superspheres intermediate shapes between spheres and polyhedrasuperspheres中间形状的球体和多面体之间.pdf

Symmetry 2012, 4, 336-343; doi:10.3390/sym4030336 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Article Superspheres: Intermediate Shapes between Spheres and Polyhedra Susumu Onaka Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259-J2-63 Nagatsuta, Yokohama 226-8502, Japan; E-Mail: onaka.s.aa@m.titech.ac.jp; Tel.: +81-45-924-5564; Fax: +81-45-924-5566 Received: 16 May 2012; in revised form: 20 June 2012 / Accepted: 25 June 2012 / Published: 3 July 2012 Abstract: Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, where {100}, {111} and {110} are the Miller indices of crystals with cubic structures. The three parameters p , a and b are included to describe the {100}- {111}-{110} near-polyhedral shapes, where p describes the degree to which the shape is a polyhedron and a and b determine the ratios of the {100}, {111} and {110} surfaces. Keywords: supersphere; particle; precipitate; materials science; crystallography 1. Introduction Small crystalline precipitates often form in alloys and have near-polyhedral shapes with round edges. Figure 1 is a transmission electron micrograph showing an example of this where the dark regions, which have shapes between a circle and a square, are Co-