Tag Archives: mathematics

Physicists solve quantum tunneling mystery

An international team of scientists studying ultrafast physics have solved a mystery of quantum mechanics, and found that quantum tunneling is an instantaneous process.

The new theory could lead to faster and smaller electronic components, for which quantum tunneling is a significant factor. It will also lead to a better understanding of diverse areas such as electron microscopy, nuclear fusion and DNA mutations.

“Timescales this short have never been explored before. It’s an entirely new world,” said one of the international team, Professor Anatoli Kheifets, from The Australian National University (ANU).

“We have modelled the most delicate processes of nature very accurately.”

At very small scales quantum physics shows that particles such as electrons have wave-like properties – their exact position is not well defined. This means they can occasionally sneak through apparently impenetrable barriers, a phenomenon called quantum tunneling.

Quantum tunneling plays a role in a number of phenomena, such as nuclear fusion in the sun, scanning tunneling microscopy, and flash memory for computers. However, the leakage of particles also limits the miniaturisation of electronic components.

Professor Kheifets and Dr. Igor Ivanov, from the ANU Research School of Physics and Engineering, are members of a team which studied ultrafast experiments at the attosecond scale (10-18 seconds), a field that has developed in the last 15 years.

Until their work, a number of attosecond phenomena could not be adequately explained, such as the time delay when a photon ionised an atom.

“At that timescale the time an electron takes to quantum tunnel out of an atom was thought to be significant. But the mathematics says the time during tunneling is imaginary – a complex number – which we realised meant it must be an instantaneous process,” said Professor Kheifets.

“A very interesting paradox arises, because electron velocity during tunneling may become greater than the speed of light. However, this does not contradict the special theory of relativity, as the tunneling velocity is also imaginary” said Dr Ivanov, who recently took up a position at the Center for Relativistic Laser Science in Korea.

The team’s calculations, which were made using the Raijin supercomputer, revealed that the delay in photoionisation originates not from quantum tunneling but from the electric field of the nucleus attracting the escaping electron.

The results give an accurate calibration for future attosecond-scale research, said Professor Kheifets.

“It’s a good reference point for future experiments, such as studying proteins unfolding, or speeding up electrons in microchips,” he said.

The research is published in Nature Physics.

Source: ANU

Wrinkle predictions:New mathematical theory may explain patterns in fingerprints, raisins, and microlenses.

By Jennifer Chu


CAMBRIDGE, Mass. – As a grape slowly dries and shrivels, its surface creases, ultimately taking on the wrinkled form of a raisin. Similar patterns can be found on the surfaces of other dried materials, as well as in human fingerprints. While these patterns have long been observed in nature, and more recently in experiments, scientists have not been able to come up with a way to predict how such patterns arise in curved systems, such as microlenses.

Now a team of MIT mathematicians and engineers has developed a mathematical theory, confirmed through experiments, that predicts how wrinkles on curved surfaces take shape. From their calculations, they determined that one main parameter — curvature — rules the type of pattern that forms: The more curved a surface is, the more its surface patterns resemble a crystal-like lattice.

The researchers say the theory, reported this week in the journal Nature Materials, may help to generally explain how fingerprints and wrinkles form.

“If you look at skin, there’s a harder layer of tissue, and underneath is a softer layer, and you see these wrinkling patterns that make fingerprints,” says Jörn Dunkel, an assistant professor of mathematics at MIT. “Could you, in principle, predict these patterns? It’s a complicated system, but there seems to be something generic going on, because you see very similar patterns over a huge range of scales.”

The group sought to develop a general theory to describe how wrinkles on curved objects form — a goal that was initially inspired by observations made by Dunkel’s collaborator, Pedro Reis, the Gilbert W. Winslow Career Development Associate Professor in Civil Engineering.

In past experiments, Reis manufactured ping pong-sized balls of polymer in order to investigate how their surface patterns may affect a sphere’s drag, or resistance to air. Reis observed a characteristic transition of surface patterns as air was slowly sucked out: As the sphere’s surface became compressed, it began to dimple, forming a pattern of regular hexagons before giving way to a more convoluted, labyrinthine configuration, similar to fingerprints.

“Existing theories could not explain why we were seeing these completely different patterns,” Reis says.

Denis Terwagne, a former postdoc in Reis’ group, mentioned this conundrum in a Department of Mathematics seminar attended by Dunkel and postdoc Norbert Stoop. The mathematicians took up the challenge, and soon contacted Reis to collaborate.

Ahead of the curve

Reis shared data from his past experiments, which Dunkel and Stoop used to formulate a generalized mathematical theory. According to Dunkel, there exists a mathematical framework for describing wrinkling, in the form of elasticity theory — a complex set of equations one could apply to Reis’ experiments to predict the resulting shapes in computer simulations. However, these equations are far too complicated to pinpoint exactly when certain patterns start to morph, let alone what causes such morphing.

Combining ideas from fluid mechanics with elasticity theory, Dunkel and Stoop derived a simplified equation that accurately predicts the wrinkling patterns found by Reis and his group.

“What type of stretching and bending is going on, and how the substrate underneath influences the pattern — all these different effects are combined in coefficients so you now have an analytically tractable equation that predicts how the patterns evolve, depending on the forces that act on that surface,” Dunkel explains.

In computer simulations, the researchers confirmed that their equation was indeed able to reproduce correctly the surface patterns observed in experiments. They were therefore also able to identify the main parameters that govern surface patterning.

As it turns out, curvature is one major determinant of whether a wrinkling surface becomes covered in hexagons or a more labyrinthine pattern: The more curved an object, the more regular its wrinkled surface. The thickness of an object’s shell also plays a role: If the outer layer is very thin compared to its curvature, an object’s surface will likely be convoluted, similar to a fingerprint. If the shell is a bit thicker, the surface will form a more hexagonal pattern.

Dunkel says the group’s theory, although based primarily on Reis’ work with spheres, may also apply to more complex objects. He and Stoop, together with postdoc Romain Lagrange, have used their equation to predict the morphing patterns in a donut-shaped object, which they have now challenged Reis to reproduce experimentally. If these predictions can be confirmed in future experiments, Reis says the new theory will serve as a design tool for scientists to engineer complex objects with morphable surfaces.

“This theory allows us to go and look at shapes other than spheres,” Reis says. “If you want to make a more complicated object wrinkle — say, a Pringle-shaped area with multiple curvatures — would the same equation still apply? Now we’re developing experiments to check their theory.”

This research was funded in part by the National Science Foundation, the Swiss National Science Foundation, and the MIT Solomon Buchsbaum Fund.

Source: MIT News Office

Hands-on kirigami: With a cut and a few folds, this structure could serve as a shelter or a microfluidic channel. Credit : Penn News

Penn Research Outlines Basic Rules for Construction With a Type of Origami

Origami is capable of turning a simple sheet of paper into a pretty paper crane, but the principles behind the paper-folding art can also be applied to making a microfluidic device for a blood test, or for storing a satellite’s solar panel in a rocket’s cargo bay.

A team of University of Pennsylvania researchers is turning kirigami, a related art form that allows the paper to be cut, into a technique that can be applied equally to structures on those vastly divergent length scales.

Hands-on kirigami: With a cut and a few folds, this structure could serve as a shelter or a microfluidic channel. Credit : Penn News
Hands-on kirigami: With a cut and a few folds, this structure could serve as a shelter or a microfluidic channel. Credit : Penn News

In a new study, the researchers lay out the rules for folding and cutting a hexagonal lattice into a wide variety of useful three-dimensional shapes. Because these rules ensure the proportions of the hexagons remain intact after the cuts and folds are made, the rules apply to starting materials of any size. This enables materials to be selected based on their relevance to the ultimate application, whether it is in nanotechnology, architecture or aerospace.

The study was conducted by Toen Castle, a postdoctoral researcher in the School of Arts & Science’s Department of Physics and Astronomy; Shu Yang, a professor in the School of Engineering and Applied Science’s Department of Materials Science and Engineering; and professor Randall Kamien, also of the Department of Physics and Astronomy. Also contributing to the study were undergraduate Xingting Gong and postdoctoral researcher Daniel Sussman, members of Kamien’s research group; graduate student Euiyeon Jung, a member of Yang’s group; and postdoctoral researcher Yigil Cho, who works in both groups.

It was published in the journal Physical Review Letters.

“If you see a fancy piece of origami,” Kamien said, “it can have arbitrarily small folds. We want to make something much simpler. If there are standards for the size of folds and cuts, we can make the math apply to any length scale. We can make channels, gates, steps and other 3-D shapes without needing to know anything about the size of the sheet and then combine those building blocks into even more complex shapes.”

A hexagonal lattice may seem like an odd choice for a starting point, but the pattern has advantages over a seemingly simpler tessellation, such as one made from squares.

“The connected centers of the hexagons make triangles,” Castle said, “so, if you start with a hexagonal lattice, you get the triangles for free. It’s like two lattices in one, whereas if you start with squares, you only get squares.”

“Plus,” Yang said, “it’s easier to fill a space with a hexagonal lattice and move from 2-D to 3-D. That’s why you see it in nature, in things like honeycombs.”

Starting from a flat hexagonal grid on a sheet of paper, the researchers outlined the fundamental cuts and folds that allow the resulting shape to keep the same proportions of the initial lattice, even if some of the material is removed. This is a critical quality for making the transition from paper to materials that might be used in real-world applications.

“You can think of the sheet of paper as a template for a mesh of rods that you can lay on top of it,” Castle said. “Alternatively, you can think of the paper as the membrane that attaches to a scaffolding. Both concepts are in the theory from the start; it’s just a question of whether you want to build the rods or the material between them.”

Having a set of rules that draws on fundamental mathematical principles means the kirigami approach can be applied equally across length scales, and with almost any material.

“The rules we lay out,” Kamien said, “tell you how you make the cuts so you only have to fold on straight lines, and so that, when you fold them together, the rods remain the same length and the centers remain the same distance apart. You may have to bend [or put hinges on] some of the rods to make the folds, but you don’t have to be able to stretch them. That also means the whole structure remains rigid when you’re done folding.”

This means it’s just a matter of picking the materials with the properties you want for your application,” Yang said. “We can go from nanoscale materials like graphene to materials you would make clothing out of to materials you would see in a space station or satellite.”

The rules also guarantee that “modules,” basic shapes like channels that can direct the flow of fluids, can be combined into more complex ones. For example, iterating those folds and cuts can produce a ratcheting interface that can lock itself into place at different points. This structural feature could change the volume of a channel or even serve as an actuator for a robot.

Kirigami is particularly attractive for nanoscale applications, where the simplest, most space-efficient shapes are necessary, and self-folding materials would circumvent some of the fabrication challenges inherent in working at such small scales.

The research was supported by the National Science Foundation through its ODISSEI program, the American Philosophical Society and the Simons Foundation.

Source: Penn News