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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 a2 - a3 a1 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] a 2 Adapted from Fig. 3.8(a), Callister & Rethwisch 8e. ex: ½, ½, -1, 0 -a3 a2 2 => [ 1120 ] a3 dashed red lines indicate projections onto a1 and a2 axes a1 2 a1 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 ] a2 - a3 a1 1 u = (2 u ' - v ') 3 1 v = (2 v ' - u ') 3 t = - (u +v ) w = w' Fig. 3.8(a), Callister & Rethwisch 8e. [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