# 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
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.

Step 3

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

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