EMCH 371_Th_Lecture 7

# EMCH 371_Th_Lecture 7 - Chapter 3 Crystalline Structure...

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Issues to address for the lecture Angle between directions? Linear and planar density? How to use X-ray diffraction to study crystalline structures? Chapter 3 Crystalline Structure - Continued

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For a cubic system, the angle between two directions [uvw] and [u'v'w'] is given by ( 29 ( 29 ( 29 2 2 2 2 2 2 ' ' ' ' ' ' cos w v u w v u ww vv uu + + + + + + = δ Angle Between Two Directions Example: Determine the angle between the directions and in a cubic unit cell. ] 1 00 [ ] 1 01 [ ( 29 2 2 2 2 2 2 1 1 0 ) 1 ( 0 0 ) 1 ( ) 1 ( 1 0 0 0 cos - + + - + + - × - + × + × = 2 2 2 1 = = 45 = 0,0,0 x z ] 1 01 [ ] 1 00 [ y
Number of atoms per unit length along a given direction Linear Density Example: Calculate the linear density along and directions in a BCC material with lattice parameter of a. ] 110 [ ] 111 [ x y z ] 110 [ ] 111 [ a a a 707 . 0 2 2 2 1 ] 110 [ = = = ρ a a a 155 . 1 3 3 2 3 2 ] 111 [ = = = repeat distance along [110] is a 2 repeat distance along [111] is 2 3 a Close-packed directions in FCC - face diagonals <110>; in BCC- body diagonals <111>; in HCP – basal axis < 0 2 11 Determine the repeat distance, r, between adjacent atoms; The linear density is the inverse, r -1 . If atoms are uniformly spaced along a given direction, then

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Planar Density The number of atoms per unit area in a given plane Close-packed planes in FCC – {111}, in HCP- basal plane (0001) in BCC, there is no close-packed planes, but {110} plane has the highest density compared with all the other planes in BCC. z
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EMCH 371_Th_Lecture 7 - Chapter 3 Crystalline Structure...

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