We theoretically study the spectrum of radial vibrational modes in composite metal nanostructures such as bimetallic core-shell particles and metal nanoshells with dielectric core in an environment. We calculate frequencies and damping rates of fundamental (breathing) modes for these nanostructures along with those of two higher-order modes. For metal nanoshells, we find that the breathing mode frequency is always lower than the one for solid particles of the same size, while the damping is higher and increases with a reduction in the shell thickness. We identify two regimes that can be characterized as weakly damped and overdamped vibrations in the presence of external medium. For bimetallic particles, we find periodic dependence of frequency and damping rate on the shell thickness with period being determined by the mode number. For both types of nanostructures, the frequency of higher modes is nearly independent of the environment, while the damping rate shows a strong sensitivity to the outside medium
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