Particle Shape in IR Spectra

The main advantage in interpreting the optical response of small particles is that the retardation can be neglected, and, consequently, the electric field can be simply computed within the framework of electrostatics. As for ultrathin films with plane-parallel boundaries (Section 3.2.2), this approach uses the electrostatic polarizability of a body to describe the IR absorption by a macroscopicaUy small, but microscopically large, particle of the same shape. The major assumption is that the applied electric field is uniform across the particle. 

Sphere. A complete description of the coupling of an electromagnetic wave and the eigenmodes of an isolated sphere of any size, given by polariton theory based on Mie s formalism (Section 1.10), indicates that all modes of a sphere-shaped crystal are radiative [293, 298], These modes are called surface modes since their origin lies in the finite size of the sample [297]. For very small spheres, there is only the lowest order surface mode (the Frohlich mode), which is neither transverse nor longitudinal [293]. Its frequency (the Frohlich frequency) is given by 

As the radius of the sphere is increased, the n ro band of the bulk mode appears in the spectrum. Eor example, for ZnS particles this occurs at r 2 p,m. As the radius increases further, (i) both the surface- and bulk-mode absorption bands broaden and split due to the appearance of higher order surface modes, (ii) the band maxima shift toward lower frequencies, and (iii) the ratio of the intensities of the surface to bulk modes decreases [293]. As can be deduced from Eq. (3.35), an increase in the dielectric constant of the surrounding medium will also cause the surface modes to shift toward the red, as does an increase in the particle dimensions. The explanation is the same as for plane-parallel films (Section 3.3.1). The presence of a dispersion in particle size causes the two absorption bands to broaden, the fine sfiucture to disappear, and the a io band to shift to lower frequencies. An analogous effect can be observed if the damping constant y of the sphere material increases. 

Ellipsoids, Cylinders, and Disks. Eor ellipsoids, the solution for a sphere 

Other Shapes. For other shapes of particles and finite cylinders, mathematical difficulties arise due to the presence of the edges and comers [302]. The surface modes in small cubes were calculated by Fuchs [303] and Napper [304]. 

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