Chapter 2.2 Direct methods

Reciprocal space

Second Online Edition (2010)

Part 2. Reciprocal space in crystal-structure determination

  1. C. Giacovazzo

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000764

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Giacovazzo, C. 2010. Direct methods. International Tables for Crystallography. B:2:2.2:215–243.

Author Information

  1. Dipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy

Publication History

  1. Published Online: 1 JUN 2010


Direct methods are essentially reciprocal-space techniques, developed historically to solve small-molecule crystal structures. Their success in this area (in practice, they solve the phase problem) is based on numerous theoretical achievements which concern origin specification (Section 2.2.3), the concepts of structure invariants and seminvariants, normalization of the structure factors (Section 2.2.4), inequalities among structure factors, and probabilistic phase relationships (Section 2.2.5). Probabilistic phase relationships are at the core of direct methods: triplet (via Cochran and via the P10 formula), quartet (according to Hauptman and to Giacovazzo) and quintet invariant phase estimates are described, along with estimates of one-phase and of two-phase structure seminvariants (via representation theory). Determinantal formulas are also quoted. Techniques for estimating invariants by using some prior information and their real-space counterparts are discussed. The success of direct methods is intimately connected with the phasing procedures. The most important tools of the procedures (e.g. the tangent formula, magic integers, random-start approaches and figures of merit for recognizing the correct solution) are analysed. The integration of direct methods with macromolecular crystallography is discussed in Section 2.2.10: in particular we refer to the ab initio methods for solving protein structures (in combination with direct-space techniques like electron-density modification, envelope determination, histogram matching etc.) as well as to the combination of direct methods with SIR–MIR and SAD–MAD techniques (both in ‘one-step’ procedures and in the two-step method, the latter requiring the prior determination of the heavy-atom or anomalous-scatterer substructure).


  • direct methods;
  • ab initio phase determination;
  • structure seminvariants;
  • origin specification;
  • phase relationships;
  • phase determination;
  • normalized structure factors;
  • quasi-normalized structure factors;
  • triplet relationships;
  • quartet phase relationships;
  • quintet phase relationships;
  • determinantal formulae;
  • Sayre’s equation;
  • tangent formula;
  • magic-integer methods;
  • macromolecular crystallography;
  • isomorphous replacement;
  • multiple isomorphous replacement with anomalous scattering;
  • multiwavelength anomalous diffraction;
  • single isomorphous replacement with anomalous scattering;
  • MIRAS;
  • MAD;