# Determine the indices for the directions shown in the following hexagonal unit cells First projection structure

Here the value of direction projections along a1, a2, z axis are
1, 1/2 and 1/2. When multiplied with 2, we get, 2, 1, 1. With u, v,
t and w directions for projection of hexagonal,

That is,

u' = 2

v' = 1

w' = 1

Feed values in Hexagonal indices determination equation we
get,

u = 1/3(2u' - v') = 1/3[2 x 2 - 1] = 1

v = 1/3(2v' - u') = 1/3[2 x 1 - 2] = 0

t = -(u + v) = -(1 + 0) = -1

w = w' = 1

Therefore the directions in the four directions in the four
indices scheme becomes, 101'(with a bar on top for
-ve)1

with, 1' = 1 with a bar on top for -ve

Second projection structure on right on top

Here the value of a1, a2, z are a/2, a and 0 (or 1/2, 1, 0).
When multiplied with 2, we get, 1, 2, 0. With u, v, t and w
directions for projection of hexagonal,

That is,

u' = 1

v' = 2

w' = 0

Feed values in Hexagonal indices determination equation we
get,

u = 1/3(2u' - v') = 1/3[2 x 1 - 2] = 0

v = 1/3(2v' - u') = 1/3[2 x 2 - 1] = 1

t = -(u + v) = -(0 + 1) = -1

w = w' = 0

Therefore the directions in the four directions in the four
indices scheme becomes, 011'(with a bar on top for
-ve)0

with, 1' = 1 with a bar on top for -ve

First Structure from below row on left

Here the value of directions projection along axis are -1, -1,
1/2. When multiplied with 2, we get, -2, -2, 1. With u, v, t and w
directions for projection of hexagonal,

That is,

u' = -2

v' = -2

w' = 1

Feed values in Hexagonal indices determination equation we
get,

u = 1/3(2u' - v') = 1/3[2 x -2 - (-2)] = -2/3

v = 1/3(2v' - u') = 1/3[2 x -2 - (-2)] = -2/3

t = -(u + v) = -(-2/3 - 2/3) = 4/3

w = w' = 1

Multiply with 3

Therefore the directions in the four directions in the four
indices scheme becomes, 2'2'(with a bar on top for -ve)43

with, 2' = 2 with a bar on top for -ve

First Structure from below row on left

Here the value of directions projection along axis are 0, -1, 0.
With u, v, t and w directions for projection of hexagonal,

That is,

u' = 0

v' = -1

w' = 0

Feed values in Hexagonal indices determination equation we
get,

u = 1/3(2u' - v') = 1/3[2 x 0 - (-1)] = 1/3

v = 1/3(2v' - u') = 1/3[2 x -1 - 0] = -2/3

t = -(u + v) = -(1/3 - 2/3) = 1/3

w = w' = 0

Multiply with 3

Therefore the directions in the four directions in the four
indices scheme becomes, 12'(with a bar on top for -ve)10

with, 2' = 2 with a bar on top for -ve