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

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 344-352   | 1 | 2 |

Section 2.5.3.4. Symmetry determination of incommensurate crystals

M. Tanakaf

2.5.3.4. Symmetry determination of incommensurate crystals

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2.5.3.4.1. General remarks

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Incommensurately modulated crystals do not have three-dimensional lattice periodicity. The crystals, however, recover lattice periodicity in a space higher than three dimensions. de Wolff (1974[link], 1977[link]) showed that one-dimensional displacive and substitutionally modulated crystals can be described as a three-dimensional section of a (3 + 1)-dimensional periodic crystal. Janner & Janssen (1980a[link],b[link]) developed a more general approach for describing a modulated crystal with n modulations as (3 + n)-dimensional periodic crystals (n = 1, 2, …). Yamamoto (1982[link]) derived a general structure-factor formula for n-dimensionally modulated crystals (n = 1, 2, …), which holds for both displacive and substitutionally modulated crystals. Tables of the (3 + 1)-dimensional space groups for one-dimensional incommensurately modulated crystals were given by de Wolff et al. (1981[link]), where the wavevector of the modulation was assumed to lie in the c direction. Later, some corrections to the tables were made by Yamamoto et al. (1985[link]). The analysis of incommensurately modulated crystals using (3 + 1)-dimensional space groups has become familiar in the field of X-ray structure analysis.

Fung et al. (1980[link]) applied the CBED method to the study of incommensurately modulated transition-metal dichalcogenides. Steeds et al. (1985[link]) applied the LACBED method (Tanaka et al., 1980[link]) to the study of incommensurately modulated crystals of NiGe1−xPx. Tanaka et al. (1988[link], pp. 74–81) examined the symmetries of the incommensurate and fundamental reflections appearing in the CBED patterns obtained from the incommensurately modulated crystals of Sr2Nb2O7 and Mo8O23. Terauchi & Tanaka (1993) clarified theoretically the interrelation between the symmetries of CBED patterns and the (3 + 1)-dimensional point-group symbols for incommensurately modulated crystals and verified experimentally the theoretical results for Sr2Nb2O7 and Mo8O23. Terauchi et al. (1994[link]) investigated dynamical extinction for the (3 + 1)-dimensional space groups. They clarified that approximate dynamical extinction lines appear in CBED discs of the reflections caused by incommensurate modulations when the amplitudes of the incommensurate modulation waves are small. They tabulated the dynamical extinction lines appearing in the CBED discs for all the (3 + 1)-dimensional space groups of the incommensurately modulated crystals. The tables were stored in the British Library Document Supply Centre as Supplementary Publication No. SUP 71810 (65 pp.). They showed an example of the dynamical extinction lines obtained from Sr2Nb2O7. The point- and space-group determinations of the (3 + 1)-dimensional crystals are described compactly in the book by Tanaka et al. (1994[link], pp. 156–205).

Fig. 2.5.3.17[link] illustrates (3 + 1)-dimensional descriptions of a crystal structure without modulation (a), a one-dimensional displacive modulated structure (b) and a one-dimensional substitutionally modulated structure (c). The arrows labelled a1a3 (a, b and c) and a4 indicate the (3 + 1)-dimensional crystal axes. The horizontal line labelled R3 represents the three-dimensional space (external space). In the (3 + 1)-dimensional description, an atom is not located at a point as in the three-dimensional space, but extends as a string along the fourth direction a4 perpendicular to the three-dimensional space R3. The shaded parallelogram is a unit cell in the (3 + 1)-dimensional space. The unit cell contains two atom strings in this case. In the case of no modulations, the atoms are shown as straight strings, as shown in Fig. 2.5.3.17[link](a). For a displacive modulation, atoms are expressed by wavy strings periodic along the fourth direction a4 as shown in Fig. 2.5.3.17[link](b). The width of the atom strings indicates the spread of the atoms in R3. The atom positions of the modulated structure in R3 are given as a three-dimensional (R3) section of the atom strings in the (3 + 1)-dimensional space. A substitutional modulation, which is described by a modulation of the atom form factor, is expressed by atom strings with a density modulation along the direction a4 as shown in Fig. 2.5.3.17[link](c).

[Figure 2.5.3.17]

Figure 2.5.3.17 | top | pdf |

The (3 + 1)-dimensional description of one-dimensionally modulated crystals. Atoms are shown as strings along the fourth direction a4. (a) No modulation, shown as straight strings. (b) Displacive modulation, shown as wavy strings. (c) Amplitude modulation with varying-density strings.

The diffraction vector G is written as[{\bf G}=h_1{\bf a}^*+h_2{\bf b}^*+h_3{\bf c}^*+h_4{\bf k},]where a set of h1h2h3h4 is a (3 + 1)-dimensional reflection index, and a*, b* and c* are the reciprocal-lattice vectors of the real-lattice vectors a, b and c of the average structure. The modulation vector k is written as[{\bf k}=k_1{\bf a}^*+k_2{\bf b}^*+k_3{\bf c}^*,]where one coefficient ki (i = 1–3) is an irrational number and the others are rational. Fig. 2.5.3.18[link](a) shows a diffraction pattern of a crystal with an incommensurate modulation wavevector k1a* (k2 and k3 = 0). Large and small black spots show the fundamental reflections and incommensurate reflections, respectively, only the first-order incommensurate reflections being shown. It should be noted that the diffraction pattern of a modulated crystal is obtained by a projection of the Fourier transform of the (3 + 1)-dimensional periodic structure. Fig. 2.5.3.18[link](b) is assumed to be the Fourier transform of Fig. 2.5.3.17[link](b). Incommensurate reflections are obtained by a projection of the reciprocal-lattice points onto [{\bf R}_3^*].

[Figure 2.5.3.18]

Figure 2.5.3.18 | top | pdf |

(a) Schematic diffraction pattern from a modulated crystal. As an example, the wave number vector of modulation is assumed to be k1a*, k1 being an irrational number. Large and small spots denote fundamental and incommensurate reflections, respectively. (b) Incommensurate reflections are obtained by a projection of the Fourier transform of a (3 + 1)-dimensional periodic structure.

The displacive modulation is expressed by the atom displacement [u_i^\mu] with x4. The structure factor [F(h_1 h_2 h_3 h_4 )] for the (3 + 1)-dimensional crystal with a displacive modulation is given by de Wolff (1974[link], 1977[link]) as follows:[\eqalignno{F(h_1 h_2 h_3 h_4 ) &= \textstyle\sum\limits_{\mu = 1}^N {f_\mu \exp [2\pi i(h_1 \bar x_1^\mu + h_2 \bar x_2^\mu + h_3 \bar x_3^\mu )}] &\cr &\quad\times \displaystyle\int\limits_0^1 \exp \left\{2\pi i\left[ \textstyle\sum\limits_{i = 1}^3 (h_i + h_4 k_i )u_i^\mu + h_4 \bar x_4^\mu \right]\right\}\, {\rm d}\bar x_4^\mu ,&\cr &&(2.5.3.6)} ]where[\bar x_4^\mu = (\bar x_1^\mu + n_1 )k_1 + (\bar x_2^\mu + n_2 )k_2 + (\bar x_3^\mu + n_3 )k_3 .][f_\mu ] and [\bar x_i^\mu](i = 1–3) are, respectively, the atom form factor and the ith component of the position of the μth atom in the unit cell of the average structure. The symbol [u_i^\mu ] is the ith component of the displacement of the μth atom. Since the atom in the (3 + 1)-dimensional space is continuous along a4 and discrete along R3, the structure factor is expressed by summation in R3 and integration along a4 as seen in equation (2.5.3.6[link]). The integration implies that the sum for the atoms with displacements is taken over the infinite number of unit cells of the average structure. That is, equation (2.5.3.6[link]) is the structure factor for a unit cell with the lattice parameter of an infinite length in R3 along the direction of the modulation wavevector k.

CBED patterns are obtained from a finite area of a specimen crystal. For the symmetry analysis of CBED patterns obtained from modulated structures, the effect of the finite size was considered by Terauchi & Tanaka (1993[link]). The integration over a unit-cell length along a4 in equation (2.5.3.6[link]) is rewritten in the following way with the summation over a finite number of three-dimensional sections of the atom strings:[\eqalignno{F'(h_1 h_2 h_3 h_4 ) &= \textstyle\sum\limits_{\mu = 1}^N f_\mu \exp [2\pi i(h_1 \bar x_1^\mu + h_2 \bar x_2^\mu + h_3 \bar x_3^\mu )] &\cr&\quad \times \displaystyle\sum\limits_{n_1 } \displaystyle\sum\limits_{n_2 } \displaystyle\sum\limits_{n_3 } \exp \left\{2\pi i\left[ \textstyle\sum\limits_{i = 1}^3 (h_i + h_4 k_i )u_i^\mu + h_4 \bar x_4^\mu \right]\right\},&\cr&& (2.5.3.7)}]where [N_1\,\lt\,n_1\le N_1'], [N_2\,\lt\,n_2\le N_2'] and [N_3\,\lt\,n_3\le N_3'], [N'=] [(N_1'-N_1)(N_2'-N_2)(N_3'-N_3)] being the number of unit cells of the average structure included in a specimen volume from which CBED patterns are taken.

The substitutional modulation arises from a periodic variation of the site-occupation probability of the atoms. This modulation is expressed by a modulation of the atom form factor [f_\mu ] with x4. The structure factor [F'(h_1 h_2 h_3 h_4 )] for a finite-size crystal is written as[\eqalignno{F'(h_1 h_2 h_3 h_4 ) &= \textstyle\sum\limits_{\mu = 1}^N {\exp [2\pi i\,(h_1 x_1^\mu + h_2 x_2^\mu + h_3 x_3^\mu )]}&\cr &\quad \times \textstyle\sum\limits_{n_1 } {\textstyle\sum\limits_{n_2 } {\textstyle\sum\limits_{n_3 } {f_\mu \exp (2\pi i h_4 x_4^\mu )} } } ,&(2.5.3.8)}]where [x_4^\mu = \textstyle\sum_i {(x_i^\mu + n_i )k_i }].

2.5.3.4.2. Point-group determination

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The symmetries of the CBED patterns can be determined by examination of the symmetries of the structure factor [F'(h_1 h_2 h_3 h_4 )] in equations (2.5.3.7)[link] or (2.5.3.8)[link]. We consider a displacive modulated structure, which has a modulation wavevector k = k3c* and belongs to (3 + 1)-dimensional space group [\mathop P\nolimits_{\;\;1\,\;\bar 1}^{P2/m} ]. This space-group symbol implies the following.

  • (1) The modulation wavevector k exists inside the first Brillouin zone of the average structure (P).

  • (2) The average structure belongs to space group P2/m, the twofold rotation axis being parallel to the c axis.

  • (3) The symmetry subsymbol 1, which is written beneath symmetry symbol 2, indicates that the modulation wavevector k is transformed into itself by symmetry operation 2 of the average structure. The symmetry subsymbol beneath symmetry symbol m indicates that the wavevector k is transformed into −k by symmetry operation m. The modulated structure has a twofold rotation axis, which is common to the average structure, but does not have mirror symmetry, which is possessed by the average structure.

For the twofold rotation axis (symbol 2) of this space group, the structure factor [F'(h_1 h_2 h_3 h_4 )] is written as[\eqalignno{&F'(h_1 h_2 h_3 h_4 )&\cr &\quad= \textstyle\sum\limits_{\mu = 1}^N {f_\mu \exp [2\pi i(h_1 \bar x_1^\mu + h_2 \bar x_2^\mu + h_3 \bar x_3^\mu )]} &\cr & \quad\quad \times \textstyle\sum\limits_{n_3 } {\exp \{2\pi i[ h_1 u_1^\mu + h_2 u_2^\mu + (h_3 + h_4 k_3 )u_3^\mu + h_4 \bar x_4^\mu ] \}} &\cr & \quad\quad + \textstyle\sum\limits_{\mu = 1}^N {f_\mu \exp [2\pi i( - h_1 \bar x_1^\mu - h_2 \bar x_2^\mu + h_3 \bar x_3^\mu )]} &\cr & \quad\quad \times \textstyle\sum\limits_{n_3 } {\exp \{2\pi i[ - h_1 u_1^\mu - h_2 u_2^\mu + (h_3 + h_4 k_3 )u_3^\mu + h_4 \bar x_4^\mu ] \}}, &\cr &&(2.5.3.9)} ]where [x_4^\mu = (x_3^\mu + n_3 )k_3 ]. It is found from equation (2.5.3.9)[link] that two structure factors [F'(h_1 h_2 h_3 h_4 )] and [F'(\bar h_1 \bar h_2 h_3 h_4 )] are the same when reflections [h_1h_2h_3h_4] and [\bar h_1\bar h_2h_3h_4] are equivalent with respect to the twofold rotation axis of the average structure. Thus, not only fundamental reflections (h4 = 0) from the average structure but also the satellite reflections (h4 ≠ 0) from the incommensurate structure show twofold rotation symmetry about the c* axis.

For the mirror plane (symbol m), the structure factor is written in a similar manner to the case of the twofold rotation axis. It is found that [F'(h_1 h_2 h_3 h_4 )] is not equal to [F'(h_1 h_2 \bar h_3 \bar h_4 )] for the incommensurate reflections h4 ≠ 0. Hence, the incommensurate reflections do not show mirror symmetry with respect to the mirror plane of the average structure. For the fundamental reflections (h4 = 0), [F'(h_1 h_2 h_3 h_4 )] is equal to [F'(h_1 h_2 \bar h_3 \bar h_4 )], indicating the existence of mirror symmetry. It should be noted that the mirror symmetry can be destroyed by the dynamical diffraction effect between the fundamental and incommensurate reflections. In most modulated structures, however, the amplitude of the modulation wave [u_i^\mu ] is not so large as to destroy the symmetry of the fundamental reflections. Thus, the fundamental reflections ought to show the symmetry of the average structure, while the incommensurate reflections lose the symmetry.

The problem of the finite size of the illuminated area is discussed using equations (2.5.3.7)[link] and (2.5.3.8)[link] in a paper by Terauchi & Tanaka (1993[link]) and in the book by Tanaka et al. (1994[link], pp. 156–205). The results are as follows: Even if the size and position of an illuminated specimen area are changed, the intensity distribution in a CBED pattern changes but the symmetry of the pattern does not. To obtain the symmetries of incommensurate crystals, it is not necessary to take CBED patterns from an area whose diameter is larger than the period of the modulated structure. The symmetries of the modulated structure can appear when more than one unit cell of the average structure is illuminated for displacive modulations. For substitutional modulations, a specimen volume that produces the average atom form factor is needed, namely a volume of about 1 nm diameter area and 50 nm thick.

Table 2.5.3.13[link] shows the point-group symmetries (third column) of the incommensurate reflections for the two point-group subsymbols. For symmetry subsymbol 1, both the fundamental and incommensurate reflections show the symmetries of the average structure. For symmetry subsymbol [\bar 1], the fundamental reflections show the symmetries of the average structure but the incommensurate reflections do not have any symmetry. These facts imply that the symmetries of the incommensurate reflections are determined by the point group of the average structure and the modulation wavevector k. In other words, observation of the symmetries of the incommensurate reflections is not necessary for the determination of the point groups, although it can ascertain the point groups of the modulated crystals.

Table 2.5.3.13| top | pdf |
Wavevectors, point- and space-group symbols and CBED symmetries of one-dimensionally modulated crystals

Wavevector transformationPoint-group symbolSymmetry of incommensurate reflectionSpace-group symbolDynamical extinction lines
[{\bf k} \rightarrow {\bf k}] 1 Same symmetry as average structure 1, s (1/2), t (±1/3), q (±1/4), h (±1/6) Yes for s, q and h
[{\bf k} \rightarrow -{\bf k}] [\bar 1] No symmetry [\bar 1] No

An example of point-group determination is shown for the incommensurate phase of Sr2Nb2O7. Many materials of the A2B2O7 family undergo phase transformations from space group Cmcm to Cmc21 and further to P21 with decreasing temperature. An incommensurate phase appears between the Cmc21 phase and the P21 phase. Sr2Nb2O7 transforms at 488 K from the Cmc21 phase into the incommensurate phase with a modulation wavevector k = (½ − δ)a* (δ = 0.009–0.023) but does not transform into the P21 phase. The space group of Sr2Nb2O7 was reported as [\mathop P\nolimits_{\;\,\,{\kern 1pt} {\kern 1pt} \bar 1{\kern 1pt} s{\kern 1pt} \bar 1}^{C{\kern 1pt} mc2_1 } ] (Yamamoto, 1988[link]). (Since the space-group notation Cmc21 is broadly accepted, the direction of the modulation is taken as the a axis.) The point group of the phase is [_{\,\bar 1\,1\,\bar 1}^{mm2} ]. The modulation wavevector k is transformed to −k by the mirror symmetry operation perpendicular to the a axis ([_{\,\bar 1}^m ]) and by the twofold rotation symmetry operation about the c axis ([_{\bar 1}^2 ]). The wavevector is transformed into itself by the mirror symmetry operation perpendicular to the b axis ([_{{\kern 1pt} 1}^m ]).

Fig. 2.5.3.19[link](a) shows a CBED pattern of the incommensurate phase of Sr2Nb2O7 taken with the [010] incidence at an accelerating voltage of 60 kV. The reflections indicated by arrowheads are the incommensurate reflections. Other reflections are the fundamental reflections. Since the pattern is produced by the interaction of the reflections in the zeroth-order Laue zone, symmetry operations ([_{\,\bar 1}^m ]) and ([_{\bar 1}^2 ]) act the same. These symmetries are confirmed by the fact that the fundamental reflections show mirror symmetry perpendicular to the a* axis (twofold rotation symmetry about the c* axis) but the incommensurate reflections do not. Fig. 2.5.3.19[link](b) shows a CBED pattern of the incommensurate phase of Sr2Nb2O7 taken with the [201] incidence at 60 kV. The reflections in the two rows indicated by arrowheads are the incommensurate reflections and the others are the fundamental reflections. Symmetry symbol ([_{{\kern 1pt} 1}^m ]) implies that both the fundamental and incommensurate reflections display mirror symmetry perpendicular to the b* axis. Fig. 2.5.3.19[link](b) exactly exhibits the symmetry.

[Figure 2.5.3.19]

Figure 2.5.3.19 | top | pdf |

CBED patterns of the incommensurate phase of Sr2Nb2O7 taken at 60 kV. (a) [010] incidence: fundamental reflections show a mirror symmetry perpendicular to the a* axis but incommensurate reflections do not [symmetry ([_{\,1}^m ])]. (b) [201] incidence: incommensurate reflections show mirror symmetry perpendicular to the b* axis [symmetry ([_{\,\bar 1}^m ])]. The wave number vector of the modulation is k = (½ − δ)a*.

2.5.3.4.3. Space-group determination

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Table 2.5.3.13[link] shows the space-group symbols (fourth column) of the modulated crystals. When a glide (screw) component τ4 between the modulation waves of two atom rows is 0, 1/2, ±1/3, ±1/4 or ±1/6, symbol 1, s, t, q or h is given, respectively (de Wolff et al., 1981[link]). Such glide components are allowed for point-group symmetry 1 but are not for point-group symmetry [\bar 1]. Dynamical extinction occurs for glide components s, q and h but does not for glide component t. When the average structure does not have a glide component, dynamical extinction due to a glide component τ4 appears in odd-order incommensurate reflections. When the average structure has a glide component, dynamical extinction due to a glide component τ4 appears in incommensurate reflections with hi + h4 = 2n + 1, where hi and h4 are the reflection indices for the average structure and incommensurate structure, respectively. Details are given in the paper by Terauchi et al. (1994[link]).

Fig. 2.5.3.20[link](a) illustrates mirror symmetry ([_{\,1}^m ]) between atom rows A and B, which is perpendicular to the b axis with no glide component (τ4 = 0). Here, the wave number vector of the modulation is assumed to be k = k3c* following the treatment of de Wolff et al. (1981[link]). Fig. 2.5.3.20[link](b) illustrates glide symmetry ([_{\,s}^m ]) with a glide component τ4 = ½. The structure factor [F(h_1 h_2 h_3 h_4 )] is written for the glide plane ([_{\,s}^m ]) of an infinite incommensurate crystal as[\eqalignno{& F(h_1 h_2 h_3 h_4 ) &\cr&\quad= \textstyle\sum\limits_{\mu = 1}^N {f_\mu \exp [2\pi i(h_1 \bar x_1^\mu + h_2 \bar x_2^\mu + h_3 \bar x_3^\mu )]} &\cr & \quad\quad \times \textstyle\int\limits_0^1 {\exp \{2\pi i[ h_1 u_1^\mu + h_2 u_2^\mu + (h_3 + h_4 k_3 )u_3^\mu + h_4 \bar x_4^\mu ] \}} \,{\rm d}\bar x_4^\mu & \cr & \quad\quad + \exp (h_4 \pi i)\textstyle\sum\limits_{\mu = 1}^N {f_\mu \exp [2\pi i(h_1 \bar x_1^\mu - h_2 \bar x_2^\mu + h_3 \bar x_3^\mu )]} &\cr & \quad\quad \times \textstyle\int\limits_0^1 {\exp\{ 2\pi i[ h_1 u_1^\mu - h_2 u_2^\mu + (h_3 + h_4 k_3 )u_3^\mu + h_4 \bar x_4^\mu ]\}\,{\rm d}\bar x_4^\mu .} &\cr&&(2.5.3.10)} ]Thus, the following phase relations are obtained between the two structure factors:[\eqalignno{F(h_1 h_2 h_3 h_4 ) &= F(h_1 \bar h_2 h_3 h_4 ) \quad\hbox{for }h_4 \hbox{ even},&\cr F(h_1 h_2 h_3 h_4 )&= -F(h_1 \bar h_2 h_3 h_4 ) \quad\hbox{for }h_4 \hbox{ odd}. &\cr &&(2.5.3.11)}]These relations are analogous to the phase relations between the two structure factors for an ordinary three-dimensional crystal with a glide plane. The relations imply that dynamical extinction occurs for the glide planes and screw axes of the (3 + 1)-dimensional crystal with an infinite dimension along the direction of the incommensurate modulation wavevector k. Terauchi et al. (1994[link]) showed that approximate dynamical extinction occurs for an incommensurate crystal of finite dimension.

[Figure 2.5.3.20]

Figure 2.5.3.20 | top | pdf |

(a) Mirror symmetry of modulation waves ([_{\;1}^m ]) τ4 = 0. (b) Glide symmetry of modulation waves ([_{\,s}^m ]) τ4 = ½. The wave number vector of modulation is k3c*.

Fig. 2.5.3.21[link](a) and (b) illustrate a spot diffraction pattern and a CBED pattern, respectively, expected from a modulated crystal with a (3 + 1)-dimensional space group [\mathop P\nolimits_{\;\,\,{\kern 1pt} \bar 1\,s\,\bar 1}^{P2mm} ] (k = k3c*) at the [100] incidence. The large and small spots in Fig. 2.5.3.21[link](a) designate the fundamental (h4 = 0) and incommensurate reflections (h4 ≠ 0), respectively. The 00h3h4 (h4 = odd) reflections shown by crosses are kinematically forbidden by the glide plane ([_{\,s}^m ]) perpendicular to the b axis. Umweganregung paths a, b and c in the ZOLZ to a kinematically forbidden reflection are drawn. The two paths a and b are geometrically equivalent with respect to the line mm perpendicular to the b axis. Since every Umwegan­regung path to a kinematically forbidden reflection contains an odd number of F(0h2,ih3,ih4,i) with odd h4,i, the following equation is obtained.[\eqalignno{&F(0h_{2,1}h_{3,1}h_{4,1})F(0h_{2,2}h_{3,2}h_{4,2})\ldots F(0h_{2,n}h_{3,n}h_{4,n})\quad\hbox{for path }a&\cr &\quad=-F(0\bar h_{2,1}h_{3,1}h_{4,1})F(0\bar h_{2,2}h_{3,2}h_{4,2})\ldots F(0\bar h_{2,n}h_{3,n}h_{4,n})&\cr&\quad\quad\hbox{for path }b,&(2.5.3.12)}]where [\textstyle\sum_{i = 1}^n {h_{2,i} } = 0], [\textstyle\sum_{i = 1}^n {h_{3,i} } = h_3 ] and [\textstyle\sum_{i = 1}^n {h_{4,i} } = h_4 ] (h4 = odd).

[Figure 2.5.3.21]

Figure 2.5.3.21 | top | pdf |

(a) Umweganregung paths a, b and c to the 0011 forbidden reflection. (b) Expected dynamical extinction lines are shown, the 0011 reflection being excited. The wave number vector of modulation is k3c*.

When reflection 00h3h4 (h4 = odd) is exactly excited, the two paths a and c are symmetric with respect to the bisector m′–m′ of the diffraction vector of the reflection and have the same excitation error. The waves passing through these paths have the same amplitude but different signs. Thus the following relation is obtained.[\eqalignno{&F(0h_{2,1}h_{3,1}h_{4,1})F(0h_{2,2}h_{3,2}h_{4,2})\ldots F(0h_{2,n}h_{3,n}h_{4,n})\quad\hbox{for path }a&\cr &\quad =-F(0\bar h_{2,n}h_{3,n}h_{4,n})F(0\bar h_{2,n-1}h_{3,n-1}h_{4,n-1})\ldots F(0\bar h_{2,1}h_{3,1}h_{4,1})&\cr&\quad\quad\hbox{for path }c,&(2.5.3.13)}]where [\textstyle\sum_{i = 1}^n {h_{2,i} } = 0], [\textstyle\sum_{i = 1}^n {h_{3,i} } = h_3 ] and [\textstyle\sum_{i = 1}^n {h_{4,i} } = h_4 ] (h4 = odd).

Therefore, dynamical extinction occurs in kinematically forbidden reflections of incommensurate crystals. Fig. 2.5.3.21[link](b) schematically shows the extinction lines in odd-order incommensurate reflections, where the 0011 reflection is exactly excited.

We consider the dynamical extinction from Sr2Nb2O7 whose space group is [\mathop P\nolimits_{\;\;\; \bar 1{\kern 1pt} \,s{\kern 1pt} \,\bar 1}^{C{\kern 1pt} mc2_1 } ]. The glide plane ([_s^c ]) is perpendicular to the b axis with a glide vector (c + a4)/2. The wave number vector of the modulation is k = (½ − δ)a*. (Since space-group notation Cmc21 is broadly accepted, the direction of the modulation is taken as the a axis.) The reflections h10h3h4 with h3 + h4 = 2n + 1 (n = integer) are kinematically forbidden. Fig. 2.5.3.22[link] shows a schematic diffraction pattern of Sr2Nb2O7 at the [001] incidence. The large and small spots indicate the fundamental (h4 = 0) and incommensurate (h4 ≠ 0) reflections, respectively. Umweganregung paths a and b to the kinematically forbidden 0001 reflection via a fundamental reflection in the ZOLZ are drawn.

[Figure 2.5.3.22]

Figure 2.5.3.22 | top | pdf |

Schematic diffraction pattern at the [001] incidence of Sr2Nb2O7. Umweganregung paths a and b via fundamental reflections to the 0001 incommensurate reflection. Large and small spots denote fundamental and incommensurate reflections, respectively. The wave number vector of modulation is k = (½ − δ)a*.

Fig. 2.5.3.23[link](a) shows a spot diffraction pattern of the incommensurate phase of Sr2Nb2O7 taken with the [001] incidence at 60 kV. The incommensurate reflections in which dynamical extinction lines appear at this incidence are those with the indices h1,even00h4,odd because h3 = 0 and h1 + h2 = 2n due to the lattice type C of the average structure.

[Figure 2.5.3.23]

Figure 2.5.3.23 | top | pdf |

Diffraction pattern of Sr2Nb2O7 taken with [001] incidence at 60 kV. (a) Spot diffraction pattern. Kinematically forbidden 0001 and [200\bar 1] incommensurate reflections exhibit definite intensity. (b) Zone-axis CBED pattern showing dynamical absence of 0001 and [200\bar 1] incommensurate reflections. (c) CBED pattern taken at an incidence with a small tilt from the zone axis to the b* direction. The kinematically forbidden incommensurate reflections have intensity due to incomplete cancellation of two waves through the Umweganregung paths. The wave number vector of modulation is k = (½ − δ)a*.

The reflections in the four columns indicated by black arrowheads are incommensurate reflections. The reflections 0001, [000\bar 1], [200\bar 1] and [\bar 2001] designated by white arrowheads are kinematically forbidden but exhibit certain intensities, which are caused by multiple diffraction. Other reflections are fundamental reflections due to the average structure.

Fig. 2.5.3.23[link](b) shows a CBED pattern corresponding to Fig. 2.5.3.23[link](a), taken from a specimen area 3 nm in diameter. The excitation errors of two Umweganregung paths a and b are the same at this electron incidence. The reflections 0001, [000\bar 1], [200\bar 1] and [\bar 2001] indicated by white arrowheads show no intensity. Dynamical extinction does not appear as a line in the present case because the width of the extinction line exceeds the disc size of the reflections. Fig. 2.5.3.23[link](c) shows a CBED pattern taken at an incidence slightly tilted toward the b* axis from that for Fig. 2.5.3.23[link](b) or the [001] zone-axis incidence. The excitation errors are no longer the same for the two Umweganregung paths. Thus, it is seen that the kinematically forbidden reflections indicated by white arrowheads have intensities due to incomplete cancellation of waves coming through different paths, which is an additional proof of the dynamical extinction.

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