The Forbidden Symmetry

How Quasicrystals Broke Two Centuries of Crystallography

For nearly two centuries, the foundational law of solid-state physics and mineralogy was the crystallographic restriction theorem, formulated mathematically by René Just Haüy and Auguste Bravais in the nineteenth century. By definition, a crystal was a solid in which atoms pack inside a small three-dimensional box—a unit cell—that repeats identically by periodic translation along three spatial axes to fill space without leaving empty gaps. Because the lattice must repeat periodically at integer multiples of its shortest interatomic spacing $a$, if the crystal also possesses rotational symmetry of order $n$ (meaning rotating the crystal by $\theta = \frac{2\pi}{n}$ leaves the lattice unchanged), then rotating a lattice vector by $+\theta$ and $-\theta$ must yield two new lattice sites whose separation is itself an exact integer multiple of $a$.

That geometric requirement reduces to a single trigonometric constraint:

Because $\cos\theta$ lies between $-1$ and $+1$, the quantity $2\cos\left(\frac{2\pi}{n}\right)$ can take only five integer values: $-2, -1, 0, 1,$ or 2. Those five integers correspond to rotations of $180^\circ$ ($n=2$), $120^\circ$ ($n=3$), $90^\circ$ ($n=4$), $60^\circ$ ($n=6$), and $360^\circ$ ($n=1$). Five-fold rotational symmetry ($n=5$, or $\theta = 72^\circ$) and ten-fold symmetry ($n=10$, or $\theta = 36^\circ$) are mathematically forbidden in any periodic lattice, because $2\cos(36^\circ)$ equals the irrational golden ratio $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.61803$, which is not an integer. Just as regular pentagons cannot tile a flat floor without leaving diamond-shaped holes, crystallography textbooks taught that no crystal could ever exhibit five-fold or ten-fold diffraction symmetry.

On the morning of April 8, 1982, at the National Bureau of Standards (now NIST) in Gaithersburg, Maryland, Israeli materials scientist Dan Shechtman—on sabbatical from the Technion in Haifa—sat alone operating a transmission electron microscope. Shechtman was examining a melt-spun ribbon of aluminum alloyed with 14 atomic percent manganese ($\text{Al}_{86}\text{Mn}_{14}$), chilled against a rotating copper wheel at one million degrees Celsius per second.

Inserting metallic foil sample 1725 into the electron beam in Selected Area Electron Diffraction mode, Shechtman looked at the fluorescent screen. On Plate 1725, the Bragg diffraction spots were needle-sharp, proving the atoms were locked in long-range order across micrometers of metal. Yet as Shechtman counted the bright dots arranged in concentric circles around the central beam, he counted ten equally spaced spots separated by exactly 36 degrees, with radial ring spacings scaling by the irrational golden ratio $\phi$. Rotating the specimen revealed six five-fold axes, ten three-fold axes, and fifteen two-fold axes—the exact three-dimensional symmetry of an icosahedron. Counting the dots a second time, Shechtman noted in his microscope logbook beside exposure 1725 a ten-fold pattern with three question marks.

Colleagues in the hallway dismissed the plate as multiple twinning—an artifact in which five or ten ordinary cubic crystals grow fused together like slices of a pie. Using dark-field imaging and a twenty-nanometer micro-beam, Shechtman proved that no twin boundaries existed inside the micron-sized grain. Unconvinced, his NBS division head dropped a copy of the International Tables for Crystallography onto Shechtman's desk, told him to read the textbook, and expelled him from the research group for "bringing disgrace" on the team. When Shechtman and Technion colleague Ilan Blech submitted their paper to the Journal of Applied Physics in summer 1984, editors rejected it without review as uninteresting to physicists, until NBS thermodynamicist John Cahn and French crystallographer Denis Gratias helped publish a condensed report in Physical Review Letters on November 12, 1984.

Five weeks later, physicists Paul Steinhardt and Dov Levine coined the term quasicrystal and explained how the alloy bypassed the crystallographic restriction theorem. Extending British mathematician Roger Penrose's two-tile non-periodic floor tilings into three dimensions using acute and obtuse golden rhombohedra packed in ratios of $\phi = \frac{1 + \sqrt{5}}{2}$, they showed that an atomic lattice can be quasiperiodic—lacking periodic translational repetition while maintaining strict long-range orientational order that generates needle-sharp icosahedral diffraction peaks.

Two-time Nobel laureate Linus Pauling led a decade-long public campaign against the discovery, arguing that Shechtman's alloys were twinned cubic crystals with massive unit cells and declaring before American Chemical Society audiences that there were no quasicrystals, only quasi-scientists. Pauling maintained his opposition until his death in 1994, by which time metallurgists had grown millimeter-sized stable aluminum-copper-iron quasicrystals whose single-crystal X-ray synchrotron patterns refuted twinning, followed in 2009 by the discovery of natural icosahedrite ($\text{Al}_{63}\text{Cu}_{24}\text{Fe}_{13}$) inside the Khatyrka meteorite. In 1992, the International Union of Crystallography formally deleted periodic translation from its definition of a crystal, and in October 2011 Dan Shechtman was awarded the unshared Nobel Prize in Chemistry.