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Basically without math, certain metals have elecrons that are "free" silver is one of these metals. These free electrons are necessary for a condition called a surface plasmon to exist. Now silver particles in an aqueous solution will have two parts to its refractive index, that of the water and that of the silver nanoparticles. Next is the interaction of light. Photons can be thought as particles or waves, in this case we will think of them as waves. When an electromagnetic wave interacts with the surface plasmon mode it causes the free electrons that are part of the surface plasmon to oscillate, usually in anti phase to the electromagnetic wave. This essentially causes certain wavelengths of light to be blocked from propagating, causing them to be reflected and giving the colour that you see.

Im actuallydoing some research on silver nanoparticles at the moment, and we can change this colour from yellow to almost anything that we want. For a yellow colour we see a plasmon resonance peak ca. 420nm but based on the size and shape of the silver nanoparticles we can tune the plasmon resonance peak.

Mie theory can be used to calcuate the extinction co-efficents. The reason why this is also called the surface plasmon is that the light waves will not penetrate more than 50nm so the interactions at the surface are the most important.

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