Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 2.2, p. 210   | 1 | 2 |

Section 2.2.2. Introduction

C. Giacovazzoa*

aDipartimento Geomineralogico, Campus Universitario, I-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, a number of effective, highly automated packages for the practical solution of the phase problem are today available to the scientific community.

The ab initio crystal structure solution of macromolecules seems not to exceed the potential of direct methods. Many efforts will certainly be devoted to this task in the near future: a report of the first achievements is given in Section 2.2.10[link].

This chapter describes both the traditional direct methods tools and the most recent and revolutionary techniques suitable for macromolecules.

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 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 techniques suitable for the ab initio crystal structure solution of macromolecules are described in Section[link]. The integration of direct methods with isomorphous-replacement and anomalous-dispersion techniques is briefly described in Sections[link] and[link].

The reader will find full coverage of the most important aspects of direct methods in the recent books by Giacovazzo (1998[link]) and Woolfson & Fan (1995[link]).


Giacovazzo, C. (1988b). Direct phasing in crystallography. New York: IUCr, Oxford University Press.
Woolfson, M. & Fan, H.-F. (1995). Physical and non-physical methods of solving crystal structures. Cambridge University Press.

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