Laue equations

In crystallography, the Laue equations give three conditions for 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:

$\mathbf {a} \cdot \mathbf {\Delta k} =2\pi h$

$\mathbf {b} \cdot \mathbf {\Delta k} =2\pi k$

$\mathbf {c} \cdot \mathbf {\Delta k} =2\pi l$

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  $\mathbf {\Delta k} \cdot (\mathbf {a} +\mathbf {b} +\mathbf {c} )=2\pi (h+k+l)$. Whence we have  $\mathbf {\Delta k} =\mathbf {G}$  or  $\mathbf {k} _{o}-\mathbf {k} _{i}=\mathbf {G}$.

From this we get the diffraction condition:

$\mathbf {k} _{o}-\mathbf {k} _{i}=\mathbf {G}$

↓

$(\mathbf {k} _{i}+\mathbf {G} )^{2}=\mathbf {k} _{o}^{2}$

↓

${k_{i}}^{2}+2\mathbf {k} \cdot \mathbf {G} +G^{2}={k_{o}}^{2}$

↓   (k_i² = k₀²)

$2\mathbf {k} _{i}\cdot \mathbf {G} =G^{2}$.

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