DIFFRACTION PATTERNS OF APERIODIC STRUCTURES WITH FROZEN IN PHONONS AND PHASONS
J. Wolny and S. Kapral
Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Krakow, Poland
It has been already shown that using a concept of reference lattice [1-2] one can easily describe the structure factor for any periodic series of scattering vectors as a Fourier transformation of an average unit cell
where u describes the shortest distance between the atomic position and the corresponding reference lattice of planes; P(u) is the probability distribution of distances u; u0= p/k0 (k0 is a chosen scattering vector) and k=m k0, i.e. it is higher harmonic of k0; fa is an atomic form factor.
For modulated structures the above approach can be even extended to an arbitrary combination of the main reflections and their satellites:
\where k is higher harmonic of a chosen k0; q - any higher harmonic of q0; u and v are the shortest distances of atomic positions to the reference lattices described either by k0 or q0; P(u,v) is an appropriate probability distribution; u0= p/k0 and v0= p/q0, and they describe the limits of the so called average unit cell in the parameter space. The discussed approach has been successively used to an analysis of the diffraction patterns of different aperiodic structures, among them are quasicrystals, modulated structures, structures with singular continuous diffraction pattern and some other structures. For the so called (3)1/2 * (3)1/2 hexagonal lattice with some frozen in phasons and phonons, it was possible to calculate the intensities of diffraction peaks from the obtained distributions of displacements as well in physical space as in perpendicular (phason) space.