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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 the Bragg law.

Equations

Take ${𝐤}_i$ to be the wavevector for the incoming (incident) beam and ${𝐤}_o$ to be the wavevector for the outgoing (diffracted) beam. ${𝐤}_o-{𝐤}_i=𝜟𝐤$ is the scattering vector and measures the change between the two wavevectors.

Take $𝐚,𝐛,𝐜$ 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:

$𝐚\cdot 𝜟𝐤=2\pi h$

$𝐛\cdot 𝜟𝐤=2\pi k$

$𝐜\cdot 𝜟𝐤=2\pi l$

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  $𝐆=h𝐀+k𝐁+l𝐂$  is the reciprocal lattice vector, we know  $𝐆\cdot (𝐚+𝐛+𝐜)=2\pi (h+k+l)$. The Laue equations specify  $𝜟𝐤\cdot (𝐚+𝐛+𝐜)=2\pi (h+k+l)$. Hence we have  $𝜟𝐤=𝐆$  or  ${𝐤}_o-{𝐤}_i=𝐆$.

From this we get the diffraction condition:

$\begin{array}{cc}{𝐤}_0-{𝐤}_i& =𝐆\\ ({𝐤}_i+𝐆{)}^2& ={𝐤}_0^2\\ {k_i}^2+2{𝐤}_i\cdot 𝐆+G^2& ={k_0}^2\end{array}$

Since $({𝐤}_0{)}^2=({𝐤}_i{)}^2$ (considering elastic scattering) and $𝐆=-𝐆$ (a negative reciprocal lattice vector is still a reciprocal lattice vector):

$2{𝐤}_i\cdot 𝐆=G^2$.

The diffraction condition  $2{𝐤}_i\cdot 𝐆=G^2$  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