The Continued Fraction and Concatenation Series
on the Farey Tree on Two Dimensional Atomic Arrays
on Periodic Substrate Potentials

Abstract

In one-dimensional Discommensuration (DC) structure of an atomic array on a periodic substrate potential in the case of commensurate relations, the DC structure is interpreted from two points of view: (i) a DC structure generally consists of a hierarchical superposition of several simple DC structures with different wave lengths. These simple DC structures correspond to ``the continued fraction'' of the structural parameter. (ii) a DC structure is generally composed of simple DC structures; this composition is called ``concatenation''. In this paper, we study these concepts on two dimensional systems. In particular, we find that a concatenation in two dimensional systems which changes a structure, corresponds to changing of a terminal partial quotient of the two dimensional continued fraction of a structural parameter. Structures generated by the concatenations coincide with a line of a succeeding phase transition on real structures.