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G. 0. Brunner ETH, Federal institute of Technology lnstitute of Crystallography and Petrography 8092 Zurich. Switzerland
Teaching the (Crystallographic) Point Groups
T h e r e are various teaching aids to demonstrate t h e nature of symmetry a n d t o train t h e s t u d e n t t o determine point croups. . . Framework units' to build molecules. wooden models ~,i crystnl.;, a n d stereoscupic views w n h lenses2 are some examples. T h e a ~ d sunnest~rihrre u n a b l ~ tsh e s r ~ l d e n to t i)roduce a n d t o handle a n inexpensive a n d unbreakable objeciexhihiting a certain symmetry a n d to obtain his personal s e t of reference objects.
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rods havine a cross Usine a circular saw with a tiltable h&e olate. , .~~ re8ucm in tiw twm d a paralldunram, recranglr, iquarr.and hemgon nrr p r ~ p a r ~ Hy d . ruruny ~ I I L lrum P ~ thwr r0d9, suitable hlocki are easily obtained in quantity. A final finish is applied to them by tumbling (rotation in a drum). The shape of the blocks is ehasen accordinc to the symmetry of the point group to be represented, e.g. for atetragonal group, a square block will he needed. An initiating mark is "painted" by the student on the upper right hand m n e r of a side face of the block. The ooerations of svmmetrv are then aoolied and the ~
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respect to morphology, the hlocks resemble crystals and the marks (if imagined as generally inclined faces) represent the general farm lhkll of a crystal class. The hlocks are space filling and by packing like hlocks together, crystal structures may he built. With the exception of the hexaeonal orism blocks. the blocks mav he reearded as unit cells
or 4 portions. Non-crystallographic point groups as well a s positions generated by screw axes m a y h e demonstrated in a n analogous manner. Group relations m a y h e demonstrated since all marks of a block of a sub-group are present in a block of a super-group (thoughJhe orientation of t h e axes m a y be different, e.g. in t h e case 3 - m3). 'Available e.g. from Prentiee-Hall Ine. Englewood Cliffs, N.J. 2Bernal, Ivan, Hamilton, Walter C., and Ricei, John S., "Symmetry," W. H. Freeman and Co., San Francisco, 1972. 3"lnternational Tables for X-Ray Crystallography," Vol. I, The Kynoch Press, Birmingham, England, 1952.
D2 222
C2" mm2
DZh mmm
onal crystals have the same volume as the conventional 120° rhombic shaped hexagonal unit cell and the translations from center to center of adjacent prism blocks correspond to the edges of the conventional 120' unit cell. The centers of the marks correspond to the "general positions" of the crystal classes (for non-hexagonal blocks, the parameters for a mark in the upper right-hand corner of a front faceare x 0.99, y i;. 0.8, a D 0.8; for hexagonal prism blacks with reference to the conventional 120' unit cell they a E x % 0r_~/8,Y 72/s or %, z 3 0.8). In the ease of the groups 3m, 32,3m, 6m2,42m it is possible to paint the marks in a way differing from the figure. The paintings according to the figure represent the first space group of each crystal class according to the enumeration of Int. Tables? if like blocks are packed together.
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Depending o n t h e aim of a course, a few representative hlocks will be sufficient t o demonstrate t h e nature of symmetry. I n a n extended course where blocks for all 32 crystal classes will h e painted, i t is convenient to divide t h e m i n t o 3
164 1 Journal of Chemical Education
The crystallographic point groups represented by suitably shaped and marked blocks (back marks of cubes omined for clarity). Arrangement of blocks according to crystal system and Laue-class.
