Question

Consider a cation in a trigonal hole. What size ion will just fit in the hole if the packed spheres have radius R?

Short answer

Cations with a radius of 0.155R will just fit in the hole.

Step-by-step solution

Step 1: Diagram  

It has been asked to consider a cation in a triangle hole. Let the radius of packed spheres be R. Let the radius of the triangle hole be x.

Determine radius of ion

$$\begin{gathered} \cos 30°=\frac{R}{R+x} \\ \frac{\sqrt{3}}{2}=\frac{R}{R+x} \\ 0.866=\frac{R}{R+x} \end{gathered}$$

$$\begin{gathered} 0.866x=R-0.866R \\ x=\frac{0.134R}{0.866} \\ x=0.155R \end{gathered}$$