Critical angle

In GISAXS, the critical angle for a thin film is the incident angle below which one gets total external reflection of the x-ray beam.

Below the critical angle, the beam is fully reflected from the film. The x-ray field probes a short distance into the film surface (due to the evanescent wave); on the order of a few nanometers. Thus, a GISAXS measurement below the critical angle is inherently probing only the film surface. A measurement well above the critical angle, by comparison, penetrates through the film and thus measures the average of the structure through the whole film. Close to the critical angle, the refracted beam is nearly parallel to the film interface; in other words the beam is coupled into waveguide modes. This increases the effective path-length of the beam through the sample, which thereby increases the intensity of the scattering.

Normally in GISAXS experiments, it is useful to do measurements both below and above the critical angle; by comparing the two patterns, you can ascertain whether the structures observed at the surface (e.g. as seen by AFM or SEM) are representative of the entire film. Measurements close to the critical angle are useful because of the intensity enhancement.

Calculating

The critical angle for a material can be calculated in a variety of ways.

From SLD

If the Scattering Length Density (SLD) is known, the critical angle can be computed from it. The critical scattering vector is:

$q_{c}={\sqrt {16\pi \mathrm {SLD} }}$

In reflection-mode, the scattering vector is:

${\begin{alignedat}{2}\mathbf {q} &=\mathbf {k} _{o}-\mathbf {k} _{i}\\|\mathbf {q} |&=q=2|k|\sin \theta _{i}={\frac {4\pi }{\lambda }}\sin \theta _{i}\\\end{alignedat}}$

So:

${\begin{alignedat}{2}\theta _{c}&=\arcsin \left({\frac {q_{c}\lambda }{4\pi }}\right)\\&=\arcsin \left({\frac {\lambda {\sqrt {16\pi \mathrm {SLD} }}}{4\pi }}\right)\end{alignedat}}$