Nanoparticle in a quantum crystal with a narrow vacancy band

Vacancy-assisted diffusion of a neutral probe nanoparticle with a radius R_p of a few lattice constants in a quantum crystal with a narrow vacancy band is considered. The diffusion coefficient of the probe D_p(T) in such a crystal should fall exponentially near T_melt, and it can go through a maximum at temperatures T_tr, where the transition from thermally activated hopping of localized vacancies to a proper band motion of delocalized vacancions takes place, under the condition that the mean free path of the vacancions l_v(T) at T_tr is less than R_p and increases with lowering the temperature quicker than the inverse value of the relative concentration of vacancies X_v(T). Below T_tr, where l_v is much longer than the probe diameter, the value of D_p should fall proportionally to X_v(T).