Chapters / Chapter 1 · Solid state / Topic 7 of 14

Voids and the radius ratio rule

Chapter 1 · Solid state and crystal structures

A tetrahedral void is bounded by four spheres; an octahedral void by six.

The empty spaces left between packed spheres are called interstitial voids. Two types matter.

Void	Surrounded by	Per sphere	Largest sphere that fits
Tetrahedral	Four spheres at the corners of a tetrahedron	2	0.225 R
Octahedral	Six spheres at the corners of an octahedron	1	0.414 R

R is the radius of the packed sphere. An fcc unit cell has 4 atoms, so it has 8 tetrahedral voids and 4 octahedral voids.

CsCl geometry: the cation touches anions along the body diagonal.

Radius ratio rule

The radius ratio r⁺/r⁻ predicts which hole a cation occupies, and therefore its coordination number. The limits are the ratios at which the cation just touches the surrounding anions.

r⁺/r⁻	Coordination	Geometry	Example
0.225 to 0.414	4	Tetrahedral	ZnS
0.414 to 0.732	6	Octahedral	NaCl
0.732 to 1.000	8	Cubic	CsCl

Worked limits, as written in the notes

CsCl, 8:8. Side of cube a = 2r⁻. Body diagonal √3 a = 2r⁺ + 2r⁻. Hence √3 = (r⁺ + r⁻)/r⁻, so r⁺/r⁻ = √3 − 1 = 1.732 − 1 = 0.732.

NaCl, octahedral, 6:6. Face diagonal √2 a = 2r⁺ + 2r⁻, with a = 2r⁻. Hence r⁺/r⁻ = √2 − 1 = 0.414.

ZnS, tetrahedral, 4:4. Half the body diagonal gives r⁺ + r⁻ = √3 a/2, with the face diagonal √2 a = 2r⁻. The limiting ratio is 0.225.