Crystallographic Notation and Stereograms

Surfaces and thin films course

Prepared as a project by Monika Fleischer with John Venables,
October-December 1998

CPES, University of Sussex, Brighton, UK.


  • Notation

  • Stereograms

  • Properties of stereograms and Algebraic relations

  • Stereographic projection of surface planes

  • References

  • Notation

  • is a direction (in real space)

  • describes a class of directions (symmetry related)

  • gives the direction of plane normals (Miller indices)

  • refers to class of (symmetry related) plane normals
  • Stereograms

  • Planes and directions of a crystal can be represented on a Stereogram, shown below.
  • Measurement of angles can be performed with a Wulff net.

    If the stereogram and the net are reproduced at exactly the same size, angles between directions or plane normals can be read off graphically. Most of this work would now be done using algebra, as given below, or using computer programs or databases, some of which are listed referenced at the end of this page.

  • Properties of Stereograms

  • Great and small circles on the sphere project into circles on the stereogram.
  • Angular relationships are preserved and can be measured along great circles using the Wulff net.
  • Algebraic relations

  • The plane (hkl) contains the direction [uvw] if the scalar product (hkl)*[uvw] = 0, i.e. hu + kv + lw = 0.

  • Similarly any plane satisfying (hkl)*[uvw] = 0 is in the zone [uvw], i.e. the family of planes containing [uvw].

  • The plane containing two directions [uvw]1 and [uvw]2 is given by the vector product [uvw]1 /\ [uvw]2.

    This can be worked out as

    e.g. the plane containing [110] and [211] is

  • Beware! Plane normals (hkl) and axes [hkl] are only the same in the cubic system!

  • The angle q between [uvw]1 and [uvw]2 (or (hkl)1 and (hkl)2) is given by:

    (For non-cubic crystals, see Kelly and Groves, appendix 3.)
  • Stereographic projection of surface planes

    In many cases, one needs to be able to read stereograms to plot and interpret data, such as surface energies as a function of crystallographic plane orientation. For surfaces and thin films, the structure and reconstructions of particular surface planes can be important. There are several web-based sources which are useful, which we are currently searching. To get an idea of the power of these tools, try


    Some of the oldest books are the best on this topic, so try A. Kelly and G.W. Groves (1970) Crystallography and crystal defects (Longman), chapters 1-3.
    Latest version of this document 12th November 2009, ex 20th Jan 2006.