Beauty, Form and Function: An Exploration of Symmetry Asset No. 33 Lecture III-6 Platonic Solids and Atomic Bonding PART III Symmetry in Crystals

By the end of this lecture, you will be able to: correlate the fractional co-ordinates of atoms with the general position diagrams in the International Tables of Crystallography discover and plot atomic bonds in crystal structures using ATOMS define and plot Platonic solids (tetrahedra and octahedra) in crystal structures Objectives

Discovering Bond Lengths & Polyhedra Plot bonds between nearest neighbor atoms (the coordination sphere) Find first nearest neighbor atoms (the atomic coordination) Define and plot atomic polyhedra Validate atomic co-ordinates conform to the General position diagram

Representing the Crystal Structure of Quartz SiO 2 P (152) Type Wyckoff Coordinates xyz Si1 3 a /3 Si2 3 a /3 Si3 3 a O1 6 c O2 6 c O3 6 c O4 6 c O5 6 c O6 6 c b/y a/x Si1 Si2 Si3 O1 O2 O3 O4 O5 O6 Note that there are 3 formula units Si 3 O 6 inside the unit cell

Representing the Crystal Structure of Quartz SiO 2 P (152) 1/3 2/

Using ATOMS to Find Bond Lengths

Using ATOMS to Find Polyhedra

Quartz Bonds Central: 1 Si1 - Type: 14 No. Label Type Distance 1 O Å 2 O Å 3 O Å 4 O Å ATOMS output (Abbreviated) 1 Ångstrom (Å) = 1 x metres Å Å Å

Quartz (SiO 2 ) Polyhedra Solid SiO 4 tetrahedraTransparent SiO 4 tetrahedra

Stishovite Bonds Central: 1 Si - Type: 14 No. Label Type Distance 1 O Å 2 O Å 3 O Å 4 O Å 5 O Å 6 O Å ATOMS output (Abbreviated) Å Å Å [Note: 2 bonds of Å superimpose in this projection]

Polyhedra Type Wyckoff Coordinates xyz Si 1 2 a Si 2 2 a 1/2 1/2 1/2 O1 4 f O2 4 f /2 O3 4 f O4 4 f /2 Si1 Si2 O1 O2 O3 O4 b a Note that there are 2 formula units Si 2 O 8 inside the unit cell P 4 2 / m n m (136)

Stishovite (SiO2) Symmetry Diagram P 4 2 / m n m (136) plane glide double glide 4 2 screw axis with inversion centre 2 fold rotation axis with inversion centre mirror plane

Stishovite (SiO2) Symmetry Diagram P 4 2 / m n m (136) 0, 1 1/2 0, 1 0 1/2 0, 1 0, 1 0, 1 0, 1 0, 1 0, 1 1/2

Stishovite (SiO 2 ) Octahedra Solid SiO 6 octahedraTransparent SiO 6 octahedra

 Atoms often arrange themselves in geometries that approximate coordination polyhedra  Individual polyhedra and the arrangement of polyhedra must conform with the space group symmetry of the compound  The chemical composition or ‘formula unit’ of a compound is the total content of the different atoms in a unit cell divide by an integer. Summary