# Coordination Number and Geometry

1. May 29, 2005

### mathwurkz

I am trying to understand a sample problem in this text I have about materials science. The question is to calculate the minimum radius ration for a coordination number of 8. The coordination geometry is cubic. What I don't understand in this problem is one of the two equations they use to solve the for the ratio.

$$R$$ is the larger ion radius, $$r$$ is the smaller ion radius and $$l$$ is the cube length edge.

Now what I don't get is how the book comes up with this relationship.

$$2R + 2r = \sqrt{3}\ l$$

The second expression $$l = 2R$$, I understand since the two large ions are touching eachother, their radius will make up the length of the cube edge. They substitute this equation in the other and solve for $$\frac{r}{R}$$ It's just that I don't understand where that first equation comes from.

2. May 29, 2005

### OlderDan

$$\sqrt{3}\ l$$ is the diagonal of the cube. Each corner is occupied by the larger ion, with the smaller ion fitting in the space between the large ions.

3. May 29, 2005

### mathwurkz

Ok. Great I understand it now. Thanks a lot OlderDan