Geometry:WAXS 3D

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In wide-angle scattering (WAXS), one cannot simply assume that the detector plane is orthogonal to the incident x-ray beam. Converting from detector pixel coordinates to 3D q-vector is not always trivial, and depends on the experimental geometry.

Area Detector on Goniometer Arm

Consider a 2D (area) detector connected to a goniometer arm. The goniometer has a center of rotation at the center of the sample (i.e. the incident beam passes through this center, and scattered rays originate from this point also). Let be the in-plane angle of the goniometer arm (rotation about -axis), and be the elevation angle (rotation away from plane and towards axis).

The final scattering vector depends on:

  • : Pixel position on detector (horizontal).
  • : Pixel position on detector (vertical).
  • : Sample-detector distance.
  • : Elevation angle of detector.
  • : In-plane angle of detector.

Note that and are defined relative to the direct-beam. That is, for and , the direct beam is at position on the area detector.

Central Point

The point can be thought of in terms of a vector that points from the source-of-scattering (center of goniometer rotation) to the detector:

This vector is then rotated about the -axis by :

And then rotated about the -axis by :

The point on the detector probes the total scattering angle , which is simply the angle between and :

Thus:

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