M. F. C. Ladd University of Surrey Guildford, Surrey England
Alternating andInversion Symmetry Elements: Common Errors
There are two common notations for symmetry elements and symmetry groups; they are those of Schoenflies (S) and Hermann-Mauguin (H-M). The symmetry elements of finite bodies are set out in the table . They are not all independent: for example, a point group consisting of only a center of symmetry may be written as S2 (S), also called Ci(S) , or as I (H-M); again, point group S 1 (S) may be written also as Cs (S), as m (H-M) or as 2 (H-M). The descriptions of the rotation axis and reflexion plane symmetry elements are uniform among current expositions of symmetry and symmetry groups. Differences exist between the operations designated SR and R, and it is in their de scriptions that misconceptions have arisen . There are relatively few errors among definitions of the inversion axis, possibly because this symmetry element is used mainly in crystallograph y, wherein a definitive work on symmetry 1 exists. Occasionally, however, one reads the general prescription for R of "rotation thr ough 360/R degrees followed by inversion through the cent er ," or worse, " t hr ough th e center of symm etry." The italicized words in this sta tement are incorrect : they should read "a p oint on th e axis (conceptual line) around whi ch th e rotation is carr ied out ." This point is, conveniently, the origin of the reference axes for the symmetry operations. In particular , it may be noted t hatR is equivalent to R combined with I only where R is an odd number (Fig. 1). The more serious confusion exists over the alternating axis. An SR axis implies self-coincidence for an R-fold (360/R degree) rotation followed by reflexion across a plane normal to the SR axis. This statement should then be reinforced to the extent of saying that this plane is not necessarily a symmetr y plane . Definitions can be found in the textbook literature that state that the plane is a "reflexion plane," a "mirror plane, " or a "symmetry plane." Further confusion is added when, for example, 8 4 is "demonstrated" by a combination of C4 and
Symmetry Elements of Finite Bodies, and Their Notations
2).
The successful study of symmetry concepts, particularly
1 Henry , N. F. M., and Lonsdale , K. , (Ed it ors), " International Tables for X-ray Crystallography ," Vol. I, 2nd Ed . Kynoch Pre ss, Birmingham , England, 1965. 2 Ladd , M. F . C., (1976), Int. J. Math. Edu c. S ci. T echnol ., 7, 395. 3 Ladd, M. F. C., and Palmer, R. A., "Structure Determination by X-ray Crystallography," Plenum Press, New York, 1977, 1978.
636 I Journal of Chemical Education
00 )
m
(]
R(R = 1,2, . .. oo) SR(R=
1,2, . . .oo)
0 +
a
c
b
0 +
•
d
Figure 1. Stereograms showing combinations of symmetry elements : (a) 3 (C:i) (d) 2 (C 2)
ah.
It is most important to define the symmetry element SR precisely, particularly 8 4 (Fig. 2). This symmetry element occurs in objects with tetrahedral symmetry . Furthermore, it cannot b~ generated by any combination of other symmetry elements. It may be noted that if the plane that is normal to the S 4 axis in point group S 4 is made a mirror symmetry plane, then the order of the group is raised and it becomes C4h (Fig.
Hermann-Mauguin R(R = 1,2, .. .oo)
Schoenfl ies
CR (R = 1,2, ...
Rotation axis Reflexion plane Inversion axis Alternating axis
(b) (e)
1 (S 2 ) 1 (S2)
3
(c) (Se) (I) ~ (C2h)
3
+
°ti
0 4
• a
b
Figure 2. Stereograms of S4 (4) and C4 " (~): (a) The sequence 1, 2, 3, 4 corresponds to a succession of righthanded S4 operations. The ~quence 1, 4, 3, 2. corresponds to a succession of right-handed 4 operations . (b) A horizontal mirror symmetry plane added to S4 produces C4 h -
in the most important case of three dimensions, requires considerable practical involvement. A scheme which has proved successful in teaching practice over several years has been described elsewhere .2 ,3
