 

m′, (2′) 

i 
m′, (i) 
Zoneaxis pattern 
Brightfield pattern 
3 
6 
3m 
3m 
6mm 
6 
3 
6 
Wholefield pattern 
3 
3 
3 
3m 
3m 
6 
3 
6 
Hexagonal sixbeam pattern 
O 
1 
1 
1 
m_{2} 
1 
m_{v} 
m_{2} 
m_{v} 
1 
1 
1 
G 
1 
1_{R} 
m_{2} 
1 
1 
m_{v} 
1_{R} 
1_{R}m_{v}(m_{2}) 
1 
1 
1_{R} 
F 
1 
1 
m_{2} 
1 
1 
1 
1 
m_{2} 
1 
1 
1 
S 
1 
1 
1 
m_{2} 
1 
1 
m_{2} 
1 
1 
1 
1 
FF′ 
1 
3_{R} 
1 
1 
1 
m_{v} 
3_{R} 
3_{R}m_{v} 
1 
1 
3_{R} 
SS′ 
1 
1 
1 
1 
1 
m_{v} 
1 
m_{v} 
1 
6_{R} 
6_{R} 
A pair of symmetrical sixbeam patterns

±O 
1 
1_{R} 
m_{2} 
1 
m_{v} 
1 
m_{v}1_{R} 
1_{R}m_{2} 
2 
1 
2(1_{R}) 
±G 
1 
1 
1 
m_{R} 
m_{v} 
1 
m_{v}m_{R} 
1 
2 
2_{R} 
21_{R} 
±F 
1 
1 
1 
1 
m_{v} 
1 
m_{v} 
1 
1 
6_{R} 
6_{R} 
±S 
1 
3_{R} 
1 
1 
m_{v} 
1 
3_{R}m_{v} 
3_{R} 
1 
1 
3_{R} 

1 
1 
1 
m_{R} 
1 
1 
m_{R} 
1 
2 
1 
2 

1 
1 
m_{R} 
1 
1 
1 
1 
m_{R} 
2 
1 
2 
Point group 
23, 3 

432, 32 
, 3m 

6 
m3, 3 
6/m 
 Projection diffraction group 
6mm1_{R} 
Diffraction group 
6m_{R}m_{R} 
6mm 
6_{R}mm_{R} 
6mm1_{R} 
Twodimensional symmetry 
6 
6mm 
3m 
6mm 
Threedimensional symmetry 
2′ 

i, (2′) 
m′, (i, 2′) 
Zoneaxis pattern 
Brightfield pattern 
6mm 
6mm 
3m 
6mm 
Wholefield pattern 
6 
6mm 
3m 
6mm 
Hexagonal sixbeam pattern 
O 
m_{2} 
m_{v} 
1 
m_{v}(m_{2}) 
m_{v}(m_{2}) 
G 
m_{2} 
m_{v} 
m_{2} 
m_{v} 
1_{R}m_{v}(m_{2}) 
F 
m_{2} 
1 
m_{2} 
1 
m_{2} 
S 
m_{2} 
1 
1 
m_{2} 
m_{2} 
FF′ 
1 
m_{v} 
1 
m_{v} 
3_{R}m_{v} 
SS′ 
1 
m_{v} 
6_{R} 
6_{R}m_{v} 
6_{R}m_{v} 
A pair of symmetrical sixbeam patterns

±O 
2m_{2} 
2m_{v′} 
m_{v}(m_{2}) 
1 
2(1_{R})m_{v′}(m_{2}) 
±G 
2m_{R} 
2m_{v′} 
2_{R}m_{v} 
2_{R}m_{R} 
21_{R}m_{v′}(m_{R}) 
±F 
1 
m_{v′} 
6_{R}m_{v} 
6_{R} 
6_{R}m_{v′} 
±S 
1 
m_{v′} 
m_{v} 
1 
3_{R}m_{v′} 

2m_{R} 
2 
1 
m_{R} 
2m_{R} 

2m_{R} 
2 
m_{R} 
1 
2m_{R} 
Point group 
622 
6mm 
m3m, 
6/mmm 

 Projection diffraction group 
41_{R}  4mm1_{R} 
Diffraction group 
4 
4_{R} 
41_{R} 
4m_{R}m_{R} 
4mm 
4_{R}mm_{R} 
4mm1_{R} 
Twodimensional symmetry 
4 
(2) 
4 
4 
4mm 
(2mm) 
4mm 
Threedimensional symmetry 


m′, (i, ) 
2′ 

, 2′ 
m′, (i, 2′, ) 
Zoneaxis pattern 
Brightfield pattern 
4 
4 
4 
4mm 
4mm 
4mm 
4mm 
Wholefield pattern 
4 
2 
4 
4 
4mm 
2mm 
4mm 
Square fourbeam pattern 

O 
1 
1 
1 
m_{2} 
m_{v} 
m_{2} 
m_{v} 
m_{v}(m_{2}) 
G 
1 
1 
1_{R} 
m_{2} 
m_{v} 
m_{2} 
m_{v} 
1_{R}m_{v}(m_{2}) 
F 
1 
1 
1 
m_{2} 
1 
1 
m_{2} 
m_{2} 
FF′ 
1 
4_{R} 
4_{R} 
1 
m_{v} 
4_{R} 
4_{R}m_{v} 
4_{R}m_{v} 
Two pairs of square fourbeam patterns

AB 
±O 
2 
2 
2(1_{R}) 
2m_{2} 
2m_{v′} 
2m_{2} 
2m_{v′} 
2(1_{R})m_{v′}(m_{2}) 
±G 
2 
2 
21_{R} 
2m_{R} 
2m_{v′} 
2m_{R} 
2m_{v′} 
21_{R}m_{v′}(m_{R}) 
FF′ 
2 
2 
2 
2m_{R} 
2 
2 
2m_{R} 
2m_{R} 
±F 
1 
4_{R} 
4_{R} 
1 
m_{v′} 
4_{R} 
4_{R}m_{v′} 
4_{R}m_{v′} 
AC 
OO′ 
4 
4 
4 
4m_{2} 
4m_{v′′} 
4m_{v} 
4m_{2} 
4m_{v′′}(m_{2}) 
GG′ 
4 
4_{R} 
41_{R} 
4m_{R} 
4m_{v′′} 
4_{R}m_{v} 
4_{R}m_{R} 
41_{R}m_{v′′}(m_{R}) 
FS 
4 
1 
4 
4m_{R} 
4 
m_{R} 
1 
41_{R}m_{v′′}(m_{R}) 
FS′ 
1 
1 
1_{R} 
1 
m_{v′′} 
m_{v} 
1 
1_{R}m_{v′′} 
Point group 
4 

4/m 
432, 422 
4mm 
, 
m3m, 4/mmm 

 Projection diffraction group 
m1_{R}  2mm1_{R} 
Diffraction group 
m_{R} 
m 
m1_{R} 
2m_{R}m_{R} 
2mm 
2_{R}mm_{R} 
2mm1_{R} 
Twodimensional symmetry 

m 
m 
2 
2mm 
m 
2mm 
Threedimensional symmetry 
2′ 

m′, 2′ 
2′ 

2′, i 
m′, 2′, i 
Zoneaxis pattern 
Brightfield pattern 
m 
m 
2mm 
2mm 
2mm 
m 
2mm 
Wholefield pattern 
1 
m 
m 
2 
2mm 
m 
2mm 
Rectangular fourbeam pattern 

O 
1 
1 
1 
1 
1 
1 
1 
G 
1 
1 
1_{R} 
1 
1 
1 
1_{R} 
F 
m_{2} 
1 
m_{2} 
m_{2} 
1 
m_{2} 
m_{2} 
S 
1 
1 
1 
m_{2} 
1 
1 
m_{2} 
Three pairs of rectangular fourbeam patterns

AB 

m_{2} 
1 
m_{2} 
m_{2} 
m_{v} 
m_{v}(m_{2}) 
m_{v}(m_{2}) 

1 
1 
1 
m_{R} 
m_{v} 
m_{v} 
m_{v}m_{R} 

1 
1 
1 
1 
m_{v} 
2_{R}m_{v} 
2_{R}m_{v} 
SS′ 
1 
1 
1_{R} 
1 
m_{v} 
m_{v} 
m_{v}1_{R} 
AC 
O_{G}O_{H} 
1 
m_{v} 
m_{v} 
m_{2} 
m_{v′} 
1 
m_{v′}(m_{2}) 
GH 
m_{R} 
m_{v} 
m_{v}m_{R} 
m_{R} 
m_{v′} 
m_{R} 
m_{v′}m_{R} 
FF′ 
1 
m_{v} 
m_{v}1_{R} 
1 
m_{v′} 
1 
m_{v′}1_{R} 

1 
m_{v} 
m_{v} 
1 
m_{v′} 
2_{R} 
2_{R}m_{v′} 
AD 

1 
1 
1_{R} 
2 
2 
1 
2(1_{R}) 
GG 
1 
1 
1 
2 
2 
2_{R} 
21_{R} 

1 
1 
1 
2m_{R} 
2 
1 
2m_{R} 

m_{R} 
1 
m_{R} 
2m_{R} 
2 
m_{R} 
2m_{R} 
Point group 
2, 222, mm2, 4, , 422, 4mm, , 32, 6, 622, 6mm, , 23, 432, 
m, mm2, 4mm, , 3m, , 6mm, , 
mm2, 4mm, 42m, 6mm, , 
222, 422, , 622, 23, 432 
mm2, 
2/m, mmm, 4/m, 4/mmm, , , 6/mmm, m3, m3m 
mmm, 4/mmm, m3, m3m, 6/mmm 

By referring to Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7, the characteristic features of the SMB method are seen to be as follows. CBED symmetry m_{2} due to a horizontal twofold rotation axis can appear in every disc of an SMB pattern. Symmetry 1_{R} due to a horizontal mirror plane, however, appears only in disc G or H of an SMB pattern. In the hexagonal sixbeam case, an inversion centre i produces CBED symmetry 6_{R} between discs S and S′ due to the combination of an inversion centre and a vertical threefold rotation axis (and/or of a horizontal mirror plane and a vertical sixfold rotation axis). This indicates that one hexagonal sixbeam pattern can reveal whether a specimen has an inversion centre or not, while the method of Buxton et al. (1976) requires two photographs for the inversion test. All the diffraction groups in Table 2.5.3.5 can be identified from one sixbeam pattern except groups 3 and 6. Diffraction groups 3 and 6 cannot be distinguished from the hexagonal sixbeam pattern because it is insensitive to the vertical axis. In the square fourbeam case, fourfold rotary inversion produces CBED symmetry 4_{R} between discs F and F′ in one SMB pattern, while Buxton et al.'s method requires four photographs to identify fourfold rotary inversion. Although an inversion centre itself does not exhibit any symmetry in the square fourbeam pattern, it causes symmetry 1_{R} due to the horizontal mirror plane produced by the combination of an inversion centre and the twofold rotation axis. Thus, symmetry 1_{R} is an indication of the existence of an inversion centre in the square fourbeam case. All of the seven diffraction groups in Table 2.5.3.6 can be identified from one square fourbeam pattern. One rectangular fourbeam pattern can distinguish all the diffraction groups in Table 2.5.3.7 except the groups m and 2mm. It is emphasized again that the inversion test can be carried out using one sixbeam pattern or one square fourbeam pattern.
Fig. 2.5.3.8 shows CBED patterns taken from a [111] pyrite (FeS_{2}) plate with an accelerating voltage of 100 kV. The space group of FeS_{2} is . The diffraction group of the plate is 6_{R} due to a threefold rotation axis and an inversion centre. The zoneaxis pattern of Fig. 2.5.3.8(a) shows threefold rotation symmetry in the BP and WP. The hexagonal sixbeam pattern of Fig. 2.5.3.8(b) shows no symmetry higher than 1 in discs O, G, F and S but shows symmetry 6_{R} between discs S and S′, which proves the existence of a threefold rotation axis and an inversion centre. The same symmetries are also seen in Fig. 2.5.3.8(c), where reflections , , , , and are excited. Table 2.5.3.5 indicates that diffraction group 6_{R} can be identified from only one hexagonal sixbeam pattern, because no other diffraction groups give rise to the same symmetries in the six discs. When Buxton et al.'s method is used, three photographs or four patterns are necessary to identify diffraction group 6_{R} (see Table 2.5.3.3). In addition, if the symmetries between Figs. 2.5.3.8(b) and (c) are examined, symmetry 2_{R} between discs G and and symmetry 6_{R} between discs F and are found. All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.5.

Figure 2.5.3.8
 top  pdf  CBED patterns of FeS_{2} taken with the [111] incidence. (a) Zoneaxis pattern, (b) hexagonal sixbeam pattern with excitation of reflection +G, (c) hexagonal sixbeam pattern with excitation of reflection −G. Symmetry 6_{R} is noted between discs S and S′ and discs and .

Fig. 2.5.3.9 shows CBED patterns taken from a [110] V_{3}Si plate with an accelerating voltage of 80 kV. The space group of V_{3}Si is Pm3n. The diffraction group of the plate is 2mm1_{R} due to two vertical mirror planes and a horizontal mirror plane, a twofold rotation axis being produced at the intersection line of two perpendicular mirror planes. The zoneaxis pattern of Fig. 2.5.3.9(a) shows symmetry 2mm in the BP and WP. The rectangular fourbeam pattern of Fig. 2.5.3.9(b) shows symmetry 1_{R} in disc H due to the horizontal mirror plane and symmetry m_{2} in both discs and F′ due to the twofold rotation axes in the [001] and [110] directions, respectively. The same symmetries are also seen in Fig. 2.5.3.9(c), where reflections , S′ and are excited. Table 2.5.3.7 implies that the diffraction group 2mm1_{R} can be identified from only one rectangular fourbeam pattern, because no other diffraction groups give rise to the same symmetries in the four discs. When Buxton et al.'s method is used, two photographs or three patterns are necessary to identify diffraction group 2mm1_{R} (see Table 2.5.3.3). One can confirm the theoretically predicted symmetries between Fig. 2.5.3.9(b) and Fig. 2.5.3.9(c). All the experimental results agree exactly with the theoretical results given in Fig. 2.5.3.7 and Table 2.5.3.7.
These experiments show that the SMB method is quite effective for determining the diffraction group of slabs. Buxton et al.'s method identifies twodimensional symmetry elements in the first place using a zoneaxis pattern, and threedimensional symmetry elements using DPs. On the other hand, the SMB method primarily finds many threedimensional symmetry elements in an SMB pattern, and twodimensional symmetry elements from a pair of SMB patterns, as shown in Tables 2.5.3.5, 2.5.3.6 and 2.5.3.7. Therefore, the use of a ZAP and SMB patterns is the most efficient way to find as many crystal symmetry elements in a specimen as possible.
When the point group of a specimen crystal is determined, the crystal axes may be found from a spot diffraction pattern recorded at a highsymmetry zone axis, using the orientations of the symmetry elements determined in the course of pointgroup determination. Integralnumber 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.
There are three spacegroup symmetry elements of diperiodic plane figures: (1) a horizontal screw axis , (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the pointgroup symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten pointgroup 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) and was discussed by Miyake et al. (1960) and Cowley et al. (1961). Goodman & Lehmpfuhl (1964) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965) 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 symmetryrelated multiplescattering paths. Based on the results of Gjønnes & Moodie (1965), Tanaka, Sekii & Nagasawa (1983) 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).
Fig. 2.5.3.10(a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a bglide plane exists perpendicular to the a axis and/or a 2_{1} screw axis exists in the b direction. Let us consider an Umweganregung path a in the zerothorder 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 2_{1} screw axis, the following relations hold between the crystal structure factors:That is, the structure factors of reflections hk0 and have the same phase for even k but have opposite phases for odd k.

Figure 2.5.3.10
 top  pdf  Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a bglide plane and a 2_{1} 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:whereand 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(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: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(b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zerothorder Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A_{2} and B_{2} lines, subscript 2 indicating that the Umweganregung paths lie in the zerothorder 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 higherorder 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 2_{1} screw axis and a glide plane,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), but extinction lines B are not produced because equation (2.5.3.4) holds only for the 2_{1} screw axis. In the case of the 2_{1} screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4), forming extinction lines B only. We call these lines A_{3} and B_{3} dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higherorder Laue zones.
It was predicted by Gjønnes & Moodie (1965) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965). Table 2.5.3.8 summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 2_{1} screw axis. The three spacegroup symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.
Symmetry elements of planeparallel specimen  Orientation to specimen surface  Dynamical extinction lines 
Twodimensional (zeroth Laue zone) interaction  Threedimensional (HOLZ) interaction 
Glide planes 
perpendicular: g 
A_{2} and B_{2} 
A_{3} 
parallel: g′ 
— 
intersection of A_{3} and B_{3} 
Twofold screw axis 
perpendicular: 2_{1} 
— 
— 
parallel: 
A_{2} and B_{2} 
B_{3} 

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A_{3} and B_{3}. It is, however, not easy to observe the extinction lines A_{3} and B_{3} because broad extinction lines A_{2} and B_{2} appear at the same time. The presence of the extinction lines A_{3} and B_{3} 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 A_{3} and B_{3} (Tanaka, Sekii & Nagasawa, 1983). That is, if HOLZ lines form lines A_{3} and B_{3}, HOLZ lines are symmetric with respect to the extinction lines A_{2} and B_{2}. If HOLZ lines do not form lines A_{3} and B_{3}, HOLZ lines are asymmetric with respect to the extinction lines A_{2} and B_{2}. When the HOLZ lines are symmetric about the extinction lines A_{2}, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B_{2}, a 2_{1} 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 shows CBED patterns taken from (a) thin and (b) thick areas of FeS_{2}, whose space group is , at the 001 Bragg setting with the [100] electronbeam incidence. In the case of the thin specimen (Fig. 2.5.3.11a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the oddorder discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11b). The HOLZ lines are symmetric with respect to both A_{2} and B_{2} extinction lines. This fact proves the presence of the extinction lines A_{3} and B_{3}, or both the c glide in the (010) plane and the 2_{1} screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9. Fig. 2.5.3.12 shows a [110] zoneaxis CBED pattern of FeS_{2}. A_{2} extinction lines are seen in both the 001 and discs. Fine HOLZ lines are symmetric with respect to the A_{2} extinction lines in the disc but asymmetric about the A_{2} extinction line in the 001 disc, indicating formation of the A_{3} extinction line only in the disc. This proves the existence of a 2_{1} 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) 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
 top  pdf  CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS_{2}. (a) Dynamical extinction lines A_{2} and B_{2} are seen. (b) Extinction lines A_{3} and B_{3} as well as A_{2} and B_{2} are formed because HOLZ lines are symmetric about lines A_{2} and B_{2}.

Another practical method for distinguishing between glide planes and 2_{1} screw axes is that reported by Steeds et al. (1978). 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 2_{1} screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10(a), extinction lines A_{3} 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 B_{3} originating from a 2_{1} 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.
We now describe a spacegroup determination method which uses the dynamical extinction lines caused by the horizontal screw axis and the vertical glide plane g of an infinitely extended parallelsided 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.) The 2_{1}, 4_{1}, 4_{3}, 6_{1}, 6_{3 }and 6_{5} screw axes of crystal space groups that are set perpendicular to the incident beam act as a symmetry element because two or three successive operations of 4_{1}, 4_{3}, 6_{1}, 6_{3} and 6_{5} screw axes make them equivalent to a 2_{1} screw axis: (4_{1})^{2} = (4_{3})^{2} = (6_{1})^{3} = (6_{3})^{3} = (6_{5})^{3} = 2_{1}. The 4_{2}, 3_{1}, 3_{2}, 6_{2} and 6_{4} screw axes that are set perpendicular to the incident beam do not produce dynamical extinction lines because the 4_{2} screw axis acts as a twofold rotation axis due to the relation (4_{2})^{2} = 2, the 3_{1} and 3_{2} screw axes give no specific symmetry in CBED patterns, and the 6_{2 }and 6_{4} screw axes are equivalent to 3_{1} and 3_{2} screw axes due to the relations (6_{2})^{2} = 3_{2} and (6_{4})^{2} = 3_{1}. Modifications of the dynamical extinction rules were investigated by Tanaka, Sekii & Nagasawa (1983) 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 A_{2}, A_{3}, B_{2} and B_{3} expected from all the possible crystal settings for all the space groups were tabulated.
Table 2.5.3.9 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 lefthand column of the pair gives the reflection indices and the symmetry elements causing the extinction lines and the righthand column gives the types of the extinction lines. The (second) suffixes 1, 2 and 3 of a 2_{1} screw axis in each column distinguish the first, the second and the third screw axis of the space group (as in the symbols 2_{11} and 2_{12} of space group P2_{1}2_{1}2). The glide symbols in the [001] column for space group P4/nnc have two suffixes (n_{21} and n_{22}). 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] electronbeam incidences, for convenience. The cglide planes of space group Pcc2 are distinguished with symbols c_{1} and c_{2} (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 incidentbeam 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] electronbeam 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 incidentbeam directions . The indices of the reflections in which extinction lines appear are odd if no remark is given. For cglide planes of space groups R3c and and for dglide planes, the reflections in which extinction lines appear are specified as 6n + 3 and 4n + 2 orders, respectively.
Space group  Incidentbeam direction 
[h0l] 
3 
P2 



4 
P2_{1} 
0k0 
A_{2} 
B_{2} 
2_{1} 

B_{3} 
5 
C2 



6 
Pm 



7 
Pc 
h0l_{o} 
A_{2} 
B_{2} 
c 
A_{3} 

8 
Cm 



9 
Cc 
h_{e}0l_{o} 
A_{2} 
B_{2} 
c 
A_{3} 

10 
P2/m 



11 
P2_{1}/m 
0k0 
A_{2} 
B_{2} 
2_{1} 

B_{3} 
12 
C2/m 



13 
P2/c 
h0l_{o} 
A_{2} 
B_{2} 
c 
A_{3} 

14 
P2_{1}/c 
0k0 
A_{2} 
B_{2} 
2_{1} 

B_{3} 
h0l_{o} 
A_{2} 
B_{2} 
c 
A_{3} 

15 
C2/c 
h_{e}0l_{o} 
A_{2} 
B_{2} 
c 
A_{3} 

Space group  Incidentbeam direction 
[100]  [010]  [001]  [hk0]  [0kl]  [h0l] 
16 
P222 


















17 
P222_{1} 
00l 
A_{2} 
B_{2} 
00l 
A_{2} 
B_{2} 



00l 
A_{2} 
B_{2} 






2_{1} 

B_{3} 
2_{1} 

B_{3} 



2_{1} 

B_{3} 






18 
P2_{1}2_{1}2 
0k0 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 



h00 
A_{2} 
B_{2} 
0k0 
A_{2} 
B_{2} 
2_{12} 

B_{3} 
2_{11} 

B_{3} 
2_{11} 

B_{3} 



2_{11} 

B_{3} 
2_{12} 

B_{3} 






0k0 

















2_{12} 











19 
P2_{1}2_{1}2_{1} 
0k0 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
0k0 
A_{2} 
B_{2} 
2_{12} 

B_{3} 
2_{11} 

B_{3} 
2_{11} 

B_{3} 
2_{13} 

B_{3} 
2_{11} 

B_{3} 
2_{12} 

B_{3} 
00l 


00l 


0k0 











2_{13} 


2_{13} 


2_{12} 











20 
C222_{1} 
00l 
A_{2} 
B_{2} 
00l 
A_{2} 
B_{2} 



00l 
A_{2} 
B_{2} 






2_{1} 

B_{3} 
2_{1} 

B_{3} 



2_{1} 

B_{3} 






21 
C222 


















22 
F222 


















23 
I222 


















24 
I2_{1}2_{1}2_{1} 


















Space group  Incidentbeam direction 
[100]  [010]  [001]  [hk0]  [0kl]  [h0l] 
25 
Pmm2 


















26 
Pmc2_{1} 
00l 
A_{2} 
B_{2} 
00l 





00l 
A_{2} 
B_{2} 



h0l_{o} 
A_{2} 
B_{2} 
c, 2_{1} 
A_{3} 
B_{3} 
2_{1} 

B_{3} 



2_{1} 

B_{3} 



c 
A_{3} 

27 
Pcc2 
00l 


00l 








0kl_{o} 
A_{2} 
B_{2} 
h0l_{o} 
A_{2} 
B_{2} 
c_{2} 
A_{3} 

c_{1} 
A_{3} 







c_{1} 
A_{3} 

c_{2} 
A_{3} 

28 
Pma2 






h00 
A_{2} 
B_{2} 






h_{o}0l 
A_{2} 
B_{2} 






a 
A_{3} 







a 
A_{3} 

29 
Pca2_{1} 
00l 


00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
00l 
A_{2} 
B_{2} 
0kl_{o} 
A_{2} 
B_{2} 
h_{o}0l 
A_{2} 
B_{2} 
2_{1} 

B_{3} 
c, 2_{1} 
A_{3} 
B_{3} 
a 
A_{3} 

2_{1} 

B_{3} 
c 
A_{3} 

a 
A_{3} 

30 
Pnc2 
00l
c 
A_{3} 

00l
n 
A_{3} 

0k0
n 
A_{2}
A_{3} 
B_{2} 



0kl: k + l = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
h0l_{o}
c 
A_{2}
A_{3} 
B_{2} 
31 
Pmn2_{1} 
00l
n, 2_{1} 
A_{2}
A_{3} 
B_{2}
B_{3} 
00l
2_{1} 

B_{3} 
h00
n 
A_{2}
A_{3} 
B_{2} 
00l
2_{1} 
A_{2} 
B_{2}
B_{3} 



h0l: h + l = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
32 
Pba2 






h00 
A_{2} 
B_{2} 



0k_{o}l 
A_{2} 
B_{2} 
h_{o}0l 
A_{2} 
B_{2} 






a 
A_{3} 




b 
A_{3} 

a 
A_{3} 







0k0 

















b 











33 
Pna2_{1} 
00l
2_{1} 

B_{3} 
00l
n, 2_{1} 
A_{2}
A_{3} 
B_{2}
B_{3} 
h00
a
0k0
n 
A_{2}
A_{3} 
B_{2} 
00l
2_{1} 
A_{2} 
B_{2}
B_{3} 
0kl: k + l = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
h_{o}0l
a 
A_{2}
A_{3} 
B_{2} 
34 
Pnn2 
00l
n_{2} 
A_{3} 

00l
n_{1} 
A_{3} 

h00
n_{2}
0k0
n_{1} 
A_{2}
A_{3} 
B_{2} 



0kl: k + l = 2n + 1
n_{1} 
A_{2}
A_{3} 
B_{2} 
h0l: h + l = 2n + 1
n_{2} 
A_{2}
A_{3} 
B_{2} 
35 



















36 

00l 
A_{2} 
B_{2} 
00l 





00l 
A_{2} 
B_{2} 



h_{e}0l_{o} 
A_{2} 
B_{2} 
c, 2_{1} 
A_{3} 
B_{3} 
2_{1} 

B_{3} 



2_{1} 

B_{3} 



c 
A_{3} 

37 

00l 


00l 








0k_{e}l_{o} 
A_{2} 
B_{2} 
h_{e}0l_{o} 
A_{2} 
B_{2} 
c_{2} 
A_{3} 

c_{1} 
A_{3} 







c_{1} 
A_{3} 

c_{2} 
A_{3} 

38 



















39 













0k_{o}l_{o} 
A_{2} 
B_{2} 















b 
A_{3} 




40 







h00 
A_{2} 
B_{2} 






h_{o}0l_{e} 
A_{2} 
B_{2} 






a 
A_{3} 







a 
A_{3} 

41 







h00 
A_{2} 
B_{2} 



0k_{o}l_{o} 
A_{2} 
B_{2} 
h_{o}0l_{e} 
A_{2} 
B_{2} 






a 
A_{3} 




b 
A_{3} 

a 
A_{3} 

42 
Fmm2 


















43 

00l: l = 4n + 2
d_{2} 
A_{3} 

00l: l = 4n + 2
d_{1} 
A_{3} 

h00: h = 4n + 2
d_{2}
0k0: k = 4n + 2
d_{1} 
A_{2}
A_{3} 
B_{2} 



0k_{e}l_{e}: k_{e} + l_{e} = 4n + 2
d_{1} 
A_{2}
A_{3} 
B_{2} 
h_{e}0l_{e}: h_{e} + l_{e} = 4n + 2
d_{2} 
A_{2}
A_{3} 
B_{2} 
44 



















45 













0k_{o}l_{o} 
A_{2} 
B_{2} 
h_{o}0l_{o} 
A_{2} 
B_{2} 












b 
A_{3} 

a 
A_{3} 

46 
















h_{o}0l_{o} 
A_{2} 
B_{2} 















a 
A_{3} 

Space group  Incidentbeam direction 
[100]  [010]  [001]  [hk0]  [0kl]  [h0l] 
47 
P2/m2/m2/m 


















48 
P2/n2/n2/n 
00l
n_{2}
0k0
n_{3} 
A_{3} 

00l
n_{1}
h00
n_{3} 
A_{3} 

0k0
n_{1}
h00
n_{2} 
A_{3} 

hk0: h + k = 2n + 1
n_{3} 
A_{2}
A_{3} 
B_{2} 
0kl: k + l = 2n + 1
n_{1} 
A_{2}
A_{3} 
B_{2} 
h0l: h + l = 2n + 1
n_{2} 
A_{2}
A_{3} 
B_{2} 
49 
P2/c2/c2/m 
00l 


00l 








0kl_{o} 
A_{2} 
B_{2} 
h0l_{o} 
A_{2} 
B_{2} 
c_{2} 
A_{3} 

c_{1} 
A_{3} 







c_{1} 
A_{3} 

c_{2} 
A_{3} 

50 
P2/b2/a2/n 
0k0
n 
A_{3} 

h00
n 
A_{3} 

0k0
b
h00
a 
A_{3} 

hk0: h + k = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
0k_{o}l
b 
A_{2}
A_{3} 
B_{2} 
h_{o}0l
a 
A_{2}
A_{3} 
B_{2} 
51 
P2_{1}/m2/m2/a 



h00 
A_{2} 
B_{2} 
h00 


h_{o}k0 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 






2_{1}, a 
A_{3} 
B_{3} 
2_{1} 

B_{3} 
a 
A_{3} 

2_{1} 

B_{3} 



52 
P2/n2_{1}/n2/a 
00l
n_{2} 
A_{3} 

00l
n_{1}
h00
a 
A_{3} 

0k0
n_{1}, 2_{1} 
A_{2}
A_{3} 
B_{2}
B_{3} 
h_{o}k0
a 
A_{2}
A_{3} 
B_{2} 
0kl: k + l = 2n + 1
n_{1} 
A_{2}
A_{3} 
B_{2} 
h0l: h + l = 2n + 1
n_{2} 
A_{2}
A_{3} 
B_{2} 
0k0 





h00 








0k0 
A_{2} 
B_{2} 
2_{1} 

B_{3} 



n_{2} 
A_{3} 







2_{1} 

B_{3} 
53 
P2/m2/n2_{1}/a 
00l
n, 2_{1} 
A_{2}
A_{3} 
B_{2}
B_{3} 
h00
a 
A_{3} 

h00
n 
A_{3} 

h_{o}k0
a 
A_{2}
A_{3} 
B_{2} 



h0l: h + l = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 



00l 





00l 
A_{2} 
B_{2} 









2_{1} 

B_{3} 



2_{1} 

B_{3} 






54 
P2_{1}/c2/c2/a 
00l 


00l 


h00 


h_{o}k0 
A_{2} 
B_{2} 
0kl_{o} 
A_{2} 
B_{2} 
h0l_{o} 
A_{2} 
B_{2} 
c_{2} 
A_{3} 

c_{1} 
A_{3} 

2_{1} 

B_{3} 
a 
A_{3} 

c_{1} 
A_{3} 

c_{2} 
A_{3} 




h00 
A_{2} 
B_{2} 






h00 
A_{2} 
B_{2} 






a, 2_{1} 
A_{3} 
B_{3} 






2_{1} 

B_{3} 



55 
P2_{1}/b2_{1}/a2/m 
0k0 


h00 


0k0 
A_{2} 
B_{2} 



0k_{o}l 
A_{2} 
B_{2} 
h_{o}0l 
A_{2} 
B_{2} 
2_{12} 

B_{3} 
2_{11} 

B_{3} 
b, 2_{12} 
A_{3} 
B_{3} 



b 
A_{3} 

a 
A_{3} 







h00 





h00 
A_{2} 
B_{2} 
0k0 
A_{2} 
B_{2} 






a, 2_{11} 





2_{11} 

B_{3} 
2_{12} 

B_{3} 
56 
P2_{1}/c2_{1}/c2/n 
00l
c_{2} 
A_{3} 

00l
c_{1} 
A_{3} 

0k0
2_{12}
h00
2_{11} 

B_{3} 
hk0: h + k = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
0kl_{o}
c_{1} 
A_{2}
A_{3} 
B_{2} 
h0l_{o}
c_{2} 
A_{2}
A_{3} 
B_{2} 
0k0 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 






h00 
A_{2} 
B_{2} 
0k0 
A_{2} 
B_{2} 
2_{12}, n 
A_{3} 
B_{3} 
2_{11}, n 
A_{3} 
B_{3} 






2_{11} 

B_{3} 
2_{12} 

B_{3} 
57 
P2/b2_{1}/c2_{1}/m 
00l 
A_{2} 
B_{2} 
00l 


0k0 
A_{2} 
B_{2} 
00l 
A_{2} 
B_{2} 
0k_{o}l 
A_{2} 
B_{2} 
h0l_{o} 
A_{2} 
B_{2} 
c, 2_{12} 
A_{3} 
B_{3} 
2_{12} 

B_{3} 
b, 2_{11} 
A_{3} 
B_{3} 
2_{12} 

B_{3} 
b 
A_{3} 

c 
A_{3} 

0k0 














0k0 
A_{2} 
B_{2} 
2_{11} 

B_{3} 












2_{11} 

B_{3} 
58 
P2_{1}/n2_{1}/n2/m 
00l
n_{2} 
A_{3} 

00l
n_{1} 
A_{3} 

0k0
n_{1}, 2_{12}
h00
n_{2}, 2_{11} 
A_{2}
A_{3} 
B_{2}
B_{3} 



0kl: k + l = 2n + 1
n_{1} 
A_{2}
A_{3} 
B_{2} 
h0l: h + l = 2n + 1
n_{2} 
A_{2}
A_{3} 
B_{2} 
0k0 


h00 








h00 
A_{2} 
B_{2} 
0k0 
A_{2} 
B_{2} 
2_{12} 

B_{3} 
2_{11} 

B_{3} 






2_{11} 

B_{3} 
2_{12} 

B_{3} 
59 
P2_{1}/m2_{1}/m2/n 
0k0
n, 2_{12} 
A_{2}
A_{3} 
B_{2}
B_{3} 
h00
n, 2_{11} 
A_{2}
A_{3} 
B_{2}
B_{3} 
0k0
2_{12}
h00
2_{11} 

B_{3} 
hk0: h + k = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
h00
2_{11} 
A_{2} 
B_{2}
B_{3} 
0k0
2_{12} 
A_{2} 
B_{2}
A_{3} 
60 
P2_{1}/b2/c2_{1}/n 
00l
c, 2_{12} 
A_{2}
A_{3} 
B_{2}
B_{3} 
h00
n, 2_{11} 
A_{2}
A_{3} 
B_{2}
B_{3} 
0k0
b 
A_{3} 

hk0: h + k = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
0k_{o}l
b 
A_{2}
A_{3} 
B_{2} 
h0l_{o}
c 
A_{2}
A_{3} 
B_{2} 
0k0 


00l 


h00 


00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 



n 
A_{3} 

2_{12} 

B_{3} 
2_{11} 

B_{3} 
2_{12} 

B_{3} 
2_{11} 

B_{3} 



61 
P2_{1}/b2_{1}/c2_{1}/a 
00l 
A_{2} 
B_{2} 
00l 


0k0 
A_{2} 
B_{2} 
h_{o}k0 
A_{2} 
B_{2} 
0k_{o}l 
A_{2} 
B_{2} 
h0l_{o} 
A_{2} 
B_{2} 
c, 2_{13} 
A_{3} 
B_{3} 
2_{13} 

B_{3} 
b, 2_{12} 
A_{3} 
B_{3} 
a 
A_{3} 

b 
A_{3} 

c 
A_{3} 

0k0 


h00 
A_{2} 
B_{2} 
h00 


00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
0k0 
A_{2} 
B_{2} 
2_{12} 

B_{3} 
a, 2_{11} 
A_{3} 
B_{3} 
2_{11} 

B_{3} 
2_{13} 

B_{3} 
2_{11} 

B_{3} 
2_{12} 

B_{3} 
62 
P2_{1}/n2_{1}/m2_{1}/a 
00l
2_{13}
0k0
2_{12} 

B_{3} 
00l
n, 2_{13}
h00
a, 2_{11} 
A_{2}
A_{3} 
B_{2}
B_{3} 
0k0
n, 2_{12} 
A_{2}
A_{3} 
B_{2}
B_{3} 
h_{o}k0
a 
A_{2}
A_{3} 
B_{2} 
0kl: k + l = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
0k0
2_{12} 
A_{2} 
B_{2}
B_{3} 






h00 


00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 









2_{11} 

B_{3} 
2_{13} 

B_{3} 
2_{11} 

B_{3} 



63 
C2/m2/c2_{1}/m 
00l 
A_{2} 
B_{2} 
00l 





00l 
A_{2} 
B_{2} 



h_{e}0l_{o} 
A_{2} 
B_{2} 
c, 2_{1} 
A_{3} 
B_{3} 
2_{1} 

B_{3} 



2_{1} 

B_{3} 



c 
A_{3} 

64 
C2/m2/c2_{1}/a 
00l 
A_{2} 
B_{2} 
00l 





h_{o}k_{o}0 
A_{2} 
B_{2} 



h_{e}0l_{o} 
A_{2} 
B_{2} 
c, 2_{1} 
A_{3} 
B_{3} 
2_{1} 

B_{3} 



a 
A_{3} 




c 
A_{3} 










00l 
A_{2} 
B_{2} 















2_{1} 

B_{3} 






65 
C2/m2/m2/m 


















66 
C2/c2/c2/m 
00l 


00l 








0k_{e}l_{o} 
A_{2} 
B_{2} 
h_{e}0l_{o} 
A_{2} 
B_{2} 
c_{2} 
A_{3} 

c_{1} 
A_{3} 







c_{1} 
A_{3} 

c_{2} 
A_{3} 

67 
C2/m2/m2/a 









h_{o}k_{o}0 
A_{2} 
B_{2} 















a 
A_{3} 







68 
C2/c2/c2/a 
00l 


00l 





h_{o}k_{o}0 
A_{2} 
B_{2} 
0k_{e}l_{o} 
A_{2} 
B_{2} 
h_{e}0l_{o} 
A_{2} 
B_{2} 
c_{2} 
A_{3} 

c_{1} 
A_{3} 




a 
A_{3} 

c_{1} 
A_{3} 

c_{2} 
A_{3} 

69 
F2/m2/m2/m 


















70 
F2/d2/d2/d 
00l: l =
4n + 2
d_{2}
0k0: k =
4n + 2
d_{3} 
A_{3} 

h00: h =
4n + 2
d_{3}
00l: l =
4n + 2
d_{1} 
A_{3} 

0k0: k =
4n + 2
d_{1}
h00: h =
4n + 2
d_{2} 
A_{3} 

h_{e}k_{e}0: h_{e} + k_{e} = 4n + 2
d_{3} 
A_{2}
A_{3} 
B_{2} 
0k_{e}l_{e}: k_{e} + l_{e} = 4n + 2
d_{1} 
A_{2}
A_{3} 
B_{2} 
h_{e}0l_{e}: h_{e} + l_{e} = 4n + 2
d_{2} 
A_{2}
A_{3} 
B_{2} 
71 
I2/m2/m2/m 


















72 
I2/b2/a2/m 












0k_{o}l_{o} 
A_{2} 
B_{2} 
h_{o}0l_{o} 
A_{2} 
B_{2} 












b 
A_{3} 

a 
A_{3} 

73 
I2_{1}/b2_{1}/c2_{1}/a 









h_{o}k_{o}0 
A_{2} 
B_{2} 
0k_{o}l_{o} 
A_{2} 
B_{2} 
h_{o}0l_{o} 
A_{2} 
B_{2} 









a 
A_{3} 

b 
A_{3} 

c 
A_{3} 

74 
I2_{1}/m2_{1}/m2_{1}/a 









h_{o}k_{o}0 
A_{2} 
B_{2} 















a 
A_{3} 







Space group  Incidentbeam direction 
[hk0] 
75 
P4 



76 
P4_{1} 
00l 
A_{2} 
B_{2} 
4_{1} 

B_{3} 
77 
P4_{2} 



78 
P4_{3} 
00l 
A_{2} 
B_{2} 
4_{3} 

B_{3} 
79 
I4 



80 
I4_{1} 



81 




82 




83 
P4/m 



84 
P4_{2}/m 



85 
P4/n 
hk0: h + k = 2n + 1 
A_{2} 
B_{2} 
n 
A_{3} 

86 
P4_{2}/n 
hk0: h + k = 2n + 1 
A_{2} 
B_{2} 
n 
A_{3} 

87 
I4/m 



88 
I4_{1}/a 
h_{o}k_{o}0 
A_{2} 
B_{2} 
a 
A_{3} 

Space group  Incidentbeam direction 
[hk0]  [0kl] 
89 
P422 






90 
P42_{1}2 



h00 
A_{2} 
B_{2} 



2_{1} 

B_{3} 
91 
P4_{1}22 
00l 
A_{2} 
B_{2} 



4_{1} 

B_{3} 



92 
P4_{1}2_{1}2 
00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
4_{1} 

B_{3} 
2_{1} 

B_{3} 
93 
P4_{2}22 






94 
P4_{2}2_{1}2 



h00 
A_{2} 
B_{2} 



2_{1} 

B_{3} 
95 
P4_{3}22 
00l 
A_{2} 
B_{2} 



4_{3} 

B_{3} 



96 
P4_{3}2_{1}2 
00l 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
4_{3} 

B_{3} 
2_{1} 

B_{3} 
97 
I422 






98 
I4_{1}22 






Space group  Incidentbeam direction 
[100]  [001]  [110]  [h0l]  [hhl] 
99 
P4mm 















100 
P4bm 



h00 
A_{2} 
B_{2} 



h_{o}0l 
A_{2} 
B_{2} 






a_{2} 
A_{3} 




a 
A_{3} 







0k0 














b_{1} 











101 
P4_{2}cm 
00l 








h0l_{o} 
A_{2} 
B_{2} 



c_{2} 
A_{3} 







c 
A_{3} 




102 
P4_{2}nm 
00l 


h00 
A_{2} 
B_{2} 



h0l: h + l = 2n + 1 
A_{2} 
B_{2} 



n_{2} 
A_{3} 

n_{2} 
A_{3} 




n 
A_{3} 







0k0 














n_{1} 











103 
P4cc 
00l 





00l 


h0l_{o} 
A_{2} 
B_{2} 
hhl_{o} 
A_{2} 
B_{2} 
c_{12} 
A_{3} 




c_{2} 
A_{3} 

c_{1} 
A_{3} 

c_{2} 
A_{3} 

104 
P4nc 
00l 


h00 
A_{2} 
B_{2} 
00l 


h0l: h + l = 2n + 1 
A_{2} 
B_{2} 
hhl_{o} 
A_{2} 
B_{2} 
n_{2} 
A_{3} 

n_{2} 
A_{3} 

c 
A_{3} 

n 
A_{3} 

c 
A_{3} 




0k0 














n_{1} 











105 
P4_{2}mc 






00l 





hhl_{o} 
A_{2} 
B_{2} 






c 
A_{3} 




c 
A_{3} 

106 
P4_{2}bc 



h00 
A_{2} 
B_{2} 
00l 


h_{o}0l 
A_{2} 
B_{2} 
hhl_{o} 
A_{2} 
B_{2} 



a_{2} 
A_{3} 

c 
A_{3} 

a 
A_{3} 

c 
A_{3} 




0k0 














b_{1} 











107 
I4mm 















108 
I4cm 









h_{o}0l_{o} 
A_{2} 
B_{2} 












c 
A_{3} 




109 
I4_{1}md 



hh0, 
A_{2} 
B_{2} 
00l: l = 4n + 2 





hhl_{e}: 2h + l_{e} = 4n + 2 
A_{2} 
B_{2} 



d 
A_{3} 

d 
A_{3} 




d 
A_{3} 

110 
I4_{1}cd 



hh0, 
A_{2} 
B_{2} 
00l: l = 4n + 2 


h_{o}0l_{o} 
A_{2} 
B_{2} 
hhl_{e}: 2h + l_{e} = 4n + 2 
A_{2} 
B_{2} 



d 
A_{3} 

d 
A_{3} 

c 
A_{3} 

d 
A_{3} 

Space group  Incidentbeam direction 
[100]  [001]  [110]  [h0l]  [hhl] 
111 
















112 







00l 





hhl_{o} 
A_{2} 
B_{2} 






c 
A_{3} 




c 
A_{3} 

113 

0k0 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 



0k0 
A_{2} 
B_{2} 



2_{12} 

B_{3} 
2_{11} 

B_{3} 



2_{1} 

B_{3} 






0k0 














2_{12} 











114 

0k0 
A_{2} 
B_{2} 
h00 
A_{2} 
B_{2} 
00l 


0k0 
A_{2} 
B_{2} 
hhl_{o} 
A_{2} 
B_{2} 
2_{12} 

B_{3} 
2_{11} 

B_{3} 
c 
A_{3} 

2_{1} 

B_{3} 
c 
A_{3} 




0k0 














2_{12} 











115 
















116 

00l 








h0l_{o} 
A_{2} 
B_{2} 



c_{2} 
A_{3} 







c 
A_{3} 




117 




h00 
A_{2} 
B_{2} 



h_{o}0l 
A_{2} 
B_{2} 






a_{2} 
A_{3} 




a 
A_{3} 







0k0 














b_{1} 











118 

00l 


h00 
A_{2} 
B_{2} 



h0l: h + l = 2n + 1 
A_{2} 
B_{2} 



n_{2} 
A_{3} 

n_{2} 
A_{3} 




n 
A_{3} 







0k0 














n_{1} 











119 
















120 










h_{o}0l_{o} 
A_{2} 
B_{2} 












c 
A_{3} 




121 
















122 




hh0, 
A_{2} 
B_{2} 
00l: l = 4n + 2 





hhl_{e}: 2h + l_{e} = 4n + 2 
A_{2} 
B_{2} 



d 
A_{3} 

d 
A_{3} 




d 
A_{3} 

Space group  Incidentbeam direction 
[100]  [001]  [110]  [h0l]  [hhl]  [hk0] 
123 
P4/mmm 


















P4/m2/m2/m 


















124 
P4/mcc 
00l 





00l 


h0l_{o} 
A_{2} 
B_{2} 
hhl_{o} 
A_{2} 
B_{2} 



P4/m2/c2/c 
c_{12} 
A_{3} 




c_{2} 
A_{3} 

c_{1} 
A_{3} 

c_{2} 
A_{3} 




125 
P4/nbm
P4/n2/b2/m 
0k0
n 
A_{3} 

h00
a_{2}
0k0
b_{1} 
A_{3} 




h_{o}0l
a 
A_{2}
A_{3} 
B_{2} 



hk0: h + k = 2n + 1
n 
A_{2}
A_{3} 
B_{2} 
126 
P4/nnc
P4/n2/n2/c<  