Analytical Methods for Materials Chapters 2 and 6 in Waseda Lesson 8 Lattice Planes and Directions Suggested Reading 192
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 1 1 2 1 3 a b c 0 0 0 1 2 1 1 3 6 [ 3 6 2] Point P Origin 193
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
<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
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
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
FAMILY OF PLANES ALL MEMBERS HAVE SAME ARRANGEMENT OF LATTICE POINTS {hkl} k We use Miller indices to denote planes 198
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
MILLER INDICES FOR A SINGLE PLANE z x y z Intercept 1 1 Reciprocal 1/1 1/1 1/ Clear 1 1 0 INDICES 1 1 0 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
MILLER INDICES FOR A SINGLE PLANE cont d x y z z Intercept 1 1 1 Reciprocal 1/1 1/1 1/1 Clear 1 1 1 INDICES 1 1 1 y x y z Intercept -1-1 -1 x Reciprocal -1/1-1/1-1/1 Clear -1-1 -1 ( 1 INDICES 1 1 1 ( 111 ) 11 ) Slide 201 201
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/ 2 2 0 2 2 0 (220) x y Planes and their multiples are not identical ( 220) (110) 202
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
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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) 110 001 1 0 1 0 0 1 0 001 lies on 1 10 220 001 2 0 2 0 0 1 0 001 lies on 220 y 205
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 1 1 0 1 1 2 2 0 2 2 u 1 0 v 0 2 w 1 2 v 1 0 u 0 2 w 1 2 [0 04] [ 01] 0 206
Indexing in Hexagonal Systems The regular 3 index system is not suitable. c a 3 120 120 120 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
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
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
c a 3 a 3 100 2110 110 1120 a 2 a 1 010 12 10 Some typical directions in an HCP unit cell using three- and four-axis systems. 210
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
Interplanar Spacing cont d 1 h k l CUBIC: 2 2 d a HEXAGONAL: TETRAGONAL: RHOMBOHEDRAL: ORTHORHOMBIC: 2 2 2 1 4h hk k l d 3 a c 2 2 2 2 2 2 1 h k l d a c 1 d 2 2 2 2 2 2 2 2 2 2 h hk k sin 2hk kl hlcos cos a 1 3cos 2cos 2 2 2 3 1 h k l d a b c 2 2 2 2 2 2 2 1 1 2 2 2 2 h k sin l 2hlcos a b c ac MONOCLINIC: 2 d 2 2 2 2 sin 1 1 2 2 2 TRICLINIC*: S 2 2 11h S22k S3l 2S12hk 2S23kl2S13hl d V S b c sin ; S a c sin ; S a b sin 2 2 2 2 2 2 2 2 2 11 22 33 cos cos cos ; cos cos cos ; cos cos cos S abc S a bc S ab c 2 2 2 12 23 13 2 2 2 V abc 1 cos cos cos 2 cos cos cos 212