# Precise definition of "nearest neighbours" in solid-state physics

I am reading Ashcroft and Mermin and, to define coordination number, they use the notion of nearest neighbours which they do not define. I'm sure it's a very trivial definition, but they had been so precise up until now so I wanted to be sure. After searching "definition of nearest neighbours" in Google and looking at Kittel and Simon, I still don't have an answer.

How are "nearest neighbours" defined in a Bravais lattice? I assume a possible definition is the naive one: fix an arbitrary point in the lattice, call it $$x$$. Given the set of all other points in the Bravais lattice, there will exist points such that $$d(x,y)$$ is minimized (where $$d$$ is the usual metric and $$y$$ is any point other than $$x$$); call this minimum value $$\alpha$$. Then a nearest neighbour is any point $$y$$ in the lattice with $$d(x,y)=\alpha$$.

The definition can doubtless be made more slick, but I'm just hoping for confirmation that it's correct.

Yes, that's about it. Lattice point $$x$$ is a nearest neighbor to lattice point $$y$$ if there are no lattice points $$z\neq y$$ such that $$d(z,y).

• Thanks very much. Just before I accept this, would you happen to have a source confirming this? I just want to make sure I'm set on the correct definition.
– EE18
Jun 11, 2021 at 21:25
• @1729_SR Ashcroft & Mermin, Chapter 4, Coordination Number. "The points in a Bravais lattice that are closest to a given point are called its nearest neighbors." Jun 11, 2021 at 21:30
• Fair enough, I guess we both interpreted A&M in the same way! Thanks again.
– EE18
Jun 11, 2021 at 21:33
• @1729_SR I just realized that's the passage you quoted as being unclear in your question - is there any specific reason you find that phrasing ambiguous? Jun 11, 2021 at 21:33
• I suppose I have an unhealthy desire for things to be phrased precisely in terms of mathematical structure (sets, metrics, etc.). When it's just written in words, I'm ambivalent about whether or not I am interpreting the author correctly.
– EE18
Jun 11, 2021 at 21:35

Nearest neighbours in a crystal are the atoms a given atom is directly bonded with. For instance, in diamond, each carbon atom is covalently bonded to four others in a tetrahedral structure. Hence, it has four nearest neighbours.

• Thanks for the answer, but I'm not sure I'm with you. A Bravais lattice is a purely mathematical construct, and so should not make reference to any notion of bonding.
– EE18
Jun 11, 2021 at 20:47
• This answer is correct @1729_SR. Not on the Bravais lattice (ie. momentum space), but on the actual (spatial) lattice of atoms. Jun 11, 2021 at 20:54
• @QuantumEyedea Fair play, but it seems to me that it's not a sufficiently general definition. After all, lattice points need not be constituted of atoms in general.
– EE18
Jun 11, 2021 at 21:00
• not sure I agree with this answer... what about lattices with more than one type of element (i.e. diatomic lattices), where a given atom not only has bonds to its own kind, but also bonds to the other type of atom. Wouldn't just be those atoms be counted as nearest neighbors that truly are "nearest"? This answer currently implies that also atoms that are further away would be counted as nearest neighbors, as long as there is a chemical bond...
– rfl
Jun 11, 2021 at 21:07
• @1729_SR As a purely mathematical question, what you've asked is ill-posed. You'd need to provide further criteria. The definition you gave in the question is one clear choice. In some contexts you're looking for a certain number of nearest neighbors, then you'd choose a ball around $x$ that is a small as possible while containing your target number. In some contexts where your definition of "lattice" includes the edges connecting point (not just the vertices), you take all points with a single-hop connection. Jun 11, 2021 at 21:07