Dependence of the crystal lattice constant on isotopic composition: Theory and ab initio calculations for C, Si, and Ge

We consider the problem of the dependence of the lattice constant of a crystal upon its isotopic purity. After giving a brief account of the theoretical framework, we report on first-principles calculations for diamond, silicon, and germanium, performed by treating nuclear vibrations by density-functional perturbation theory. Our results are in good agreement with existing experimental data for diamond and germanium, and provide reliable predictions for silicon, which are relevant for metrological applications