International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2006). Vol. E, ch. 5.2, pp. 394-398   | 1 | 2 |

Section 5.2.2.  The basic concepts of the scanning

V. Kopskýa* and D. B. Litvinb

aDepartment of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail:  kopsky@fzu.cz

5.2.2. The basic concepts of the scanning

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If a crystal with a symmetry of the space group [\cal G] is transected by a crystallographic1 plane, called a section plane, then the subgroup of all elements of the space group [\cal G] which leave the plane invariant is a layer group, which is called a sectional layer group, of this section plane under the action of the group [\cal G]. Analogously, if the crystal is penetrated by a crystallographic1 straight line, called the penetration straight line, then the subgroup of all elements of the space group [\cal G] which leave the straight line invariant is a rod group, which is called the penetration rod group, of this penetration straight line under the action of the group [\cal G].

Sectional layer groups are therefore symmetries of crystallographic section planes and penetration rod groups are symmetries of crystallographic penetration straight lines under the action of space groups. In this sense they are analogous to site symmetries of Wyckoff positions. In addition, analogous to points, the section planes and penetration straight lines form orbits under the action of the space group [\cal G]. Planes or straight lines belonging to the same orbit have, with reference to their respective coordinate systems, the same sectional layer symmetry or penetration rod symmetry and the crystal is described in the same way with reference to any of these coordinate systems.

While every sectional layer group is, by definition, a subgroup of the corresponding space group, not every subgroup of the space group which is a layer group is necessarily a sectional layer group. Analogously, a penetration rod group is a subgroup of the corresponding space group but not every rod subgroup of a space group is a penetration rod group. It can be shown that every sectional layer group is either a maximal layer subgroup of the space group or a halving subgroup of a maximal layer subgroup, see Section 5.2.2.6[link].

We shall consider explicitly only the sectional layer groups, although the penetration rod groups can also be deduced from the scanning tables, see the example in Section 5.2.2.2[link].

5.2.2.1. The scanning for sectional layer groups

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A plane in a three-dimensional space is associated with a two-dimensional vector space [V({\bf a}',{\bf b}')] which is called the orientation of the plane. If the plane of this orientation also contains a point [P+{\bf r}], we shall denote it by a symbol [(P+{\bf r}, V({\bf a}',{\bf b}'))]. A straight line is associated with a one-dimensional vector space [V({\bf d})] which is called the direction of the straight line. If the straight line of this direction also contains a point [P+{\bf r}], we shall denote it by a symbol [(P+{\bf r}, V({\bf d}))]. We assume in what follows that the vector [{\bf d}] is not a linear combination of vectors [{\bf a}'], [{\bf b}'].

Then the set of all parallel planes with a common orientation [V({\bf a}',{\bf b}')] contains planes [(P+s{\bf d};V({\bf a}',{\bf b}'))]. Points [P+s{\bf d}] along a straight line [(P;V({\bf d}))] specify the location of individual planes as the points in which the planes intersect with the straight line [(P;V({\bf d}))] (Fig. 5.2.2.1[link] left). On the other hand, the set of all straight lines with a common direction [V({\bf d})] contains straight lines [(P+x'{\bf a}'+y'{\bf b}';V({\bf d}))]. The location of individual straight lines of the set is defined by their intersection points [P+x'{\bf a}'+y'{\bf b}'] with the plane [(P;V({\bf a}',{\bf b}'))] (Fig. 5.2.2.1[link] right).

[Figure 5.2.2.1]

Figure 5.2.2.1 | top | pdf |

Sets of parallel planes (left) and sets of parallel straight lines (right).

We consider now a space group [\cal G], with a point group G and translation subgroup [T_{G}], described by a symmetry diagram or by symmetry operations with reference to a crystallographic coordinate system [(P; {\bf a},{\bf b},{\bf c})] (as listed, for example, in IT A ). We want to solve the following two problems:

  • (1) Find the sectional layer groups [{\cal L}(P + s{\bf d}; V({\bf a}', {\bf b}'))] which contain all those elements of [\cal G] which leave the planes ([P + s{\bf d}]; [V({\bf a}', {\bf b}')]) invariant.

  • (2) Find the orbit of planes, generated by the plane ([P + s{\bf d}; V({\bf a}', {\bf b}'))] under the action of the space group [\cal G].

The general goal is to describe all possible cases, classify and systemize them. Since the first part of the problem may be described as a search for the change of the sectional layer symmetry as a plane of a given orientation changes its position so that one of its points moves along a straight line ([P; V({\bf d})]), we shall call this procedure the scanning of the space group [\cal G] for sectional layer groups of planes with the orientation [V({\bf a}', {\bf b}')] along the scanning line [(P; V({\bf d}))]. We shall use also abbreviated expressions in different contexts; for example the scanning of the space group [\cal G] (for layer groups) will mean the determination of the sectional layer groups for the space group [\cal G] and all possible orientations.

An analogous procedure is the scanning of the space group [\cal G] for penetration rod groups [{\cal R}(P+x'{\bf a}'+y'{\bf b}';V({\bf d}))] of straight lines with the direction [V({\bf d})] along the scanning plane [(P; V({\bf a}', {\bf b}'))].

Crystallographic orientations of planes are characterized by Miller (or Bravais–Miller) indices [(hkl)] [or [(hkil)]]. These indices determine a two-dimensional vector space, the orientation, all vectors of which leave the section planes with given Miller indices invariant. Those vectors of the translation group [T_{G}] (the lattice of [\cal G]) which lie in this space constitute a two-dimensional translation subgroup [T_{G1} = T({\bf a}',{\bf b}')] with a certain basis [({\bf a}',{\bf b}')]. This is the group of all those translations from [T_{G}] that leave the section planes with given Miller indices invariant. This group is therefore a common translation subgroup of all sectional layer groups of section planes with these Miller indices. The vectors [{\bf a}'], [{\bf b}'] can be taken as the basis vectors of the two-dimensional vector space [V({\bf a}',{\bf b}')] and hence [T({\bf a}',{\bf b}') = T_{G} \cap V({\bf a}',{\bf b}')].

The scanning line [(P;V({\bf d}))] and the scanning direction [V({\bf d})] are defined by a vector d. This vector can be, quite generally, chosen as any vector complementary to the orientation [V({\bf a}', {\bf b}')], i.e. as an arbitrary vector, noncollinear with [{\bf a}'], [{\bf b}'], which needs not even define a crystallographic direction. Since, for a given space group [\cal G] and orientation [V({\bf a}', {\bf b}')], the sectional layer group [{\cal L}(P + s{\bf d}; V({\bf a}', {\bf b}'))] depends only on the distance of the plane from the origin P, it might seem to be of advantage to choose the direction d always perpendicular to [V({\bf a}', {\bf b}')], as in the example below. This, however, is not always the most suitable choice. We shall subordinate the choice of vector [{\bf d}] to a strict convention, see Section 5.2.2.3[link], and call it the scanning vector.

Example

Consider a crystal whose space-group symmetry is [Pbcm], [D_{2h}^{11}] (No. 57). The sectional layer symmetries of planes with an [(001)] orientation depend on the location of the plane along the line [P+z{\bf c}], where the basis vector [{\bf c}] is chosen as the scanning vector [{\bf d}]. If [z = 0,{{1}\over{2}}], the sectional layer group is [p2/b11] (L16), if [z = {{1}\over{4}},{{3}\over{4}}], the sectional layer group is [pb2_{1}m] (L29). The same holds if we add an integer n to the coordinate z. All these layer groups are maximal subgroups of the group [Pbcm]. The sectional layer symmetry of any other plane is a layer group [pb11] (L12). The symbol of a layer group for a section plane located at [P+z{\bf c}] is given with reference to the coordinate system [(P+z{\bf d};{\bf a},{\bf b},{\bf d} = {\bf c})]. Notice that there are an infinite number of section planes with [z = n] and [(n+{{1}\over{2}})] or [z = (n+{{1}\over{4}})] and [(n+{{3}\over{4}})] and an infinite number of corresponding sectional layer groups which can be written as [p2/b11] ([n{\bf d}]) and [p2/b11] [[(n+{{1}\over{2}}){\bf d}]] (L16) or [pb2_{1}m] [[(n+{{1}\over{4}}){\bf d}]] and [pb2_{1}m] [[(n+{{3}\over{4}}){\bf d}]] (L29) with reference to the coordinate system [(P;{\bf a},{\bf b},{\bf d} = {\bf c})]. All these groups are maximal layer subgroups of the group [Pbcm]. There are also an infinite number of section planes with other values of z which change continuously between the previously given discrete values of z; to all these section planes there corresponds one sectional layer group, [pb11] (L12), whose Hermann–Mauguin symbol does not depend on z. This group is said to be floating in the direction [{\bf c}] and it is a halving subgroup of all previously given sectional layer groups, see Section 5.2.2.6[link].

5.2.2.2. The scanning group and the scanning theorem

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The main step in the solution of the scanning problem and in its tabular presentation is the introduction of the scanning group (Kopský, 1990[link]), which is a central concept of scanning. This group is an intermediate product in the process for scanning of the sectional layer groups and for the penetration rod groups. We shall introduce this group with reference to the scanning for sectional layer groups. The scanning group is then a space group which depends on the scanned space group [\cal G] and on the orientation [V({\bf a}',{\bf b}')]. The prominent status of the scanning group is seen from: (i) the scanning theorem, which facilitates determination of the sectional layer groups as well as penetration rod groups in more complicated cases, and (ii) the convention for the choice of vectors [{\bf a}'], [{\bf b}'] and [{\bf d}], see Section 5.2.2.3[link], which standardizes the description of the scanning.

Definition of the scanning group: Let [\cal G] be a space group with a point group G and [V({\bf a}', {\bf b}')] an orientation of planes, defined by Miller indices [(hkl)]. Further let H be that subgroup of the point group G of the space group [\cal G] that contains all those elements of G that leave the orientation [V({\bf a}', {\bf b}')] invariant, so that [HV({\bf a}', {\bf b}') = V({\bf a}', {\bf b}')].

The space group[{\cal H} = {\cal H}({\cal G}, (hkl)) = {\cal H}({\cal G}, V({\bf a}', {\bf b}')), \eqno(5.2.2.1)]which is an equitranslational subgroup of the space group [\cal G] corresponding to the point group H, is called the scanning group for the space group [\cal G] and for the orientation [V({\bf a}', {\bf b}')] with Miller indices [(hkl)].

The importance of the scanning group for the scanning process is due to the following theorem (Kopský, 1990[link]):

The scanning theorem: The scanning of the space group [\cal G] for the sectional layer groups of section planes with an orientation [V({\bf a}', {\bf b}')] is identical with the scanning of the scanning group [{\cal H}({\cal G}, (hkl)) = {\cal H}({\cal G}, V({\bf a}', {\bf b}'))] for the sectional layer groups of section planes with the same orientation [V({\bf a}', {\bf b}')].

The scanning group [\cal H] has, by definition, the same lattice [T_{G}] as the scanned group [\cal G]. However, the scanning group frequently belongs to a lower system than the group [\cal G], because its point group H is a subgroup of the point group G, and its conventional basis may be different from the conventional basis of [\cal G]. In addition, the scanning group is always a reducible space group (Kopský, 1988[link], 1989a[link],b[link], 1990[link]) because its point group H leaves the subspace [V({\bf a}',{\bf b}')] invariant.

Example

Consider the cubic space group [P432] ([O^{1}]) and section planes of orientation defined by the Miller indices (001). The scanning group for this orientation is the group [P422] ([D_{4}^{1}]) with reference to a basis [{\bf a}' = {\bf a}], [{\bf b}' = {\bf b}], [{\bf d} = {\bf c}]. Compare, in Part 6[link] , Table [P432] ([O^{1}]), the blocks headed Linear orbit and Sectional layer group [{\cal L}(s{\bf d})] for the original scanned group and for the scanning group to see that they are identical. Moreover, we receive the same results for orientations defined by the Miller indices (100) and (010) where the scanning group is denoted by the same Hermann–Mauguin symbol [P422] with reference to appropriate bases.

In addition, the diagram of the scanning group provides immediate information about the penetration rod groups of penetration straight lines with the direction [{\bf c}] characterized by the direction indices [001]. Indeed, the spatial distribution of these rod groups is immediately seen from the diagram of the scanning group [P422] in the basis [({\bf a},{\bf b},{\bf c})]. For directions corresponding to the indices [100] and [010] we obtain the same results with reference to respective bases [({\bf b},{\bf c},{\bf a})] and [({\bf c},{\bf a},{\bf b})].

5.2.2.3. The conventional basis of the scanning group

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In the Scanning tables of Part 6[link] , we follow the usual crystallographic practice to define the orientation of planes by their Miller indices (Bravais–Miller indices in hexagonal cases). This itself already guarantees that the orientations considered are crystallographic. The choice of vectors [{\bf a}'], [{\bf b}'] and [{\bf d}] is governed by a convention in which we distinguish the cases of orthogonal and inclined scanning.

Convention: Given the orientation of planes by Miller or Bravais–Miller indices, we choose vectors [{\bf a}'], [{\bf b}'] and the vector d of the scanning direction according to the following rules:

  • (i) Orthogonal scanning: If the scanning group [\cal H] is of orthorhombic or higher symmetry, or if it is monoclinic with the direction of its unique axis orthogonal to the orientation of the planes, we call the scanning orthogonal and the vectors [{\bf a}'], [{\bf b}'], [{\bf d}] are chosen in such a way that the triplet [({\bf a}',{\bf b}',{\bf d})] constitutes a conventional right-handed basis of the scanning group [\cal H].

  • (ii) Inclined scanning: If the scanning group is either triclinic or monoclinic with its unique axis parallel to the section planes, we call the scanning inclined. In this case we choose vectors [{\bf a}'], [{\bf b}'] in such a way that they constitute a conventional basis of the vector lattice [T({\bf a}',{\bf b}')], common to all sectional layer groups, while the scanning vector d is chosen as the shortest complementary vector.

Note that, in cases of orthogonal scanning, the first two vectors [{\bf a}'], [{\bf b}'] of the conventional basis of the scanning group [\cal H] automatically constitute a conventional basis of the lattice [T({\bf a}',{\bf b}')] and d is orthogonal to the orientation [V({\bf a}',{\bf b}')]. In cases of inclined scanning it is always possible to choose the vectors [{\bf a}'], [{\bf b}'] so that they constitute a conventional basis of the vector lattice [T({\bf a}',{\bf b}')]. However, it is generally impossible to choose all three vectors [{\bf a}'], [{\bf b}'] and d as a strictly conventional basis of the scanning group because the first two vectors must lie in the space defined by Miller (Bravais–Miller) indices, which usually leads to a clash with the metric conditions as they are given, for example, in Section 9.1.4 (vi[link] ) and (vii[link] ) of IT A (2005[link]).

The choice of the scanning direction d as that of a vector of the basis of the scanning group guarantees the periodicity d of the scanning. As a result, it is sufficient to describe the scanning for a given orientation, i.e. the sectional layer groups and orbits of planes, only in the interval with [0 \leq s \,\lt\, 1] on the scanning line [P+s{\bf d}]. Indeed, the crystal structure of symmetry [\cal G] is periodically repeated with periodicity d in the scanning direction. The sectional layer groups are, however, repeated in the scanning direction with the periodicity of the translation normalizer of [\cal G]. This is identical with the periodicity of the translation normalizer of the scanning group [\cal H] (see the examples in Section 5.2.5.1[link]). We recall that the translation normalizer of the space group [\cal G], as defined by Kopský (1993b[link],c[link]), is the translation subgroup of the Cheshire group (Euclidean normalizer) of [\cal G] [see Hirschfeld (1968[link]) and Koch & Fischer in Part 15[link] of IT A, 1987 edition or later].

In the application of the convention we note the following:

  • Item 1. If [G = H] for a certain orientation of planes so that this orientation is invariant under all elements of the point group G of the space group [\cal G], then [{\cal G} = {\cal H}], i.e. the scanning group [\cal H] coincides with the original space group [\cal G].

    The typical cases of this relationship are orientations (001) for the monoclinic, orthorhombic and tetragonal groups and the orientations (0001) for the trigonal and hexagonal groups. In these cases, the conventional basis of the original space group [\cal G] also coincides with the conventional basis of the scanning group [\cal H] and the group [\cal H] is therefore represented by the same Hermann–Mauguin symbol as the group [\cal G].

  • Item 2. The conventional basis of the scanning group [\cal H] may differ from the conventional basis of the original group [\cal G] even if these groups are identical. In this case the group is generally denoted by different Hermann–Mauguin symbols. This always happens in the cases of monoclinic and very frequently in cases of orthorhombic groups for other orientations than (001) because the conventional vectors [{\bf a}'], [{\bf b}'], d of the scanning group [\cal H] cannot be made identical with the conventional basis vectors a, b, c of the group [\cal G].

    Example

    Consider the space group [{\cal G} = Pmmm] ([D_{2h}^{1}]) and the orientations described by the Miller indices (001), (100), (010). The scanning group [{\cal H} = {\cal G}] is identical with the scanned group and its Hermann–Mauguin symbol [Pmmm] is the same for all three orientations.

    If, however, the scanned group is the group [{\cal G} = Pmma] ([D_{2h}^{5}]), then again the scanning group [\cal H] is identical with the scanned group [\cal G] for the three orientations, but the Hermann–Mauguin symbols of the scanning group are now different: they are [Pmma], [Pmcm] and [Pbmm] for the orientations (001), (100) and (010), respectively.

  • Item 3. If [H \subset G], so that the point group H is a proper subgroup of the point group G, then the conventional basis of the scanning group [\cal H] is usually different from the conventional basis of the original group [\cal G], although the groups are equitranslational, i.e. have the same translation subgroup. The conventional basis of the scanning group [\cal H] in the case when [H \subset G] actually coincides with the conventional basis of the space group [\cal G] only in the cases of the orientations (001), (100) and/or (010) if [\cal G] is cubic of lattice type P or I and hence [\cal H] is tetragonal of the same lattice type. The centring type of the scanning group [\cal H] is also frequently different from the centring type of the original group [\cal G].

5.2.2.4. The types of scanning

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It is useful to characterize various scanning tasks using the names of the crystallographic systems of the scanned group and of the scanning group. The scanning tables are naturally built up from lower to higher symmetries, according to the standard sequence of space groups. In this process, some already-considered space groups of lower crystallographic systems appear as scanning groups for those orientations which are not invariant under the point group G of the scanned space group [\cal G] of a higher crystallographic system. In the first column of Table 5.2.2.1[link], the crystallographic systems are listed in their usual hierarchy and to the right of each system are listed the lower systems from which some groups appear as scanning groups. We use terms such as tetragonal/monoclinic scanning when a tetragonal space group [\cal G] is considered and the scanning group [\cal H] is monoclinic. Simple expressions such as orthorhombic scanning will mean that the scanning group [\cal H] is orthorhombic, to distinguish it from the expression scanning of orthorhombic groups, which means that the original space group [\cal G] is orthorhombic. The lattice of trigonal scanning groups in the case of cubic/trigonal scanning is always rhombohedral as indicated in parentheses.

Table 5.2.2.1| top | pdf |
Various types of scanning

Triclinic          
Monoclinic triclinic        
Orthorhombic monoclinic triclinic      
Tetragonal orthorhombic monoclinic triclinic    
Trigonal orthorhombic monoclinic triclinic    
Hexagonal orthorhombic monoclinic triclinic    
Cubic tetragonal trigonal (rhombohedral) orthorhombic monoclinic triclinic

5.2.2.5. Orientation orbits

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The point group G of the scanned group [\cal G] acts on the orientations defined by Miller indices [(hkl)] or Bravais–Miller indices [(hkil)]. The set of all orientations [V({\bf a}_{i}', {\bf b}_{i}')] obtained from a given orientation [V({\bf a}_{1}', {\bf b}_{1}')] by the action of the elements of the group G is called the orientation orbit. The point group [H_{1} \subseteq G] which leaves the orientation [V({\bf a}_{1}', {\bf b}_{1}')] invariant is the point group of the scanning group [{\cal H}_{1}] for this orientation. From the coset resolution[G = H_{1} \cup g_{2}H_{1} \cup \ldots \cup g_{p}H_{1} \eqno(5.2.2.2)]we obtain orientations of the orbit by the action of cosets on the first orientation: [V({\bf a}_{i}',{\bf b}_{i}') = g_{i}H_{1}V({\bf a}_{1}',{\bf b}_{1}') = g_{i}V({\bf a}_{1}',{\bf b}_{1}')]. In general, the number of orientations in the orbit is equal to the index [p = [G:H_{1}]] of the subgroup [H_{1}] in G. The point group [H_{i} \subset G] which leaves the orientation [V({\bf a}_{i}',{\bf b}_{i}')] invariant is the conjugate subgroup [H_{i} = g_{i}H_{1}g_{i}^{-1}] of the point group [H_{1}] in the group G. If [H_{1} = H = G], then the scanning group [\cal H] is identical with the scanned group [\cal G] and the orientation orbit contains just one orientation.

In the general case, to each orientation [V({\bf a}_{i}',{\bf b}_{i}')] there corresponds a scanning group [{\cal H}_{i}], conjugate to the scanning group [{\cal H}_{1}]. The elements of a coset [g_{i}H_{1}] send the scanning vector [{\bf d}_{1}] for the first orientation into scanning vectors [{\bf d}_{i} = g_{i}H_{1}{\bf d}_{1} = g_{i}{\bf d}_{1}] for orientations [V({\bf a}_{i}',{\bf b}_{i}')].

The set of the conjugate scanning groups [{\cal H}_{i}] is obtained from the coset resolution of the space group, which corresponds to the coset resolution (5.2.2.2)[link] of the point group:[{\cal G} = {\cal H}_{1} \cup \{g_{2}|{\bf s}_{2}\}{\cal H}_{1} \cup \ldots \cup \{g_{p}|{\bf s}_{p}\}{\cal H}_{1}. \eqno(5.2.2.3)]The scanning groups [{\cal H}_{i} = \{g_{i}|{\bf s}_{i}\}{\cal H}_{1}\{g_{i}^{-1}|-g_{i}^{-1}{\bf s}_{i}\}] are related in the same way to the respective conventional bases [({\bf a}_{i}',{\bf b}_{i}',{\bf d}_{i}) = (g_{i}{\bf a}_{1}',g_{i}{\bf b}_{1}',g_{i}{\bf d}_{1})] and hence they are expressed by the same Hermann–Mauguin symbols. However, the operations in the three-dimensional Euclidean space, which correspond to operations [g_{i}] on the vector space, often contain additional translations [{\bf s}_{i}]. Quite generally, the scanning for an orientation [V({\bf a}_{i}',{\bf b}_{i}')] is described in the same manner with reference to the coordinate system [(P+{\bf s}_{i};V({\bf a}_{i}',{\bf b}_{i}',{\bf d}_{i}))] as the scanning for the orientation [V({\bf a}_{1}',{\bf b}_{1}')] is described with reference to a coordinate system [(P; V({\bf a}_{1}',{\bf b}_{1}',{\bf d}_{1}))].

In analogy with Wyckoff positions, see Section 8.3.2[link] of IT A, we distinguish three types of orientations and of orientation orbits:

  • (1) special orientations and special orientation orbits with fixed parameters;

  • (2) special orientations and special orientation orbits with variable parameter; and

  • (3) general orientations and general orientation orbits.

The type of the orbit is the same as the type of each of its orientations. Orientations and orientation orbits have the following characteristic properties:

  • (1) An orientation [V({\bf a}', {\bf b}')] is a special orientation with fixed parameters if its symmetry H is either at least orthorhombic or if it is monoclinic with the vector of its unique axis orthogonal to the orientation.

  • (2) An orientation [V({\bf a}', {\bf b}')] is a special orientation with variable parameter if its symmetry H is monoclinic and if it contains the vector of the unique axis.

  • (3) An orientation [V({\bf a}', {\bf b}')] is a general orientation if its symmetry H is triclinic.

Example 1

Orientations defined by the Miller indices (001) are special orientations with fixed parameters for monoclinic groups with unique axis c as well as for orthorhombic and tetragonal groups. Bravais–Miller indices (0001) also define special orientation with fixed parameters. In each of these cases, the orientation orbit contains just one orientation.

Orientations (010) and (100) are special orientations with fixed parameters for all orthorhombic groups and each such orientation constitutes the orientation orbit.

Orientations (001), (010) and (100) are special orientations with fixed parameters for cubic groups and they belong to the same orientation orbit.

Example 2

Orientations [(mn0)] are special orientations with variable parameter for monoclinic groups with unique axis c. Each such orientation constitutes an orientation orbit.

For cubic groups, the orientations [(mn0)] [with the exclusion of cases [m = \pm 1], [n = 0] and [m = 0], [n = \pm 1] in groups of Laue class [m{\bar 3}] ([T_{h}]) and also of cases [m = \pm 1], [n = \pm 1] in groups of Laue class [m{\bar 3}m] ([O_{h}])] are special orientations with variable parameter. The orientation orbits contain six equivalent orientations in groups of Laue class [m{\bar 3}] ([T_{h}]) and 12 in groups of Laue class [m{\bar 3}m] ([O_{h}]), see Section 5.2.4.6[link].

Orientation orbits are correlated with orbits of crystal faces, see Part 10[link] of IT A. If the group H does not contain elements that change the sign of the normal, then the orientation orbit is characterized by the same set of Miller indices as the set of equivalent crystal faces. Generally, the group H contains a halving subgroup [H_{o}] whose elements leave the normal to the orientation [V({\bf a}', {\bf b}')] invariant while elements of the coset change its sign. In this case, the number of equivalent crystal faces is twice the number of orientations in the orbit. The group [H_{o}] is identical with the point symmetry of a crystal face of orientation [V({\bf a}', {\bf b}')]. Such a face located at a point [P+s{\bf d}] is sent to a face of the same orientation located at a point [P-s{\bf d}] by those elements of H which are not contained in [H_{o}]. These are the same elements which change the direction of the scanning.

5.2.2.6. Linear orbits

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We consider a section plane with orientation [V({\bf a}',{\bf b}')] and location [P+s{\bf d}]. The orbit of planes generated by the action of the scanned group [\cal G] on this section plane splits into subsets of planes with the same orientation. The suborbit of planes with the same orientation is identical with the orbit under the action of the scanning group for this orientation. This suborbit is called the linear orbit of planes. If the orientation orbit contains only one orientation (scanning group = scanned group), then the linear orbit contains all planes of the orbit. If there are several orientations in the orientation orbit, then to each of these orientations there corresponds its own linear orbit. As shown in the previous section, the description of the scanning with reference to corresponding coordinate systems is identical for different orientations of the orientation orbit. The separation of planes and their sectional layer symmetries are the same in each of these orbits. In other words, the spatial distribution of layer symmetries is the same for all orientations of the orientation orbit; the scanning, however, begins generally at a point [P+{\bf s}_{i}] for the orientation [V({\bf a}_{i}',{\bf b}_{i}')]. We shall concentrate our attention now to one linear orbit.

The parameter s in the description of linear orbits defines the position of the section plane by its intersection [P+s{\bf d}] with the scanning line. The parameter therefore specifies the distance of the section plane from the origin P in units of d and is referred to as the level at which the section plane is located. Intersections at [P+(s+n){\bf d}], [n \in Z] (integer) are translationally equivalent to an intersection at [P+s{\bf d}] where [0 \leq s \,\lt\, 1]. The section planes at levels [P+(s+n){\bf d}] form an orbit under the translation group [T({\bf d})] generated by the scanning vector [{\bf d}]. The set of these planes is called the translation orbit. Each translation orbit has exactly one representative plane in the interval [0 \leq s \,\lt\, 1]. The linear orbit consists of one or several translation orbits.

We distinguish two types of locations and linear orbits:

  • (1) Special locations of section planes and special linear orbits.

  • (2) General locations of section planes and general linear orbits.

With reference to parameter s, the special locations always correspond to a fixed parameter, the general locations to a variable parameter. Special locations are singular in the sense that in the infinitesimal vicinity of a section plane at a special location there are only section planes of general location. The sectional layer groups corresponding to these locations have the following properties:

  • (1) The sectional symmetry of a plane in a special location is a layer group which contains operations changing the direction of the normal to the plane.

  • (2) The sectional symmetry of a plane in a general location is a layer group which does not contain operations changing the direction of the normal to the plane.

  • (3) The sectional symmetries of planes in special locations are always maximal layer subgroups of the space group [\cal G] as well as of the scanning group [\cal H]. The sectional symmetry of a plane in a general location is a common halving subgroup of all sectional layer groups for special locations. We say that such a sectional layer group is floating in the scanning direction.

Comment: If the point group H of the scanning group [\cal H] does not contain elements that change the normal to section planes, then all locations are general locations and there is only one sectional layer group common to all locations of section planes. The scanning group with this property is also called floating in the scanning direction.

The number of planes in a translation orbit: The total number of planes in a translation orbit is infinite because the index of the sectional layer group in the scanning group is. We can, however, count the number of planes in a translation orbit in an interval [0 \leq s \,\lt\, 1]. If the point group of the scanning group is H and the point group of the sectional layer group for a given translation orbit is L, then the number of planes in this orbit in the interval [0 \leq s \,\lt\, 1] equals the index [[H:L]] when the centring of the scanning group is P or C. When the centring of the scanning group is A, B, I or F, this number is [2[H:L]]; when the centring type of the scanning group is R, this number is [3[H:L]].

The number f of planes in an orbit with a general parameter s per unit interval also defines the length of the fundamental region of the space group [\cal G] as well as of the scanning group [\cal H] in this interval. This length [s_{o}] is a fraction of unit interval, [s_{o} = {{1}\over{f}}], where [f = [H:L]], [2[H:L]] or [3[H:L]] according to the centring of the scanning group and L is the point group of sectional layer groups corresponding to a general orbit.

5.2.2.7. Orthogonal, inclined and triclinic scanning

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It is convenient for future reference to refine the basic categories of orthogonal and inclined scanning as follows:

  • (1) Orthogonal scanning. We call the scanning orthogonal if the scanning group is orthorhombic, tetragonal, trigonal or hexagonal.

  • (2a) Monoclinic/orthogonal scanning. This term is used if the scanning group is monoclinic and the vector d defines its unique axis.

    In both cases the vector d is orthogonal to the vectors [{\bf a}'] and [{\bf b}'] and they occur whenever the orientation orbit is a special orbit with fixed parameters.

    The absolute value [d = |{\bf d}|] of the scanning vector is, in cases of orthogonal scanning, equal to the interplanar distance defined by the Miller indices of the orientation.

  • (2b) Monoclinic/inclined scanning. The scanning is called monoclinic/inclined if the scanning group is monoclinic and its unique axis is one of the vectors [{\bf a}'], [{\bf b}']. The vector d is actually not necessarily inclined to the orientation [V({\bf a}'], [{\bf b}')]. It may be orthogonal owing to special metric conditions of the lattice which are determined by the scanned group [\cal G]. It is, however, a vector of a monoclinic basis which lies in the plane orthogonal to the unique axis. This case occurs when the orientation orbit is a special orbit with one variable parameter.

    The interplanar distance d in the case of inclined scanning is [d = |{\bf d}|\cos \varphi] where [\varphi] is the angle of the vector d with the normal to the plane.

  • (3) Triclinic scanning. The scanning is called triclinic or trivial if the scanning group is triclinic. This case occurs when the orientation orbit is a general orbit.

The difference between monoclinic/orthogonal and monoclinic/inclined scanning is illustrated in Fig. 5.2.2.2[link]. The orientation in the first case is fixed, while the second case applies to various orientations containing the monoclinic unique axis. The orientation can be defined by one free parameter, the angle [\varphi]; we use instead Miller indices [(mn0)].

[Figure 5.2.2.2]

Figure 5.2.2.2 | top | pdf |

Monoclinic/orthogonal (left) and monoclinic/inclined (right) scanning.

References

Hirschfeld, F. L. (1968). Symmetry in the generation of trial structures. Acta Cryst. A24, 301–311.
International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]
Kopský, V. (1988). Reducible space groups. Lecture Notes in Physics, 313, 352–356. Proceedings of the 16th International Colloquium on Group-Theoretical Methods in Physics, Varna, 1987. Berlin: Springer Verlag.
Kopský, V. (1989a). Subperiodic groups as factor groups of reducible space groups. Acta Cryst. A45, 805–815.
Kopský, V. (1989b). Subperiodic classes of reducible space groups. Acta Cryst. A45, 815–823.
Kopský, V. (1990). The scanning group and the scanning theorem for layer and rod groups. Ferroelectrics, 111, 81–85.
Kopský, V. (1993b). Translation normalizers of Euclidean motion groups. I. Elementary theory. J. Math. Phys. 34, 1548–1556.
Kopský, V. (1993c). Translation normalizers of Euclidean motion groups. II. Systematic calculation. J. Math. Phys. 34, 1557–1576.








































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