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the free encyclopedia
Quasicrystals are a
peculiar form of solid in which
the
atoms of the solid are
arranged in a seemingly regular,
yet non-repeating structure. They
were first observed by
Dan Shechtman in 1982.
Patterns in quasicrystals
In a normal
crystalline solid the
positions of atoms are arranged in
a
periodic
crystal lattice of points,
which repeats itself in space the
same way that a honeycomb
structure repeats itself in the
plane: each cell has an identical
pattern of cells surrounding it.
In a quasicrystal, the pattern of
atoms is only
quasiperiodic. The local
arrangements of atoms are fixed,
and in a regular pattern, but are
not periodic throughout the entire
material: each cell has a
different configuration of cells
surrounding it.
Quasicrystals are remarkable in
that some of them display
five-fold
symmetry. In an ordinary
crystal, only 1-, 2-, 3-, 4-, and
6-fold symmetries are possible.
This is a geometrical consequence
of filling space with congruent
solids—these are the only
symmetries that can fill space.
Prior to the discovery of
quasicrystals, it was thought that
five-fold crystal symmetry could
never occur, because there are no
space-filling periodic
tilings, or
space groups, which have
five-fold symmetry. Quasicrystals
helped to redefine the notion of
what makes a crystal, since they
do not have a repeating unit cell
but do display sharp diffraction
peaks.
There is a strong analogy
between the quasicrystal and the
Penrose tiling of
Roger Penrose. In fact, some
quasicrystals can be sliced such
that the atoms on the surface
follow the exact pattern of the
Penrose tiling.
The geometric interpretation
For a periodic pattern, if you
fill all of
space with the pattern, you
can slide the pattern a certain
distance in a certain direction,
and every
atom will lie exactly where an
atom lay in the original pattern.
For a quasiperiodic pattern, if
you fill space with it, there is
no distance you can slide the
pattern to make every atom lie
exactly where an atom lay in the
original pattern. However, you
can take a bounded region, no
matter how large, and slide it to
match up exactly with some other
part of the original pattern.
There is actually a simple
relationship between periodic and
quasiperiodic patterns. Any
quasiperiodic pattern of points
can be formed from a periodic
pattern in some higher
dimension.
For example, to create the
pattern for a
three-dimensional quasicrystal,
you can start with a regular grid
of points in six-dimensional
space. Let the 3D space be a
linear subspace that passes
through 6D space at an angle. Take
every point in the 6D space that
is within a certain distance of
the 3D subspace.
Project those points into the
subspace. If the angle is an
irrational number such as the
golden mean, the pattern will
be quasiperiodic.
Every quasiperiodic pattern can
be generated this way. Every
pattern generated this way will be
either periodic or quasiperiodic.
This geometric approach is a
useful way to analyze physical
quasicrystals. In a crystal, flaws
are locations where the pattern is
interrupted. In a quasicrystal,
flaws are locations where the 3D
"subspace" is bent, or wrinkled,
or broken as it passes through the
higher-dimensional space.
Resources
- D. P. DiVincenzo and P. J.
Steinhardt, eds. 1991.
Quasicrystals: The State of the
Art. Directions in Condensed
Matter Physics, Vol 11.
ISBN 9810205228.
