Bloch's theorem for amorphous matter

Bloch's theorem states that scattering of a wave from an ordered lattice structure is insignificant:

https://en.wikipedia.org/wiki/Bloch%27s_theorem

We demonstrated in our April 1, 2022 blog post that destructive interference in such a case cancels almost all the scattered waves. We also argued that the scattering from an amorphous material is insignificant, provided that its density is very precisely constant for zones which produce scattered waves of a specific phase φ

When can scattering be significant? Destructive interference does not cancel scattering from random obstacles. Also, if the obstacles are ordered and the incoming wave "resonates" with them, there is very significant scattering.

Bloch's theorem for amorphous matter. Let us have an observer very far away from the amorphous material which is being tested. Let Z be the zone which produces a scattered wave whose phase is in the range [φ₁, φ₂). If the density of the material in each such zone is "very precisely" the same, then the scattered wave is insignificant.

The phrase "very precisely" means that the variance of the density must be much less than what would result from a totally random replacement of atoms, where we would also allow the overlapping of atoms.

The wave must not lose energy when it is scattered from the material. The quantum of the wave must not be able to kick an atom of the material from its place. The material must not have a resonant frequency at the frequency of the wave.

The theorem is essentially another way to say that destructive interference wipes out almost all of the scattered wave.

Examples

1. Why visible light is not scattered by glass?

Let us consider a 500 nm laser beam entering a block of glass. The atom spacing is only some 0.1 nm in glass. The wave interacts primarily with volumes of the material where the size of a volume is a half of the wavelength.

Since each volume contains a huge number of atoms, the volumes are very similar with respect to their scattering capability.

We can now apply the ordinary version of Bloch's theorem: the "lattice" of the volumes is ordered and does not cause scattering.

Even though glass is amorphous, that fact is not significant at the level of 250 nanometers. The material is very uniform at such a scale. We do not need Bloch's theorem for amorphous materials in this case.

2. Scattering of a conducting electron from atoms (ions) of liquid metal. The Fermi velocity of a conducting electron is something like 1,000 km/s. Its de Broglie wavelength is 0.7 nm. Since the spacing of atoms is 0.1 nm, we cannot claim that the material is uniform at the scale 0.35 nm.

We can resort to Bloch's theorem for amorphous matter. Let us have an observer very far away. We divide the metal in zones which produce each phase φ of the scattered wave. The zones are very much like each other and cause an almost total destructive interference of the scattered wave.

The resistivity of a liquid metal just above its melting point is roughly double the solid metal. The free path of a conducting electron is about a half of that in the solid. Bloch's theorem for amorphous matter explains in part why scattering of conducting electrons does not explode when a metal melts.

Conclusions

Bloch's theorem about scattering from a periodic potential is just a special case of a more general theorem: if the zones, which produce different phases of the scattered wave, are "very similar", then destructive interference wipes out almost all the scattering.

If we place scattering objects totally randomly in space, then the zones are not "very similar".

If we have a material which is essentially "uniform" at the scale of the wavelength, then scattering is insignificant.