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I wonder if there is a close formula for the sequence of distances of neighbor shells in an FCC lattice, i.e. a formula of the form $d_n = a f(n)$ where $a = \frac{A}{\sqrt{2}}$ is the nearest-neighbor distance, and $A$ is the unit cell side length. It's easy to retrieve such distances (visually) for the first 4 shells: $$ d_1 = a\\ d_2=a\sqrt{2}\\ d_3=a\sqrt{3}\\ d_4=a\sqrt{4}\\ $$

The pattern is similar to the sequence $a\sqrt{n}$, but some values of $n$ are not present in the real sequence of distances. Is there a known formula for this ?

I couldn't find any.

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Three primitive basis vectors for FCC are:

$$ \mathbf{a}_1 = \frac{A}{2}(0, 1, 1); \\ \mathbf{a}_2 = \frac{A}{2}(1, 0, 1); \\ \mathbf{a}_3 = \frac{A}{2}(1, 1, 0); \\ $$

Therefore, a lattice vector $\mathbf{R}$ has form:

$$ \mathbf{R} = n \mathbf{a}_1 + l \mathbf{a}_2 + m \mathbf{a}_3 = \frac{A}{2} (l+m, n+m, n+l); $$

The distance from the center lattice point:

$$ d(n, l ,m) = \frac{A}{2} \sqrt{(l+m)^2 +(n+m)^2 + (n+l)^2} $$

Rewrite as three integers $x$, $y$, and $z$, with constrain $x+y+z = even$:

$$ \frac{d(x, y ,z)}{A/\sqrt{2}}=\frac{d(x, y ,z)}{a} = \frac{1}{\sqrt{2}} \sqrt{x^2 +y^2 + z^2}, \\ x+y+z = 2(n+l+m) = \text{an even integer}. $$ where I adopt your notation, $ a = A/\sqrt{2} $

\begin{matrix} x & 1 & 2 & 2 & 2 & 3 & 2 & 3 & 4 & 3 (4) &4\\ y & 1 & 0 & 1 & 2 & 1 & 2 & 2 & 0 & 3 (1) &0\\ z & 0 & 0 & 1 & 0 & 0 & 2 & 1 & 0 & 0 (1) &2\\ x^2+y^3+z^2 & 2 & 4 & 6 & 8 & 10 & 12 & 14 &16 &18 & 20\\ d(a) & 1 & \sqrt{2} & \sqrt{3} & 2 & \sqrt{5} & \sqrt{6} & \sqrt{7}& \sqrt{8} &\sqrt{9} &\sqrt{10} \end{matrix}

According to the link in Michael Seifert's comment, the even value of $x^2+y^2+z^2$ only misses $28$ and $60$ for $x^2+y^2+z^2 < 83$.

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    $\begingroup$ Note that the integers which can be expressed as the sum of three squares can be found as sequence 378 in the OEIS. You would need to drop the odd numbers in that sequence ($x + y + z \text{ even} \iff x^2 + y^2 + z^2 \text{ even}$), but all the possible distances would be there. Alternately, the distances squared appear to be sequence 401, though I'm not sure from the description how it connects to lattice distances. $\endgroup$ Feb 22, 2021 at 13:24
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    $\begingroup$ There's also a comment on sequence 401 that all numbers in the list there are not of the form $4^k(16n+14)$. But I don't know if that's just an observation, or if it's been proven that the only numbers that are missed are of that form. $\endgroup$ Feb 22, 2021 at 13:49
  • $\begingroup$ The missing numbers of these two sequences are different, one (378) misses 28, the other (401) 30. I might have to write a code to check it. $\endgroup$
    – ytlu
    Feb 22, 2021 at 13:59
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    $\begingroup$ Sorry, I wasn't clear. If you take the even integers from sequence 378 and divide them all by 2, you get sequence 401. So the fact that 28 is missing from sequence 378 is reflected by the fact that 14 is missing from sequence 401. $\endgroup$ Feb 22, 2021 at 14:58
  • $\begingroup$ Thank your for your answer ! I suspected there wouldn't be a closed formula wrt $n$, but this is clearly enough for coding the list of neighbors. $\endgroup$
    – Toool
    Feb 25, 2021 at 10:47

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