extinction rule

Fig. 1. Extinction of reflections due to the type of the crystal lattice.

Electron diffraction pattern of Cu taken at the [001] electron incidence, where Cu takes the face-centered cubic lattice. In the case of the face-centered cubic lattice, for the reflections h, k, l having even- and odd-numbered mixed indices, the crystal structure factor takes zero or F (h k l) = 0. Thus, the diffraction spots vanish. In this figure, it is seen that the diffraction spots of 100 and 110, etc., vanish. The pattern was taken with the JEM-2100Plus at an accelerating voltage of 200 kV.

Fig. 2. Extinction of reflections due to a symmetry (d-glide plane) element of the space group.

Electron diffraction patterns of Si taken at the [01-1] electron incidence, Si crystal having d-glide planes. (a) a sufficiently thin specimen and (b) a thick specimen. Extinction of the reflections due to the d-glide planes occurs when reflection indices take h+k+l = 4n+2 (n is integer). The 200 reflection (indicated by an arrow in Fig. 2(a) and (b)) is forbidden kinematically but has a certain intensity due to "Umweganregung" or successive excitation of the 1-1-1 reflection and the 111 reflection as an example. In the case of a thin specimen (a), "Umweganregung" is weak and thus, the 200 reflection is weak. In the case of the thick specimen (b), "Umweganregung" is strong and thus, the reflection is strong. The patterns were taken with the JEM-F200 at an accelerating voltage of 200 kV.

Fig. 3. Partial extinction (dark lines) in the reflections excited by "Umweganregung". (Data courtesy: Professor Kenji Tsuda, Tohoku University)

Diffraction patterns of FeS₂. (a) a selected-area diffraction pattern taken at the [100] electron incidence, (b) a convergent-beam electron diffraction (CBED) pattern taken at the [100] electron incidence and (c) a CBED pattern taken at the electron incident slightly tilted from the [001] zone axis to excite the 100 reflection.
FeS₂ belongs to the space group of P2_1/a3 and has a-glide plane in the (010) plane and 2₁ screw axis parallel to the a-axis. Owing to these symmetry elements, the odd-order h00 reflections have certain intensities due to the dynamical effect ("Umweganregung"). However in those reflections, dark lines are created due to dynamical extinction (shown by arrowheads in Fig.(b) and (c)). The "dynamical extinction" should be referred to the description in Fig. 4. The patterns were taken with the JEM-2010 at an accelerating voltage of 100 kV.

Fig. 4. "Umweganregung" via the zeroth-order Laue zone reflections (ZOLZ reflections) and dynamical extinction (dark lines) appearing in the CBED patterns. (a), (c) Enlarged figures of Fig. 3 (b), (c). (b), (d) Schematic illustrations of dynamical extinction.

We assume that a-glide planes exist in the (010) planes and 2₁ screw axes exist parallel to the a-axis. Then, the odd-order h00 reflections vanish kinematically. However, these reflections have certain intensities due to the dynamical diffraction effect ("Umweganregung"). It should be noted that dynamical extinction (dark lines) is produced in the CBED disks excited by "Umweganregung" (dark lines in the odd-order reflections).
In Fig. (a), the dark line A is seen horizontally in the odd-order h00 reflections. The "Umweganregung" paths a and b, shown in Fig. (b), are symmetric to both the glide plane and the screw axis. Taking account of the phases of the reflections in each path, the resultant reflections passing through the paths a and b cancel out each other, forming the dark line A. The dark line A appears in all the odd-order reflections.
Fig. (c) shows the case where the odd-order 100 reflection satisfies the Bragg condition, exhibiting the single dark line B perpendicular to the dark line A in the 100 reflection. Fig. (d) shows the "Umweganregung" paths a and c. Both the paths are symmetric to the dotted line m’-m’, and the reflections passing through these two paths cancel out each other, forming the dark line B. For more details, refer to the following references.

(Proofread by Professor Kenji Tsuda, Tohoku University)

References:

M. Tanaka, M. Terauchi, K, Tsuda, "Introduction to Electron Diffraction and Elementary Crystallography (in Japanese)", (Kyoritsu Printing Co., Ltd.) p121

M. Tanaka, "International Tables for Crystallography", Vol. B, 2^nd online ed., §2.5.3, (International Union of Crystallography, Chester, 2010)

M. Tanaka and M. Terauchi, "Convergent Beam Electron Diffraction" (JEOL-Maruzen, Tokyo,1985) p50 (http://www2.tagen.tohoku.ac.jp/lab/terauchi/html/activity/cbedbook.html )