Spheroids surface modes

Figure 12.5 Effect of shape on the position of the lowcst-order surface mode of small spheroids. Arrows next to the various shapes show the direction of the electric field.

Theoretical considerations have shown that the spheroidal modes (even /) can be Raman active while the torsional modes (odd /) are Raman inactive [165]. A further point, which can help to recognize the low-frequency modes, is that spheroidal modes with / = 0 are observed only in the polarized geometry, whereas the quadrupolar modes with I = 2 are observed in polarized as well as in depolarized geometry. The spheroidal as well as the surface modes show resonance enhancement when the laser energy matches the region of the first absorption band [166]. 

The high fraction of surface atoms in thin ionic slabs [141,173-175] and quantum dots causes the appearance of new modes on the low energetic side of the LO mode. These modes are due to vibrations at the crystallite surface and are therefore called surface optical (SO) phonons. In the case of the slabs, the calculation of SO modes corresponds to the problem of the IF modes. For quantum dots, the situation is different Here, the surface mode cannot be represented by a plane wave but by modes with spheroidal or torsional character, like the LF modes (see Sec. IILB.3). In samples containing semiconductor quan-... 

The spherical modes (/ = 0) of the dot are excited with parallel polarization of the exciting laser light and have no surface contribution. The three degenerate modes for / = 1 are called Frohlich modes and correspond to an uniform polarization of the sphere. For I = 2, one gets the so-called spheroidal quadrupolar modes. The modes with / > 1 are usually called surface modes. Theoretical calculations of the one-phonon scattering in nanocrystallites predict that the surface mode is only allowed for / = 1, whereas the 1 = 2 modes are forbidden within the dipole approximation [178]. 

The most essential properly of acoustic vibrations in a nanoparticle is the existence of minimum size-quantized frequencies corresponding to acoustic resonances of the particle. In dielectric nanociystals, the Debye model is not valid for evaluation of the PDOS if the radius of the nanociystal is less than 10 nm. The vibrational modes of a finite sphere were analyzed previously by Lamb (1882) and Tamura (1995). A stress-free boundary condition at the surface and a finiteness condition on both elastic displacements and stresses at the center are assumed. These boundary conditions yield the spheroidal modes and torsional modes, determined by the following eigenvalue equations ... 

As defined, dendritic networks are considered to result from the one-, two-, and three-dimensional orientation of dendrimers thus ordering can be geometrically likened to rods surfaces or sheets and cubes, tetrahedrons, or spheres, respectively. Due to the broad scope and breadth of potential macromolecular architectures that can be obtained by application of different modes of connectivity, we will herein concentrate on networks constructed from the simplest dendritic structures, namely those that are pseudospheri-cal or globular. The principles that are presented here pertaining to network formation should be easily adaptable to non-spheroidal dendritic structures as well as macromolecular assemblies possessing only limited dendritic character. 

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