Mie scattering near the Fröhlich mode

We consider the scattering of electromagnetic radiation by a spherical particle, known as Mie scattering. Of particular interest is the case where the particle is small, compared to an optical wavelength. A series expansion in the particle radius then gives an approximation for the otherwise very complicated scattering coefficients. However, for certain values of the permittivity of the particle, the approximation for electric multipole radiation breaks down. This is known as the Fröhlich resonance. On one hand, the scattering coefficients diverge, and on the other hand, they violate conservation of energy. We derive a new expression for the small-radius limit, which does not have these problems. We find that there exists a resonant radius for certain parameters, and the resonance is shifted as compared to the common expression. We show that our approximation is excellent by comparing it to the exact solution.