Diffraction from single crystals with monochromatic radiation

In the present version of the notes, we will only briefly touch upon the important topic of single crystal scattering.

Rotation of a crystal in the beam

Consider again the situation of a crystal in a monochromatic beam, shown by Figure xx--CrossReference--fig:scatteringplane--xx. We assume that the crystal is oriented so that the reciprocal lattice vector under consideration is kept within the scattering plane, which is spanned by \({\bf k}_{\rm i}\) and \({\bf k}_{\rm f}\). Scattering occurs when the crystal is oriented so that Bragg's law is fulfilled, \({\bf q} = \boldsymbol\tau\).

The plot of the scattered intensity as a function of crystal orientation, \(\omega\), is known as a rocking curve. For a real crystal, this curve is not shaped as a \(\delta\)-function, but has a non-zero width. There are a number of contributions to this broadening, including beam divergence, imperfect monochromaticity, imperfect sample (mosaicity), and wave-mechanical effects (dynamical diffraction). The two latter effects will be discussed in the subsections below.

The broadening of the peak makes it appropriate to calculate the integrated area under the rocking curve, also known as the scattering power:

\begin{equation}\label{dummy302014135} P \equiv \Psi \displaystyle\int {\boldsymbol\sigma}_{\rm {\boldsymbol\tau}} d\omega . \end{equation}

This integral can be solved using equation \eqref{eq:bragg_half_int}:

\begin{align} \label{eq:Pk} P &= N \dfrac{(2\pi)^3}{V_0} \Psi |F_{\rm N}({\boldsymbol\tau})|^2 \displaystyle\int \dfrac{2}{k_{\rm i}} \delta(\tau^2 - 2k_{\rm i}\tau\cos(\omega)) d\omega \\ &= N \dfrac{(2\pi)^3}{V_0} \Psi \dfrac{|F_{\rm N}|^2}{k_{\rm i}^3 \sin\theta} .\nonumber \end{align}

It is seen that the unphysical \(\delta\)-function has been integrated out. Equation \eqref{eq:Pk} is alternatively written

\begin{equation}\label{dummy428611923} P = \Psi \dfrac{V}{V_0^2} \dfrac{\lambda^3}{\sin\theta} |F_{\rm N}({\boldsymbol\tau})|^2 . \end{equation}

It is seen that the scattering power depends quite strongly upon the wavelength of the neutron beam.

Crystal mosaicity; secondary extinction

Most crystals used for neutron scattering experiments are imperfect. They can often be described by an assembly of small crystallites, aligned randomly around a mean orientation. We then talk about a mosaic crystal. Often the crystallite orientations can be well described by a normal distribution, and one talks about a Gaussian mosaic. The mosaicity of the crystal is given as the FWHM of this Gaussian curve, and is often denoted by \(\eta\). The mosaicity will serve to broaden the rocking curve, which to first approximation will obtain the same integrated intensity as calculated above, and with a width equal to \(\eta\).

Due to the scattering from the crystallites, the beam flux is attenuated while traveling through the crystal. On the other hand, the beam scattered inside the crystallite due to the reciprocal lattice vector, \({\boldsymbol\tau}\), can be scattered back to the original beam by a reciprocal lattice vector, \(- {\boldsymbol\tau}\). This complex dampening of the original beam is denoted secondary extinction and will be described in a later volume of these notes.

Experimental considerations

When a crystal is used as a monochromator, some amount of mosaicity is required, in order to let the crystal reflect a larger part of the incoming beam. Due to the secondary extinction, the peak reflectivity is often similar for a low-mosaic as for a high-mosaic crystal. Real mosaic crystals used for monochromators often have a mosaicity of \(0.1^\circ - 1^\circ\). The smaller the mosaic, the smaller the divergence, wavelength spread, and total flux of the outgoing beam.

Extinction can result in the effect that the width of the rocking curve can be somewhat different from the true crystalline mosaicity, \(\eta\). For monochromators, one uses the width of the rocking curve to describe the mosaicity.

Perfect crystals; primary extinction

Some materials can be produced with perfect crystallinity, i.e. with zero mosaicity over several millimeters - or even centimeters. Typical examples are the semiconductors Si and Ge. Even in these cases, the rocking curve is not truly shaped as a \(\delta\)-function. A detailed calculation of the wave propagation within the crystal shows that the wave is strongly attenuated within the crystal. This effect is denoted primary extinction and it ruins the assumption of interference between nuclei of the whole crystal. This again produces an effective mosaicity of the crystal of the order a fraction of an arc minute.

The calculation of primary extinction, also known as dynamical diffraction, can be found in most textbooks on X-ray diffraction; for example in . However, it is often not important for neutron scattering, due to its smaller scattering cross section, and hence longer attenuation depth within the crystal. In this version of the notes, we will not dig deeper into the topic of primary extinction.