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I'm aware that refractive index usually correlates with density. I'm curious to know if there are strong counter examples to this rule.

In particular I'm interested to know if there is a pair of materials, one with greater density and lesser refractive index, the other with lesser density but greater refractive index.

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Yes.

In the visible, diamond has an exceptionally high index of refraction. Wikpededia says 2.42 at 589 nm. Its density is 3.51 g/cc.

Moissanite (a naturally occuring Silicon Carbide) has an even higher index at 2.65, with a density of 3.22

The index of refraction of glasses is usually around 1.5 with denser glasses having a higher index. Densities vary quite a bit, with Crown glass at 2.5 g/cc and the densest flint glass at 7.2 g/cc. The index of refraction of flint glass is 1.69.

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  • $\begingroup$ Great example, thank you $\endgroup$
    – alcorn
    Commented Oct 5, 2020 at 2:01

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