Page 142 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Quasicrystals and Momentum Space

J.L. CALAIS
Quantum Chemistry Group, University of Uppsala
Box 518, S - 75120 - Uppsala, Sweden

1. Introduction

In November 1984 the world of crystallography was thoroughly shaken by the news that
"forbidden" peaks characteristic of icosahedral symmetry had been recorded in electron
diffraction diagrams of an Al-Mn-alloy [1]. According to "classical" crystallography long
range order is compatible with rotations through multiples of but not with
rotations through . The point group of a space group must be one of the 32
crystallographic point groups and the icosahedral group is certainly not one of them.
Scientific results of that nature are among the most interesting ones, since they open up
qualitatively new perspectives.
A crystal is an extended system with (in principle) perfect long range order, which is
invariant under all operations of a certain space group. At the other extreme we have
disordered systems with a "completely" random arrangement of its constituent atoms.
Intermediate cases with more or less short range order have been known for a long time [3].
What was unexpected in the paper by Shechtman et al. [1] was the combination of long
range order and a non crystallographic point group. Already in 1902 the French
mathematician Esclangon [4] pointed out, however, that arrangements which are aperiodic
but non random are possible. And even though the paper by Shechtman et al. [1] must be
regarded as the one which opened up this new field of crystallography, it seems that some
Japanese results 20 years earlier [5] should also be interpreted as providing experimental
evidence for the existence of quasi-periodic structures.

During the nearly ten years which have passed since the appearance of the "Shechtman
paper" a large amount of both experimental and theoretical research has been carried out on
quasiperiodic structures. For more material about quasicrystals we refer to a paper in La
Recherche by the French collaborator in the Shechtman team [6], to a thesis by Dulea [7J,
and to a survey paper with a large number of references [8].
Last year a magnificent paper by Mermin appeared in the Reviews of Modern Physics [9],
as the (so far) crowning contribution to a series of papers describing nothing less than a
reformulation of crystallography [10 - 18]. Emphasising reciprocal space concepts Mermin
and his collaborators have been able to treat both "classical" crystals and quasicrystals with
the same method. As is often the case with truly original work this first of all throws new
light on the theory of the "ordinary" space groups, which leads to a deeper understanding of
notions and relationships believed to be well known. Then it provides a straightforward
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Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 127–138.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.