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Let's start with something that isn't white. A piece of quartz is highly
transparent. It does not absorb light. It does have an index of
refraction. A light beam will partially reflect off the air/quartz
interface due to the index mismatch. If you take a zillion identical small
quartz spheres and arrange them in a regular lattice like a crystal, you
might think that all those little reflections would add up to make
something quite non-transparent, but in fact a funny thing happens. If the
lattice really is perfect, the scattering is coherent. The coherent
forward scattering reconstructs the incoming wave perfectly, making the
lattice transparent (although it has a refractive index of its own). You
can visualize this in terms of the Huygens construction if you want.
For more on this see _The Feynman Lectures on Physics_ volume III
chapter 13,
or any solid-state physics text.