Directions and planes in three-dimensions (3-D) are not easy to visualize, and electron stereographic projection or stereogram is a method to helps analysis of crystal structures. This method is used to present 3-D orientation relationships on a two-dimensional (2-D) figure. When constructed accurately, angles between directions in any crystal system can be quantitatively measured. In general, in the stereographic projection, all directions to be studied are considered as originating from the center of the sphere. The point at which each of these radii intersects the sphere is then projected back through the equatorial plane.
Like drawing a map of the earth, you can imagine a crystal located inside a sphere as shown in Figure 4124a. There are two steps to obtain a stereographic projection. Firstly, draw lines normal to corresponding crystal planes from the center of the sphere to intersect the sphere at point Ns (N1 to Nn) in the northern hemisphere (There are 5 Ns shown in the crystal example). Similarly, there are 5 Ss (S1 to S5) shown in the southern hemisphere and there are 6 Ps (P1 to P6) shown on the equatorial plane. This projection on the sphere is called spherical projection.
Figure 4124a. The spherical projection. The crystal is located at the center of the sphere. Poles to planes are normally labeled as the indices with the brackets removed.
Secondly, as shown in Figure 4124b, draw lines from the south pole (S3) to the points Ns and from the north pole (N3) to the points Ss. These new lines cut the equatorial plane (in grey) at the blue points P's. The disk of the equatorial plane is the stereographic projection and the points (Ps and P’s) uniquely represents the crystalline plane. The black circumference of the disk is called primitive great circle. Note a great circle is one whose plane passes through the center of the sphere so that it always passes through opposite ends of a diameter in the projection. Circles on the sphere which do not contain the center of the sphere are smaller, and thus are called small circles.
Figure 4124b. The stereographic projection.
It is hard to recognize the pattern in the grey disk in Figure 4124b, but Figure 4124c gives you a better idea. Figure 4124c shows the stereographic projection (the great disk in Figure 4124b) of the isometric crystal
(in Figure 4124a) with the faces identified by Miller indices. The red and blue dots in Figure 4124c are the corresponding red and blue dots in Figure 4124b, respectively (Some dots are not shown in Figure 4124b).
Figure 4124c. Stereographic projection of the isometric crystal
(in Figure 4124a).
The stereographic projection is an analogous and powerful tool for working problems that involve relative orientations between two different crystals. With stereographic projections, we can easily visualize crystallographic features:
i) Crystal symmetry.
ii) Slip planes and directions.
iii) Crystal planes and orientation relationships.
iv) Grain orientations.
Although these problems can be solved with rotation matrices, stereographic projections are quick and easy, refer to Wulff Net. Assuming [UVW] is the electron beam direction, the poles (plane normals) at the UVW zone axis represent the possible diffraction planes for this zone. Therefore, when [UVW] is in the center of the projection, all the hkl reflections should be around the circumference of the projection. This is the reason that electron stereographic projection can be used for interpretation of electron diffraction patterns and determination of crystalline orientations.