Short-Range Order and the Electronic Structure of Binary Alloys and Liquid Metals: A Comparison of Different Approaches

Abstract

A number of exact results concerning the effects of short-range order on the energy spectrum of disordered systems are derived. In a single-band model of the substitutional alloy, it is shown that the height and width of the minority subband must scale as ${[\frac{x}{(1+\ensuremath{\epsilon})}]}^{\frac{1}{2}}$ and ${[x(1+\ensuremath{\epsilon})]}^{\frac{1}{2}}$, respectively. Here $x$ is the impurity concentration and $\ensuremath{\epsilon}$ is a parameter that specifies the degree of short-range order, i.e., the enhanced probability that impurity atoms will be nearest neighbors. These results depend only on the form of the (nearest-neighbor) two-site distribution function and it is, therefore, rather surprising that they are not reproduced exactly by any of the existing theories of short-range order. In each case, the difficulty can be traced to an improper factorization of the three-site distribution function. As a simple model of the liquid metal, we consider a close-packed array of nonoverlapping spherical-well potentials of depth $\ensuremath{\delta}$, radius ${R}_{m}$, and mean density $n$. This model is used to compare alternate formalisms: the quasicrystalline approximation of Lax, and the more recent approaches of Gyorffy, and of Schwartz and Ehrenreich. The fact that the short-range order guarantees that the potentials are nonoverlapping allows us to extract from each order in perturbation theory the leading contribution to the moments of the electron self-energy. The result is simply expressed in terms of the renormalized cumulants ${Q}^{(p)}(x)$, where $x$ is the dimensionless density parameter $x=\frac{n4\ensuremath{\pi}{{R}_{m}}^{3}}{3}$. These contributions isolate just those terms of higher order in $x$ that are most important for the moments of the electronic spectrum and are reproduced to all orders by only the equations of Schwartz and Ehrenreich.