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I came across the following property of the reciprocal lattices.

Let be $\Lambda$ a Bravais lattice and $\Lambda^*$ its reciprocal lattice; let be $\vec{G} \in \Lambda^*$ and $\vec{G}_0$ the shortest vector in $ \Lambda^*$ that is parallel to $\vec{G}$. Then, the relation between the two vectors is: $\vec{G}=n\vec{G}_0$, where n is integer.

I looked for a proof, but I noticed that for example on Ashcroft-Mermin (chapter 5) the proof is left as a problem to the reader. The hint is: suppose the contrary is true (this means "suppose n is not integer", I think) and show that there is a vector parallel to $\vec{G}$ that is shorter than $\vec{G}_0$. Despite the hint, I'm not able to prove this statement. Does anyone please know this proof?

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    $\begingroup$ How about the following argument: if $n$ is not an integer, then $\mathbf{G}-[n]\mathbf{G}_0=\{n\}\mathbf{G}_0$ is still parallel to $\mathbf{G}$ ($[n]$ is the integer part of $n$, and $\{n\}=n-[n]$), and still belongs to $\Lambda^*$, but with $0<\{n\}<1$, so the length of $\{n\}\mathbf{G}_0$ is smaller than that of $\mathbf{G}_0$. $\endgroup$
    – Meng Cheng
    Commented Sep 14, 2022 at 13:41
  • $\begingroup$ You convinced me: if you turn this into an answer I can approve it. $\endgroup$ Commented Sep 14, 2022 at 13:48

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Suppose $n$ is not an integer. Denote by $[n]$ the integral part of $n$, and $\{n\}=n-[n]$, so $0<\{n\}<1$. Consider $\vec{G}-[n]\vec{G}_0=\{n\}\vec{G}_0$, which is still parallel to $\vec{G}$, and belongs to $\Lambda^*$. But the length of $\{n\}\vec{G}_0$ is strictly smaller than $\vec{G}_0$, so a contradiction.

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