Crystallography; from hcp to fcc

[hzx]

Homework Statement

Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all (111) planes, what is the translation vector r in terms of unit vectors [100], [010], and [001] of the fcc lattice?

Homework Equations

The Attempt at a Solution

Unit vectors [100], [010], and [001] of the fcc lattice can be expressed as a, b, and c (any arbitrary origin point O would work).
The problem can be converted to finding relative position of one atom from hcp from an arbitrary origin point O defined using fcc lattice.

 

[Valkarie]

From the diagram of hcp lattice, it can be seen that in fcc, there is a shift of c/2 and c/√3 along x and y axis respectively. Therefore, the translation vector r = (c/2)[100] + (c/√3)[010]