# Chapters 2 and 6 in Waseda. Lesson 8 Lattice Planes and Directions. Suggested Reading

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1 Analytical Methods for Materials Chapters 2 and 6 in Waseda Lesson 8 Lattice Planes and Directions Suggested Reading 192

2 Directions and Miller Indices Draw vector and define the tail as the origin. z Determine the length of the vector projection in unit cell dimensions a, b, and c. Remove fractions by multiplying by the smallest possible factor. [201] O [111] P a c y Enclose in square brackets Negative indices are written with a bar over the number.. b [110] x a b c [ 3 6 2] Point P Origin 193

3 Families of Directions (i.e., directions of a form) In cubic systems, directions that have the same indices are equivalent regardless of their order or sign. z y [010] [001] [100] x [010] [100] [001] We enclose indices in carats rather than brackets to indicate a family of directions The family of <100> directions is: [10 0], [100] [010], [0 10] [001], [001] All of these vectors have the same size and # lattice points/length 194

4 <100> CUBIC <aaa> [100] [010] [001] [100] [010] [001] TETRAGONAL <aac> [100] [010] [100] [010] In non-cubic systems, directions with [100] [100] the same indices may not be equivalent. ORTHORHOMBIC <abc> 195

5 Directions in Crystals Directions and their multiples are identical [110] Ex.: z y [030] [020] [220] [220] 2 [110] x [010] Vectors and multiples of vectors have the same # lattice points/length 196

6 Miller Indices for Planes Specific crystallographic plane: (hkl) Family of crystallographic planes: {hkl} Ex.: (hkl), (lkh), (hlk) etc. In cubic systems, planes having the same indices are equivalent regardless of order or sign. In hexagonal crystals, we use a four index system (hkil) k i l). We can convert from three to four indices h+k = -i 197

7 FAMILY OF PLANES ALL MEMBERS HAVE SAME ARRANGEMENT OF LATTICE POINTS {hkl} k We use Miller indices to denote planes 198

8 PROCEDURES FOR INDICES OF PLANES (Miller indices) 1. Identify the coordinate intercepts of the plane (i.e., the coordinates at which the plane intersects the x, y, and z axes). If plane is parallel to an axis, the intercept is taken as infinity (). If the plane passes through the origin, consider an equivalent plane in an adjacent unit cell or select a different origin for the same plane. 2. Take reciprocals of the intercepts. 3. Clear fractions to the lowest integers. 4. Cite specific planes in parentheses,(hkl), placing bars over negative indices. 199

9 MILLER INDICES FOR A SINGLE PLANE z x y z Intercept 1 1 Reciprocal 1/1 1/1 1/ Clear INDICES y (110) x Slide 200 The {110} family of planes (110), (011), (101), (110), (011), (101) (110), (1 10), (101), (10 1), (01 1), (0 11) 200

10 MILLER INDICES FOR A SINGLE PLANE cont d x y z z Intercept Reciprocal 1/1 1/1 1/1 Clear INDICES y x y z Intercept x Reciprocal -1/1-1/1-1/1 Clear ( 1 INDICES ( 111 ) 11 ) Slide

11 MILLER INDICES FOR A SINGLE PLANE cont d z x y z Intercept Reciprocal Clear INDICES 1/2 1/2 2/1 2/1 1/ (220) x y Planes and their multiples are not identical ( 220) (110) 202

12 Planes in Unit Cells Some important aspects of Miller indices for planes: 1. Planes and their negatives are identical. This was NOT the case for directions. 2. Planes and their multiples are NOT identical. This is opposite to the case for directions. 3. In cubic systems,, a direction that has the same indices as a plane is to that plane. This is not always true for non-cubic systems. 203

13 204

14 Planes of a Zone A zone is a direction [uvw] z (1 10) ZONE AXIS [uvw] = [001] (220) Planes belonging to a particular zone are parallel to one direction known as the zone axis. hkl uvw 0 (010) lies on lies on 220 y 205

15 How to Determine the Zone Axis Take the cross product of the intersecting planes. z (1 10) ZONE AXIS [uvw] = [001] (220) (010) (h 1 k 1 l 1 ) (h 2 k 2 l 2 ) = [uvw] y u v w u v u 1 0 v 0 2 w 1 2 v 1 0 u 0 2 w 1 2 [0 04] [ 01] 0 206

16 Indexing in Hexagonal Systems The regular 3 index system is not suitable. c a Planes with the same a 1 indices do not necessarily look like. a 3 a 2 [1 10] [001] (0001) c 4 index system introduced. Miller-Bravais indices (1100) a 1 (10 11) [100] (1210) a 2 [010] 207

17 Indexing in Hexagonal Systems Planes: (hkl) becomes (hkil) i = -(h+k) Directions: [UVW] becomes [uvtw] U = u-t ; u = (2U V)/3 V = v-t ; v = (2V U)/3 W=w ; t=-(u+v) 208

18 PLANES Miller Indices Miller-Bravais Indices a 3 (hkl) a 3 (hkil) (100) (1010) (010) (110) (0 110) (1100) a 2 a 2 (110) (010) (1 100) (0110) a 1 (100) a 1 (10 10) DIRECTIONS (UVW) (uvtw) a 3 a 3 [120] [110] [1100] [0 110] a 2 a 2 [110] [100] [1120] [2110] a 1 [210] a 1 [1010] 209

19 c a 3 a a 2 a Some typical directions in an HCP unit cell using three- and four-axis systems. 210

20 Inter-planar Spacings z y (100) (110) x Assuming no intercept on z-axis (210) d 210 The inter-planar spacing in a particular direction is the distance between equivalent planes of atoms. Each material has a set of characteristic inter-planar spacings. They are directly related to crystal size (i.e. lattice parameters) and atom location. a 211

21 Interplanar Spacing cont d 1 h k l CUBIC: 2 2 d a HEXAGONAL: TETRAGONAL: RHOMBOHEDRAL: ORTHORHOMBIC: h hk k l d 3 a c h k l d a c 1 d h hk k sin 2hk kl hlcos cos a 1 3cos 2cos h k l d a b c h k sin l 2hlcos a b c ac MONOCLINIC: 2 d sin TRICLINIC*: S h S22k S3l 2S12hk 2S23kl2S13hl d V S b c sin ; S a c sin ; S a b sin cos cos cos ; cos cos cos ; cos cos cos S abc S a bc S ab c V abc 1 cos cos cos 2 cos cos cos 212

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