CLASSIFICATION OF SUPERSTRUCTURES BY SYMMETRY

RYOICHI SADANAGA, TOSHIYUKI SAWADA and KAZUMASA OHSUMI*

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

and

KAZUHIDE KAMIYA

Chemical Research Laboratories, Central Research Division,

Takeda Chemical Industries, Ltd., Jusohonmachi, 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 sub-structures 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

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 struc-

tural modifications (polymorphs or poly-types) 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 con-ceived in which the substructure is also

composed of sub-substructures, but these multiply complex structures will not be

(Manuscript recieved August 6, 1979) * Present address : Department of Crystallography and Structure Sciences , University of Bern, CH-

3012 Bern, Sahlistrasse 6, Switzerland.

considered here because no new principle is

required for the interpretation of these cases. From this substructure-framework con-

struction 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 opera-

tion. It is an ordinary space-group opera-tion 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 sub-space 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 super-

position upon by an appropriate motion p, but no triangle has been given

upon which AA'B'C' is brought to superposi-tion by the motion p. Because the sym-metry 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. How-ever, for the sake of convenience, we shall employ in this paper the term 'partially

symmetric' for the description of a con-figuration 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 substruc-

tures to superpose upon another. A com-

plete set of local operations which bring one of the substructures to superpose upon

itself obviously constitutes a group. Ac-

cording 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

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,

PRINCIPLE OF THE CLASSIFICATION

According to the definition previously

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

tions. In some cases such as organic crystals

consisting of molecules of complex conforma-

tion, each of the operations tends to bear 'pseudo' character

, their symmetry relation

In (1), hi indicates an element of the hull which brings the i th substructure X to

superpose upon the substructure X, re-

presenting 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-

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.

holds only approximately, and part A of the substructure is only nearly equal in con-figuration to part B in the case of partial

operation (Kamiya, 1979). However, even in these approximate cases, a clear distinc-

tion can be drawn between the local opera-tion 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 be-come 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

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 sub-structure 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 pseudo-

symmetry operations and partial pseudo-operations 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 struc-tures, 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 frame-

work are shown respectively in (A) and

(B) in Fig. 2. The symmetry of the result-ing superstructure given in (C) contains

local two-fold rotation axes, global reflec-tion 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 in-

clined 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 21 with 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 C6H60 forms 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 ap-parent 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 sub-structures (Ito, 1950; Ito and Sadanaga, 1976), the latter seems to suggest the pos-

sibility that it has grown from a liquid crystal in a nematic state, its substructure

being one-dimensional and pseudo-sym-metric.

A number of examples of super-symmetric structures have been found in

organic compounds with simple molecules

(Sawada, 1978; 1979) and those with com-

plex molecules (Kamiya, 1979). It is to be emphasized the fact that though these examples have been picked up quite arbi-

trarily 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 mole-cules to each other and to the entire structure. All the examples of super-

symmetry 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 sub-

structure 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 OD- Strukturen 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 Crystallo-

graphic Space Groupoid. Proc. Japan Acad., 52, 119-121.

*Kamiya, K. (1979), In the preparation. *Loewy, L. (1927), UUber abstrakt definierte Trans-

mutationssysteme 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 Sym- metry 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 mole- cular crystal structures. Acta Cryst., A34, Si.

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