Now, the Nobel Prize in chemistry is out and it has been given for the

[DrDu]

Now, the Nobel Prize in chemistry is out and it has been given for the discovery of quasi-crystals.
I consider quasi-crystals to be quite a fascinating kind of matter although I know very little about it.
Maybe we can discuss some points.
What I find quite striking although it doesn't seem to be duely recognized in many articles is the fact that the diffraction pattern is, at least ideally, a pure point pattern which is nevertheless everywhere dense. Basically, the diffraction pattern is the Fourier transform of the distance distribution function.
Consider a quasi-crystal which is quasi-crystalline only in one dimension. Then we could integrate the diffraction pattern twice from - infinity to x. The resultant function will be everywhere continuous but nowhere differentiable, i.e. it will be a fractal.
Does every fractal in one dimension correspond to a quasi-crystal? How about 3 dimensions?
How is all this related to quasi-crystals being representable as projections of a higher dimensional regular lattice?

Now we could assume that the diffraction pattern is the spectrum of some fictive hamiltonian.
It will form part of the "essential spectrum" which comprises both the dense point spectrum and the continuous spectrum of the operator.
A continuous diffraction spectrum would correspond to a material which only possesses near order.
Now there are some theorems that the continuous spectrum of an operator is not stable even under infinitely small perturbations which can transform it into a pure point spectrum. An example is Anderson localization.
Is quasi-crystalinity some kind of Anderson localization in momentum space?

 

[Vanadium 50]

Consider a quasi-crystal which is quasi-crystalline only in one dimension.

I don't believe that is possible. Quasicrystals are the results of tilings.

 

[DrDu]

Well, Fibonacci series are often mentioned as examples of "one-dimensional" quasicrystals, see e.g.:
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SSG_note/mchap05_SP.pdf

 

[rhody]

I am happy that a Nobel was awarded to Daniel Shechtman for this http://www.nature.com/news/2011/111005/full/news.2011.572.html". I distinctly remembering after reading Penrose's book, "The Emperor's New Mind", about impossible quasi periodic tiling patterns, quasicrystals. This concept I distinctly remember being amazed by. Cheers to Penrose and others for predicting them.

Given the relative simplicity of making these materials, it's certain that the five-fold patterns had been seen by numerous scientists before Shechtman, who dismissed them because they didn't fit the rigid rules of crystallography," says Elser.

Indeed, such 'aperiodic' five-fold structures had been described by mathematicians many decades before — most famously by British mathematician Roger Penrose. Related complex designs are found in Islamic art and architecture.

I find it curious these impossible five-fold structures found their way into human consciousness later expressed in Islamic art and architecture long before being predicted and then physically discovered in the later part of the twentieth century. Amazing.

Rhody...

 

[PAllen]

Anyone else remember how Linus Pauling (great though he was - I was floored by Nature of the Chemical Bond on first encounter) insisted quasi-crystals couldn't be a new phenomenon, they were just crystal twinning? This was a well deserved Nobel.

 

[DrDu]

Another question: Why and when are the projection from higher dimensional space approach and the parketting approach equivalent? Any crisp argument?

 

[read]

Consider a quasi-crystal which is quasi-crystalline only in one dimension. Then we could integrate the diffraction pattern twice from - infinity to x. The resultant function will be everywhere continuous but nowhere differentiable, i.e. it will be a fractal.
Does every fractal in one dimension correspond to a quasi-crystal? How about 3 dimensions?
How is all this related to quasi-crystals being representable as projections of a higher dimensional regular lattice?

In the modern crystallography there is a notation of aperiodic crystals. They are crystals with normal basis a,b,c and a set of propagation (or wave) k-vectors. The atomic positions (or/and occupancies) are modulated according to x(t_n)=x(0)+Ʃ_k cos(k.t_n+phi), where x0 is the position in zeroth cell, t is a vector pointing to an n_th cell. The extra dimensions are simply the phases/2pi for each propagation k-vector. In this respect there is no principal difference between the situation with one k-vector and the case with two or three k-vectors that occurs in quasicrystals. For instance icosahedral phase of AlMn has 3 k-vectors that corresponds to 3+3 Bragg indices, i.e. 6D space.

 

[rhody]

I thought I would share this: http://www.solid.phys.ethz.ch/ott/staff/beeli/Structural_investigation.html"

High-resolution transmission electron microscopy in combination with further electron microscopy techniques is employed for the structural characterisation of quasicrystalline alloys such als icosahedral and decagonal Al-Mn-Pd as well as decagonal Al-Co-Ni.

Is quasi-crystalinity some kind of http://en.wikipedia.org/wiki/Anderson_localization" in momentum space?

In condensed matter physics, Anderson localization, also known as strong localization, is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first one to suggest the possibility of electron localization inside a semiconductor, provided that the degree of randomness of the impurities or defects is sufficiently large

DrDru,

Since I am a mere mortal with a curiosity of properties of quasi-crystals, could you explain in layman's term's, or provide a graphic or analogy so that I can wrap my head around this. Thanks...

Rhody...

 

[DrDu]

Anderson localisation means that some disorder in a crystal can transform the continuous eigenfunctions (plane waves) of an electron into localized eigenfunctions. As localized eigenfunctions are proper functions in Hilbert space (as opposed to plane wave solutions), the hamiltonian also has a corresponding proper eigenvalue. So when Anderson localisation takes place, the continuous spectrum is transformed into a dense point spectrum. I was speculating whether something analogous may be the case in quasi-crystals.