A systematic method to identify the space group from PED and CBED patterns part I - theory

Ultramicroscopy 121 (2012) 42–60

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A systematic method to identify the space group from PED and CBED patterns part I - theory J.P. Morniroli a,n, G. Ji b, D. Jacob b a b

Universite´ Lille 1 and Ecole Nationale Supe´rieure de Chimie de Lille, Cite´ Scientifique, 59655 Villeneuve d’Ascq, France Unite´ Mate´riaux et Transformations(UMET) CNRS UMR 8207, Universite´ Lille 1, Bˆ at C6, 59655 Villeneuve d’Ascq, France

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 January 2012 Received in revised form 18 April 2012 Accepted 28 April 2012 Available online 19 May 2012

This systematic method allows the unambiguous identification of the extinction and diffraction symbols of a crystal by comparison of a few experimental Precession Electron Diffraction (PED) patterns with theoretical patterns drawn for all the extinction and diffraction symbols. The method requires the detection of the Laue class, of the kinematically forbidden reflections and of the shift and periodicity differences between the reflections located in the First-Order Laue Zone (FOLZ) with respect to the ones located in the Zero-Order Laue Zone (ZOLZ). The actual space group can be selected, among the possible space groups connected with each extinction symbol or diffraction symbol, from the identification of the point group. This point group is available from observation of the 2D symmetry of the ZOLZ on Convergent-Beam Electron Diffraction (CBED) patterns. & 2012 Elsevier B.V. All rights reserved.

Keywords: Electron diffraction Precession electron diffraction Space groups

1. Introduction In a previous paper [1], it was reported that a few possible space groups of a crystal can be inferred from electron microdiffraction Zone-Axis Patterns (ZAP) through the identification of the extinction symbol. The extinction symbol, introduced by Buerger [2] for X-ray diffraction, connects the integral, zonal and serial reflection conditions due to the Bravais lattices, the glide planes and the screw axes with the possible space groups. There are 219 different extinction symbols (if for the monoclinic, orthorhombic and trigonal crystal systems, all the cell choices and settings are considered) written in agreement with the HermannMauguin notation. There are also 242 diffraction symbols which consist of the Laue class followed by the extinction symbol. The extinction and diffraction symbols are listed, for each crystal system, in the International Tables for Crystallography [3]. A systematic method was proposed based on observations of Whole Pattern (WP) ZAPs. A WP must display at least Zero-Order Laue Zone (ZOLZ) and First-Order Laue Zone (FOLZ) reflections. It took into account the ‘net’ symmetry and the shift and periodicity differences between the reflections located in the FOLZ with respect to the ones located in the ZOLZ. For brevity we will use ‘FOLZ/ZOLZ shift’ to refer to the relative displacement between the net of reflections in the ZOLZ and the net of reflections in the FOLZ. Similarly ‘FOLZ/ZOLZ periodicity difference’ is used to refer to the difference in spacing between the two nets. The ‘net’

n

Corresponding author. Tel.: þ33 (0)3 20 64 02 90; fax: þ 33 (0)3 20 33 61 48. E-mail address: [email protected] (J.P. Morniroli).

0304-3991/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2012.04.008

symmetry considers only the position of the reflections on a diffraction pattern. It is easy to observe experimentally since it does not require a perfect ZAP alignment. It is connected with the seven crystal system via the crystal lattices (Table 1). The FOLZ/ ZOLZ shifts are connected with the Bravais lattices and the FOLZ/ ZOLZ periodicity differences with the presence of glide planes. Some difficulties were encountered with this method: – The screw axes were not identified because the corresponding kinematically forbidden reflections are usually not distinguishable from the allowed reflections due to the double diffraction phenomenon (dynamical interactions). Thus, the extinction symbols which contain screw axes were not detectable. – It was sometimes difficult to observe enough reflections in the FOLZ to surely infer the FOLZ/ZOLZ shifts and periodicity differences.

Furthermore, the ‘‘ideal’’ symmetry which considers both the position and the intensity of the reflections was usually not available unless the ZAP is extremely well aligned. This symmetry is important since, depending on its measurement accuracy, it is connected with the Laue classes [4] or with the point groups of the crystal [5]. Despite these difficulties, this systematic method was successfully applied by many authors to different crystal structures, such as Bi14O20(SO4) [6], Ba8Co7O21 [7], Ni10Sn5P3 [8], Mn-N nitride in Fe-Mn-N alloys [9], SmMn2GeO7 [10] and Th4Fe3Al32 [11], etc. Electron precession proposed by Vincent and Midgley [12] is a new electron diffraction technique which has many advantages with respect to conventional Selected-Area Electron Diffraction

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Table 1 Connection between the ZOLZ and WP ‘net’ symmetries of spot, microdiffraction and precession patterns and the 7 crystal systems. In this table, the ZOLZ symmetries are given between parentheses. ‘Net’ symmetries of spot, microdiffraction and PED patterns. The ‘net’ symmetries are connected with the 7 crystal systems. ‘Net’ symmetry (ZOLZ) WP (6mm) 6mm (6mm) 3m (4mm) 4mm (2mm) 2mm

[0001] [0001] [100] [010] [001]

(2mm)

m

[u0w]

[uv0]

(2)

2

(2)

1

[uvw]

[010] [01¯0] [uvw] Unique axis b

[001] [001¯] [uvw] Unique axis b

Triclinic

Monoclinic

Crystal system

[0vw] [u0w] [uv0]

[001] [100] [010] [110] [11¯0] [u0w] [0vw] [uv0] [uuw]

/112¯0S /11¯00S

/uu¯0wS

/111S /001S /110S /001S for Pa3¯

/uvt0S /uu¯0wS /uutwS

/uv0S /uuwS

/uvtwS hP hexagonal lattice Trigonal or Hexagonal

/uvwS

/112¯0S [uvw]

[uvw]

/uvtwS hR rhomboedral lattice

Orthorhombic

Tetragonal

Trigonal

(SAED) or electron microdiffraction. Three features are especially useful for the present analysis: – The precession patterns display a larger number of reflections in the ZOLZ but also in the FOLZ and this number increases with the precession angle a. As a result, the FOLZ/ZOLZ shifts and periodicity differences are easy to detect and to identify without ambiguity [13]. – At large precession angle (a 421), a ‘few beam’ or a ‘systematic row’ behavior prevails during the precession movement of the incident beam [12,14]. This means that the double diffraction routes to kinematically forbidden reflections (and also to the allowed reflections) are strongly reduced. Therefore, it becomes possible to identify the forbidden reflections due to the glide planes but also those connected with screw axes. Note that the kinematically forbidden reflections due to special Wyckoff positions remain visible on precession patterns for all precession angles [12,14,15]. The diffracted intensities are integrated over a large domain on each side of the exact Bragg orientation so that a precession pattern looks always well aligned even if the crystal is not perfectly aligned at the zone axis. As a result, the diffracted intensities can be taken into account so that the ‘‘ideal’’ symmetry also becomes available. In principle, this ‘‘ideal’’ symmetry is connected with the point group as reported by Buxton et al. [5]. Nevertheless, for the noncentrosymmetric point groups, the identification of the actual ‘‘ideal’’ symmetry is delicate. These point groups exhibit, for some specific ZAPs, a symmetry lowering with respect to the corresponding centrosymmetric point groups (the Laue classes). This symmetry lowering produces weak differences of intensity between some rotation- or mirror-related reflections. Despite this weak effect, typical differences of aspects are produced inside the disks of ZOLZ CBED patterns (see for example, the encircled disks in Fig. 1a and b). The precession patterns being made of spots (with a parallel incident beam) or small diameter disks (with nearly parallel and focused incident beam) displaying a uniform intensity do not reveal clearly these very weak differences of intensity (see the examples in Fig. 1a0 and b0 ) and therefore displays the symmetry of the centrosymmetric point group i.e., the symmetry of the Laue class. [4]. Thus, it is more realistic to

Cubic

consider that the ‘‘ideal’’ symmetry of the precession patterns is connected with the 11 Laue classes instead of the 32 point groups as given in Table 2. Note that with unconventional precession patterns obtained with a small precession angle and without ‘descan’ it becomes possible to detect both the centro and non centrosymmetric point groups [4] but this method requires some experimental abilities and is more complex to perform than CBED.

2. Description of the method Taking into account these new and original PED features, the systematic method proposed in 1992 [1] which was based on microdiffraction can be greatly improved as described in the present paper. This improved method involves three steps: – First step: identification of the crystal system, via the Laue class, from PED patterns, – Second step: identification of a few possible space groups, via the extinction symbol or the diffraction symbol1, from PED patterns, – Third step: selection of the actual space group from CBED patterns.

2.1. First step: identification of the crystal system, via the Laue class, from PED patterns In the method described previously, the crystal system was identified from observations of the microdiffraction patterns displaying the highest WP ‘‘net’’ symmetry (upper part of 1 The diffraction symbol (Laue class followed by the extinction symbol) is useful when the possible space groups associated with an extinction symbol belong to different Laue classes. This could occur for some tetragonal (Laue classes 4/m and 4/mmm), trigonal (Laue classes 3¯ and 3¯/m), hexagonal (Laue classes 6/m and 6/mmm) and cubic (Laue classes m3¯ and m3¯m) extinction symbols. In these cases, the number of possible space groups is reduced if the diffraction symbol is identified. An example will be given in Section 2.2.2. In the present paper, we will not consider the monoclinic and orthorhombic diffraction symbols since the monoclinic and orthorhombic crystal systems have only one Laue class (2/m and mmm, respectively).

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Fig. 1. Examples of simulated CBED and PED patterns of non centrosymmetric crystals. a et a’ - [110] ZAP of a crystal belonging to the noncentrosymmetric cubic point group 4¯3m. The symmetry (m) typical of this point group is evidenced by the difference of aspects between the encircled mirror-related CBED disks. The corresponding PED pattern displays a (2mm) ‘ideal’ symmetry typical of the Laue class m3¯m. b et b’ - [111] ZAP of a crystal belonging to the noncentrosymmetric cubic point group 432. The symmetry (3m) typical of this point group is evidenced by the difference of aspects between the encircled rotation-related CBED disks. The corresponding PED pattern displays a (6mm) ‘ideal’ symmetry typical of the Laue class m3¯m. Patterns simulated with Jems [18]. The precession patterns were simulated with a precession angle of 31.

Table 3). Experimentally, the ‘‘net’’ symmetry which is connected to the crystal system is very easy to identify from any spot pattern since it is not very sensitive to a crystal misorientation. Nevertheless, the method had main disadvantages: – except for the cubic and tetragonal crystal systems, it required observation of at least some FOLZ reflections which may not be readily accessible especially with thick crystals using these methods. – in the case of the hexagonal lattice, it did not allow to make the distinction between the hexagonal and the trigonal crystal systems.

As indicated in the introduction, the ‘‘ideal’’ symmetry of PED patterns is connected with the Laue class and it can be easily identified. In addition, the PED patterns display more FOLZ reflections. Therefore, it is recommended to identify the crystal systems, via the Laue class, by observing the PED WP displaying the highest ‘‘ideal’’ symmetry and by using the lower part of Table 3. Note that the Laue classes m3¯m and 4/mmm require only the observation of the ZOLZ. Note also that the triclinic, monoclinic and orthorhombic crystal systems which have only one Laue class can be simply identified from their ‘net’ symmetries. These features constitute a great experimental simplification.

2.2. Second step: identification of a few possible space groups via the extinction and diffraction symbols from PED patterns The extinction symbol requires the knowledge of three features: the Bravais lattice, the glide planes and the screw axes. The diffraction symbol requires also the knowledge of the Laue class which can be identified from the ‘ideal’ symmetry of PED patterns as indicated in the previous section. The Bravais lattice can be inferred from the shifts between the Laue zones. The glide planes can be identified from the ZOLZ/FOLZ periodicity differences. The ZOLZ extinctions due to both glide planes and screw axes can be identified from PED patterns [14]. It is pointed out that the forbidden reflections displaying a Gjønnes and Moodie (GM) line [16] can also be identified from CBED patterns. The present method is based on observations of the smallest possible number of ZAPs. Therefore, the zone axes are selected in order to give typical and unambiguous information about the Bravais lattice, the glide planes and the screw axes. For the Bravais lattice determination, the useful ZAPs are directed along the main symmetry elements, i.e. along the 2, 3, 4 and 6 fold-rotation axes, depending on the crystal system. The axial a, b, c, diagonal n and diamond d glide planes are parallel to specific (hkl) lattice planes and they affect only the reflections of the type hk0, h0l, 0kl or hhl (zonal reflection conditions). General hkl reflections are never involved. This means that the corresponding

J.P. Morniroli et al. / Ultramicroscopy 121 (2012) 42–60

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Table 2 Connection between the ZOLZ and WP ‘ideal’ symmetries of precession electron diffraction patterns and the 11 Laue classes. ‘Ideal’ symmetries of PED patterns. The ‘ideal’ symmetries are connected with the 11 Laue classes. )Ideal * symmetry (ZOLZ) WP (6mm) 6mm (6mm) 3m (6) 6 (6) 3 (4mm) 4mm (4) 4 (2mm) 2mm

[0001] [0001]

/111S

[0001] [0001] /111S

[0001]

/001S

[001] [001] [100] [010] [001]

(2mm)

m

[u0w]

[uv0]

(2)

2

(2)

1

[010] [01¯0] /uvwS

[001] [001¯] /uvwS

[uvw]

Laue class

1¯

Crystal system

Tricl.

12/m1 112/m 2/m Unique Unique Axis b Axis c Monoclinic

[uv0] [u0w] [0vw]

[100] [010] [110] [11¯0] [uv0]

[100] [010] [110] [11¯0] [u0w] [uv0] [uuw]

[uvw]

[u0w] [uuw] [uvw]

[uvw]

/112¯0S /11¯00S /uvt0S /uutwS % /uu0wS /uvtwS

mmm

4/m

4/mmm

3¯

Ortho.

Tetragonal

forbidden hkl nodes of the reciprocal lattice are all located in the (uvw)n layer of the reciprocal lattice which is parallel to the (hkl) glide plane and which contains the origin On of the reciprocal lattice (Fig. 2a). The other (uvw)n layers with na0 are not affected. To identify the glide planes, the easiest way consists in observing the [uvw] zone axis which is perpendicular to the (hkl) glide plane. In this case, the ZOLZ is directly connected with the (uvw)n layer with n¼0 and the FOLZ with the (uvw)n layer with n¼1 so that a typical FOLZ/ ZOLZ periodicity difference is observed. Double diffraction routes to these forbidden reflections are not possible by double diffraction of ZOLZ reflections (although double diffraction into those reflections does occur when double diffraction involves reflections in higherorder zones, such contributions are extremely weak). Thus this identification is easy and reliable since it is hard to mistake the change in spacing between the ZOLZ and the High-Order Laue Zones (HOLZs). The 21, 31, 32, 41, 42, 43, 61, 62, 63, 64 and 65 screw axes are directed along specific [uvw] rows of the direct lattice. They affect only the h00, 0k0 or 00l nodes (serial reflection conditions) located along the hkl reciprocal row simultaneously parallel to the screw axes and containing the origin On of the reciprocal lattice (Fig. 2b). Therefore, the forbidden reflections due to screw axes can be observed on any zone axis perpendicular to the screw axis. In this case, the ZOLZ displays the hkl systematic row containing the forbidden reflections. Experimentally, the most symmetrical zone axes are usually selected. Taking into account these features, the useful zone axes which give useful information about the Bravais lattices, the glide planes and the screw axes can be deduced. They are listed, for each crystal system, in Table 4. We have simulated, for 201n extinction symbols associated with 221n diffraction symbols, the required zone-axis WP. n The numbers of extinction and diffraction symbols are different from the numbers 219 and 242 given in the International

Trigonal

/11¯00S /uu¯0wS

/112¯0S /uutwS

/112¯0S

/11¯00S

/uvt0S /uutwS /uvtwS

/uvt0S % /uu0wS /uvtwS

3¯m1 3¯m

3¯1m

/112¯0S /11¯00S

/001S for Pa3¯

/110S

/112¯0S /11¯00S /uvt0S

/uvt0S /uutwS /uu¯0wS

/110S /uv0S

/uu0S /uuwS

/uutwS % /uu0wS /uvtwS

/uvtwS

/uuwS /uvwS

/uvwS

6/m

6/mmm

m3¯

m3¯m

Hexagonal

Cubic

Table for Crystallography because we have considered only the two most used descriptions for the monoclinic crystal system i.e., the b and c unique axis descriptions and, for the rhombohedral lattice, only the description by a triple obverse hexagonal unit cell. The triclinic crystal system is not studied since its corresponding diffraction patterns are not typical enough to be recognized. PED kinematic simulations were performed with the software ‘Electron Diffraction’ [17] while the dynamical PED and CBED patterns were simulated with the ‘‘Jems’’ software from P. Stadelmann [18]. The symbols used for the diffracted reflections on these simulations are given in Table 5. All these simulations are available from the web site: www.Electron-Diffraction.fr Three examples are given in the following section. They will be used in the part II of this paper dedicated to the experimental applications.t 2.2.1. First example This first example concerns the monoclinic crystal system. The monoclinic system is rather complex due to three possible descriptions named ‘unique axis a’, ‘unique axis b’ or ‘unique axis c’ depending on the lattice parameter a, b or c which is selected along the main monoclinic two-fold rotation axis (Fig. 3a, b and c). In addition, for each description, three different monoclinic cells can be selected as shown in Fig. 3d which corresponds to the ‘unique axis b’; the cell with b o1201 is usually chosen. In this paper, we take into account only the two unique-axis b and c descriptions. They are referred by means of the subscripts b and c. For the sake of simplicity, only the most frequently used description, the ‘unique axis b’ description, is considered in the present example. Let us consider the extinction symbol C1c1b (with diffraction symbol 12/m1 C1c1b) in agreement with the two possible space groups C1c1 and C12/c1. According to Table 4, the [010]b or the

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Table 3 Deduction of the crystal system and the Laue class from observation of the zone-axis PED pattern displaying the highest ‘net’ and ‘ideal’ symmetries. This table is deduced from Tables 1 and 2.

[01¯0]b WP and some [u0w]b ZOLZ patterns (for example, [100]b, [101]b and [001]b) must be considered for the identification of the monoclinic extinction symbol. Note that for the monoclinic crystal, the [010]b and the [01¯0]b patterns do not superimpose but are mirror related. The selected patterns, located around the [010]b and [01¯0]b zone axes are schematically drawn and arranged in a coherent way in Fig. 4a and b, respectively. To simplify the drawings of the [010]b and [01¯0]b WP, only the parts of the FOLZ located along the main ZOLZ directions are shown. On these WPs: – The FOLZ/ZOLZ periodicity differences are characterized by means of shaded parallelograms which represent the ‘unit cell’ of the reflections located in the ZOLZ and in the FOLZ. On the present example, the ZOLZ parallelogram is two times larger along one direction than the FOLZ one indicating the occurrence of a periodicity difference due to the a glide plane (in this case an axial c glide plane). The ZOLZ and the FOLZ parallelograms must have parallel sides. – The FOLZ reflections located along the main ZOLZ directions are circled in order to indicate the absence of FOLZ/ZOLZ shift along these directions. On Fig. 4a and b, such circled FOLZ reflections are located along three main ZOLZ directions while

there is ZOLZ/FOLZ shift along a fourth one. This shift is due to the C Bravais lattice. – The kinematically forbidden reflections are indicated by a large white circle. Some of them may display a dark line which corresponds to a GM line [16] (line of dynamical absence). In the present example, the 001b (and 003b, 005by) and 001¯b (and 003¯b, 005¯by) reflections in the [100]b ZAP are forbidden and they display a GM line. These forbidden reflections alternate with the allowed reflections. – The hkl indices of the reflections as well as the bnb angle are indicated. They can be used to define the values of the ab, bb, cb and bb lattice parameters.

Each pattern is given a specific notation. In the example shown in Fig. 4a and b, the features of the [010]b and [01¯0]b WPs (FOLZ/ZOLZ shift and periodicity difference) are named mS[010]b4 and mS[01¯0]b4, respectively. The letter m stands for monoclinic, the letter S for the A, B or I Bravais lattice and the number (4) means that other kinds of [010]b and [01¯0]b patterns (1, 2, 3, 4, 5 and 6) are also encountered for the other monoclinic extinction symbols (see Fig. 5a, b and c). In the same way, the patterns of the three other zone axes [100]b, [101]b and [001]b of the type [u0w]b are named m[u0w]b2per or m[u0w]b4. The

J.P. Morniroli et al. / Ultramicroscopy 121 (2012) 42–60

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Fig. 2. Arrangement of the forbidden nodes in the reciprocal lattice. a - Forbidden nodes due to a glide plane parallel to a set of (hkl) lattice planes. The corresponding forbidden reflections are all located in the (uvw)n layer of the reciprocal lattice containing the origin On of the reciprocal lattice. b - Forbidden nodes due to a screw axis directed along a [uvw] row. The corresponding forbidden reflections are all located along the hkl reciprocal row containing the origin On.

Table 4 Zone axes useful for the identification of the extinction symbol as a function of the crystal system. Zone axes useful for the identification of the extinction symbol Crystal System

Bravais Lattices

Possible glide planes

Possible screw axes

Zone axes useful for the identification of the extinction symbol

Triclinic Monoclinic Unique axis b Monoclinic Unique axis c Orthorhombic

aP mP mS (mC, mA, mI)

– c, a, n // (010)

– 21 // [010]

mP mS (mA, mB, mI)

a, b, n // (001)

21 // [001]

oP oS (oC, oA, oB) oI oF tP tI

b, c, n, d // (100) c, a, n, d // (010) a, b, n, d // (001)

21 // [100] 21 // [010] 21 // [001]

– [010]b WP or [01¯0]b WP [001]c WP or [001¯]c WP [100] WP

b, c, n // (100) c, a, n // (010) a, b, n // (001) c, n, d // (110), (11¯0)

Trigonal

hP hR

c // {112¯0} c // {11¯00}

Hexagonal

hP

Cubic

cP cI

21 // [100] 21 // [010] 41 // [001] 42 // [001] 43 // [001] 31 // [001] 32 // [001] 61 // [001] 62 // [001] 63 // [001] 64// [001] 65 // [001] 21, 42, 41, 43 // [100] 21, 42, 41, 43 // [010] 21, 42, 41, 43 // [001]

Tetragonal

cF

b, c, n, d // (100) c, a, n, d // (010) a, b, n, d // (001) c, n, d // (110), (11¯0) a, n, d // (011), (011¯) b, n, d // (101), (1¯01)

subscript ‘per’ means that the GM line is oriented perpendicularly with respect to the [010]b or [01¯0]b zone axis. The small stereogram drawn at the upper right corner of Fig. 4a and b indicates the location of the glide planes and screw axes. Glide

[u0w]b ZOLZ [uv0]c ZOLZ [010] WP

[001] WP

[001] WP

[100] or [010] WP

[110] or [11¯0] WP

/112¯0S WP

/11¯00S WP

[0001] (optional)

/001S WP

/110S WP

/111S (optional)

planes and screw axes are represented by the graphical symbols given in the International Tables for Crystallography [3]. Thus, in the present case, the c glide plane of the C1c1b extinction symbol is parallel to the (010) lattice plane and it is indicated by a dotted circle.

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Table 5 Symbols used to describe the different types of reflections present on the simulated electron diffraction patterns. Diffracted reflections Kinematical forbidden reflections Kinematical forbidden reflection displaying a Gjønnes and Moodie lines FOLZ reflection located along a ZOLZ mirror

FOLZ kinematical forbidden reflection located along a ZOLZ mirror

FOLZ kinematical forbidden reflection displaying a Gjønnes and Moodie line and located along ZOLZ mirror

Fig. 3. Description of the monoclinic crystal system. a - Unique axis a description. a - Unique axis b description. c - Unique axis c description. d - Description of the monoclinic lattice by means of three monoclinic cells in the case of the unique axis b.

The [010]b or [01¯0]b, [001]b, [101]b and [100]b ZAPs discussed here are also indicated by dark points. This stereogram is used to predict the FOLZ/ZOLZ shifts as well as the GM lines. A FOLZ/ZOLZ shift is observed provided the ZAP in question is perpendicular to a glide plane (Fig. 6a). This is the case for the [010]b and [01¯0]b ZAPs. About the GM lines, in the general case, they are observed on ZOLZ patterns whose ‘unit cell’ is a non-centered rectangle or square when: – a glide plane a, b, c, n or d is parallel to the incident beam provided the associated glide vector is not parallel to the incident beam (Fig. 6b)

– a 21, 41, 43, 61, 63 or 65 screw axis is perpendicular to the incident beam (Fig. 6c). The stereogram indicates that these conditions are observed for the non-centered [100]b pattern where GM lines are actually present since a cb glide plane is parallel to the [100]b zone axis (Fig. 6d). The table at the bottom of Fig. 4a and b summarizes all the features of the C1c1b extinction symbol. This extinction symbol is distinguished from the 14 other monoclinic extinction symbols by means of the number mS2a and mS2a0 . The hyphen (‘) refers to the patterns around the [01¯0]b zone axis and the subscript (a) is used to make the distinction with two other

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extinction symbols mS2b and mS2c. As a matter of fact, the three extinction symbols mS2a, mS2b and mS2c correspond to the three possibilities of monoclinic cell choice mentioned earlier (Table 6). Note that a [010]b pattern can also be considered as a [01¯0]b pattern and vice versa. This comes down to exchange the ab and cb lattice parameters and to reverse the bb axis. If all the monoclinic extinction symbols are taken into account, then ten different types of [010]b or [01¯0]b WP and five types of [u0w]b ZOLZ patterns are obtained. The ones corresponding to the S Bravais lattices are given, in accordance with their notation, in Fig. 5a–d. The patterns for the P Bravais lattice are available from the web site www.Electron-Diffraction.fr Table 6 connects all these [010]b, [01¯0]b, and [u0w]b types of diffraction patterns with their corresponding extinction symbols and with their possible space groups. Some types of diffraction

49

patterns are typical of a unique extinction symbol (shaded cells). In this case a single pattern is required for the identification of the extinction symbol and this fact constitutes a great experimental simplification. In Table 6, the extinction symbols associated with the three monoclinic cell choices are separated by dotted lines. Table 6 is used in conjunction with Fig. 5 for the experimental identification of the extinction symbol (see part II of this paper for an application).

2.2.2. Second example The second example concerns the extinction symbol P- - (with two diffraction symbols m3¯ P- - - and m3¯m P- - -) belonging to the cubic crystal system and in agreement with five possible space groups: P23, Pm3¯, P432, P4¯3m and Pm3¯m.

Fig. 4. Schematic description of the zone axes typical of the C1c1b (unique axis b) A11ac (unique axis c) extinction symbol belonging to the monoclinic crystal system. The cb ac glide plane and the selected ZAPs are described by means of a bold circle and dark spots, respectively on a stereographic projection. The CBED and PED symmetries are also indicated. a – [010] ZAP and some surrounding [u0w] ZAPs. b – [01¯0] ZAP and some surrounding [u0w] ZAPs.

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Fig. 4. (continued)

For the cubic crystal system, two WP zone axes are required: /001S and /110S (Table 4). The /111S zone axis is optional and is mainly useful for the identification of the crystal system and the Laue class (see Table 3). The schematic drawings of the [001], [101] and [111] WP zone axes are displayed coherently on Fig. 7. To simplify the drawings, only the FOLZ reflections located on or close to the ‘net’ ZOLZ mirrors are shown (‘net’ mirrors are concerned with the positions of the reflections and not with their intensity). – The FOLZ/ZOLZ periodicity differences are deduced from the shaded square (for the [001] ZAP), the rectangular (for the [101] ZAP) or the lozenge (for the [111] ZAP) ‘unit cells’ drawn in the FOLZ and in the ZOLZ. On this example, there is no FOLZ/ ZOLZ periodicity difference on any of these three ZAPs. – The FOLZ/ZOLZ shifts are identified by means of the circled FOLZ reflections. Circled FOLZ reflections located along a ‘net’ ZOLZ mirror indicate the absence of shift along this direction. This is the case on the [001] pattern where circled FOLZ

reflections are present on the two m1 and m2 ‘net’ mirrors. Contrarily, a FOLZ/ZOLZ shift is observed on the [101] and [111] patterns along the m1 mirrors. – As for the previous monoclinic example, the kinematically forbidden reflections (potentially displaying a GM line) are indicated by a large open circle. No such forbidden reflections are present for this extinction symbol. The notations cP/001S1 and cP/110S1 were attributed to the features of the [001] and [101] ZAPs. The letters c and P refer to the cubic crystal system and to the P Bravais lattice, respectively. A specific double notation cP/111S1a - (cP/111S1) is used for the features of the [111] ZAP. The exponent (a) means that other /111S patterns with b, c and d exponents also exists for the other extinction symbols and these patterns differ from each other only by the kinematically forbidden reflections located in the FOLZ, the forbidden reflections located in the ZOLZ being identical (see Fig. 9c). If these FOLZ forbidden reflections are not taken

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into account, then all the patterns with the notations cP/111S1a, cP/111S1b, cP/111S1c and cP/111S1d look similar and therefore are all noted between parentheses (cP/111S1). This distinction is based on two experimental considerations: – the identification of the forbidden reflections located in the ZOLZ is much easier than the identification of the ones located in the FOLZ. Most of the time, the FOLZ forbidden reflections are not required for the extinction symbol analysis. – the observation of the forbidden reflections located in the FOLZ may allow the identification of the extinction symbol from a unique diffraction pattern. This feature is especially useful when the required zone axes are located at 901 from each other and are not easily obtained from the same specimen. This is especially the case for the orthorhombic, tetragonal, trigonal and hexagonal crystal systems. This also means that it is not useful to observe the FOLZ forbidden reflections for the patterns which are described by a simple notation. For this extinction symbol P- - -, the two diffraction symbols m3¯ P- - - and m3¯m P- - - exist since the five possible space groups belong to the two different Laue classes m3¯ (P23 and Pm3¯) and m3¯m (P432, P4¯3m and Pm3¯m). These Laue classes are indicated in Fig.7. If both the Laue class and the extinction symbol are experimentally identified, then the diffraction symbol is known and the five possibilities of

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space groups are reduced to two for the m3¯ P- - - diffraction symbol and to three for the m3¯m P- - - diffraction symbol. One practical example is given in the second part of this article. 2.2.3. Third example This last example is also taken from the cubic crystal system and concerns a special extinction symbol: the Pa- extinction symbol (with diffraction symbol m3¯ Pa- -) which is in agreement with a unique space group: Pa3¯ (with full notation P21/a3¯). As in the previous case, the schematic drawings of the [001], [101] and [111] ZAPs are given on Fig. 8. The [001] ZAP displays the following features: – a FOLZ/ZOLZ periodicity difference is observed since the shaded rectangular ‘unit cell’ which characterizes the ZOLZ reflections is two times larger than the square ‘unit cell’ of the FOLZ reflections. This is due to the axial a glide plane. Note that the ‘net’ symmetry of this [001] ZOLZ is only (2mm) and not (4mm) as is the case for all the other cubic extinction symbols. This singularity is connected with the lack of a 4 foldrotation axis for the Pa3¯ space group. – the absence of FOLZ/ZOLZ shift. – the 010, 030, 050, 01¯0, 03¯0, 05¯0y are ZOLZ forbidden reflections displaying a GM line. Some forbidden reflections with GM lines are also observed in the FOLZ along the two ZOLZ

Fig. 5. Schematic descriptions of the different types of diffraction patterns in the case of the monoclinic crystal system for the S Bravais lattices

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Fig. 5. (continued)

‘net’ mirrors m1 and m2. On the m1 mirror they alternate with allowed reflections while on the m2 mirror all the reflections are forbidden. For the [101] ZAP: – there is no FOLZ/ZOLZ periodicity difference. – a FOLZ/ZOLZ shift is observed along the ‘net’ m1 ZOLZ mirror. – forbidden reflections with GM lines are observed in the ZOLZ along two perpendicular systematic rows. Some forbidden reflections with and without GM lines are also present in the FOLZ.

For the [111] ZAP: – there is no FOLZ/ZOLZ periodicity difference. – a FOLZ/ZOLZ shift is observed along the m1 ZOLZ ‘net’ mirror. – forbidden reflections without GM lines are observed in the ZOLZ and in the FOLZ. On the corresponding stereogram (Fig. 8), the dashed circle and the dashed and dotted lines describe the a, b and c axial glide planes, respectively and the semi-arrows, the 21 screw axes. This stereogram clearly indicates that a FOLZ/ZOLZ periodicity difference occurs only for the three equivalent [100], [010] and

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Fig. 6. Schematic descriptions of the formation of FOLZ/ZOLZ periodicity differences and Gjønnes and Moodie (GM) lines. a - A FOLZ/ZOLZ periodicity difference occurs if a glide plane is perpendicular to the incident beam. b - A GM line is observed if a glide plane a, b, c, n or d is parallel to the incident beam provided the glide vector is not parallel to the incident beam. c - A GM line is observed if a screw axis 21, 41, 43, 61, 63, 65 is perpendicular to the incident beam. d - Arrangement of the glide plane cb with respect to the [100] ZAP in the case of the extinction symbol C1c1b. e - Arrangement of the glide planes and screw axes with respect to the [001] ZAP in the case of the extinction symbol Pa- -. e - Arrangement of the glide planes and screw axes with respect to the [101] ZAP in the case of the extinction symbol Pa- -.

[001] ZAPs since these three ZAPs are perpendicular to the axial glide planes a, b and c, respectively. GM lines are observed both in the /001S and /110S ZAPs. Thus, the [001] ZAP is parallel to the two axial glide planes b and c

and perpendicular to the two - [100] and [010] - 21 screw axes. According to Fig. 6e, GM lines are expected on the forbidden reflections located along the two main perpendicular systematic rows. Actually, they are observed only along one systematic row

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Table 6 Connection between the types of diffraction patterns and the extinction symbols in the case of the monoclinic crystal system. In the column ‘Number’, the hyphen ‘ refers to the patterns around the [01¯0]b zone axis while the subscripts (a) (b) and (c) correspond to the three possibilities of monoclinic cell choice. The shaded areas indicate typical pattern types which can be used alone to identify the extinction symbol.

for the forbidden reflections 010, 030, 01¯0, 03¯0y and are due both to the b glide plane and to the [010] 21 screw axis. The forbidden reflections 100, 300, 100, 300y. located along the other perpendicular systematic row are not visible. This is due to the a glide plane perpendicular to [001] which is responsible of the FOLZ/ZOLZ periodicity difference. The [101] ZAP is parallel to the c glide plane and perpendicular to the [010] 21 screw (Fig. 6f). These features explain the occurrence of GM lines along the two main perpendicular systematic rows. There are 17 different extinction symbols for the cubic crystal system (8 for the P Bravais lattice, 5 for the I lattice and 4 for the F lattice). All the possibilities of /001S, /110S and /111S diffraction patterns as well as their notations are given on Fig. 9a, b and c for the case of the P Bravais lattice. The other I and F Bravais lattices are available from the web site www.Electron-Diffraction.fr. Table 7 makes the connection between these types of patterns, the 17 extinctions symbols and the 35 cubic space groups. 2.3. Third step: selection of the actual space group among the possible space groups Depending on the extinction symbol and/or the diffraction symbol, one or a few possible space groups belonging to different point groups are encountered. Therefore, the selection of the actual space group among the possible ones, requires the

knowledge of the point group. As indicated in Section 1, the point group is connected with the CBED ‘‘ideal’’ symmetries. To this aim, we indicate in the figures describing each extinction symbol, the ZOLZ and WP CBED symmetriesn. If we consider the first example on Fig. 4a and b, the two possible space groups C1c1 and C12/c1 display different ZOLZ CBED symmetries for the [010]b or [01¯0]b and [u0w]b ZAPs. Therefore, the easiest way to identify the actual space group is to obtain a [010], [01¯0] or [u0w] CBED pattern and to observe its ZOLZ ‘‘ideal’’ symmetry. The GM lines can also be detected on the [u0w]b CBED patterns whose ZOLZ ‘unit cell’ is a non-centered rectangle and thus produce additional validation. The precession ‘‘ideal’’ symmetries which are connected with the 11 Laue classes are not adapted to this determination since both space groups C1c1 and C12/c1 belong to the same Laue class. n The ZOLZ CBED symmetries refer only to bi-dimensional (2D) features present in CBED patterns. If three-dimensional (3D) features are also present (HOLZ) lines), they must not be taken into account. The WP CBED symmetries refer to patterns which display at least one HOLZ and therefore contain 3D features (HOLZ reflections and HOLZ lines). The Bright-Field (BF) symmetries are not considered here because they are connected with the 11 Laue classes and not with the 32 point groups. The case of the second example P- - - is more complex since this extinction symbol is in agreement with the five possible

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Fig. 7. Schematic description of the zone axes typical of the P- - - extinction symbol belonging to the cubic crystal system.

space groups P23, Pm3¯, P432, P4¯3m and Pm3¯m belonging to five different point groups 23, m3¯, 432, 4¯3m and m3¯m (Fig. 6). If the diffraction symbol m3¯ P- - - or m3¯m P- - - is known, then the number of possible space groups is reduced to 2 or 3, respectively. Nevertheless the observation of the ZOLZ on the /001S or/and /110S CBED patterns allows the identification of the actual point group and therefore the identification of the space group. One practical example will be given in the second part of this paper.

3. Experimental aspects The strategy developed in this method starts with examinations of PED WPs which are easy to perform and ends up with

ZOLZ CBED patterns which are more delicate to obtain and requires good quality patterns. 3.1. First step The first step involves a search for the zone axis displaying the highest ‘‘ideal’’ symmetry. A WP displaying several FOLZ reflections is required. PED is very well adapted to this step. It is recommended to perform it with a nearly parallel and focused incident beam in order to reduce the size of the diffracted area to a few tens of nanometers. With beam sensitive materials, SAED can also be used provided that a small diffracted area is chosen to avoid artefacts due to thickness or orientation variation within the diffracted area.

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Fig. 8. Schematic description of the zone axes typical of the Pa- - extinction symbol belonging to the cubic crystal system.

Some experimental simplifications are available in the case of the triclinic, monoclinic and orthorhombic crystal systems. They have only one Laue class and therefore can be more simply identified from their ‘net’ symmetry. The identification of the Laue classes 4/mmm and m3¯m requires only the observation of the ZOLZ. 3.2. Second step Once the crystal system is obtained, the identification of the extinction symbol requires the observation of a few specific zone axes, depending on the crystal system (see Table 4). These patterns can be found from their ‘net’ and ‘ideal’ symmetries (see Tables 1 and 2). The FOLZ/ZOLZ shifts and periodicity differences are easily and surely identified from PED patterns performed with a moderate

precession angle in order to obtain several FOLZ reflections and avoid the superimposition of the FOLZ with the ZOLZ. PED patterns are also well adapted to the identification of the ZOLZ forbidden reflections due to glide planes and screw axes. Two PED experiments should be performed at small and large precession angles for a sure identification of these reflections. The ZOLZ kinematically forbidden reflections displaying GM lines can be easily identified from CBED patterns if these patterns are available. Those located in the FOLZ requires more delicate experiments. 3.3. Third step The third step involves CBED patterns. Most of the time, observation of two-dimensional (2D) features present in the ZOLZ of a unique ZAP is sufficient and this fact constitutes a great

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experimental simplification since WP CBED patterns displaying clear 3D features are usually difficult to obtain and sometimes even impossible. For noncentrosymmetric space groups, special attention should be focused on the features present inside pairs of rotation- or mirror- related diffracted disks. In order to observe details inside the diffracted disks the spot size should be adapted to the crystal quality of the diffracted area. Most of the times, a spot size in the range 20 to 50 nanometers gives good results. Observations at low temperature and energy-filtered patterns could be useful and add some confidence to the results. Unconventional precession patterns obtained with a small precession angle and without ‘descan’’ can also be used to identify the point groups [4].

– –

–

– 4. Main difficulties and advantages with respect to the previous method based on microdiffraction patterns The main drawbacks of the method are the following: – It is implied that any zone axis could be obtained experimentally. Except for the cubic crystal system, the required zone axes are located at 901 from each other which is very large compared with the tilt capabilities of the transmission electron microscopes. In most cases, two crystals with different orientations should be studied. Usually, there is no problem with equiaxed polycrystalline specimens containing crystals with different orientations. With textured specimens or with single crystals, two thin foils having different orientations may be

–

– –

57

necessary. Observation of Kikuchi lines helps to find the required ZAPs. Note that just one pattern may be required if the extinctions located in the FOLZ can be detected. The method also assumes that FOLZ reflections are visible on the ZAPs. For various reasons, this is not always the case. The best solution to overcome this difficulty consists in tilting the specimen, or preferably the incident beam, along the main directions or the ‘net’ mirrors present in the ZOLZ until FOLZ reflections appear [1]. Diffraction patterns with m, 2mm, 4mm, 3m or 6mm symmetries are easy to recognize. Rotation axes when present in patterns are generally more difficult to detect. Difficulties may be encountered with the monoclinic and the orthorhombic crystal systems due to the various possible settings (6 settings for the orthorhombic crystal system, 3 settings for the monoclinic crystal system and three cell choices for each of the 3 settings). The superposition of the Laue zones which occurs with a large precession angle for crystals having large lattice parameters constitutes another difficulty. Difficulties may also occur with crystals showing a small symmetry departure from a high symmetry. 26 space groups cannot be uniquely identified using the present procedure since they have the same 2D and 3D CBED symmetries and belong to the same extinction and diffraction symbols. They are: I222 and I212121 I23 and I213

Fig. 9. a, b and c - Schematic descriptions of the different types of diffraction patterns in the case of the cubic crystal system for the P Bravais lattice.

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Fig. 9. (continued)

P41 and P43 P4122 and P4322 P41212 and P43212 P4332 and P4132 P31 and P32 P3112 and P3212 P3121 and P3221 P62 and P64 P6222 and P6422 P61 and P65 P6122 and P6522 If only GM lines are used (i.e. if forbidden reflections that do not give GM lines are not identified), then there are 53 space groups that cannot be uniquely determined [19].

The present method has many advantages with respect to the previous method based on microdiffraction patterns. Among them: – The possibility to detect the kinematically forbidden reflections due to both glide planes and screw axes. The latter were not detectable with the microdiffraction method due to dynamical interactions. – The possibility to observe simultaneously many reflections in the FOLZ. This feature allows an easy and very sure identification of the FOLZ/ZOLZ shifts and periodicity differences. With the microdiffraction method, the number of reflections visible in the FOLZ is usually small unless a very thin crystal is observed. – The possibility to observe the ‘‘ideal’’ symmetry on PED patterns giving access to the Laue class. With the microdiffraction method

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Table 7 The couples of space groups marked with an arrow cannot be distinguished by CBED. Connection between the types of diffraction patterns and the extinction symbols in the case of the cubic crystal system. The shaded areas indicate typical pattern types which can be used alone to identify the extinction symbol. For the light shaded areas, the FOLZ forbidden reflections must be considered.

only the ‘‘net’’ symmetry is usually available since the ‘‘ideal’’ symmetry requires an extremely well aligned crystal. – Experimentally, a good quality zone-axis PED pattern is much easier to obtain than a conventional microdiffraction pattern because it does not require perfect alignment. – The interesting and typical features of the diffraction patterns are indicated on the drawings of the theoretical patterns for each of the 201 considered extinctions symbols (www.Electro n-Diffraction.fr). Therefore, the experiments can be selected and focused on the locations where these features are visible.

– 201 extinctions symbols (14  2 monoclinic, 111 orthorhombic, 31 tetragonal, 6 trigonal, 7 hexagonal and 18 cubic) can be identified unambiguously from the present method while only 57 different types of diffraction patterns (10 monoclinic, 16 orthorhombic, 14 tetragonal, 2 trigonal, 4 hexagonal, 11 cubic) could be identified from the previous method. The procedure described in the present paper starts with the identification of the crystal system from PED and finishes with CBED. The theoretical diffraction patterns drawn for each of the 201 extinction symbols can also be very useful if the experimental

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analysis starts with the identification of the point group from CBED symmetries using the procedures proposed by Buxton et al. [5], Eades [20] and Tanaka et al. [21]. In a second step, the possible space groups and their corresponding extinctions symbols are inferred from the point group. Finally, the occurrence of GM lines in the experimental patterns is compared with the pattern of appearances of GM lines in the tabulated theoretical patterns for the possible space groups. Thus the space group is determined. In most cases, the analysis is successful provided that 3D features are available on CBED patterns, that the kinematically forbidden reflections contain a GM line (which is not always the case) and assuming that the Bravais lattice is known. If not, additional PED experiments should be performed. The GM lines can also be analyzed using the procedures proposed by Tanaka [22] and Eades [23].

The theoretical figures describing each of the 201 extinction symbols can be downloaded from the web site: www.ElectronDiffraction.fr

Acknowledgments The authors would like to acknowledge Pierre Stadelmann (CIME-EPFL, Switzerland) for his help in the dynamical simulation of precession electron diffraction patterns. The TEM facility in Lille (France) is supported by the Conseil Re´gional du Nord-Pas de Calais and the European Regional Development Fund (ERDF). References

5. Conclusion The systematic method described in the present paper involves three steps: – First step: identification of the crystal system, via the Laue class, from observation of the highest symmetrical zone-axis PED pattern. – Second step: identification of the extinction and diffraction symbols from PED patterns. This step is based on comparisons of experimental electron precession patterns with theoretical patterns drawn for all the extinction symbols. On these patterns, three typical features are considered: – the presence or the absence of kinematically forbidden reflections in the ZOLZ and perhaps in the FOLZ. – the shifts between the reflections located in the FOLZ with respect to the ones located in the ZOLZ. – the periodicity differences between the reflections located in the FOLZ with respect to the ones located in the ZOLZ. It is shown that these features allow the unambiguous identification of all the extinction symbols from observation of one, two or three typical PED ZAPs, depending on the crystal system. If the Laue class is known from the first step, then the diffraction symbol can also be considered. Each extinction or diffraction symbol is compatible with a few possible space groups belonging to different point groups. – Third step: selection of the actual space group among the possible ones from the identification of the point group by means of CBED patterns. In most cases, the observation of the ZOLZ symmetry is the unique requirement. This method represents a great improvement with respect to a previous method which was based on electron microdiffraction patterns when PED was not available.

[1] J.-P. Morniroli, J.W. Steeds, Ultramicroscopy 45 (1992) 219. [2] N.J. Buerger, Z. Kristallogr. 91 (1935) 255. [3] Th. Hahn, International Tables for Crystallography, fifth ed., Springer, Dordrecht, 2005. [4] J.-P. Morniroli, P. Stadelmann, G. Ji, S. Nicolopoulos, J. Microsc. 237 (2010) 511. [5] B.F. Buxton, J.A. Eades, J.W. Steeds, G.M. Rackham, Proc. Roy. Soc. (London) A 281 (1976) 171. [6] M.G. Francesconi, A.L. Kirbyshire, C. Greaves, O. Richard, G. Van Tendeloo, Chem. Mater. 10 (1998) 626. [7] K. Boulahya, M. Parras, J.M.G. Gonza´lez-Calbet, A. Vegas, J. Solid State Chem. 151 (2000) 77. [8] F.J. Garcı´a-Garcı´a, A.K. Larsson, S. Furuseth, J. Solid State Chem. 166 (2002) 352. [9] M. Goune´, A. Redjaı¨mia, T. Belmonte, H. Michel, J. Appl. Crystallogr. 36 (2003) 103. [10] E.A. Juarez-Arellano, J.M. Ochoa, L. Bucio, J. Reyes-Gasga, E. Orozco, Acta Crystallogr B61 (2005) 11. [11] D. Enidjer, A. Venkert, J. Bernstein, M. Talianker, J. Alloys Compd. 474 (2009) 169. [12] R. Vincent, P.A. Midgley, Ultramicroscopy 53 (1994) 271. [13] J.-P. Morniroli, A. Redjaı¨mia, J. Microsc. 227 (2007) 157. ¨ [14] J.-P. Morniroli, G. Ji, in: P. Moeck, S. Hovmoller, S. Nicolopoulos, S. Rouvimov, V. Petkov, M. Gateshki, P. Fraundorf (Eds.), Materials Research Society Symposium Proceedings V1184, MRS Spring meeting, San Francisco, CA USA, April 13–17, 2009, pp.41–48. [15] C.S. Own, L.D. Marks, W. Sinkler, Acta Crystallogr A62 (2006) 434. [16] J. Gjønnes, A.F. Moodie, Acta Crystallogr 19 (1965) 65. [17] J.-P. Morniroli, Electron Diffraction, a Software to Simulate Electron Diffraction Patterns, USTL & ENSCL, Lille, 2002. [18] P.A. Stadelmann, JEMS (Java version electron microscopy software); software available at /http://cimewww.epfl.ch/people/stadelmann/jemsWebSite/jems. htmlS. [19] K. Tsuda, K. Sitoh, M. Terauchi, M. Tanaka, P. Goodman, Acta Cryst A56 (359) (2000) 25. [20] J.A. Eades, EUREM 88; IOP Conf. Series 93 (1988) Vol. 1, 3–12. [21] M. Tanaka, R. Saito, H. Sekii, Acta Cryst A39 (1983) 357. [22] M. Tanaka, H. Sekii, T. Nagasawa, Acta Cryst A39 (1983) 825. [23] J.A. Eades, Microbeam Analysis, D. E. Newbury Ed., (1988) 75–80.