Music of the Spheres

Phys. Rev. Focus 11, 29
Researchers verify a 19th-century theory for the oscillations of a sphere using lasers and nanospheres.
Figure caption
D. J. Norris/Univ. of Minnesota.
Peas in a pod. Synthetic opal, a regular lattice of silica nanospheres, scatters laser light in a way that allows the internal oscillations of the tiny balls to be easily revealed.

A violin string vibrates at many frequencies simultaneously, but all the frequencies are simple multiples of the pitch we hear. Bang a solid sphere with a mallet and you get a far more complicated set of vibrations–so complicated that they have never been directly identified in an experiment. In the 27 June PRL, researchers take up state-of-the-art methods, scattering laser light off silica nanospheres, to verify a century-old theoretical prediction of these vibrations. They identify the frequencies associated with both twisting and bulging motions, a result made possible by the uniformity of the tiny spheres they studied.

In 1882 the English mathematical physicist Horace Lamb analyzed the acoustic oscillations of a solid sphere and predicted two types, or “modes,” of vibrations: spheroidal and torsional. The simplest spheroidal mode is pulsation or “breathing” of the entire sphere; higher frequencies correspond to oscillations in which segments of the sphere’s surface pulse in and out with opposite phase.

In the simplest torsional mode, opposite hemispheres twist back and forth alternately, somewhat “like a dog shaking itself dry,” as Meng Hua Kuok of the National University of Singapore puts it. Lamb’s analysis forms the basis for the theory of seismic waves, but the Earth is a distinctly non-uniform sphere, and seismic waves are essentially surface phenomena.

Researchers have tried to detect the modes in the lab using very small spheres, which have higher oscillation frequencies and greater separation between the frequencies than larger spheres. But experiments have been inconclusive, chiefly because samples consisted of nanospheres of less than perfect shape and with a range of sizes. Such samples display a large number of acoustic modes, with little separation in frequency.

Kuok and his colleagues bought commercially-made synthetic opals consisting of regular arrays of silica (CHEMSiO2) spheres. Different opals had nanospheres ranging from 204 to 340 nanometers in diameter, but in each sample the diameters varied by no more than 3 percent. The nanospheres were of almost perfect spherical form.

The researchers then shined green laser light on these opals. Some of the light bouncing off the spheres was shifted in frequency by an amount equivalent to the frequency of acoustic oscillations, a phenomenon known as Brillouin scattering. Vibrations of the nanospheres arose through natural thermal excitation, as well as stimulation by the incident light. In these experiments, each nanosphere in the sample acts as an independent, freely oscillating sphere, but because they are so uniform in size, their interaction in bulk with the laser radiation creates a series of sharply defined peaks in the spectrum of scattered light.

Comparing their results with the predictions of Lamb’s theory, Kuok and his team identified seven distinct oscillations of the nanospheres, each characterized by a pair of “quantum numbers” corresponding to a specific spheroidal or torsional oscillation. Kuok emphasizes, though, that this is “basically a test of classical theory–there is no quantum mechanics.”

This is “an ideal system for such experiments,” says David Norris of the University of Minnesota in Minneapolis. Both Norris and Kuok have to scratch their heads to think of applications of the new results, since it pertains to such an idealized set-up. But “it’s a beautiful textbook demonstration,” Norris says.

–David Lindley

David Lindley is a freelance science writer in Alexandria, Virginia.

Subject Areas

Interdisciplinary Physics

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