Miller indices and planes in crystals

In reality this third index is not needed, as it is simply derived from the known values of h and k. However, it does help to bring out the relationship between the planes. Using four indices, the planes are A, (1120), B, (1210) and C, (2110). Because it is a redundant index, the value of i is sometimes replaced by a dot, to give indices (hk.l). This nomenclature emphasises that the hexagonal system is under discussion without actually including a value for i. 

The response of a crystal to an external stimulus such as a tensile stress, electric field, and so on, is frequently dependent upon the direction of the applied stimulus. It is therefore important to be able to specify directions in crystals in an unambiguous fashion. Directions are written generally as [mvw] and are enclosed in square brackets. Note that the symbol [wvw] includes all parallel directions, just as (MZ) specifies a set of parallel planes. 

The direction [100] is parallel to the a-axis, the direction [010] is parallel to the b-axis and [001] is parallel to the c-axis. The direction [111] lies along the body diagonal of the unit cell. Negative values of u, v and w are written u, (pronounced 

Directions in a crystal are specified in the same way. In these instances the integers u, v, w are applied to the unit cell vectors, a-, b- and c-. As with Miller indices, it is sometimes convenient to group together all directions that are identical by virtue of the symmetry of the structure. These are represented by the notation (uvw). In a cubic crystal the symbol (100) represents the six directions [100], [100], [010], [0T0], 

A zone is a set of planes, all of which are parallel to a single direction. A group of planes which intersect along a common line therefore forms a zone. The direction that is parallel to the planes, which is the same as the line of intersection, is called the zone axis. The zone axis [uvw] is perpendicular to the plane (uvw) in cubic crystals but not in crystals of other symmetry. 

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