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This exercise considers a crystal whose unit cell has base vectors that are notnecessarily 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-ejand Hj are the elements of the matrix HG1. Show that the vectorsf , Hije, are the reciprocal vectors and that Hj f f(b) If the vectors u and v are given by01-1obtain 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

Question
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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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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Step 1

To discuss the properties of the reciprocal basis to the given basis of vectors

Step 2

We are given a system of basis vectors e(1),e(2),e(3) and the matrices G and H are defined as above.

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Step 3

By definition, the reciprocal system of vectors f1, f2,f3 are characterized b...

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