Binary self-similar chemical structures

The recursive series (f_{n+1} =lcdot f_n +mcdot f_{n-1} ) (l, m = 1,2,3...) defines the generators of a chain of components L and S, (left{ {begin{array}{l} Lrightarrow underbrace{LL{ldots }L}_lunderbrace{SS{ldots }S}_m Srightarrow L end{array}} right} ), such that L and S have a geometric dimension equal to (d = 0, 1, 2, 3,{ldots }). If L, S are the atoms of two different elements A, B (dimension (d = 0)) or two self-similar structures with (d > 0) that are composed of these elements, then such a generator forms a self-similar binary structure with the dimension (d+1) (a quasicrystal), composed of (hbox {A}_{x^{prime }}hbox {B}), where (x^{prime }) depends solely on the parameters l and m. In this study, the stoichiometric coefficient (x^{prime }) was calculated for about 20 of such quasicrystals. The generator was found to enable, for certain values of l and m, the formation of structures with degenerate symmetry, that is, the transition from self-similar to translational symmetry. Thus, the generated structures can be divided according to l and m into three groups: structures that contain only one type of homobonds, A–A, with self-similarity as the only permissible symmetry; structures that at least sometimes contain both types of homobonds, A–A and B–B, with self-similarity as the only permissible symmetry; and structures that sometimes show translational symmetry. All of the researched structures in the first group have the same estimated values of internal energy and configuration entropy determined by the isotopic composition. All structures in the second group have the same configuration entropy but different internal energies. In the third group, the configuration entropy of structures that show translational symmetry is lower than that of the self-similar structures. On the other hand, internal energy favours (that is, is lower in) structures with translational symmetry over self-similar structures only when the energy of the A–B bonds is higher than the mean energy of A–A and B–B bonds. In other words, self-similar systems can be energetically more stable compared to crystals as long as the energy of the A–B heterobonds is low. The method for generating self-similar objects proposed in this paper seems to be the first method in which the means of generation do not change with the geometric dimension d of the generated structure.