Phase determination from the Karle-Hauptman determinant. II. Connexion between inequalities and probabilities

The probabilistic properties of a Karle-Hauptman determinant are investigated, with particular reference to the case where all elements are assumed to be known except one. In previous papers it has been shown that the matrix associated with a Karle-Hauptman determinant can be interpreted as a covariance matrix, and also that the probability law associated with one unknown element is a complex Gaussian law centred at the expected value given by the regression-plane equation. These results are now extended to the case where several structure factors are unknown. Furthermore, the connexion between inequalities, the Sayre-Hughes equation and probability relations is discussed. It appears that the Karle-Hauptman inequality defines the allowed domain as a hyperellipsoid, the centre of which corresponds to the most probable set of structure-factor phases. Factors concerned in the selection of a determinant suitable for efficient phase determination are given.