This module will be in English because it is also used for other Hyperscripts. | |||

One of the more exciting (or frightening, depending on one's perspective to science)
developments in understanding quasicrystals was the insight that quasicrystals can be constructed by projecting a perfect lattice in 6-dimensional space onto a properly chosen
3-dimensional subspace (which is the kind of space you and I know). | |||

The gist of this outrageous statement is actually far easier to understand than it appears
at first. For that we look at a simple analogue: We project a perfect
on a 2-dimensional lattice. How this is done is shown in the figure below.1-dimensional subspace |
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Starting from a simple 2-dimensional cubic lattice, we draw a straight line x
at an angle _{p}a to the x-direction. The decisive point is that tan a
must be an irrational number, e.g. (5) or ^{½}p/3
or whatever. This makes sure that x will _{p}not touch another lattice
point ever. | |||

We than define an area (yellow) by drawing a line parallel to x at some
distance _{p}T , which can be an arbitrary number. | |||

Some lattice points will now be found within the yellow area, we project their position onto
x. _{p} | |||

The sequence of points obtained that way (shown at the bottom as the sequence
of green "diamonds") is - by definition if you think about it - aperiodic; it will never
repeat. | |||

But it is not completely random either. There are only two different distances between points, their sequence just does not follow a periodic pattern. | |||

We have actually produced a one-dimensional quasicrystal. |

Now lets take a six-dimensional space and
construct a cubic primitive lattice. No mathematician has the slightest problem doing that - you simply get a "hypercube"
elementary cell with 64 corners and so on. | ||

Now lets take a regular three-dimensional space. We make sure that the three-dimensional space
is oriented relative to the six-dimensional space in such a way, that the six base vectors of the hypercube are projected
onto the three-dimensional space with the fivefold symmetry of an icosahedra. (An icosahedra
is one of those regular "eders" with triangular faces, where always five triangles sort of group around an axis
with five-fold symmetry. | ||

Interestingly, instead of some tan a the number N = [1
+ (5) appears - which is the "magic number of the ^{½}]golden
ratio". This may or may not mean something special. | ||

Now you define some neighborhood around your three-dimensional space and start projecting - you will get the exact arrangement of atoms in a real three-dimensional quasicrystal. | ||

Don't worry if you can not imagine all this - nobody can. | ||

Only worry if you can't see that there is a sophisticated, but nevertheless rather clear-cut mathematical procedure of how to construct a three-dimensional point sequence by all this hypercube projection stuff. | ||

If you did not worry so far, you should now, pondering the question: What
does it mean? | ||

Again, who knows for sure? But we do know that there is some meaning. Lets just
look at one example. | ||

In three-dimensional quasicrystals we find some entities that look and behave exactly like
dislocations (a one-dimensional defect in three-dimensional
lattices). These whatever-they-are entities come into being after some deformation like
real dislocations, they move through the quasicrystal like real
dislocation, they look like real dislocations in the electron microscope - but they
simply cannot be real dislocations because real
dislocations can only exist in periodic lattices. |
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Turns out they are dislocations in the six-dimensional periodic lattice
- no problem to define dislocations there. What we see is sort of what is left in the three-dimensional space where atoms
live (not to mention you and me). | ||

And while real dislocations are always characterized by
their Burgers vector - just a regular vector with three
components - the quasicrystal sort-of-dislocations need a vector with six components for their characterization.
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Mercifully enough, it turns out that the particulars of the projection scheme always allow
to decompose the one six-dimensional Burgers vector into two
three dimensional ones. These two regular vectors have a precise meaning as to what this dislocation-liek thing does to
the quasicrystal when it moves through it. | ||

All in all - quasicrystals, quite unexpectedly, link materials science with rather
involved and rather unexpected math. It will be exciting to witness what will come from all this during the next 10
or 20 years. | ||

© H. Föll (MaWi 1 Skript)