A periodically Ordered Structures in One Dimension

Sammanfattning: This thesis deals with different aspects of aperiodically ordered structures in one dimension. The approach is entirely theoretical, and the models used have various relevance for real systems. The main objective is to make a general study of how the solutions to some well-known equations behave when they are used to describe physical phenomena in structures which are aperiodically ordered.To obtain structures that are in some sense generic for aperiodic order, we use different sequences such as the Fibonacci sequence, the Thue-Morse sequence, the period-doubling sequence, the Rudin-Shapiro sequence, etc. We let some physical entities, such as on-site potentials or atomic masses, be ordered according to these sequences.One system studied is a tight-binding Schrödinger equation where the on-site potentialis modulated by some different circle sequences. It is found that for almost all such sequences related to the precious means, the electron spectra are purely singular continuous and of zero Lebesgue measure. A closely related study is for chains of aperiodically ordered quantum dots. The systems are modelled by a Kronig-Penney potential, and we calculate their conductance as a function of the Fermi level for realistic dimensions of such chains. We find some signs that we interpret as fingerprints of the singular continuous spectra these models possess in the limit of an infinite number of dots.Inspired by the fact that proteins can be considered as aperiodic chains and the somewhat successful attempts to classify their folding properties by use of the methods of block variables, we apply the same concept to some different deterministic aperiodic sequences. There turns out, however, to be no direct correspondence between the behaviour of the block variables and other properties of the sequences.The rest of the studies are about phenomena that need nonlinear terms in the equations for their description. One study is for how the wavefunction for an initially localized polaron in an aperiodically ordered lattice will spread. The governing equation we use is the discrete nonlinear Schrödinger equation (DNLS). We find that for a large enough nonlinearity, the probability of finding the quasiparticle at the initial site will always be nonzero and the participation number finite for all systems under study (self-trapping). For potentials yielding a singular continuous energy spectrum in the linear limit, selftrapping seems to appear for arbitrarily small nonlinearities. We also find that the root-mean-square width of the wave packet will increase infinitely with time for those of the studied systems which have a continuous part in their linear energy spectra, even when self-trapping has occurred.Another study is for arrays of Josephson junctions, described by the perturbed discrete sine-Gordon equation. The junctions are of two different kinds and ordered aperiodically but equidistantly. We study the dynamics of a single fluxon in such a lattice.With the use of an effective potential we explain the behaviour of the fluxon when it gets pinned in different arrays. The potential also gives a qualitative understanding of the deviation of the velocity of a propagating fluxon compared with an earlier obtained formula. It turns out that the self-similarity of the underlying sequences is important for the detailed dynamics, but not for the speed of a propagating fluxon.Finally there are investigations of various aspects of lattice dynamics when the atoms in the lattice have two different masses ( diatomic lattices) and are ordered according to some of the sequences mentioned above. This time we use classical mechanics to obtain the governing equations of motion.A study of solitary wave propagation in aperiodically ordered diatomic Tod a lattices reveals that the damping is considerable less for these systems than for a random lattice. The short range correlation between the atoms in the aperiodic lattices seems to be of main importance for how much the wave is damped. We suggest therefore that the entropy according to Shannon might be a relevant measure for the properties of the lattices in this case. It is shown that this measure yields at least an approximative agreement with what is actually achieved by our numerical experiments.The anharmonic terms in the potentials used can give rise to localized modes in lattices which only have extended modes in the harmonic approximation. This intrinsic localization is often referred to as discrete breathers. We find such modes numerically in some aperiodically ordered diatomic lattices in two different ways. The first method is known as the rotating wave approximation, which essentially means that we discard all harmonics except the one of lowest frequency. The second method is the use of the anti-continuous limit, which in this case refers to when the larger mass-value is infinite. In this limit, the system is integrable and one can find a solution to the equations of motion. This solution is then continued by the implicit function theorem to finite values of the larger mass.In analogy with the polaron study mentioned above, we also study the same phenomena for these classical models and energy localization. This means that we excite a few sites in the lattice and follow how the energy will spread in the system as time proceeds. We find the propagation to be correlated to the underlying dispersion relation and to the nature of the phase space for a reduced system.