Fd-3m (227) Space Group
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Table 1903 shows the extinctions (also called forbidden spots) in the diffraction patterns of the crystals with space group Fd-3m 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. Fd-3m (227) space group.

Name in the International Tables for Crystallography Lattice type Patterson space group Fd-3m 227 m-3m Cubic `192` d-spacing ratios of allowed Bragg reflections 1, √3, √8, √11, √12, √16, √19, √24, √27, √32, √43, ... As fcc, but if all even and h + k + l ≠ 4n, then absent (n is integer) 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) +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 Centrosymmetric Si, Ge, Sn - diamond cubic. Diamond Si: Si at 0, 0, 0. MgAl2O4 (spinel): Mg at 0, 0, 0; Al at 5/8, 5/8, 5/8; and O at 0.387, 0.387, 0.387. Cu2Mg (a Laves structure): Mg at 0, 0, 0 and Cu at 5/8, 5/8, 5/8. γ - Al2O3: O is cubic close-packed and Al occupy octahedral and tetrahedral sites. SiO2: Si at 0, 0, 0 and O at 1/8, 1/8, 1/8. KOs2O6: Fd-3m (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 β-pyrochlore oxide superconductor KOs2O6 along the [001] zone axis. [1] The square array with small dark disks near the center is zero-order Laue zone (ZOLZ) and the surrounding circle formed by the highly contrasted disks is first-order Laue zone (FOLZ). The magnified image of the inset presents a four-fold rotational symmetry along the c* axis and two mirror symmetries ma and mb, 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 4mm1R. The former is consistent with a non-centrosymmetric PG of 4mm (tetragonal structure), and the latter with a centrosymmetric PG of m-3m (cubic structure) or 4/mmm (tetragonal structure). Assuming the crystal system of KOs2O6 is cubic (confirmed by XRD), the PG can only m-3m.

Given the KOs2O6 crystal is in Fd-3m 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 200-type reflections that are forbidden in Fd-3m. 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 KOs2O6 Fd-3m crystal, the presence of the dynamical extinction lines provides direct evidence of the existence of a d-glide symmetry. [2]

Figure 1903b. (a) CBED pattern taken from a KOs2O6 crystal along [001] zone axis. (b) Magnified pattern of the ZOLZ from (a). [1]

[1] Jun-Ichi Yamaura, Zenji Hiroi, Kenji Tsuda, Koichi Izawa, Yasuo Ohishi, Satoshi Tsutsui, Re-examination of the crystal structure of the β-pyrochlore oxide superconductor KOs2O6 by X-ray and convergent-beam electron diffraction analyses, Solid State Communications 149 (2009) 31-34.
[2] M. Tanaka, M. Terauchi, Convergent-Beam Electron Diffraction, JEOL-Maruzen, Tokyo, 1985.

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