Direct methods

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Méthodes directes (Fr). Metodi diretti (It). 直接法 (Ja).


A family of methods for estimating the phases of the Fourier transform of the scattering density from the corresponding magnitudes in crystal structure determination. The phases of scattered diffraction beams cannot be directly observed. However, they can be estimated from probability relationships applied to the phases of the most intense diffraction peaks. The facts that scattering centres in a crystal are discrete atoms (i.e. sources of electron density) and that the electron density must be non-negative are the types of constraints that restrict the possible values of the phases, and allow initial estimates of some of them.


The first mathematical relationships capable of giving phase information were obtained in the form of inequalities by D. Harker and J. S. Kasper [(1948). Acta Cryst. 1, 70–45. Phases of Fouriercoefficients directly from crystal diffraction data], and further developed by J. Karle and H. Hauptman [(1950). Acta Cryst. 3, 180–187. The phases and magnitudes of the structure factors]. In 1953 H. Hauptman and J. Karle established the basic concepts and the probabilistic foundations of the direct methods (The solution of the phase problem. ACA Monograph), for which they received the 1985 Nobel Prize in Chemistry. An important relation is the Sayre equation, which is a mathematical relationship that allows one to calculate probable values for the phases of some diffracted beams [D. Sayre (1952). Acta Cryst. 5, 60–65 . The squaring method: A new method for phase determination].

See also

  • Direct methods. C. Giacovazzo. International Tables for Crystallography (2008). Vol. B, ch. 2.2, pp. 215-243