Now questions of studying processes of interaction of electromagnetic radiation with various systems of small particles (nanoparticles) near to a surface of solid state in view of their physical and chemical structure, the sizes and the form, and also their spatial arrangement have got the big urgency in connection with designing and creation of selective coatings and films with beforehand set electrodynamic properties for their functional use in physics, chemistry and biology. In this connection we had been developed the theory of the electrostatic response of small system of spherical particles (SP) above a surface in an external electric field in view of them multipole interactions among themselves and with a substrate.
The basic theory in dipole approximation, was received a number of new analytical results and laws of influence of a substrate on polarization and optical properties SP (nanoparticles) the spherical form were investigated. The general expressions for tensor à polarizability separate SP and two SP above a substrate with Lorenz dielectric functions SP and substrates in approximation were received. Expressions for resonant frequencies of superficial modes and forces corresponding to them "oscillators" were found. It was shown, that dipole interaction between SP and a substrate results in splitting and displacement of frequency of superficial modes separate SP, and generally we have four new modes which differ from each other both frequency, and optical activity (forces "oscillators"). The case of two SP (nanoparticles), located on distance d (centre to centre spheres) from each other in an external electric field was considered and analysed also. It was shown, as in this case there are four superficial modes which frequencies are calculated in work generally different sizes SP and their dielectric functions. The case of two identical particles of different radius was in details investigated and was shown, that here too arises four superficial fashions, resonance cooperating with external fields.
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