Infrared activity in elemental crystals

In a previous paper, Zallen [Phys. Rev. 173, 824 (1968)] reported a group-theoretical analysis of the competition between unit-cell complexity and crystal symmetry in determining the presence or absence of infrared-active phonons in an elemental crystal. Here we correct an error in that paper’s treatment of certain hexagonal space groups. Our results modify the minimum-complexity condition for infrared activity: For 228 of the 230 space groups, a necessary and sufficient condition for the existence of symmetry-allowed infrared-active modes in an elemental crystal is the presence of three or more atoms in the primitive unit cell. The two exceptional space groups are P6/mmm (${\mathit{D}}_{6\mathit{h}}^1$) and P$6_3$/mmc (${\mathit{D}}_{6\mathit{h}}^4$); for each of these symmetries, there exists one structure with four atoms per cell and no infrared modes. The P$6_3$/mmc structure includes, as special cases, Lonsdaleite (or ‘‘wurtzite silicon’’) as well as a c-axis-aligned hcp arrangement of diatomic molecules which is relevant to models of solid molecular hydrogen at high pressure.