Frequencies of tilt and twist boundaries among random grain boundaries

There are a number of classifications of homophase grain boundaries. It is quite common to divide them into twist, tilt and general boundaries. As in the case of the classification into coincident lattice (CSL) boundaries and non-CSL boundaries, one may ask about the possible frequencies of incidence of tilt and twist boundaries in a set of "random" boundaries. The probabilities of occurrence of these particular boundary types are clearly defined if small deviations from pure twist and tilt conditions are allowed. We estimated the probabilities numerically for the cases of cubic and hexagonal holohedries. For a given randomly generated boundary, a computer program searched for the nearest pure-tilt and pure-twist boundaries. All symmetrically equivalent representations of the random boundary were processed, and the smallest distance was taken as the result. The distance was based on both the difference in misorientations and the deviation between boundary inclinations. The findings concerning tilt boundaries turned out to be striking. For instance, if the allowed deviation from pure-tilt conditions is only 1 degrees, then as many as 39.0% and 21.2% of random boundaries have near-tilt character for the cubic and hexagonal cases, respectively. If the limiting deviation is raised to 5 degrees, the frequencies of near-tilt boundaries reach 98.6% and 77.0%, respectively.