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

• Kittel, C. (1976). Introduction to Solid State Physics, New York: John Wiley & Sons. ISBN 0-471-49024-5