Fractal Behaviour of Diffraction Pattern of Thue–Morse Sequence

The Thue–Morse sequence is an example of an aperiodic structure with a singular continuous contribution to the diffraction pattern. Many satellites observed near the main diffraction peaks have been indexed and their intensities have been numerically calculated. Details of such a calculation are presented for satellite peaks indexed by 1/3. An average-unit-cell approach was successfully used to describe probabilities of atomic displacements from the points of the reference lattice. For a certain parameterisation of such probabilities, the results of fairly simple integer calculations could be easily generalised. For some defined sets of points, the analytical expressions for diffraction intensities were found and tested numerically on a wide range of scattering vectors and for up to 232 atoms.