PHASE TRANSFORMATION

Microscopic mechanisms ofequilibrium melting of a solidAmit Samanta,1,2* Mark E. Tuckerman,3,4,5* Tang-Qing Yu,6 Weinan E7,8*

The melting of a solid, like other first-order phase transitions, exhibits an intrinsictime-scale disparity: The time spent by the system in metastable states is orders ofmagnitude longer than the transition times between the states. Using rare-eventsampling techniques, we find that melting of representative solids—here, copper andaluminum—occurs via multiple, competing pathways involving the formation and migrationof point defects or dislocations. Each path is characterized by multiple barrier-crossingevents arising from multiple metastable states within the solid basin. At temperaturesapproaching superheating, melting becomes a single barrier-crossing process, and at thelimit of superheating, the melting mechanism is driven by a vibrational instability. Ourfindings reveal the importance of nonlocal behavior, suggesting a revision of theperspective of classical nucleation theory.

Theoretical work on the melting of a soliddates back to Lindemann, who in 1910 en-visioned the melting transition as a vibra-tional instability (1). In 1939, Born postulatedthe macroscopic instability criteria of a

solid in terms of the elastic constants (2–4). Mostother theoretical models are centered aroundthe role of defects, including point defects suchas vacancies and interstitials and line defects suchas dislocations, that proliferate in the solid closeto the melting point (5–8).Contrary to these melting theories, classical

notions of homogeneous melting envision theformation of an initial liquid nucleus that isaided by thermal fluctuations without any pref-erential nucleating sites (9, 10). Within the frame-work of the classical nucleation theory (CNT),the radius r of the liquid nucleus serves as areaction coordinate, and the Gibbs free energyDG(r) is a balance between the free energy gainedin forming a liquid nucleus of volume 4pr3/3and the work needed to create an interface be-tween the solid and such a nucleus

DGðrÞ ¼ 4

3pr3rDmþ 4pr2gs ð1Þ

Here, Dm = ml − ms < 0 is the chemical potentialdifference between the liquid and solid phases,r is the liquid density, 4pr2 is the surface area

of the nucleus, and gs is its surface tension. Thecritical nucleus size r* = −2gs/(rDm) maximizesthis free energy [DGðr*Þ ¼ 16pg3s=3ðrDmÞ2] (11)and determines the length scale beyond whichgrowth of the cluster becomes favorable. At thesolid-liquid coexistence point, Dm = 0, and thetheory predicts an infinite free-energy barrierand corresponding suppression of the nuclea-tion rate. This picture involves numerous sim-plifying assumptions and fails to account for thepotentially important role of defects, dislocations,and multiple barriers along potential meltingpaths (11).Close to the coexistence point, the melting of

a solid involves activation from a metastablelocalminimum. Consequently, a theoretical anal-ysis of the melting mechanisms using standardatomistic simulation methods is not feasiblebecause melting is a rare barrier-crossing eventwith mean first passage time many orders ofmagnitude greater than the vibrational frequen-cy of atoms. State-of-the-art rare-event samplingtechniques now render possible the computa-tional study of equilibrium melting and theextraction of dominant pathways and free en-ergetics. We used adiabatic free-energy dynam-ics (AFED) (12) together with the string method(13, 14) in order to explore the multidimensionalfree-energy surface (FES) efficiently and to con-struct a microscopic picture of the melting processfor two commonly studied prototypical systems:copper (Cu) and aluminum (Al) (15, 16).We began by analyzing the equilibrium melt-

ing process using as collective variables the vol-ume (V) of the system and the two Steinhardtorder parameters Q4 and Q6 (17); this combina-tion captures the positional and orientationalordering. In Fig. 1, A and B, we show a projectionof the Gibbs FES onto the V-Q4 subspace for Cuat 1350 K, close to the melting point of 1358 K,and 1 atm pressure. Contrary to the simple pic-ture from CNT of a smooth FES possessingmeta-stable solid and liquid basins separated by anindex-1 saddle point, the FES obtained here has

multiple locally stable states characterized by dif-ferent defects—primarily vacancy-interstitial pairs,dislocations, and interstitial clusters—and thefree-energy barriers separating these states rangeover multiple energy scales. The FES for Al (fig.S1), close to the melting point of 933 K, exhibits asimilar structure. How vacancy-interstitial pairscan form, diffuse, cluster, and annihilate is illus-trated in Fig. 1C.The existence of a multitude of metastable

states suggests that there exists an ensemble ofmultiple competing transition pathways. Onesuch melting channel and the associated freeenergy profile are shown in Fig. 2A. Along thispath, as the system moves out of the solid basin,the volume of the solid and the number of va-cancy and interstitial pairs increases. Thus, thisparticular melting pathway proceeds via the for-mation of point defects, which is entropicallyfavorable but energetically costly. The competi-tion between entropic and enthalpic contribu-tions causes the free energy to reach a maximumvalue (Fig. 2A, “S2”) at a certain defect concen-tration. After the saddle S2, the system lowersits free energy by forming defect clusters atthe expense of isolated defects. Cluster forma-tion is enabled via the defect kinetics. In both Cuand Al, diffusion barriers of interstitial defectsare quite small (0.09 and 0.13 eV, respectively);barriers for vacancy diffusion are somewhatlarger (0.70 and 0.65 eV, respectively). Previously,Couchman and Reynolds had conjectured thata relationship exists between the melting pointand vacancy concentration, thus suggesting adirect role for vacancy diffusion in the meltingprocess (18). At 1350 K in Cu, these defects aremobile and diffuse over long distances in thesolid. Occasionally, some of these interstitial andvacancy migration paths meet, and defect clus-ters can form. Inside these defect clusters, thereis a notable loss of crystalline order (the localSteinhardt parameter q6 ∼0:1), which coincideswith an enhanced diffusivity of atoms inside thecluster. After S2, the system moves to a shallowmetastable state, and the defect cluster evolvesinto a liquid nucleus, the existence of which dur-ing melting also constitutes a deviation fromthe CNT.As the system moves out of the metastable

state toward the saddle S1 (fig. S3), the size ofthe liquid nucleus increases. This process is some-times observed to be aided by the coalescenceof a few smaller liquid nuclei. A test of the sys-tem size scaling of the free-energy barrier (fig.S3E) reveals that the S1 barrier scales roughlyas N2/3. After the liquid nucleus attains a crit-ical size, the free energy along the path decreases.This corresponds to the second important bot-tleneck, with a barrier of ∼125 eV relative to thesolid basin. From here, the liquid nucleus simplyincreases in size until the entire system trans-forms to the liquid state.Our enhanced sampling calculations suggest

the existence of a melting path along which dis-location activity aids the formation of an initialliquid nucleus. The corresponding free-energyprofile and key saddle structures are shown in

SCIENCE sciencemag.org 7 NOVEMBER 2014 • VOL 346 ISSUE 6210 729

1Condensed Matter and Materials Division, Lawrence LivermoreNational Laboratory, Livermore, CA 94550 USA. 2Program inApplied and Computational Mathematics, Princeton University,Princeton, NJ 08544, USA. 3Department of Chemistry, NewYork University (NYU), New York, NY 10003, USA. 4CourantInstitute of Mathematical Sciences, NYU, New York, NY 10012,USA. 5NYU–East China Normal University Center forComputational Chemistry at NYU Shanghai, Shanghai 200062,China. 6Courant Institute of Mathematical Sciences, NYU, NewYork, NY 10012, USA. 7Beijing International Center forMathematical Research and School of Mathematical Sciences,Peking University, Beijing, China. 8Department of Mathematicsand Program in Applied and Computational Mathematics,Princeton University, Princeton, NJ 08544, USA.*Corresponding author. E-mail: asamanta@math.princeton.edu(A.S.); weinan@math.princeton.edu (W.E.); mark.tuckerman@nyu.edu (M.E.T.)

RESEARCH | REPORTS

on

Janu

ary

19, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Janu

ary

19, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Janu

ary

19, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Janu

ary

19, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

Fig. 2B. Along this path, we first observe theformation of defect clusters from point defectsfollowed by the heterogeneous nucleation ofdislocations from the defect clusters. The dis-location density increases as the system movestoward the saddle S1, and the first liquid em-bryo is formed heterogeneously from these dis-locations. This process of initiating meltingfrom dislocations is reminiscent of recent obser-vations of heterogeneous nucleation in colloidalsystems (19) and embedded lead (Pb) nano-particles in an Al matrix (20) and differs con-siderably from the notion of proliferation ofdislocations (5, 7).From our analysis of the escape pathways

from the solid basin, we conclude that meltinghas a high probability of being coupled to defectactivity. Nevertheless, motivated by recent re-ports of melting without the aid of defects (21),we were inspired to ask whether it is possible formelting to occur in this manner. By calculatingVoronoi volumes associated with each of theatoms, we are able to show that close to themelting point, the free volume associated withvacancies is spatially delocalized, and it is difficultto identify individual defects. Previous exper-iments have shown that on-the-fly determinationof defects at high temperatures is nontrivial (22).Furthermore, under equilibrium conditions, the

possibility that Cu at 1350 K is defect-free seemsremote because of the presence of nearly de-generate defect states, with transition barrierson the order of 1 to 4 eV. Nevertheless, we ex-plored the possibility of a melting pathway con-necting the defect-free solid minimum and theliquid basin that does not pass through the de-fect states. On the FES, a possible pathway formelting without the aid of defects has a notablyhigher free-energy barrier (∼150 eV) than thebarriers along the defect-mediated melting path-ways. Along such a defect-free path, the forma-tion of the initial liquid nucleus is correlatedwith thermal fluctuations and is not aided byany stable defects. However, if this path is al-lowed to relax in path space, it eventually con-verges to one of the two defect-mediated meltingpathways.Temperature plays an important role in deter-

mining the characteristics of the FES. Thus, wealso sought to compare equilibrium melting tomelting of a superheated solid. As the system issuperheated beyond themelting point, the liquidbasin engulfs larger portions of the FES, and themetastable state without defects vanishes (Fig.2E). This change in the FES affects the meltingmechanisms: Compared with the FES at 1350 Kin Cu, at 1550 K (and at 1200 K for Al) there arefewer locally stable states present inside the

solid basin, and there is only one important bot-tleneck formelting—namely, the barrier to escapethe solid basin. This barrier has a substantialentropic contribution (fig. S4). The absence of ametastable state without defects indicates thatmelting at these superheated conditions alwaysoriginates from a solid with preexisting pointdefects or defect clusters.With further increase in temperature, the

melting barrier along the minimum free-energypath vanishes at∼1600K in Cu and ∼1275 K in Al.Above these temperatures, atoms execute large-amplitude vibrations about their equilibriumpositions, and the solid-to-liquid transition is aconsequence of an instability (Fig. 2C, inset,and fig. S5). The crucial role of the vibrationalamplitude in this process is realized only byseparating the diffusive motions of the atomsfrom their vibrational motions. This mecha-nism is reminiscent of Lindemann’s vibrationalinstability (1) and is in agreement with recentexperimental observations of melting in super-heated aluminum (23). The fact that Lindemann’scriterion corresponds to an instability of thesolid and not to equilibrium melting leads toan overestimation of the melting point. This isrelevant because Lindemann’s criterion is stillcommonly used to determine the melting pointof a solid (24–26).

730 7 NOVEMBER 2014 • VOL 346 ISSUE 6210 sciencemag.org SCIENCE

Q4

A

B Solid

S1

Liquid basin0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

39 40 41 42 43

Fig. 1. The FES of a solid shows multiscale characteris-tics and is sensitive to temperature. (A and B) The FESsat 1350 K, 1 atm for Cu.The solid basin contains two valleys[(A), “A” and “B”] separated by a shallow ridge. Valley Acontains local minima pertaining to the solid with or withoutpoint defects, whereas Valley B contains local minimapertaining mainly to defect clusters.Within valley A, atomicmotions in the state without defects are highly correlatedand have a waiting time of ∼300 ps [analyzed by calculatingthe number of atoms displaced more than nearest-neighborspacings, (D), inset], whereas point defect–related diffusivemotions of atoms have a waiting time of ∼0.10 ps [(D) andfig. S2]. Inside valley B, as the volume of the systemincreases, the defect cluster develops liquidlike character-istics. (C) A time evolution of defects in the solid basin. Thedominant atomic processes are vacancy-interstitial forma-tion, vacancy diffusion (indicated with blue arrows), intersti-tial diffusion (black arrows), interstitial cluster formation,and vacancy-interstitial annihilation (red double-ended ar-row). Processes such as vacancy-interstitial pair formation and cluster formation are circled in red.

RESEARCH | REPORTS

Theories such as CNT and that suggested bythe present study address homogeneous meltingin the bulk. The system-size scaling dependenceof the associated free-energy barrier, however,indicates that as N grows, melting is more likelyto be initiated from regions that break thetranslational and orientational order of the bulksolid. These include defect clusters, voids, grainboundaries, and dislocations, all of which in-crease the energy of a solid and lead to het-erogeneousmelting.We sought to investigate thiscase by initiating trajectories from preexistingdislocations and from grain boundaries obtainedby rapidly cooling the liquid [(17), section XII, fig.S6]. In both cases, the resultingmetastable stateslie closer to the saddle S1 on the FES, resulting ina decrease in the melting barrier. As shown inFig. 2D, melting in a solid with grain boundariesis initiated at these boundaries, and the barriercan be as low as 6 eV at the melting point (fig.S7). The atoms in the grain boundaries are fluid-like, and hence, melting occurs via gradualgrowth of these liquid-like regions. In a solidwith multiple dislocations, the elastic inter-actions between the dislocations can decreasethe melting barrier by as much as a factor of sixbelow the barrier for melting of a Cu samplewithout preexisting defects.The sensitivity of the free-energy surface to

temperature is shown in Fig. 2C. There exist threedifferent melting regimes: (i) Close to the meltingpoint [temperature (T) < 1525K for Cu,T < 1200K

for Al], the transition occurs via multiple barriercrossings, and elastic interactions play a domi-nant role in determining the melting mecha-nisms. (ii) At superheated temperatures (1525to 1590 K for Cu, 1200 to 1265 K for Al), the solid-to-liquid transition is a single barrier-crossingevent, and the barrier contains enthalpic aswell as entropic contributions. (iii) At the limitof instability, melting is initiated by the prop-agation of large vibrations of the atoms in ashort span of time across the crystal. This largevariation in the underlying melting mechanismssuggests a strong coupling between the orienta-tional order, density, and temperature: At a giventemperature, how Q6 and Q4 change with Vdetermines the structure of defects present in thesolid, locations of the saddle points, and thebasins of attraction.Notwithstanding the above differences in the

melting mechanisms, we find that melting isaided by mobile defects both at the thermody-namic melting point and at conditions of super-heating. In both of these states, defects constituteless than 1/1000 of the total lattice sites. Conse-quently, melting and lattice instabilities are nottriggered by the proliferation of defects (6, 27, 28),which contrasts with theories based on the asser-tion that “melting is correlatedwith the achieve-ment of a critical vacancy concentration” (29).Further, our enhanced sampling calculations re-veal that the size of a liquidnucleus doesnot changesmoothly along the minimum free-energy pathway.

It is clear that multiple, competing pathwaysand multiple barrier-crossing events along eachpath are crucial for the formation of the initialliquid embryo and constitute a qualitative de-parture from the simplifying assumptions ofCNT. An important element that is clearly miss-ing in the DG of CNT is the contribution from thestrain energy. Along the dislocation-mediatedmelting pathway (Fig. 2B), the liquid nucleus isformed preferentially on the dislocations via adecrease in the strain energy associated with thedislocations. The change in free energy can beexpressed as DG(r) = −arlogr/ro + Dm4pr3/3 +4pr2gs, where a =Gb2/4p(1− n), ro is a constant, bis the Burgers vector, n is Poisson’s ratio, and−arlogr/ro is the decrease in the elastic energy ofthe dislocation owing to the formation of theliquid nucleus (30). At the melting point, Dm =0; hence, the critical radius of the liquid nu-cleus is r∗ ∼ Gb2/32p2 gs(1 − n). Similarly, for-mation of dislocations from point defects anddefect clusters involves a competition betweenthe elastic energy of dislocation and the inter-action energy between the dislocation and thedefects, likely requiring the addition of nonlocalenergy terms in any mean-field type of descrip-tion of melting. This physical interplay of pro-cesses with multiple length and time scalesindicates that melting of a solid cannot beviewed through the simple lens of CNT butshould be regarded as a complex “multiscale”phenomenon.

SCIENCE sciencemag.org 7 NOVEMBER 2014 • VOL 346 ISSUE 6210 731

Fig. 2. Melting in a solid without preexisting de-fects can proceed via multiple competing path-ways as shown for Cu at 1350 K. (A) Along thepoint defect–mediated melting pathway, there aretwo important saddles: S2 (formation of defect clus-ter) and S1 (formation of liquid nucleus of criticalsize). (B) Along the dislocation-mediated meltingpathway, there are multiple barriers, including de-fect cluster formation, dislocation nucleation, andliquid nucleus formation. The atoms are coloredaccording to the local orientation order parameterq4. (C) (Inset) The sharp discontinuity in the vi-brational amplitude (av) of atoms in the solid co-incides with the vanishing of the free-energy barrier. av is extracted from the root-mean-squared-displacement (RMSD) after subtracting the diffusioncontributions, and hence, the sharp discontinuity observed in av is not observed in the usual RMSD profiles [for example, figure 2 in (25)]. (D) The effect ofgrain boundaries and dislocations on the melting barriers.The solid in GB1 has only two grains,whereas GB2 has about four grains. (E) FES at 1550 K for Cu,illustrating that melting is a single barrier-crossing event that involves enthalpic as well as entropic barriers at superheated conditions.

RESEARCH | REPORTS

REFERENCES AND NOTES

1. F. A. Lindemann, Phys. Z. 11, 609 (1910).2. M. Born, J. Chem. Phys. 7, 591 (1939).3. N. F. Mott, Nature 145, 801–802 (1940).4. J. L. Tallon, Nature 299, 188 (1982).5. D. Kuhlmann-Wilsdorf, Phys. Rev. 140, A1599–A1610

(1965).6. T. Gorecki, Z. Metallkunde Mat. Res. Adv. Tech. 65, 426 (1974).7. L. Gómez, A. Dobry, Ch. Geuting, H. T. Diep, L. Burakovsky,

Phys. Rev. Lett. 90, 095701 (2003).8. X. M. Bai, M. Li, Phys. Rev. B 77, 134109 (2008).9. M. Volmer, A. Weber, Z. Phys. Chem. 119, 277 (1926).10. R. Becker, W. Döring, Ann. Phys. 416, 719–752 (1935).11. R. P. Sear, Int. Mater. Rev. 57, 328–356 (2012).12. T. Q. Yu, M. E. Tuckerman, Phys. Rev. Lett. 107, 015701

(2011).13. W. E, W. Ren, E. Vanden-Eijnden, J. Phys. Chem. B 109,

6688–6693 (2005).14. L. Maragliano, A. Fischer, E. Vanden-Eijnden, G. Ciccotti,

J. Chem. Phys. 125, 024106 (2006).15. R. Lynden-Bell, J. Van Duijneveldt, D. Frenkel, Mol. Phys. 80,

801–814 (1993).

16. M. Forsblom, G. Grimvall, Nat. Mater. 4, 388–390(2005).

17. Materials and methods are available as supplementarymaterials on Science Online

18. P. R. Couchman, C. L. Reynolds Jr., Philos. Mag. 34, 327–329(1976).

19. A. M. Alsayed, M. F. Islam, J. Zhang, P. J. Collings, A. G. Yodh,Science 309, 1207–1210 (2005).

20. H. Rösner, B. Freitag, G. Wilde, Philos. Mag. Lett. 87, 341–347(2007).

21. Z. Wang, F. Wang, Y. Peng, Z. Zheng, Y. Han, Science 338,87–90 (2012).

22. E. Fukushima, A. Ookawa, J. Phys. Soc. Jpn. 10, 970–981(1955).

23. B. J. Siwick, J. R. Dwyer, R. E. Jordan, R. J. D. Miller, Science302, 1382–1385 (2003).

24. J. J. Gilvarry, Phys. Rev. 102, 308–316 (1956).25. Z. H. Jin, P. Gumbsch, K. Lu, E. Ma, Phys. Rev. Lett. 87, 055703

(2001).26. K. Sokolowski-Tinten et al., Nature 422, 287–289 (2003).27. R. M. J. Cotterill, J. U. Madsen, Nature 288, 467–469

(1980).

28. H. J. Fecht, W. L. Johnson, Nature 334, 50–51 (1988).29. R. W. Cahn, Nature 334, 17–18 (1988).30. J. W. Cahn, Acta Metall. 5, 169–172 (1957).

ACKNOWLEDGMENTS

Work by W.E and A.S. at Princeton was supported by the U.S.Department of Energy (DOE) (grant DE-SC0009248) and in Beijingby the National Natural Science Foundation of China (9130005).This work was partially performed under the auspices of DOE byLawrence Livermore National Laboratory under contractDE-AC52-07NA27344. Work by M.E.T. was supported by NSF grantCHE-1301314.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/346/6210/729/suppl/DC1Materials and MethodsFigs. S1 to S7References (31–41)

24 March 2014; accepted 10 October 201410.1126/science.1253810

EARLY UNIVERSE

On the origin of near-infraredextragalactic backgroundlight anisotropyMichael Zemcov,1,2 Joseph Smidt,3,4 Toshiaki Arai,5,6 James Bock,1,2* Asantha Cooray,4

Yan Gong,4 Min Gyu Kim,7 Phillip Korngut,2,1 Anson Lam,8,1 Dae Hee Lee,9

Toshio Matsumoto,5,10 Shuji Matsuura,5 Uk Won Nam,9 Gael Roudier,2

Kohji Tsumura,11 Takehiko Wada5

Extragalactic background light (EBL) anisotropy traces variations in the total productionof photons over cosmic history and may contain faint, extended components missed ingalaxy point-source surveys. Infrared EBL fluctuations have been attributed to primordialgalaxies and black holes at the epoch of reionization (EOR) or, alternately, intrahalolight (IHL) from stars tidally stripped from their parent galaxies at low redshift. We reportnew EBL anisotropy measurements from a specialized sounding rocket experiment at1.1 and 1.6 micrometers. The observed fluctuations exceed the amplitude from knowngalaxy populations, are inconsistent with EOR galaxies and black holes, and are largelyexplained by IHL emission. The measured fluctuations are associated with an EBL intensitythat is comparable to the background from known galaxies measured through numbercounts and therefore a substantial contribution to the energy contained in photons inthe cosmos.

At near-infraredwavelengths, where the largezodiacal light foreground complicates ab-solute photometrymeasurements, the extra-galactic background light (EBL)may be bestaccessed by anisotropy measurements. On

large angular scales, fluctuations are producedby the clustering of galaxies, which is driven bythe underlying distribution of dark matter. EBLanisotropymeasurements can probe emission fromepoch of reionization (EOR) galaxies (1–3) anddirect-collapse black holes (4) that formed dur-ing the EOR before the universewas fully ionizedby exploiting the distinctive Lyman cutoff featurein the rest-frame ultraviolet (UV), thus probingthe UV luminosity density at high redshifts (5).However, large-scale fluctuations may also arisefrom the intrahalo light (IHL) created by starsstripped from their parent galaxies during tidal

interactions (6) at redshift z < 3. A multiwave-length fluctuation analysis can distinguish amongthese scenarios and constrain the EOR star for-mation rate.A search for such background components

must carefully account for fluctuations producedby known galaxy populations. Linear galaxy clus-tering is an important contribution to fluctuationson scales much larger than galaxies themselves.On fine scales, the variation in the number ofgalaxies produces predominantly Poissonian fluc-tuations, with an amplitude that depends on theluminosity distribution. Anisotropymeasurementssuppress foreground galaxy fluctuations bymask-ing known galaxies from an external catalog.The first detections of infrared fluctuations in

excess of the contribution from known galaxieswith the Spitzer Space Telescope (7–9) were in-

terpreted as arising from a population of faintfirst-light galaxies at z > 7. The Hubble SpaceTelescope was used at shorter wavelengths (10)to carry out a fluctuation study in a small deepfield but did not report fluctuations in excess ofknown galaxy populations. Measurements withthe AKARI satellite (11) show excess fluctuationswith a blue spectrum rapidly rising from 4.1 mmto 2.4 mm. Fluctuation measurements in a largesurvey field (6) with Spitzer agreed with earliermeasurements (7–9) but were instead interpretedas arising from tidally stripped stars at z ∼ 1 to 3.Most recently, a partial correlation has been re-ported (12) between Spitzer and soft x-ray im-ages, which Yue et al. (4) interpret as arising fromdirect-collapse black holes at z > 12.We have developed and flown the specialized

Cosmic Infrared Background Experiment [CIBER(13)], a rocket-borne instrument specifically de-signed to study the spatial and spectral propertiesof the EBL. The imaging instrument (14) mea-sures fluctuations in Dl/l = 0.5 bands centeredat 1.1 and 1.6 mmusing two 11-cm telescopes eachwith a 2° by 2° field of view. Here, we report ouranalysis of data from two flights in 2010 and 2012.The CIBER imager data are reduced from raw

telemetered time streams, flat-field–correctedbased

732 7 NOVEMBER 2014 • VOL 346 ISSUE 6210 sciencemag.org SCIENCE

1Department of Physics, Mathematics, and Astronomy,California Institute of Technology, Pasadena, CA 91125, USA.2Jet Propulsion Laboratory (JPL), National Aeronautics andSpace Administration (NASA), Pasadena, CA 91109, USA.3Theoretical Division, Los Alamos National Laboratory, LosAlamos, NM 87545, USA. 4Department of Physics andAstronomy, University of California, Irvine, CA 92697, USA.5Department of Space Astronomy and Astrophysics,Institute of Space and Astronautical Science (ISAS), JapanAerospace Exploration Agency (JAXA), Sagamihara, Kanagawa252-5210, Japan. 6Department of Physics, Graduate School ofScience, The University of Tokyo, Tokyo 113-0033, Japan.7Department of Physics and Astronomy, Seoul NationalUniversity, Seoul 151-742, Korea. 8Department of Physicsand Astronomy, University of California, Los Angeles, CA90095, USA. 9Korea Astronomy and Space Science Institute(KASI), Daejeon 305-348, Korea. 10Institute of Astronomyand Astrophysics, Academia Sinica, Taipei 10617, Taiwan,Republic of China. 11Frontier Research Institute forInterdisciplinary Science, Tohoku University, Sendai, Miyagi,980-8578, Japan.*Corresponding author. E-mail: jjb@astro.caltech.edu

RESEARCH | REPORTS

DOI: 10.1126/science.1253810, 729 (2014);346 Science

et al.Amit SamantaMicroscopic mechanisms of equilibrium melting of a solid

This copy is for your personal, non-commercial use only.

clicking here.colleagues, clients, or customers by , you can order high-quality copies for yourIf you wish to distribute this article to others

  here.following the guidelines

can be obtained byPermission to republish or repurpose articles or portions of articles

  ): January 19, 2015 www.sciencemag.org (this information is current as of

The following resources related to this article are available online at

http://www.sciencemag.org/content/346/6210/729.full.htmlversion of this article at:

including high-resolution figures, can be found in the onlineUpdated information and services,

http://www.sciencemag.org/content/suppl/2014/11/05/346.6210.729.DC1.html can be found at: Supporting Online Material

http://www.sciencemag.org/content/346/6210/729.full.html#relatedfound at:

can berelated to this article A list of selected additional articles on the Science Web sites

http://www.sciencemag.org/content/346/6210/729.full.html#ref-list-1, 4 of which can be accessed free:cites 40 articlesThis article

http://www.sciencemag.org/content/346/6210/729.full.html#related-urls1 articles hosted by HighWire Press; see:cited by This article has been

http://www.sciencemag.org/cgi/collection/mat_sciMaterials Science

subject collections:This article appears in the following

registered trademark of AAAS. is aScience2014 by the American Association for the Advancement of Science; all rights reserved. The title

CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience

on

Janu

ary

19, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om