Accurate determination of the radii and order parameter of the layers of a binary alloy, liquid or amorphous, from the eigenstates of the exchange operator of its components

Abstract In this work, a rigorous mathematical method to calculate the order parameter in a binary amorphous or liquid alloy, is established even when the alloy is made up of very differently sized atoms. This method, based on the study of the eigenstates of the exchange operator of the alloy components, allows revealing different spherical layers whose inner and outer radii are solutions of a certain equation. These layers have no constant thickness but contain numbers of atoms of each species that are exactly in the ratio of the concentrations of these two species. Thus, each layer has exactly the average concentration of the alloy, allowing to properly formalize the order parameter. Thus the order parameter is formalized in each of these layers. This method makes possible to retrieve the subtle Bhatia-Thornton formalism [Phys. Rev., B2, 3004 (1970)] used for the description of binary alloys. We discuss the link between the zero q value of concentration-concentration Bhatia-Thornton partial structure factor and the numbers of neighbours. We apply our method to study the order in the Bi0.3Ga0.7 liquid alloy. We are able to determine eight layers and their order parameter on the interatomic distance-range [0, 15 A].