How many 4th nearest neighbours are there in an hcp lattice?

My teacher told us that there are 18 4th nearest neighbors in an hcp lattice, at a distance of √3a . I have only been able to identify 12.


Can you help me find all the 4th nearest neighbors?




With HCP, things get hairy pretty quickly.


This is the first coordination sphere, with 12 atoms at the distance $a$ from the origin:


[![Sphere 1](https://i.stack.imgur.com/gSXTU.png)](https://i.stack.imgur.com/gSXTU.png)


These are the second nearest neighbors, i.e. the 6 atoms at $\sqrt{2}a$ from the origin, each touching 4 atoms of the first sphere:


[![Sphere 2](https://i.stack.imgur.com/AfaYL.png)](https://i.stack.imgur.com/AfaYL.png)


These are the third nearest neighbors, which are the 2 atoms at $\sqrt{8\over3}a$ from the origin, each touching 3 atoms of the first sphere:


[![Sphere 3](https://i.stack.imgur.com/yh79N.png)](https://i.stack.imgur.com/yh79N.png)


And finally, here are your 18 atoms which are the $4^{th}$ nearest neighbors of the atom at the origin. Each touches 2 atoms of the first sphere:


[![Sphere 4](https://i.stack.imgur.com/4Idta.png)](https://i.stack.imgur.com/4Idta.png)


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