How many lattice points are there in one unit cell of each of the following lattice? 1. Face-centred cubic  2. End-centred monoclinic  3. Body-centred

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How many lattice points are there in one unit cell of each of the following lattice?

1. Face-centred cubic

2. End-centred monoclinic

3. Body-centred

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1. A face-centred cubic unit cell has lattice points at the corners of the cube and at face centres. There are eight comers and six faces of the cube. Each atom at corner makes a contribution of $\frac{1}{8}$ while each atom at face centre makes a contribution of $\frac{1}{2}$ to the unit cell. Therefore, the number of atoms per unit cell

= 8 × $\frac{1}{8}$ + 6 × $\frac{1}{2}$ = 1 + 3 = 4

2. A end-centred monoclinic unit cell has lattice points at the face centres of only one set (two) of faces, in addition to the lattice points at the comers of the unit cell. Therefore, the number of atoms per unit cell

= 8 × $\frac{1}{8}$ + 2 × $\frac{1}{2}$ = 1 + 1 = 2

3. A body centred cubic unit cell has lattice points at the comers of the cube and at the body centre. Therefore, the number of atoms per unit cell

= 8 × $\frac{1}{8}$ + 1 × 1 = 1 + 1 = 2