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The Casimir effect, a two-body interaction via vacuum fluctuations, is afundamental property of quantum systems. In solid state physics it emerges as along-range interaction between two impurity atoms via virtual phonons. In theclassical limit for the impurity atoms in $D$ dimensions the interaction isknown to follow the universal power-law $U(r)\sim r^{-D}$. However, for finitemasses of the impurity atoms on a lattice, it was predicted to be $U(r)\simr^{-2D-1}$ at large distances. We examine how one power-law can change intoanother with increase of the impurity mass and in presence of an externalpotential. We provide the exact solution for the system in one-dimension. Atlarge distances indeed $U(r)\sim r^{-3}$ for finite impurity masses, while forthe infinite impurity masses or in an external potential it crosses over to$U(r)\sim r^{-1}$ . At short distances the Casimir interaction is not universaland depends on the impurity mass and the external potential.

Phonon-mediated Casimir interaction between finite mass impurities