  We study phonon-mediated damping of mechanical vibrations in a finite
quantum-mechanical atomic-chain model. Our study is motivated by the quest to
understand the quality factors (Q) of nanomechanical resonators and
nanoelectromechanical systems (NEMS), as well as actual experiments with
suspended atomic chains and molecular junctions. We consider a finite atomic
chain which is coupled to a zero-temperature outer environment, modeled as two
additional semi-infinite chains, thus inducing "clamping-losses". Weak coupling
to the outer environment ensures that the clamping losses are small, and that
the initially discrete nature of the phonon spectrum is approximately
maintained. We then consider a phonon damping process known as "Landau-Rumer
damping", where phonons in the excited mode of vibration decay into other modes
through anharmonic phonon-phonon interaction. The approximately discrete nature
of the phonon spectrum leads to sharp nonmonotonic changes in Q as parameters
are varied, and to the appearance of resonances in the damping. The latter
correspond to the existence of decay processes where the participating phonons
approximately conserve energy. We explore means to control the damping by
changing either the number of atoms in the chains or the ratio between the
longitudinal and transverse speeds of sound, thereby suggesting future
experiments to observe this resonance-like behavior.
