Hexagonal Crystals In the three-axis coordinate system, a problem arises for hexagonal crystals in that some crystallographic equivalent directions will not have the same set of indices. Hence, a four-axis (Miller-Bravais) coordinate system is used. Three-axis Four-axis Chapter 3 -

HCP Crystallographic Directions z Algorithm a 2 - a 3 Adapted from Fig. 3. 8(a), Callister & Rethwisch 8 e. a 1 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a 1, a 2, a 3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] a 2 -a 3 a 2 2 ex: ½, ½, -1, 0 => [ 1120 ] a 3 dashed red lines indicate projections onto a 1 and a 2 axes a 1 2 a 1 Chapter 3 - 2

HCP Crystallographic Directions • Hexagonal Crystals – 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i. e. , u'v'w') as follows. z [ u 'v 'w ' ] ® [ uvtw ] a 2 - a 3 a 1 1 u = (2 u ' - v ') 3 1 v = (2 v ' - u ') 3 t = - ( u +v ) w = w' Fig. 3. 8(a), Callister & Rethwisch 8 e. [010] [110] Chapter 3 - 3

Conversion of Directional Indices [111] [112‾ 3] Chapter 3 - 4

Determine the Miller-Bravais directional indices for the direction shown in green. Directional Indices Chapter 3 - 5

3. 35 Determine indices for the directions shown in the following hexagonal unit cells: Chapter 3 - 6