Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-03 GTBL042-Callister-v2 September 6, 2007 15:33

48 • Chapter 3 / Structures of Metals and Ceramics

Solution

For this coordination, the small cation is surrounded by three anions to form

an equilateral triangle as shown here, triangleABC; the centers of all four ions

are coplanar.

Cation

Anion

rA

rC



B

O

C

A

P

This boils down to a relatively simple plane trigonometry problem. Consid-

eration of the right triangleAPOmakes it clear that the side lengths are related

to the anion and cation radiirAandrCas

AP=rA

and

AO=rA+rC

Furthermore, the side length ratioAP/AOis a function of the angleαas

AP

AO

=cosα

The magnitude ofαis 30◦, since lineAObisects the 60◦angleBAC. Thus,

AP

AO

=

rA

rA+rC

=cos 30◦=

√

3

2

Solving for the cation–anion radius ratio, we have

rC

rA

=

1 −

√

3 / 2

√

3 / 2

= 0. 155

Table 3.4 Ionic Radii for Several Cations and

Anions (for a Coordination Number of 6)

Cation Ionic Radius (nm) Anion Ionic Radius (nm)

Al^3 + 0.053 Br− 0.196

Ba^2 + 0.136 Cl− 0.181

Ca^2 + 0.100 F− 0.133

Cs+ 0.170 I− 0.220

Fe^2 + 0.077 O^2 − 0.140

Fe^3 + 0.069 S^2 − 0.184

K+ 0.138

Mg^2 + 0.072

Mn^2 + 0.067

Na+ 0.102

Ni^2 + 0.069

Si^4 + 0.040

Ti^4 + 0.061