Many atoms in one solid: two frontier answers
Back on the close-packing rung you learned nature's tidiest trick: identical atoms, treated as hard spheres, stack like a grocer's orange pyramid into close-packed arrays — FCC and HCP — where every atom touches twelve others, the coordination number is 12, and 74 percent of space is filled. The last two guides then broke the rules: quasicrystals gave long-range order with no periodicity at all, and incommensurate structures stretched a periodic backbone until a second rhythm no longer fit inside it. This guide stays inside the periodic world — every structure here is a bona-fide crystal with a repeating unit cell — but asks a different question: what happens when the atoms are not all the same?
Two very different things can happen, and they are almost mirror images of each other. If the atoms come in a few sharply different SIZES, the crystal often abandons simple close packing and builds something intricate — a Frank-Kasper phase or a giant-celled complex intermetallic, where a beautifully ordered arrangement can run to hundreds or even more than a thousand atoms before it repeats. But if you throw in five or more elements of SIMILAR size in near-equal amounts, the opposite happens: they collapse into one absurdly simple lattice — FCC or BCC — with every site occupied at random. That second family is the high-entropy alloy. Complexity dialled all the way up in one, and all the way down in the other, from the very same starting problem.
Frank's frustration: the icosahedron that will not tile
To see why different atom sizes lead somewhere so intricate, start with a puzzle spotted by Charles Frank in 1952. Take one atom and pack twelve neighbours around it as tightly as possible. The FCC answer arranges those twelve into a cuboctahedron; but the tightest huddle of all is an icosahedron — a twenty-faced ball with the neighbours sitting a hair closer to the centre, a genuinely lower-energy cluster. So a liquid metal, cooling, would love every atom to sit at the heart of an icosahedron. There is just one catastrophe: the icosahedron has 5-fold symmetry axes, and — as the crystallographic restriction told you two rungs ago — no periodic lattice can have 5-fold symmetry. The locally best packing simply cannot tile space.
The same clash shows up in the tetrahedron, the densest way to pack four spheres. You would love to fill a solid entirely with regular tetrahedra, but they refuse: the angle along a tetrahedron's edge is arccos(1/3) = 70.53 degrees, and five of them crammed around one edge span 5 times 70.53 = 352.6 degrees, leaving a 7.4-degree gap that will not close. (Twenty tetrahedra sharing a corner make an icosahedron — with exactly those gaps built in.) This is geometric frustration: the local rule the atoms want to obey and the global demand of filling space are flatly incompatible. A quasicrystal resolves it by giving up periodicity. A Frank-Kasper phase resolves it the other way — by keeping periodicity and paying with distortion and complexity.
COORDINATION SHELLS ALLOWED IN A FRANK-KASPER PHASE
(all faces triangular; faces = 2*CN - 4, by Euler)
CN 12 icosahedron 20 triangular faces <- smaller atom
CN 14 (Z14 polyhedron) 24 triangular faces
CN 15 (Z15 polyhedron) 26 triangular faces
CN 16 Friauf polyhedron 28 triangular faces <- larger atom
* ONLY these four ever appear -- never CN 13
* space is carved into (slightly squashed) tetrahedra:
"tetrahedrally close packed"
* average CN sits between 12 and ~13.5 -- a compromise
the pure-12 packing of FCC / HCP can never reachFrank-Kasper phases and giant unit cells
A Frank-Kasper phase is a crystal built entirely from those four coordination shells — 'tetrahedrally close packed', a touch less rigidly stacked than FCC yet often just as dense, because two atom sizes together fill space more cleverly than one size can. Because it needs a big atom for the high-coordination sites and a small one for the icosahedra, it lives in alloys of unlike atoms. You have already met its relatives without knowing it: the Laves phases (MgCu2, MgZn2), whose ideal size ratio is exactly sqrt(3/2) = 1.225; the brittle sigma phase, 30 atoms in a tetragonal cell, the villain that embrittles stainless steels when chromium and iron segregate the wrong way; and the A15 phase of Nb3Sn and V3Si — the workhorse superconductors coiled inside MRI magnets. Ordinary metallurgy is riddled with Frank-Kasper structures.
Push the size mismatch and the chemistry further and the unit cell balloons. In compounds like NaCd2 a single unit cell holds more than a thousand atoms — around 1150, packed into a cube some 30.6 angstrom on a side — an ordered arrangement so large it took decades to solve. These giant-celled crystals build themselves from nested atomic clusters: an icosahedron wrapped in a larger shell wrapped in a larger shell still, a clean example of structural hierarchy, structure organised at several sizes at once. And here is the lovely twist that ties the whole rung together: many of these complex intermetallics are approximants of quasicrystals. They pack from the very same icosahedral clusters a quasicrystal arranges aperiodically — only stacked in a (very large) repeating cell instead. The giant crystal is periodicity doing its level best to imitate a forbidden order.
How would diffraction betray a giant-celled crystal? A huge real-space cell means a tiny reciprocal-cell spacing, so its reciprocal lattice is dense and its powder pattern a thicket of hundreds of closely spaced sharp peaks — the very opposite of the handful of peaks a simple metal gives. Two honest notes. First, 'tetrahedrally close packed' does not mean these phases beat 0.74 in every case; the tetrahedra are slightly distorted (that 7.4-degree gap again), so the packing is efficient but bought with strain. Second, this complexity is rarely welcome in service: the sigma phase in particular is hard and brittle, and metallurgists spend real effort keeping it from forming in stainless steels and nickel superalloys.
High-entropy alloys: complexity hidden in a simple cell
Now swing to the opposite response. A conventional alloy is one host metal — iron in steel, aluminium in an airframe — lightly seasoned with a few percent of other elements. A high-entropy alloy throws out that hierarchy: it mixes five or more elements in near-equal amounts, each between about 5 and 35 atomic percent, with no single 'base'. The name comes from the configurational entropy of such a random mix. For an equiatomic n-component solid solution the mixing entropy is R times ln n; for n = 5 that is R times 1.61 = 13.4 joules per mole per kelvin — far more than a dilute alloy carries. In the free energy G = H minus T times S, a large positive S lowers G, and the founding idea was that this entropy could stabilise a simple random solid solution instead of the tangle of intermetallic compounds you might otherwise expect.
The structural punchline is startling. Take the original Cantor alloy, equal parts cobalt, chromium, iron, manganese and nickel, and it crystallises as a single, plain FCC metal; the refractory alloy NbMoTaW does the same as BCC. Five elements, one simple lattice — and every lattice site is occupied at random by whichever atom turns up. It is a substitutional solid solution pushed to its very limit: the lattice is as simple and periodic as pure copper, but the motif decorating it is a random cocktail. This is the exact mirror of the Frank-Kasper answer. Faced with chemical complexity, one family builds an enormous ordered cell; the other keeps the tiniest cell it can and hides all the complexity in the randomness of who sits where.
- Count the principal elements and take them equiatomic: the Cantor alloy has n = 5, so each element's atomic fraction is x = 1/5 = 0.2.
- The configurational entropy of an ideal random mix is S = minus R times the sum, over all the elements, of x times ln x.
- With five equal fractions this collapses to S = R times ln n = R times ln 5 = R times 1.61.
- Put in R = 8.31 joules per mole per kelvin and you get S = 13.4 joules per mole per kelvin, about 1.6R — the 'high entropy' the family is named after.
- At 1300 kelvin the entropy's pull on the free energy is T times S = about 17 kilojoules per mole — enough to compete with, though not always to beat, the ordering energy of an intermetallic compound. That 'not always' is the whole catch of the next paragraph.
Two frontiers, one lesson
Line the whole rung up and a single map appears. Guide 1's quasicrystals abandon periodicity outright; guide 2's incommensurate and modulated structures keep a periodic backbone but let a second wave drift out of step with it; the Frank-Kasper phases and complex intermetallics of this guide keep strict periodicity yet blow the unit cell up to a thousand atoms; and high-entropy alloys keep the smallest of cells but randomise its contents. Four ways to be more than a simple crystal, and each one pays back in properties: the sigma phase embrittles, Laves phases strengthen, the A15 phase superconducts, and high-entropy alloys can be strong, ductile and corrosion-resistant at once — a direct reminder that the structure-property link runs all the way out to the frontier.
Both frontiers also break the old way of doing structure science, and that sets up the final guide. You cannot solve a 1150-atom cell by hand, and you certainly cannot try every five-element mixture by experiment — five elements chosen from sixty, at a dozen compositions each, is already millions of alloys. So the field leans ever harder on computation: crystal-structure prediction from first principles, to guess what will form before you melt anything, and materials informatics — machine learning turned loose on huge structure databases, the 'Materials Genome' — to steer the search. And because the real structure of a high-entropy alloy is the LOCAL one, hidden under the tidy average, the frontier increasingly means watching structure as it forms and works, operando, and reading the disorder directly with pair-distribution-function analysis. That is exactly where guide 5 goes next.