Accessible reciprocal space

Every diffraction experiment can be described as a scan in the reciprocal space. In the case of surface scattering the accessible reciprocal space is defined by two kinds of half spheres in the reciprocal space shown in figure [18]. The larger sphere reflects the energy conservation during a typical diffraction experiment, i.e. the incident k$_I$ and final wave vector k$_R$ have the same length (elastic scattering). Another inaccessible area are the two smaller spheres, called Laue zones, which are a result of the fact that the the k$_I$ and k$_R$ can not penetrate the sample during an experiment.

Figure 2.1: Accessible reciprocal space for the wave vector transfer Q. The wave vector transfer Q is limited by the energy conservation (elastic scattering is considered - larger half sphere) and the fact that the incident and the exit wave vector can not penetrate the sample (Laue zones - two smaller half spheres.) Figure adapted from adapted from [18].

The chosen scan type determines the properties of the sample which can be investigated. The most important parameter during a scattering experiment is the wave vector transfer

$$\textbf{Q} = \textbf{k$_R$}-\textbf{k$_I$},$$ (1)

where k$_I$ and k$_R$ are the incident and reflected wave vectors of the radiation with the wavelength $\lambda$. In this chapter only elastic scattering is considered, i.e.

$$\vert\textbf{k$_I$}\vert = \vert\textbf{k$_R$}\vert=\frac{2\pi}{\lambda}.$$ (2)

The length of Q defines the size of structures (lattice spacing, layer thicknesses, roughnesses, ...) which can be investigated. The components of Q define the direction in which these structures can be investigated. There are at least three experimental methods (X-Ray Reflectometry - XRR, Grazing Incidence Small Angle Scattering - GISAX, Grazing Incidence Diffraction - GID) which scan different areas in the above mentioned region in the reciprocal space with the wave vector transfer Q during the measurement and are therefore sensitive to different structural properties of the sample. In the following subsection the X-Ray Reflectivity important for understanding of this work is introduced. An introduction to the other methods can be found in references [19,20].