Diamond consists of a giant 3D lattice of carbon atoms arranged tetrahedrally, and can be contaminated by only a few impurities. There are strong covalent bonds between the atoms, so it's the hardest naturally-occuring substance, and has a high optical density. This means when light travels from air into a diamond, the light slows down more than it would slow down if it were travelling into, for example, glass or water. The refractive index of diamond is about 2.4, which means light travels 2.4 times as fast in a vacuum as it does in diamond.
When light enters the diamond, it not only changes speed, but changes direction too: this is called refraction. White light's made up of the full spectrum of colours, but violet travels slower in diamond than red travels in diamond. As a result, violet refracts more and red refracts less, and the white light is split into a rainbow. That's dispersion, the reason why diamonds shine with many different colours.
|Dispersion, from TutorVista|
Reflection of light within a material is called total internal reflection. This happens when the light ray's angle of incidence is greater than the critical angle of the material. If you plug in diamond's very high refractive index from earlier into an equation, you can work out diamond's critical angle, which is very low - just 24.4°. So when the light's angle of incidence is greater than just 24.4°, which is quite often, the light is totally internally reflected.
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