Equations for determining tetartohedral twin fractions

In most cases of merohedral twinning, two different twin-domain orientations are present. A rarer type of merohedral twinning exists in which there are four different twin-domain orientations. The former case is referred to as hemihedral twinning, while the latter more complex type is referred to as tetartohedral twinning. In tetartohedral twinning, each observed reflection is the weighted sum of four twin-related but otherwise independent reflection intensities. The weights that determine how the true crystallographic intensities combine to give the observed intensities are described by four twin fractions representing the fractional volumes of the four different domain orientations within the specimen. Here, equations are developed to determine values for the four tetartohedral twin fractions based on a statistical comparison of quadruplets of twin-related reflections. Knowledge of the twin fractions is important in working backwards to obtain values for the true crystallographic intensities. Use of the equations is demonstrated with synthetic intensity data simulated according to given values of the twin fractions.