Extended Superspheres for Shape Approximation of Near Polyhedral Nanoparticles and a Measure of the Degree of Polyhedrality

Crystalline nanoparticles or nanoprecipitates with a cubic structure often have near polyhedral shapes composed of low-index planes with {100}, {111} and {110}. To consider such near polyhedral shapes, algebraic formulas of extended superspheres that can express intermediate shapes between spheres and various polyhedra have been presented. Four extended superspheres, (i) {100} regular-hexahedral; (ii) {111} regular-octahedral (iii) {110} rhombic-dodecahedral and (iv) {100}-{111}-{110} rhombicuboctahedral superspheres are treated in this study. A measure ∏ to indicate the degree of polyhedrality is presented to discuss shape transitions of the extended superspheres. As an application of ∏ superspherical coherent precipitate is shown