Analytical results for enhancement factor (EF) of surface enhanced Raman spectroscopy (SERS) for two metallic spheres and nano-shells

Interactions between symmetric two metallic spheres and an electromagnetic (EM) field polarized in the symmetric axis are described. Spherical symmetries of the present systems are exploited by the use of bi-spherical coordinates. Boundary conditions are applied, which assume a very large negative value for the dielectric constant of special metals. The EM field in small gaps (in hotspots) is amplified by many orders of magnitude relative to the external field. Analytical results for the maximal enhancement factor in surface enhanced Raman spectroscopy for two metallic spheres and two nano-shells are obtained as a function of various parameters. The calculations are based on potentials derived by using an electrostatic model, which is valid for very small particles.