Explanation Given: For the Figure (I) The tail of the x -axis direction a ″ 1 vector , a ″ 1 = 0 a . The tail of the y -axis direction a ″ 2 vector, a ″ 2 = 0 a . The tail of the z -axis direction z ″ vector, z ″ = 0 c . The vector head coordinate of the x -axis direction a ′ 1 , a ′ 1 = − a . The vector head coordinate of the y -axis direction a ′ 2 , a ′ 2 = − a . The vector head coordinate of the z -axis direction z ′ , z ′ = c . For the Figure (II) The tail of the x -axis direction a ″ 1 vector, a ″ 1 = 0 a . The tail of the y -axis direction a ″ 2 vector, a ″ 2 = 0 a . The tail of the z -axis direction z ″ vector, z ″ = 0 c . The vector head coordinate of the x -axis direction a ′ 1 , a ′ 1 = − a . The vector head coordinate of the y -axis direction a ′ 2 , a ′ 2 = 0 a . The vector head coordinate of the z -axis direction z ′ , z ′ = 0 c . For the Figure (III) The tail of the x -axis direction a ″ 1 vector, a ″ 1 = 0 a . The tail of the y -axis direction a ″ 2 vector, a ″ 2 = 0 a . The tail of the z -axis direction z ″ vector, z ″ = 0 c . The vector head coordinate of the x -axis direction a ′ 1 , a ′ 1 = a . The vector head coordinate of the y -axis direction a ′ 2 , a ′ 2 = a / 2 . The vector head coordinate of the z -axis direction z ′ , z ′ = 0 c . For the Figure (IV) The tail of the x -axis direction a ″ 1 vector, a ″ 1 = 0 a . The tail of the y -axis direction a ″ 2 vector, a ″ 2 = 0 a . The tail of the z -axis direction z ″ vector, z ″ = 0 c . The vector head coordinate of the x -axis direction a ′ 1 , a ′ 1 = a . The vector head coordinate of the y -axis direction a ′ 2 , a ′ 2 = 0 a . The vector head coordinate of the z -axis direction z ′ , z ′ = c / 2 . Explanation: Write the expression of direction index referenced to x -axis. U = n ( a ′ 1 − a ″ 1 a ) (I) Here, the factor that may be required to reduce u , v , and w to integers is n Write the expression of direction index referenced to y -axis. V = n ( a ′ 2 − a ″ 2 a ) (II) Write the expression of direction index referenced to z -axis. W = n ( z ′ − z ″ c ) (III) Write the expression for conversion from three-index system to the four-index system [ U V W ] = [ u v t w ] Follow the formula given below: u = 1 3 ( 2 U − V ) (IV) v = 1 3 ( 2 V − U ) (V) t = − ( u + v ) (VI) w = W (VII) Conclusion: For the Figure (I) Draw the diagram of the indices for the direction in hexagonal unit cell. Assume the factor that may be required to reduce u , v , and w to integers is n = 1 . Substitute 1 for n , − a for a ′ 1 and 0 a for a ″ 1 in Equation (I) U = ( 1 ) ( ( − a ) − ( 0 a ) a ) = − 1 Substitute 1 for n , − a for a ′ 2 and 0 a for a ″ 2 in Equation (II) V = ( 1 ) ( ( − a ) − ( 0 a ) a ) = − 1 Substitute 1 for n , c for z ′ and 0 c for z ″ in Equation (III) W = ( 1 ) ( ( c ) − ( 0 c ) c ) = 1 Substitute − 1 for U and − 1 for V in Equation (IV) u = 1 3 ( 2 ( − 1 ) − ( − 1 ) ) = − 1 3 Substitute − 1 for U and − 1 for V in Equation (V) v = 1 3 ( 2 ( − 1 ) − ( − 1 ) ) = − 1 3 Substitute − 1 3 for u and − 1 3 for v in Equation (VI) t = − ( − 1 3 − 1 3 ) = 2 3 Substitute 1 for W in Equation (VII) w = 1 Here, for reduce the values to the lowest set of integers. Multiply each by the factor 3. The value of the u , v , t and w are − 1 , − 1 , 2 , and 3 . Thus, the vector represents the ( 1 ¯ 1 ¯ 23 ) _ direction. For the Figure (II) Draw the diagram of the indices for the direction in hexagonal unit cell. Assume the factor that may be required to reduce u , v , and w to integers is n = 1

9th Edition

ISBN: 9781118324578

Author: Jr. William D. Callister

Publisher: WILEY

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Question

Chapter 3.17, Problem 41QP

To determine

The indices for directions shown in the following hexagonal unit cell.

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Chapter 3.17, Problem 40QP

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Materials Science and Engineering

Ch. 3.17 - Prob. 1QP Ch. 3.17 - Prob. 2QP Ch. 3.17 - Prob. 3QP Ch. 3.17 - Prob. 4QP Ch. 3.17 - Prob. 5QP Ch. 3.17 - Prob. 6QP Ch. 3.17 - Prob. 7QP Ch. 3.17 - Prob. 8QP Ch. 3.17 - Prob. 9QP Ch. 3.17 - Prob. 10QP

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