Page 150 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUASICRYSTALS AND MOMENTUM SPACE 135
The symmetry properties of the momentum space wave functions can be obtained either
from their position space counterparts or more directly from the counterpart of the
Hamiltonian in momentum space.

4. Momentum space characteristics of quasicrystals

One of the main points in the papers by Mermin and his collaborators [9 - 18] is the
insistence on the primacy of reciprocal space. The properties of the Fourier transform of the
density rather than the density itself determine those properties which are of importance for
"generalized" crystallography. As pointed out by Mermin that point view was stressed in a
paper by Bienenstock and Ewald already in 1962 [26].

Irrespective of the type of extended system we are interested in we impose periodic
boundary conditions in position space - "the large period": BK. Such conditions imply a
discretisation of momentum and reciprocal space |27] which means that integrations are
replaced by summations:

The discrete momenta can be written as

where the are positive or negative integers or zero, and the very large even integer G
characterizes the BK region The reciprocal basis vectors do not require any
actual physical lattice, but can be seen as just providing a suitable framework. We have
used (IV. 1) several times in the previous section, but there we had lattices both in direct and
in reciprocal space, and then this procedure may have seemed more natural. In the present
section there is definitely no lattice in direct space and the "lattice" in reciprocal space may
be of a different nature from the ordinary ones. Because of the periodic boundary
conditions, (IV. 1) should still be used, however.

The Fourier expansion of the density in an extended system which does not have any
particular symmetry is

This sum over all reciprocal space vectors of the form (IV.2) should be carefully
distinguished from the expansion (III.4) of the density of a periodic crystal. If the density
has the "little period", the expansion (IV.3) reduces to a sum over all reciprocal lattice
vectors. The general case (IV.3) and the periodic case (III.4) actually represent two
extreme cases. The presence of "more and more symmetry" in the density can be gauged
by the disappearance of more and more Fourier components in (IV.3). If some of the
Fourier components in (IV.3) vanish, but not necessarily all which do not correspond to a
set of reciprocal lattice vectors, we have a Fourier expansion of a density with another
type of long range order than the one known from traditional crystals. There are
quasicrystals, incommensurately modulated crystals or incommensurately modulated
quasicrystals [9].