VOLUME 78, NUMBER 24                   P H Y S I C A L R E V I E W L E T T E R S                                    16 JUNE 1997

Benassi et al. Reply Longitudinal collective modes asso-        [4]. The derived value, vco   4 meV, leads the authors
ciated to density fluctuations are thought to have a propa-     of the Comment to propose that the modes become
gating nature whenever there exists a linear relations          localized already at Qco   1 nm21.
between their excitation energy (i.e., their frequency V)         As the authors of the Comment explicitly recognize,
and their momentum (i.e., their wave vector Q): V               however, also with their model one finds a linear disper-
yQ. The proportionality constant y is the propagation           sion of the measured excitations in the whole Q region
velocity, generally referred to as the speed of sound [1].      where an excitation can be observed, i.e., up to energies
  Recently we reported an inelastic x-ray scattering (IXS)      more than 3 times larger than the derived crossover fre-
[2] study on the high frequency collective dynamics of          quency vco. This dispersion gives again for the peak en-
vitreous silica  y-SiO2 . There, the IXS spectra show           ergy a relation V Q    yQ, with y corresponding to the
inelastically scattered intensity strongly dispersing with      speed of sound in the Q   0 limit. Consequently, ac-
momentum transfer  Q  in the 1 4 nm21 range. At each            cording to the definition of propagation reported at the be-
Q value, the shape of the inelastic signal was successfully     ginning of this Reply, and to the definition of localization
summarized by an energy position V and a linewidth              given by the authors of the Comment elsewhere [4], there
G extracted using a damped harmonic oscillator (DHO)            is a marked contradiction among the objective existence
line shape. This data analysis gives a linear dispersion        of a linear dispersion in the whole considered data set,
for V, V Q    yQ, and a quadratic dispersion for G,             and the localization of the acoustic modes above 1 nm21
G Q    dQ2. On the basis of the previous definition of          proposed by the authors of the Comment.
propagation, and on the derived linear relation V   yQ,           In conclusion, we disagree with an interpretation that
this collective mode was recognized to be propagating           neglects a dispersion directly observable in the raw data,
in the whole considered Q region. The corresponding             and gives excessive emphasis to a specific theory in the
energy region extends up to  15 meV, a value well above         attempt to demonstrate that the collective dynamics in
the Boson peak energy of  4 6 meV. We concluded,                y-SiO2 become localized at wavelengths of 2p Qco  
therefore, that these propagating modes contribute to the       6 nm.
Boson peak found in incoherent scattering measurements.
  As it was extensively repeated, we did not want to
give a special physical meaning to the DHO model,               P. Benassi,1 M. Krisch,2 C. Masciovecchio,2 V. Mazzacurati,1
although it is more appropriate than other symmetric            G. Monaco,1 G. Ruocco,1 F. Sette,2 and R. Verbeni2
                                                                 1Universitá di L'Aquila
line shapes as the linewidth approaches the peak energy           and Istituto Nazionale di Fisica della Materia
value. The DHO was chosen only to extract spectroscopic           I-67100, L'Aquila, Italy
shape parameters without the bias of imposing theories.          2European Synchrotron Radiation Facility
Similar results are also obtained using other line shapes as      B.P. 220, F-38043 Grenoble, Cedex France
Gaussians and Lorentzians.
  In the present Comment [3] we learn that another              Received 10 December 1996            [S0031-9007(97)03362-0]
line shape can also fit our IXS data with an appropriate        PACS numbers: 63.10.+a, 61.10.Eq, 63.50.+x, 78.70.Ck
choice of the parameters. Among them, there is the
crossover frequency v                                            [1] See, for example, the recent discussion in J. Fabian and
                         co that should mark the transition          P. B. Allen, Phys. Rev. Lett. 77, 3839 (1996).
from propagation to localization. In this context, a             [2] P. Benassi et al., Phys. Rev. Lett. 77, 3835 (1996).
localized mode must have a basically Q independent               [3] M. Foret et al., preceding Comment, Phys. Rev. Lett. 78,
energy, according to the definition of localization given by         4669 (1997).
the authors of the present Comment in their previous work        [4] M. Foret et al., Phys. Rev. Lett. 77, 3831 (1996).
















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