Phonons and low thermal expansion in sodalite zinc borate Zn4B6O12X(X=O,

We examine isotropic near-zero thermal expansion in the cubic materials ${\mathrm{Zn}}_{4}{\mathrm{B}}_{6}{\mathrm{O}}_{12}X(X=\mathrm{O},\mathrm{S},\mathrm{Se})$ using density functional theory calculations of the lattice dynamics within the Gr\"uneisen approximation. The calculated thermal expansion is in excellent agreement with experiment. The near-zero thermal expansion is shown to arise from a balance between two specific groups of vibrations that favor positive and negative thermal expansion, respectively. This gives a clear general indicator of how crystal structures can be engineered to give near-zero thermal expansion.