Solution (source code)

= Solution

A <Bravais lattice> is an infinite periodic set of equivalent points $\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3$, with integer $n_i$ and independent primitive translations $\mathbf a_i$. It describes translation symmetry, without assigning a chemical species to a point. An <atomic basis of a crystal> is the motif of atoms, with their species and relative positions, repeated identically at every <lattice point>. The resulting <crystal structure> consists of the lattice decorated by that basis; the basis can contain several atoms even when the lattice is primitive.

A <primitive unit cell> tiles space under lattice translations and contains exactly one <lattice point>, counting shared boundary points fractionally. Its <volume> is $|\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)|$. A <conventional unit cell> is chosen to display useful crystal symmetry and may contain several <lattice points>. Thus a centered conventional cell is not primitive, but the crystal still has a primitive cell. Neither cell size nor the number of dots drawn on its boundary should be confused with the number of atoms in the crystal basis.