Q.Calculate the number of atoms in a fcc unit cell.

There are exactly seven primitive crystal systems, distinguished by how their three edge lengths and three angles relate: cubic (a=b=c, all angles 90 deg), tetragonal (a=b not-eq c, all 90 deg), orthorhombic (a not-eq b not-eq c, all 90 deg), hexagonal (a=b not-eq c, alpha=beta=90 deg, gamma=120 deg), rhombohedral (a=b=c, all angles equal but not 90 deg), monoclinic (a not-eq b not-eq c, alpha=gamma=90 deg, beta not-eq 90 deg), and triclinic (a not-eq b not-eq c, no angle equal to 90 deg or to another). Once possible centrings (body-centred, face-centred, base-centred) of these seven are also counted, Bravais showed only 14 distinct lattices are geometrically possible -- the 14 Bravais lattices (Table 6.2): 3 cubic + 2 tetragonal + 1 hexagonal + 4 orthorhombic + 2 monoclinic + 1 trigonal + 1 triclinic = 14.

For the three cubic unit cells, the number of atoms belonging to one cell is found by weighting each shared lattice position: a corner atom (shared by 8 cells) contributes 1/8; a face-centred atom (shared by 2 cells) contributes 1/2; a body-centred atom (unshared) contributes a full 1.

• Simple cubic (SC): atoms at 8 corners only. Atoms/cell = 8x(1/8) = 1. Coordination number 6 (touch along edges only, a = 2r). …