Discovery of Quasicrystals; Icosahedral Phase

The existence of quasicrystals in an icosahedral phase
was first conjectured and investigated in 1981
as a candiate for the blue phase in liquid crystals by

H. Kleinert and K. Maki,
Lattice Textures in Cholesteric Liquid Crystals
Fortschr. Phys. 29, 1 (1981). See p. 242 and 243.


A phase with these properties was discovered experimentally three years later by Shechtman at al. in 1984:
D. Shechtman at al., Phys. Rev. Lett. 53, 1951 (1984)
in a rapid solidification of the alloy Al86Mn14.
For more details on this system click on this link! See also the citations of Kleinert and Maki in the review article on Liquid Crystals by Tamar Seideman.

A picture of the square of the icosahedral order parameter in Eq. (4,22) of above Kleinert and Maki paper can easily be plotted with the help of this short Mathematica Pragram:
nfold=5; k[n_]:=2 Pi *n/nfold
phi=Sum[E^(I (Cos[k[n]]*x+Sin[k[n]]*y)),{n,0,nfold-1}]
ContourPlot[Abs[phi]^2,{x,-60,60},{y,-60,60}]

Icosahedral order parameter

By changing the number nfold, the program draws a picture of the density
of other two-dimensional quasicrystal symmetries, for instance a picture of the nfold=7 heptagonal order parameter:

heptagonal order parameter

Read also this note.
And see the mathematical history here.

Note added in 2011: Shechtman's experimental discovery just earned him the Nobel Prize 2011 for a ``Paradigm Shift in Crystallography". Chemists had a hard time believing in the existence of a five-fold symmetric crystal structure discussed in our 1981 paper with Maki.
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