# Chapter 3: The Structure of Crystalline Solids

Crystallographic Planes For cubic crystals, planes and directions having the same Chapter 3 indices are perpendicular to one another. Crystallographic Planes Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices. Algorithm 1a. If the plane passes through the selected origin, choose another plane or move the origin. 1b. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts *3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl) *3 On occasion, index reduction is not carried out, as you will see later. Chapter 3 - 2 Crystallographic Planes z example 1. Intercepts 2. Reciprocals

3. Reduction a 1 1/1 1 1 4. Miller Indices (110) example 1. Intercepts 2. Reciprocals 3. Reduction 4. Miller Indices a

1/2 1/ 2 0 1 0 b 1 1/1 1 1 c 1/ 0 0 c b a x y

z b c 1/ 1/ 0 0 c a (100) b x Chapter 3 - 3 y Crystallographic Planes z example 1. Intercepts 2. Reciprocals

3. Reduction 4. Miller Indices a 1/2 1/ 2 6 b 1 1/1 1 3 (634) c c 3/4 1/ 4/3

4 a x b y Family of Planes {hkl} Ex: {100} = (100), (010), (001), (100), (010), (001) Chapter 3 - 4 Determine the Miller indices for the plane shown. Chapter 3 - 5 Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used z example 1. Intercepts 2. Reciprocals 3. Reduction

a1 1 1 1 1 a2 1/ 0 0 a3 -1 -1 -1 -1 c 1 1 1 1 a2 a3

4. Miller-Bravais Indices (1011) a1 Adapted from Fig. 3.8(b), Callister & Rethwisch 8e. Chapter 3 - 6 3.50 Determine the indices for the planes shown in the hexagonal unit cells below: Chapter 3 - 7 3.50 Determine the indices for the planes shown in the hexagonal unit cells below: Chapter 3 - 8

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