A topological definition of a Wigner–Seitz cell and the atomic scattering factor

Abstract

An atom is defined as a region of space bound by a surface of local zero flux in the gradient vector field of the electron density. The same boundary condition defines a proper open system, one whose observables and their equations of motion are defined by quantum mechanics. Applied to a crystal, this boundary condition coincides with the original definition of the atomic cell in metallic sodium given by Wigner & Seitz. It is proposed that it be used to generalize the concept of a Wigner–Seitz cell, defining it as the smallest connected region of space bounded by a `zero-flux surface' and exhibiting the translational invariance of the crystal. This definition, as well as removing the arbitrary nature of the original method of construction of the cell in the general case, maximizes the relation of the cell and the derived atomic form factors to the physical form exhibited by the charge distribution of its constituent atoms. The topology of the electron density, as summarized in terms of its critical points, also defines the atomic connectivity and structure within a cell. Attention is drawn to the correspondence of the symmetries of the structural elements determined by the critical points with the site symmetries tabulated in International Tables for Crystallography. The atomic scattering factor is defined for an atom in a crystal and determined in ab initio calculations for diamond and silicon. The transferable nature of atomic charge distributions is demonstrated. It enables one to estimate a structure factor and its phase in a crystal using the density of an atom or functional group obtained in a molecular calculation. Atoms in a crystal, along with defects and vacancies, are identifiable with bounded regions of real space. Their properties are additive and are defined by quantum mechanics.