## 4.5 Reciprocity of F and I lattices

Let us consider a face-centered lattice. It is well known that the basic vectors a, b, c, of the elementary cell are given in terms of the vectors a, b, c of the face centered cell by (Fig. 5):

 (4.4)

In a similar way, the basic vectors a,b, c of the elementary cell of a body centered lattice are given in terms of the basic vectors of the multiple cell by (Fig. 6):

 (4.5)

Let us now look for the reciprocal lattice of the face-centered lattice. Its unit cell vectors are given by, using (2.1) and (4.4):

 (4.6)

Noting that the face-centered cell is of the fourth order, we find:

We may thus express a* in terms of the basic vectors of the reciprocal lattice of the lattice of vectors a, b, c:

This may also be written:

This relation shows that the reciprocal lattice of a face-centered lattice is a body centered lattice whose multiple cell is defined by 2a*, 2b*, 2c*. If we index the reciprocal lattice defined by a*, b*, c*, that is the reciprocal lattice of the multiple lattice defined by a, b, c, we find that only the nodes such that

belong to the reciprocal lattice of the face-centered lattice. This shows that the only Bragg reflexions on a face-centered lattice have indices which are all of the same parity.

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