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1、27. F. Banhart, J. Mater. Sci. 41, 4505 (2006).28. V. H. Crespi, N. G. Chopra, M. L. Cohen, A. Zettl,S. G. Louie, Phys. Rev. B 54, 5927 (1996).29. The extent of displacement may vary depending on the tiltangle, because the local thickness of the specimen alongthe axis of the excitation may change. H
2、owever, the skindepth of the MWNT ring specimen for the 532-nm light isdeduced to be 2 mm absorption coefficient a =1.0104cm1(35), which exceeds the largest local thickness alongthe ring specimen at a tilt angle of 35. In addition, theabsorption cross section of MWNTs is reported to be weaklydepende
3、nt on the polarization of the incident beam forthick tubes (36). To further suppress any polarizationdependence,wesetthepolarizationoftheopticalexcitationbeam so that it was not along the long axis of the tube.Consequently, the heat gradient and thermal stress areuniform for the tilt angles recorded
4、 in this study.30. P. Poncharal, Z. L. Wang, D. Ugarte, W. A. de Heer,Science 283, 1513 (1999).31. L. Meirovich, Elements of Vibration Analysis (McGraw-Hill,New York, ed. 2, 1986).32. X.-L. Wei, Y. Liu, Q. Chen, M.-S. Wang, L.-M. Peng,Adv. Funct. Mater. 18, 1555 (2008).33. M. M. J. Treacy, T. W. Ebb
5、esen, J. M. Gibson, Nature 381,678 (1996).34. G. V. Hartland, Annu. Rev. Phys. Chem. 57, 403(2006).35. T. Nakamiya et al., Thin Solid Films 517, 3854(2009).36. C. Ni, P. R. Bandaru, Carbon 47, 2898 (2009).37. S. Jonic, C. Vnien-Bryan, Curr. Opin. Pharmacol. 9, 636(2009).38. Supported by NSF (grant D
6、MR-0964886) and Air ForceOffice of Scientific Research (grant FA9550-07-1-0484)in the Physical Biology Center for Ultrafast Science andTechnology supported by Gordon and Betty MooreFoundation at Caltech. A patent application has beenfiled by Caltech based on the methodology presentedherein.Supportin
7、g Online Materialwww.sciencemag.org/cgi/content/full/328/5986/1668/DC1Movies S1 to S35 April 2010; accepted 19 May 201010.1126/science.1190470Measurement of the InstantaneousVelocity of a Brownian ParticleTongcang Li, Simon Kheifets, David Medellin, Mark G. Raizen*Brownian motion of particles affect
8、s many branches of science. We report on the Brownian motionof micrometer-sized beads of glass held in air by an optical tweezer, over a wide range of pressures,and we measured the instantaneous velocity of a Brownian particle. Our results provide directverification of the energy equipartition theor
9、em for a Brownian particle. For short times, theballistic regime of Brownian motion was observed, in contrast to the usual diffusive regime. Wediscuss the applications of these methods toward cooling the center-of-mass motion of a bead invacuum to the quantum ground motional state.In 1907, Albert Ei
10、nstein published a paper inwhich he considered the instantaneous ve-locity of a Brownian particle (1, 2). By mea-suring this quantity, one could prove that “thekinetic energy of the motion of the centre of grav-ity of a particle is independent of the size andnature of the particle and independent of
11、 thenature of its environment.” This is one of thebasic tenets of statistical mechanics, known astheequipartitiontheorem.However,becauseofthe very rapid randomization of the motion,Einstein concluded that the instantaneous veloc-ity of a Brownian particle would be impossible tomeasure in practice.We
12、 report here on the measurement of theinstantaneous velocity of a Brownian particle in asystem consisting of a single, micrometer-sizedSiO2bead held in a dual-beam optical tweezer inair, over a wide range of pressures. The velocitydata were used to verify the Maxwell-Boltzmannvelocity distribution a
13、nd the equipartition theoremfor a Brownian particle. The ability to measureinstantaneous velocity enables new fundamentaltests of statistical mechanics of Brownian par-ticles and is also a necessary step toward the cool-ing of a particle to the quantum ground motionalstate in vacuum.The earliest qua
14、ntitative studies of Brownianmotion were focused on measuring velocities,and they generated enormous controversy (3, 4).The measured velocities of Brownian particles(3) were almost 1000-fold smaller than what waspredicted by the energy equipartition theorem.Recentexperimentswithfastdetectorsthatstud
15、iedBrownian motion in liquid (57) and gaseous(810) environments observed nondiffusive mo-tion of a Brownian particle.Einsteins theory predicts that Dx(t)2 2Dt,whereDx(t)2 is the mean square displace-ment (MSD) in one dimension of a free Brown-ian particle during time t,andD is the diffusionconstant
16、(11). The diffusion constant can be cal-culated by D kBT=g, where kBis Boltz-manns constant, T is the temperature, and g isthe Stokes friction coefficient. The mean veloc-ity measured over an interval of time t is v Dx(t)2p/t 2Dp/tp. This diverges as t ap-proaches 0 and therefore does not represent thereal velocity of the particle (1, 2).The equation Dx(t)2 2Dt, however, isvalid only when t tp; that is, in the diffusiveregime. Here, t
