Spherical-periodic order and relevant short-range structural units in simple crystal structures

Since crystals grow from liquid and glassy states, the liquid structure feature may also exist in the pair-correlation functions of crystalline systems. The present article addresses the spherical-periodic order derived from Friedel oscillations in the pair-correlation functions of simple crystal structures such as face-centered cubic, hexagonal close-packed, and body-centered cubic structures. In these simple crystal structures, the resonance lattice planes, corresponding to the strong peaks in reciprocal space, give the Friedel wavelengths using their interplanar spacings, which are {111} and {200} for the face-centered cubic structure, {100}, {002}, and {101} for the hexagonal close-packed structure, and {110} for the body-centered cubic structure. After being scaled with the Friedel wavelengths corresponding to the most intense diffraction peaks, the pair-correlation functions all show atomic density maxima within the spherical-periodic zones. From such a spherical-periodic picture of any simple crystal structure, it is possible to identify a charge-neutral and mean-density local atomic entity that serves as the molecule-like structural unit of the whole structure. Examples in Cu-Zn, Co-Cr, and β-Ti alloys are provided to show how to interpret the compositions of simple-crystal-based industrial alloys.