# Laue diffraction/equations: Path difference, why a minus sign and not a plus?

The Laue diffraction (that reduces to the Bragg refraction in the end) has for the path difference of the incoming and outgoing beam

$$\Delta s = \vec R \cdot ( \vec k_0/k_0 - \vec k^\prime/k^\prime )$$

with $\vec R$ being the translation vector of the lattice and the two $\vec k$ being the incoming and outgoing wave vector.

My question now: Why is there a minus and not a plus between the wave vectors? In the Bragg way of describing such diffraction you add the two paths contributing (incoming + outgoing) to get $\Delta s = d\sin \Theta + d\sin\Theta=2d\sin\Theta$ (with scattering angle $\Theta$ and lattice distance $d$), but on the Laue way you take the difference between the scalar products. For pictures see Wikipedia, for example.

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