VOLUME 77, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 28 OCTOBER 1996
Evidence of High Frequency Propagating Modes in Vitreous Silica
P. Benassi,1 M. Krisch,2 C. Masciovecchio,2 V. Mazzacurati,1 G. Monaco,1 G. Ruocco,1 F. Sette,2 and R. Verbeni2
1Universitá di L'Aquila and Istituto Nazionale di Fisica della Materia, I-67100, L'Aquila, Italy
2European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble, Cedex France
(Received 11 June 1996)
High energy resolution inelastic x-ray scattering measurements in the 1 6 nm21 momentum transfer
(Q) region in vitreous silica (y-SiO2) at T 1050 K show the existence of collective excitations
propagating with a sound velocity of 5800 6 200 m s up to Q 3.5 nm21. The linewidths of the
excitations, which are found to obey a Q2 law, are consistent with previous determinations made at low
Q and low T. The picture of the atomic dynamics in y-SiO2 emerging from this study indicates that
these propagating modes must contribute to the boson peak. [S0031-9007(96)01466-4]
PACS numbers: 63.10.+a, 61.10.Eq, 63.50.+x, 78.70.Ck
The detailed understanding of the dynamical properties to its origin. Moreover, the comparison of the present
of topologically disordered systems, like glasses, is still data with light-scattering data at T 300 K shows that the
an open question. At low momentum transfer (Q) val- linewidth of these excitations is temperature independent,
ues, the existence of propagating collective excitations is suggesting that the peaks broadening is not associated with
demonstrated by the sharp Brillouin lines observable in dynamical processes, but to the ill-definition of Q as a good
light-scattering experiments. This is straightforward from quantum number.
an intuitive point of view since at low Q's one samples The experiment was carried out at the new very high en-
particle-particle correlations on a long time and large space ergy resolution inelastic x-ray scattering beam line (BL21-
scales with respect to interatomic motions and distances. ID16) at the European Synchrotron Radiation Facility.
In the mesoscopic time-space domain the situation is This instrument is based on backscattering from high or-
more complicated, and the existence of collective dynam- der reflections in perfect silicon crystals, and in this work
ics is doubted. The evidence of modes in glasses in the we used the Si(999) reflection at 17.794 keV. The to-
mesoscopic region comes from incoherent neutron scatter- tal instrumental resolution function was measured using a
ing and light scattering studies [1], where a broad band Plexiglas scatterer at the maximum of its static structure
is found around 2 to 10 meV, almost independently from factor where the scattering is dominated by the elastic com-
the material. This band has been named boson peak be- ponent; the energy resolution, full width at half maximum
cause its intensity scales with temperature approximately (FWHM), was 2.8 6 0.2 meV. The momentum transfer,
according to the Bose-Einstein statistics. The relaxational Q 2k0 sin us 2 (where k0 and us are the wave vector of
or vibrational character of the excitations giving rise to the the incident photon and the scattering angle, respectively),
boson peak is highly debated [1], especially in view of the was selected between 1 and 6 nm21. The Q resolution
fact that the extrapolation of the dispersion relation found was set to 0.3 nm21 by an aperture in front of the ana-
at small Q to the mesoscopic Q region would give exci- lyzer crystal. Energy scans were performed by varying the
tation energies similar to those of the boson peak. In this relative temperature between the monochromator and ana-
region, the experimental determination of the dynamical lyzer crystals. Each scan took about 120 min, and each Q
structure factor S Q, E became only recently possible, point was obtained by typically averaging four scans. The
thanks to the development of inelastic x-rays scattering data were normalized to the intensity of the incident beam.
(IXS) with meV energy resolution. It was shown for some Further details on the beam line are reported elsewhere
"intermediate" [2] and "fragile" [3] glasses that propagat- [2,69].
ing excitations exist up to energies comparable to that of The SiO2 suprasil sample, purchased from Goodfellow,
the boson peak. Similar determinations have not been at- was a 2 mm diameter rod. Its dimension was comparable
tempted yet on "strong" network-forming glasses. Among to the x-ray photoabsorption length and gave negligible
them, vitreous silica, y-SiO2, is probably the archetype [4]. multiple scattering. A preliminary experiment performed
In this Letter we present the measurement of the S Q, E at room temperature on y-SiO2 showed very weak IXS in-
of y-SiO2 at T 1050 K, in the 1 6 nm21 momentum tensity, strongly merged into the tails of the central peak,
transfer range. We report the existence of collective modes thus not allowing a determination of the spectral shape
in the whole investigated Q range. These modes are with the necessary accuracy [10]. To increase the expected
found to propagate with a velocity of sound y 5800 6 inelastic scattering signal we performed the measurements
200 m s up to Q 3.5 nm21 and E 13 meV. Hence, at about 1000 K. This gave an enhancement of the excita-
the energies spanned by these excitations cover the boson tions with E 10 meV by a factor of 3.5. The y-SiO2
peak region [5], thus indicating that they must contribute rod was placed inside a graphite tube ( 20 mm in length,
0031-9007 96 77(18) 3835(4)$10.00 © 1996 The American Physical Society 3835
VOLUME 77, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 28 OCTOBER 1996
0.2 mm wall thickness, and 3 mm outer diameter), which and width, we have fitted the spectra to the convolution of
was resistively heated in vacuum. A 1 mm diameter hole the experimental resolution function with the model func-
orthogonal to the graphite cylinder axis allowed the pas- tion F Q, E given by
sage of the incoming and scattered beams without scat-
tering from the graphite. The temperature of the sample, F Q, E Io Q d E 1 n E 1 1 I Q
monitored with an optical pyrometer, was 1050 6 50 K. EG Q V Q
3 . (1)
The glassy structure of the sample was checked several V Q 2 2 E2 2 1 G Q 2E2
times measuring in situ at T 1050 K the static structure
factor S Q , which was found to be in agreement with pre- It consists of a d function to account for the elastic scatter-
vious determinations [4]. ing, and a damped harmonic oscillator (DHO) model [11]
A representative inelastic x-ray scattering spectrum of
y-SiO2 at T 1050 K is reported in Fig. 1 together
with the fit discussed in the following. The spectrum is
dominated by an intense elastic peak, whose shape is de-
termined by the instrumental resolution. At the sides of
the central peak, well above the tails of the elastic scat-
tering broadened by the resolution function, the inelas-
tic signal is clearly visible. The inset of Fig. 1 shows a
close-up of the region where the inelastic intensity is bet-
ter visible, together with the residual of the fit reported in
standard deviation units. This inelastic signal is due to
collective density fluctuations, and it is observed at all in-
vestigated Q points. The inelastic x-ray scattering spectra
taken in the 1 # Q # 4 nm21 region are shown in Fig. 2.
In each spectrum, the dashed line representing the resolu-
tion function is shown aligned with the central peak. The
energy position of the inelastic intensity is characterized
by a clear Q dependence, indicating the propagating na-
ture of the excitations. To determine their energy position
FIG. 1. Inelastic x-ray scattering spectrum of y-SiO2 at T
1050 K and Q 1.5 nm21 (open circles). The dashed line FIG. 2. X-ray spectra of y-SiO2 at T 1050 K taken at
represents the instrumental resolution function, shown aligned different Q values. The data are shown together with their
with the central peak to emphasize the presence of inelastic best fits (full line) and the individual contributions to the fitting
scattering. The full line is the best fit to the data as discussed function: elastic peak (dashed line), and inelastic components
in the text. In the inset we report (a) the expansion of the (dotted line). To emphasize the presence of the inelastic
region where the inelastic signal is better visible and (b) the contribution, the vertical scale has been chosen to show the
residual of the fit in standard deviation units. tails of the central peak.
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VOLUME 77, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 28 OCTOBER 1996
for the two side peaks. Here Io Q and I Q are related to magnitude smaller, provides a convincing indication that
the intensities of the central peak and of the inelastic con- (i) the Q2 dependence of G is genuine, and (ii) the tem-
tributions, respectively, V Q and G Q are the excitation perature dependence of the excitations linewidths is negli-
energy and width, and n E is the Bose factor. The DHO gible in the whole 3001000 K temperature range.
model function has already been successfully applied to We have measured the S Q, E of y-SiO2 also in the
describe the shape of the S Q, E of disordered systems in Brillouin light-scattering (BLS) region to further increase
previous inelastic neutron [12] and x-ray [2,6,9] scattering the explored Q range. The experiment was carried out us-
studies, and in a simulated model glass [13]. A standard ing the 514.5 nm line of an Argon ion laser in backscatter-
x2 minimization procedure has been utilized to determine ing geometry and a SOPRA DMDP2000 monochromator
the relevant fitting parameters, namely, V Q , G Q , and operating with 3.5 meV total energy resolution [16]. The
R Q , which is the ratio between inelastic and elastic inte- room temperature spectrum of the same y-SiO2 sample
grated intensities. utilized in the IXS measurement is reported in the inset
The values of V Q obtained from the fit are reported in of Fig. 5. In this figure, we also show the Stokes Bril-
Fig. 3 for Q # 3.5 nm21. We observe a linear dispersion louin line in detail, together with the instrumental resolu-
in Q with a sound velocity y 5800 6 200 m s (dotted tion function. The FWHM of the Brillouin line measured
line), a value consistent with the 6050 m s one derived at Q 0.036 nm21 is reported in Fig. 4 (open square): its
from elastic constant measurements at T 1050 K [14] value is also consistent with the G Q ~ Q2 law. Another
(dot-dashed line). At Q larger than 4 nm21, the fitting important consistency between the high-Q (IXS) and low-
procedure was no longer stable probably due to the large Q (BLS) measurements is found for the inelastic to elastic
broadening of the excitations. It was therefore not possible intensity ratio, R Q , as seen in Fig. 6, where values mea-
to determine a reliable value for V Q . To estimate G Q sured with IXS (full dots) and BLS (open dot) are reported.
at Q $ 4 nm21, we imposed to V Q in the 4 # Q # In conclusion, we have shown for y-SiO2 that: (i) Propa-
6 nm21 region two extreme values, namely, V Q yQ gating modes exist in the 1 3.5 nm21 Q region, and
and V Q V Q 3.5 nm21 13 meV. The corre- their sound velocity is 5800 6 200 m s at T 1050 K;
sponding two values obtained for G Q at each Q were this value is consistent with low-Q measurements [14].
equivalent within their error bars. The values of G Q are (ii) At Q $ 4 nm21, i.e., for wavelengths l # 1.5 nm,
reported in Fig. 4 in a log-log scale (open circles). The the width of the excitations becomes comparable to their
dotted line represents the best fit to the IXS data, and it energy. (iii) The width of the excitations follows a Q2
has a slope of 1.95, thus indicating that G Q ~ Q2 in law, matching the POT and BLS measurements at lower
the whole examined Q range. As observed in Fig. 4, this Q. (iv) The widths of the excitations do not change in the
Q2 behavior overlaps with the extrapolation in the IXS Q 3001050 K temperature range. This last result confirms
range of the linewidths obtained at low Q using the pi- the T independence of the linewidths already observed in
cosecond optical technique (POT) on y-SiO2 at room tem- the 100 300 K region [15].
perature [15]. The excellent consistency between our data,
taken at T 1050 K and at Q $ 1 nm21, with the data of
Ref. [15], taken at T 300 K and at Q values 1 order of
FIG. 4. Full width at half maximum of the excitations
measured with different techniques. Open circles represent the
parameter G Q of the DHO model [Eq. (1)] obtained from
the fit of the 1050 K IXS data. The error bars for the data
FIG. 3. Excitations energy, V Q , from the DHO model for at Q 4, 5, and 6 nm21 represent the range of variability
the data taken up to Q # 3.5 nm21; see text. The dotted line obtained from the different choices of V Q , as discussed in
is the best fit to V Q and has a slope of 5800 m s; the dot- the text. The open diamonds refer to the 300 K POT data of
dashed line has the slope of 6050 m s; i.e., the sound velocity Ref. [15], while the open square refers to the 300 K Brillouin
derived from the elastic constant at zero frequency measured at light-scattering (BLS) data. The dashed line, with a slope of
1050 K [14]. 1.95, is the best fit to the IXS data.
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VOLUME 77, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 28 OCTOBER 1996
In this picture, at decreasing wavelengths, the disorder re-
duces the definition of Q as a good quantum number. It
is interesting to point out that the Q2 law of the linewidth
found here does not have, to our knowledge, a theoretical
explanation. (iv) For Q $ 4 nm21, it is no longer possi-
ble to establish the propagating or nonpropagating nature
of the excitations. It is worth mentioning, to this regard,
that Q 4 nm21 corresponds to a wavelength value com-
parable to the structural correlations expected in SiO2 at
mesoscopic lengths: therefore, the disappearance of exci-
tations with a definite E-Q relation could mark the onset
of a new dynamical regime localized in strongly bonded
nanovolumes of the material [4,5].
This scenario is very similar to that found in other more
FIG. 5. Brillouin light-scattering measurement of the phonon
broadening in y-SiO fragile glasses [2,3], and maybe one can start to speculate
2 at 300 K. The longitudinal Stokes
Brillouin line is reported (open circles) together with the fit on the universality of the dynamics of topologically disor-
(full line) of the data to the convolution of the resolution dered systems.
function (dashed line) with a Lorentzian modeling the S Q, E . We acknowledge O. Consorte, W. Galli, B. Gorges,
EL 141 meV is the energy position of the Brillouin line, as K. Martel, and J. F. Ribois for their technical assistance,
indicated in the inset where the whole spectrum is also shown. and M. Sampoli and G. Signorelli for useful discussions.
The scenario for the mesoscopic dynamics of y-SiO2
emerging from this work can then be summarized as fol- [1] See Dynamics of Disordered Materials II, edited by
lows: (i) From long wavelengths down to l 1.5 nm A. J. Dianoux, W. Petry, and D. Richter (North-Holland,
there are density fluctuations described by phononlike, Amsterdam, 1993).
propagating, quasiharmonic modes with nonlocalized ei- [2] C. Masciovecchio, G. Ruocco, F. Sette, M. Krisch,
R. Verbeni, U. Bergmann, and M. Soltwisch, Phys. Rev.
genvectors. This result complements earlier inelastic neu- Lett. 76, 3356 (1996).
tron work showing that the dynamics at energies above [3] L. Borjesson et al. (to be published).
2 meV and Q above 10 nm21 have a dominant harmonic [4] A. C. Wright, J. Non-Cryst. Solids 179, 84 (1994).
component [17]. (ii) The propagating dynamics extends [5] F. Terki, C. Levelut, M. Boissier, and J. Pelous, Phys.
to the same energy range as the boson peak, and there- Rev. B 53, 1 (1996).
fore it must contribute to this debated spectral feature, and [6] F. Sette, G. Ruocco, M. Krisch, U. Bergmann, C. Mas-
maybe it is at its origin. (iii) The temperature indepen- ciovecchio, V. Mazzacurati, G. Signorelli, and R. Verbeni,
dence of the excitations linewidth is likely to be related to Phys. Rev. Lett. 75, 850 (1995).
the structural disorder in y-SiO [7] C. Masciovecchio, U. Bergmann, M. Krisch, G. Ruocco,
2, also temperature inde-
pendent in this region, rather than to dynamical processes. F. Sette, and R. Verbeni, Nucl. Instrum. Methods Phys.
Res., Sect. B 111, 181 (1996).
[8] R. Verbeni, F. Sette, M. Krisch, U. Bergmann, B. Gorges,
C. Halcoussis, K. Martel, C. Masciovecchio, J. F. Ribois,
G. Ruocco, and H. Sinn, J. Synchrotron Radiat. 3, 62
(1996).
[9] F. Sette, G. Ruocco, M. Krisch, C. Masciovecchio, and
R. Verbeni, Phys. Scr. (to be published).
[10] F. Sette and G. Ruocco (private communication).
[11] B. Fak and B. Dorner, Institute Laue Langevin, Grenoble,
France, Report No. 92FA008S, 1992.
[12] J. Teixeira, M. C. Bellissant-Funel, S. H. Chen, and
B. Dorner, Phys. Rev. Lett. 54, 2681 (1985).
[13] G. Ruocco and M. Sampoli (private communication).
[14] Handbook of Glass Properties, edited by N. P. Bansal and
R. H. Doremus (Academic Press, London, 1986).
[15] T. C. Zhu, H. J. Maris, and J. Tauc, Phys. Rev. B 44, 4281
FIG. 6. The inelastic to elastic ratio R Q is reported for (1991).
the IXS data in y-SiO [16] V. Mazzacurati, P. Benassi, and G. Ruocco, J. Phys. E 21,
2 at T 1050 K (full circles) and
for the BLS experiment at Q 0 (open circle). The value 798 (1988).
R Q 0.04, measured with BLS at room temperature, is [17] U. Buchenau, H. M. Zhou, N. Nucker, K. S. Gilroy, and
multiplied for 1050 300 to account for the linear temperature W. A. Phillips, Phys. Rev. Lett. 60, 1318 (1988), and
dependence of the inelastic signal. references therein.
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