 Solution

A Bravais lattice is the set of positions generated by integer combinations of independent primitive translation vectors, $\mathbf{R}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3$. Every lattice point has the same translational environment. A crystal may additionally have a basis of several atoms attached to each lattice point; those atoms are not automatically separate points of its Bravais lattice.

A unit cell is a region whose translated copies fill the crystal without gaps or overlaps. It can be chosen for convenient symmetry, and a conventional cell may contain several lattice points. A primitive unit cell is a smallest such repeating cell, containing exactly one lattice point when boundary sharing is counted. Its shape is not unique, but its volume $\Omega =|{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)|$ is fixed for a given lattice. For a monatomic Bravais crystal, each primitive cell contains one atom.