What are the indices for the directions indicated by the two

What are the indices for the directions indicated by the two vectors in the sketch below? Determine indices for the directions shown in the following unit cells: What are the indices for the two planes drawn in the sketch below Below are shown three different crystallographic planes for a unit cel of some hypothetical metal. The circles represent atoms:

Solution

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