Chapter 18: Metallic Structures II: Topologically Close-Packed Phases

The geometric foundation contrasts traditional close-packed structures, which achieve twelve-fold coordination in face-centered cubic and hexagonal arrangements, with icosahedral coordination identified by Frank and Kasper as energetically favorable due to its exclusive tetrahedral interstices despite incompatibility with long-range translational periodicity. The chapter derives the four primary Frank-Kasper coordination polyhedra with coordination numbers twelve, fourteen, fifteen, and sixteen, which satisfy Euler's topological requirements and serve as fundamental building blocks in complex intermetallic compounds. Detailed examination of the A15 class illustrates how constituent atoms form infinite chains of edge-sharing icosahedra, exemplified by technologically significant phases like the superconducting Nb3Sn and Cr3Si prototype. The Laves phase family, stabilized by critical atomic size ratios, exhibits three polymorphic variants characterized by the Friauf polyhedron as a key structural unit. Investigation of shear-related structures addresses the brittle sigma phase prevalent in transition metal alloys and steels, alongside the mu phase, employing Kagome net representations and specialized stacking notations to describe their complex periodic arrangements. The chapter concludes by bridging these giant unit cell structures to quasicrystal phenomena through crystalline approximant phases such as alpha-Al-Mn-Si and Mg32(Al,Zn)49, which embed large icosahedral clusters like the Mackay icosahedron and Bergman polyhedron within periodic lattice frameworks, demonstrating the continuum between ordered crystalline phases and aperiodic quasicrystalline arrangements.