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?
- 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.
- 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.)