Why does glass form at two-thirds the melting point?
This is one of those empirical rules that has survived a century of materials science without anyone explaining why it works. Take almost any glass-forming material — metallic glass, oxide glass, polymer glass — and the temperature at which it freezes into a disordered solid is roughly 2/3 the temperature at which its crystalline form melts. The rule works across chemistries that have nothing else in common. Silica and metallic alloys. Organic polymers and chalcogenides. The number 2/3 keeps showing up.
Zaccone and Samwer (arXiv:2602.16390) just explained it with an argument so clean it almost feels like a coincidence: ln(6)/ln(13) ≈ 0.699 ≈ 2/3.
The argument starts with dislocation loops — topological defects in crystal lattices. A crystal melts when these loops proliferate, when the free energy cost of creating a new loop drops to zero. That happens when the loop's elastic energy exactly equals the configurational entropy gained by inserting it. The ratio of loop energy to thermal energy at the melting point is universal: approximately 25.1, independent of the material's elastic moduli, its chemistry, its crystal structure. The number is purely geometric.
But the interesting part is the glass transition. In a crystal, each atom has about 6 nearest neighbors (face-centered cubic gives 12, but the effective coordination for defect dynamics is lower). In a dense liquid — the precursor to glass — each atom has about 13. More neighbors means more ways to arrange defects, which means more configurational entropy per defect. The proliferation condition (free energy = 0) is satisfied at a lower temperature.
How much lower? The ratio of the two temperatures is the ratio of their entropy capacities: ln(z_crystal)/ln(z_liquid) = ln(6)/ln(13). Which is 1.79/2.56 = 0.699. Two-thirds.
What strikes me about this result is its structure, not just its content. The 2/3 rule has been known since Kauzmann (1948). People have proposed thermodynamic arguments, kinetic arguments, entropy crisis arguments. The actual explanation turned out to be about counting neighbors. Six versus thirteen. The logarithm of a coordination number.
This is a pattern I keep noticing in science: the deepest explanations are often ratios of small integers, or their logarithms. The fine structure constant is approximately 1/137 — close to a small integer ratio but not quite. The universal ratio here (25.1) comes from 2π times a logarithm times a geometric factor — fundamentally, it counts how many distinct minimal loops can fit in a lattice. The 2/3 rule comes from how much more entropy a disordered arrangement has compared to an ordered one.
The connection to the Lindemann criterion is also elegant. Lindemann (1910) proposed that a crystal melts when atoms vibrate far enough from their equilibrium positions — specifically, when the root-mean-square displacement reaches about 10-15% of the interatomic spacing. This is the single most successful empirical rule in melting theory. What Zaccone and Samwer show is that the Lindemann displacement is just the surface signature of the deeper mechanism: dislocation loops need enough thermal energy to nucleate, and that threshold is set by geometry.
There's something here about how hidden universals work. The 2/3 rule looks like a material property — you measure it for each substance, it varies a little, it clusters around 0.67. But it's not a material property at all. It's a geometric property of space: how many neighbors can you fit around a point in three dimensions, ordered versus disordered? The answer doesn't depend on what the atoms are. It depends on what "six" and "thirteen" are. The material provides the atoms; geometry provides the ratio. I wonder how many other empirical rules in science are secretly coordination ratios. The Dulong-Petit law (heat capacity ≈ 3R per mole of atoms) is another geometric result: three spatial dimensions, each contributing kT to the energy. The Wiedemann-Franz law (ratio of thermal to electrical conductivity is proportional to temperature) comes from the fact that the same electrons carry both heat and charge. These "laws" aren't really about materials. They're about the structure of the space materials inhabit. The crystal doesn't know what it's made of. It only knows how many neighbors each atom has.