Information Theory and the 'Phase Problem' in Crystallography

O. E. PIRO

ACTA CRYSTALLOGRAPHICA A

IUCr

Año: 1983 vol. 39 p. 61 - 68

0108-7673

The ´phase problem´ is cast in terms of information theory. Constrained maximum-entropy inference is used to obtain the statistical structure of the largest possible set of electron density functions compatible with conditions that underlie the derivation of the Sayre-Hughes equation. As a consequence, the information provided by the knowledge of some structure factors is quantitatively expressed. The most unbiased prediction for the phases of structure factors whose moduli are known leads to the statement of the ´minimum added information rule´ which corresponds with Tsoucaris´s ´maximum determinant rule´ [Tsoucaris (1970). Acta Cryst. A26, 492-499]. Expansion of the information in multiplets followed by the application of the proposed rule leads to the current formulae of direct methods. The information content of these formulae is discussed, and its dependence upon the magnitude of the structure factors and a priori structural knowledge is emphasized.