Determine the indices for the directions shown in the following hexagonal unit cells

Please show how you get the answer

3.57 Determine indices for the directions shown in the following hexagonal unit cells: on 2 ell, sketch the following (e) 11 (g) [123] IT221 (h) [103] an (a) r the directions shown (b) (d) 3.58 sketch the [1123] and [1010) directions in a hexagonal unit cell

3.57 Determine indices for the directions shown in the following hexagonal unit cells: on 2 ell, sketch the following (e) 11 (g) [123] IT221 (h) [103] an (a) r the directions shown (b) (d) 3.58 sketch the [1123] and [1010) directions in a hexagonal unit cell

Answer

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

Answer = 101'1

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

Answer = 011'0

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

Answer = 2'2'43

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

Answer = 12'10