§ 1.6. Pure point spectra in diffraction from quasicrystals.

Sharp diffraction peaks in scattering are found not only in crystals but also in quasicrystals. The reason for this is given in Harald Bohr's theory of quasiperiodic functions.

Bohr [2], [3] from his concept of restrictions to irrational sections $E^{m}_{{\parallel}}<E^{n},m<n$ eq. 8 of periodic functions derived the Fourier properties of quasiperiodic functions. This requires the notion of a Z-module.

Under projection, the parallel projected lattice points of $\Lambda$ fall on a $Z$ -module. The points of this $Z$ -module are the integral linear combinations of the projections of the lattice basis vectors from $E^{n}$ to $E^{m}_{{\parallel}}$ . The number of lattice basis vectors from $\Lambda\in E^{n}$ is n and so is the number of projected basis vectors of the $Z$ -module in $E^{m},m<n$ . These n projections cannot be linear independent on $E^{m}$ w.r.t. real coefficients. They are nevertheless linear independent w.r.t. integral linear combinations. As a consequence, the set of these module points is discrete and countable but dense on $E^{m}$ .

The periodic functions on $E^{n}$ , as we have seen in section 2.3, have their Fourier coefficients on a reciprocal lattice $\Lambda^{R}$ in k-space $E^{n}$ . By Fourier-analysing the restriction of the periodic function it can be shown that the Fourier spectrum of a quasiperiodic function is discrete. Taking into account the projection, the Fourier coefficients of a quasiperiodic function can be assigned to a $Z$ -module on a k-space $E^{m}_{{\parallel}}$ , with module points the parallel projections of the reciprocal lattice $\Lambda^{R}$ from the k-space $E^{n}$ to a k-space $E^{m}_{{\parallel}}$ . Again the module points are dense. Fortunately, the observed Fourier coefficients vary systematically in intensity. The strongest diffraction peaks already suffice to assign them to a discrete module. They clearly display the non-crystallographic (5fold, icosahedral,…) point symmetry which signalled their discovery [55]. It is now standard in quasicrystallography to assign the diffraction peaks to reciprocal lattice points in k-space $E^{n}$ or to their parallel projections.