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This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal (a) The basis vectors of the unit cell of a crystal, with the origin O at one corner, are denoted by ei, e2 , ез. The matrix G has elements Gijs where Gij ei-ej and Hj are the elements of the matrix HG1. Show that the vectors f , Hije, are the reciprocal vectors and that Hj f f (b) If the vectors u and v are given by 0 1-1 obtain expressions for lul, vl, and u v. T/3, write down G and hence obtain H. points p ei, qe2, es, and (ii) the angle between this normal and e,. (c) If the basis vectors are each of length a and the angle between each pair is (d) Calculate (i) the length of the normal from O onto the plane containing the

This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal (a) The basis vectors of the unit cell of a crystal, with the origin O at one corner, are denoted by ei, e2 , ез. The matrix G has elements Gijs where Gij ei-ej and Hj are the elements of the matrix HG1. Show that the vectors f , Hije, are the reciprocal vectors and that Hj f f (b) If the vectors u and v are given by 0 1-1 obtain expressions for lul, vl, and u v. T/3, write down G and hence obtain H. points p ei, qe2, es, and (ii) the angle between this normal and e,. (c) If the basis vectors are each of length a and the angle between each pair is (d) Calculate (i) the length of the normal from O onto the plane containing the

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal (a) The basis vectors of the unit cell of a crystal, with the origin O at one corner, are denoted by ei, e2 , ез. The matrix G has elements Gijs where Gij ei-ej and Hj are the elements of the matrix HG1. Show that the vectors f , Hije, are the reciprocal vectors and that Hj f f (b) If the vectors u and v are given by 0 1-1 obtain expressions for lul, vl, and u v. T/3, write down G and hence obtain H. points p ei, qe2, es, and (ii) the angle between this normal and e,. (c) If the basis vectors are each of length a and the angle between each pair is (d) Calculate (i) the length of the normal from O onto the plane containing the
Transcribed Image Text:This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal (a) The basis vectors of the unit cell of a crystal, with the origin O at one corner, are denoted by ei, e2 , ез. The matrix G has elements Gijs where Gij ei-ej and Hj are the elements of the matrix HG1. Show that the vectors f , Hije, are the reciprocal vectors and that Hj f f (b) If the vectors u and v are given by 0 1-1 obtain expressions for lul, vl, and u v. T/3, write down G and hence obtain H. points p ei, qe2, es, and (ii) the angle between this normal and e,. (c) If the basis vectors are each of length a and the angle between each pair is (d) Calculate (i) the length of the normal from O onto the plane containing the
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