International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 318323
Section 2.5.3.3.2. Identification of screw axes and glide planes
M. Tanaka^{f}

There are three spacegroup symmetry elements of diperiodic plane figures: (1) a horizontal screw axis , (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the pointgroup symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten pointgroup symmetry elements form 80 space groups.)
The ordinary extinction rules for screw axes and glide planes hold only in the approximation of kinematical diffraction. The kinematically forbidden reflections caused by these symmetry elements appear owing to Umweganregung of dynamical diffraction. Extinction of intensity, however, does take place in these reflections at certain crystal orientations with respect to the incident beam (i.e. in certain regions within a CBED disc). This dynamical extinction was first predicted by Cowley & Moodie (1959) and was discussed by Miyake et al. (1960) and Cowley et al. (1961). Goodman & Lehmpfuhl (1964) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965) discussed the dynamical extinction in a more general way considering not only ZOLZ reflections but also HOLZ reflections. They completely clarified the dynamical extinction rules by considering the exact cancellation which may occur along certain symmetryrelated multiplescattering paths. Based on the results of Gjønnes & Moodie (1965), Tanaka, Sekii & Nagasawa (1983) tabulated the dynamical extinctions expected at all the possible crystal orientations for all the space groups. These were later tabulated in a better form on pages 162 to 172 of the book by Tanaka & Terauchi (1985).
Fig. 2.5.3.10(a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a bglide plane exists perpendicular to the a axis and/or a 2_{1} screw axis exists in the b direction. Let us consider an Umweganregung path a in the zerothorder Laue zone to the 010 forbidden reflection and path b which is symmetric to path a with respect to axis k. Owing to the translation of one half of the lattice parameter b caused by the glide plane and/or the 2_{1} screw axis, the following relations hold between the crystal structure factors:That is, the structure factors of reflections hk0 and have the same phase for even k but have opposite phases for odd k.
Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold:whereand the functions including the excitation errors are omitted because only the cases in which the functions are the same for all the paths are considered. The excitation errors for paths a and b become the same when the projection of the Laue point along the zone axis concerned, L, lies on axis k. Since the two waves passing through paths a and b have the same amplitude but opposite signs, these waves are superposed on the 0k0 discs (k = odd) and cancel out, resulting in dark lines A in the forbidden discs, as shown in Fig. 2.5.3.10(b). The line A runs parallel to axis k passing through the projection point of the zone axis.
In path c, the reflections are arranged in the reverse order to those in path b. When the 010 reflection is exactly excited, two paths a and c are symmetric with respect to the bisector m′–m′ of the 010 vector having the same excitation errors. The following equation holds:Since the waves passing through these paths have the same amplitude but opposite signs, these waves are superposed on the 010 discs and cancel out, resulting in dark line B in this disc, as shown in Fig. 2.5.3.10(b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zerothorder Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A_{2} and B_{2} lines, subscript 2 indicating that the Umweganregung paths lie in the zerothorder Laue zone.
The dynamical extinction effect is analogous to interference phenomena in the Michelson interferometer. That is, the incident beam is split into two beams by Bragg reflections in a crystal. These beams take different paths, in which they suffer a relative phase shift of π and are finally superposed on a kinematically forbidden reflection to cancel out.
When the paths include higherorder Laue zones, the glide plane produces only extinction lines A but the screw axis causes only extinction lines B. These facts are attributed to the different relations between structure factors for a 2_{1} screw axis and a glide plane,In the case of the glide plane, extinction lines A are still formed because two waves passing through paths a and b have opposite signs to each other according to equation (2.5.3.5), but extinction lines B are not produced because equation (2.5.3.4) holds only for the 2_{1} screw axis. In the case of the 2_{1} screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4), forming extinction lines B only. We call these lines A_{3} and B_{3} dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higherorder Laue zones.
It was predicted by Gjønnes & Moodie (1965) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965). Table 2.5.3.8 summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 2_{1} screw axis. The three spacegroup symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A_{3} and B_{3}. It is, however, not easy to observe the extinction lines A_{3} and B_{3} because broad extinction lines A_{2} and B_{2} appear at the same time. The presence of the extinction lines A_{3} and B_{3} can be revealed by inspecting the symmetries of fine defect HOLZ lines appearing in the forbidden reflections instead of by direct observation of the lines A_{3} and B_{3} (Tanaka, Sekii & Nagasawa, 1983). That is, if HOLZ lines form lines A_{3} and B_{3}, HOLZ lines are symmetric with respect to the extinction lines A_{2} and B_{2}. If HOLZ lines do not form lines A_{3} and B_{3}, HOLZ lines are asymmetric with respect to the extinction lines A_{2} and B_{2}. When the HOLZ lines are symmetric about the extinction lines A_{2}, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B_{2}, a 2_{1} screw axis exists. It should be noted that a relatively thick specimen area has to be selected to observe HOLZ lines in ZOLZ reflection discs.
Fig. 2.5.3.11 shows CBED patterns taken from (a) thin and (b) thick areas of FeS_{2}, whose space group is , at the 001 Bragg setting with the [100] electronbeam incidence. In the case of the thin specimen (Fig. 2.5.3.11a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the oddorder discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11b). The HOLZ lines are symmetric with respect to both A_{2} and B_{2} extinction lines. This fact proves the presence of the extinction lines A_{3} and B_{3}, or both the c glide in the (010) plane and the 2_{1} screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9. Fig. 2.5.3.12 shows a [110] zoneaxis CBED pattern of FeS_{2}. A_{2} extinction lines are seen in both the 001 and discs. Fine HOLZ lines are symmetric with respect to the A_{2} extinction lines in the disc but asymmetric about the A_{2} extinction line in the 001 disc, indicating formation of the A_{3} extinction line only in the disc. This proves the existence of a 2_{1} screw axis in the [001] direction and an a glide in the (001) plane. The appearance of HOLZ lines is easily changed by a change of a few hundred volts in the accelerating voltage. Steeds & Evans (1980) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV.
Another practical method for distinguishing between glide planes and 2_{1} screw axes is that reported by Steeds et al. (1978). The method is based on the fact that the extinction lines are observable even when a crystal is rotated with a glide plane kept parallel and with a 2_{1} screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10(a), extinction lines A_{3} produced by a glide plane remain even when the crystal is rotated with respect to axis h but the lines are destroyed by a rotation of the crystal about axis k. Extinction lines B_{3} originating from a 2_{1} screw axis are not destroyed by a crystal rotation about axis k but the lines are destroyed by a rotation with respect to axis h.
References
Cowley, J. M. & Moodie, A. F. (1959). The scattering of electrons by atoms and crystals. III. Singlecrystal diffraction patterns. Acta Cryst. 12, 360–367.Cowley, J. M., Moodie, A. F., Miyake, S., Takagi, S. & Fujimoto, F. (1961). The extinction rules for reflections in symmetrical electron diffraction spot patterns. Acta Cryst. 14, 87–88.
Gjønnes, J. & Moodie, A. F. (1965). Extinction conditions in dynamic theory of electron diffraction patterns. Acta Cryst. 19, 65–67.
Goodman, P. & Lehmpfuhl, G. (1964). Verbotene elektronenbeugungsreflexe von CdS. Z. Naturforsch. Teil A, 19, 818–820.
Miyake, S., Takagi, S. & Fujimoto, F. (1960). The extinction rule for reflexions in symmetrical spot patterns of electron diffraction by crystals. Acta Cryst. 13, 360–361.
Steeds, J. W. & Evans, N. S. (1980). Practical examples of point and space group determination in convergent beam diffraction. Proc. Electron Microsc. Soc. Am. pp. 188–191.
Steeds, J. W., Rackham, G. M. & Shannon, M. D. (1978). On the observation of dynamically forbidden lines in two and three dimensional electron diffraction. In Electron Diffraction 1927–1977. Inst. Phys. Conf. Ser. No. 41, pp. 135–139.
Tanaka, M., Sekii, H. & Nagasawa, T. (1983). Space group determination by dynamic extinction in convergent beam electron diffraction. Acta Cryst. A39, 825–837.
Tanaka, M. & Terauchi, M. (1985). ConvergentBeam Electron Diffraction. Tokyo: JEOL Ltd.
Tanaka, M., Terauchi, M. & Sekii, H. (1987). Observation of dynamic extinction due to a glide plane perpendicular to an incident beam by convergentbeam electron diffraction. Ultramicroscopy, 21, 245–250.