Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.2, p. 215   | 1 | 2 |

Section 2.2.2. Introduction

C. Giacovazzoa*

aDipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy
Correspondence e-mail:

2.2.2. Introduction

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Direct methods are today the most widely used tool for solving small crystal structures. They work well both for equal-atom molecules and when a few heavy atoms exist in the structure. In recent years the theoretical background of direct methods has been improved to take into account a large variety of prior information (the form of the molecule, its orientation, a partial structure, the presence of pseudosymmetry or of a superstructure, the availability of isomorphous data or of data affected by anomalous-dispersion effects, …). Owing to this progress and to the increasing availability of powerful computers, the phase problem for small molecules has been solved in practice: a number of effective, highly automated packages are today available to the scientific community.

The combination of direct methods with so-called direct-space methods have recently allowed the ab initio crystal structure solution of proteins. The present limit of complexity is about 2500 non-hydrogen atoms in the asymmetric unit, but diffraction data at atomic resolution (~1 Å) are required. Trials are under way to bring this limit to 1.5 Å and have shown some success.

The theoretical background and tables useful for origin specification are given in Section 2.2.3[link]; in Section 2.2.4[link] the procedures for normalizing structure factors are summarized. Phase-determining formulae (inequalities, probabilistic formulae for triplet, quartet and quintet invariants, and for one- and two-phase s.s.'s, determinantal formulae) are given in Section 2.2.5.[link] In Section 2.2.6[link] the connection between direct methods and related techniques in real space is discussed. Practical procedures for solving small-molecule crystal structures are described in Sections 2.2.7[link] and 2.2.8[link], and references to the most extensively used packages are given in Section 2.2.9.[link] The integration of direct methods, isomorphous replacement and anomalous-dispersion techniques is briefly discussed in Section 2.2.10[link].

The reader interested in a more detailed description of the topic is referred to a recent textbook (Giacovazzo, 1998[link]).


Giacovazzo, C. (1998). Direct Phasing in Crystallography. New York: IUCr, Oxford University Press.

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