Recently

it has been shown that quantum sizs effects may be observed at low temperature in comercial optical glass filters.It is also reported that small crystallites of CdS are synthesized via colloidal chemical techniques

and their optical properties as a function cf crystallite size are studied at extreme dilution. This is a new type of quantum well heterstructures. Unlike layered heterstures

in which there is confinement in only one dimension

they are

in space

confined in all three dimensions.The optical properties of a micro-crystallites embeded in another dielectric strongly depend on the thermal vibration of the crystal. The presence of the crystal boundary leads specific surface vibration of the crystal. These vibration are localized near the boundary. Thermal excitation of surface vibrations is also accompanied by the occurrence of alternating dipole moments and

therefore

cf an electromagnetic field. The present paper deals with the theoretical investigation of the bulk and surface vibrational states for a microcrystallites embeded in another dielectric. It is an important problem for understanding the optical phonomena occurred in these new types of heterstructures.In the sixties

a phenomenological describtion of the vibrational states for any shaped dielectric imbeded in another dielectric had been given by Englman and Ruppin. However

they did not consider the influence of the electronic polarizability. Recently

Wendler and Pen Jinsheng deduced the equations of motion for the polarization eigenmode of the dielectrie bilayer system from the microscopic dynamical equations. Here we shall extend these results to discuss the vibration states of a dielectric sphere or other shaped crystallites embeded in another polar or non-polar dielectric.of course

the influence of the electronic polarizability is also taken into consideration.