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331HW2Solution - Homework 2 Solution Winter 2011...

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Homework 2 Solution Winter 2011 Prof.Ringel 3.41 Determine the Miller indices for the planes shown in the following unit cell: Solution For plane A we will leave the origin at the unit cell as shown; this is a (403) plane, as summarized below. x y z Intercepts a 2 b 2c 3 Intercepts in terms of a , b , and c 1 2 2 3 Reciprocals of intercepts 2 0 3 2 Reduction 4 0 3 Enclosure (403) For plane B we will move the origin of the unit cell one unit cell distance to the right along the y axis, and one unit cell distance parallel to the x axis; thus, this is a ( 1 1 2) plane, as summarized below. x y z Intercepts a b c 2 Intercepts in terms of a , b , and c – 1 – 1 1 2 Reciprocals of intercepts – 1 – 1 2 Reduction (not necessary) Enclosure ( 1 1 2)
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Homework 2 Solution Winter 2011 Prof.Ringel 3.43 Determine the Miller indices for the planes shown in the following unit cell: Solution For plane A since the plane passes through the origin of the coordinate system as shown, we will move the origin of the coordinate system one unit cell distance to the right along the y axis; thus, this is a (3 2 4) plane, as summarized below. x y z Intercepts 2 a 3 b c 2 Intercepts in terms of a , b , and c 2 3 – 1 1 2 Reciprocals of intercepts 3 2 – 1 2 Reduction 3 – 2 4 Enclosure (3 2 4) For plane B we will leave the origin at the unit cell as shown; this is a (221) plane, as summarized below. x y z Intercepts a 2 b 2 c Intercepts in terms of a , b , and c 1 2 1 2 1 Reciprocals of intercepts 2 2 1 Reduction not necessary Enclosure (221)
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Homework 2 Solution Winter 2011 Prof.Ringel 3.52 (a) Derive linear density expressions for FCC [ 100 ] and [ 111 ] directions in terms of the atomic radius R. Solution (a) In the figure below is shown a [100] direction within an FCC unit cell. For this [100] direction there is one atom at each of the two unit cell corners, and, thus, there is the equivalent of 1 atom that is centered on the direction vector. The length of this direction vector is just the unit cell edge length, 2 R 2 (Equation 3.1).
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