Chapter 1.1 Reciprocal space in crystallography

Reciprocal space

Second Online Edition (2010)

Part 1. General relationships and techniques

  1. U. Shmueli

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000758

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Shmueli, U. 2010. Reciprocal space in crystallography. International Tables for Crystallography. B:1:1.1:2–9.

Author Information

  1. School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel

Publication History

  1. Published Online: 1 JUN 2010


After a brief introduction in Section 1.1.1, Section 1.1.2 of this chapter presents a formal definition of dual (reciprocal) bases and a brief overview of applications of the reciprocal lattice basis to lattice geometry, diffraction conditions and Fourier synthesis of functions with the periodicity of the crystal. In Section 1.1.3, the fundamental relationships between direct and reciprocal bases are derived and summarized. Section 1.1.4 introduces the basics of tensor notation and the representation of the above relationships in this notation, which is particularly well adapted to analytical considerations as well as to computer programming. Several examples from various areas of crystallographic computing follow this introduction and a detailed derivation of the finite rotation operator is presented in this context. This is followed by a section on transformation of basis vectors (Section 1.1.5), of importance in many areas of crystallography. The chapter is concluded with brief mentions of analytical aspects of the concept of reciprocal space in crystallography (Section 1.1.6). The purpose of this chapter is to introduce the reader to or remind them of the fundamentals of concepts which are employed throughout this volume.


  • direct lattice;
  • reciprocal lattice;
  • direct bases;
  • reciprocal bases;
  • reciprocal space;
  • tensor-algebraic formulation;
  • transformations of coordinates;
  • metric tensors;
  • Bloch’s theorem