A Correlation in a Series of +1 and −1 and an Application to a Layer Structure: Laves Phase

Abstract

In a series consisting of +1 and −1, a pair of the i-th and j-th elements, a(i) and a(j), is characterized by two integers; one is the separation m=j−i (j>i) and the other is the sum d of m−1 numerals between the two elements. If the pair is denoted by [a(i), a(j); m, d], all pairs in a series belong to one of the following four types for given parameters m and d: [1, 1; m, d], [1, \bar1; m, d], [\bar1, 1; m, d] and [\bar1, \bar1; m, d], where 1 and \bar1 are used instead of +1 and −1. Out of these four types, it has been verified that two types of pairs [1, \bar1; m, d] and [\bar1, 1; m, d] appear alternately in any series and therefore the total numbers of pairs are equal to each other for these two types or are different at most by unity. This correlation is applied to a layer structure such as Laves phase, and specific relations are derived.