Laue diffraction

In Laue diffraction, one places a crystal with fixed orientation within a polychromatic beam of neutrons. Ideally, for each reciprocal lattice point, there exists a neutron wavelength that fulfil the Bragg law. In this way, all reciprocal lattice points will contribute with scattering peaks for the same experiment. The position of the diffracted peaks will appear as though each set of lattice planes reflects (one wavelength of) the beam as a mirror.

Qualitative Laue investigations

One important use of Laue diffraction is to define the orientation and quality of a crystal to be used for other types of neutron scattering experiments (e.g. monochromatic diffraction or inelastic scattering). The positions of the Laue reflections directly show the crystal orientation with respect to one of the main crystal axes, and the symmetry and relative strengths of the Laue reflections show important information of the space group of the crystal.

One will often use backscattering Laue diffraction for this type of Laue investigations.

Experimental consideration

By use of the backscattering Laue technique, it is possible to screen the quality of a large set of crystals, using only seconds to minutes of measurement time per crystal.

Quantitative Laue investigations

Laue diffraction can used also to refine the structure of a crystal by measuring the nuclear structure factor, \(F_{\rm N}({\bf q})\), for a series of scattering vectors equalling reciprocal lattice vectors, \({\bf q} = {\bf \tau}\).

As a start, we need to calculate the scattering power of the Bragg reflections. We define

\begin{align} P &= \int \Psi(\lambda) \sigma_{{\rm tot},{\boldsymbol\tau}} d\lambda \label{dummy1650580835}\\ &= N \frac{(2\pi)^3}{V_0} \left| F_{\rm N}({\boldsymbol\tau}) \right|^2 \int \frac{2}{k}\delta(\tau^2 - 2 k\tau\cos(\omega)) \Psi(\lambda) d\lambda . \nonumber \end{align}

By using the substitution \(x=2 k_{\rm i}\tau\cos(\omega)\), we reach the result

\begin{equation} \label{dummy1252261532} P = N\dfrac{(2\pi)^3}{V_0} \dfrac{\pi\Psi(\lambda_{\rm i})|F_{\rm N}({\boldsymbol\tau})|^2}{k_{\rm i}^4\cos^2(\omega)} = \dfrac{V}{V_0^2} \Psi(\lambda_{\rm i})\dfrac{\lambda^4}{2\sin(\theta)}|F_{\rm N}({\boldsymbol\tau})|^2 . \end{equation}

This expression will equal the observed count rate in a particular Laue reflection. The number should, however, like any other theoretical parameter be corrected by a proper experimental normalization factor.

We will not at this point in time go deeper into Laue diffraction. This section may later be expanded through input from from Mogens Christensen, Univ. Århus.