The model described above does not predict damping of the acoustic
modes. Yet damping is experimentally observed, and it would therefore be
desirable to be able to calculate damping times of nanoparticles embedded in
a homogeneous medium. The parameter that determines the acoustic inter-
action of a sphere and a matrix is the acoustic impedance of both materials
Z = v L, where is the material density and v L is the longitudinal sound
velocity of the material. The acoustic impedance governs the reflection of
sound waves at a material interface. If the impedance of a sphere equals
the impedance of a matrix in which it is embedded, sound waves are not re-
flected but propagate out of the sphere immediately, and the mode will be
highly damped. If there is a large impedance mismatch between the sphere
and the matrix, the sound waves are strongly reflected and will remain con-
fined in the sphere much longer. In this case, the acoustic mode will have a
very low damping.