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.
