CLASSIFICATION OF SUPERSTRUCTURES BY SYMMETRY

CLASSIFICATION

OF

SUPERSTRUCTURES

BY

RYOICHI SADANAGA, TOSHIYUKI SAWADA and Mineralogical

SYMMETRY

KAZUMASA OHSUMI*

Institute, Faculty of Science, University of Tokyo, Hongo, Tokyo, Japan and

Chemical

Research

KAZUHIDE

KAMIYA

Laboratories,

Central

Takeda Chemical Industries, Ltd., Jusohonmachi,

Research

Division,

Yodogawa-ku, Osaka, Japan

A superstructure is a structure consisting of substructures which are either strictly equal or nearly equal to each other in configuration, and the way of arrangement of substructures in a superstructure is here called a framework. Symmetry operations are divided into two, global and local, according respectively as they are effective everywhere in a crystal space or only within a subspace of it. A partial operation is defined as an operation of superposition of A upon B (B#A), A and B being subspaces of a crystal space. The substructure is either locally or partially symmetric, and the framework is either globally or partially symmetric. Therefore, superstructures are classified by their symmetry features into four types, a typical example of each being mentioned as : the first type with a locally symmetric substructure and a globally symmetric framework (polysynthetic structure), the second type with a partially symmetric substructure and a globally symmetric framework (supersymmetric structure), the third type with locally symmetric substructure and a partially symmetric framework (OD-structure), and the fourth type with a partially symmetric substructure and a partially symmetric framework (quasi-crystalline state such as a melt).

INTRODUCTION A superstructure

is defined in this

paper as a structure consisting of substructures which are either strictly equal or nearly equal to each other in configuration. A superstructure is therefore determined by the substructure and the configuration of points which represents the way of arrangement of the substructures. For the sake of concise description, we shall call this way of arrangement of the substructures the framework (of the superstructure). The substructure used in the above definition of superstructure is composed of a number of atoms, more than one molecule (Manuscript recieved August 6, 1979) * Present address : Department of Crystallography 3012 Bern, Sahlistrasse 6, Switzerland.

in the case of a molecular crystal. A part of the entire structure is recognized as a substructure if it has a certain kind of symmetry feature which is independent from the space group or other symmetry feature of the entire structure, or if it appears as a structural unit common to different structural modifications (polymorphs or polytypes) of the substance even if all the symmetry elements of the part merge into the symmetry of the entire structure to form its space group. In this connection, cases may be conceived in which the substructure is also composed of sub-substructures, but these multiply complex structures will not be and Structure

Sciences , University

of Bern, CH-

considered here because no new principle is required for the interpretation of these cases. From this substructure-framework construction of the superstructure, we can derive a reasonable method of classifying superstructures by means of the symmetry feature of the substructure and that of the framework and can thus arrive at a deeper understanding of the nature of crystalline and quasi-crystalline states as demonstrated in this paper. SPACE GROUPOID As we have done elsewhere (Sadanaga and Ohsumi, 1975; 1979),we consider in this paper three types of operation, each of which superposes a configuration of points upon itself or another. The first is a symmetry operation which is effective everywhere in a crystal space and is called a global operation. It is an ordinary space-group operation which brings the entire structure to superpose upon itself. The second is a symmetry operation called a local operation which is effective only within a certain subspace of a crystal space and brings the subspace to superpose upon itself. The third type of operation will be specified as partial. It operates only on a subspace A of a crystal space to bring it to superposition upon another B. Though this partial operation p brings A to superpose upon B, namely, p (A) = B, it cannot operate on B, p(B) being geometrically meaningless ; when AABC and AA'B'C' are given as AABC AA'B'C', AABC will be brought to superposition upon by an appropriate motion p, but no triangle has been given upon which AA'B'C' is brought to superposition by the motion p. Because the symmetry operation is a motion which brings a certain configuration of points to super-

position upon itself, the partial operation .7) will not be a symmetry operation. However, for the sake of convenience, we shall employ in this paper the term 'partially symmetric' for the description of a configuration of points with such subspaces A and B as given above. When a structure X consists of a finite number of substructures in such local symmetries as with space groups isomorphic with each other (or with the same space group which is a mathematically special but practically general case), the set of all operations, each of which brings each of the substructures to superpose upon itself or another, forms a groupoid defined by Brandt (1926) and introduced into crystallography by Dornberger-Schiff (1957). We call the structure I specified above a space-groupoid structure. A finitely extended model of it is illustrated in Fig. 1 in which substructures are regular hexagons of equal size distributed in a three-dimensional space, just as benzene molecules in a liquid or vapour state. A space groupoid consists of two classes of operations of superposition : one class is composed of local operations, each bringing each of the substructures to superpose upon. itself, and the other is composed of opera.bons, each bringing each of the substructures to superpose upon another. A complete set of local operations which bring one of the substructures to superpose upon

itself obviously constitutes a group. According to Loewy (1927), we shall call this group K, the kernel of the groupoid, and the set H of all the operations of the second class the hull of the groupoid, in which

PRINCIPLE According

OF THE

CLASSIFICATION

to the

definition

previously

given, the substructure has either a set of local operations or a set of partial opera-

partial operations are necessarily contained. The groupoid M can be decomposed (Loewy, 1927) as follows, where the plus sign stands for a set-theoretical union,

tions. In some cases such as organic crystals consisting of molecules of complex conforma-

In (1), hi indicates an element of the hull which brings the i th substructure X to superpose upon the substructure X, representing the kernel K0. Since each of the elements of Ko brings X, to superpose upon itself, Kok expresses the set of all such elements in the hull that bring Xi to super-

holds only approximately, and part A of the substructure is only nearly equal in configuration to part B in the case of partial operation (Kamiya, 1979). However, even in these approximate cases, a clear distinction can be drawn between the local operation and the partial one. If the substructure is locally symmetric, the superstructure will be either an ordinary space-group structure or a space-groupoid structure according respectively as all the symmetry elements of the substructure become part of the space group of the superstructure or some of the symmetry elements of the former are kept independent of the symmetry of the latter. On the other hand, if the substructure is only

pose upon X0, and hilKohi the set of all such elements in the hull that bring Xi to superpose upon X,. Hence, the union of all the off-diagonal terms in (1) constitutes the hull if The diagonal term hi-I-Kok brings Xi to superpose upon itself and is a group isomorphic with the kernel Ko. It will now be obvious that a space groupoid can describe the structures of the liquid and vapour states of a molecular substance, where conditions imposed upon the elements of the hull will in general be weak ones. When a space-groupoid structure is in a solid state, these conditions will become stronger because of a smaller degree of freedom in the orientations and positions of the substructures in it.

tion, each of the 'pseudo' character

operations , their

tends

symmetry

to bear relation

partially symmetric, the superstructure will be neither a space-group structure nor a space-groupoid one. Next, let us proceed to the examination of the operations of superposition in the framework. If the superstructure under examination is ideally crystalline, the

symmetry of its framework will naturally be one of the 230 space groups, according to which the substructures are distributed over the entire superstructure. This means that if a space-groupoid structure is ideally crystalline, the hull of the space groupoid must contain a space group, though even in this case the hull must also contain partial operations in the forms of Kok and h1-1K0 where hi expresses a global operation. From the above consideration, four types of superstructure will be derived. The first type is the case in which the substructure is locally symmetric and the framework is globally symmetric. The typical examples of this type will be the polysynthetic structures discovered and described by Ito (1950). In the second type, the substructure is partially symmetric and the framework is globally symmetric. The superstructures of this type fall in the category of the supersymmetric tructure proposed by Zorkii (1978). The above two types represent the cases of ideal crystal. The third type is that while the substructure is locally symmetric, the framework is partially symmetric. The OD-structures studied by Dornberger-Schiff (1956; 1964) in which stacking faults are found in their layer structures come under this class. The fourth type will be the case in which both the substructure and the framework are partially symmetric. The state to which superstructures of this type belong may be specified as 'quasi-crystalline'. The above way of classification is not considered to be a clear-cut one, especially because of frequent appearance of pseudosymmetry operations and partial pseudooperations which necessarily give rise to intermediate types. However, since the present classification offers those four typical cases, we are convinced that it can claim a

proper place in the study of superstructures. EXAMPLES (1) Polysynthetic structures Examples of this type of superstructures have been amply demonstrated by Ito (1950). Therefore, in order only to facilitate the understanding of space-groupoid structures, we shall show a simple but fictitious example in Fig. 2. A substructure in the form of a slab and with two-fold rotation axes and the global symmetry of a framework are shown respectively in (A) and

(B) in Fig. 2. The symmetry of the resulting superstructure given in (C) contains local two-fold rotation axes, global reflection and glide planes, and partial glide planes shown by the horizontal broken lines. (2) Supersymmetric structures Zorkii (1978) discovered a peculiar symmetry feature that if a molecular crystal contains more than one molecule in an asymmetric unit of its space group, these molecules can be brought to superposition upon each other by an operation which includes a rotation of 360°/n and a shift along the rotation axis, and this axis has some special orientation with respect to the crystal lattice. He called such an effect `supersymmetry' . Independently of Zorkii's work, we have been engaged in the study of the same phenomenon for some time, and we shall mention in this paper two salient examples : one is the structure of picric acid and the other that of phenol. In Fig. 3, two molecules of picric acid C8H3N307 in an asymmetric unit of its structure are shown in (A), each in the form of a paper kite with its plane inclined to the plane of projection. One of these two molecules in an asymmetric unit is brought to superpose upon the other by a partial pseudo-42 operation in the direction of the b axis of the crystal. The asymmetric units, each built up of two molecules as above, will then be arranged according to the space group Pca 21with a=9.26, b= 19.13 and c=9.71 A as shown in (B) to form a supersymmetric structure in (C). In this superstructure, an asymmetric unit in row I in (C) is brought to superposition upon a unit adjacent to it in the same row by a partial pseudo-48 operation, and the same relation applies to the asymmetric units in row II by a partial pseudo-41 operation, both these axes being parallel to the b axis of

the crystal. These partial pseudo-43 and -41 operations are the results of product between the partial pseudo-42 operation of the substructure and the global operations of the framework. Phenol C6H60forms a monoclinic crystal with a=6.02, b=9.04, c= 15.18 A, y=90.0° and the space group P 21. Its structure projected onto (100) is shown in Fig. 4

(Gillier-Pandraud, 1967). As will be apparent in this figure, three molecules in an asymmetric unit of the space group are arranged according to a local pseudo-3, axis parallel to the a axis of the crystal. Because this supersymmetry is local though approximately, the superstructure may also be looked upon as a space-groupoid structure. However, a subtle difference seems to exist between those structures strictly according to a space groupoid and this case of phenol; while the former are the results of polysymmetric synthesis on strictly symmetric two-dimensional substructures (Ito, 1950; Ito and Sadanaga, 1976), the latter seems to suggest the possibility that it has grown from a liquid crystal in a nematic state, its substructure being one-dimensional and pseudo-symmetric. A number of examples of supersymmetric structures have been found in organic compounds with simple molecules (Sawada, 1978; 1979) and those with complex molecules (Kamiya, 1979). It is to be emphasized the fact that though these examples have been picked up quite arbitrarily from the results of the structure

determinations carried out by these workers and those found in literatures, any of them has never failed to reveal supersymmetry. Therefore, we can now draw a conclusion that if more than one molecule is contained in the asymmetric unit of a molecular crystal, supersymmetry will appear to relate these crystallographically unrelated molecules to each other and to the entire structure. All the examples of supersymmetry so far discovered are confined to molecular crystals, but it is expected for this type of symmetry to be found in inorganic structures upon closer examinations in future. (3) OD-structures Since structures of this type have been intensively studied by Dornberger-Schiff (1956; 1964), no further description seems to be necessary for the present purpose. In Fig. 5, a case is illustrated in which the substructure is locally symmetric and the framework is partially symmetric with

stacking vectors in an irregular sequence. (4) Quasi-crystalline states Those cases in which both the substructure and the framework are partially symmetric have not been worked out in any details. However, the growth in symmetry of the substructure followed by that of the framework into partial operations and then into local and global ones will become an important subject in the future structural investigations of melts, solutions and liquid crystals. REFERENCES Brandt, * H. (1926), Ober eine Verallgemeinerung des Gruppenbegriffes. Math. Ann., 96, 360-366. Dornberger-Schiff, K. (1956), On Order-Disorder * Structures (OD-Structures). Acta Cryst., 9, 593-601. *--. (1957), On symmetry in structures with stacking disorder (order-disorder structures) consisting of layers of one kind only. Acta Cryst., 10, 820.

. (1964), Grundziige einer Theorie der ODStrukturen aus Schichten. Abhdl. Deuts. Akad. Wiss. Berlin, 3, Akademie-Verlag, Berlin. Gillier-Pandraud, H. (1967), Structure cristalline * du phenol. Bull. Soc. Chim. France, 6, 1988-1995. Ito, T. (1950), X-Ray Studies on Polymorphism, * Maruzen, Tokyo. *Ito, T. and Sadanaga, R. (1976), On the Crystallographic Space Groupoid. Proc. Japan Acad., 52, 119-121. Kamiya, *K. (1979), In the preparation. Loewy, L. (1927), UUber abstrakt * definierte Transmutationssysteme oder Mischgruppen. J. Math., 157, 239-254. Sadanaga, * R. and Ohsumi, K. (1975), Symmetric Relations between the Vector Set and the Crystal Structure. Proc. Japan Acad., 51, 179— 183. . (1979), Basic Theorems of Vector Symmetry in Crystallography. Acta Cryst., A35, 115-122. Sawada, * T. (1978), On the crystal structure and its pseudo symmetry of picric acid. Thesis for Dector of Science submitted to the University of Tokyo. *--. (1979), In the preparation. Zorkii, P.M. (1978), The *supersymmetry of molecular crystal structures. Acta Cryst., A34, Si.

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