# Coordination numbers and radius ratio

According to the image above (source), for a radius-ratio between 0.732 and 1, the coordination number is 8. However, for a radius-ratio of 1, the coordination is 12. This is the same for the boundary values between the other coordination numbers, but I’m focusing on the radius-ratio of 1 for now. I don’t understand how the same boundary value could correspond to two different coordination numbers. If you have a monoatomic metal, then all the atoms constituting it have the same radius and the radius-ratio is 1. In order to achieve the closest packing possible, the metal atoms would form a CCP or HCP structure, wouldn’t they? Those two crystal structures have the greatest coordination number. Yet other monoatomic metals, with a radius-ratio of 1, form can BCC structures, with a coordination number of 8. Why would they form such a structure when their radius-ratio can also correspond to a CN of 12? How can one radius-ratio correspond to two CNs?

## 1 Answer

1. Coordination number is the number of the nearest neighbours of an atom, and it is related to the Kepler's conjecture. For CN12 (12 is the 'kissing number' of a 3D sphere), there are two possible ways for the spheres to be 'packed' which are HCP and CCP. Space-filling efficiency they two are exactly the same, and it is why one coordination number can correspond to two crystal structures.
2. The crystal structure of metal may depend on many features, and the space-filling efficiency is just one of them. Even by the thermal expansion, the crystal structure can change. (Tin is the most familiar example.)