  The interaction between two metallic spheres with radius R with external
electromagnetic (EM) field polarized in the symmetric z direction is described.
Solutions of Laplace equation with bi-spherical coordinates are developed. Hot
spots are obtained under the condition that the shortest distance between the
two spheres surfaces is very small relative to their radius. Boundary
conditions are applied which assume very large real negative value for the
dielectric constant of the metallic spheres which is valid for atoms like that
of AG or Au at certain frequencies. Under these conditions the EM field is
amplified by many orders of magnitudes relative to the incident EM field.
Analytical results for maximal Raman enhancement factor (EF ) are obtained as
function of various parameters. The present study can be applied to
surface-enhanced Raman spectroscopy (SERS) and two-photon induced illumination
(TPI-PL) in which the amplification is proportional to the fourth power of the
incident EM field.
