Occupation and diffusion of interstitial solutes in dilute alloys in perspective of the Gauss-Legendre three-square theorem

In the example of the diffusion of C, N, O in dilute ferric iron alloys, it is shown that the polyhedron consisting of equivalent occupation of interstitial solutes in dilute alloys can be classified into seven groups altogether, i.e. cube, octahedron, cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron and truncated cuboctahedron. No more polyhedron can be found. The notation and the migration paths of C, N, O in dilute alloys are well described by the Gauss-Legendre three-square theorem. The abstraction of the migration paths gives rise to the Wythoffian operations illustrated with cube and octahedron. The occupation and migration paths of the diffuser in this work can be generalized to the diffusion of vacancy and interstitial atoms in other host materials with low concentration of substitutional solutes.