I'm wondering for what the points in the reciprocal lattice physically stand for, I know that they are the k-vectors, that are the fourier-transformed vectors of the lattice in the real space. Do those k-vectors have special physical things? Are they the vectors who give good diffraction? Or is it : if a k-vector has it endpoint onto the plane that's perpendicular to a line spanned by 2 reciprocal lattice points then the k-vector gives good diffraction?

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    $\begingroup$ Maybe this definition can help: a reciprocal lattice vector is a vector $k$ such that a plane wave $e^{ikr}$ has the periodicity of the lattice. This definition is closely related to the Von-Laue formulation of diffraction. $\endgroup$ Commented Feb 23, 2020 at 20:00

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If $\Lambda \subset \mathbb{R}^d$ is a lattice, the reciprocal lattice $\Lambda^* \subset \mathbb{R}^d$ may be defined as the lattice of vectors $a$ such that for all $v \in \Lambda$, $v \cdot a \in \mathbb{Z}$.

The point of the reciprocal lattice is that if we take the Fourier transform of a function $f(v)$ valued on $\Lambda$, meaning we take $$\tilde f(k) = \sum_{v \in \Lambda} f(v) e^{2\pi i v \cdot k},$$ then $\tilde f(k)$ will be a periodic function in $k \in \mathbb{R}^d$. The periodicity is phrased precisely in terms of the reciprocal lattice, such that if $k$ and $k'$ differ by a element of the reciprocal lattice, then $\tilde f(k) = \tilde f(k')$. This follows from the definition and the Fourier transform. Thus if we think about phonons and band electrons, these carry a conserved crystal momentum which lives in $\mathbb{R}^d/\Lambda^*$, known as the Brillouin zone.

So in $k$-space, the reciprocal lattice itself is those set of $k$ vectors which are equivalent to the origin. You can therefore think of the reciprocal lattice as the set of physical momenta which represent trivial crystal momenta.

Consider scattering, where a wave of with some wavenumber $k$ passes through a crystal lattice $\Lambda$. This wave carries momentum and can transfer it to the crystal or receive it from the crystal but it does not carry crystal momentum, ie. it can only couple to the phonon zero mode which causes the center of mass motion of the crystal. Thus, all possible momenta it may transfer to the crystal lie in the reciprocal lattice. This is why the reciprocal lattice shows up as the diffraction pattern.


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