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“Golden ratio” hints at hidden atomic symmetry

Jan. 7, 2010
Courtesy Helmholtz Association 
of German Research Centres
and World Science staff

A hith­er­to un­dis­cov­ered sym­me­try can be found in sol­id mat­ter at very small scales, phys­i­cists are re­port­ing. The sym­me­try, they say, in­volves the gold­en ra­tio fa­mous from art and ar­chi­tec­ture.

Par­t­i­cles at the atom­ic, or quan­tum, scale be­have in un­ex­pected and seem­ingly irra­t­ional ways. New prop­er­ties emerge that stem from an ef­fect known as Hei­sen­berg’s Un­cer­tain­ty Prin­ci­ple. 

Di­a­gram of the neu­tron scat­ter­ing tech­nique used in the ex­per­i­ments. The ar­rows rep­re­sent the spin ax­es of par­t­i­cles in the linked chain of atoms. A mag­net­ic field is used to tune the chains of spins to a "quantum critical" state. The res­o­nant modes (“notes”) are de­tected by scat­ter­ing neu­trons, which are beamed at the set­up in the di­rec­tion shown. While pass­ing through the sys­tem, these scat­ter with the char­ac­ter­is­tic fre­quen­cies of the spin chains. (Cred­it: Ten­nan­t/HZB)


The re­search­ers in the new study fo­cused on the mag­net­ic ma­te­ri­al co­balt nio­bate. It con­sists of linked mag­net­ic at­oms, which form chains like a very thin ba­r mag­net, but only one at­om wide. They are con­sid­ered a use­ful mod­el for de­scrib­ing mag­netism at ti­ny scales in sol­id state mat­ter. 

When a mag­net­ic field is ap­plied to the chain at right an­gles to an aligned “spin” of its par­t­i­cles, the mag­net­ic chain trans­forms in­to a new state called quan­tum crit­i­cal, ac­cord­ing to the phys­i­cists. This can be thought of as a quan­tum ver­sion of a frac­tal pat­tern, a pat­tern that looks the same at any scale. 

Such a state is al­so a type of “quan­tum un­cer­tain” or “Schrödinger cat” state, in which cer­tain prop­er­ties of the par­t­i­cles are in­de­ter­mi­nate, said Al­an Ten­nant of the Helm­holtz Cen­ter Ber­lin for Ma­te­ri­als and En­er­gy, one of the re­search­ers. 

By tun­ing the sys­tem the re­search­ers found that the chain of at­oms acts like a gui­tar string whose ten­sion comes from in­ter­ac­tion be­tween the spins of the con­stit­u­ent par­t­i­cles. “For these in­ter­ac­tions we found a se­ries,” or “scale,” of “res­o­nant notes,” said Radu Coldea of Ox­ford Uni­vers­ity, who led the re­search.

“The first two notes show a per­fect rela­t­ion­ship with each oth­er,” added Col­dea, prin­ci­ple au­thor of a pa­per on the find­ings to ap­pear in the Jan. 8 is­sue of the re­search jour­nal Sci­ence.

The “pitch” of these notes, or their fre­quen­cies of vibra­t­ion, are in a ra­tio of about 1.618, the same “the gold­en ra­tio fa­mous from art and ar­chi­tec­ture,” he con­tin­ued. If two num­bers are re­lat­ed by the gold­en ra­tio, their sum is al­so re­lat­ed to the larg­er of them by the gold­en ra­tio. In oth­er words, if A di­vid­ed by B is that spe­cial num­ber, then A+B di­vid­ed by A is the same num­ber. 

Artists and ar­chi­tects have have used the gold­en ra­tio for cen­turies—for ex­am­ple, rectan­gles 1.618 times high­er than they are wide—be­cause it sup­posedly pro­vides es­thet­ic­ally pleas­ing forms. The gold­en ra­tio is irra­t­ional, like pi, mean­ing its dec­i­mals go on for­ev­er.

In the “quan­tum un­cer­tain” state of mat­ter, the ra­tio “re­flects a beau­ti­ful prop­er­ty of the quan­tum sys­tem – a hid­den sym­me­try,” Col­dea said. It is “ac­tually quite a spe­cial one called E8 by math­e­mati­cians, and this is its first ob­serva­t­ion in a ma­te­ri­al.” The find­ings dra­mat­ic­ally il­lus­trate how math­e­mat­i­cal the­o­ries de­vel­oped for par­t­i­cle phys­ics may find ap­plica­t­ion in sci­ence at the nano­scale—the scale of a few at­oms—and ul­ti­mately in fu­ture tech­nol­o­gy, he added.

“Such dis­cov­er­ies are lead­ing phys­i­cists to spec­u­late that the quan­tum, atom­ic- scale world may have its own un­der­ly­ing or­der,” said Ten­nant. “Si­m­i­lar sur­prises may await re­search­ers in oth­er ma­te­ri­als in the quan­tum crit­i­cal state.” 

The re­search­ers reached their re­sults by us­ing a spe­cial probe called neu­tron scat­ter­ing that al­lows phys­i­cists to see the atom­ic-scale vibra­t­ions of a sys­tem. “Us­ing neu­tron scat­ter­ing gives us un­ri­valled in­sight in­to how dif­fer­ent the quan­tum world can be from the every­day,” said re­searcher Elisa Wheel­er, who has worked at both Ox­ford and the Ber­lin cen­ter on the proj­ect.


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A hitherto undiscovered symmetry can be found in solid matter at infinitesimally small scales, physicists are reporting. The symmetry, they say, involves the golden ratio famous from art and architecture. Particles at the atomic, or quantum, scale behave in unexpected and seemingly irrational ways. New properties emerge which are the result of an effect known as the Heisenberg’s Uncertainty Principle. The researchers in the new study focused on the magnetic material cobalt niobate. It consists of linked magnetic atoms, which form chains like a very thin bar magnet, but only one atom wide. They are considered a useful model for describing magnetism at tiny scales in solid state matter. When a magnetic field is applied to the chain at right angles to an aligned “spin” of its particles, the magnetic chain transforms into a new state called quantum critical, according to the physicists. This can be thought of as a quantum version of a fractal pattern, a pattern that looks the same at any scale. Such a state is also called a “quantum uncertain” or “Schrödinger cat” state, in which certain properties of the particles are indeterminate, said Alan Tennant of the Helmholtz Center Berlin for Materials and Energy, one of the researchers. By tuning the system the researchers found that the chain of atoms acts like a guitar string whose tension comes from interaction between the spins of the constituent particles. “For these interactions we found a series,” or “scale,” of “resonant notes,” said Radu Coldea of Oxford University, who led the research. “The first two notes show a perfect relationship with each other,” added Coldea, principle author of a paper on the findings to appear in the Jan. 8 issue of the research journal Science. The “pitch” of these notes, or their frequencies of vibration, are in a ratio of about 1.618, the same “the golden ratio famous from art and architecture,” he continued. If two numbers are related by the golden ratio, their sum is also related to the larger of them by the golden ratio. In other words, if a divided by b is that special number, then a+b divided by a is the same number. Artists and architects have have used the golden ratio for centuries—for example, rectangles 1.618 times higher than they are wide—because it supposedly provides esthetically pleasing forms. The golden ratio is irrational, like pi, meaning its decimals go on forever. In the “quantum uncertain” state of matter, the ratio “reflects a beautiful property of the quantum system – a hidden symmetry,” Coldea said. It is “actually quite a special one called E8 by mathematicians, and this is its first observation in a material.” The findings dramatically illustrate how mathematical theories developed for particle physics may find application in science at the nanoscale—the scale of a few atoms—and ultimately in future technology, he added. “Such discoveries are leading physicists to speculate that the quantum, atomic scale world may have its own underlying order,” said Tennant. “Similar surprises may await researchers in other materials in the quantum critical state.” The researchers reached their results by using a special probe called neutron scattering that allows physicists to see the atomic-scale vibrations of a system. “Using neutron scattering gives us unrivalled insight into how different the quantum world can be from the every day,” said researcher Elisa Wheeler, who has worked at both Oxford and the Berlin center on the project.