Electronic Structure of One-Dimensional Binary Alloys

The electronic structure of one-dimensional binary alloys is studied in terms of exact mathematical expressions. It is shown by counter examples that the Saxon-Hutner theorem and its converse do not necessarily hold for all potentials. The exact phase transfer theory is used. The present approach has the advantage that it can examine with the same ease both the Saxon-Hutner theorem and its converse. Various sufficient conditions of validity are found. The physical content of these conditions for potentials that are localized and symmetric is analyzed by means of the one-dimensional scattering phase shifts of the individual constituent potentials. As an example, it is shown that both the Saxon-Hutner theorem and its converse are valid if the phase shifts of the two localized symmetric potentials forming the binary alloy are solutions belonging to a certain class $T$, in which the even and odd phase shifts of type-$A$ and type-$B$ symmetric potentials indicated by ${\alpha }_{+}$, ${\beta }_{+}$, and ${\alpha }_{-}$, ${\beta }_{-}$, respectively, satisfy the condition $\left[\frac{sin({\alpha }_{+}+{\alpha }_{-})}{sin({\alpha }_{+}-{\alpha }_{-})}\right]=\left[\frac{sin({\beta }_{+}+{\beta }_{-})}{sin({\beta }_{+}-{\beta }_{-})}\right]=f$, where $f$ is a constant. The analysis can be trivially extended to the study of alloys composed of more than two elements.