  Vacancies are prevalent point defects in crystals, but their thermal
responses are elusive. Herein, we formulate a simple theoretical model to shed
light on the vacancy evolution during heating. Vibrational excitations are
thoroughly investigated via moment recurrence techniques in quantum statistical
mechanics. On that basis, we carry out numerical analyses for Ag, Cu, and Ni
with the Sutton-Chen many-body potential. Our results reveal that the
well-known Arrhenius law is insufficient to describe the proliferation of
vacancies. Specifically, anharmonic effects lead to a strong nonlinearity in
the Gibbs energy of vacancy formation. Our physical picture is well supported
by previous simulations and experiments.
