In sulfur hexafluoride,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

\mathrm{SF_6}

, the six fluorine atoms sit at the vertices of a regular octahedron centred on the sulfur atom. The molecule's full symmetry group is

O_h = O \times \{E, i\}

, of order 48, where

O

is the octahedron's 24 proper rotations and

i

is inversion through the sulfur atom. ((a)) Using the fact that

\{E, i\}

is a factor of this direct product, find the size of the conjugacy class containing

i

. ((b)) Given that the six fluorine atoms form a single orbit under

O_h

, use the orbit-stabiliser theorem to find the order of the subgroup that fixes one fluorine atom in place. ((c)) Label the fluorines

F_1

F_6

so that

F_1

and

F_2

lie on one

\mathrm{S{-}F}

axis, and rotate by

90^\circ

about that axis. This fixes

F_1

and

F_2

and cycles the remaining four fluorine atoms

F_3 \to F_4 \to F_5 \to F_6 \to F_3

. Write this permutation of the six fluorine atoms in cycle notation and give its sign.