Screwy symmetry revealed

Math trick reverses spirals, providing new ways to explore materials

Physicists have put a new twist on the humble corkscrew. Just as a butterfly appears identical to its mirror image, objects made of structures that tilt, twist or spiral possess a symmetry now recognized for the first time.

NEW TWIST Molecular structures that tilt clockwise (orange) or counterclockwise (blue) — as well spirals of either direction — have revealed a new kind of symmetry that could help scientists design new materials. Gopalan Laboratory, Ryan Haislmaier/Penn State Univ.

The new discovery is based on a mathematical operation that transforms a clockwise helix into a counterclockwise one, or vice versa.

“Normally, a helix flips when you put a mirror up to it,” says Venkatraman Gopalan, a materials scientist at Pennsylvania State University in University Park. “We’ve developed a special kind of mirror with this math woven into it.” Seen in this mirror, an object with a spiral shape will look just like itself.

This symmetry joins a list of other, long-known ways to move or manipulate an object and leave it looking the same afterwards. A snowflake has what’s called rotation symmetry: Turn it 60 degrees, and its appearance doesn’t change. A piece of wallpaper with a repeating pattern looks identical when moved a bit to the right or left, demonstrating translational symmetry.

“This new symmetry we’re playing around with has not been taken into account up to now,” says Daniel Litvin, a physicist at Penn State Berks in Reading and coauthor with Gopalan of an April 3 paper in Nature Materials. “It gives you a finer classification of materials.”

Centuries ago, scientists figured out that symmetries limit the number of ways atoms can be arranged into crystals. They calculated 230 possible patterns. Sixty years ago, another kind of symmetry called time reversal was included, extending the number of possible arrangements in magnetic materials to 1,421. With Gopalan and Litvin’s new “rotation-reversal” symmetry in hand, crystallographers now have 17,807 patterns to look for when trying to characterize a material’s 3-D structure.

The patterns aren’t just eye candy. Symmetries dictate the properties of matter, all the way down to atoms, electrons and the smallest building blocks of the universe. A piece of rock salt, for instance, is made of cube-shaped crystals with symmetries that allow light to pass through the material unperturbed. If the symmetries were those of a prism, the light would be split into a rainbow of colors.

“People have looked at certain materials and wondered why they have certain properties,” says Manfred Fiebig, a physicist at the University of Bonn in Germany who was not involved in the research. “Now they may be able to argue that it’s because of this new symmetry.”

Seen with this new symmetry, commonplace, well-studied materials may hold new surprises. Quartz, found in watches and jewelry, is known to twist light clockwise or counterclockwise, depending on whether its crystal structure is of the right-handed or left-handed variety. But Gopalan and Litvin’s approach makes a new testable prediction for this crystal: Because of symmetry, squeezing quartz or exposing it to an electric field should change its interaction with light.

“This is quite an important result,” says Mike Glazer, a crystallographer at the University of Oxford in England who also was not involved in the research. “Understanding symmetry is always an important first step to discovering new materials.”

Glazer hopes to apply the new symmetry to his favorite group of materials: the perovskites. These naturally occurring minerals are made of octahedra stacked such that some bits lean to the right and some to the left. Depending on the structure, some perovskites produce electricity when vibrated. Others change their magnetic properties when a voltage is applied — a useful trick for electronics. A better understanding of these materials’ underlying symmetries could help scientists predict when these properties emerge and find new, technologically useful materials.

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