STRUCTURE FACTOR OF 1D SYSTEMS (SUPERLATTICES) BASED ON 2-LETTER SUBSTITUTION RULES .1. DELTA (BRAGG) PEAKS

The recent generalization to the case of arbitrary tile lengths and ar bitrary scattering factors of the calculation of the structure factor of 1D substitutional systems is studied in detail. This method makes i t easy to find all the peaks in the diffraction spectrum of a system. The well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations o f two incommensurate frequencies, respectively, were found to be the l = 0 subsets of two more general types of spectra, infinite-periodic ( or limit-periodic) and infinite-quasiperiodic (or limit-quasiperiodic) characterized by rational numbers of the type m/n(l), l = 0,...,infin ity in place of the above integers. Substitution rules that produce qu asicrystalline quasiperiodic and infinite-quasiperiodic spectra give t he same type of spectrum for all values of the ratio rho = rho(a)/rho( b) of the two tile lengths rho(a) and rho(b). This is not the case for the other rules. Thus the same substitution rule (such as the copper- mean rule) can give an infinite-periodic spectrum for a single rationa l ratio rho = rho(a)/rho(b) of the two tile lengths rho(a) and rho(b). a periodic-like spectrum for other rational rho, and a spectrum in so me aspects similar to that of a random system when rho is an irrationa l number. On the other hand, a Thue-Morse system diffracts as a period ic crystal when rho not equal 1 but has no non-trivial delta peaks whe n rho = 1. Other Thue-Morse-like systems can have infinite-periodic sp ectra for all rho.