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

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