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. ${\displaystyle \mathbf {k} _{o}-\mathbf {k} _{i}=\mathbf {\Delta k} }$ is the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.

Take ${\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} }$ 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:

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

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

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

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

From this we get the diffraction condition:

${\displaystyle {\begin{aligned}\mathbf {k} _{o}-\mathbf {k} _{i}&=\mathbf {G} \\(\mathbf {k} _{i}+\mathbf {G} )^{2}&=\mathbf {k} _{o}^{2}\\{k_{i}}^{2}+2\mathbf {k} _{i}\cdot \mathbf {G} +G^{2}&={k_{o}}^{2}\end{aligned}}}$

Since ${\displaystyle (\mathbf {k} _{o})^{2}=(\mathbf {k} _{i})^{2}}$ (considering elastic scattering) and ${\displaystyle \mathbf {G} =-\mathbf {G} }$ (a negative reciprocal lattice vector is still a reciprocal lattice vector):

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

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

References

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