The crystallographic quaternions and their product law

Unit quaternions are widely used in science to encode rotations because the quaternion product is more efficient than matrix product and more stable than Rodrigues product to calculate the composition of two rotations. However, quaternions in their usual form refers to a Cartesian basis; they cannot be used in crystallography as they are. The usual way to solve this issue to apply back-and-forth coordinate changes from the crystal basis to a Cartesian basis attached to the crystal with the help of the structure tensor. Here, we show that actually quaternions can be used directly in the crystal basis by generalizing the quaternion product law. In that aim, we introduced a matrix that we called cross tensor. It allows the calculation of the cross product in the crystal basis, a bit like the metric tensor allows it for the scalar product. We also show the cross tensor is proportional to the inverse of the metric tensor. The formula of crystallographic quaternion product is then given; it depends uniquely on the metric tensor. The application of the crystallographic quaternions to Electron Back Scatter Diffraction is discussed.