= Laue equations
{c}
{title2=$a_i\cdot(\mathbf s_{\rm out}-\mathbf s_{\rm in})=m_i\lambda$}
{wiki}
= Laue condition
{c}
{synonym}
For incident and outgoing unit propagation vectors $\mathbf s_{\rm in}$ and $\mathbf s_{\rm out}$, <wavelength> $\lambda>0$, and <primitive lattice translation vectors> $a_1,a_2,a_3$, the <Laue equations> are
$$
a_1\cdot(\mathbf s_{\rm out}-\mathbf s_{\rm in})=h\lambda,\qquad
a_2\cdot(\mathbf s_{\rm out}-\mathbf s_{\rm in})=k\lambda,\qquad
a_3\cdot(\mathbf s_{\rm out}-\mathbf s_{\rm in})=l\lambda,
\qquad (h,k,l)\in\mathbb Z^3.
$$
All three must hold for the same <wavelength> and outgoing direction. The integers label a <reciprocal lattice> vector; they need not be coprime. For <wavevectors> $\mathbf k_{\rm in,out}=(2\pi/\lambda)\mathbf s_{\rm in,out}$, equivalently
$$
\mathbf G=\mathbf k_{\rm out}-\mathbf k_{\rm in}=h b_1+k b_2+l b_3,\qquad a_i\cdot b_j=2\pi\delta_{ij}.
$$
This uses angular <wavenumbers>; the crystallographic convention without $2\pi$ instead uses $\mathbf s/\lambda$.<Online Dictionary of Crystallography by International Union of Crystallography/Laue equations>[Online Dictionary of Crystallography by International Union of Crystallography; Definition: three scalar conditions and reciprocal form]{ref}
For a short proof, translate a scatterer through $a_i$. Its incident <wave phase> changes by $\mathbf k_{\rm in}\cdot a_i$ and propagation to the same distant observation direction contributes $-\mathbf k_{\rm out}\cdot a_i$. The relative <wave phase> factor is therefore $\exp(-i\mathbf G\cdot a_i)$. It equals one for all <lattice translations> exactly when $\mathbf G$ belongs to the <reciprocal lattice>. A finite <crystal lattice structure factor> broadens the ideal maxima; the <crystal basis structure factor> can cancel them. Geometric phase matching is necessary for a sharp reflection, but does not guarantee a visible spot.
For <elastic scattering>, the additional constraint is
$$
|\mathbf k_{\rm in}+\mathbf G|^2=|\mathbf k_{\rm in}|^2,
\qquad 2\mathbf k_{\rm in}\cdot\mathbf G+|\mathbf G|^2=0.
$$
It is an <Ewald sphere> centered at $-\mathbf k_{\rm in}$ in the space of transfer vectors $\mathbf G$. The parent <elastic Bragg scattering condition> uses $q=\mathbf k_{\rm in}-\mathbf k_{\rm out}=-\mathbf G$, so its displayed condition has the opposite sign. Both give <Bragg's law> for the same <lattice planes>.
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