%0 Journal Article
%T On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems
%A Enrique Maci¨¢
%J ISRN Condensed Matter Physics
%D 2014
%R 10.1155/2014/165943
%X The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interest in the last few years. In this work we present a review on the notion of critical wave functions and, in the light of the obtained results, we suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transport properties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related to both fractal and quasiperiodic distribution of atoms through the space. 1. Basic Notions 1.1. Orderings of Matter The notion of order is one of the most fundamental ones. In fact, order inspires the best human civilization achievements in politics, ethics, arts, and sciences [1]. Order pervades also most workings of Nature as the universe unfolds creating symmetric patterns and stable structures. Among them, solid matter arrangements were initially categorized in a dichotomist way, namely, as either ordered or disordered matter forms. In this way, ordered matter was identified with periodic arrays of atoms through the three-dimensional space, while disordered matter was related to random atomic distributions instead. Thus, the notions of crystalline matter and spatial periodicity were born interwoven from the very beginning, just as amorphous matter was conceptually related to randomness in a natural way (Figure 1(a)). Figure 1: In 1992 the notion of crystal was widened beyond mere periodicity. This conceptual diagram presents the position of aperiodic crystals, no longer based on the notion of periodic translation symmetry, among the different orderings of matter. The diverse aperiodic crystal families are arranged according to the dimension (see ( 2)) of their embedding hyperspaces (numerical labels). Nevertheless, the unexpected finding of incommensurate phases during the 1960s and 1970s, followed by the discovery of quasicrystalline alloys in 1982, opened up a discussion forum on the very crystal notion in the crystallographic, condensed matters physics and materials science communities. Indeed, initially it was thought that quasicrystals (short for quasiperiodic crystals) corresponded to a somewhat intermediate order form between that of crystals and amorphous materials [2]. However, it was soon realized that quasicrystals (QCs), exhibiting long-range order along with orientational symmetries not compatible with periodic translations, actually represented a new order style, which should be properly interpreted as a natural
%U http://www.hindawi.com/journals/isrn.cmp/2014/165943/