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Question

How many lattice points are there in one unit cell of each of the following lattice?
(i) Face-centred cubic
(ii) Face-centred tetragonal
(iii) Body-centred


Solution

(i) Face-centred cubic
$$8$$ lattice points at corners and $$6$$ lattice points at face centres are present in face-centred cubic.
A corner lattice point contributes one eight to the unit cell and a lattice point at face centre contributes one half to the unit cell.
Total number of lattice points per unit cell: $$\displaystyle 8 \times \frac {1}{8} + 6  \times \frac {1}{2} = 1 + 3 = 4$$
(ii) Face-centred tetragonal
$$8$$ lattice points at corners and $$6$$ lattice points at face centres.
A corner lattice point contributes one eight to the unit cell and a lattice point at face centre contributes one half to the unit cell.
Total number of lattice points per unit cell: $$\displaystyle 8 \times \frac {1}{8} + 6  \times \frac {1}{2} = 1 + 3 = 4$$
(iii) Body-centred
$$8$$ lattice points at corners and $$1$$ lattice point at body centre.
A corner lattice point contributes one eight to the unit cell and a lattice point at body centre contributes one  to the unit cell.
Total number of lattice points per unit cell: $$\displaystyle 8 \times \frac {1}{8} + 1  \times 1 = 1 + 1 = 2$$

Chemistry
NCERT
Standard XII

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