Surface-enhanced Raman scattering in a two-oscillator electromagnetic model

A two-oscillator model is considered to investigate the effect of a metal substrate of dielectric function $\epsilon \left(\omega \right)$ on the Raman scattering from a molecule absorbed on the metal surface. In the presence of the metal and an external electric field, the linear motion of the electronic and ionic oscillators in the molecule get coupled, in general. For obtaining Raman scattering at the Stokes frequency, a phenomenological nonlinear force term, which is bilinear in the oscillator amplitudes, is introduced in the equation of motion. The whole problem is considerably simplified when we use the fact that the ionic mass is much larger than the electronic mass and the ionic vibration frequency is much smaller than the electronic and optical frequencies. It is shown that because of different renormalization factors the frequency dependence of the enhancement factor $F$, taken to be the ratio of Raman intensity with and without the metal, is quite different from that calculated by using the familiar polarizability-derivative theory. Applying the well-known fluctuation-dissipation theorem, the new Raman line shape is also calculated to contrast it with the corresponding line shape in the absence of the metal.