Laue equations

Laue equation

In crystallography, the Laue equations give three conditions for incident waves to be diffracted by a crystal lattice. They are named after physicist Max von Laue (1879 — 1960). They reduce to Bragg's law.

Equations

Take \mathbf{k}_i to be the wavevector for the incoming (incident) beam and \mathbf{k}_o to be the wavevector for the outgoing (diffracted) beam. \mathbf{k}_o - \mathbf{k}_i = \mathbf{\Delta k} is the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.

Take \mathbf{a}\, ,\mathbf{b}\, ,\mathbf{c} to be the primitive vectors of the crystal lattice. The three Laue conditions for the scattering vector, or the Laue equations, for integer values of a reflection's reciprocal lattice indices (h,k,l) are as follows:

\mathbf{a}\cdot\mathbf{\Delta k}=2\pi h
\mathbf{b}\cdot\mathbf{\Delta k}=2\pi k
\mathbf{c}\cdot\mathbf{\Delta k}=2\pi l

These conditions say that the scattering vector must be oriented in a specific direction in relation to the primitive vectors of the crystal lattice.

Relation to Bragg Law

If  \mathbf{G}=h\mathbf{A}+k\mathbf{B}+l\mathbf{C}  is the reciprocal lattice vector, we know  \mathbf{G}\cdot (\mathbf{a}+\mathbf{b}+\mathbf{c})=2\pi (h+k+l). The Laue equations specify  \mathbf{\Delta k}\cdot (\mathbf{a}+\mathbf{b}+\mathbf{c})=2\pi (h+k+l). Hence we have  \mathbf{\Delta k}=\mathbf{G}  or  \mathbf{k}_o - \mathbf{k}_i = \mathbf{G}.

From this we get the diffraction condition:

\begin{align} \mathbf{k}_o - \mathbf{k}_i &= \mathbf{G}\\ (\mathbf{k}_i + \mathbf{G})^2 &= \mathbf{k}_o^2\\ {k_i}^2 + 2\mathbf{k}_i\cdot\mathbf{G} + G^2 &= {k_o}^2 \end{align}

Since (\mathbf{k}_o)^2=(\mathbf{k}_i)^2 (considering elastic scattering) and \mathbf{G} = -\mathbf{G} (a negative reciprocal lattice vector is still a reciprocal lattice vector):

2\mathbf{k}_i\cdot\mathbf{G}=G^2.

The diffraction condition  \;2\mathbf{k}_i\cdot\mathbf{G}=G^2  reduces to the Bragg law  \;2d\sin\theta =n\lambda.

References