a Frank-Kasper phase
/ FRANK KAS-per /
Ordinary metals pack their atoms like stacked oranges — the simple close packings (FCC and HCP) where every atom has 12 neighbours. But suppose you want to pack atoms even more efficiently, filling space almost entirely with tetrahedra (the tightest little cluster four spheres can make). It turns out you cannot do it with perfect tetrahedra alone — they leave gaps, a kind of geometric frustration — so nature compromises by allowing atoms with MORE than 12 neighbours. A Frank-Kasper phase is a metallic crystal built on exactly this principle: space filled by slightly distorted tetrahedra, with atoms whose coordination numbers are 12, 14, 15 or 16.
The rule, worked out by Frank and Kasper in 1958-59, is beautifully specific. In these 'topologically close-packed' structures every atom sits at the centre of one of just four allowed coordination polyhedra: the icosahedron (12 neighbours), or larger cages with 14, 15, or 16 neighbours. The larger-coordination sites form networks of 'major skeleton' lines threading through the structure. Because icosahedral (five-fold) local order is favoured but five-fold symmetry cannot tile space periodically, the crystal reconciles the two by adopting a large, complicated unit cell containing many atoms of several different sizes — the sigma phase, the Laves phases, the A15 phase (like Nb3Sn), and mu and chi phases are all Frank-Kasper structures.
These phases matter a great deal in real engineering. They are common in transition-metal alloys, and their appearance is usually bad news in high-temperature superalloys: brittle, plate-like sigma, mu and Laves particles precipitate from the nickel matrix and embrittle turbine blades, so alloy designers work hard to keep the composition out of the Frank-Kasper danger zone. On the useful side, the A15 Frank-Kasper compounds Nb3Sn and Nb3Ge are important superconductors. Conceptually, Frank-Kasper phases are the bridge between simple crystals and quasicrystals: they show how a solid handles the frustration of favoured icosahedral order by building a giant but still periodic cell — a quasicrystal is what you get when it gives up periodicity altogether.
The sigma phase (e.g. in Fe-Cr steels) has a tetragonal unit cell of 30 atoms sitting on five distinct sites with coordination numbers 12, 14 and 15 — no atom has the simple 12-only environment of FCC. That mix of high-coordination cages is the hallmark of a Frank-Kasper phase, and sigma's brittleness is why stainless steels are formulated to avoid it.
Frank-Kasper phases: space filled by distorted tetrahedra, coordination 12/14/15/16, giant cells — the crystalline cousin of quasicrystals.
Frank-Kasper phases are still ordinary PERIODIC crystals, just with big complicated cells and unusual coordination — do not confuse them with quasicrystals, which give up periodicity entirely. They are the frustrated-packing relative, not the aperiodic one.