Enhancement Factors for a Spherical Geometry

For a molecule adsorbed on the surface of a metal sphere r = a) with its vibrational mode oriented normal to the surface, the average DSP enhancement is 

Note that in this treatment of the EM enhancement, we have not considered the effect which the induced dipole in the metal particle has on the local field felt by the molecule. This is the so-called image enhancement effect and contributes an additional factor so that the total EM enhancement, G , becomes 

From Eq. (12) it is apparent that when efcjif = -2so, the value of g becomes infinite, which would maximize G. This situation will occur at a given value of (Ol thus the system may be tuned into resonance by changing the excitation frequency, (o. However, the dielectric functions are complex variables so that a zero in the denominator of g giving an infinite enhancement is, in fact, not possible since Si should be expressed as e, = + is2- Assuming 

It should be recognized for the small-sphere model that if the molecule is located at a distance from the surface of the particle, the enhancement would be diminished by the factor (a /r ), and near the dipolar surface plasmon resonance 

if n = 500 nm and r = 550 nm so that the molecule is 50 nm from the surface, the enhancement only decreases by 0.32. It can be seen that the EM enhancement decays fairly slowly with distance from the metal sphere. 

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