- 1. Introduction
- 2. Space-group symmetry
- 3. Symmetry relations between space groups
- 4. 3D crystallographic point groups
- 5. Applications
- 6. Conclusions
- A1. Crystallographic symmetry operations
- A2. Coordinate transformations: basic results
- A3. Group–subgroup relations of space groups
- A4. Normalizers of space groups
- Supporting information
- References
- 1. Introduction
- 2. Space-group symmetry
- 3. Symmetry relations between space groups
- 4. 3D crystallographic point groups
- 5. Applications
- 6. Conclusions
- A1. Crystallographic symmetry operations
- A2. Coordinate transformations: basic results
- A3. Group–subgroup relations of space groups
- A4. Normalizers of space groups
- Supporting information
- References
teaching and education
The International Tables Symmetry Database
aKarlsruhe Institute of Technology, Institute of Applied Geosciences, Karlsruhe, Germany, beFaber Soluciones Inteligentes SL, Bilbao, Spain, cDepartment of Chemistry, St Olaf College, Northfield, USA, and dDepartamento de Física, Universidad del País Vasco UPV/EHU, Spain
*Correspondence e-mail: gemma.delaflor@kit.edu
The International Tables Symmetry Database (https://symmdb.iucr.org/), which is part of International Tables for Crystallography, is a collection of individual databases of crystallographic space-group and point-group information with associated programs. The programs let the user access and in some cases interactively visualize the data, and some also allow new data to be calculated `on the fly'. Together these databases and programs expand upon and complement the symmetry information provided in International Tables for Crystallography Volume A, Space-Group Symmetry, and Volume A1, Symmetry Relations between Space Groups. The Symmetry Database allows users to learn about and explore the space and point groups, and facilitates the study of group–subgroup relations between space groups, with applications in determining crystal-structure relationships, in studying phase transitions and in domain-structure analysis. The use of the International Tables Symmetry Database in all these areas is demonstrated using several examples.
Keywords: International Tables for Crystallography; Symmetry Database; space-group symmetry; symmetry relations between space groups; point groups; subgroups; supergroups.
1. Introduction
The International Tables Symmetry Database (https://symmdb.iucr.org/), which we will refer to as the Symmetry Database from here on, provides access to databases of information on the crystallographic point and space groups, including information on the symmetry relations between space groups (Kroumova et al., 2021). These component databases expand upon and complement the symmetry information provided in International Tables for Crystallography Volume A, Space-Group Symmetry (abbreviated as ITA, 2016), and Volume A1, Symmetry Relations between Space Groups (abbreviated as ITA1, 2010). The information stored in the databases can be either retrieved directly or generated `on the fly' using a range of auxiliary programs. Some programs facilitate the analysis of group–subgroup relations between space groups and three provide different kinds of interactive visualizations. All these web applications (referred to as programs or visualizers from here on) have user-friendly menus and help pages with brief explanations of the crystallographic information that is displayed and of the functionality of the programs, with links to more details as provided in the relevant volumes of International Tables for Crystallography. The Symmetry Database has been developed specifically for the International Union of Crystallography and access is via a subscription to the online version of International Tables for Crystallography (https://it.iucr.org/).
An important advantage of the Symmetry Database is that the different programs can communicate with each other, so that the output of some programs is used directly as the input for others. In this way, the Symmetry Database becomes a working environment with, for example, the appropriate programs for addressing crystallographic problems related to the study of group–subgroup relations between space groups.
The aim of this contribution is to present the programs and visualizers of the Symmetry Database and to provide examples showing how they can be used. Basic definitions and explanations of the notation used in the description of symmetry operations, coordinate transformations, and subgroups and supergroups of space groups can be found in Appendix A.
The programs, visualizers and component databases of the Symmetry Database use the standard or default settings of the space groups. These are specific settings of the space groups that coincide with the conventional space-group descriptions found in ITA. For space groups with more than one description in ITA, the following settings are chosen as standard: unique axis b setting, cell choice 1 for monoclinic groups, hexagonal axes setting for rhombohedral groups, and the origin choice 2 description (i.e. with the origin at a centre of inversion) for those centrosymmetric groups that are listed with respect to two origins in ITA.
The Symmetry Database is arranged into three parts (see Fig. 1). The first part provides access to the crystallographic space-group database, which includes the generators, general and special Wyckoff positions, and affine, Euclidean and chirality-preserving Euclidean normalizers. This part also includes an interactive visualizer allowing the user to explore the general-position diagrams and symmetry elements of the space groups. The data that are provided in this part are in fact valid for all space groups that belong to the same space-group type. (For the definition and a detailed discussion of the term space-group type, see Section 1.3.4.1 of ITA.) The second part of the Symmetry Database provides access to a database of the maximal subgroups and the minimal supergroups of the space groups together with an interactive visualizer allowing the study of the group–subgroup relations between space groups. The third part focuses on the three-dimensional crystallographic point groups, providing access to databases of generators, general positions and Wyckoff positions. This part also includes interactive visualization of point-group symmetry elements. These three parts are described in detail in Sections 2, 3 and 4, respectively.
The Symmetry Database provides 14 server-side crystallographic programs and three interactive visualizers. There are 12 main programs and two auxiliary programs. The main programs, which carry out a variety of calculations based on the input submitted by the user, include six programs for space groups and point groups (Generators, General position and Wyckoff positions) and six programs for space groups only (Normalizers, Maximal subgroups, Series of isomorphic subgroups, Minimal supergroups, Group–subgroup relations and Supergroups). In addition, two auxiliary programs, coset decomposition and Wyckoff-position splittings, are used to perform some calculations using the data stored in the databases as input. The three interactive visualization programs include a 3D space-group general position and visualizer that uses JSmol (Interactive 3D general-position visualizer), a JavaScript-based 3D crystallographic point-group symmetry visualizer (Interactive visualization), and a graph-based application for exploring the group–subgroup relationships between space groups (Graph of maximal subgroups).
2. Space-group symmetry
This part of the Symmetry Database hosts the data for all 230 space groups in their conventional settings. In addition, crystallographic data for the 530 settings for the monoclinic and orthorhombic space groups listed in Table 1.5.4.4 of ITA (`the ITA settings') are also available. The data for the generators, general and special Wyckoff positions, and affine, Euclidean and chirality-preserving Euclidean normalizers can be explored here. A 3D interactive visualizer of the points of the general position and the symmetry elements that relate them completes this part. The input for the programs in this part is the space-group numbers according to ITA, which can be selected directly from a table that lists them along with their associated Hermann–Mauguin symbols.
2.1. Generators and general position
The generators and general position of a Generators and General position, respectively. The generators and the general-position entries are given as coordinate triplets, as matrix–column representations of the corresponding symmetry operations and as geometric interpretations (see Appendix A1).
are shown by the programs(i) The list of coordinate triplets (x, y, z) reproduces the data from the General position blocks of the space-group tables found in ITA. The coordinate triplets may also be interpreted as shorthand descriptions of the matrix forms of the corresponding symmetry operations [see (ii) below].
(ii) For the matrix–column representations, the symmetry operations of the space groups are described by (3 × 4) matrix–column pairs (W, w) with reference to a coordinate system consisting of an origin O and a basis (a1, a2, a3).
(iii) The geometric interpretation of the symmetry operations is given (a) following the conventions in ITA [including the symbol of the its glide or screw component (if relevant), and the location of the related geometric element] and (b) using Seitz notation [see Glazer et al. (2014)].
Fig. 2 shows the general position for the Pba2 (No. 32) in the standard setting.
The programs Generators and General position list the generators and general-position entries of the space groups in the standard setting as well as in any of the ITA settings. Clicking on `Change settings' gives a list of the ITA settings of the the corresponding data can be calculated with respect to one of these settings by choosing it directly from this list. If a particular setting is not in this list (i.e. it is not one of the ITA settings), it can be obtained by clicking on `Change basis' and specifying the coordinate transformation that relates the basis of the non-conventional setting (a, b, c)non-conv to that of the standard setting (a, b, c)stand (for details on coordinate transformations, see Appendix A2). The coordinate transformation is described by a matrix–column pair (P, p) and consists of two parts: a linear part P given by a (3 × 3) matrix, which describes the change of direction and/or length of the basis vectors, (a, b, c)non-conv = (a, b, c)standP, and an origin shift p = (p1, p2, p3) given by a (3 × 1) column, whose coefficients describe the position of the non-conventional origin with respect to the standard one. A matrix–column pair
is often written in the following concise form:
2.2. Wyckoff positions
The program Wyckoff positions lists the Wyckoff positions for a designated The listing follows that of ITA (see Fig. 3): the Wyckoff-position block starts with the general position at the top, followed downwards by the various special Wyckoff positions with decreasing multiplicity and increasing The data for each include (i) the multiplicity, i.e. the number of equivalent positions in the conventional (ii) the Wyckoff letter, which is an alphabetical label; (iii) the described by the isomorphic to the site-symmetry group; and (iv) a set of coordinate triplets of the equivalent Wyckoff-position points in the shown under `Coordinates'. For centred space groups, the centring translations are listed above the coordinate triplets. The point groups isomorphic to the site-symmetry groups (see the third column in Fig. 3) are described using oriented symbols: these are modified Hermann–Mauguin point-group symbols that show how the symmetry elements of a site are related to the symmetry elements of the (for more details, see Section 2.1.3.12 of ITA). An explicit listing of the symmetry operations of the site-symmetry group of a point is obtained by clicking directly on its coordinate triplet. The symmetry operations of the site-symmetry group of an arbitrary point (specified by its coordinates but not necessarily within the unit cell) can be also be calculated using the option `Specific orbit'. Fig. 4 shows the list of the symmetry operations of the site-symmetry group for the point 2, 5/4, 1/2 of the Pmma (No. 51). This point belongs to the 4h and its site-symmetry group is `.2.'.
The program Wyckoff positions provides a list of the Wyckoff positions in different space-group settings, either by specifying the coordinate transformation (P, p) to a new basis (`Change basis'), or by selecting one of the ITA settings of the corresponding directly, using (`Change setting').
2.3. Normalizers
The normalizers of space groups play an important role in a number of applications (see Chapter 3.5 of ITA). For example, the number of different but equivalent structure descriptions (even after fixing the space-group setting and origin choice) is determined by the Euclidean of the corresponding For more details see Section 5.1. Following ITA, the program Normalizers shows the Euclidean, chirality-preserving Euclidean and affine normalizers of the space groups (see Appendix A4). The normalizers are described with respect to the standard space-group settings. For triclinic and monoclinic groups (whose affine normalizers are not isomorphic to groups of motions), parametric representations of the matrix–column pairs of the mappings of the affine normalizers are shown together with the appropriate restrictions on the coefficients. Fig. 5 shows the output of the program Normalizers for the Euclidean of the F432 (No. 209). (Note that the type of can be selected by clicking on the tabs near the top of the page, below the space-group symbol.) The output of the program is organized into three blocks. In the first block of the output, general information about the is displayed: the Hermann–Mauguin symbol of the the basis of the described in terms of the basis of the and the index of in the The second block shows the additional generators, i.e. the additional symmetry operations that can generate the successively from the (see ITA, Tables 3.5.2.3, 3.5.2.4 and 3.5.2.5). These additional symmetry operations are represented by their coordinate triplets, their matrix–column representations and their corresponding geometric interpretations. For example, the Euclidean of the F432(a, b, c) is with basis vectors (1/2a, 1/2b, 1/2c) and index 4 (see Fig. 5). There are two additional generators that can be used to generate (1/2a, 1/2b, 1/2c) from F432: a translation t(1/2, 1/2, 1/2) and an inversion through the point 0, 0, 0. The representatives of the decomposition of the with respect to the are listed in the last block of the output. Each representative is specified by its coordinate triplet, matrix–column representation and geometrical interpretation (see Fig. 5). The affine of the F432 coincides with the Euclidean one, so the output of the program for the F432 is the same for the Euclidean and affine normalizers.
2.4. Interactive 3D visualization
The visualizer in this section allows the user to explore the symmetry operations that relate the various coordinate triplets of the general position of a JSmol, the JavaScript implementation of Jmol (Hanson, 2013). The starting point for interaction is a table showing the general position of the along with an interactive panel where the three-dimensional general-position diagram is displayed (see Fig. 6; the video in the supporting information also gives a detailed explanation of how to use this visualizer).
It usesThe table of the general-position triplets and their interpretation as symmetry operations is interactive: clicking on a specific i.e. those that would change the handedness of a chiral object, such as inversion. The colours of the points distinguish their heights along the b axis for triclinic and monoclinic space groups, and along the c axis for the other space groups. The action of the can be visualized by clicking on the `animate' button.
shows its action on the initial general-position point depicted within the yellow ring on the general-position diagram. If the image point is outside the its equivalent point in the (as reached by a lattice translation) is also indicated. Points in red rings are obtained by symmetry operations of the second kind,The general-position diagram is also interactive: the x, y, z to that position. The geometric description of the (type, orientation, screw or glide component, location) is displayed at the top of the diagram. The corresponding symmetry-equivalent position and the related (modulo lattice translation) is highlighted in the general-position table. The option `animate' can be used to visualize its action.
can be dragged with the mouse in order to see it from different perspectives. Clicking on a general-position point shows the operation that transforms the initial general-position pointFig. 6 shows a screenshot of the output of Interactive 3D general-position visualizer for the P42nm (No. 102). Clicking on the sixth coordinate triplet −x + 1/2, y + 1/2, z + 1/2 in the general-position table shows the user the corresponding n-glide reflection with glide vector 0, 1/2, 1/2, located at 1/4, y, z (the blue plane in the diagram). The animation shows the action of the selected on the initial general-position point to a symmetry-equivalent point inside the unit cell.
The potential for Interactive 3D general-position visualizer in teaching crystallography is enhanced by the ability to animate the action of the different symmetry operations, in particular those with an intrinsic translation part different from zero, such as screw rotations or glide reflections.
3. Symmetry relations between space groups
This part of the Symmetry Database focuses on the study of group–subgroup relations between space groups. The data in Volume A1 of International Tables for Crystallography on maximal subgroups of space groups of indices 2, 3 and 4 are extended to include the series of all isomorphic subgroups for indices up to 27 (125 for some cubic groups). In contrast to ITA1, where only space-group types of supergroups are indicated, our database contains individual information for each minimal including the transformation matrix that relates the conventional bases of the group and the Additional programs in this part allow a detailed study of supergroups of space groups of any index (not just minimal supergroups) and the generation of interactive graphs of chains of maximal subgroups.
3.1. Maximal subgroups
All maximal non-isomorphic and maximal isomorphic subgroups of a Symmetry Database using the program Maximal subgroups. The maximal-subgroup information is presented in blocks of translationengleiche and klassengleiche subgroups (Hermann, 1929) as defined in Appendix A3.2 (see Section 2.2.4 of ITA1, and Fig. 7). The maximal subgroups are distributed into classes of conjugate subgroups and the classes are distinguished by different background colours. Each is specified by (i) the index of in ; (ii) the Hermann–Mauguin symbol of , referred to the coordinate system of (this symbol is a link to the class of conjugate subgroups); (iii) the conventional short Hermann–Mauguin symbol and the space-group number of (the space-group number is a link to a list of maximal subgroups of ); (iv) the coordinate triplets of the symmetry operations of that are used as generators for the ; and (v) the transformation matrix–column pair (P, p) that relates the standard bases of and (see Appendix A3.2). For certain applications it is necessary to represent the subgroups as subsets of elements of . This is achieved by clicking on the transformation matrix, which is a link to an auxiliary tool that transforms the general-position representatives of to the coordinate system of . In addition, for each the splittings of all Wyckoff positions of to that of and the decomposition of with respect to can be calculated by the auxiliary programs Wyckoff-position splittings and coset decomposition.
of indices 2, 3 and 4 can be retrieved from the3.1.1. Wyckoff-position splittings
The relations between the Wyckoff positions of a Wyckoff-position splittings, used by many of the programs of this part, calculates the splitting of the Wyckoff positions for a given group–subgroup pair and the transformation matrix–column pair (P, p) that relates the standard bases of and . The splitting of the Wyckoff positions of into the Wyckoff positions of is specified by their multiplicities and Wyckoff letters. This program provides further information on Wyckoff-position splittings that is not listed in ITA1, namely the relations between the unit-cell representatives of the orbit of and the corresponding representatives of the suborbits of . For example, Fig. 8 shows the Wyckoff-position splitting schemes for the group–maximal-subgroup pair I23 (No. 197) > P23 (No. 195), [here I is the three-dimensional unit matrix and o is the (3 × 1) column matrix containing zeros as coefficients]. The 2a of I23 splits into two independent positions of P23 with no site-symmetry reduction:
and those of its subgroups can involve splittings of the Wyckoff positions, reduction of the site symmetries or both. The auxiliary programThe relations between the coordinate triplets of the Wyckoff positions are displayed under the link `show relations'. These relations are presented in a table form showing the Wyckoff-position coordinate triplets with respect to the standard group basis and the corresponding triplets referred to the basis of the .
Each of these triplets is further interpreted as a coordinate triplet of a As an example, see the yellow table shown at the bottom of Fig. 8The link `Splitting for a specific orbit' permits the calculation of the Wyckoff-position splitting schemes for an arbitrary orbit of specified by the coordinate triplet of any point of the orbit.
3.1.2. decomposition
The auxiliary program coset decomposition performs the left or right decomposition of a group with respect to a (see Appendix A3.1):
where n is the index of in and (Wi, wi) with i = 1,…, n are the representatives. The representative (W1, w1) is assumed to be the identity element (I, o) of . By default, the program calculates the right decomposition of the group with respect to the and the representatives that are shown are described in the basis of the To calculate the left decomposition, one has to click on the option `Left cosets'. Clicking on `show complete cosets' lists the members of the , where is decomposed with respect to its translational .
As an example, Fig. 9 shows the output of coset decomposition for the right decomposition of the I23 (No. 197) with respect to one of its maximal subgroups, R3 (No. 146), = , , ; 0, 0, 0.
3.2. Series of isomorphic subgroups
The maximal subgroups of index higher than 4 with indices p, p2 and p3, where p is a prime number, are isomorphic subgroups and infinite in number. In ITA1 the isomorphic subgroups of indices higher than four, [i] > 4, are listed not individually but as members of series under the heading Series of maximal isomorphic subgroups. The program Series of isomorphic subgroups provides access to the database of maximal isomorphic subgroups. Apart from the parametric descriptions of the series, the program provides an individual listing of all maximal isomorphic subgroups of indices as high as 27 for all space groups, except for the cubic ones, where for some groups the isomorphic subgroups of indices up to 125 are shown. For each series, the Hermann–Mauguin symbol of the the selected generators, the restrictions on the parameters describing the series, and the transformation matrix relating and are listed [see Fig 10(a)]. The option `show series' generates maximal isomorphic subgroups of any allowed index `on the fly'. The format and content of the data are similar to those of the program Maximal subgroups (see Section 3.1). Fig. 10(b) shows the maximal isomorphic subgroups of index 3 of the P6322 (No. 182) generated using this option. For each of the indices, a list of the maximal subgroups is presented together with links to the auxiliary programs Wyckoff-position splittings (see Section 3.1.1) and coset decomposition (see Section 3.1.2).
3.3. Minimal supergroups
The program Minimal supergroups lists the isomorphic and non-isomorphic minimal supergroups of indices 2, 3 and 4 (see Appendix A3.3) for any For triclinic and monoclinic space groups, this list includes minimal isomorphic supergroups of indices up to 7. The minimal-supergroup information is presented in blocks of translationengleiche and non-isomorphic klassengleiche supergroups, as defined in Appendix A3.3 (see also Section 2.1.6 of ITA1). For each minimal the index, the Hermann–Mauguin symbol, the space-group number and the transformation matrix (P, p) relating the standard bases of the and the group are specified (see Fig. 11). As in other programs of the Symmetry Database, the transformation matrix is a link to an auxiliary tool that calculates the general position of the with respect to the basis of the group. For each group–supergroup relation, links to the auxiliary programs Wyckoff-position splittings (see Section 3.1.1) and coset decomposition (see Section 3.1.2) are also provided.
3.4. Group–subgroup relations
A group–subgroup pair is specified by the group , P, p) relating the standard basis of the group to that of the For a given group–subgroup pair, the program Group–subgroup relations calculates (i) the Wyckoff-position splitting scheme of the group with respect to the , (ii) the left and right decomposition of with respect to , and (iii) the transformation of the general position of with respect to the basis of . As input, the group, the and the transformation matrix relating the standard basis of the group to that of the have to be specified. The group and the can be specified by typing in either the space-group number or the Hermann–Mauguin symbol. Before performing any calculation, the program checks the matrix–column pair (P, p) and if it is not valid an error message is displayed. When the transformation matrix for the group–subgroup pair is not known, the auxiliary program Step-by-step calculation of the program Group–subgroup relations can be used to find it.
and transformation matrix–column pair (3.5. Graph of maximal subgroups
Group–subgroup symmetry relations can be visualized as graphs of chains of maximal subgroups using Graph of maximal subgroups. For a given group–subgroup pair , the program calculates all possible group–subgroup chains of intermediate maximal subgroups : = and displays the results in an interactive graph. The auxiliary program Step-by-step calculation can be used to find the transformation matrix–column pairs (P, p) defining the group–subgroup relations when the correct transformation matrices are not known. This can also be used to calculate the shortest maximal-subgroup path between the group and the (Aljazzar & Leue, 2011). The input to Graph of maximal subgroups is the group and the ; the index of in is optional. If the index of the in is specified, the group–subgroup relations are shown in a contracted graph (see Fig. 12); if not, the group–subgroup relations are represented by a general contracted graph (see Fig. 13). The general contracted graph contains all possible groups that appear as intermediate maximal subgroups between and . It is important to note that in both the contracted and general contracted graphs a node specified by a space-group symbol indicates a space-group type, i.e. a node might represent different space groups belonging to the same space-group type.
As one can see in Fig. 12, the output of Graph of maximal subgroups is divided into three main sections. The first section hosts the auxiliary programs Change index and Classify. While with Change index one can repeat the calculations with a different index, Classify is used to display the conjugacy classes of maximal subgroups for the given type and index. For each of the conjugacy classes of subgroups, the program indicates the corresponding Hermann group and the factorization of the index [i] = [it] · [ik], where the index [it] indicates the reduction of point-group symmetry while [ik] reflects the loss of translation symmetry (see Appendix A3.2; for more details, see Section 1.2.7 of ITA1). For example, in the case of the group–subgroup relation (No. 122) > C2 (No. 5) of index 4, there are three different subgroups C2 distributed between two classes of conjugate subgroups. The three different subgroups correspond to the three sets of twofold rotation axes in : those pointing along [100] and [010] of the tetragonal cell give rise to the two conjugate subgroups, and the twofold axes of the third one (which forms a class of conjugate subgroups by itself) are along the tetragonal axis. The three subgroups C2 are translationengleiche subgroups of , i.e. the Hermann group for the pair of index 4 coincides with the C2 with [i] = [it] = 4 and [ik] = 1.
The transformation matrices for the different chains of maximal subgroups for the selected group–subgroup pair are listed on the right-hand side of the second section of the output, entitled `Chains of maximal subgroups'. On the left-hand side, the interactive group–subgroup graph is displayed. When hovering over any graph node, the corresponding
with its maximal subgroups and minimal supergroups is shown. When clicking on any node, the chains that go through this node are highlighted on the right-hand side. In the last section of the output there are three tabs with more detailed information regarding the graph and the chains of the maximal subgroups. All the chains are listed under the tab `Chains of maximal subgroups'. For each chain, the space groups involved are shown and all transformations that correspond to the chain are displayed. The transformation matrices are links and calculate the general position of the in the basis of the group . The tab `Group–maximal pairs' contains all the transformation matrix–column pairs for each group–maximal pair in the graph. The other tab shows all subgroups of involved in the graph.3.6. Supergroups
The last program available in this part of the Symmetry Database is the program Supergroups, which calculates all supergroups of a specific space-group type and index (see Appendix A3.3). In contrast to ITA1, where only the space-group types of supergroups are indicated, in the Symmetry Database each is listed individually and specified by the transformation matrix that relates the conventional bases of the group and the As input, the group, and index have to be specified. The program lists all the transformation matrices that specify different supergroups, and it is possible to calculate the general position of the group in the basis of each The decomposition and Wyckoff-position splittings of each group–supergroup relation can also be calculated.
4. 3D crystallographic point groups
For those interested in molecular symmetry and in the physical properties of materials, the generators, general positions and Wyckoff positions of the three-dimensional crystallographic point groups are available in the third part of the Symmetry Database. These data can be transformed to different settings, enhancing and extending the data presented in Chapter 3.2 of ITA. Moreover, simple, clear and instructive interactive visualization of the symmetry elements and the stereographic projections of the three-dimensional crystallographic point groups is also available. The input for the programs in this part is the which can be selected directly from a table that lists the Hermann–Mauguin and Schönflies symbols for all 32 three-dimensional crystallographic point groups.
4.1. Data for the crystallographic point groups
In this section of the Symmetry Database, there are three programs called Generators, General position and Wyckoff positions (analogous to the programs for the space groups) that list the generators, general positions and Wyckoff positions for the 32 three-dimensional crystallographic point groups, respectively. The crystallographic data retrieved by these programs can be transformed to different settings by clicking on the options `Change basis' or `Change setting'. The output of these three programs is similar to the output of the programs Generators, General position and Wyckoff positions for the space groups (see Sections 2.1 and 2.2 for more details).
4.2. Interactive 3D visualization
The program Interactive visualization for point groups (Arribas et al., 2014), which was inspired by classical wooden crystallographic models, is an interactive visualizer of the symmetry elements and their stereographic projections for the crystallographic point groups. Interactive three-dimensional polyhedra are used to represent idealized crystals and their corresponding symmetry elements (see Fig. 14). The user can explore different shapes of polyhedra compatible with the selected and view them from different angles, and can also selectively enable or disable the visualization of the symmetry elements. It is an excellent program for learning about essential symmetry concepts and the international Hermann–Mauguin notation of point-symmetry groups.
5. Applications
Besides providing access to the crystallographic databases for the point and space groups, the programs and visualizers of the Symmetry Database are very useful in the study of symmetry relations between space groups. Information about symmetry relations is essential for the resolution of a number of crystallographic problems, such as the analysis of domain structures, the study of phase transitions and the prediction of new crystal structures. The use of some of these programs and visualizers is demonstrated in this section with several illustrative examples.
5.1. Equivalent crystal-structure descriptions
In general, crystal structures are described through their space-group symmetry, the unit-cell parameters and the atomic positions in the i.e. on the chosen setting of the Even after fixing the setting of the in most cases there are several equivalent possible but different descriptions of the same (Koch & Fischer, 2006).
The description of a structure is not unique and depends on the coordinate systems,Consider, for example, the simple and well known structure of caesium chloride (CsCl), which crystallizes in the cubic ):
(No. 221). The origin can be chosen to be at the centre of a Cs or a Cl atom, which results in two different sets of coordinates (see Fig. 15The equivalence of the two descriptions is easy to recognize [see Fig. 15: they are related by a translation of the type t(1/2, 1/2, 1/2)]. But how many equivalent descriptions of the CsCl structure exist? How can we calculate all possible symmetry-equivalent descriptions of a given crystal structure?
The Euclidean normalizer plays an important role in the determination of all symmetry-equivalent descriptions of a given structure with . The number of different equivalent coordinate descriptions of a is equal to the index [i] of the group in its Euclidean :
There are only two space groups, (No. 229) and (No. 230), that give rise to only one description of a
This is due to the fact that the Euclidean coincides with the . For all other space groups, there exist at least two equivalent descriptions.The transformations generating the set of symmetry-equivalent descriptions of a given
can be calculated easily by the decomposition of with respect to :where n is the index of in , (Wi, wi) with i = 1,…, n represents the representatives and (W1, w1) is assumed to be the identity element (I, o). The transformation of the initial structure by the representatives in equation (3) yields all symmetry-equivalent descriptions:
where X represents the coordinates of the atoms in the initial structure, X′ corresponds to the coordinates of the atoms of the equivalent descriptions and (, ) acts as the transformation matrix.
The symmetry-equivalent descriptions of a given structure and their generation can be easily obtained using the program Normalizers (see Section 2.3). In the case of CsCl, the Euclidean of the (a, b, c) is with basis vectors (1/2a, 1/2b, 1/2c). The index of in is two; this means that there are only two different symmetry-equivalent descriptions for the structure of CsCl. The decomposition of the Euclidean with respect to the space group,
shows that the second equivalent description of CsCl is generated from the initial description by a translation of the type t(1/2, 1/2, 1/2).
Crystal structures with space groups that are polar have infinitely many symmetry-equivalent descriptions, since they are not fixed by symmetry (for more details, see Appendix A4 and Chapter 3.5 of ITA). This is the case for the co-crystallization of caffeine and 4-chloro-3-nitrobenzoic acid (Ghosh & Reddy, 2012), which gives a structure with a polar Fdd2 (No. 43). This structure is available from the Cambridge Crystallographic Data Centre as CCDC reference 88530 (BEDYIU). The Euclidean of Fdd2(a, b, c) is P1ban(1/2a, 1/2b, εc), a that contains continuous translations in one direction. This is indicated by εc for the third basis vector of the and by the superscript 1 to the Bravais letter (see Chapter 3.5 of ITA). The index of P1ban in Fdd2 is 2 · ∞; this means that there are two symmetry-equivalent descriptions, and for each one, infinitely many symmetry-equivalent descriptions exist. The translations of the type t(0, 0, t) give rise to infinite equivalent descriptions that differ only in their z coordinates.
5.2. Symmetry relations between space groups
5.2.1. Phase transitions
Consider two phases of the same compound whose symmetries are described by the parent structure) and (derivative structure), such that . The relationship between these two structures can be characterized by a global distortion that is decomposed into a strain, describing the global distortion of the lattice of the derivative structure relative to the parent structure, and an atomic displacement field, representing the displacements of the atoms of the low-symmetry phase from their positions in the high-symmetry phase. The global distortion can be characterized if the description of the parent structure is transformed by an appropriate transformation (P, p) to an equivalent description that is most similar to that of the derivative structure. This new description of the parent structure is usually called the reference description of the parent structure relative to the derivative structure. The metric deformation accompanying the can be determined by the comparison of the bases of the derivative and the reference structures. Similarly, the difference between the atomic positions of the reference and the derivative structure provides information related to the atomic displacements occurring during the phase transition.
(As an illustrative example consider the structural 2. At room temperature CaCl2 crystallizes in the orthorhombic Pnnm (No. 58). At about 235°C a structural takes place to a high-symmetry tetragonal phase of the rutile type. The following descriptions of the two phases of CaCl2 (Howard et al., 2005) are taken from the Inorganic Database (Fachinformationszentrum Karlsruhe and National Institute of Standards and Technology, https://www.fiz-karlsruhe.de/icsd.html, abbreviated as ICSD):
of CaCl(a) ICSD No. 246417. The symmetry of the high-temperature phase is described by the tetragonal P42/mnm (No. 136) with cell parameters at = 6.38310 (10) Å and ct = 4.20409 (7) Å. The coordinates of the atoms in the are given as
Ca: 2a 0, 0, 0,
Cl: 4f 0.3047, 0.3047, 0.
(b) ICSD No. 246416. The low-symmetry structure is described by the orthorhombic Pnnm with cell parameters aorth = 6.43763 (5) Å, borth = 6.28765 (5) Å and corth = 4.17823 (1) Å. The coordinates of the atoms in the are given as
Ca: 2a 0, 0, 0,
Cl: 4g 0.3249, 0.2822, 0.
The Pnnm is a maximal of index 2 of the group P42/mnm. The transformation matrix relating the group with the determined by the program Maximal subgroups (see Section 3.1), is the identity matrix, i.e. (P, p) = (a, b, c; 0, 0, 0). This means that the lattice parameters of the reference and parent structures are equal (for details of the unit-cell lattice transformation see Appendix A2). From the comparison of the lattice parameters of the reference and the derivative structures, the lattice deformation accompanying the can be determined.
The coordinates of the representative Ca atom occupying the position 2a 0, 0, 0 in P42/mnm are transformed to 2a 0, 0, 0, while those of Cl are transformed from 4f 0.3047, 0.3047, 0 in P42/mnm to 4g 0.3047, 0.3047, 0 (results provided by the auxiliary program Wyckoff-position splittings, see Section 3.1.1). A comparison between the transformed atomic coordinates and the atomic coordinates of the low-symmetry phase, Ca 0, 0, 0 and Cl 0.3249, 0.2822, 0, reveals the corresponding atomic displacements associated with the which, in this case, are related to the displacements of the Cl atoms.
5.2.2. Domain structure analysis
A displacive or order–disorder parent or prototypic phase, is transformed to a crystalline phase, known as the daughter or distorted phase, with such that . The distorted phase is usually inhomogeneous, and consists of an infinite number of homogeneous regions called domains. Note that domains differ in their location in space, in their size and/or orientation, and potentially in their space groups , which, however, belong to the same space-group type .
often results in the formation of domain structures. A homogeneous single phase with , known as theThe domains in the distorted phase can be classified into a finite and small number of domain states. Two domains belong to the same domain state if their crystal patterns are identical, i.e. if they occupy regions of space that are part of the same The number and types of domain states that might form during the can be predicted by the analysis of the group–subgroup relation between the prototype and the distorted phases. The domain states can be classified into two types:
(i) Twin domain states are formed if the of the distorted phase is a of the of the parent phase, . For example, twin domain states occur if is a translationengleiche (equi-translational) of the (see Appendix A3.2) of the prototype phase.
(ii) Antiphase domain states are formed if the translation of the distorted phase is a of the translation of the parent phase, . For example, antiphase domain states occur if the of the distorted phase is a klassengleiche (equi-class) of (see Appendix A3.2).
Twin and antiphase domain states can also be simultaneously observed if is a general of (see Appendix A3.2).
The number of single domain states (Si) can be determined by the index [i] of in . Following Hermann's theorem, the index [i] can also be factorized as
where [it] is the t-index, the translationengleich index, and [ik] is the k-index, the klassengleich index (see Appendix A3.2); these represent the number of different twin and antiphase domain states that might be formed during the respectively.
Let us consider BaTiO3, which exhibits a ferroelectric from a prototype (non-polar) phase with cubic (No. 221) to a distorted polar phase with tetragonal of the type P4mm (No. 99). The type and number of domain states that might occur in the low-symmetry phase can be calculated using Group–subgroup relations (see Section 3.4) or Graph of maximal subgroups (see Section 3.5). The Hermann group in this example is P4mm (No. 99) with it = 6 and ik = 1. Since P4mm is a translationengleiche of of index 6, six different twin domain states are expected to be formed during the For the there are three different (but conjugate in P4mm, see Appendix A3.1) subgroups isomorphic to P4mm of index 6 [with their fourfold axes directed along the a, b and c directions, i.e. with (P, p) = (b, c, a), often written as (b, c, a), (c, a, b) and (a, b, c)].
Different domain states can exhibit different tensor properties and different diffraction patterns and can differ in other physical properties. The tensor properties of the crystal can be used to distinguish the different domain states if the relation between the domain states is known, i.e. if the is known. This can be obtained from the decomposition of the of the prototype phase with respect to , where each of the representatives might act as a In this example, the decomposition of with respect to is calculated using the auxiliary program coset decomposition (see Section 3.1.2) within Group–subgroup pair or Graph of maximal subgroups (see Table 1).
|
Let us analyse the polarization P, a tensor of rank one, in the low-symmetry phase of BaTiO3. The polarization in each domain state Si can be determined by
where i = 1, …, 6 represents the number of domain states Si and gi are the corresponding operations. The polarization vector P compatible with is of the type
Table 1 shows the polarization Pi in the different domain states calculated by equation (7).
5.2.3. Hierarchical trees
The structural relationships between group–subgroup-related crystal structures can be presented in a concise and comprehensive form using Bärnighausen trees (Bärnighausen, 1980). Starting from a high-symmetry structure type (aristotype) and reducing the symmetry, by distortions or substitutions of atoms, lower-symmetry structure types (hettotypes) are derived. Detailed information on how to construct Bärnighausen trees is available in Chapter 1.6 of ITA1 and in the book by Müller (2013).1 The programs Maximal subgroups (see Section 3.1), Graph of maximal subgroups (see Section 3.5) and Wyckoff-position splittings (see Section 3.1.1) are very useful for obtaining the information needed to construct Bärnighausen trees.
Such hierarchical trees can also be constructed by searching in the opposite direction (Kitaev et al., 2015), i.e. the highest possible structure types (archetypes) are reached by starting from a hettotype or root structure. In an ascending group–supergroup tree the of the high-symmetry structure is a of the of the root structure. Taking into account that any group–supergroup relation can be represented by a chain of minimal supergroups, , the search for possible aristotypes can be performed as a stepwise procedure over a chain of minimal supergroups of the of the root structure. Starting form the of the root structure, a list of the minimal supergroups can be fetched, along with the transformation matrices (P, p) relating the with the group, via the program Minimal supergroups (see Section 3.3). The validity of each (P, p) has to be evaluated with respect to the splitting of the supergroup's Wyckoff positions into the starting . For each transformation matrix, the splitting of the Wyckoff positions is directly calculated by the auxiliary program Wyckoff-position splittings within the program Minimal supergroups, and those supergroups that split their Wyckoff positions into the occupied positions of the root structure are selected. Note that the calculated splitting of the Wyckoff positions has to be checked against the occupied positions in the root structure along with all its symmetry-equivalent structures as calculated by applying the Euclidean using the program Normalizers (see Section 2.3). The search then continues upward until the highest possible structure type whose space groups do not have any supergroups fulfilling the conditions for the splitting of the Wyckoff positions is reached. In addition to these symmetry conditions, one has to take into account that a higher-symmetry phase can only be reached if the strain and the atomic displacement relating the two structures are small enough [see Capillas et al. (2007)]. More detailed information about the procedure for deriving and constructing group–supergroup trees is provided by Kitaev et al. (2015).
Let us construct the group–supergroup tree ascending from a root structure of KCuF3 belonging to the I4/mcm (No. 140) with the atoms occupying the 4a, 4d, 4b and 8h Wyckoff positions. The I4/mcm has three minimal supergroups: (No. 226), P4/mmm (No. 123) and I4/mcm (No. 140). I4/mcm does not satisfy the Wyckoff-position splitting conditions, and thus the upward transition into a structure type with I4/mcm is forbidden, as shown in Fig. 16. Upward in this tree, there is only one for the and three for P4/mmm. Among these four supergroups only (No. 221) meets the splitting conditions of the Wyckoff positions for both and P4/mmm. In the next step, the two supergroups of the group do not satisfy the splitting conditions of the Wyckoff positions. Therefore the : 1a, 1b, 1c structure type should be considered as the archetype for the I4/mcm: 4a, 4d, 4b, 8h structure type.
Note that there are cases in which the use of non-standard settings is more convenient for describing the symmetry relationships between crystal structures clearly. The results of the programs of the Symmetry Database should be used with care, as the programs used the standard setting as a default.
6. Conclusions
The Symmetry Database forms part of International Tables for Crystallography and hosts (i) crystallographic databases and visualizers for the point and space groups and (ii) programs and an interactive visualizer for studying the symmetry relations between space groups. These programs do not need local installation; the only requirement is to be subscribed to the online version of International Tables for Crystallography.
The Symmetry Database provides access to the generators, general positions and Wyckoff positions for the point and space groups, and enables transformation of this information to different settings. Databases of the affine, Euclidean and chirality-preserving Euclidean normalizers of the space groups and of the maximal subgroups and minimal supergroups of the space groups are included. There are programs for calculating the supergroups of space groups of any index and for generating interactive graphs of maximal subgroups. Programs for visualizing the symmetry elements and the stereographic projections of the point groups and the general-position diagrams of the space groups are also provided and may be useful for teaching.
APPENDIX A
Some definitions and information on notation
This appendix is based on Chapters 1.3, 1.5 of 1.7 of the Teaching Edition of International Tables for Crystallography (2021) and Chapter 3.5 of ITA (see also Chapters 8.1, 8.2 and 8.3 of the 5th edition of International Tables for Crystallography Volume A, 2006).
A1. operations
The notation for the space-group symmetry operations in the Symmetry Database closely follows the conventions adopted in ITA.
In order to describe the symmetry operations analytically, one introduces a coordinate system consisting of a set of basis vectors and an origin O. A can be regarded as an instruction for how to calculate the coordinates of the image point from the coordinates x, y, z of the original point X.
The equations are
These equations can be written using the matrix formalism:
Here, the symmetry operations (W, w) are given in a matrix–column form consisting of a (3 × 3) matrix (linear) part W and a (3 × 1) column (translation) part w:
The vertical line in equation (9) is to guide the eye and has no mathematical meaning.
Apart from the matrix–column pair representation of (W, w), a shorthand notation for the symmetry operations is often used. It is obtained from the left-hand side of equation (8) by omitting the terms with coefficients 0 and writing the three different rows of equation (8) in one line, separated by commas. For example, consider the last in the general position list obtained by General position for the P41 (No. 76):
would be = 0x+1y+0z, = , = 0x+0y+1z+3/4. The shorthand notation for (W, w) thus reads: .
A2. Coordinate transformations: basic results
Let a coordinate system be given with a basis (a1, a2, a3) and an origin O. The general (affine) transformation between two coordinate systems consists of two parts, a linear part and a shift of the origin.
(1) The linear part is described by a (3 × 3) matrix
that relates the primed (`new') basis to the unprimed (`old') basis (a1, a2, a3) according to
(2) A shift of the origin is defined by the shift vector
The basis vectors a1, a2, a3 are fixed at the origin O while the new basis vectors are fixed at the new origin O′. The new origin has the coordinates (p1, p2, p3) in the old coordinate system.
The general affine transformation of the coordinates of a point X in given by the column
is given by the following formula:
The G of the of the is transformed by the matrix P as follows:
where represents the P.
of the new and is the transposed matrix ofThe volume of the V changes with the transformation. The volume of the new V′ is obtained by
with det(P) being the determinant of the matrix P.
The matrix–column pairs of the symmetry operations are also changed by a change of the coordinate system. If a W, w) and in the `new'(primed) coordinate system by the pair , then the relation between the pairs (W, w) and is given by
is described in the `old' (unprimed) coordinate system by the matrix–column pair (This equation may be written more explicitly as follows:
A3. Group–subgroup relations of space groups
A3.1. Basic definitions
A subset of elements of a group is called a subgroup of () if it fulfils the group postulates with respect to the law of composition of . In general, the group itself is included among the set of subgroups of , i.e. . If is fulfilled, then the is called a proper of .
Let be a 2 is a subset of the elements of with the property that all its elements are different, and that the sets and have no element in common. Thus, the set also contains elements of . If there is another element that belongs neither to nor to , one can form another set . All elements of are different and none occurs already in or in . This procedure can be continued until each element of belongs to one of these sets. In this way the group can be partitioned into sets (called cosets) such that each element belongs to exactly one of these cosets.
of of order . Because is a proper of there must be elements that are not elements of . Let be one of them. Then the set of elementsThe partition just described is called a decomposition ( : ) into left cosets of the group relative to the group .
The sets , are called left cosets because the elements are multiplied with the new elements from the left-hand side. The procedure is called a decomposition into right cosets if the elements are multiplied with the new elements gs from the right-hand side.
The elements gp or gs are called the coset representatives. The number of cosets is called the index of in . The representative g1 is always taken as the identity element .
Two subgroups are called conjugate if there is an element such that holds. In this way, the subgroups of of the same space-group type and index are distributed into classes of conjugate subgroups that are also called conjugacy classes of subgroups. Different conjugacy classes of subgroups may contain different numbers of subgroups, i.e. have different lengths.
A normal (invariant) if it is identical to all of its conjugates, , for all , i.e. if its consists of the one only (sometimes called also a self-conjugated subgroup).
of a group is aA3.2. Type of subgroups of space groups
The following types of subgroups of space groups are to be distinguished (Hermann, 1929):
(i) A translationengleiche (equi-translational) subgroup or a t-subgroup of if the set () of translations is retained, i.e. , but the number of cosets of the decomposition [], i.e. the order of the , is reduced.
of a is called a(ii) A klassengleiche (equi-class) subgroup or a k-subgroup if the set of all translations of is reduced to but all linear parts of are retained. Then the number of cosets of the decompositions () and () is the same, i.e. the order of the is the same as that of .
of a is called a(iii) A klassengleiche or k-subgroup is called isomorphic or an isomorphic subgroup if it belongs to the same affine space-group type (isomorphism type) as .
(iv) A general or a general subgroup if it is neither a translationengleiche nor a klassengleiche It has lost translations as well as linear parts, i.e. and .
of a is calledAny
of a group is related to a specific subset of elements of and this subset defines the uniquely: different subgroups of , even those isomorphic to , correspond to different subsets of the elements of .In the Symmetry Database any of a is specified by its space-group number in ITA, the index in the group and the transformation matrix–column pair (P, p) that relates the standard bases of and of :
The column p = (p1, p2, p3) of coordinates of the origin of is referred to the coordinate system of .
The Symmetry Database, i.e. the space-group type of and the transformation matrix (P, p), are completely sufficient to define the uniquely: the transformation of the coordinate triplets of the general position of (in the standard setting) to the coordinate system of by (P, p)−1 yields exactly the subset of elements of corresponding to .
data listed in theA very important result on group–subgroup relations between space groups is given by Hermann's theorem: For any group–subgroup chain between space groups there exists a uniquely defined Hermann group, with , where is a translationengleiche of and is a klassengleiche of . The decisive point is that any group–subgroup relation between space groups of index [i] can be split into a translationengleiche chain between the space groups and and a klassengleiche chain between the space groups and . This implies that the index [i] can be expressed as [i] = [ik] · [it], where [ik] and [it] are the indices characterizing the klassengleiche and translationengleiche chains, respectively.
, called theIt may happen that either or holds. In particular, one of these equations must hold if is a maximal translationengleiche or a klassengleiche never a general subgroup.
of . In other words, a maximal of a is either aIf the maximal subgroups are known for each i] is related to the original group through a chain of maximal subgroups , , , such that , where is a maximal of of index [ij] with j = 1,…, k. The number k is finite and the relation holds, i.e. the total index [i] is the product of the indices [ij]. [In general, several chains of maximal subgroups relating the group and its could exist. However, the total indices of in calculated over the different chains should coincide with the index of in . The transformation matrices (P, p) for the symmetry reduction calculated over the different chains could differ up to a matrix belonging to the of .]
then in principle each non-maximal of a with finite index can be obtained from the data for the maximal subgroups. A non-maximal of finite index [In a similar way, one can express the transformation matrix (P, p) for the symmetry reduction as a product of the transformation matrices characterizing each of the intermediate steps : = [here the matrices relate the bases of and , i.e. ].
A3.3. Supergroups of space groups
In the previous section, the relation was seen from the viewpoint of the group . In this case was a supergroup of . As for the subgroups of , different kinds of supergroups of may be distinguished. The following definitions are obvious:
of . However, the same relation may be viewed from the group . In that case is a(i) Let be a maximal minimal supergroup of .
of . Then is called a(ii) If is a translationengleiche of , then is a translationengleiche supergroup (t-supergroup) of .
(iii) If is a klassengleiche of , then is a klassengleiche supergroup (k-supergroup) of .
(iv) If is an isomorphic isomorphic supergroup of .
of , then is an(v) If is a general general supergroup of .
of , then is aFollowing Hermann's theorem, a minimal translationengleiche (t-supergroup) or a klassengleiche (k-supergroup).
of a is either aIn general, the search for supergroups of space groups is much more difficult than the search for subgroups. One of the reasons for this is that the search for subgroups is restricted to the elements of the i]. On the other hand, there may be not only an infinite number of supergroups of a for a finite index [i] but even an uncountably infinite number of supergroups of . As an example, consider the group . Then there is an infinite number of t-supergroups of index 2 because there is no restriction for the sites of the centres of inversion and thus of the conventional origin of .
itself, whereas the search for supergroups has to take into account the whole (continuous) group of all isometries. For example, there are only a finite number of subgroups of any for any given index [A4. Normalizers of space groups
For any group–subgroup pair it is possible to define an intermediate group , called the normalizer of with respect to , as the set of all elements that map onto itself by conjugation:
The
is an intermediate group between and . It is the largest intermediate group that contains as a The may coincide either with or with .For most crystallographic applications it is enough to consider the normalizers with respect to three supergroups of the space groups: the Euclidean group is the set of all transformations that preserve the distances and angles, which are also known as isometries; the group is the set of all chirality-preserving Euclidean mappings, i.e. the isometries of the first kind (all translations and proper rotations, including screw rotations); and the affine group is the set of all affine transformations – transformations that preserve angles but not distances.
A4.1. Euclidean normalizer
The Euclidean
of a is defined as the set of symmetry operations of the Euclidean group that leave the invariant by conjugation:Each element of the Euclidean
of a maps the onto itself (as a set) by conjugation.For most space groups, the Euclidean normalizers are also space groups and can be designated by Hermann–Mauguin symbols. In the case of polar groups their normalizers contain continuous translations due to the fact that their origins cannot be fixed by symmetry. The additional continuous translations can be in one, two or three independent lattice directions and they are used for the designation of the normalizers of polar groups by modified Hermann–Mauguin symbols. These symbols contain a superscript to the lattice symbol (e.g. P1mmm), which indicates the number of independent directions of continuous translations.
A4.2. Chirality-preserving Euclidean normalizer
The group is the group of all chirality-preserving Euclidean mappings and a i.e. the chirality-preserving Euclidean is a of the Euclidean .
of the Euclidean group ,This Sohncke space group, in other words if only contains symmetry operations of the first kind (Flack, 2003).
is defined if is aAmong the 230 space-group types there are 65 that belong to a Sohncke space-group type. The Sohncke space-group types are composed of 11 pairs of enantiomorphic or chiral space-group types and the other 43 space-group types are known as achiral. A is defined as chiral if its Euclidean does not have any of the second kind. The chirality-preserving Euclidean coincides with the Euclidean for the 11 pairs of enantiomorphic space-group types. This means that their normalizers only contain operations of the first kind. For the rest of the space-group types is a non-centrosymmetric of index 2 of the Euclidean normalizer.
A4.3. Affine normalizer
The affine group is the group of all affine mappings, a
of the Euclidean group and of all space groups. The affine is constructed for each in the following way:Either the affine i.e. .
of a group is a of the Euclidean or they coincide,For cubic, hexagonal, trigonal, tetragonal and some orthorhombic space groups the Euclidean and affine normalizers coincide. For the remaining 38 orthorhombic space groups the affine ITA).
is isomorphic to the Euclidean of highest possible symmetry. The affine normalizers of monoclinic and triclinic space groups are not isomorphic to any and cannot be characterized by a modified Hermann–Mauguin symbol. In these cases, the affine normalizers are listed in parametric form (see Table 3.5.2.6 ofSupporting information
A guide to using the program Interactive 3D general-position visualizer. DOI: https://doi.org/10.1107/S1600576723009068/dv5006sup1.mp4
Footnotes
1The book by Müller is also available in German (Symmetriebeziehungen zwischen verwandten Kristallstrukturen; Vieweg+Teubner, 2012) and in Spanish (Relaciones de simetría entre estructuras cristalinas; Síntesis, 2013).
2The formulation means is the set of the products g2hj of g2 with all elements .
Acknowledgements
The Symmetry Database has been developed as part of an ongoing project between the International Union of Crystallography, eFaber Soluciones Inteligentes SL (Bilbao) and the Bilbao Crystallographic Server team. Most of the additional crystallographic data for the space groups, their subgroups and supergroups, and program algorithms have been provided by the Bilbao Crystallographic Server (https://www.cryst.ehu.es). Open access funding enabled and organized by Projekt DEAL.
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