Scaling the Temperature-dependent Boson Peak of Vitreous Silica with the high-frequency Bulk Modulus derived from Brillouin Scattering Data

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by  B. Rufflé

Scaling the Temperature-dependent Boson Peak of Vitreous Silica
with the high-frequency Bulk Modulus derived from Brillouin Scattering Data
1 1 1 1 1 2 2
B. Ruffl´e , S. Ayrinhac , E. Courtens , R. Vacher , M. Foret , A. Wischnewski , and U. Buchenau
1Laboratoire des Collo¨ıdes, Verres et Nanomat´eriaux, UMR 5587 CNRS
Universit´e Montpellier II, F-34095 Montpellier Cedex 5, France
2Institut fu¨r Fesko¨rperforschung, Forschungszentrum Ju¨lich, Postfach 1913, D-52425 Ju¨lich, Germany
(Dated: September18, 2009)
The position and strength of the boson peak in silica glass vary considerably with temperature
0 T. Such variations cannot be explained solely with changes in the Debye energy. New Brillouin
1 scattering measurements are presented which allow determining the T-dependence of unrelaxed
0 acoustic velocities. Usinga velocity based on thebulk modulus,scaling exponentsarefound which
2 agreewiththesoft-potentialmodel. Theunrelaxedbulkmodulusthusappearstobeagoodmeasure
for thestructural evolution of silica with T and to set theenergy scale for the soft potentials.
n
a
J PACSnumbers: 63.50.Lm,64.70.kj,78.35.+c,78.70.Nx

5
2
The natureof collectivevibrations inglassesandtheir silica indicate a progressive and reversible polyamorphic

relationto structuraldisorderaretopicsofactivediscus- transformation related to the reorientations of the –Si–
]
n sion and considerable interest. The reduced density of O–Si– bonds forming ring structures, this without bond
n vibrational states, g(ν)/ν2, where ν = ω/2π is the fre- breaking or reconstruction [15]. The BP evolution pre-
-
s quency, generally shows an excess over the Debye level sumablyrelatesto thattransformation. Agoodmeasure
di gD(ν)/ν2 calculated from the acoustic velocities. This for the degree of transformation might be a suitably de-
. excess, I(ν) (g gD)/ν2, is called the boson peak fined elastic modulus. One should recall that the elastic
t ≡ −
a (BP). It is generally agreed that the BP must bear re- propertiesofglassesatultrasonicfrequenciesareaffected
m
lation to the strong scattering of acoustic modes leading by thermally activated relaxations (TAR) of structural
- to the plateauin the temperature (T)dependence of the defects, as known for over half-a-century [16]. Further-
d
thermal conductivity, a feature universally observed in more,theanharmoniccouplingofsoundwiththethermal
n
o dielectric glasses [1]. Two broad categories of processes bath depresses the sound velocities with increasing T,
c aremainlyinvokedtoexplaintheseanomaliesassumma- alsoobservedlong ago[17]. A suitably defined structure-
[ rized e.g. in [2, 3]: 1) the interaction of acoustic modes dependent modulus should not include these viscoelas-

1 with structural or elastic disorder, and 2) the presence tic effects. The particular case of silica was recently re-
v of additional vibrations and their resonant coupling to visited on the basis of available and new measurements
7 acoustic waves, as described e.g. by the soft-potential of sound velocity and attenuation covering a very broad
4
model(SPM)[4]orrelateddevelopments[5]. Severalau- range of ν and T [18]. The relaxations can be described
3
thorsrecentlyattempted scalingBPdatainterms ofthe by double-well potentials with a distribution of barriers
4
. Debye density of states, e.g. [3, 6–9]. They advocated and asymmetries [19]. The appropriate distribution and
1
that such a scaling supports the view that BP modes theeffectofanharmonicityweredeterminedin[18]. This
0
0 are strictly acoustic. We examine here the case of silica, allowed extracting an unrelaxed or bare velocity for the
1 a prototypical glass of high technical interest which ex- longitudinal acoustic mode, v∞LA, which was found to in-
: hibits the strongest known BP excess [10]. We find that crease considerably with T [18]. As shown below, the
v
TA
i scaling with the Debye velocity is inappropriate, while same is now observedon the transverse mode, v∞ . The
X
scalingin terms ofthe bulk modulus leads bothto a sat- bare bulk modulus, K∞, also increases strongly with T.
ar isfactory master curve and to exponents that are com- Changes in the corresponding “velocity”, v∞K ∝ √K∞,
patiblewiththeSPM.Weproposethatanappropriately couldprovideameasurefortheprogressinthepolyamor-
determinedbulkmodulusisagoodmeasureforthestruc- phictransformationwithincreasingT. InthisLetter,we
tural evolution of silica with T. Further, the relation to showthatindeedtheBPofsilicasuccessfullyscalesunto
the SPM implies that the second category of models is a single master curve with the use of vK(T), this with
∞
here the relevant one. exponents that are non-trivial. Our results indicate that
the bare modulus is a good measure for the structural
Silica is a good candidate for a meaningful scaling of
evolution in function of T, and suggest that the latter
the BP in function of T, since both the BP position,
affects the strength and position of the BP.
νBP, and strength, IBP I(νBP), vary significantly with
≡
T [11]. The glass exhibits anomalous thermomechanical The symbols in Figure 1 present Brillouin-scattering
properties that are typical for tetrahedral networks [12]. results that are new for LA waves at elevated T as well
Among them, the elastic modulidecreaseunder pressure as for TA waves over the entire range. The data were
[13] and harden with increasing T [14]. Simulations of obtained on a high-quality silica sample of low OH con-
2

7000 14
6800 v LA a-SiO ν) 6
666246000000 vv∞ 1 3LL 4TAA GHHzz (a) -3Hz) 1102 2 ν) / g(g(D2345
m/s) 56800000 ν) (T 8 1
ocities ( 444012000000 vv ∞ TTAA 3×10 I( 46 11505105 K KK 00.0 0.1 ν0 /.2 νD 0.3 0.4
el 1 THz 271 K
und v 33890000 v 1T5A GHz (b) 2 334152833 KKK 568277333 KKK 1104 K
o 0
S 3700 0 1 2 3 4
5200
5000 v K Frequency ν (THz)
∞
4800 v D
4600 ∞ FIG. 2: The neutron scattering BP of silica at 11 different
4400 v D temperatures. TheDebyelevelat 1 THz calculated from the
4200 1 THz (c) velocityinFig. 1cisalreadysubtracted. Theinsetillustrates
4000 a scaling of the entire g(ν) using the Debye frequency at 1
3800 THz, theBP frequency.
0 200 400 600 800 1000
Temperature (K)
frequencies. It should be remarked that in the absence
FIG. 1: The T-dependence of the sound velocities measured
in silica with Brillouin scattering (points) and renormalized of structural changes with T and with negligible density
to1THz(dashedlines)andtoinfinitefrequency(solidlines): changes, the bare velocities should be independent of T.
(a)theLAmode;(b)theTAmode;(c)thecalculated Debye The observed dependence is thus a signature of the pro-
velocities vD at two frequencies compared to the bare bulk- gressive polyamorphic transformation [15]. As already
modulus “velocity” v∞K. mentioned in [18, 20], the velocities v∞ might be hard
to directly observe. The reason is the interaction with
theBP,asdescribede.g. in[2,23]. However,atconstant
densitythesevelocitiesdirectlyrelatetomicroscopicelas-
centration( 100ppm)usingthehigh-resolutiontandem
interferomet≤er described in [20]. The Brillouin shifts Ω tic stiffnesses. Based on v∞LA and v∞TA one can construct
otherquantities. Iftheinterestisinthedensityofacous-
andhalf-widthsΓaremeasurednearbackscattering(LA)
◦ tic modes, one considers the unrelaxed Debye velocity
or at 90 (TA). Care is taken to eliminate the spectral
vD given by 3/(vD)3 = 1/(vLA)3+2/(vTA)3. If instead
broadening due to the finite aperture, which is an im- ∞ ∞ ∞ ∞
◦ the interest is in the average rigidity of the structure at
portant correction at 90 . The shifts are converted to
sound velocities using the known T-dependence of the short distances, one can consider a “velocity” v∞K given
by (vK)2 =(vLA)2 4(vTA)2 since the bulk modulus K
refractive index [21]. Following [18], the internal fric- ∞ ∞ − 3 ∞ 4
tion Q−1 = 2Γ/Ω allows calculating the contributions relates to the elastic constants byDK =CK11− 3C44. The
of TAR and anharmonicity to the velocities, δvTAR and very different T-dependence of v∞ and v∞is emphasized
inFig. 1c. Wenowexploretherelationbetweenthebare
δvANH, respectively [22]. Correcting the data points in
velocities and the BP position and strength.
Fig. 1 for these velocity shifts, the solid lines repre-
senting the bare velocities for both LA and TA modes Figure 2 shows measurements of the excess density of
are obtained. Near and above room T, it is the anhar- vibrational states of silica in the BP region. The data
monic term δvANH = vQ−1/Ωτth which dominates by were obtained with neutron scattering as described in
−
far the velocity corrections needed to obtain the bare [11]. For selection rule reasons, it is most important to
values [18]. The principal source of uncertainty is in the usehereneutrondataratherthanRamanscatteringones
mean thermal relaxation time τth. ¿From [18, 21] we as available e.g. in [24]. Indeed, SiO4-libration modes
estimate that the uncertainty in lnτth is at most 0.1, that are inactive in Raman scattering are important to
±
which leads to the same uncertainty on δvANH/v. The the BP [25], as also confirmed in a hyper-Raman study
dashed lines show the velocities calculated at the inter- [26]. The ordinate of Fig. 2 shows the excess I(ν), ob-
mediate frequency of 1 THz corresponding to the ap- tained by subtracting from the various curves the Debye
proximate position of the BP maximum, νBP. The T- level, gD(ν)/ν2. For this calculation we used a constant
dependence of the bare velocities, v∞LA and v∞TA, is con- DebyewavevectorkD =1.576 1010m−1asthevariation
×
siderably stronger than observed at Brillouin-scattering ofthe atomic density with T is comparativelynegligible.
3

1.2 14
a×(v K )α β a-SiO 51 K
THz) 1.1 ν ∞ 1K)] 12 2 110505 KK
( BP 5 10 271 K
νBP1.0 α = 0.96 ± 0.08 v K (∞ 331583 KK
8
/ 423 K
0.9 T) α = 1 523 K
-3z) 13 β = -1.29 ± 0.11 v K (∞ 6 β = -4/3 687733 KK
TH 12 / [ 4 1104 K
( ) ν
I BP 11 IBP (I 2
× 10
3 b×(v K )β 0
10 9 ∞ 0 1 2 3 4
0 200 400 600 800 1000 1200 ν / [v K (T) / v K (51K)]α
∞ ∞
Temperature (K)
FIG. 4: The BP data of Fig. 2 scaled with exponents α=1
FIG.3: Theexperimentally determinedBPpositions andin- and β =−4/3.
tensities in function of T, adjusted topowers of v∞K.
solute values do not affect the scaling exponents to be
TheDebyelevelisthen3/νD3,withνD =vDkD/2π,where determined below. As observed in Fig. 3, it is remark-
vD istakenfromFig. 1cat1THz. OnenoticesthatI(ν) able that νBP increases by as much as 24%and that IBP
does not approach 0 at ν = 0. The reason lies in quasi- decreasesby39%overthisT range. Overthesamerange
elasticscattering(QES),alowfrequencystronglyanhar- of T, v1DTHz only increases by 7%, while v∞D increases by
monic excess that scatters in addition to the harmonic 11%. Here, a Debye scaling implies that νBP νD vD
−3 D −3 ∝ ∝
BP. The two components can be separated based on the and that IBP νD (v ) . While the latter happens
anharmonicity, revealing that QES “decreases with in- to be approxi∝mately∝verified with v∞D, the former can-
creasing ν and is undetectable above 600 GHz at room not. This shows that checking the validity of the Debye
T and below” [27]. The QES contribution also goes scaling can be a delicate matter to which we return in
through a maximum below room T, while it decreases the final discussion. At any rate this scaling does not
anddoesnotshowanyadditionalbroadeningatelevated work in silica. This is not so surprising. Indeed, there is
T [28, 29]. In view of this, and as it would be difficult nowample evidence thatthe BP ofsilica doesnot derive
subtracting QES from the data, we rather leave it but its strengthfromacousticmodes,asalreadyknownfrom
do not insist that scaling applies below 0.65 THz. The neutron scattering [25] and hyper-Raman [26] results.
inset of Fig. 2 zooms on the BP region, using a scaled The data points of Fig. 3 can be adjusted to a bare
abscissa, ν/νD, and a scaled ordinate, g/gD. This Debye velocity v∞ with
scaling does not lead to a satisfactory master curve, not
νBP(T)=a[v∞(T)]α , IBP(T)=b[v∞(T)]β . (1)
so much because of a poor scaling of the intensities, but
mainly because the BP positions do not superpose. This We remark that if this works for one particular type of
situation is not significantly improved if one used v∞D in v∞,itwillworkforall. Indeed,overtherestrictedranges
place of v1DTHz, as will become clear below. of interest here, we observe the approximate relations
We now obtain from Fig. 2 the T-dependence of νBP v∞K (v∞LA)1.43 (v∞D)2.13 (v∞TA)2.27. We note that
K ∼ ∼ ∼
and IBP that are shown in Fig. 3. To this effect, the v∞ increases by 25% over the range of T, rather similar
successive curves are scaled to the first one taken as ref- to the increase in νBP. It seems thus appropriateto first
erence. The data at 51 K are indeed least affected by try v∞K in (1). This gives the solid lines traced in Fig.
QES. Specifically, the curve at T is scaled by replacing 3 with the exponents shown there. The scatter in the
ν by ν/x and I by I/y. Its difference with the 51 K experimentalpointsdoesnotresultfromthescalingpro-
curveisthenminimizedbyaleast-squareprocedureover cedure described in the previous paragraph, but rather
the range from νBP 0.35 THz to νBP +1 THz. The from the neutron data themselves. The uncertainty in
BP parameters at T−are then νBP(T) = x νBP(51 K) v∞K related to τth leads to variations in both α and β
and IBP(T) = y IBP(51 K). To obtain the values on that are about half the error bars given in Fig. 3. The
an absolute scale, it remains to estimate νBP(51K) and exponents are nearly α = 1 and β = 4/3, well within
−
IBP(51K). This is done by fitting the 51 K data to a these error bars. Using the latter values, the entire data
log-normal, IBPexp[ (logν/νBP)2/2σ2]. Although this scales as shown in Fig. 4. Except for the region below
−
is somewhat ad hoc, it is of no real importance since ab- 0.75 THz which is affected by QES, the scaling is ob-
∼
4
viouslyverysatisfactory. Itis nowofinteresttoconsider Our results show that for silica in function of T a De-
the meaning of these exponents within the SPM. byescalingofthe largeexcursionsinνBP(T)andIBP(T)
The progressive polyamorphic transformation of silica is not possible. A scaling can be performed in terms of
occurs without any change in the network connections unrelaxed velocities v∞. The exponents that are found
[15]. Hence, the number of defects producing quasi-local usingabarevelocitybasedonthebulkmodulus,vK,are
∞
vibrationsshouldnotchange. Itistheenvironmentofthe remarkably compatible with the existence of quasi-local
soft harmonic oscillators which is modified. The latter vibrations described by the soft-potential model. It thus
are characterized by an energy 0 = Mv2 [4], where M seems that the unrelaxed bulk modulus provides a good
E
is the mean atomic mass. The unspecified velocity v measure for the T-dependent polyamorphic transforma-
entering 0 is certainly anunrelaxedv∞. Assuming that tion of silica and that it plays a key role in setting the
E
only 0 changes with T, and using Eqs. (1.5) of [4], one scale for the the soft potentials.
E
obtains
ηL v∞−2/3 , W v∞2/3 . (2)
∝ ∝
Here, ηL is the small parameter that scales the ki- [1] R.C. ZellerandR.O.Pohl, Phys.Rev.B4,2029 (1971).
netic energy of the soft potential Hamiltonian, and W [2] B. Ruffl´e, D.A. Parshin, E. Courtens, and R. Vacher,
is the crossover energy between vibrational and tunnel- Phys. Rev.Lett. 100, 015501 (2008).
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strength of its low frequency (ν νBP) onset. This is [4] D.A. Parshin, Fiz. Tverd. Tela (Leningrad) 36, 1809
≪ (1994) [Sov. Phys. Solid State36, 991 (1994)].
seenbycomparingEqs. (5.12)and(5.18)of[5]. Onecan
[5] D.A. Parshin, H.R. Schober, and V.L. Gurevich, Phys.
thus use a well-known expression for the onset, which is Rev. B 76, 064206 (2007).
that I(ν) ν2/W5 [30], up to ν = νBP to derive the [6] A. Monaco et al.,Phys. Rev.Lett. 96, 205502 (2006).
∝
scaling. This gives [7] A. Monaco et al.,Phys. Rev.Lett. 97, 135501 (2006).
[8] K. Niss et al., Phys.Rev.Lett. 99, 055502 (2007).
2 5
νBP/IBP W . (3) [9] S. Caponi et al.,Phys. Rev.Lett. 102, 027402 (2009).
∝
[10] A.P. Sokolov et al.,Phys. Rev.Lett. 78, 2405 (1997).
Introducing (1) and (2) in (3), one obtains
[11] A. Wischnewski, U. Buchenau, A.J. Dianoux, W.A.
Kamitakahara,andJ.L.Zarestky,Phys.Rev.B57,2663
2α β = 10/3 . (4)
− (1998).
[12] J.T. KrauseandC.R. Kurkjian,J.Am.Ceram. Soc.51,
Thispreciselyagreeswiththevaluesα=1andβ = 4/3
foundaboveusingv∞K forscaling. This suggeststha−tthe [13] 2M2.6R(.1V96u8k)e.vich,J. Non-Cryst.Solids, 11, 25 (1972).
bulk modulus gives in the present case a sufficiently ap- [14] A. Polian et al.,Europhys. Lett.57, 375 (2002).
propriatemeasurefortheaverageinteractionsofthesoft- [15] Liping Huang and J. Kieffer, Phys. Rev. B 69, 224203
potentialswiththeirenvironment. 0beingcontrolledby (2004).
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38, 125 (1955).
mostly hydrodynamic-like on the average.
[17] T.N. Claytor and R.J. Sladek, Phys. Rev. B 18, 5842
ComparedtosilicainfunctionofT,inthesilicatesthat
(1978).
were investigated for scaling,the relative range of νBP is [18] R.Vacher,E.Courtens, and M.Foret, Phys.Rev.B72,
smaller. It is about 6% for the three curves that scale 214205 (2005).
in [7], less in [6], and nil in [3], while it is 24% presently. [19] K.S. Gilroy and W.A. Phillips, Philos. Mag. 43, 735
That makes checking for the validity of a Debye scaling (1981).
all the more demanding. It would require a stringent [20] E. Rat et al.,Phys. Rev.B 72, 214204 (2005).
[21] S.Ayrinhac,Doctoralthesis,Univ.ofMontpellier2,Nov.
analysis both of the peak positions and of the intensity
2008 (unpublished).
which is in excess over the Debye level. In particular
[22] With the new Γ values of the present measurements,
in [3], there is no change in νD and thus no possibility theconstant C scaling theTARcontribution in [18] now
to check the scaling law. By comparison, there exists equals 1.9×10−3 [23], thesame for LA and TA modes.
one report of a failure of the Debye scaling tested on a [23] A. Devoset al.,Phys.Rev. B 77, 100201(R) (2008).
polymerunderpressure[8]. Asimilarconclusionwasan- [24] A. Fontana et al.,Europhys.Lett. 47, 56 (1999).
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a successful Debye scaling of Raman scattering data on
[27] U. Buchenau et al., Phys.Rev.Lett. 60, 1318 (1988).
a reactive mixture during polymerization [9]. However, [28] A.P. Sokolov et al.,Europhys. Lett.38, 49 (1997).
this is a complicated physico-chemical situation so that [29] A.Fontanaetal.,J.Non-Cryst.Solids351,1928(2005).
the significance of the result is momentarily not under- [30] M.A. Ramos et al.,phys.sta. sol. (a) 135, 477 (1993).
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available scaling evidence that the origin of boson peaks
in glasses is necessarily acoustic.