ABSTRACT
Sketch illustrating di¤erent models considered in the literature: a) binary transfer model, b) binary on-site model, c) ternary mixed model [13], and d) Fibonacci binary mixed model.
atoms (and thereof the value of the spring constant representing the bond) will depend on the nature of the involved atoms. In this case, the aperiodic distribution of masses in the system induces a (generally di¤erent) aperiodic distribution of spring constants in the chain (Fig. 5.1c-d). Therefore, in most physical situations of interest, one must consider the so-called mixed models. Earlier studies focused on the study of either transfer[1, 2, 3] or on-site models. [4, 5] Subsequently, more general models were considered.[6, 7, 8, 9, 10, 11] Ternary models, de…ned in terms of two di¤erent spring constants and three di¤erent kinds of masses aperiodically distributed have been also considered (Fig. 5.1c).[12, 13] Since most stable QCs found to date are ternary alloys this class of models makes a closer connection with the chemical complexity of these materials.[14] More general models, considering three di¤erent types of bonds connecting …ve di¤erent types of masses have also been considered in the literature.[15] A standard way to study the acoustic and thermal properties of a lattice is
to consider a nearest-neighbor harmonic chain given by the Lagrangian
L = 1
Kn;n+1(n n+1)2; (5.1)
where n is the displacement of the nth atom from its equilibrium position; mn, with n = A;B, is the corresponding mass, Kn;n1 denotes the strength of the harmonic coupling between neighbor atoms, and N is the number of particles in the system. The dynamical equation for the normal modes n =
Tight-binding chain model describing the electron dynamics in terms of on-site energies "n and transfer integrals tn;n1.
