International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 318-323   | 1 | 2 |

Section 2.5.3.3.2. Identification of screw axes and glide planes

M. Tanakaf

2.5.3.3.2. Identification of screw axes and glide planes

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There are three space-group symmetry elements of diperiodic plane figures: (1) a horizontal screw axis [2_1'], (2) a vertical glide plane g with a horizontal glide vector and (3) a horizontal glide plane g′. These are related to the point-group symmetry elements 2′, m and m′ of diperiodic plane figures, respectively. (It is noted that these symmetry elements and ten point-group 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[link]) and was discussed by Miyake et al. (1960[link]) and Cowley et al. (1961[link]). Goodman & Lehmpfuhl (1964[link]) first observed the dynamical extinction as dark cross lines in kinematically forbidden reflection discs of CBED patterns of CdS. Gjønnes & Moodie (1965[link]) 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 symmetry-related multiple-scattering paths. Based on the results of Gjønnes & Moodie (1965[link]), Tanaka, Sekii & Nagasawa (1983[link]) 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[link]).

Fig. 2.5.3.10[link](a) illustrates Umweganregung paths to a kinematically forbidden reflection. The 0k0 (k = odd) reflections are kinematically forbidden because a b-glide plane exists perpendicular to the a axis and/or a 21 screw axis exists in the b direction. Let us consider an Umweganregung path a in the zeroth-order 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 21 screw axis, the following relations hold between the crystal structure factors:[\eqalignno{F(h,k) &= F(\bar{h},k)\quad \hbox{for }k=2n,&\cr F(h,k) &= -F(\bar{h},k)\quad\hbox{for }k=2n+1. &(2.5.3.1)}]That is, the structure factors of reflections hk0 and [\bar{h}k0] have the same phase for even k but have opposite phases for odd k.

[Figure 2.5.3.10]

Figure 2.5.3.10 | top | pdf |

Illustration of the production of dynamical extinction lines in kinematically forbidden reflections due to a b-glide plane and a 21 screw axis. (a) Umweganregung paths a, b and c. (b) Dynamical extinction lines A are formed in forbidden reflections 0k0 (k = odd). Extinction line B perpendicular to the lines A is formed in the exactly excited 010 reflection.

Since an Umweganregung path to the kinematically forbidden reflection 0k0 contains an odd number of reflections with odd k, the following relations hold:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_1,k_1)F(\bar{h}_2,k_2)\ldots F(\bar{h}_n,k_n)\quad\hbox{for path }b,&\cr&&(2.5.3.2)}]where[\textstyle\sum\limits_{i=1}^n h_i=0,\quad \textstyle\sum\limits_{i=1}^n k_i = k \,\,(k={\rm odd})]and 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[link](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:[\eqalignno{&F(h_1,k_1)F(h_2,k_2)\ldots F(h_n,k_n) \quad\hbox{for path }a&\cr &\quad = -F(\bar{h}_n,k_n)F(\bar{h}_{n-1},k_{n-1})\ldots F(\bar{h}_1,k_1)\quad\hbox{for path }c.&\cr&&(2.5.3.3)}]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[link](b). Line B appears perpendicular to line A at the exact Bragg positions. When Umweganregung paths are present only in the zeroth-order Laue zone, the glide plane and screw axis produce the same dynamical extinction lines A and B. We call these lines A2 and B2 lines, subscript 2 indicating that the Umweganregung paths lie in the zeroth-order 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 higher-order 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 21 screw axis and a glide plane,[\eqalignno{F(hkl)&=(-1)^kF(\bar{h}k\bar{l})\,\,\,\hbox{for a }2_1\hbox{ screw axis in the [010] direction,}&\cr &&(2.5.3.4)\cr F(hkl)&=(-1)^kF(\bar{h}kl)\,\,\,\hbox{for a }b\hbox{ glide in the (100) plane.}&\cr&&(2.5.3.5)}%fd2o5o3o5]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)[link], but extinction lines B are not produced because equation (2.5.3.4)[link] holds only for the 21 screw axis. In the case of the 21 screw axis, only the waves passing through paths a and c have opposite signs according to equation (2.5.3.4)[link], forming extinction lines B only. We call these lines A3 and B3 dynamical extinction lines, suffix 3 indicating the Umweganregung paths being via higher-order Laue zones.

It was predicted by Gjønnes & Moodie (1965[link]) that a horizontal glide plane g′ gives a dark spot at the crossing point between extinction lines A and B (Fig. 2.5.3.10[link]b) due to the cancellation between the waves passing through paths b and c. Tanaka, Terauchi & Sekii (1987[link]) observed this dynamical extinction, though it appeared in a slightly different manner to that predicted by Gjønnes & Moodie (1965[link]). Table 2.5.3.8[link] summarizes the appearance of the dynamical extinction lines for the glide planes g and g′ and the 21 screw axis. The three space-group symmetry elements can be identified from the observed extinctions because these three symmetry elements produce different kinds of dynamical extinctions.

Table 2.5.3.8| top | pdf |
Dynamical extinction rules for an infinitely extended parallel-sided specimen

Symmetry elements of plane-parallel specimenOrientation to specimen surfaceDynamical extinction lines
Two-dimensional (zeroth Laue zone) interactionThree-dimensional (HOLZ) interaction
Glide planes perpendicular: g A2 and B2 A3
parallel: g intersection of A3 and B3
Twofold screw axis perpendicular: 21
parallel: [2_1^\prime] A2 and B2 B3

In principle, a horizontal screw axis and a vertical glide plane can be distinguished by observations of the extinction lines A3 and B3. It is, however, not easy to observe the extinction lines A3 and B3 because broad extinction lines A2 and B2 appear at the same time. The presence of the extinction lines A3 and B3 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 A3 and B3 (Tanaka, Sekii & Nagasawa, 1983[link]). That is, if HOLZ lines form lines A3 and B3, HOLZ lines are symmetric with respect to the extinction lines A2 and B2. If HOLZ lines do not form lines A3 and B3, HOLZ lines are asymmetric with respect to the extinction lines A2 and B2. When the HOLZ lines are symmetric about the extinction lines A2, the specimen crystal has a glide plane. When the HOLZ lines are symmetric with respect to lines B2, a 21 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[link] shows CBED patterns taken from (a) thin and (b) thick areas of FeS2, whose space group is [P2_1/a\bar{3}], at the 001 Bragg setting with the [100] electron-beam incidence. In the case of the thin specimen (Fig. 2.5.3.11[link]a), only the broad dynamical extinction lines formed by the interaction of ZOLZ reflections are seen in the odd-order discs. On the other hand, fine HOLZ lines are clearly seen in the thick specimen (Fig. 2.5.3.11[link]b). The HOLZ lines are symmetric with respect to both A2 and B2 extinction lines. This fact proves the presence of the extinction lines A3 and B3, or both the c glide in the (010) plane and the 21 screw axis in the c direction, this fact being confirmed by consulting Table 2.5.3.9[link]. Fig. 2.5.3.12[link] shows a [110] zone-axis CBED pattern of FeS2. A2 extinction lines are seen in both the 001 and [\bar{1}10] discs. Fine HOLZ lines are symmetric with respect to the A2 extinction lines in the [\bar{1}10] disc but asymmetric about the A2 extinction line in the 001 disc, indicating formation of the A3 extinction line only in the [\bar{1}10] disc. This proves the existence of a 21 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[link]) demonstrated for spinel changes in the appearance of HOLZ lines in the ZOLZ discs at accelerating voltages around 100 kV.

[Figure 2.5.3.11]

Figure 2.5.3.11 | top | pdf |

CBED patterns obtained from (a) thin and (b) thick areas of (001) FeS2. (a) Dynamical extinction lines A2 and B2 are seen. (b) Extinction lines A3 and B3 as well as A2 and B2 are formed because HOLZ lines are symmetric about lines A2 and B2.

[Figure 2.5.3.12]

Figure 2.5.3.12 | top | pdf |

CBED pattern of FeS2 taken with the [110] electron-beam incidence. In the 001 and [00\bar 1] discs, HOLZ lines are asymmetric with respect to extinction lines A2, indicating the existence of a 21 screw axis parallel to the c axis. In the [\bar 1 10] and [1\bar10] discs, HOLZ lines are symmetric with respect to extinction lines A2, indicating existence of a glide plane perpendicular to the c axis.

Another practical method for distinguishing between glide planes and 21 screw axes is that reported by Steeds et al. (1978[link]). 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 21 screw axis perpendicular to the incident beam. With reference to Fig. 2.5.3.10[link](a), extinction lines A3 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 B3 originating from a 21 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. Single-crystal 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). Convergent-Beam 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 convergent-beam electron diffraction. Ultramicroscopy, 21, 245–250.








































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