Localized modes and the vibrations of nanostructures

Due to rapid developments in nanotechnology and the importance of nanomaterial characterization, the study of the acoustic response of small structures is of significant current interest. Frequency domain techniques such as Raman scattering and direct time-resolved pump-probe experiments are nondestructive and complementary tools to detect and identify these acoustic vibrations. A possible application of these experimental works is the determination of the geometry of a nanostructure.;For this to be practical, one needs to find a way to go from a measurement of the set of frequencies of the normal modes to a determination of the values of whatever set of parameters are used to describe the shape. M. Kac drew attention to this general class of problem in a famous paper in 1966 entitled "Can one hear the shape of a drum?" However, in fact, the problem is more difficult than it appears because it turns out that the frequencies vary with the dimensions in a surprisingly complicated way as we will discuss in this thesis.;"Level repulsion" related mode localization is our primary interest. In 1956 Shaw observed a prominent vibration resonance localized at the edge of a thick barium titanate disk. It is known that for certain special values of Poisson's ratio these modes are perfectly localized, are uncoupled to bulk modes, and thus do not lose energy by acoustic radiation. In this thesis we consider the conditions for mode localization in different structures, and show that regardless of the value of Poisson's ratio it is often possible to design a structure with an end shape such that a perfectly localized mode appears. This localization has interesting effects on the way that the vibrational patterns and frequencies of the normal modes of a structure are changed when the dimensions of the structure are altered.