Quasi Crystals 

History and Basics  
You don't have to be a genius to realize that you can't completely tile your twodimensional bathroom floor or walls with regular fivesided tiles or pentagons. Hexagons  yes. Equilateral triangles, squares, and rectangles  yes; see the figure below.  
You don't have to be a genius either to realize that you cannot "tile" threedimensional space (akin to filling out a volume) with regular fivesided bricks (called dodecahedrons) either. Rectangular bricks  yes. Tetrahedrons  yes. Others? Who knows. But dodecahedrons  definitely no.  


You don't have to be a genius either to know that the diffraction pattern of regular crystals therefore can never have a fivefold symmetry. But you need to know a bit about crystals and Fourier transformations though, or you must now look at this module.  
When in 1982 Dan Shechtman observed a diffraction pattern like the one below, he was flabbergasted, to put it mildly. His specimen was a rapidly cooled alloy of aluminum (Al) and Manganese (Mn), nothing particularly exciting. The presence of a diffraction pattern proved that the material was crystalline, i.e. with a regular arrangement of its atoms, and the 5 or 10 fold symmetry of the diffraction pattern proved that there must be an arrangement of building blocks that was simply impossible according to common and well justified believe.  
Of course, nobody believed Shechtman at first. His boss suggested that he quit, extremely famous Nobel prize winner Linus Pauling pontificated that "there are no quasicrystals, only quasiscientists"; journals refused to print his papers. In 2011 Dan Shechtman was awarded the Nobel prize.  


The breakthrough occurred in 1984 after some openminded scientists adopted Shechtman's results. It also became clear that the solution to the obvious problem was already at hand, since more mathematically inclined scientists had already dealt with the problem theoretically years before Shechtman's discovery.  
In particular, famous Roger Penrose had shown in 1974 how to tile a
plane in a way that provided regularity and
a fivefold symmetry. What was lost was the usual
translation
symmetry of normal crystals, i.e. the property that nothing changes in the
pattern if you just move it in the plane. The figure below illustrates this "Penrose tiling". 



Penrose used 4 differently shaped tiles but you
can get away with just two types as shown. Just connect the centres of the old
tiles as illustrated in the overlapping part. There are a lot of fivefold structures in this tiling, and there is a lot of regularity. Now do the whole thing with properly chosen bricks in three dimensions, and you have the structure of the quasicrystals that Shechtman observed. Scientists have made a lot of quasiscrystals by now; even Mother Nature makes them. They have rather peculiar properties but we still have to find major applications for this new class of materials 

Boggling the Mind  
Now lets give the difficult part a quick look. First, we ask ourselves if, maybe, you could tile the plane in a quasicrystal pattern with just one tile? It is clear to you that this question has a defined answer: yes or no!  
It is probably not so clear to you that this
question belongs to the class of perfectly legitimate questions with a definite
yes / no answer for which it is impossible to find the answer in a systematic way, i.e. by an algorithm. Mathematicians have proved that with
mathematical rigor. You may happen to find the answer somehow, for example by
lucky guessing, but you will never be able to program a computer to find the
answer. So far, by the way, nobody has found the answer to the question above. It is rather unlikely that one tile will do, but who knows? 

Now to the really weird part. Quasicrystals actually result from perfectly legitimate real crystals with translation symmetry and everything  if you construct these crystals in six dimensions. Mathematicians have no problems to do that with equations, it just boggles the mind a bit to imagine it.  
All you need to do is to construct a nice
sixdimensional crystal. Then you project some of its lattice points onto our
common threedimensional space in some proper way. You can't imagine that? I
can't either. But it's just like projecting threedimensional objects onto twodimensional space, just a bit more involved. Take a sphere, for example. Project it onto a plane and you have a circle for all projection geometries. Project a cube and you can get squares, rectangles and hexagons, depending on the projection geometry. 

Let's do it in an even simpler way. We take a lattice in two dimensions and project it onto onedimensional space, i.e. on a line. The figure below illustrates how it is done:  


The general recipe is simple:


Look at the sequence of projected points on the
red line above. There is clearly some regularity but the pattern never repeats;
it is a onedimensional quasicrystal. If you have a lot of free time, you can now do the same thing for a threedimensional crystal projected onto a twodimensional space, i.e. a plane. Please send me the drawings after you are done. 

Now you are thinking that this is pretty crazy but just what you would expect of the nerdiest of the nerds, the mathematicians. Surly, this stuff has no bearing on real quasicrystals that you can buy and touch?  
Not so. Lattice defects like dislocations, one should think, can only exist in real threedimensional crystals. Amazingly enough, with an electron microscope, we see things that look like dislocations and behave like dislocations also in quasi crystals. These things are dislocations. But, as it turns out, only if we associate a sixdimensional vector with them, in contrast to dislocations in normal crystals, where a threedimensional vector is sufficient.  
If you now admit that Dan Shechtman got his Nobel prize for rather good reasons, I will rest my case.  
Experimental Techniques for Measuring Diffusion Parameters
The Story of SelfInterstitials in Silicon
© H. Föll (Iron, Steel and Swords script)