International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 |
International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 90-96
Section 2.1.8. The subgroup graphs^{a}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and ^{b}Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain |
The group–subgroup relations between the space groups may also be described by graphs. This way is chosen in Chapters 2.4 and 2.5 . Graphs for the group–subgroup relations between crystallographic point groups have been published, for example, in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and in IT A (2005), Figs. 10.1.3.2 and 10.1.4.3 . Three kinds of graphs for subgroups of space groups have been constructed and can be found in the literature:
A complete collection of graphs of the first two kinds is presented in this volume: in Chapter 2.4 those displaying the translationengleiche or t-subgroup relations and in Chapter 2.5 those for the klassengleiche or k-subgroup relations. Neither type of graph is restricted to maximal subgroups but both contain t- or k-subgroups of higher indices, with the exception of isomorphic subgroups, cf. Section 2.1.8.3 below.
The group–subgroup relations are direct relations between the space groups themselves, not between their types. However, each such relation is valid for a pair of space groups, one from each of the types, and for each space group of a given type there exists a corresponding relation. In this sense, one can speak of a relation between the space-group types, keeping in mind the difference between space groups and space-group types, cf. Section 1.2.5.3 .
The space groups in the graphs are denoted by the standard HM symbols and the space-group numbers. In each graph, each space-group type is displayed at most once. Such graphs are called contracted graphs here. Without this contraction, the more complex graphs would be much too large for the page size of this volume.
The symbol of a space group is connected by uninterrupted straight lines with the symbols of all its maximal non-isomorphic subgroups or minimal non-isomorphic supergroups. In general, the maximal subgroups of are drawn on a lower level than ; in the same way, the minimal supergroups of are mostly drawn on a higher level than . For exceptions see Section 2.1.8.3. Multiple lines may occur in the graphs for t-subgroups. They are explained in Section 2.1.8.2. No indices are attached to the lines. They can be taken from the corresponding subgroup tables of Chapter 2.3 , and are also provided by the general formulae of Section 1.2.8 . For the k-subgroup graphs, they are further specified at the end of Section 2.1.8.3.
Let be a space group and () the normal subgroup of all its translations. Owing to the isomorphism between the factor group and the point group , see Section 1.2.5.4 , according to the first isomorphism theorem, Ledermann & Weir (1996), t-subgroup graphs are the same (up to the symbols) as the corresponding graphs between point groups. However, in this volume, the graphs are not complete but are contracted, displaying each space-group type at most once. This contraction may cause the graphs to look different from the point-group graphs and also different for different space groups of the same point group, cf. Example 2.1.8.2.1.
One can indicate the connections between a space group and its maximal subgroups in different ways. In the contracted t-subgroup graphs one line is drawn for each conjugacy class of maximal subgroups of . Thus, a line represents the connection to an individual subgroup only if this is a normal maximal subgroup of , otherwise it represents the connection to more than one subgroup. The conjugacy relations are not necessarily transferable to non-maximal subgroups, cf. Example 2.1.8.2.2. On the other hand, multiple lines are possible, see the examples. Although it is not in general possible to reconstruct the complete graph from the contracted one, the content of information of such a graph is higher than that of a graph which is drawn with simple lines only.
The graph for the space group at its top also contains the contracted graphs for all subgroups which occur in it, see Example 2.1.8.2.3.
Owing to lack of space for the large graphs, in all graphs of t-subgroups the group , No. 1, and its connections have been omitted. Therefore, to obtain the full graph one has to supplement the graphs by at the bottom and to connect by one line to each of the symbols that have no connection downwards.
Within the same graph, symbols on the same level indicate subgroups of the same index relative to the group at the top. The distance between the levels indicates the size of the index. For a more detailed discussion, see Example 2.1.8.2.2. For the sequence and the numbers of the graphs, see the paragraph after Example 2.1.8.2.2.
Example 2.1.8.2.1
Compare the t-subgroup graphs in Figs. 2.4.4.2 , 2.4.4.3 and 2.4.4.8 of , No. 52, , No. 53, and , No. 64, respectively. The complete (uncontracted) graphs would have the shape of the graph of the point group with at the top (first level), seven point groups^{7} (, , , , , and ) in the second level, seven point groups (, , , , , and ) in the third level and the point group at the bottom (fourth level). The group is connected to each of the seven subgroups at the second level by one line. Each of the groups of the second level is connected with three groups of the third level by one line. All seven groups of the third level are connected by one line each with the point group 1 at the bottom.
The contracted graph of the point group would have at the top, three point-group types (, and ) at the second level and three point-group types (, m and ) at the third level. The point group 1 at the bottom would not be displayed (no fourth level). Single lines would connect with 222, with 2, with 2, with m and with ; a double line would connect with m; triple lines would connect with , with and 222 with 2.
The number of fields in a contracted t-subgroup graph is between the numbers of fields in the complete and in the contracted point-group graphs. The graph in Fig. 2.4.4.2 of , No. 52, has six space-group types at the second level and four space-group types at the third level. For the graph in Fig. 2.4.4.3 of , No. 53, these numbers are seven and five and for the graph in Fig. 2.4.4.8 of , No. 64 (formerly ), the numbers are seven and six. However, in all these graphs the number of connections is always seven from top to the second level and three from each field of the second level downwards to the ground level, independent of the amount of contraction and of the local multiplicity of lines.
Example 2.1.8.2.2
Compare the t-subgroup graphs shown in Fig. 2.4.1.1 for , No. 221, and Fig. 2.4.1.5 , , No. 225. These graphs are contracted from the point-group graph . There are altogether nine levels (without the lowest level of ). The indices relative to the top space groups and are 1, 2, 3, 4, 6, 8, 12, 16 and 24, corresponding to the point-group orders 48, 24, 16, 12, 8, 6, 4, 3 and 2, respectively. The height of the levels in the graphs reflects the index; the distances between the levels are proportional to the logarithms of the indices but are slightly distorted here in order to adapt to the density of the lines.
From the top space-group symbol there are five lines to the symbols of maximal subgroups: The three symbols at the level of index 2 are those of cubic normal subgroups, the one (tetragonal) symbol at the level of index 3 represents a conjugacy class of three, the symbol , No. 166, at the level of index 4 represents a conjugacy class of four subgroups.
The graphs differ in the levels of the indices 12 and 24 (orthorhombic, monoclinic and triclinic subgroups) by the number of symbols (nine and seven for index 12, five and three for index 24). The number of lines between neighbouring connected levels depends only on the number and kind of symbols in the upper level.
However, for non-maximal subgroups the conjugacy relations may not hold. For example, in Fig. 2.4.1.1 , the space group has three normal maximal subgroups of type and is thus connected to its symbol by a triple line, although these subgroups are conjugate subgroups of the non-minimal supergroup .
Example 2.1.8.2.3
The t-subgroup graphs in Figs. 2.4.1.1 and 2.4.1.5 contain the t-subgroups of the summits and and their relations. In addition, the t-subgroup graph of includes the t-subgroup graphs of the cubic summits , , and , those of the tetragonal summits , , , , , , and , those of the rhombohedral summits , , , and etc., as well as the t-subgroup graphs of several orthorhombic and monoclinic summits. The graph of the summit includes analogously the graphs of the cubic summits , , and , of the tetragonal summits , etc., also the graphs of rhombohedral summits etc. Thus, many other graphs are included in the two basic graphs and can be extracted from them. The same holds for the other graphs displayed in Figs. 2.4.1.2 to 2.4.4.8 : each of them includes the contracted graphs of all its subgroups as summits. For this reason one does not need 229 or 218 different graphs to cover all t-subgroup graphs of the 229 space-group types but only 37 ( can be excluded as trivial).
The preceding Example 2.1.8.2.3 suggests that one should choose the graphs in such a way that their number can be kept small. It is natural to display the `big' graphs first and later those smaller graphs that are still missing. This procedure is behind the sequence of the t-subgroup graphs in this volume.
For the index of a maximal t-subgroup, Lemma 1.2.8.2.3 is repeated: the index of a maximal non-isomorphic subgroup is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups . The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups . The index is 2, 3 or 4 for cubic space groups .
There are 29 graphs for klassengleiche or k-subgroups, one for each crystal class with the exception of the crystal classes 1, and with only one space-group type each: , No. 1, , No. 2, and , No. 174, respectively. The sequence of the graphs is determined by the sequence of the point groups in IT A, Table 2.1.2.1 , fourth column. The graphs of , and are nearly trivial, because to these crystal classes only two space-group types belong. The graphs of with 22, of with 28 and of with 20 space-group types are the most complicated ones.
Isomorphic subgroups are special cases of k-subgroups. With the exception of both partners of the enantiomorphic space-group types, isomorphic subgroups are not displayed in the graphs. The explicit display of the isomorphic subgroups would add an infinite number of lines from each field for a space group back to this field, or at least one line (e.g. a circle) implicitly representing the infinite number of isomorphic subgroups, see the tables of maximal subgroups of Chapter 2.3 .^{8} Such a line would have to be attached to every space-group symbol. Thus, there would be no more information.
Nevertheless, connections between isomorphic space groups are included indirectly if the group–subgroup chain encloses a space group of another type. In this case, a space group may be a subgroup of a space group and a subgroup of , where and belong to the same space-group type. The subgroup chain is then – – . The two space groups and are not identical but isomorphic. Whereas in general the label for the subgroup is positioned at a lower level than that for the original space group, for such relations the symbols for and can only be drawn on the same level, connected by horizontal lines. If this happens at the top of a graph, the top level is occupied by more than one symbol (the number of symbols in the top level is the same as the number of symmorphic space-group types of the crystal class).
Horizontal lines are drawn as left or right arrows depending on the kind of relation. The arrow is always directed from the supergroup to the subgroup. If the relation is two-sided, as is always the case for enantiomorphic space-group types, then the relation is displayed by a pair of horizontal lines, one of them formed by a right and the other by a left arrow. In the graph in Fig. 2.5.1.5 for crystal class , the connections of with and with are displayed by double-headed arrows instead. Furthermore, some arrows in Fig. 2.5.1.5 , crystal class , and Fig. 2.5.1.6 , , are dashed or dotted in order to better distinguish the different lines and to increase clarity.
The different kinds of relations are demonstrated in the following examples.
Example 2.1.8.3.1
In the graph in Fig. 2.5.1.1 , crystal class 2, a space group may be a subgroup of index 2 of a space group by `Loss of centring translations'. On the other hand, subgroups of in the block `Enlarged unit cell', belong to the type , see the tables of maximal subgroups in Chapter 2.3 . Therefore, both symbols are drawn at the same level and are connected by a pair of arrows pointing in opposite directions. Thus, the top level is occupied twice. In the graph in Fig. 2.5.1.2 of crystal class m, both the top level and the bottom level are each occupied by the symbols of two space-group types.
Example 2.1.8.3.2
There are four symbols at the top level of the graph in Fig. 2.5.1.4 , crystal class 222. Their relations are rather complicated. Whereas one can go (by index 2) from directly to a subgroup of type and vice versa, the connection from directly to is one-way. One always has to pass on the way from to a subgroup of the types or . Thus, the only maximal subgroup of among these groups is . One can go directly from to but not to etc.
Because of the horizontal connecting arrows, it is clear that there cannot be much correspondence between the level in the graphs and the subgroup index. However, in no graph is a subgroup positioned at a higher level than the supergroup.
Example 2.1.8.3.3
Consider the graph in Fig. 2.5.1.6 for crystal class . To the space group , No. 65, belong maximal non-isomorphic subgroups of the 11 space-group types (from left to right) , No. 72, , No. 63, , No. 74, , No. 59, , No. 55, , No. 50, , No. 51, , No. 53, , No. 66, , No. 47, and , No. 71. Although all of them have index 2, their symbols are positioned at very different levels of the graph.
The table for the subgroups of in Chapter 2.3 lists 22 non-isomorphic k-subgroups of index 2, because some of the space-group types mentioned above are represented by two or four different subgroups. This multiplicity cannot be displayed by multiple lines because the density of the lines in some of the k-subgroup graphs does not permit this kind of presentation, e.g. for . The multiplicity may be taken from the subgroup tables in Chapter 2.3 , where each non-isomorphic subgroup is listed individually.
Consider the connections from , No. 65, to , No. 55. There are among others: the direct connection of index 2, the connection of index 4 over , No. 72, the connection of index 8 over , No. 74, and , No. 51. Thus, starting from the same space group of type one arrives at different space groups of the type with different unit cells but all belonging to the same space-group type and thus represented by the same field of the graph.
The index of a k-subgroup is restricted by Lemma 1.2.8.2.3 and by additional conditions. For the following statements one has to note that enantiomorphic space groups are isomorphic.
There are no graphs for plane groups in this volume. The four graphs for t-subgroups of plane groups are apart from the symbols the same as those for the corresponding space groups: –, –, – and –, where the graphs for the space groups are included in the t-subgroup graphs in Figs. 2.4.1.1 , 2.4.3.1 , 2.4.2.1 and 2.4.2.3 , respectively.
The k-subgroup graphs are trivial for the plane groups , , , , and because there is only one plane group in its crystal class. The graphs for the crystal classes and consist of two plane groups each: and , and . Nevertheless, the graphs are different: the relation is one-sided for the tetragonal plane-group pair as it is in the space-group pair – and it is two-sided for the hexagonal plane-group pair as it is in the space-group pair (81)– (82). The graph for the three plane groups of the crystal class m corresponds to the space-group graph for the crystal class 2.
Finally, the graph for the four plane groups of crystal class has no direct analogue among the k-subgroup graphs of the space groups. It can be obtained, however, from the graph in Fig. 2.5.1.3 of crystal class by removing the fields of , No. 15, and , No. 11, with all their connections to the remaining fields. The replacements are then: , No. 12, by , No. 9, , No. 10, by , No. 6, , No. 13, by , No. 7, and , No. 14, by , No. 8.
If a subgroup is not maximal then there must be a group–subgroup chain – of maximal subgroups with more than two members which connects with . There are three possibilities: may be a t-subgroup or a k-subgroup or a general subgroup of . In the first two cases, the application of the graphs is straightforward because at least one of the graphs will permit one to find the possible chains directly. If is a k-subgroup of , isomorphic subgroups have to be included if necessary. If is a general subgroup of one has to combine t- and k-subgroup graphs.
There is, however, a severe shortcoming to using contracted graphs for the analysis of group–subgroup relations, and great care has to be taken in such investigations. All subgroups with the same space-group type are represented by the same field of the graph, but these different non-maximal subgroups may permit different routes to a common original (super)group.
Example 2.1.8.5.1
An example for translationengleiche subgroups is provided by the group–subgroup chain – of index 12. The contracted graph may be drawn by the program Subgroupgraph from the Bilbao Crystallographic Server, http://www.cryst.ehu.es/ . It is shown in Fig. 2.1.8.1; each field represents all occurring subgroups of a space-group type: , No. 139, represents three subgroups, , No. 166, represents four subgroups, and , No. 12, represents nine subgroups belonging to two conjugacy classes. Fig. 2.1.8.1 is part of the contracted total graph of the translationengleiche subgroups of the space group , which is displayed in Fig. 2.4.1.5 . With Subgroupgraph one can also obtain the complete graph between and the set of all nine subgroups of the type . It is too large to be reproduced here.
More instructive are the complete graphs for different single subgroups of the type of . They can also be obtained with the program Subgroupgraph with the exception of the direction indices. In Fig. 2.1.8.2 such a `complete' graph is displayed for one of the six subgroups of type of index 12 whose monoclinic axes point in the directions of . Similarly, in Fig. 2.1.8.3 the complete graph is drawn for one of the three subgroups of of index 12 whose monoclinic axes point in the directions of . It differs markedly from the contracted graph and from the first complete graph. It is easily seen that it may be very misleading to use the contracted graph or the wrong individual complete graph instead of the right individual complete graph.
In a contracted graph, no basis transformations and origin shifts can be included because they are often ambiguous. In the complete graphs the basis transformations and origin shifts should be listed if these graphs display structural information and not just group–subgroup relations. The group–subgroup relations do not depend on the coordinate systems relative to which the groups are described. On the other hand, the coordinate system is decisive for the coordinates of the atoms of the crystal structures displayed and connected in a Bärnighausen tree. Therefore, for the description of structural relations in a Bärnighausen tree knowledge of the transformations (matrix and column) is essential and great care has to be taken to list them correctly, see Chapter 1.6 and Example 2.1.8.5.4. If one wants to list the transformations in subgroup graphs, one can use the transformations which are presented in the subgroup tables.
The use of the graphs of Chapters 2.4 and 2.5 is advantageous if general subgroups, in particular those of higher indices, are sought. As stated by Hermann's theorem, Lemma 1.2.8.1.2 , a Hermann group always exists and it is uniquely determined for any specific group–subgroup pair . If the subgroup relation is general, the group divides the chain into two subchains, the chain between the translationengleiche space groups and that between the klassengleiche space groups . Thus, however long and complicated the real chain may be, there is always a chain for which only two graphs are needed: a t-subgroup graph for the relation between and and a k-subgroup graph for the relation between and .
For a given pair of space-group types and a given index [i], however, there could exist several Hermann groups of different space-group types. The graphs of this volume are very helpful in their determination. The index [i] is the product of the index , which is the ratio of the crystal class orders of and , and the index of the lattice reduction from to , . The graphs of t-subgroups (Chapter 2.4 ) are used to find the types of subgroups of with index , belonging to the crystal class of . From the k-graphs (Chapter 2.5 ) it can be seen whether can be a supergroup of with index and thus a possible Hermann group . The following two examples illustrate this two-step procedure for the determination of the space-group types of Hermann groups.
Example 2.1.8.5.2
Consider a pair of the space-group types with index [i] = 24. The factorization of the index [i] into and follows from the crystal classes of and . From the graph of t-subgroups of (Fig. 2.4.1.1 ) one finds that there are two space-group types, namely P2/m and C2/m, of the relevant crystal class (2/m) and index . Checking the graph (Fig. 2.5.1.3 ) of the k-subgroups of the space groups of the crystal class 2/m confirms that space groups of both space-group types P2/m and C2/m have (maximal) subgroups, i.e. space groups of both space-group types are Hermann groups for the pair of index [i] = 24, depending on the individual space group .
Example 2.1.8.5.3
The determination of the space-group types of the Hermann groups for the pair (No. 229) > Cmcm (No. 63) of index [i] = 12 follows the same procedure as in the previous example. The index [i] = 12 is factorized into and taking into account the orders of the point groups of and . The graph of t-subgroups of (Fig. 2.4.1.9 ) shows that the subgroups of the space-group types Fmmm and Immm are candidates for Hermann groups. Reference to the graph of k-subgroups of the crystal class mmm (Fig. 2.5.1.6 ) indicates that Immm has no maximal subgroups of Cmcm type, i.e. only space groups of Fmmm type can be Hermann groups for the pair of index [i] = 12.
Apart from the chains that can be found by the above considerations, other chains may exist. In some relatively simple cases, the graphs of this volume may be helpful to find such chains. However, one has to take into account that the tabulated graphs are contracted ones. In particular this means that they contain nothing about the numbers of subgroups of a certain kind and on their relations, for example conjugacy relations.
The following practical example may display the situation. It is based on the combination of the graphs of Chapters 2.4 and 2.5 with the subgroup tables of Chapters 2.2 and 2.3 .
Example 2.1.8.5.4
Let be a space group of type , No. 221. What are its subgroups of type I4/mcm, No. 140, and index 6?
As the order of the crystal class is reduced from cubic (48) to tetragonal (16) by index 3, the reduction of the translation subgroup must have index 2. To find the Hermann group , we look in the subgroup table of for tetragonal t-subgroups and find one class of three conjugate maximal t-subgroups of crystal class 4/mmm: P4/mmm, No. 123. By each of the three conjugate subgroups one of the axes a, b or c is distinguished. As this distinguished direction is kept in the other steps, one can take one of the conjugates as the representative and can continue with the consideration of only this representative. For the representative direction we choose the c axis, because this is the standard setting of P4/mmm. The relations of the other conjugates can then be obtained by replacing c by a or b.
The coordinate systems of and P4/mmm are the same, but is possible for P4/mmm. In the subgroup table of P4/mmm one looks for subgroups of type I4/mcm and index 2 and finds four non-conjugate subgroups with the same basis but different origin shifts. There can be no other subgroups of type I4/mcm because P4/mmm is the only possible Hermann group . Are there other chains from to the subgroups ?
Such a new chain of subgroups must have two steps. The first one leads from to a k-subgroup. In the graph for k-subgroups of one finds two subgroups of index 2, namely and . One finds from the corresponding graphs of t-subgroups or from the subgroup tables that only has subgroups of space-group type I4/mcm. The subgroup tables of and show that there are two non-conjugate subgroups of type which each have one conjugacy class of three subgroups of type I4/mcm. For the preferred direction c only one of the conjugate subgroups is relevant. Therefore, there are two subgroups I4/mcm of index 2 belonging to chains passing . It follows that two of the four subgroups obtained from Hermann's group are also subgroups of and two are not.
The two common subgroups are found by comparing their origin shifts from , which must be the same for both ways. The use of (4 × 4) matrices is convenient. The relevant equations for the bases and origin shifts are:leading to an origin shift of andleading to an origin shift of , which is equivalent to because an integer origin shift means only the choice of another conventional origin.
The result is Fig. 2.1.8.4, which is a complete graph, i.e. each field of the graph represents exactly one space group. The names of the substances belonging to the different subgroups show that the occurrence of such unexpected relations is not unrealistic. The crystal structures of KCuF_{3} and LT-SrTiO_{3} both belong to the space-group type I4/mcm. They can be derived by different distortions from the same ideal perovskite ABX_{3} structure, space group (Figs. 2.1.8.4 and 2.1.8.5).
The subgroup realized by LT-SrTiO_{3} is not a subgroup of . The other subgroup, which is realized by KCuF_{3}, is a subgroup of . This cannot be concluded from the contracted graphs, but can be seen from the combination of the graphs with the tables or from the complete graph (Billiet, 1981; Koch, 1984; Wondratschek & Aroyo, 2001).
The remaining two subgroups do not form symmetries of distorted perovskites. The listed orbits of the Wyckoff positions, 4a, 4b, 4c, 4d and 8e, are all extraordinary orbits, i.e. they have more translations than the lattice of I4/mcm, see Engel et al. (1984). In Strukturbericht 1 (1931) the possible space groups for the cubic perovskites are listed on p. 301; there are five possible space groups from P23 to . The latter space-group symbol is framed and is considered to be the true symmetry of cubic perovskites. The other space groups are t-subgroups of . They would be taken if for some reason the site symmetries of the orbits would contradict the site symmetries of . Similarly, the true symmetry of these tetragonal perovskite derivatives would be P4/mmm with the original (only tetragonally distorted) lattice and not I4/mcm.^{9} For the two (empty) subgroups I4/mcm a distorted variant of the perovskite structure does not exist. Other special positions or the general position of the original cubic space group have to be occupied if the space group I4/mcm shall be realized. This example shows clearly the difference between the subgroup graphs of group theory and the Bärnighausen trees of crystal chemistry.
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