A new method for phase determination. The `maximum determinant rule'

The joint probability of a set of structure factors involved in a Karle-Hauptman determinant is evaluated. For equal atoms, under conditions to be specified, the theory leads to the conclusion; among all combinations of phases compatible with the condition of non-negativity of a Karle-Hauptman determinant, the most probable combination is that which maximizes this determinant. This `maximum determinant rule' can be used as a basis for the practical determination of phases. Special cases and further properties of the determinants are also obtained from the main expression for the joint probability.