The used geometry - X-Ray Reflectivity (XRR)

An XRR measurement is a $\theta-2\theta$ scan as known from conventional X-ray diffraction experiments performed in a very small incident angle $\alpha$ region (typically $0~\leq~\alpha~\leq~4$). The incident wave vector k$_I$ and the outgoing wave vector k$_R$ are in the same plane (coplanar geometry) as shown in figure . If the incident angle $\alpha$ and the exit angle $\alpha_{R}$ are the same, the wave vector transfer Q has only a component perpendicular to the surface Q=(0,0,Q$_z$). It means, that kind of measurements is sensitive to vertical modulations in the sample. Usually the intensity of the scattered radiation drops drastically when increasing the incident angle (I $\propto$ Q$_z$$^{4}$), that is why only small values for the length of Q can be reached. It is clear that this method can not be used to investigate atomic structure but it is suitable to receive information about layer thickness, interface and surface roughness or modulation of the chemical composition [21].

Figure: Sketch of the XRR geometry. The incident wave vector k$_I$ and outgoing wave vector k$_R$ are in the same plane (coplanar geometry). The incident angle $\alpha$ and the exit angle $\alpha_{R}$ are typically a few degrees. The symmetric scan ($\alpha$ = $\alpha_{R}$) used for determination of the layer thickness and the surface roughness [21] is sketched. Due to the coplanar geometry Q has only a vertical component Q=(0,0,Q$_z$).

Using an asymmetric scan where $\alpha \neq \alpha_{R}$ one can investigate lateral parameters of the sample, there is, however, always a restriction of the two Laue zones and due to the fast decrease of the intensity only very small in-plane components of Q can be reached.

$$Q_x=2 k \sin \theta \sin(\alpha-\theta)$$ (3)

with

$$2\theta=\alpha+\alpha_{R}$$ (4)

Assuming for $\alpha$ the typical value of 3 degrees and extremely asymmetric conditions ($\alpha_{R}$=0, i.e. the wave vector transfer Q touches the Laue zone) the maximum value of $Q_x$ can be written as

$$Q_{x,max} \approx 2k\sin^{2}(1.5°) \approx 0.01~\textnormal{\AA}^{-1}$$ (5)

Therefore the typical size which can be investigated laterally is

$$d=\frac{2\pi}{Q_{x,max}} \approx 600~\textnormal{\AA}$$ (6)

for 14.4 keV radiation.