Coordination Number and Geometry

[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 each other, 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.

 

[OlderDan]

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 each other, 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.

$$\sqrt{3}\ l$$ 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.

 

[mathwurkz]

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