# Geometry:WAXS 3D

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.

## Contents

# 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 :

### Total scattering

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

Thus:

### Components

The momentum transfer vector is (for elastic scattering):

This vector is of course the surface of the Ewald sphere.

## Arbitrary Point

For other points on the detector face, we can combine the above result with the known results for the Geometry of TSAXS. The incident beam is:

For and , we can compute the vector onto the detector face:

This vector is then rotated about the -axis by :

The vector is then rotated about the -axis by :

### Components

Where:

### Total magnitude

Thus:

#### Check

We define:

And calculate:

Grouping and rearranging:

# Area Detector on Goniometer Arm, with offsets

In the most general case, the sample may not sit at the exact center of the goniometer rotation. In such a case, corrections must be applied.

TBD