Miller indices for hexagonal structures. Why and how we use 4 indices.
Miller indices are a super useful way of identifying points, directions, and planes in crystal structures. Miller indices can also denote families of equivalent planes and directions. In non-cubic systems its easy to identify all members of the family by identifying permutations of the miller indices including both positive and negative values. For example, the 100 family includes -100, 100, 010, 0-10, 001, and 00-1. The problem with hexagonal crystal structures CANNOT generate all family members by using permutations when we use the traditional 3 indix Miller notation. Instead, we need a new system with 4 indices based on 4 axes. This video shows how this works.