# 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:

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:

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:

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:

• I found one of the early Olex2 advertising cards mentioned. A small scale photo is here in the chatroom. Aug 24, 2021 at 6:52
• I'm sorry but I did not understand the figures for the 2nd and 4th nearest neighbors. Has the atom at the center been taken as the origin? The atoms linked by the black lines represent the required neighbor? Aug 26, 2021 at 12:40
• Yes and yes. Also, note that the currently added atoms on each stage are shaded, while the previously added atoms are empty. Aug 26, 2021 at 14:11
• In the figure for second nearest atom, there are 18 atoms linked by the black lines. However, there are only 6 second nearest neighbors. Similarly, in the the figure for 4th nearest neighbors, there are more than 18 atoms linked by the black lines. In the figure for 1st and 3rd nearest neighbors, I can make out the required atoms. Could you please clarify the required neighbor in the 2nd and 4th one. Aug 29, 2021 at 13:08
• When a new atom is linked to an old atom, they both are linked by a black line. There is no going around it. Just compare any picture to the previous one to see which atoms are new. Aug 29, 2021 at 14:01