Structure of Materials An Introduction to Crystallography, Diffraction and Symmetry

In this chapter, we introduce a few important tools for crystallography. We begin with the stereographic projection, an important graphical tool for the description of 3-D crystals. Then, we discuss briefly the vector cross product, which we used in Chapter 6 to define the reciprocal lattice. We introduce general relations between different lattices (coordinate transformations), a method to convert crystal coordinates to Cartesian coordinates, and we conclude the chapter with examples of stereographic projections for cubic and monoclinic crystals.

The stereographic projection

In Chapter 5, we defined the Miller indices as a convenient tool to describe lattice planes. We also defined the concept of a family. Since real crystals are 3-D objects, we should, in principle, make 3-D drawings to represent planes and plane normals. This is tedious, in particular for the lower-symmetry crystal systems, such as the triclinic and monoclinic systems. Miller devised a graphical tool to simplify the representation of 3-D objects such as crystals. This tool is known as stereographic projection.

A stereographic projection is a 2-D representation of a 3-D object located at the center of a sphere. Figure 7.1 shows a sphere of radius R; to obtain the stereographic projection (SP) of a point on the sphere, one connects the point with the south pole of the sphere and then determines the intersection of this connection line with the equatorial plane. The resulting point is the SP of the original point. The point on the sphere could represent the normal to a crystal plane, as shown in the figure. The stereographic projection itself is then only the equatorial plane of Fig. 7.1. The projection is represented by a circle, corresponding to the equatorial circle. Inside the circle, the projections from points in the northern hemisphere are represented by small solid circles.