# The concept of nearest neighbours

I'm having trouble understanding the concept of "nearest neighbours" and "next nearest neighbours." I understand that the number of nearest neighbours is the number of atoms per unit cell, but that's all I know (and I'm not sure why).

I'd like someone to show me how to calculate the number of nearest, second nearest, third nearest, etc. neighbours and the nearest neighbour distance for either a BCC or FCC structure.

For example, I've read that there are 6 nearest neighbours, 12 second nearest neighbours, etc. for a simple cubic lattice, with nearest neighbour distance 1, sqrt(2), etc. I don't know how this is found for SC or any other structure. Please explain thoroughly, and visually if possible.

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#### Solution Preview

Please see the complete explanation in the attached file.

Here we determine the number of nearest neighbors, next nearest neighbors, and third nearest neighbors of the SC, BCC, and FCC lattices. For simplicity, in everything that follows, we set a equal to 1.

First consider the SC case, which is probably the simplest case. Here, the lattice points are all of the form , where are integers. Consider the lattice point The distance from to is given by

The nearest neighbors to O are the lattice points not equal to O such that OP is minimal. It is easy to see that the minimum nonzero value of is 1, which holds if and only if exactly one of the coordinates has absolute value 1 and the others are equal to zero. This gives a total of six possibilities, namely . Thus we see that the lattice point O has 6 nearest neighbors, each a distance of 1 from O. Since there is nothing special about O, we see that every lattice point has 6 nearest neighbors, each a distance of 1 from the original lattice point.

What about next nearest neighbors? For this case, we need to consider the second smallest nonzero value of , which is 2, achieved when ...

#### Solution Summary

The solution computes the separation of nearest neighbors, next nearest neighbors, and third nearest (next next nearest) neighbors for SC, FCC, and BCC lattices.