GI missing wedge

Grazing-incidence measurements give rise to a 'missing wedge' of data near the q_z axis, owing to the curvature of the Ewald sphere.

Explanation

Consider a sample coordinate system where z points vertical (film normal), y points along the beam, and thus x is orthogonal to both of those. The sample's reciprocal space is similar: q_z is the film normal, q_y is along the beam, and q_x is across the sample (both q_x and q_y are in the sample plane). We can also define:

$q_{r}={\sqrt {q_{x}^{2}+q_{y}^{2}}}$

which is the total q-vector in the sample plane.

The detector is a flat plane, but in reciprocal-space, what the detector probes is a curved slice through the 3D reciprocal-space, known as the Ewald sphere. For small-angle scattering, it's a reasonable approximation to ignore this curvature and treat the detector image as a flat slice through reciprocal-space. But this becomes a poor approximation at wide-angles, where the curvature of the Ewald sphere is significant.

The curvature of the Ewald sphere means that any given pixel is not probing (qx,qz), but actually probing (qx,qy,qz) in the 3D space, where you can't assume qy=0 (again, unless you're at sufficiently small angle). So the problem is: how do you represent this 3D slice on a 2D graph for showing on a piece of paper? The options are:

1. Just ignore the qy component, and plot qz on the vertical, and qx on the horizontal. This will look nice (no missing wedge). However it can be misleading. For instance, scattering rods that are vertical in reciprocal space will look curved in this view. (Higher orders along qz will not line up correctly.) Also, it looks like the image is showing the qz axis of reciprocal-space (where qx=0), but this is a lie: in reality the experiment never probed the "true qz", since the Ewald sphere was curving away from this axis. So, actually, you never probed the qz axis.

2. Alternatively, you can plot qz along the vertical, but plot qr along the horizontal. This introduces a 'missing wedge', which is physically correct in the sense that you never actually measured the qz axis in reciprocal space (the area where both qx=0 and qy=0). The downside is that you've essentially assumed that qx and qy are equivalent: i.e. that the film is an in-plane powder. For in-plane aligned samples, you may need to be careful about interpreting the image.

Either of the above options for displaying data is "okay", as long as you're clear about what you're presenting to people, and as long as you keep in mind the things that a particular image mis-represents.

So, if you want to have an image without a missing wedge, you can plot (qx,qz). (I'm not sure if view.gtk can do this easily, but the Python code has this as an option.)