The linewidth of longitudinal acoustic waves in densified silica glass is
obtained by inelastic x-ray scattering. It increases with a high power alpha of
the frequency up to a crossover where the waves experience strong scattering.
We find that \alpha is at least 4, and probably larger. Resonance and
hybridization of acoustic waves with the boson-peak modes seems to be a more
likely explanation for these findings than Rayleigh scattering from disorder.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Letter

A. Matic

B. Dreyfus

B. Rufflé

E. Courtens

G. Monaco

K. Trachenko

M. Foret

M. Rothenfusser

R. Vacher

R. C. Zeller

V. G. Karpov

Y. Inamura

English

arXiv

Crossref

Observation of the Onset of Strong Scattering on High Frequency Acoustic Phonons in Densified Silica Glass

4 pages, 4 figuresThe linewidth of longitudinal acoustic waves in densified silica glass is obtained by inelastic x-ray scattering. It increases with a high power alpha of the frequency up to a crossover where the waves experience strong scattering. We find that \alpha is at least 4, and probably larger. Resonance and hybridization of acoustic waves with the boson-peak modes seems to be a more likely explanation for these findings than Rayleigh scattering from disorder

Ruffle, B.

Foret, M.

Courtens, E.

Vacher, R.

Monaco, G.

HAL Descartes

Observation of the onset of strong scattering on high frequency acoustic phonons in densified silica glass

Recently, a near-field optical technique which allows spectroscopic measurements with 20 nm spatial resolution. As such, the method makes use of the strongly enhanced electric field close to a sharp metal tip under laser illumination and relies on the detection of two-photon excited fluorescence. This paper shows the chemical specificity of this near-field technique and demonstrate an unprecedented spatial resolution

Ruffle B.

Foret M.

Courtens E.

Vacher R.

Monaco G.

Archivio istituzionale della ricerca - Università di Padova

P H Y S I C A L R E V I E W L E T T E R S week ending7 MARCH 2003VOLUME 90, NUMBER 9Observation of the Onset of Strong Scattering on High Frequency Acoustic Phononsin Densified Silica GlassB. Rufflé,1 M. Foret,1 E. Courtens,1 R. Vacher,1 and G. Monaco21Laboratoire des Verres, UMR 5587 CNRS, Université Montpellier 2, F-34095 Montpellier, France2European Synchrotron Radiation Facility, Boı̂te Postale 220, F-38043 Grenoble, France(Received 30 August 2002; published 3 March 2003)095502-1The linewidth of longitudinal acoustic waves in densified silica glass is obtained by inelastic x-rayscattering. It increases with a high power  of the frequency up to a crossover where the wavesexperience strong scattering. We find that  is at least 4, and probably larger. Resonance andhybridization of acoustic waves with the boson-peak modes seems to be a more likely explanationfor these findings than Rayleigh scattering from disorder.DOI: 10.1103/PhysRevLett.90.095502 PACS numbers: 63.50.+x, 78.35.+c, 61.43.–j, 78.70.Cksized that inelastic x-ray scattering (IXS) still is a diffi-cult spectroscopy with severe limitations on resolutiondard slit of 20 60 mm (width  height), using the sig-nal from the first sharp diffraction peak of polymethylA controversy regarding the fate of very highfrequency acousticlike excitations in glasses arose inrecent years [1,2]. Acoustic waves propagate in glassesup to rather high frequencies, for example, up to at least=2 ’ 0:4 THz in vitreous silica, v-SiO2 [3,4].However, silica shows a pronounced ‘‘plateau’’ in thetemperature (T) dependence of its thermal conductivityT [5]. This feature was explained by postulating theexistence of a rapid crossover of acoustic waves into aregime where they experience strong scattering [6]. Inv-SiO2, this crossover is expected at a frequencyco=2 ’ 1 THz [7]. For the observed horizontal plateauto occur, besides strong scattering of thermal phonons,phonons of lower  must also experience a sufficientlystrong T-independent scattering [8]. Several models pre-dict that the inhomogeneous width of acoustic waves canincrease with a high power of , typically in 4 [9,10],leading to this rapid crossover. However, the origin of thehigh power remained debated, whether Rayleigh scatter-ing from structural defects [6], or resonance with localmodes [10–12]. Rayleigh scattering seems generally tooweak in materials as homogeneous as dense glasses to beable to account for the relatively lowco suggested by theposition of the plateau [6,8]. The local modes are seen inthe ‘‘boson peak’’ (BP) and produce an excess over theDebye specific heat [13]. Successful models based onresonance or on soft potentials were developed to de-scribe the observed properties, e.g., in [10,14,15].It might seem, judging from some recent literature [2],that the above picture should have been revised after itbecame possible to observe in glasses the spectrum oflongitudinal acoustic (LA) waves at THz frequencieswith x-ray Brillouin scattering (BS) [16]. This new ex-perimental tool is important here, as it potentially allowsone to check some of the previous conjectures. It led tomany publications claiming to prove that sound propa-gates at frequencies much above the earlier expectationsfor co, e.g., [17,18]. In this respect, it must be empha-0031-9007=03=90(9)=095502(4)$20.00 and intensity. First, it is practically not possible withcurrent instruments to investigate scattering vectors Qbelow 1 nm1, a value which happens to coincide withthe expected wave vector q at crossover, qco, in v-SiO2 [7]and in many other glasses. Second, the narrowest instru-mental profile allowing for sufficient intensity [19] stillhas an energy width around 1.5 meV (or ’ 0:4 THz) andextended Lorentzian-like wings. Owing to the relativelystrong elastic scattering of glasses, this tends to mask theweaker Brillouin signal. For these reasons it is still notpossible to investigate with IXS the LAwaves of v-SiO2 atfrequencies where their linewidth might grow in 4[20,21]. Finally, the signal-to-noise ratio being also small,the subtle changes in BS line shapes that indicate strongscattering might easily go unnoticed [22]. To alleviateseveral of these difficulties, we investigated permanentlydensified silica glass, d-SiO2, in which co=2 ’ 2 THzand qco ’ 2 nm1 [23,24]. A clear demonstration of arapid increase of the LA width as  approaches cofrom below is still missing. This is a crucial check forthe onset of strong scattering. It is all the more importantthat many recent Letters advocated acoustic linewidthsthat vary as Q2 up to very high Q values, denying theonset of strong scattering and the existence of the corre-sponding crossover, e.g., [17,18,25–29]. In this Letter, weuse the window between the smallest accessible Q and theqco of d-SiO2 to study the approach of the crossover. Wefind a width in  with  at least equal to 4 and probablylarger, corroborating the early intuition of a rapid cross-over to strong scattering.The experiments were performed on the high-resolution IXS spectrometer ID16 at the EuropeanSynchrotron Radiation Facility in Grenoble, France.The x-ray energy is 21.7 keV. The main components ofthe spectrometer are a T-scanned monochromator, thesample, a battery of five T-stabilized spherical analyzers,and the corresponding detectors. The instrumental lineshape was determined for each analyzer behind its stan-2003 The American Physical Society 095502-102040 (a) Q = 1.17 nm-1DHO020 (b) Q = 1.17 nm-1DHO   010 (c) Q = 1.8 nm-1DHO   010 (d)Q = 1.8 nm-1EMA   0510 (e) Q = 4.2 nm-1DHO   -30 -20 -10 0 10 20 300510 (f) Q = 4.2 nm-1EMAIntensities [Counts per channel in 90s] X-Ray Energy Loss [meV]FIG. 1. IXS spectra of d-SiO2 and their fits explained in thetext: (a) a full spectrum and its central portion 1=20;(b),(c),(e) damped harmonic oscillator (DHO) fits of the in-elastic part after subtraction of a central peak (CP) freelyadjusted in the fits; (d),(f) effective-medium-approximation(EMA) fits of the inelastic part after subtraction of a CP alsofreely adjusted.P H Y S I C A L R E V I E W L E T T E R S week ending7 MARCH 2003VOLUME 90, NUMBER 9methacrylate at 20 K. Very clean Voigt-like shapes wereobtained with half widths at half maximum of ’0:75 meV. The analyzers permit data collection at fiveQ in increments of ’ 3 nm1. As we are now interested insmall Q, the first analyzer was placed to the left of theforward direction, and the second to the right. This allowssimultaneous collection of two interesting data pointswith Q  2 nm1. The data presented here were thusacquired with only three different positions of the firstanalyzer, j Q j	 2, 1.8, and 1:6 nm1. Each spectrum wasaccumulated for nearly two full days. The sample ofd-SiO2, of density 2:62 g=cm3, is the same as in previousmeasurements [23,24]. It is heated at 565 K to increase theinelastic signal. At this T, the densified structure doesnot relax. This was subsequently checked with optical BS,the sound velocity providing a sensitive test for thedensity [23].Figure 1 illustrates typical spectra and their fits. Theindicated values of Q correspond to the center of thecollection slit which gives a spread Q ’ 0:18 nm1.The spectra consist of a strong elastic peak plus a weakerBrillouin doublet. In Fig. 1(a), the full raw spectrum isshown for the smallest usable Q, namely, Q 	 1:17 nm1[30]. The dashed line follows the elastic peak, illustratingthe relative weakness of the doublet. To extract informa-tion on the position  and half-width , the spectra arefirst adjusted to a damped harmonic oscillator (DHO)profile plus an elastic central peak (CP), convolutedwith the instrumental function. Account is taken of thebroadening due to the collection angle by summing overcontributions from surface elements of the slit, each at itsown Q, using d=dQ 	 vg. Here vg is the group velocitywhich is determined iteratively. There are four free pa-rameters in the fits: the integrated intensities of the elastic(ICP) and inelastic (IDHO) components,  and . Thesmall background shown by the baselines is fixed to thedetector noise, measured independently. Figures 1(b) and1(c) illustrate, for the two extreme values of Q measuredbelow crossover, the inelastic part that remains aftersubtraction of the adjusted elastic contribution ICP. Thesolid lines are the DHO fits. In Fig. 1(b) the width ismostly instrumental, while in 1(c) it is due in large part toa real broadening of the Brillouin signal.The values obtained for  are shown in Fig. 2 againstthose for  which agree with [23].  increases veryrapidly with . There should be a homogeneous broad-ening contribution to this width, hom. It is estimatedfrom hom / 2 by extrapolation of an optical BSdetermination, hom=2 	 26 5 MHz for =2 	41:5 GHz at 600 K [31]. Interestingly, in that BS experi-ment the linewidths of d-SiO2 are very near those ofcrystal quartz over a broad range of T, whereas the line-widths of v-SiO2 are typically 4 times larger. Thus, itseems reasonable to assume hom / 2, as, e.g., theAkhiezer mechanism might extend up to the THz regionin d-SiO2 at these elevated T values. This gives the hom095502-2drawn in Fig. 2. The total rate is then approximated byMatthiesen’s rule of adding rates due to independentprocesses,  	 hom  inh [21]. The inhomogeneousscattering width is adjusted to inh / . A fit of thefive measured points with this expression gives the solidline in Fig. 2 with  	 4:21 0:15, and 2 	 0:76. Forcomparison, if we force  	 2, the best fit shown by thedotted line gives 2 	 5:4.The inelastic intensities, IDHO, derived from these fitsare shown in the inset of Fig. 2. With the very clean095502-2FIG. 3. The DOS of the local modes (squares) extracted fromneutron scattering data [32] compared to inh (dots) and (triangles). For illustration the boson peak calculated from thisDOS is shown by the full line in 105 meV3.FIG. 2. The linewidths  extracted from DHO fits. homextrapolated from optical Brillouin scattering is shown bya full line terminated by dots. A full line ( 	 4:2) showsthe fit to hom  inh. The dashed line ( 	 5:7) is the samefit restricted to the four lowest data points. The dotted lineillustrates the best fit with  	 2, i.e., with 2. The inset showsthe total inelastic intensities (dots) normalized to 5 h accumu-lation on the best analyzer (#2) compared to a line / v2LA.P H Y S I C A L R E V I E W L E T T E R S week ending7 MARCH 2003VOLUME 90, NUMBER 9instrumental profile and the long accumulation times,these are much more significant than previous ones [23].They are compared to v2LA (line), where vLA 	 =Q isthe phase velocity determined in this experiment. Theline is scaled to coincide with the most precise low Qpoint. One notes an excess in the observed IDHO whichgrows as Q approaches qco. One should ask whether this isdue to the use of the DHO instead of the more appropriateeffective-medium-approximation (EMA) line shape [23].To investigate this we adjust all spectra to the EMA,including those obtained above qco on the other detectors.We find that "hco 	 8:5 meV gives a good overall adjust-ment with the other parameters as in [23]. With theseparameters fixed, there remain only the intensities IEMAand ICP to fit each individual spectrum. At small Q, forexample, for Q 	 1:17 nm1, the EMA and the DHO areequivalent. On the other hand, for Q  qco, the EMA issuperior as shown in Figs. 1(e) and 1(f). This particularspectrum is the average of three similar spectra acquiredon the third analyzer at Q 	 4, 4.2, and 4:4 nm1, for atotal counting time of nearly one week. The three spectrawere independently fitted and then added, reducing thesize of the error bars. Figure 1(e) shows that the DHOdemands a larger signal at small ! than the EMA inFig. 1(f). The fit adjusts this by using a smaller ICP forthe DHO, but this is not able to represent properly theprofile. Below qco, a similar comparison is shown inFigs. 1(c) and 1(d). One notices the same trend. How-ever, this is not the reason for the excess intensity.Indeed, we find that IDHO ’ IEMA well within the errorbars for all five data points. The small dip in the center ofthe EMA fits is compensated by their more extended095502-3wings. The intensity excess seems thus real. For thepresent discussion, the main virtue of the EMA is toprovide a fair estimate for co. There is, of course, no to be extracted from this EMAwhich postulates  	 4.We now remark that the highest point in Fig. 2 is nearlyat co. Owing to the Ioffe-Regel saturation of the damp-ing, one cannot expect that inh /  extends up to thispoint. The saturation of inh is an important aspect of theEMA [23]. Therefore, it is more reasonable to fit only thelowest four points to the power law. The result is shown bythe dashed line in Fig. 2, which corresponds to  	 5:71:0 with 2 	 0:28. The extrapolation of this fit to thehighest point gives a inh nearly equal to twice the ob-served one. This is precisely the saturation expected nearcrossover [23]. The value of  obtained in this mannerdoes not change by allowing for the uncertainty in hom,only its uncertainty increases slightly. If  is really solarge, the onset of strong scattering of sound cannot bedue to Rayleigh scattering from structural inhomogene-ities which give  	 4 [9].Resonance of sound with pointlike local modes, as-suming a bilinear coupling, leads to inh / ,where  is the mode density of states (DOS) [10,14].Figure 3 shows the DOS of d-SiO2 taken from [32] aftersubtraction of a Debye contribution of 1:44 105  "h2, with "h in meV. The BP, =2, with itsmaximum near co [24], is also shown. The values ofinh and  are drawn on the same plot for comparison.They increase together with  but do not seem exactlyproportional to . There can be several reasons for this.One is that an exact determination of  with neutronscattering is always a difficult task, especially for d-SiO2as a large sample can contain a few less densifiedfragments. Another possible reason is more profound. Insilica the BP relates to rigid tetrahedra rotations [33]095502-3FIG. 4. Logarithmic presentation of Q for the opticalBrillouin scattering point [31] and the IXS data obtainedhere. The straight line gives a power  	 2:47 which averagesover different regimes and which is of no further significance.P H Y S I C A L R E V I E W L E T T E R S week ending7 MARCH 2003VOLUME 90, NUMBER 9which cannot be pointlike in this connected network [34].Thus, it produces a diffuse optic branch which hybridizeswith the acoustic branch, an effect that depends both onthe frequency and size of the excitations. Hence, theremight be too few BP modes of sufficient spatial extent (afew nm) to appreciably hybridize with acoustic phononsat q 	 1:17 nm1. The anomalous BS intensity increasenear co can also result from this hybridization. Eitherthe hybridization effectively softens the LA waves in-creasing their fluctuation amplitudes over what vLA sug-gests, or the coupled local modes give themselves acoherent contribution to the BS amplitude. Further inves-tigations are needed to clarify these points.As final remark we stress that a logarithmic presenta-tion of Q over a large range of Q can be misleading.This is illustrated in Fig. 4 for the data presented here.The line is a fit with  / Q, which gives  	 2:470:08. Such plots have often been taken as evidence for theabsence of crossover phenomena, e.g., in [2,17,27]. Theabove discussion shows that this reasoning is unfounded.In Fig. 4, the lowest Q point is purely homogeneouslybroadened, the next four points are dominated by inho-mogeneous broadening, the sixth point is near crossover,and the following three points are well above crossover.Hence, that presentation averages over three distinct re-gimes: one where  	 hom, one which is dominated byinh, and finally one where  should not be extracted fromDHO fits. The rapidly growing inh dominates  only overa narrow range in  (or Q). To be able to conclude about as we do here, one needs (i) to know the position of coand (ii) to have a sufficient number of data points withsignificant values of  just below co. This was not thecase in previous experiments, e.g., in [24] or in [35].In this Letter, we reported the first spectroscopic ob-servation of the onset of strong scattering of LA waves095502-4with q < qco in a network glass. In that region of q theinhomogeneous linewidth grows with a very high powerof , at least equal to 4, and probably higher. Also theintensity shows an anomalous increase with q. It seemsthat these results could be explained by the resonance andhybridization of the LAwaves with the local BP modes. Ifso, these measurements might provide some informationabout the structure of glasses at extended length scalesabout which so little is known otherwise.The authors thank Dr. M. Arai for the excellent sampleand Professor M. Dyakonov for an interesting discussion.[1] E. Courtens et al., Solid State Commun. 117, 187 (2001).[2] G. Ruocco and F. Sette, J. Phys. Condens. Matter 13, 9141(2001).[3] M. Rothenfusser et al., in Phonon Scattering inCondensed Matter, edited by W. Eisenmenger(Springer, Berlin, 1984), p. 419.[4] T. C. Zhu et al., Phys. Rev. B 44, 4281 (1991).[5] R. C. Zeller and R. O. Pohl, Phys. Rev. B 4, 2029 (1971).[6] J. E. Graebner et al., Phys. Rev. B 34, 5696 (1986).[7] A. K. 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B 66, 024204 (2002).[25] R. Dell’Anna et al., Phys. Rev. Lett. 80, 1236 (1998).[26] M. Montagna et al., Phys. Rev. Lett. 83, 3450 (1999).[27] G. Ruocco et al., Phys. Rev. Lett. 83, 5583 (1999).[28] L. Angelani et al., Phys. Rev. Lett. 84, 4874 (2000).[29] G. Ruocco et al., Phys. Rev. Lett. 84, 5788 (2000).[30] The spectrum at Q 	 0:97 nm1 is not usable owing to atechnical problem related to near-forward scattering.[31] E. Rat, Ph.D. thesis, University of Montpellier, 1999.[32] Y. Inamura et al., Physica (Amsterdam) 263B–264B,299 (1999).[33] B. Hehlen et al., Phys. Rev. Lett. 84, 5355 (2000).[34] K. Trachenko and M. T. Dove, J. Phys. Condens. Matter14, 1143 (2002).[35] A. Matic et al., Phys. Rev. Lett. 86, 3803 (2001).095502-4

Observation of the onset of strong scattering on high frequency acoustic
  phonons in densified silica glass

Observation of the onset of strong scattering on high frequency acoustic phonons in densified silica glass

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