«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.

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.




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)