Past exam of the condensed matter physics course of the University of Oxford / 2018 / 1 / c / Solution

The beta phase has a primitive cubic lattice with an eight-atom atomic basis of a crystal, rather than a body-centred cubic lattice. A lattice translation must map the entire atomic arrangement onto itself. For example, the putative body-centering vector $(1/2,1/2,1/2)$ maps $(1/4,1/2,0)$ to $(3/4,0,1/2)$ modulo integer coordinates, where the given atomic basis of a crystal has no atom. None of the seven nonzero basis positions generates a translation that preserves all eight sites. ThereforeThe presence of atoms at both the origin and the body centre alone is insufficient to establish body-centering of the full crystal structure.