Rotary-Inversion Symmetry III. Higher-Order Axes

Course Material Index  Section Index  Previous Page  Next Page

Higher-Order Axes

So far only the two simplest forms of rotary-inversion symmetry have been discussed. The three other symmetry operators of this type that describe the crystallographic symmetries observed in either molecules or 3-dimensional diffraction patterns are the -3, -4, and -6 rotary-inversion axes. The symbols for these are shown on the next page and examples of each of these will be seen in the section on point-group symmetry.

Applying the general definition given earlier, the -3 (bar 3) symmetry operator is defined as an anticlockwise rotation of 120° about an axis followed by an inversion through a point on that axis. In the figures shown on this page, the axis is shown by the dashed red line and the point of inversion is shown by the open red circle on that line. The concept of -3 symmetry is well-illustrated with reference to the staggered conformation of the ethane molecule, C₂H₆, shown below:

-3
	View perpendicular to axis	View down axis

How many -3 operations are required to bring the hydrogen atom labelled "1" back to its original position? If you are not sure, watch the effect of repeated -3 operations.

The underlying concepts of the -4 and -6 rotary-inversion symmetry operators are similar to those of the -3. The -4 symmetry operator is a combination of 90° rotation plus inversion; the -6 has a 60° rotation plus inversion. The -6 rotary-inversion axis is shown in the figure below, which again is of ethane, but in its eclipsed conformation:

-6
	View perpendicular to axis	View down axis

Course Material Index  Section Index  Previous Page  Next Page