Laue equations

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 the Bragg law.

Equations

Take $mathbf\{k\}\_i$ to be the wavevector fo 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 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:

: $mathbf\{a\}cdotmathbf\{Delta\; k\}=2pi\; h$: $mathbf\{b\}cdotmathbf\{Delta\; k\}=2pi\; k$: $mathbf\{c\}cdotmathbf\{Delta\; k\}=2pi\; l$

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

From this we get the diffraction condition:: $mathbf\{k\}\_o\; -\; mathbf\{k\}\_i\; =\; mathbf\{G\}$: ↓: $(mathbf\{k\}\_i\; +\; mathbf\{G\})^2=\; mathbf\{k\}\_o^2$: ↓: $\{k\_i\}^2\; +\; 2mathbf\{k\}cdotmathbf\{G\}\; +\; G^2\; =\; \{k\_o\}^2$: ↓ (k_i² = k₀²): $2mathbf\{k\}\_icdotmathbf\{G\}=G^2$.

The diffraction condition $;2mathbf\{k\}\_icdotmathbf\{G\}=G^2$ reduces to the Bragg law $;2dsin\; heta\; =nlambda$.

References

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