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Lecture%2004 - EEE 352: Lecture 04 Crystal Directions; Wave...

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1 EEE 352: Lecture 04 Crystal Directions; Wave Propagation (110) (001) Electron beam generated pattern in TEM Crystal Directions We have referred to various directions in the crystal as (100), (110), and (111). What do these mean? How are these directions determined? Consider a cube- and a plane- 1/2 1 2/3 The plane has intercepts: x = 0.5 a , y = 0.667 a , z = a . Crystal Directions We want the NORMAL to the surface. So we take these intercepts (in units of a ), and invert them: Then we take the lowest common set of integers: These are the MILLER INDICES of the plane. The NORMAL to the plane is the (4,3,2) direction, which is normally written just (432). (A negative number is indicated by a bar over the top of the number.) 1 , 2 3 , 2 1 , 3 2 , 2 1 () 2 , 3 , 4 1 , 2 3 , 2 1 , 3 2 , 2 1 Crystal Directions 1 1 1 1 (111) 1 (110)
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2 Crystal Directions 1/2 1 1 (100) 1/2 (220) Crystal Directions (110) (001) Electron beam generated pattern in TEM So far, we have discussed the concept of crystal directions: 1 1 1 1 (111) 1 (110) We want to say a few more things about this. x y z Consider this plane. The intercepts are 0,0, , which leads to 1,1,0 for Miller indices. What are the normals to the plane?
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3 x y z -1 -1 The intercepts are 1, -1, , which leads to The normal direction is ( ) 0 1 1 [ ] 0 1 1 ˆ ˆ y x The intercepts are -1, 1, , which leads to The normal direction is ( ) 10 1 [ ] 10 1 ˆ ˆ + y x x y z -1 -1 We can easily shift the planes by one lattice vector in x or y x y z -1 -1 [ ] 0 1 1 [] 10 1 These are two different normals to the same plane. Directions have square brackets […] Planes have parentheses (…) x y z -1 -1 Now consider the following plane: [ ] 0 1 1 [ ] 110 This direction lies in the original plane.
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4 x y z -1 -1 Now consider the following plane: [] 001 This direction lies in the original plane. x y z -1 -1 [ ] 10 1 points into the page [ ] 110 [ ] 001 [ ] 111 Crystal Directions We want the NORMAL to the surface. So we take these intercepts (in units of a ), and invert them: Then we take the lowest common set of integers: These are the MILLER INDICES of the plane. The NORMAL to the plane is the [4,3,2] direction, which is normally written just [432]. (A negative number is indicated by a bar over the top of the number.) 1 , 2 3 , 2 1 , 3 2 , 2 1 2 , 3 , 4 1 , 2 3 , 2 1 , 3 2 , 2 1 This inversion creates units of 1/cm These “numbers” define a new VECTOR in this “reciprocal space” Crystal Directions 1/2 1 1 [100] 1/2 [220] These vectors are normal to the planes, but are defined in this new space.
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This note was uploaded on 09/19/2009 for the course EEE 352 taught by Professor Ferry during the Spring '08 term at ASU.

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Lecture%2004 - EEE 352: Lecture 04 Crystal Directions; Wave...

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