Isotope dependence of the lattice parameter of germanium from path-integral Monte Carlo simulations

The dependence of the lattice parameter upon the isotope mass for five isotopically pure Ge crystals was studied by quantum path-integral Monte Carlo simulations. The interatomic interactions in the solid were described by an empirical potential of the Stillinger-Weber type. At 50 K the isotopic effect leads to an increase of $2.3\times {10}^{-4}$ Å in the lattice parameter of ${}^{70}$Ge with respect to ${}^{76}$Ge. Comparison of the simulation results with available experimental data for ${}^{74}$Ge shows that the employed model provides a realistic description of this anharmonic effect. The path-integral results were compared to those derived from a quasiharmonic approximation of the crystal. Within this approximation, the calculated fractional change of the lattice parameter of ${}^{74}$Ge with respect to a crystal whose atoms have the average mass of natural Ge amounts to $\Delta a/a=-9.2\times {10}^{-6}$ at $T=0\mathrm{}$ K. Some limitations of the quasiharmonic approximation are shown at temperatures above 200 K.