Equivalence of two different definitions of quasicrystals

[nomadreid]

TL;DR Summary

Why is "ordered but not periodic structure" equivalent to "distribution where it and its Fourier transform both have discrete supports"?

https://en.wikipedia.org/wiki/Riemann_hypothesis#Quasicrystals a quasicrystal as "a distribution with discrete support whose Fourier transform also has discrete support."
https://en.wikipedia.org/wiki/Quasicrystal#Mathematicsdefines a quasicrystal as "a structure that is ordered but not periodic".

I would be grateful for an explanation of the equivalence between the two definitions. I read further down in the second article, in https://en.wikipedia.org/wiki/Quasicrystal#Mathematics where it gave an explanation which perhaps satisfies the better informed than me, without my seeing the connection, either formal or informal. Thanks.

 

[jvicens]

Thank you for your interest in the Riemann Hypothesis and quasicrystals. I can provide some insight into the equivalence between the two definitions you mentioned.

Firstly, let's start with the definition of a quasicrystal as "a structure that is ordered but not periodic". This means that a quasicrystal has a distinct and predictable arrangement of atoms or particles, but this arrangement does not repeat in a regular pattern like a crystal. In other words, the positions of the particles are not determined by a simple repeating unit cell, as seen in periodic crystals.

Now, let's look at the definition of a quasicrystal as "a distribution with discrete support whose Fourier transform also has discrete support." This definition is more mathematical in nature and may be a bit harder to understand for those without a background in mathematics. Essentially, it means that the distribution of particles in a quasicrystal has a finite number of points where the particles are located, and the Fourier transform of this distribution also has a finite number of points. The Fourier transform is a mathematical operation that converts a function in one domain (such as space) to a function in another domain (such as frequency).

To understand the equivalence between these two definitions, we need to consider the relationship between the structure of a quasicrystal and its diffraction pattern. When a beam of X-rays or electrons is passed through a crystal, it produces a diffraction pattern, which is a series of bright spots on a detector. The positions and intensities of these spots provide information about the arrangement of atoms in the crystal.

In the case of a periodic crystal, these spots form a regular pattern, which can be described by a mathematical concept called a lattice. However, in the case of a quasicrystal, the spots do not form a regular pattern, and therefore cannot be described by a lattice. Instead, the diffraction pattern of a quasicrystal is made up of a finite number of discrete spots, which corresponds to the discrete support mentioned in the definition.

Moreover, the Fourier transform of a crystal's diffraction pattern is also a lattice, while the Fourier transform of a quasicrystal's diffraction pattern is a discrete set of points. This is where the second part of the definition comes in, as the Fourier transform of a quasicrystal also has discrete support.

In summary, the equivalence between the two definitions lies in the