Table 1903 shows the extinctions (also called forbidden spots) in the diffraction patterns of the crystals with space group Fd3m such as diamond (C), silicon (Si), germanium (Ge), and tin (Sn) elements as a result of the destructive interference between the two interpenetrating face centred cubic (fcc) lattices displaced by a vector (1/4, 1/4, 1/4). Figure 1903a shows the diffraction pattern of a silicon (Si) crystal in [110] zone axis. The reflections marked in black are extinct in the single scattering approximation when the crystal is very thin, while the ones marked in green exist in the patterns of both thin and thick crystals. When the crystal become thicker, the multiple scattering occurs and thus these reflections can gain intensity and become visible because of more successive scatterings (even though they are probably weak if the sample is still relatively thin), for instance, electrons are indirectly scattered into the (002) reflection because of multiple scattering through the (1 1 1) and (1 1 1) scattering vectors. The red arrows represents the multiple scattering paths for forming the visible (002) reflection.
Table 1903. Fd3m (227) space group.
Name in the International Tables for Crystallography 
Fd3m 



227 

m3m 
Crystal system 
Cubic 



192 
dspacing ratios of allowed Bragg reflections 
1, √3, √8, √11, √12, √16, √19, √24, √27, √32, √43, ... 
Allowed reflections 
As fcc, but if all even and h + k + l ≠ 4n, then absent (n is integer) 
Forbidden reflections 
0 k l (k + l ≠ 4n); 0 0 l (l ≠ 4n); h, k, l are mixed odd and even; or, all even and h + k + l ≠ 4n (or defined by h + k + l = 4n + 2) 
Asymm 

Symmetry (atomic coordinates) 
+x, +y, +z;
x, y+1/2, +z+1/2;
x+1/2, +y+1/2, z;
+x+1/2, y, z+1/2;
+z, +x, +y;
+z+1/2, x, y+1/2;
z, x+1/2, +y+1/2;
z+1/2, +x+1/2, y;
+y, +z, +x;
y+1/2, +z+1/2, x;
+y+1/2, z, x+1/2;
y, z+1/2, +x+1/2;
+y+3/4, +x+1/4, z+3/4;
y+1/4, x+1/4, z+1/4;
+y+1/4, x+3/4, +z+3/4;
y+3/4, +x+3/4, +z+1/4;
+x+3/4, +z+1/4, y+3/4;
x+3/4, +z+3/4, +y+1/4;
x+1/4, z+1/4, y+1/4;
+x+1/4, z+3/4, +y+3/4;
+z+3/4, +y+1/4, x+3/4;
+z+1/4, y+3/4, +x+3/4;
z+3/4, +y+3/4, +x+1/4;
z+1/4, y+1/4, x+1/4;
x+1/4, y+1/4, z+1/4;
+x+1/4, +y+3/4, z+3/4;
+x+3/4, y+3/4, +z+1/4;
x+3/4, +y+1/4, +z+3/4;
z+1/4, x+1/4, y+1/4;
z+3/4, +x+1/4, +y+3/4;
+z+1/4, +x+3/4, y+3/4;
+z+3/4, x+3/4, +y+1/4;
y+1/4, z+1/4, x+1/4;
+y+3/4, z+3/4, +x+1/4;
y+3/4, +z+1/4, +x+3/4;
+y+1/4, +z+3/4, x+3/4;
y+1/2, x, +z+1/2;
+y, +x, +z;
y, +x+1/2, z+1/2;
+y+1/2, x+1/2, z;
x+1/2, z, +y+1/2;
+x+1/2, z+1/2, y;
+x, +z, +y;
x, +z+1/2, y+1/2;
z+1/2, y, +x+1/2;
z, +y+1/2, x+1/2;
+z+1/2, y+1/2, x;
+z, +y, +x;
+x, +y+1/2, +z+1/2;
x, y+1, +z+1;
x+1/2, +y+1, z+1/2;
+x+1/2, y+1/2, z+1;
+z, +x+1/2, +y+1/2;
+z+1/2, x+1/2, y+1;
z, x+1, +y+1;
z+1/2, +x+1, y+1/2;
+y, +z+1/2, +x+1/2;
y+1/2, +z+1, x+1/2;
+y+1/2, z+1/2, x+1;
y, z+1, +x+1;
+y+3/4, +x+3/4, z+5/4;
y+1/4, x+3/4, z+3/4;
+y+1/4, x+5/4, +z+5/4;
y+3/4, +x+5/4, +z+3/4;
+x+3/4, +z+3/4, y+5/4;
x+3/4, +z+5/4, +y+3/4;
x+1/4, z+3/4, y+3/4;
+x+1/4, z+5/4, +y+5/4;
+z+3/4, +y+3/4, x+5/4;
+z+1/4, y+5/4, +x+5/4;
z+3/4, +y+5/4, +x+3/4;
z+1/4, y+3/4, x+3/4;
x+1/4, y+3/4, z+3/4;
+x+1/4, +y+5/4, z+5/4;
+x+3/4, y+5/4, +z+3/4;
x+3/4, +y+3/4, +z+5/4;
z+1/4, x+3/4, y+3/4;
z+3/4, +x+3/4, +y+5/4;
+z+1/4, +x+5/4, y+5/4;
+z+3/4, x+5/4, +y+3/4;
y+1/4, z+3/4, x+3/4;
+y+3/4, z+5/4, +x+3/4;
y+3/4, +z+3/4, +x+5/4;
+y+1/4, +z+5/4, x+5/4;
y+1/2, x+1/2, +z+1;
+y, +x+1/2, +z+1/2;
y, +x+1, z+1;
+y+1/2, x+1, z+1/2;
x+1/2, z+1/2, +y+1;
+x+1/2, z+1, y+1/2;
+x, +z+1/2, +y+1/2;
x, +z+1, y+1;
z+1/2, y+1/2, +x+1;
z, +y+1, x+1;
+z+1/2, y+1, x+1/2;
+z, +y+1/2, +x+1/2;
+x+1/2, +y, +z+1/2;
x+1/2, y+1/2, +z+1;
x+1, +y+1/2, z+1/2;
+x+1, y, z+1;
+z+1/2, +x, +y+1/2;
+z+1, x, y+1;
z+1/2, x+1/2, +y+1;
z+1, +x+1/2, y+1/2;
+y+1/2, +z, +x+1/2;
y+1, +z+1/2, x+1/2;
+y+1, z, x+1;
y+1/2, z+1/2, +x+1;
+y+5/4, +x+1/4, z+5/4;
y+3/4, x+1/4, z+3/4;
+y+3/4, x+3/4, +z+5/4;
y+5/4, +x+3/4, +z+3/4;
+x+5/4, +z+1/4, y+5/4;
x+5/4, +z+3/4, +y+3/4;
x+3/4, z+1/4, y+3/4;
+x+3/4, z+3/4, +y+5/4;
+z+5/4, +y+1/4, x+5/4;
+z+3/4, y+3/4, +x+5/4;
z+5/4, +y+3/4, +x+3/4;
z+3/4, y+1/4, x+3/4;
x+3/4, y+1/4, z+3/4;
+x+3/4, +y+3/4, z+5/4;
+x+5/4, y+3/4, +z+3/4;
x+5/4, +y+1/4, +z+5/4;
z+3/4, x+1/4, y+3/4;
z+5/4, +x+1/4, +y+5/4;
+z+3/4, +x+3/4, y+5/4;
+z+5/4, x+3/4, +y+3/4;
y+3/4, z+1/4, x+3/4;
+y+5/4, z+3/4, +x+3/4;
y+5/4, +z+1/4, +x+5/4;
+y+3/4, +z+3/4, x+5/4;
y+1, x, +z+1;
+y+1/2, +x, +z+1/2;
y+1/2, +x+1/2, z+1;
+y+1, x+1/2, z+1/2;
x+1, z, +y+1;
+x+1, z+1/2, y+1/2;
+x+1/2, +z, +y+1/2;
x+1/2, +z+1/2, y+1;
z+1, y, +x+1;
z+1/2, +y+1/2, x+1;
+z+1, y+1/2, x+1/2;
+z+1/2, +y, +x+1/2;
+x+1/2, +y+1/2, +z;
x+1/2, y+1, +z+1/2;
x+1, +y+1, z;
+x+1, y+1/2, z+1/2;
+z+1/2, +x+1/2, +y;
+z+1, x+1/2, y+1/2;
z+1/2, x+1, +y+1/2;
z+1, +x+1, y;
+y+1/2, +z+1/2, +x;
y+1, +z+1, x;
+y+1, z+1/2, x+1/2;
y+1/2, z+1, +x+1/2;
+y+5/4, +x+3/4, z+3/4;
y+3/4, x+3/4, z+1/4;
+y+3/4, x+5/4, +z+3/4;
y+5/4, +x+5/4, +z+1/4;
+x+5/4, +z+3/4, y+3/4;
x+5/4, +z+5/4, +y+1/4;
x+3/4, z+3/4, y+1/4;
+x+3/4, z+5/4, +y+3/4;
+z+5/4, +y+3/4, x+3/4;
+z+3/4, y+5/4, +x+3/4;
z+5/4, +y+5/4, +x+1/4;
z+3/4, y+3/4, x+1/4;
x+3/4, y+3/4, z+1/4;
+x+3/4, +y+5/4, z+3/4;
+x+5/4, y+5/4, +z+1/4;
x+5/4, +y+3/4, +z+3/4;
z+3/4, x+3/4, y+1/4;
z+5/4, +x+3/4, +y+3/4;
+z+3/4, +x+5/4, y+3/4;
+z+5/4, x+5/4, +y+1/4;
y+3/4, z+3/4, x+1/4;
+y+5/4, z+5/4, +x+1/4;
y+5/4, +z+3/4, +x+3/4;
+y+3/4, +z+5/4, x+3/4;
y+1, x+1/2, +z+1/2;
+y+1/2, +x+1/2, +z;
y+1/2, +x+1, z+1/2;
+y+1, x+1, z;
x+1, z+1/2, +y+1/2;
+x+1, z+1, y;
+x+1/2, +z+1/2, +y;
x+1/2, +z+1, y+1/2;
z+1, y+1/2, +x+1/2;
z+1/2, +y+1, x+1/2;
+z+1, y+1, x;
+z+1/2, +y+1/2, +x 
Crystal symmetry 
Centrosymmetric 
Crystal examples 
Si, Ge, Sn  diamond cubic.
Diamond Si: Si at 0, 0, 0.
MgAl_{2}O_{4} (spinel):
Mg at 0, 0, 0;
Al at 5/8, 5/8, 5/8; and
O at 0.387, 0.387, 0.387.
Cu_{2}Mg (a Laves structure): Mg at 0, 0, 0 and Cu at 5/8, 5/8, 5/8.
γ  Al_{2}O_{3}: O
is cubic closepacked and Al occupy octahedral and tetrahedral sites.
SiO_{2}: Si at 0, 0, 0 and
O at 1/8, 1/8, 1/8.
KOs_{2}O_{6}: Fd3m (227) space group. 
Figure 1903a. Diffraction pattern of a Si crystal in [110] zone axis
orientation. The spot sizes represent the intensities.
Space group can be determined by CBED technique. For instance, Figure 1903b (a) shows the CBED pattern of βpyrochlore oxide superconductor KOs_{2}O_{6} along the [001] zone axis. [1] The square array with small dark disks near the center is zeroorder Laue zone (ZOLZ) and the surrounding circle formed by the highly contrasted disks is firstorder Laue zone (FOLZ). The magnified image of the inset presents a fourfold rotational symmetry along the c* axis and two mirror symmetries m_{a} and m_{b}, indicating that the whole pattern (WP) has 4mm symmetry. According to the general relationship among WPs, diffraction groups (DGs) and point groups (PGs) shown in a table in page2693, the diffraction groups for WP of 4mm symmetry is either 4mm or 4mm1_{R}. The former is consistent with a noncentrosymmetric PG of 4mm (tetragonal structure), and the latter with a centrosymmetric PG of m3m (cubic structure) or 4/mmm (tetragonal structure). Assuming the crystal system of KOs_{2}O_{6} is cubic (confirmed by XRD), the PG can only m3m.
Given the KOs_{2}O_{6} crystal is in Fd3m space group, in the magnified, indexed ZOLZ pattern in Figure 1903b, the bright broad lines (indicated by the white arrows) are suggested at the position of 200type reflections that are forbidden in Fd3m. Such lines are called dynamical extinction lines in CBED patterns, which appear at the kinematically forbidden reflections caused by glide planes or screw axes due to dynamical diffraction at certain incident directions. For the KOs_{2}O_{6} Fd3m crystal, the presence of the dynamical extinction lines provides direct evidence of the existence of a dglide symmetry. [2]
Figure 1903b. (a) CBED pattern taken from a KOs_{2}O_{6} crystal along [001] zone axis. (b) Magnified pattern of the ZOLZ from (a). [1]
[1] JunIchi Yamaura, Zenji Hiroi, Kenji Tsuda, Koichi Izawa, Yasuo Ohishi, Satoshi Tsutsui, Reexamination of the crystal structure of the βpyrochlore oxide superconductor KOs2O6 by Xray and convergentbeam electron
diffraction analyses, Solid State Communications 149 (2009) 3134.
[2] M. Tanaka, M. Terauchi, ConvergentBeam Electron Diffraction, JEOLMaruzen,
Tokyo, 1985.
