Crystallography- Hexagonal and Rhombohedral Indexing

Question 1

Indexing for rhombohedral unit cell

Answer 1

Use the bottom 3 sides [a1, a2, a3]. These are in non-cartesian directions. Miller indices for planes (h k l) as normal possible but impractical

Question 2

Hexagonal 3-index

Answer 2

Use [a1, a2, c] (first two on base and c is centre vertical). These are non-cartesian directions. The direction index is [u v w]. Miller indices (h k l)

Question 3

Hexagonal 4-index

Answer 3

Bravais (or Miller-Bravais). Use [a1, a2, a3, c] so a 4D vector basis system. a1, aw, and a3 are 120° apart. New plane indices (h k i l).
Rule of h+k+i=0. Direction index [u v t w]

Question 4

Cartesian axes in hexagonal 3 and 4-index

Answer 4

z direction is vertical. (001) or (0001)
x direction is along a1. (01bar0) or (01bar10)
y direction between a1 and a2 and perpendicular to x direction. (210) or (21bar1bar0)

Question 5

How to use (hkil)

Answer 5

For a plane through the HCP structure, find where it intersects a1, a2, a3 and c axes. Find reciprocal of each. Reduce to common factors. Put in normal brackets

Question 6

What do each types of brackets mean?

Answer 6

(hkl): one specified plane
{hkl}: set of non-parallel planes which are equivalent in symmetry
[a b c]: one specified direction
<a>: a set of non-parallel directions which are equivalent in symmetry</a>

Question 7

Example of set of non-parallel planes equivalent in symmetry

Answer 7

In cubic system:

{100}= (100),(010),(001),(-100)…

Question 8

Example of set of non-parallel directions equivalent in symmetry

Answer 8

In cubic system!

<111>=[111],[-111],[1-11],[11-1]…