International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. E, ch. 5.2, pp. 401-405
Section 5.2.3. The contents and arrangement of the scanning tables^{a}Freelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and ^{b}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA |
In the scanning tables two formats are used:
The tables are grouped according to crystallographic systems. Within each system, the standard-format tables are grouped into geometric classes in the same order as in IT A . The auxiliary tables follow the tables of standard format at the end of each Laue class.
The content and arrangement of the standard-format tables are as follows:
The standard tables for triclinic groups describe the trivial scanning where the scanning group is or . The tables for monoclinic groups describe monoclinic/orthogonal scanning and monoclinic/inclined scanning. The standard tables for the remaining groups describe only orthogonal scanning for these groups.
The headline begins with the serial number of the space-group type identical with the numbering given in IT A, followed by a short Hermann–Mauguin symbol. The Schönflies symbol is given in the upper right-hand corner.
The next line is centred and contains the full Hermann–Mauguin symbol of the specific space group for which the scanning is described in the table. This is followed by a statement of origin in those cases where two space groups of different origin are considered, or by a statement of cell choice when different cell choices are used for a monoclinic space group.
The specific space group considered in the table is that space group, including its orientation (setting) and location (origin choice), the diagram of which is presented in IT A, assuming that the upper left-hand corner of the diagram represents the origin P, its left edge downwards the vector a, its upper edge to the right the vector b, while vector c is directed upwards. In the case of orthorhombic and monoclinic groups, this is the diagram in the () setting, the so-called standard setting. For some group types, two different origins are given in IT A. Both are used to consider two specific groups of the same type with different locations in the present tables. The scanning for each of these groups is described in a separate table. In the case of monoclinic groups, one, three or six different cell choices, depending on the group type, are considered, see Section 5.2.4.2.
Each table is divided into five columns. The first column is entitled Orientation orbit or Orientation orbit . The orientations are specified by their Miller or Bravais–Miller indices. Each orientation defines a row for which the scanning is described in the next columns. Orientations which belong to the same orbit are grouped together and orientation orbits are separated by horizontal double lines across the table for space groups of the tetragonal and higher-symmetry systems and for the monoclinic groups. The vertical separation for orthorhombic groups is explained in Section 5.2.4.3.
Orientation orbits are listed in each table in the following order from top to bottom:
General orientation orbits are not included; the corresponding scanning is trivial and the presentation of these orbits would take up too much space.
The second column is entitled Conventional basis of the scanning group and it contains three subcolumns headed by the symbols of vectors , , . Next to it is the third column with the heading Scanning group . In the subcolumns, the vectors , and of the conventional bases of the scanning groups are specified in terms of the conventional basis (a, b, c) of the scanned group . The scanning groups are described in the third column by their short Hermann–Mauguin symbols.
The fourth column, headed Linear orbit , describes the linear orbits of planes for the orientation of this row and the fifth column, headed Sectional layer group , describes the corresponding sectional layer groups.
The location of the plane along the line determines a certain layer group; the symbol next to is a shorthand for the sectional layer group of the section plane passing through the point on the scanning line. , as a function of s, has a periodicity of the translation normalizer of the space group in the direction d but we list the translation orbits within , i.e. with periodicity d. This is important because the planes at levels separated by the periodicity of the normalizer do not necessarily belong to the same orbit.
The planes form orbits with fixed parameter s and with a variable parameter s. The orbits with fixed parameter s are recorded in terms of fractions of vector d; one of these fractions always lies in the interval , where is the length of the fundamental region of the scanned group along the scanning line in units of d. The fixed values of s are always given in the range . If planes at different levels belong to the same orbit, then the levels are enclosed in square brackets. The sectional layer group corresponding to a certain level s is then given in the fifth column by its Hermann–Mauguin symbol in the coordinate system . If the levels on the same line refer to the same Hermann–Mauguin symbol of a sectional layer group but are not enclosed in brackets, then they belong to different orbits. The sectional layer groups belonging to different planes of the orbit are certainly of the same type and parameters but they may be oriented or located in different ways so that their Hermann–Mauguin symbols are different because they refer to the same basis (, ). In this case, the levels corresponding to the same orbit are listed in a column, beginning and ending with brackets, and to each level is given the sectional layer group.
There is always only one row (which may, however, split for typographical reasons) corresponding to orbits with a variable parameter s and the one sectional layer group which is floating along the scanning direction and which is a common subgroup of all sectional layer groups for orbits with fixed parameters. This row always contains the term where s belongs to the fundamental region of the group along the line . Here is a fraction of 1 and the region is a fraction of the interval . These levels correspond to locations of planes of the translation orbit along the direction d within the unit interval. The levels are expressed in a compact way; as a result there appears an entry in cases when the scanning group is not polar. Since s is in the interval , is negative and hence not in the interval ; this level is equivalent to the level .
Following each Hermann–Mauguin symbol, we give the sequential number of the type to which the sectional layer group belongs, according to its numbering in Parts 1–4 of this volume.
Example 1
Orientation orbit (001) for the space groups , (No. 89), , (No. 93) and , (No. 91).
Group : The entries `' in the fourth column followed by in the fifth column indicate that there are two separate translation orbits, represented by planes passing through P and ; planes of both orbits have the same sectional layer group with reference to the respective coordinate systems.
The sectional layer symmetry at a general level is and the translation orbit contains planes at two levels (the index of the point group in the point group ), described as [, ]. It is and both levels belong to the same orbit. For positive s we can change to to get the level in the interval .
Group : The entries are now enclosed between square brackets to indicate that the planes at these levels along the line belong to the same orbit. The sectional layer symmetry is .
The sectional layer symmetry at a general level is , so that there must be four [422 () : 122 ()] levels which are described as where . Again we can change to to get the level in the interval .
Group : The entry `' in the first subrow and the entry ` in the second subrow indicate that the planes on corresponding levels all belong to the same translation orbit. The corresponding sectional layer groups and for the first and second subrow are of the same type but the orientations of their twofold axes are different. The Hermann–Mauguin symbols are therefore different because they are expressed with reference to the same basis [in this case the basis (a, b)].
The sectional layer symmetry at a general level is so that and there must be eight levels which are described as , .
Example 2
We consider the group , (No. 148) and the orientation (0001). There are three subrows in the columns for the translation orbits and the sectional layer groups. In the first row there are the entries , ; and ; in the second row , , and [(2a + b)/3]; and in the third row and [(a + 2b)/3]. This is to be interpreted as follows: the levels , and belong to one translation orbit, distinct from the orbit to which belong the levels , and . The sectional layer groups are groups on all these levels but they are located at different distances from points for different levels .
The sectional layer symmetry at a general level is . The point group 3 is of index 2 in the point group and the lattice is of the type R so there are six planes in the translation orbit per unit interval along d and . The translation orbit is described by [, , ].
Example 3
Space group , (No. 123). The scanning groups for the orientations (100) and (010) which belong to the same orientation orbit are expressed by the same Hermann–Mauguin symbol in their respective bases. The translation orbits and sectional layer groups are therefore expressed in the same block.
The scanning groups for the orientations (110) and () of the same orientation orbit under the space group , (No. 125) are expressed by the same Hermann–Mauguin symbol (d/4) in the respective bases if the scanned group is chosen according to origin choice 1 in IT A. Hence the translation orbits and sectional layer groups are expressed in one block; they are the same with reference to their corresponding bases. For origin choice 2, the locations of the scanning groups are different; we obtain the group for the orientation (110) and for the orientation . Each of these scanning groups has its own box with the translation orbits and sectional layer groups. If we compare the two boxes, we observe that the data in the second box are the same as in the first box but shifted by .
Example 4
Consider the block of the orientation orbit (111), (), (), () for space groups , (No. 212), , (No. 213) and , (No. 214). The Hermann–Mauguin symbol of the scanning group with reference to their bases is the same, , up to a shift of the origin. In the row for each orientation, therefore not only are the bases given, but also the location of the origin so that a complete coordinate system is specified in such a way that the symbol is exactly the same for each orientation. The symbol of the scanning group, the location of the orbits and the sectional layer groups are given in the last block; all this information is formally the same but for each orientation it refers to its own coordinate system.
The auxiliary tables describe cases of monoclinic/inclined scanning for groups of orthorhombic and higher symmetries. They are clustered together for groups of each Laue class, starting from Laue class – , after the tables of orthogonal scanning, i.e. after the standard-format tables for this Laue class.
All possible cases of monoclinic/inclined scanning reduce to cases where the scanned group itself is monoclinic and the orientation is defined by the Miller indices . These cases are described as a part of the standard-format tables for monoclinic groups. Two bases are used in this description:
If the scanned group is of higher than monoclinic symmetry, then the monoclinic scanning group and we use three bases:
Two types of tables from which orbits of planes and sectional layer groups can be deduced are given:
In the next two sections we describe the construction of these two types of tables and their use in detail.
The cases of monoclinic/inclined scanning occur when the orientation of the section plane:
Auxiliary basis of the scanning group. In each of these cases, there is a set of orientations for which the property (i), (ii) or (iii) is common and all orientations of this set contain the vector that defines the unique axis of a monoclinic scanning group which is also common for all orientations of the set. An auxiliary basis of this scanning group is defined with reference to that one orientation of the set which is described by Miller indices .
The first column of each table describes orientations of the orbit by Miller indices with reference to the conventional basis of the scanned group . Various possible situations can be distinguished by three criteria:
The second column assigns to each orientation the conventional basis of the monoclinic scanning group that is related to the auxiliary basis given in the third column in the same way as to the standard basis in the case of monoclinic groups.
The conventional basis is always chosen so that its first vector is the vector of the common unique axis. Vector is defined by the orientation of section planes and hence by Miller indices (either directly or indirectly through transformation to a monoclinic basis). There is the same freedom in the choice of the scanning direction d as in the cases of monoclinic/inclined scanning in the case of monoclinic groups.
Each table of orientation orbits for a certain centring type(s) is followed by reference tables which are organized by arithmetic classes belonging to this centring type(s). The scanned space groups are given in the first row by their sequential number, Schönflies symbol and short Hermann–Mauguin symbol. They are arranged in order of their sequential numbers unless there is a clash with arithmetic classes; a preference is given to collect groups of the same arithmetic class in one table. If space allows it, groups of more than one arithmetic class are described in one table.
The first column is identical with the first column of the table of orientation orbits. On the intersection of a column which specifies the scanned group and of a row which specifies the orientation by its Miller (Bravais–Miller) indices is found the scanning group, given by its Hermann–Mauguin symbol with reference to the auxiliary basis . This symbol, which may also contain a shift of origin, instructs us which monoclinic scanning table to consult. The vectors , , d that determine the lattice of sectional layer groups and the scanning direction are those given in the table of orientation orbits. Depending on the values of parameters m, n, p, q we find the scanning group in its basis and the respective sectional layer groups.