Appendix A1. Coordinate System¶
Definition of orientation¶
ReciPro uses a right-handed coordinate system with axes defined as:
| Axis | Direction |
|---|---|
| X | Right of the monitor |
| Y | Upward on the monitor |
| Z | Vertically in front of the monitor |
The beam direction corresponds to the viewing direction (looking at the monitor), which is the −Z axis.
Most operations in ReciPro involve only directions (3×3 rotation matrices) and do not require explicit origin positions. However, the Crystal Diffraction function requires explicit origin — see Appendix A2. Detector Coordinate System.
Initial crystal direction¶
ReciPro defines the initial crystal orientation as follows:
- The c-axis is aligned with the Z-axis direction.
- The b-axis lies on the YZ plane and is close to the Y-axis.
- The a-axis is determined by the b- and c-axes (right-hand rule).
In other words: - The direction in front of the monitor corresponds to the [001] crystal axis. - The right direction on the monitor corresponds to the normal of the (100) crystal plane.
Note: The c-axis always corresponds to Z, but in some crystal systems the a- and b-axes do not necessarily correspond to X and Y.
Euler angles¶
ReciPro uses Euler angles Φ, θ, and Ψ to represent crystal orientation.
When all angles are zero, the rotation axes correspond to: - Φ → Z-axis (1st, highest rotation) - θ → X-axis (2nd, middle rotation) - Ψ → Z-axis (3rd, lowest rotation)
The three angles have a hierarchy: Φ is the highest rotation, followed by θ, then Ψ. The direction of a lower axis depends on the state of the upper rotations. When Φ = θ = Ψ = 15°, for example, the Φ axis still coincides with Z, but the θ and Ψ axes generally do not align with any of X, Y, or Z.