One-Dimensional Impurity Bands

The density of states of one-dimensional crystals consisting of $\delta$ functions randomly distributed has been calculated on the IBM 650 computer. The chains contained 500-1000 impurity atoms, and the most probable error in the integrated density of states at various energies was estimated to be at most ½%. Calculations were performed for various values of the parameter $\epsilon =\frac{n}{{\kappa }_0}$, where $n$ is the density of atoms and ${\kappa }_0$ the attenuation constant appropriate to the isolated bound state. The results at different densities are compared with those obtained from various physical models. At low densities the machine results display a singularity in the density of states at the isolated atom energy. For $\epsilon \ll 1$, a simple pair theory fits the machine results quantitatively in the wings and displays a similar singularity. At high density ($\epsilon \gg 1$) the machine results are smooth and fitted well by a crude optical model, except for a tail below the band edge omitted by the latter. An optical model containing local density fluctuations provides a qualitative understanding of the tail and a fair over-all fit for $\epsilon \gtrsim 1$.