This is an archive copy of the IUCr web site dating from 2008. For current content please visit https://www.iucr.org.

The Reciprocal Lattice

A. Authier

Laboratoire de Minéralogie Cristallographie associé au C.N.R.S. - Université Pierre et Marie Curie, PARIS

Download in PDF format  

• 1. Introduction
• 2. Crystallographic Definition
  • 2.1 Definition
  • 2.2 Fundamental law of the reciprocal lattice
• 3. Reciprocal Space and Dual Space
  • 3.1 Definition
  • 3.2 The volumes of the unit cells in direct and reciprocal space are inverse
  • 3.3 Calculation of the reciprocal lattice vectors using the metric tensor
• 4. Crystallographic Calculations Using the Reciprocal Lattice
  • 4.1 Scalar product of direct and reciprocal lattice vectors
  • 4.2 Vector product of two direct lattice vectors
  • 4.3 Indices of the set of lattice planes parallel to two direct lattice rows
  • 4.4 Zone axis of two sets of direct lattice planes
  • 4.5 Reciprocity of F and I lattices
• 5. Diffraction Condition in the Reciprocal Lattice

Teaching Aims

To give a firm mathematical understanding of the reciprocal lattice, of the relationships between real and reciprocal space and of their implications for X-ray diffraction.

Level

This approach would be suitable for final year undergraduates in physics and mathematics or for initial post-graduate students in other disciplines provided that their mathematical background is adequate.

Background

A familiarity with vector manipulation is needed and, for certain sections, an understanding of tensor calculus.

Practical Resources

No specific practical resources are required.

Time Required for Teaching

If the mathematical background is already adequate this could be taught in 3 or 4 lectures. More would be required, however, if time has to be spent on mathematical equations and derivations as in places the treatment given is very concise.

Copyright © 1981, 1998 International Union of Crystallography