Point-Group Determination

Point-Group Determination

There are many schemes for the determination of point-group symmetry of an object. Most follow some sort of flow-diagram type logic scheme such as the much simplified one below, which can be used to identify triclinic, monoclinic, and orthorhombic point groups:

Is a high-order (i.e. 3, 4, 6, -3, -4, -6) symmetry axis present?		YES ←	Tetrahedral, Trigonal, Hexagonal, or Cubic Point Group
NO ↓
Is mirror symmetry present?		NO ←	Is twofold axis present?		NO ←	Is point of inversion present?
			YES ↓			NO ↓	YES ↓
			Are other twofold axes present?			1	-1
YES ↓			NO ↓	YES ↓
			2	2 2 2

Are other mirror planes present?		YES ←	Is point of inversion present?		NO ←	m m 2
NO ↓			YES ↓
Is point of inversion present?			m m m
NO ↓	YES ↓
m	2 / m

However, all such flow charts rely on the ability of the user to correctly identify the symmetry elements present in the object, and this in the opinion of the author is their biggest weakness. In particular, experience shows that proper rotation axes can usually be identified more easily than the rotary-inversion axes -3, -4, and -6. The latter two symmetry elements are particularly difficult to spot quickly since a -4 axis may appear to be only a twofold axis and a -6 axis is more easily visualized as a 3 axis with a mirror plane perpendicular to it.

Another aspect of point-group determination is the fact that the low-symmetry point groups are usually sub-sets of high-symmetry point groups. Thus identification of a mirror plane implies that the point group belongs at least to the monoclinic system: the point-group symmetry may be higher if further symmetry elements are found, but it can never be lower (i.e. belong to the triclinic crystal system in this example).

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Author(s): Jeremy Karl Cockcroft