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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 expressThis 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 2**a***,
2**b***, 2**c***. 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

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Crystallography**