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HIGH RESOLUTION NEUTRON SPECTROSCOPY OF THE CROSSOVER IN THE VIBRATIONAL
DENSITY OF STATES OF SILICA AEROGELS
H. Conrad, J. Fricke, G. Reichenauer
To cite this version:
H. Conrad, J. Fricke, G. Reichenauer. HIGH RESOLUTION NEUTRON SPECTROSCOPY OF THE
CROSSOVER IN THE VIBRATIONAL DENSITY OF STATES OF SILICA AEROGELS. Journal
de Physique Colloques, 1989, 50 (C4), pp.C4-157-C4-162. �10.1051/jphyscol:1989425�. �jpa-00229501�
REVUE D E PHYSIQUE APPLIQU~E
Colloque C4, Suppl6ment au n04, Tome 24, avril 1989
HIGH RESOLUTION NEUTRON SPECTROSCOPY OF THE CROSSOVER IN THE VIBRATIONAL DENSITY OF STATES OF SILICA AEROGELS
H. CONRAD, J. FRICKE' and G. REICHENAUER*
Institut fur Festk6perforschung der Kernforschungsanlage Jiilich GmbH, Postfach 1913, 0-5170 Jiilich, F.R.G.
'~hysikalisches Institut der Universitiit Wiirzburg, Am Hubland, 0-8700 Wiirzburg, F. R. G.
RBsumB : .La diffusion des neutrons A haute resolution a QB appliqube pour la dbterrnination des frequences de
"crossover" oco et d'exposant spectral x pour trois aerogels preparbs en catalyse basique. Les frequences de
"crossover" et I'exposant ont QB dQerrninbs par une analyse de la densit6 d'Btats des neutrons diffuses non- Blastiquement. Nous avons obsew6 directernent et dUini ainsi les frbquences de "crossover" cornme valeur limite d'une loi de Debye : g(0)aw2. Pour des frbquences w<oco nous avons trow6 une relation avec une puissance diffbrente, I'exposant &ant x = 1,O
+
0.5 pour tous les Bchantillons exarninbs. Les frbquences de "crossover" ont Qe trouvees dependantes de la densite p, eh accord avec les previsions. Les valeurs trowees sont (wco)/(2x) =0,8-1,7 et 2,2 GHz pour des densites p = 0,12-0,18 et 0,25 g . ~ m - ~ respectivement. En outre, pour les trois Bchantillons, les vitesses du son effectives ont QB calculees & partir du rapport de diffusion totale blastique et inblastique. Les valeurs trouvees sont : Ve = 92,154 et 210 m.5' pour les densites indiquees prBc6demment.Abstract
-
Neutron backscattering has been applied for the determination of the crossover frequencies w o and spectral exponent x of three base-catalyzed silica aerogels. Thencrossover frequencies and spectral exponent have been deduced from a vibrational density of states analysis of inelastically scattered neutrons. We have directly observed and thereby defined the crossover frequencies as the limiting values up to which a Debye law g(w) w2 holds. For frequencies w > wco we have found a different power law with an exponent x = 1 f 0.5 for all specimens investi- gated. The crossover frequencies, on the other hand, have been found to depend on mass density p as expected. The values are wco/2n = 0.8; 1.7 and 2.2 GHz for densi-ties P = 0.12; 0.18 and 0.25 gcm-3, respectively. Moreover, the effective sound ve- locities of the samples have been calculated from the ratio of total elastic to to- tal inelastic one-quantum scattering and found to be (with increasing density, resp.) ve = 92; 154 and 210 ms-l in reasonable agreement with ultrasonic data.
1
-
INTRODUCTIONThere is increasing evidence that silica aerogels may be viewed as fractally connected struc- tures /1,2,3/, especially as a class of fractals which is termed "self-similar". Self-similari- ty means an invariance over a certain range of length scales of certain geometrical properties of a porous structure. In other words, the structure is meant to be indistinguishable geometri- cally within certain limits. The lower limit is obviously an interatomic or intermolecular dis- tance, although it is frequently a somewhat larger one according to the "diameter" 1 of some elementary building block. The upper limit termed correlation length 5 is a distance beyond which the structure appears homogeneous. Besides the intriguing structural property that the mass within a sphere of radius R of such a porous body does not scale with the Euclidean dimen- sion D = 3 but rather with a "fractal dimension" d < 3, there are also consequences on the dy- namics of fractals to be expected. In fact, there have been predictions on dynamical properties like elasticity, diffusion or vibrational behaviour /4,5/ to deviate from ordinary (non-frac- tal) solids. Especially the vibrational density of states g(w) has been shown theoretically to be different from normal Debye behaviour. Whereas generally g(w) a @-l holds, -a crossover from this law to a different frequency dependence has been postulated with g(w) a wd-I for w > wco (wco is the crossover frequency) 151. Interesting enough, this exponent d-1 contains a funda- mentally new quantity, the "fracton dimensionality"
a,
different from both the Euclidean dimen-sion D and the "fractal dimension" d. It has been conjectured to be a universal quantity with the numerical value 8 = 413.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989425
Intuitively, the crossover from a Debye behaviour to fractal dynamics means a qualitative change in the vibrational characteristics. In the Debye range (low frequencies resp. long wavelenghts) vibrations can propagate. These propagating modes have been called phonons as they can be identified with sound waves. These modes correspond to dimensions within the fractal body larger than the correlation length 5, i.e. where the structure appears to be homogeneous. On the fractal length scale, on the other hand, the vibrations have been predicted to become loca- lized / 6 / and have been called "fractons". To probe either propagating or localized vibrations scattering techniques will be adequate methods. In fact, there is a paper / 7 / in which for the first time Brillouin light scattering has been successfully applied for the determination of the crossover characteristics in neutrally reacted silica aerogels. In a Brillouin scattering ex- periment the phonon dispersion is determined, i.e. a linear relation w = c - k in the limit of small wavenumbers k (large phonon wavelengths A = l/k). The constant c is the sound velocity, td
the frequency shift of the scattered light, which is identified with the phonon frequency. In that experiment a deviation from this linear relation has been observed with the tendency of a convex curvature with increasing k towards a constant value w = const. The authors have inter- preted this as an indication of the onset of localization of modes, i.e. the existence of frac- tons. From their data they deduce crossover frequencies Uco between 6 x lo9 cps and 7 x lolo cps for mass densities p between 0.1 gcm-3 and 0.4 gcm-3 and a fracton dimension d = 1.25 f 0.06.
2 - REMARKS ON NEUTRON SCATTERING
There is a second well-known scattering technique which is extensively used for the study of both structure and dynamics of condensed matter: neutron scattering. As far as structure is con- cerned it is analogous and/or complementary to X-ray diffraction and has indeed been applied for the structure determination of aerogels by small angle scattering as quoted above /3/. As far as dynamics is concerned it is analogous to inelastic light scattering but with the invaluable ad- vantage of not being restricted to the low wavevector regime.
Neutrons interact with the nuclei of a specimen and couple to any type of motion within a sample. There are no restrictions and selection rules besides conservation of momentum and energy. This is, on the one hand, a great advantage but may be, on the other hand, an undesired complication in data interpretation. A fundamental difference between neutron scattering and light or X-ray scattering, which is due to the nuclear interaction of neutrons, is the appea- rance of both coherent and incoherent scattering. "Incoherence" in this respect means disorder and stems from the fact that the scattering amplitude depends on both the isotope and the nuclear spin of the constituent nuclei. The relative "strength" of coherent and incoherent scat tering depends on the elements present in a sample. The strongest incoherent scatterer is light hydrogen.
As neutrons probe the motions of atoms and molecules, it is obvious now that collective exci- tations like phonons are easily observable with dominantly coherently scattering atoms, whereas individual motions are better visible with incoherently scattering elements. While inelastic co- herent neutron scattering selects via momentum and energy conservation distinctive vibrational modes thereby establishing phonon dispersion relations, inelastic incoherent scattering reflects
the whole spectrum of vibrations the individual atom is subjected to. Indeed, neutron scattering theory shows 181 that the differential cross section for inelastic incoherent one-quantum scat- tering is proportional to the density of vibrational states g(w) discussed in the introduction.
Most samples (and also silica aerogels) scatter neutrons both coherently and incoherently, which therefore normally strongly complicates the extraction of g(w), as the coherent one-phonon cross section usually does not factorize to yield g(w). But as was shown by Buchenau / 9 / , the function g(w) can be separated for amorphous solids provided certain approximations hold. We then can write the total (coherent
+
incoherent) one-phonon cross sectionwhere ki and kf are the moduli of the incoming and scattered wavevectors;
4
= + kf-?& is the mo- mentum transfer vector of the scattering event (divided by Planck's constant*); the energy transfer is defined by -hw = (3i2/2m)(kZf-kZi) with m the neutron mass; the expression in brackets is the Bose factor with +1 for energy loss scattering, kB is Boltzmann's constant; I(o)(Q) is the total elastic scattering, qo is an average over all long-wavelength eigenmodes of the amor- phous solid at the considered frequency. The integral averages over all propagation directions.Ma, is the average atomic mass of the aerogel-forming molecule Si02. An essential condition for equation 1 to be fulfilled is that the harmonic approximation holds. This has been verified in the present experiment by determining the temperature dependence of the Debye-Waller factor (see section 5).
From equation (1) folldws that in the limiting case Ziw << kBT the intensity of inelastically scattered neutrons is proportional to g(w)/w2 which is a constant for Debye solids. If, on the other hand, the density of states of a solid beyond wco varied like wd-1 with iT < 3 as mentioned above, it should be observable in the neutron inelastic regime as decreasing intensity with in- creasing energy transfer provided the crossover frequencies are close to values accessible by existing spectrometers.
If we take the recent Brillouin results cited above / 7 / , we obtain as an order of magnitude estimate Ziwo =
...
4...40.peV. These figures limit the choice to the two spectrometer types with the highest energy resolution obtainable, the backscattering spectrometer or the spin echo spectrometer. We chose the former one, which got its name from the way the incoming neutrons are monochromatized and the scattered neutrons energy-analyzed, viz. by Bragg reflections through 180" from certain single crystal planes. In this way, which is obviously the limiting case for reaching the best resolution via Bragg reflection, we obtain a resolution of 1 ueV.A remark is suitable at this point concerning the inelastic intensities to be expected in the energy transfer window of this spectrometer ( " f 20 ueV), which should contain the Debye range.
In this range, g(w)/w2 = ( v / ~ I T ~ ) - ( ~ / v ~ ~
+
2/vT3), which is very small in common solids due to the large sound velocities of the order of several LO3 ms-l.
In the silica aerogels investi- gated the sound velocities are of the order of 100 ms-l. Therefore the inelastic scattering, hardly observable in that range in common solids, is stronger by a factor of about lo4.3
-
EXPERIMENTALThe neutron spectra have been measured with the improved backscattering spectrometer BSSl in the new neutron guide hall at the Jiilich research reactor FRJ-2. The incoming neutrons have an energy centered at 2.08 meV. It is scanned in the present case by fll ueV maximum with the aid of an hydraulically moved oscillatory drive onto which the backreflecting silicon single crystals are mounted. The scattered neutrons are analyzed by backreflection from focussing plates with again [llll-oriented silicon crystals. The neutron detectors are adjusted to the analyzer foci. For the present measurement we chose 5 scattering angles such that we covered a Q-range between 0.5 and 1.9 A-l.
Three base-catalyzed silica-aerogels have been investigated at temperatures between 12 K and 360 K. The mass densities of the samples are p = 0.12; 0.18 and 0.25 gcm-3. The aerogels were baked for 40 hours at 500 "C, in order to reduce the strong incoherent scattering from phy- sisorbed H20 and chemisorbed OH-groups; under N2-atmosphere the aerogels then were packed in an indium-sealed aluminium cylinder of 3 cm diameter. Using the elastic scattering from a vanadium- standard as a reference, we determined the scattering probability for the specimens; the com- parison with the theoretical scattering probability for the pure Si02-skeleton gives an estimate of about 0.3 residual protons per Si02 unit; i.e. the incoherent scattering of the hydrogen is still about twice as strong as the averaged coherent scattering from the silica. Further evalua- tions base on the assumption that the remaining chemisorbed OH-groups, which are strongly bound to the internal Si02 surface or are part of interparticle bonds /lo/, have no influence on the low frequency motion of the aerogel building blocks.
4
-
DATA EVALUATION AND RESULTSTypical energy scans are presented in figure 1. The essential feature is the composite form of the spectra consisting of an elastic line superimposed on an inelastic distribution. The shape of this inelastic scattering depends strongly on the mass density of the aerogels. The sample with p = 0.12 gcm-3 exhibits a pronounced energy dependence suggesting an underlying spectral density of states of the form
wX
with x < 2 for w > wco. The sample with p = 0.25 g ~ m - ~ , on the contrary, only produced a nearly energy independent contribution over the inelastic energy range we employed during this measurement. This suggests that the latter sample exhibits Debye beha- viour over most of the measured range. In other words,.tiwo is close to or beyond 11 ueV.In order to evaluate the spectra quantitatively we used a scattering cross section consisting of an elastic line co6(w) and an inelastic contribution, which f o r 5 w << kBT is proportional to g(w) /w2, as already mentioned. The symmetric function g(w) is constructed such that it is gD(w) = cDw2 for IwI c wco (Debye or phonon part) and gF(w) = G~wcoZ(~/wco)X for
I w I
2 wCO(fracton part). If so defined, g~(w) and g~(w) are continuous at w = wco, which is a reasonable but not necessary choice supported by the data. In particular, there is no indication in the data of a peak near the crossover frequency pnstulated for reasons of conservation of mode num- bers /11/ or as a consequence of a flat dispersion curve /6,7/.
Figure 2 shows the scattering cross section based on a density of states function as just de- scribed. The scattering cross section has to be folded with the experimental resolution function and fitted to the data points varying the parameters co, wco, x and cl. Typical fit results are
C4-160 REVUE DE PHYSIQUE APPLIQUEE
the continuous lines through the data points in figure 1. If we now identify the empirical expo- nent x with the fractal notation, x : r-1, we obtain the spectral dimension rgiven in Table 1 together with the fitted crossover frequencies v o = wco/2~.
Fig. 1 - Typical energy scans at T = 300 K for two values of the momentum transfer Q for the three investigated aerogel specimens of different densities p . The arrows indicate the fitted positions of the crossover energies
hCo.
The curves through the data points are the results of the fitting procedure described in the text. Note that these curves in the elastic regime(&I = 0) represent the independently measured instrumental line shapes and are not just a guide
to the eye.
Fig. 2
-
Theoretical scattering cross section based on the assumption of a crossover from Debye behaviour to fractal behaviour at w = wco.One can even go one step further and calculate an effective sound velocity ve defined by l/ve3 = 1/vL3
+
2/vT3 from the values of the parameters co and cl; ve then can be compared with values deduced from ultrasonic measurements. In this way one will confirm the validity of the approximations leading to equation (1). According to equ. (I), in the Debye limit, the total scattering cross section is d20/dw& = c06 (w)+
( Q ~ / ~ M ~ ~ ) ~ ~ T G ~ . c ~ c06(w)+
cl. Taking the wellknown relation between the Debye density of states and the effective sound velocity, i.e, g(w) = ( ~ ~ ~ / 6 i ~ ~ v ~ ~ ) w ~ GDw2 with the average atomic volune Vav, we can calculate ve from the ratio of elastic to inelastic scatteringThe factor Tr enters because of the integration over the instrumental resolution,
r
being thehalf width at half maximum. The effective sound velocities calculated from the neutron data according to equation (2) are given in Table 2 together with the corresponding data from ultra- sonic measurements.
Table 1.
Crossover frequencies v o = uC0/2r and spectral dimension 3 for three aerogels. (Brackets indi- cate that the value may be not significant due to large statistical errors.)
Table 2.
Comparison of effective sound velocities vz3 = vf3
+
2 v ~ 3 from neutron scattering data according to equation 2 and from ultrasonic measurements 1121.~ ( g c m - ~ ) ve (ms-l) ve (ms-l) neutrons ultrasonic
...
...
The sound velocities calculated from the neutron data are mean values of the results obtained from sets of up to 35 spectra per sample taken at 5 different Q-values and up to 7 different temperatures between 1 2 K and 360 K. The errors quoted are single standard deviations. The agreement with ultrasonic results is reasonably good. Nevertheless we won't stress that quality too much, because the third power of the sound velocity enters the neutron scattering cross sec- tion. Therefore larger deviations of the fit parameters co and cl in equation 2 within one set of spectra are essentially smoothed.
5
-
DISCUSSION AND CONCLUSIONLet us start with the results for the effective sound velocities. The agreement of the neutron data with ultrasonic measurement is essential for the interpretation of the measured spectra in terms of vibrational density of states on which in turn the further treatment rests. So the ob- vious agreement justifies the application of equation 1 to both incoherently and coherently scattering samples like the amorphous aerogels and the conclusions about the spectra drawn thereof. On the other hand, as at a first glance one might be tempted to interpret the spectra alternatively in terms of e.g. rotational diffusion of protons of residual OH-groups, we will discuss and exclude this possibility with several arguments. Firstly, the amount of residual OH- groups from the sample preparation varies proportional to the mass density P of the samples.
Therefore, the topology of the intensity distribution in the neutron spectra should be equal for all samples. From figure 1 it is obvious that the intensity profiles differ markedly for the three specimens. Secondly, the intensity variation of both elastic and inelastic (quasielastic) scattering from rotationally diffusing protons obey Q-dependences described by spherical or cy- lindrical Bessel functions. No agreement has been found between the experimental intensities and theoretical form factors. Finally, rotational motion of an OH-group is a thermally activated process. This in turn should be observable as a step in the temperature dependence of elasti- cally scattered neutrons. In contrast we found in a log-linear plot of intensity versus tempera- ture a straight line between 12 K and 360 K as shown in Figure 3, thereby establishing the vali- dity of the harmonic approximation implied by equ. (1).
We have presented the first direct observation of a crossover from ordinary Debye vibrational characteristics to a different spectral behaviour resembling features predicted for fractal structures. Having in mind that a deviation from an u *-dependence alone does not imply fractal connectivity of the underlying structure /23/, a conclusion our investigated aerogels might be fractals were based right on this crossover. On the other hand, our values for the spectral di- mension 7 (see Table 1) appear to be far from the conjectured universal value 413 151, even if the estimated experimental errors (single standard deviation) are taken into account. An expla- nation for that fact were that the porosity of our base-catalyzed aerogels is not a well-defined
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fractal. Nevertheless, any type of porosity on an adequate length scale should be the origin of a "mode deficiency" leading to a spectral density varying as wX with x < 2 and w > wco.
Fig. 3
-
Elastic intensity as a function of temperature for two samples ( p = 0.25 gcm-3 andp = 0.12 gcm-3) and 5 momentum transfers Q. The figures at the lines are corresponding numerical values of Q in units of
X-l.
No indication of any discontinuity is observed.We would like to add a final remark on the numerical values for wc0 and 2 for the most dense sample (compare last column in Table l).'These figures need not co be taken too seriously as the reliability of the fitting procedure breaks down if the crossover reaches the edge of the dyna- mic range of the spectrometer. Nevertheless, even this case supports our interpretation.
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2696 (1986) ACKNOWLEDGEMENTThe authors are grateful to U. Buchenau for stimulating this work and for many fruitful discussions. The authors are indebted to Mrs L. Schatzler for her excellent work with the com- puter codes for data evaluation as well as to B. Huy for technical assistance. One of the authors (HC) wants to express the great pleasure he had in collaborating with R. Schatzler in setting to work the new spectrometer.
