- Email: [email protected]
Time-of-flight inelastic neutron scattering study of liquid 4He in silica aerogels
IOURNAL OF
ELSEVIER
Journal of Non-Crystalline Solids 188 (1995) 41-45
Time-of-flight inelastic neutron scattering study of liquid 4He in silica aerogels " Gerrit Coddens a,b,, Jan De Kinder a,b Ren6 Millet
a
a Laboratoire LEon Brillouin, CEA-CNRS, F-91191 Gif-sur-Yoette cddex, France b Physics Department, University ofAntwerp (U.LA.), B-2610 Wilrijk (Antwerp), Belgium Received 18 September 1994
Abstract
Neutron scattering results on the elementary excitations of superfluid 4He in two different silica aerogels are compared. One was obtained by base-catalysed reaction with a pore volume of 93.2%, a density of 0.150 g//crn 3 and a Hausdorff dimension, D = 1.8, and the other was obtained in a neutral reaction with a pore volume of 96.5%, a density of 0.080 g//cm3 and a Hausdorff dimension, D -- 2.4. Results in both samples are remarkably identical to each other and to the behaviour in bulk 4He.
1. Introduction
The study of the superfluid properties of helium in confined media, and more specifically in silica aerogels, has yielded some astounding results over the last few years. The most remarkable one comes from the work of Kim et al. [1] on the phase diagram of 4He-3He mixtures which gave evidence for the coexistence of two superfluids at temperatures of the order of 1K, one rich in 3He that does not exist in bulk mixtures at these temperatures and the other (the more conventional one) rich in 4He. Note that the superfluid transition temperature of bulk 3He at standard vapor pressure (SVP) is only 2.6 mK (as
Presented at the 4th International Symposium on Aerogels, Berkeley, CA, USA, 19-21 September 1994. * Corresponding author. Tel: +33 67 14 46 77. Telefax: +33 67 14 34 98. E-mail: [email protected].
opposed to 2.17 K for 4He). The superfluid properties of plain 4He in aerogels have been studied by Reppy and co-workers [2] mainly by two techniques, namely Andronikashvili's torsion pendulum method (for the determination of the superfluid fraction) and specific heat measurements. Their very accurate resuits gave rise to several theoretical problems that have not been resolved. The consequences of disorder on phase transitions have been studied theoretically by Harris [3]. The Harris criterion says that a continuous phase transition (such as the superfluid transition) will not be affected by uncorrelated disorder if the critical exponent, ~pure, for the specific heat is negative in the pure system. The criterion does thus not apply if the disorder is correlated, but it can be extended such that its conclusions go unscathed provided the correlations decay fast enough. The data from Reppy et al. contradict this theoretical criterion since they show that some of the critical exponents are modi-
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 0 2 2 - 4
42
G. Coddens et al. /Journal of Non-Crystalline Solids 188 (1995) 41-45
lied while apure = - 0 . 0 1 3 + 0.003 [4]. Apparently, one has to argue that a fractal harbours a long-range type of correlation in the disorder for which the Harris criterion does not apply or that apure is too weakly negative to allow one to draw firm conclusions from the criterion. The results also violate a relation due to Josephson [5] between the critical exponent, if, of the superfluid fraction, the critical exponent, a, and the dimensionality, d of the helium sample, ffd = ( 2 - a X d - 2), which has been confinned beautifully in the case of bulk 4He [4]. By contrast with those already mentioned, neutron scattering is a microscopic probe. As Landau [6] has shown by simple hydrodynamical theory, there is a direct relation between many macroscopic properties and the spectrum of the elementary excitations. As an example, we can cite the experimental verification of this for the specific heat by Bendt et al. [7]. Even if the anomalies observed by Reppy et al. in the specific heat primarily address the critical fluctuations at T x, one may hope that a difference of universality class of the superfluid might also show up as deviations from bulk behaviour in the dispersion curves for the elementary excitations well below the lambda transition. We reported [8] on a measurement done with an aerogel that had been prepared in a base-catalysed reaction (further denoted as the B sample). Results on one single type of sample always raise questions as regards to their universality. In the present paper, we compare the previous data with results obtained on a different sample with approximately the same pore volume (96.5%) but prepared in a neutral reaction (further denoted as the N sampie).
2. Experimental procedures
lium or argon atmospheres from their production until insertion into the sample holder. The pore size distribution obtained by thermoporometry by Quinson et al. [9] on our B sample peaks around 130 ,~. The Hausdorff dimension of this sample is D = 1.8, its macroscopic density is 0.150 g / c m 3 and its fractal cluster radius (correlation length), ~, is 220 ,A. The particles building up the structure are spheres of 35-40 A diameter with a smooth surface as indicated by the Q-4 Porod plot in small-angle neutron scattering. The N Sample has a Hausdorff dimension, D = 2.4, a mean panicle diameter of 10-12 ,~ and cluster radii ~ ~ 400 A; the macroscopic density is 0.080 g / c m 3, the pore volume fraction is 96.5% and the pore size distribution peaks are about 110 ,~. The samples, 30 mm in diameter and 50 mm high, were put into indium-sealed thin-walled steel containers and introduced into a top-loading cryostat. They were filled with 99.999% pure helium in situ through a capillary. Both samples were deuterated to reduce background scattering from the aerogel. The experiments on the B sample have been described elsewhere [8]. The experiments with the N sample were carried out with the time-of-flight spectrometer Mibemol of the Laboratoire IAon Brillouin in Saclay, with incident wavelengths, A, of 5 ,~ (3.27 meV) and 6.5 ~, (1.94 meV). The first choice enables one to study the whole phonon-maxon-roton curve in neutron energy loss and corresponds to the maximum incident flux for the instrument. The elastic energy resolution under these conditions is 160 ixeV full width at half maximum (FWHM). The second choice has been made to make a demonstration experiment for multiple scattering effects. The elastic resolution is here 75 p~eV and the available range of Q excludes the roton minimum. 300 3He detectors were positioned at 54 angular positions, 20, between 14° and 147 °. Spectra were taken at 1.6 K and 4He fillings, C, of 0 (for background) and 100%. A vanadium run for normalisation completed the data.
By contrast with the sample B, which was broken into a small number of pieces by mechanical shocks over the years, the N sample was monolithic, and the amount of bulk helium outside the aerogel in the neutron beam is entirely negligible. The samples 1 were manipulated in waterfree conditions under he-
3. Results
1The sampleswere kindly providedby Dr M. Foret, Professor R. Vacher and ProfessorJ. Pelous.
Fig. 1 shows a comparison between the data on the N sample obtained with incident wavelengths of h = 6.5 and 5.0 ,~. Both data sets are constant-20
43
G. Coddens et aL/Journal of Non-Crystalline Solids 188 (1995) 41-45 o
1.0
I
I
I
~t=6.5A
>" 0.8 r-
I
.
.
.
.
k=5~,
qlmaxon
E 0.6
J
:-= 0J.
/
double scattering ~]i
7°02
" >v-¢ , , :~.:~:,;r , , ++!,+ ~i i-i+ ~ ] ;V•~ , v, 0/~
0.6
0.8
1.0
' 1.2
1.4
16
Energy ( meV )
Fig. 1. Comparison of it = 5 A ( - - - ) and 6.5 .~ ( ) constant-20 scans on the N sample (T= 1.6 K). The (elastic+ roton) double scattering event (shaded area) disappears in the A= 6.5 ,~ data. scans at Q = 1/~[-1. The difference in width, which is immediately conspicuous, is entirely due to the different energy resolutions in the two experiments as was spelled out above. The aerogel samples are strong scatterers (typically 25% for the B sample). At low temperatures this scattering is purely elastic. Even if the main part is small-angle scattering some multiple scattering involving larger-Q elastic events might affect the result of S(Q, to). The most prominent inelastic scattering event is the roton minimum. We can thus expect a unique feature, namely that the two events combining in double scattering can be isolated without ambiguity: an elastic event in the aerogel combined with a roton event in the helium. By excluding the roton minimum from the available range of Q (which is the case in the 6.5 ,~ data), this effect can be completely suppressed as illustrated in Fig. 1. The fact that the roton intensity is so strong means that this effect cannot possibly avoided if the range of Q extends beyond the roton minimum. It shows up also in data of Lauter and co-workers [10] on helium films on graphite, and one should not get into the pitfall of taking it for a flat mode. This complicates the analysis of the data. Three data sets have been analysed by fitting (one or two) damped harmonic oscillator functions convoluted with the experimental resolution to them. These fits were performed on constant-20 scans, but the analytical expressions were appropriately modified. o The data sets concern the N sample at A = 5 or 6 A,
T = 1.6 K and the B sample at A = 5 A, T = 1.8 K. For the 6.5 .~ run, only one oscillator is needed and it describes the experimental data well. The 5 ,~ data required an additional oscillator. For the B sample, where the temperature is higher, this second component is purely phenomenological as it is used to describe both the multiple scattering event (at the energy of the roton) and the multiphonon contribution at higher energies by one single broad but physically meaningless contribution. A correct description would actually require three components but this is not feasible given the quality of the data. Nevertheless, the characteristics of the phononmaxon-roton excitations are well described by the first component. For the N sample, the second component represents just the multiple scattering. The peak positions of the elementary excitations as extracted from these fits are given in the dispersion curves of Fig. 2. It is obvious from this comparison that our peak positions are reliable and that the results are identical for both samples. Because of the differences in temperature, we refrained from extend-
I
1.0
I
I
I
sample
,•w•kN
=6.5~
(1B
/
_
sample_
+0/:
~0~
0.2
0/
B sample
k=5~
0.2: / 0
0
I
I
]
J
0.5
1.0
1.5
2.0
Wavevector (./~-1) Fig. 2. Dispersion curve of the elementaryexcitationsof superfluid 4He in the B sample (A = 5 A, T = 1.8 K, 0 ) and the N sample (T = 1.6 K) for A= 5 .~ (O) and it = 6.5 ,~ ( • ).
44
G. Coddens et al. /Journal of Non-Crystalline Solids 188 (1995) 41-45 1.0
I
I
. ~ 0,8 e" 0
•
roton x ' °
•
O
B sample o N sample •
g~
eO
~ O.6
0
:--= 0.4
"°2F 0
/; 0.5
7
S 1.0
1.5
2.0
Energy (meV) Fig. 3. Comparison of the roton peak (constant-2 0 scans) in the B sample (T = 1.8 K, O ) and the N sample (T = 1.6 K, O). The maxima in the intensity have been arbitrarily normalised to 1.
ing this comparison with the intensity and the width of the excitations. However, even from a qualitative inspection it is clear that in both samples the behaviour is distressingly identical to the behaviour of bulk 4He, except for the multiple scaottering effect. Fig. 3 shows a comparison at k = 5 A between the N and B samples. The data were normalized to a same peak intensity. (In reality the maximum of the intensity is much higher in the lower-temperature data obtained on the N sample,) The differences in width can be ascribed entirely to this temperature difference.
4. D i s c u s s i o n
Our results on the two samples are identical, which gives us some confidence that they also apply to other samples with the same pore volume, e.g., the Airglass samples used by Reppy et al., for which the only characterisation available in the literature is a determination of the pore size distribution by nitrogen absorption. The B sample belongs to the family of samples for which very accurate modeling of the structure has been achieved by Hasmy et al. [11]. Allowing for the polydispersity, the simulations are able to reproduce perfectly the cross-over regime between the fractal and the particle region in the small-angle scattering curves. (For the neutrally reacted samples such simulations are unfortunately not yet available
at present.) We consider the existence of this characterisation by experimentally validated computer data as very important since it allows one to obtain with some confidence information on properties that are not readily accessible to experiment, e.g., on the type of disorder (with respect to the Harris criterion). If it is has been well established by small-angle neutron scattering data [12] that silica aerogels can exhibit fractal density-density correlations over one to two decades of length scales in the matter distribution, it is much less clear that such correlations exist in the pores, whose distribution is probably the one that counts most for helium. A reliable description of these pore-size correlations for base-catalysed aerogels can be obtained from the simulations. A problem that has never been mentioned in the context of aerogels is that small pores can act as entropy filters by the fountain effect 2 such that the distribution of the superfluid fraction might be heterogeneous if the range of pore sizes is broad. In this respect aerogels could stand out markedly different against Vycor ® where the pore diameter distribution seems to be fairly monodisperse. Since the torsion pendulum experiments probe only the percolating backbone of the superfluid [2], this and other effects could complicate the interpretation of Reppy's data [13]. Some confirmation by an independent technique of this interpretation is thus very much wanted in order to validate the macroscopic techniques. Our microscopic data do not lend such support, but we do not know whether theory says that they should. A different, new approach to the problems of understanding the superfluid behaviour of 4He in silica aerogels has been undertaken by Thibault and coworkers [14] in a series of original and very precise experiments on the dilatation of the rather compliant aerogel which is elastically coupled to the superfluid. These results seem to indicate the possibility of a percolation transition in the superfluid. For a better understanding of superfluid helium in silica aerogels, it would be highly recommendable to obtain the full range of experimental results on the B sample (as a standard) and confront them with the 2 The normal component of the superfluid carries the entropy (i.e., the temperature). This normal component cannot traverse small pores and gets clamped while the superfluid component can flow through without friction.
G. Coddens et al. /Journal of Non-Crystalline Solids 188 (1995) 41-45
theories, constrained by the values of the relevant parameters obtained from the simulation.
5. Conclusions Elementary excitations of superfluid 4He were measured in two different silica aerogels with significantly different microstructures, but comparable by pore volumes. No evidence was found that the behaviour of the superfluid should be different from the bulk 4He. The authors wish to express gratitude towards Mr E. Anglaret, Dr M. Foret, Professor R. Vacher and Professor J. Pelous for providing the samples and for their continuous interest in this work. They also thank Professor L. Puech for communicating Ref. [14] prior to publication.
References [1] S.B. Kim, J. Ma and M.W.H. Chan, Phys. Rev. Lett. 71 (1993) 2268.
45
[2] J.D. Reppy, J. Low Temp. Phys. 87 (1992) 205; G.K.S. Wong, P.A. Crowell, H.A. Cho and J.D. Reppy, Phys. Rev. B48 (1993) 3858. [3] A.B. Harris, J. Phys. C7 (1974) 1671. [4] J.A. Lipa and T.C.P. Chui, Phys. Rev. Lett. 51 (1983) 2291. [5] B.D. Josephson, Phys. Lett. 21 (1966) 608. [6] L.D. Landau, J. Phys. (Moscow) 5 (1941) 71. [7] P.J. Bendt, R.D. Cowan, and J.L. Yarnell, Phys. Rev. 113 (1959) 1386. [8] J. De Kinder, G. Coddens and R. Millet, Z. Phys. B, Condens. Matter 95 (1994) 511. [9] J.F. Quinson, M. Pauthe, M. Lacroix, T. Woignier, J. Phalippou and H. Hdach, J. Non-Cryst. Solids 147&148 (1992) 699. [10] H.J. Lauter, H. Godfrin and P. Leiderer, J. Low Temp. Phys. 87 (1992) 425; H.J. Lauter, H. Godfrin, V.L.P. Franck and P. Leiderer, Phys. Rev. Lett. 68 (1992) 2484. [11] A. Hasmy, E. Anglaret, M. Foret, J. Pelous and R. Jullien, Phys. Rev. B50 (1994) 6006; R. Vacher, T. Woignier, J. Phalippou and J. Pelous, J. Non-Cryst. Solids 106 (1988) 161. [12] R. Vacher, T. Woignier, J. Pelous and E. Courtens, Phys. Rev. B37 (1988) 6500. [13] R. Maynard and G. Deutscher, Europhys. Lett. 10 (1989) 257. [14] P. Thibault, PhD thesis, University of Grenoble (1994).
ELSEVIER
Journal of Non-Crystalline Solids 188 (1995) 41-45
Time-of-flight inelastic neutron scattering study of liquid 4He in silica aerogels " Gerrit Coddens a,b,, Jan De Kinder a,b Ren6 Millet
a
a Laboratoire LEon Brillouin, CEA-CNRS, F-91191 Gif-sur-Yoette cddex, France b Physics Department, University ofAntwerp (U.LA.), B-2610 Wilrijk (Antwerp), Belgium Received 18 September 1994
Abstract
Neutron scattering results on the elementary excitations of superfluid 4He in two different silica aerogels are compared. One was obtained by base-catalysed reaction with a pore volume of 93.2%, a density of 0.150 g//crn 3 and a Hausdorff dimension, D = 1.8, and the other was obtained in a neutral reaction with a pore volume of 96.5%, a density of 0.080 g//cm3 and a Hausdorff dimension, D -- 2.4. Results in both samples are remarkably identical to each other and to the behaviour in bulk 4He.
1. Introduction
The study of the superfluid properties of helium in confined media, and more specifically in silica aerogels, has yielded some astounding results over the last few years. The most remarkable one comes from the work of Kim et al. [1] on the phase diagram of 4He-3He mixtures which gave evidence for the coexistence of two superfluids at temperatures of the order of 1K, one rich in 3He that does not exist in bulk mixtures at these temperatures and the other (the more conventional one) rich in 4He. Note that the superfluid transition temperature of bulk 3He at standard vapor pressure (SVP) is only 2.6 mK (as
Presented at the 4th International Symposium on Aerogels, Berkeley, CA, USA, 19-21 September 1994. * Corresponding author. Tel: +33 67 14 46 77. Telefax: +33 67 14 34 98. E-mail: [email protected].
opposed to 2.17 K for 4He). The superfluid properties of plain 4He in aerogels have been studied by Reppy and co-workers [2] mainly by two techniques, namely Andronikashvili's torsion pendulum method (for the determination of the superfluid fraction) and specific heat measurements. Their very accurate resuits gave rise to several theoretical problems that have not been resolved. The consequences of disorder on phase transitions have been studied theoretically by Harris [3]. The Harris criterion says that a continuous phase transition (such as the superfluid transition) will not be affected by uncorrelated disorder if the critical exponent, ~pure, for the specific heat is negative in the pure system. The criterion does thus not apply if the disorder is correlated, but it can be extended such that its conclusions go unscathed provided the correlations decay fast enough. The data from Reppy et al. contradict this theoretical criterion since they show that some of the critical exponents are modi-
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 0 2 2 - 4
42
G. Coddens et al. /Journal of Non-Crystalline Solids 188 (1995) 41-45
lied while apure = - 0 . 0 1 3 + 0.003 [4]. Apparently, one has to argue that a fractal harbours a long-range type of correlation in the disorder for which the Harris criterion does not apply or that apure is too weakly negative to allow one to draw firm conclusions from the criterion. The results also violate a relation due to Josephson [5] between the critical exponent, if, of the superfluid fraction, the critical exponent, a, and the dimensionality, d of the helium sample, ffd = ( 2 - a X d - 2), which has been confinned beautifully in the case of bulk 4He [4]. By contrast with those already mentioned, neutron scattering is a microscopic probe. As Landau [6] has shown by simple hydrodynamical theory, there is a direct relation between many macroscopic properties and the spectrum of the elementary excitations. As an example, we can cite the experimental verification of this for the specific heat by Bendt et al. [7]. Even if the anomalies observed by Reppy et al. in the specific heat primarily address the critical fluctuations at T x, one may hope that a difference of universality class of the superfluid might also show up as deviations from bulk behaviour in the dispersion curves for the elementary excitations well below the lambda transition. We reported [8] on a measurement done with an aerogel that had been prepared in a base-catalysed reaction (further denoted as the B sample). Results on one single type of sample always raise questions as regards to their universality. In the present paper, we compare the previous data with results obtained on a different sample with approximately the same pore volume (96.5%) but prepared in a neutral reaction (further denoted as the N sampie).
2. Experimental procedures
lium or argon atmospheres from their production until insertion into the sample holder. The pore size distribution obtained by thermoporometry by Quinson et al. [9] on our B sample peaks around 130 ,~. The Hausdorff dimension of this sample is D = 1.8, its macroscopic density is 0.150 g / c m 3 and its fractal cluster radius (correlation length), ~, is 220 ,A. The particles building up the structure are spheres of 35-40 A diameter with a smooth surface as indicated by the Q-4 Porod plot in small-angle neutron scattering. The N Sample has a Hausdorff dimension, D = 2.4, a mean panicle diameter of 10-12 ,~ and cluster radii ~ ~ 400 A; the macroscopic density is 0.080 g / c m 3, the pore volume fraction is 96.5% and the pore size distribution peaks are about 110 ,~. The samples, 30 mm in diameter and 50 mm high, were put into indium-sealed thin-walled steel containers and introduced into a top-loading cryostat. They were filled with 99.999% pure helium in situ through a capillary. Both samples were deuterated to reduce background scattering from the aerogel. The experiments on the B sample have been described elsewhere [8]. The experiments with the N sample were carried out with the time-of-flight spectrometer Mibemol of the Laboratoire IAon Brillouin in Saclay, with incident wavelengths, A, of 5 ,~ (3.27 meV) and 6.5 ~, (1.94 meV). The first choice enables one to study the whole phonon-maxon-roton curve in neutron energy loss and corresponds to the maximum incident flux for the instrument. The elastic energy resolution under these conditions is 160 ixeV full width at half maximum (FWHM). The second choice has been made to make a demonstration experiment for multiple scattering effects. The elastic resolution is here 75 p~eV and the available range of Q excludes the roton minimum. 300 3He detectors were positioned at 54 angular positions, 20, between 14° and 147 °. Spectra were taken at 1.6 K and 4He fillings, C, of 0 (for background) and 100%. A vanadium run for normalisation completed the data.
By contrast with the sample B, which was broken into a small number of pieces by mechanical shocks over the years, the N sample was monolithic, and the amount of bulk helium outside the aerogel in the neutron beam is entirely negligible. The samples 1 were manipulated in waterfree conditions under he-
3. Results
1The sampleswere kindly providedby Dr M. Foret, Professor R. Vacher and ProfessorJ. Pelous.
Fig. 1 shows a comparison between the data on the N sample obtained with incident wavelengths of h = 6.5 and 5.0 ,~. Both data sets are constant-20
43
G. Coddens et aL/Journal of Non-Crystalline Solids 188 (1995) 41-45 o
1.0
I
I
I
~t=6.5A
>" 0.8 r-
I
.
.
.
.
k=5~,
qlmaxon
E 0.6
J
:-= 0J.
/
double scattering ~]i
7°02
" >v-¢ , , :~.:~:,;r , , ++!,+ ~i i-i+ ~ ] ;V•~ , v, 0/~
0.6
0.8
1.0
' 1.2
1.4
16
Energy ( meV )
Fig. 1. Comparison of it = 5 A ( - - - ) and 6.5 .~ ( ) constant-20 scans on the N sample (T= 1.6 K). The (elastic+ roton) double scattering event (shaded area) disappears in the A= 6.5 ,~ data. scans at Q = 1/~[-1. The difference in width, which is immediately conspicuous, is entirely due to the different energy resolutions in the two experiments as was spelled out above. The aerogel samples are strong scatterers (typically 25% for the B sample). At low temperatures this scattering is purely elastic. Even if the main part is small-angle scattering some multiple scattering involving larger-Q elastic events might affect the result of S(Q, to). The most prominent inelastic scattering event is the roton minimum. We can thus expect a unique feature, namely that the two events combining in double scattering can be isolated without ambiguity: an elastic event in the aerogel combined with a roton event in the helium. By excluding the roton minimum from the available range of Q (which is the case in the 6.5 ,~ data), this effect can be completely suppressed as illustrated in Fig. 1. The fact that the roton intensity is so strong means that this effect cannot possibly avoided if the range of Q extends beyond the roton minimum. It shows up also in data of Lauter and co-workers [10] on helium films on graphite, and one should not get into the pitfall of taking it for a flat mode. This complicates the analysis of the data. Three data sets have been analysed by fitting (one or two) damped harmonic oscillator functions convoluted with the experimental resolution to them. These fits were performed on constant-20 scans, but the analytical expressions were appropriately modified. o The data sets concern the N sample at A = 5 or 6 A,
T = 1.6 K and the B sample at A = 5 A, T = 1.8 K. For the 6.5 .~ run, only one oscillator is needed and it describes the experimental data well. The 5 ,~ data required an additional oscillator. For the B sample, where the temperature is higher, this second component is purely phenomenological as it is used to describe both the multiple scattering event (at the energy of the roton) and the multiphonon contribution at higher energies by one single broad but physically meaningless contribution. A correct description would actually require three components but this is not feasible given the quality of the data. Nevertheless, the characteristics of the phononmaxon-roton excitations are well described by the first component. For the N sample, the second component represents just the multiple scattering. The peak positions of the elementary excitations as extracted from these fits are given in the dispersion curves of Fig. 2. It is obvious from this comparison that our peak positions are reliable and that the results are identical for both samples. Because of the differences in temperature, we refrained from extend-
I
1.0
I
I
I
sample
,•w•kN
=6.5~
(1B
/
_
sample_
+0/:
~0~
0.2
0/
B sample
k=5~
0.2: / 0
0
I
I
]
J
0.5
1.0
1.5
2.0
Wavevector (./~-1) Fig. 2. Dispersion curve of the elementaryexcitationsof superfluid 4He in the B sample (A = 5 A, T = 1.8 K, 0 ) and the N sample (T = 1.6 K) for A= 5 .~ (O) and it = 6.5 ,~ ( • ).
44
G. Coddens et al. /Journal of Non-Crystalline Solids 188 (1995) 41-45 1.0
I
I
. ~ 0,8 e" 0
•
roton x ' °
•
O
B sample o N sample •
g~
eO
~ O.6
0
:--= 0.4
"°2F 0
/; 0.5
7
S 1.0
1.5
2.0
Energy (meV) Fig. 3. Comparison of the roton peak (constant-2 0 scans) in the B sample (T = 1.8 K, O ) and the N sample (T = 1.6 K, O). The maxima in the intensity have been arbitrarily normalised to 1.
ing this comparison with the intensity and the width of the excitations. However, even from a qualitative inspection it is clear that in both samples the behaviour is distressingly identical to the behaviour of bulk 4He, except for the multiple scaottering effect. Fig. 3 shows a comparison at k = 5 A between the N and B samples. The data were normalized to a same peak intensity. (In reality the maximum of the intensity is much higher in the lower-temperature data obtained on the N sample,) The differences in width can be ascribed entirely to this temperature difference.
4. D i s c u s s i o n
Our results on the two samples are identical, which gives us some confidence that they also apply to other samples with the same pore volume, e.g., the Airglass samples used by Reppy et al., for which the only characterisation available in the literature is a determination of the pore size distribution by nitrogen absorption. The B sample belongs to the family of samples for which very accurate modeling of the structure has been achieved by Hasmy et al. [11]. Allowing for the polydispersity, the simulations are able to reproduce perfectly the cross-over regime between the fractal and the particle region in the small-angle scattering curves. (For the neutrally reacted samples such simulations are unfortunately not yet available
at present.) We consider the existence of this characterisation by experimentally validated computer data as very important since it allows one to obtain with some confidence information on properties that are not readily accessible to experiment, e.g., on the type of disorder (with respect to the Harris criterion). If it is has been well established by small-angle neutron scattering data [12] that silica aerogels can exhibit fractal density-density correlations over one to two decades of length scales in the matter distribution, it is much less clear that such correlations exist in the pores, whose distribution is probably the one that counts most for helium. A reliable description of these pore-size correlations for base-catalysed aerogels can be obtained from the simulations. A problem that has never been mentioned in the context of aerogels is that small pores can act as entropy filters by the fountain effect 2 such that the distribution of the superfluid fraction might be heterogeneous if the range of pore sizes is broad. In this respect aerogels could stand out markedly different against Vycor ® where the pore diameter distribution seems to be fairly monodisperse. Since the torsion pendulum experiments probe only the percolating backbone of the superfluid [2], this and other effects could complicate the interpretation of Reppy's data [13]. Some confirmation by an independent technique of this interpretation is thus very much wanted in order to validate the macroscopic techniques. Our microscopic data do not lend such support, but we do not know whether theory says that they should. A different, new approach to the problems of understanding the superfluid behaviour of 4He in silica aerogels has been undertaken by Thibault and coworkers [14] in a series of original and very precise experiments on the dilatation of the rather compliant aerogel which is elastically coupled to the superfluid. These results seem to indicate the possibility of a percolation transition in the superfluid. For a better understanding of superfluid helium in silica aerogels, it would be highly recommendable to obtain the full range of experimental results on the B sample (as a standard) and confront them with the 2 The normal component of the superfluid carries the entropy (i.e., the temperature). This normal component cannot traverse small pores and gets clamped while the superfluid component can flow through without friction.
G. Coddens et al. /Journal of Non-Crystalline Solids 188 (1995) 41-45
theories, constrained by the values of the relevant parameters obtained from the simulation.
5. Conclusions Elementary excitations of superfluid 4He were measured in two different silica aerogels with significantly different microstructures, but comparable by pore volumes. No evidence was found that the behaviour of the superfluid should be different from the bulk 4He. The authors wish to express gratitude towards Mr E. Anglaret, Dr M. Foret, Professor R. Vacher and Professor J. Pelous for providing the samples and for their continuous interest in this work. They also thank Professor L. Puech for communicating Ref. [14] prior to publication.
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