Diffraction of Thick TEM Specimen
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Thicker specimens have additional diffraction features as well as the usual Bragg diffractions. For instance, Table 4129 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 4129a 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 4129. Conditions of forbidden and allowed reflections (h k l) of common crystal structures
Bravais Lattice Forbidden reflections Allowed reflections Example Compounds
fcc h, k, l are mixed odd and even; or, all even and h + k + l ≠ 4n (Or defined by h + k + l = 4n + 2) As fcc, but if all even and h + k + l ≠ 4n, then absent (n is integer)
Si, Ge, Sn - diamond cubic

Figure 4129a. Diffraction pattern of a Si crystal in [110] zone axis
orientation. The spot sizes represent the intensities.

Figure 4129b shows the schematic illustration of the reflection- and forbidden-relrods of zone axis diffraction patterns at different specimen thickness. At zero tilt, the Ewald sphere is tangential to zero-order Laue zone (ZOLZ) of the reciprocal lattices in Figures 4129b (a), (b) and (d). For very thin specimens, the forbidden-relrods (D2 and D7) do not appear as shown in Figure 4129b (a), while they appear when the thickness of the specimens increases as shown in Figure 4129b (b) and (d) and the visible forbidden-relrods elongate more for thicker specimens. Note that the forbidden-relrods become visible due to double diffraction in thicker specimens. When the specimen is tilted, the loci of the intersections of Ewald sphere with the relrods does not change, and thus the projected positions of reflections do not move. However, the excitation errors (s) for the relrods change significantly, so that reflection intensities in the electron diffraction patterns change significantly. Especially, for the relatively thin specimen as shown in Figure 4129b (c), the forbidden-relrods (D2 and D7) will disappear on the recorded diffraction pattern (intersections of Ewald sphere with the relrods) because they are too short to appear with specimen-tilting. However, for very thick specimens as shown in 4129b (d), both the reflection- and forbidden-relrods are almost in the same length so that the forbidden-relrods will not disappear before reflection-relrods with specimen-tilting.

 (a) (b) (c) (d)

Figure 4129b. Schematic illustration of reflection- and forbidden-relrods of zero-order Laue zone (ZOLZ), depending on film thickness and specimen tilting: (a) At zero tilt, very thin TEM specimen for which forbidden-relrods do not exist; (b) At zero tilt and (c) at high-angle tilt, not very thin TEM specimen for which short forbidden-relrods appear; and (c) At zero tilt, very thick TEM specimen for which long forbidden-relrods appear. Here, t1 < t2 < t3.

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