Models for the Study of Cubic Crystallographic Point Groups Alcuin F. Gremillion University of Arkansas at Little Rock, Little Rock, AR 72204 The most widely used method for the study of crystal structures and of the motif in molecular crystals employs X-ray diffraction. Usually, the study of X-ray crystal structure analysis is hased upon a prior study of the theory of point groups and space groups. Early in this study, the student learns the meaning of symmetry, how to identify the symmetry elements of ohjects such as polyhedra and of the repeat pattern of a motif in wallpaper and fabrics. From the symmetry elements, one proceeds to the corresponding symmetry -
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The learning of such matters is often conducted with the aid of models. The need for such an aid increases as the symmetry of the corresponding point group increases. The cuhic point groups are the more symmetrical ones and have Schoenflies symbols T, Td,Th, 0, and Oh. The construction of models, as well as their study, is instructive. Kinds of Models The variety of models which can he employed in the study of the symmetry of the cubic point groups is limited only by the imagination. Usually such models are based upon the cube (2),the tetrahedron (31,the octahedron (31,and ball-and-stick molecular models commonly found in the chemistry classroom. When a single basis, such as the cube, is used to produce the several point group symmetries, it is often done by variation of the manner in which the surface of an object is tessellated (2). This paper describes a series of easily constructed octahedra having surface facets so that each displays one of the cuhic point group symmetries. These may be used simultaneously with the other kinds of models to provide within each point group several objects for comparison and study. Construction of the Models The basic structural unit is the octahedron which can he made from cardboard of about 1-mm thickness. Octahedra having 12-cm edges have proven especially satisfactory for small classroom discussions and individual study. Cardboard of greater thickness does not allow easy construction of models with sharp edges. Material substantially less thick is easily deformed when the models are large. Figure 1illustrates the template for making the octahedron. After it is cut from the cardboard, it is scored with a knife along the lines where it is to be folded. Edges AB and BC are brought together and held in place with several thin strips of masking tape placed at 2 to 3-cm intervals along the edges. These edges are glued together a t the points between the pieces of tape with Ducoo cement that has been thinned somewhat with amvl acetate. The same orocedure is followed with edges DE andEF. After the cement has set, the tape is removed and cement is aoolied to the remainder of the newlv formed edges of the tw;half-octahedra. The model is then folded along GH so as to bring the two halves of the octahedron together in the equatorial plane. The cementing is then completed. One set of models consists of five octahedra. To four of these, one attaches equilateral paper triangles positioned so as to ohtain the desired symmetry for each model. The color of the triangles should contrast well with that of the under194
Journal of Chemical Education
D Figure 1. Template for the octahedron.
Figure 2. Triangular facet on the face of an octahedron,
quired for one sit of models. The model with T symmetry is produced by fixing the center of a triangle to the center of each of four alternate faces of an octahedron in such manner that the mirror planes of a guest triangle do not coincide with those of its host face. This is illustrated in Figure 2, where 0" < a < 60°, and a is the same fur all triangles. The model having T d symmetry is similar to that having T symmetry with the exception that the mirror planes of a
guest triangle coincide with those of its host face, i.e., a =
no ". When constmctine the model with Th svmmetrv. a triangle is fixed to each of the eight octahedron'faces. his model cas three mutuallv oeruendicular horizontal mirrors in the edees
a, but that half will he so placed that a is positive while the other half has a negative a. In terms of the indices of octahedron faces, the settings of a are indicated in the table. This is illustrated in Figure 3 where the image of the 111 face (triangle-CDE) of the octahedron in the mirror plane ABCD is the 111face (triangle CDF). The model for the O h group is the octahedron unadorned with triangular facets. I t has the full complement of mirror plane elements. The model with 0 symmetry has no mirror symmetry elements. When constructing the model of this latter group, a triangle is attached to each octahedron face but such that all have the same value of a with the same setting. This is illustrated in Figure 4 (b) where the photograph of a model with 0 symmetry is found. Photographs of the models with T and Th symmetries are also given in Figure 4. Models of the type discussed here, as well as others, can be used in various ways. Perhaps these models give the greatest aid in point group study as they are being constructed. Such models have been employed by students to identify the essential symmetry elements of the point groups and which constitute the inJernational symbols for these groups. These symbols are 23,43m, 63,432, and $36 corresponding to the T , Td, Th, 0 and Oh groups, respectively. These models have also been used to find the complete collection of symmetry elements of a point group which follows from those elements in the international symbol.
Figure 3. A pair of mirror image faces in the T, model
The Settings of a in Terms of the Indices of Octahedron Faces Setting of the Triangle
Octahedron
Literature Cited (1) Buergei. M. J., "Elementary Crystallography an Introduction to the FundamenLd Geometrical Features ofCrystals,(l Revised printing 1963, J o b Wilesand Sonr, New Ynrk. London, Sydney, Chspter21. Principles in Solid State and Molecular Physics," John Wileyand (2) Lax. M., "S-etry Sons, New York. London. Sydney. Toronto, 1974, p. 61. (3) Cotton, F. A . "Chemical Applications of Group Theory." John Wiley and Sonr,New Ymk, London, Sydney, Toronto. 1971, pp. 3 9 4 5 .
face indices
+a -a +a -a
-1 1 1
111 111 111
Figure 4. (a) Model for the Tgroup. (b) Model far the Ogroup. (c) Model for the Th group
Volume 59
Number 3
March 1982
