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Is the interaction between metallic nanoparticles (~ 20 nm) and light in the UV-Vis-NIR range governed by Mie theory or by Rayleigh scattering theory?

Where are the Au and Ag plasmonic peaks located? Does the formula depends to $\lambda/d$ ?

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    $\begingroup$ Neither (internal electronic effects impacts scattering), and of course the plasmon resonanace depends on size. $\endgroup$
    – Jon Custer
    Commented Sep 23, 2014 at 13:16

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Sigma provides plethora of information on silver and gold nano-particles. Silver absorbs around 400nm and gold around 500nm. You can see that shapes of spectra are very complex. This figure shows dependence of peak wavelength from particle size.

It seems like Mie is used in description of metal surface plasmon resonance.

Nature review Light scattering and surface plasmons on small spherical particles also gives a lot of information.

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Mie provided a strict solution of the scattering of a plane electromagnetic wave by a homogeneous sphere. It is valid for an arbitrary size of a particle. However, the calculations can be difficult in some cases.

The Rayleigh theory is the approximation of the Mie theory in the non-resonant regime (elastic scattering), in which only the electric dipole vibration is taken into account (so called dipole approximation). This holds for non-absorbing sphere if $nx\ll1$, where $n$ is the refractive index of the particle, and $x=\frac{2\pi R}{\lambda}$ is the size parameter, in which $R$ is the radius of the particle, and $\lambda$ is the wavelength of incident light.

For example, if your particle of $R=10$nm is a water droplet ($n=1.33$) which is illuminated by green light ($\lambda=532$nm), then $nx\approx0.16$. For these particular values of $n$ and $x$ this approximation has the limit of validity below 1%. The conclusion is that you can use the Rayleigh theory in this case.

Please compare the article by Walstra for detailed explanation and the applicability of other approximations.

However, in the case of gold we have resonant scattering and Mie theory need to be used. Nevertheless, for small particles dipole (quasi-static) approximation can be applied as well. It gives the relation of scattering cross-section $\sigma_{s}\propto\frac{R^3}{\lambda^4}$ providing that the dispersion of the dielectric constant of gold is neglected. Proper formulas for both the scattering and extinction cross-sections can be found for example in: Optical Properties of Metallic Nanoparticles, F. Valee.

The Localized Surface Plasmon Resonance (LSPR) of a metal particle of $R=10$nm is around 510-570nm in the case of Au, and 355-450nm in the case of Ag. These are spectral positions in the case of particles incorporated in dielectric matrices with $n=1.0-1.8$. In general, spectral position of the LSPR depends on the size of particle (also because of dielectric and quantum confinements), material of the particle (dielectric constant of the particle), the dielectric constant of surrounding medium, distribution of size in the ensemble of particles, shape of the particle, etc.

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