Orientation matrix

In a diffraction experiment, the orientation matrix describes the orientation of the single crystal in space. The matrix relates the reciprocal axes of the crystal with the Cartesian coordinate system of the diffractometer. In order to calculate the impact position of a reflection hkl on the detector, one needs the orientation matrix and the setting angles of the goniostat motors. In most cases, the determination of the orientation matrix is the first step in a diffraction experiment and is based on a rather small number of reflections. In a later stage, the orientation matrix can be refined on a larger number of reflections.

Other commonly used names for the orientation matrix are reciprocal axes matrix and UB matrix.

The orientation matrix is a product of the matrix U and the matrix B. The matrix B transforms a reciprocal lattice vector into an orthonormal coordinate system. The matrix B has the following property: [math]B^T B = G^{-1}[/math] ([math]G^{-1}[/math] is the reciprocal space metric tensor). The matrix U is a rotation matrix and relates the orthonormal ccordinate system of the crystal to the orthonormal ccordinate system of the diffractometer.

Matrix B can be created in infinite number of ways, and different computer programs use different definitions for the diffractometer coordinate system. Care needs to be taken how the orientation matrix is defined in each individual case.

Example coordinate system

Definition of the diffractometer coordinate system in the EVAL software [Duisenberg, Kroon-Batenburg & Schreurs (2003). J. Appl. Cryst. 36, 220-229].

The single crystal is located at the origin K. The primary X-ray beam defines the X axis. The Z axis coincides with the rotation axis of the ω motor of the goniostat. The Y axis completes the right-handed coordinate system. All goniostat motors of the diffractometer are set to zero. The impact position of the diffracted beam on the detector is characterized by the coordinates [math](x_D,y_D)[/math].

Example orientation matrix

In this example definition, the reciprocal axes matrix is given as

[math] R = \left( \begin{array}{lll} a^\star_x & b^\star_x & c^\star_x \\ a^\star_y & b^\star_y & c^\star_y \\ a^\star_z & b^\star_z & c^\star_z \end{array} \right) [/math]

It is thus straightforward to extract the reciprocal axes from this matrix. [math]a^\star[/math] corresponds to the first column, [math]b^\star[/math] to the second, [math]c^\star[/math] to the third. The subscripts x, y, and z indicate the Cartesian coordinates of the diffractometer.

References

W. R. Busing & H. A. Levy (1967). Acta Cryst. 22, 457-464.
E. Prince (1994). Mathematical Techniques in Crystallography and Materials Science (2nd ed.), pp. 54-56.
P. Luger (1980). Modern X-ray Analysis on Single Crystals, pp. 176-183.