PHYSICIST Al Sievers is playing with a simple chain of pendulums. In his laboratory at Cornell University in Ithaca, New York, he kicks the chain into action, and then watches a mystifying performance. Nudge one part of a system like this and the energy should gradually spread out, like ripples on a pond. However, Sievers’s pendulums do nothing of the sort. After things settle down, a few pendulums are still swinging away like mad, while all the rest have fallen eerily quiet.
“Energy should spread out like ripples on a pond. Sievers’s pendulums do nothing of the sort”
OK, so Sievers’s pendulums are really more like tuning-fork prongs, etched onto the surface of a semiconductor wafer and only 50 micrometres across, but they mimic the workings of a huge range of systems made from parts that can vibrate or oscillate together. That’s why these experiments have physicists scratching their heads. A few years ago, most physicists would have said that such behaviour was impossible. Now they are wondering if some of their fundamental theories need rethinking.
Advertisement
Time-honoured theories of physics insist that energy in a “spatially periodic” system – anything from a line of pendulums to the atoms in a solid crystal – should naturally spread out with time. You can inject it in one place, but don’t expect it to stay there. “The typical physicist thinks localised excitations in any perfectly periodic system should be impossible,” says Sergej Flach of the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany.
The weird hotspots that Sievers sees with his pendulums have also turned up in substances ranging from helium to uranium. These strange, compact forms – sometimes known as discrete breathers for their localised fluctuating character – may just open a new chapter in the physics of solid matter.
For it seems increasingly likely that breathers contribute to almost everything that goes on in solids – from the flow of heat or electricity to the way materials expand or contract as they change temperature. Researchers even believe breathers could make some materials conduct heat differently in opposing directions – something long thought to be impossible – or might lie behind the unexplained failure of jet engine rotor blades, a mystery for 35 years. The more researchers learn, the more they suspect these exotic forms have been with us all the time.
The obvious question is why these entities have gone unnoticed for so long. The answer, at least in part, is that until fairly recently physicists had strong reasons for believing that nothing like a breather could exist. The great figures of physics more or less ruled them out decades ago while developing the basic theory of crystalline solids.
If you disturb some of the atoms or molecules in a crystalline solid, they’ll vibrate back and forth about their resting positions. As long as the perturbation is small compared with the atoms’ mutual separation, the resulting motion is quite simple, and the mathematics that describes it is called “linear”, which amounts to saying straightforward and predictable. Among other things, this mathematics shows that energy injected into a crystal – or indeed any spatially periodic system – should take the form of waves spread out in space as the atoms (or pendulums) wobble back and forth in an organised way. According to the standard theory, you will always find this behaviour if you jiggle the system weakly.
Stronger disturbances that push the atoms further from their resting points should lead to complex non-linear motion, which is much harder to calculate. However, physicists in the early 20th century argued in forceful terms that such non-linear motion shouldn’t last. Pour a lot of energy into only a handful of atoms and you’ll certainly get vigorous non-linear motion and complicated oscillations. But as the energy spreads out, non-linear waves should gradually dissolve into the more familiar waves described by linear motion.
This immensely successful perspective has been the basis of the physics of solids for nearly a century. It explains quite accurately how vibrations in a crystal lattice – known as phonons in quantum terms – impede the flow of electricity, plus hundreds of other basic facts about solids.
Physicists have long recognised that non-linear motion can be important in some special cases, such as in crystal lattices with “defects” – dislocations or other flaws due to missing or additional atoms. But in a perfect lattice with no such disordering influences, linear theory and its extended waves seemed to be enough.
In 1988, however, Sievers working with Japanese physicist Shozo Takeno of the Kyoto Institute of Technology, stumbled across something unexpected. They were trying to explain experiments that had measured vibrations around a crystal defect at a frequency different from anything predicted by conventional theory. These vibration were neither extended waves nor localised non-linear oscillations linked to defects. Having exhausted all the ideas they could think of, Sievers and Takeno entertained a more radical possibility. “We eventually began to wonder,” Sievers recalls, “if these vibrations were linked not to any defect, but to something going on in the perfect lattice itself.” Pretty soon he and Takeno had spotted an unexpected loophole – a way that non-linear energy might evade the laws laid down by conventional thinking.
The standard theory says non-linear vibrations decay by transferring their energy to linear waves with frequencies that are multiples or “harmonics” of the oscillating frequency of the non-linear wave. A non-linear wave with frequency f, for example, will decay by giving up its energy to linear waves with frequencies 2f, 3f and so on. Sievers and Takeno realised, however, that in some circumstances this might be impossible.
That’s because linear theory says that a periodic system can support waves only within a band of allowed frequencies. This range depends on the spacing between atoms and the forces that hold them in place. In principle, however, a non-linear wave might have a frequency below the allowed band, while its harmonics would all lie above it. “Once created,” says Sievers, “such a wave wouldn’t be on speaking terms with the rest of the lattice.” Unable to lose energy by normal means, the wave should persist – forming a breather.
Within two years, researchers had found hints of strange waves like this in computer simulations, yet most physicists still suspected Sievers and Takeno’s breathers were mere mathematical quirks with no relevance to the real world. Then in 1994 the idea received a boost when physicist Serge Aubry of the French Atomic Energy Commission in Saclay teamed up with mathematician Robert MacKay of the University of Warwick, UK. Using simplified one-dimensional models of a crystal, they proved that stable, localised non-linear waves of the kind described by Sievers and Takeno should probably be common and relatively easy to stir up. Even so, many physicists still remained sceptical.
“All through the 1990s,” Sievers recalls, “referees [on journals] were saying to me that 1D models are special. Show us this in 3D and we might believe.” Since then, he and others have done just that in numerous experiments with real crystals and artificial systems designed to bring breathers into the open.
In the late 1990s, Flach working with Peter Binder of the University of Erlangen in Germany and others, detected breathers in networks of superconducting devices. But it is perhaps Sievers’s micromechanical “pendulums” that demonstrate most clearly how breathers live and work. The pendulums are actually flexible strips of silicon nitride overhanging a microscale cliff edge (see Diagram). Blasting these cantilevers with high-frequency sound waves sets them oscillating back and forth. And because the bending of one exerts small forces on its neighbours, they act like a chain of pendulums or vibrating atoms with forces acting between them.
Good vibrations
In recent experiments, Sievers and Masayuki Sato, working with their Cornell colleague Bruce Hubbard, first pumped a large amount of energy into the system, producing a glut of energy spread uniformly across the array. Monitoring the cantilevers’ motion with lasers, they found this uniform energy decayed spontaneously into a small number of breathers.
Meanwhile a rapidly growing number of experiments point to similar apparitions in ordinary materials. Four years ago, Tuvy Markovich and colleagues at Technion, the Israel Institute for Technology in Haifa, used neutron scattering to probe the vibrational motion of atoms within the crystal lattice of solid helium. Conventional theory predicted three different classes of vibrational waves, and there was clear evidence for all three. But they also found breathers carrying energy at frequencies ruled out by linear theory.
Other experiments have found evidence for breathers in ultra-cold atomic gases and in optical materials. Last year, physicist Michael Manley of Lawrence Livermore National Laboratory in California found similar excitations in an ordinary chunk of uranium when he heated it to above about 175 °C. X-ray and neutron scattering experiments indicated that lots of vibrational energy was confined to a few neighbouring atoms. “These sort of things should exist all over the place,” says Manley. “It’s just really hard to see them.”
At the moment, breathers in real materials seem to lie close to the threshold of detection. Yet Sievers, Manley and other researchers fully expect that similarly exotic creatures exist in almost all crystals; spotting them is only a matter of finding the right conditions. This would not be so surprising, for as physicist Donald Campbell of Boston University points out, non-linearity is the norm in nature, despite physicists’ historical focus on linear physics. “The Polish mathematician Stanislaus Ulam used to remark that talking about non-linear science is like calling the bulk of zoology the study of non-elephants,” says Campbell.
Breathers may be the most recent non-linear animal to be captured and scrutinised, but their influence may be very important indeed. They could bring some surprises even to the mundane physics of what goes on in ordinary solids. Several years ago, for example, theorists discovered that breathers may interfere with the fundamental process of heat conduction. Standard theory predicts that when a warm object is in a cold environment the temperature difference between the two should shrink exponentially with time, causing the object to cool rapidly.
However, breathers can get in the way. Aubry and colleagues created a computer simulation of a cooling process, using a simple model of interacting atoms for the initially warm object. After giving these atoms a good dose of energy, to put the material at some relatively high temperature, they then plunged it into a low-temperature environment. If the interactions between the atoms were all linear, they found, then the material did indeed cool rapidly, as theory predicts.
Including non-linear interactions changed everything. Aubry’s team now found the initial burst of energy stirred up breathers that locked energy in place and greedily held onto it. The energy in these breathers did slowly leak away, but their influence meant that the material’s temperature dropped far more slowly than physicists would ordinarily expect. Breathers cannot foil heat flow, but they can certainly slow it down.
So far, no one is sure whether the same thing happens in real materials, not least because Aubry’s model is based on a one-dimensional line of atoms. But today physicists today routinely build systems that are effectively one-dimensional, such as carbon nanotubes, and Michel Peyrard of the Ecole Normale Supérieure in Lyon, France, thinks that breathers will be important and could be put to good use. Peyrard has already shown, for example, that it should be possible to use breathers to make thermal rectifiers – devices that let heat flow more easily in one direction than in the other. This should occur quite naturally, Peyrard argues, if you can make a material whose thermal conductivity – the ease with which heat flows through it – varies with both position and temperature.
Variation with position is achieved simply by altering the material’s composition. Variation with temperature is more difficult to engineer, but it should emerge whenever non-linearity becomes important. Increasing thermal energy should drive vibrations out of the linear range to create breathers that alter how easily heat flows. By engineering such variations, Peyrard has shown, you could make heat flow 10 times more easily in one direction than its opposite.
Related theoretical work by Italian physicist Giulio Casati at the University of Insubria in Como, Italy, suggests that this factor could be increased to 100. “I am currently looking for a laboratory that can test this idea,” he says. The ultimate result of such work would be devices that control the flow of heat in much the same way as today’s electronics devices control that of electricity.
This is not the only area in which breathers might have important practical ramifications. For more than three decades, aeronautical engineers have been puzzled by how often the fan blades that compress air in jet engines fail abruptly. It is clear that vibrations cause the damage and that the vibrations are often strangely focused in small areas of the blade, concentrating damage there. Engineers hunting for answers typically find local defects or flaws in the blade material that might focus the energy. But not always – sometimes there is no apparent cause.
Sievers suggests that breathers may provide a natural explanation for this notorious puzzle. “From our work,” he says, “it’s clear that even perfect blades with no flaws might give rise to large localised vibrations.” As in his system of tiny pendulums, non-linearity can focus the energy all by itself, without the help of any imperfections. If this is the case, then searching for imperfections might be a wild goose chase. Managing blade failures will instead require finding some way to control the formation of breathers, and perhaps learning how to drive them out of the system.
Although the idea is speculative, it is a possibility that engineers won’t rule out. “Most of the failures we see can be readily attributed to geometrical or chemical weaknesses in the metal microstructure,” says engineer Peter Spittle of jet-engine manufacturer Rolls-Royce in the UK. “But this is an interesting idea that might conceivably come into play in some cases.”
The most exciting possibility, everyone agrees, is that breathers turn out to be more important than anyone would have guessed, and even ubiquitous in solids. “I think in the end they’ll turn out to be widespread,” says Sievers. It will just take better equipment and more imagination to find them.
