Planes

The properties of crystalline materials are anisotropic, meaning that they are different when measured in different directions in or on the crystal. Example include the elastic constants (which determine the speed of sound), mechanical strength, and optical properties. Surface properties also vary depending on which side of the crystal the surface lies. All of these very important issues can be described and understood by considering the orientations of geometric planes. Obviously crystal surfaces form planes, and thus the specification of a crystal surface can be made by describing the orientation of the surface plane, assuming the crystal itself is in some standard orientation. Less obviously, the anisotropic behavior of the bulk crystal is most effectively described by considering the behavior of all atoms lying in the plane perpendicular to the direction in which the property (e.g., sound speed) is measured.  

A plane can be specified by selecting an orientation (i.e., the orientation of a vector normal to the plane), and a lattice site that is in the plane (to locate the plane in space). One can then identify a specific set of atoms, defined as all the atoms that lie in this plane (so, if the plane defines one face of the crystal surface, then these atoms would be all the surface atoms for that crystal face). If the plane is then translated in the direction of its normal (keeping its orientation fixed), it will not encounter another similar set of co-planar atoms until it has been traveled some discrete distance through the lattice. This distance depends on the plane's orientation, and the anisotropy of crystal properties is in part a consequence of this geometric fact.

The following figure presents a 2-dimensional lattice with planes showing two orientations (in 2-D the planes are now just lines). Two adjacent planes are shown for each orientation. The red vectors are the normal vectors for each orientation, and one can see how the spacing between the planes is different for each orientation.