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Neutron scattering studies of liquid 3He and 3He-4He mixtures
ELSEVIER
Physica B 197 (1994) 206-214
Neutron scattering studies of liquid 3He and 3He-4He mixtures B. Ffik a'*, R. Scherm b aCEA, DOpartement de Recherche Fondamentale sur la MatiOre Condensde, SPSMS/ MDN, Grenoble Cedex, France blnstitut Laue-Langevin, Grenoble Cedex, France
Abstract
Density and spin fluctuations in liquid 3He and 3He-4He mixtures have been studied by neutron inelastic scattering. We discuss the similarities between the density fluctuations in normal liquid 3He and superfluid 4He (anomalous dispersion and the 'roton minimum' that reflects the local order) as well as damping mechanisms that could explain the large width of the 3He zero-sound mode. A line-shape analysis shows that existing theories cannot describe the spin fluctuations observed in liquid 3He. In 3He-4He mixtures, the shift and broadening of the phonon-roton mode is critically analyzed in terms of Landau damping. The 3He quasiparticle dispersion, and therefore the effective mass, obtained from neutron scattering measurements on 3He-4He mixtures, shows a clear temperature dependence. Although this is in contrast to what is normally assumed, the specific heat calculated from this dispersion is in excellent agreement with thermodynamic measurements.
1. Introduction
T h e q u a n t u m liquids 3He and 4He, which are m o d e l systems of strongly interacting fermions and bosons, continue to reveal new fascinating physics, and attract a broad interest among condensed matter physicists [1]. The properties of these liquids are best understood in terms of e l e m e n t a r y excitations. Inelastic scattering of t h e r m a l (or cold) neutrons is a unique tool for studies of q u a n t u m liquids, since the m o m e n t u m and energy of the excitations can be measured simultaneously. In a neutron scattering experim e n t , neutrons with a well defined incident wave
* Corresponding author.
vector k i are scattered by the sample, and the wave vector kf of the scattered neutrons is determined. Hence, the neutron transfers a m o m e n t u m Q = k i - k f and an energy E = ( k ~ k ~ ) h 2 / 2 m n to the sample. In this way, one (or several) excitation(s) with wave vector Q and energy E can be created. At nonzero t e m p e r a tures, an excitation can also be annihilated, thereby increasing the energy of the scattered neutron. Since neutrons interact with the nuclei via the strong interaction, it is the density fluctuations of the system that are observed. If the nuclei studied have a spin, as for 3He, nuclear spin fluctuations are observed in addition to the density fluctuations, the latter being p r o b e d by the spin-averaged coherent scattering. The intensity of the scattered neutrons is proportional to the dynamic structure factor S * ( Q , E ) , which for a mixture of 3He and 4He can be written as
0921-4526/94/$07.00 O 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0470-2
B. Fdk, R. Scherm / Physica B 197 (1994) 206-214
o'S*( Q, E) = (1 - x3)o'4S44(Q , E) + x 3 o r 3c S 3 3 ( Q ,
E) + x3o'3S33(Q i I , E)
-[- ~ / X 3 ( 1 -- x 3 ) o 3 4 8 3 4 ( O , E )
(1)
where x 3 is the molar concentration of 3He, the o-'s are scattering cross-sections for the different nuclei, Sj~(Q, E) describes density correlations between atoms of type j and k, and S~3(Q , E) describes spin correlations between the 3He nuclear spins. Equation (1) includes the limits of pure 3He and 4He as well. Note that there is a contribution from both density and spin fluctuations from 3He, whereas only density fluctuations are possible in 4He, since its nuclear spin is zero. Although neutron scattering is an ideal probe to study both density and spin fluctuations of 3He, the enormous absorption cross-section makes these experiments very difficult. In addition, the heat produced by the absorption of neutrons by 3He nuclei limits the lowest temperature that can be achieved. The properties of superfluid 4He was at large explained by Landau in 1947, who introduced the concept of elementary excitations. The dispersion of these excitations, the phonon-roton curve, was obtained phenomenologically by Landau. The first microscopic calculation of the dispersion was made by Feynman in 1954. The dispersion was measured a few years later for the first time, using neutron inelastic scattering. However, the nature of the excitations proposed by Landau and by Feynman remained an enigma, and the temperature dependence of the excitations was never well understood. Recently, Glyde and Griffin [2] proposed a new interpretation, where the phonon part corresponds to collective density fluctuations, and the roton part to single-particle excitations that are coupled to the density fluctuation spectrum through the Bose condensate. Hence, it is the Bose-broken symmetry that is responsible for the fact that a sharp mode is observed in superfluid 4He at low temperatures (--1K). The interpretation by Glyde and Griffin is the first to address the temperature dependence of the scattering, as well as providing a microscopic basis for the nature of the excitations. The effective interac-
207
tions in liquid 4He can be probed by varying the density of the system, through the application of a pressure of up to 24 bars. This provides a unique opportunity to test theoretical models. The first experiment on a strongly absorbing 3He-aHe mixture in 1973 [3] was made possible by improvements in neutron scattering techniques. A few years later, the first experiments were done on pure 3He by groups at the Institut Laue-Langevin (ILL) [4] and at Argonne National Laboratory [5]. In this paper, we will summarize the results obtained from the secondgeneration neutron scattering experiments on liquid 3He [6] and 3He-nile mixtures [7], performed in 1986 and 1987 at the ILL, and still producing new results. We will also compare these results with recent theories, where appropriate. Finally, we will discuss along which lines future experiments and data analysis might develop.
2. Liquid 3He Figure 1 shows the measured total dynamic structure factor S*(Q, E) as a function of wave vector Q and energy E for normal liquid 3He at saturated vapor pressure (SVP) and a temperature of T = 0.12 K (which is well below the Fermi temperature -1 of --1.5 K). At wave vectors above 2k F = 1.57 A (k F is the Fermi wave vector) the scattering consists of a broad peak resembling that from weakly interacting particle-hole pairs. For wave vectors smaller than 1 A - l , two excitations are distinguished: at low energies a continuum of spin-fluctuations, and at higher energies the zero-sound mode. The latter is a collective density fluctuation similar to phonons in solids. In fact, the resemblance between the dispersion of the broadened zero-sound mode in 3He and the sharp phonon-roton excitations in superfluid 4He is striking (Fig. 2). In the figure, the wave vectors have been rescaled with density, so that the first Bragg reflection would occur at Q* = 1, if the liquids were BCC solids. Since the zero-sound mode is heavily damped for wave vectors larger than 1 A-~, its 'dispersion' was
208
B. Fdk, R. Scherm / Phvsica B 197 (1994) 206-214
I <=2.5
30
J~
2.5
5
'
7.5
lO
I
'
I
,
>~2.5 " -2
I
12." sv 25
20
QJ C bJ
15
10
,
1
I 2
Momentum
0 / A-'
Fig. 1. Dynamical structure factor S*(Q E) in liquid 3He (p = 0 bar T = 0.12 K). Low intensity is represented by light regions, higher intensity with darker regions. For small wave vectors (Q < 1/~ ), the low-energy scatterin_gz(E- 1 K) corresponds to spin fluctuations, while the high-energy scattering (E = 5-15 K) to the zero-sound mode. At Q = 1.3 ,~- the two modes merge, and at even higher wave vectors the scattering resembles that expected for particle-hole pairs.
taken as the 'mountain ridge' of the broad scattering, obtained by finding the maximum of S*(Q, E) as a function of energy for each Q. According to the polarization potential theory by Pines and c o w o r k e r s [8], this agreement is maybe not so surprising, since the collective density fluctuations in 3He and 4He have the same origin, namely the restoring forces, which depend on the strong particle interaction, being similar in both liquids. On the other hand, the
'roton minimum' at a wave vector Q0 reflects the local order in the liquid, which is related to the maximum of the static structure factor S(Q). The quantitative agreement between the two liquids, shown in Fig. 2, is spoiled at higher pressures, although the position of the minimum of the dispersion curves occurs at the same wave vector Q*. Even nearly classical liquids like D 2 have collective excitations whose dispersion follow a p h o n o n - m a x o n - r o t o n curve [9].
B. Ftik, R. Scherm / Physica B 197 (1994) 206-214
i
31te
I
o SVP [] 20 bars
4H e
SVP
. . . . 18 bars
o
Z0 [] rl / f . . . . ".
[]
OCt#
07
t. O
'O
[]
.09 °
// o j t
I 1
Q* = Qdtlo/2~ Fig. 2. Comparison of the position of the maximum of the broad zero-sound and particle-hole scattering from 3He with the sharp p h o n o n - r o t o n mode in superfluid 4He. The wave vector Q* is scaled with respect to density.
For small wave vectors where the zero-sound mode is well defined, the dispersion has anomalous character (upward dispersion), the phase velocity exceeding the ultrasonic zero-sound velocity by more than 10% at SVP. At higher pressures, the anomalous character diminishes, and disappears completely at pressures somewhat smaller than 20 bars. A similar behavior is observed in 4He, where the anomalous character of the dispersion (4% at SVP) disappears at about the same pressure. Pines and co-workers have shown that the reduction of the anomalous character of the dispersion in both 3He [10] and 4He [11] is a natural result of the polarization potential theory. Their calculations are based on the random-phase approximation, in which the dynamic susceptibility x(Q, E) of 3He is given by
Xsc(Q,E)
x(Q, E) - 1 - V(Q, E)Xsc(Q, E)
(2)
where Xsc(Q, E) is the screened susceptibility of
209
the quasiparticles, and V(Q, E) is the effective quasiparticle-quasihole interaction. The density response is obtained by using the spin-symmetric interaction VS, while the spin fluctuations are described by using the spin-antisymmetric interaction Va. The dynamic structure factor S(Q, E) is proportional to the imaginary part of X(Q, E). In the Landau picture, the effective mass m*/ m3, which enters the screened susceptibility, is 2.8 at SVP. The above approximation was first introduced by Pines and co-workers [12], who reproduced the zero-sound dispersion obtained by Sk61d et al. [5], and later [10] the pressure dependence observed by Scherm et al. [6]. More simplified calculations [13], using the Landau interaction in Eq. (2), did not reproduce the anomalous dispersion and the flattening of the dispersion at large Q, thereby showing the importance of the specific form of the polarization potentials used by Pines and coworkers. The main disadvantage with the RPA model is that the damping of the zero-sound mode cannot be included in a straightforward way. The zero-sound mode in 3He is strongly damped, its line width is of the order of 0.2 meV at T = 0 . 1 2 K . This is in contrast to 4He, where the phonon-roton mode is extremely sharp (its width at T = 1 K is only 2 ixeV). There are three possible damping mechanisms: (1) the zerosound can decay into two other zero-sound modes, i.e. a three-phonon process; (2) the zerosound can decay by exciting a particle-hole pair (so-called Landau damping): (3) the zero-sound can decay by exciting several particle-hole pairs, i.e. damping due to multipairs. Process 1 is kinematically allowed only in the absence of anomalous dispersion, and can hence not explain the broadening observed at low pressures. For wave vectors below 1 A-~, process 2 is excluded since no overlap between the zero-sound mode and the particle-hole continuum is visible in neutron scattering data. Consequently, it was concluded in Ref. [6] that the damping due to multipairs (process 3) was the dominating decay mechanism for the zero-sound mode. There has been several calculations of the damping of the zero-sound mode due to multipairs. Glyde and Khanna [13] predicted a zero-sound damping
210
B. Fgtk, R. Scherm / Physica B 197 (1994) 206-214
much larger than the one observed. Holas and Singwi [14] made an extension to the RPA model to calculate the zero-sound width due to multipair damping, using the polarization potentials of Hess and Pines [10]. They found that the inclusion of the damping spoiled the good agreement with the zero-sound dispersion obtained in Ref. [10], although the agreement could be restored by modifying the polarization potentials. It is not clear whether these new polarization potentials also correctly describe the transport properties of liquid He. The calculated zero-sound width reproduces quite well the Q dependence of the measured width, but not its pressure dependence. Most recently, Stierstorfer [15] in a more exact approach obtained similar results for the Q dependence at SVP. An alternative explanation of the large zero-sound width has been forwarded by Glyde and collaborators [16]. Their calculations based on Green function formalism suggest that the Lindhard function describing the particle-hole pairs in strongly interacting 3He may develop a tail well outside the 'traditional' particle-hole band. The overlap of this tail with the zero-sound mode would then make Landau damping possible (process 2). However, the calculations in Ref. [16] were performed on spin-polarized 3He, so any quantitative estimates of the damping have to await that these complicated calculations can be performed on normal liquid 3He. The spin fluctuations can be treated on an equal footing to the density fluctuations, using the RPA approximation (2) with the spin-antisymmetric particle-hole interaction. Existing theories use either an effective mass m * / m 3 of 2.8 (Landau picture) or 1 (paramagnon model [17]) in the Lindhard function Xsc. However, it was shown recently that none of these models agrees with recent high-resolution neutron scattering data [18]. Landau-type models (e.g. Refs. [12-14]) using an effective mass of about 2.8 cannot describe the tail of the spin-fluctuation scattering. This failure is independent of the interaction potential used in (2); it is related to the upper edge of the particle-hole band, which is inversely proportional to the effective mass. The paramagnon model [17], which uses the bare
mass ( m * / m 3 = 1) in Eq. (2) and a single-parameter interaction V ( Q , E ) = J , is in fair agreement with the experimental data at low pressures. At higher pressures, the model clearly fails to describe the low-energy enhancement of the spin fluctuations. In fact, fits of the model to the data give a value of J > 1; this would correspond to a ferromagnetic instability, which is not observed in liquid 3He. By introducing a Qdependent effective mass m * ( Q ) , a good description of the spin-fluctuation scattering is obtained for all wave vectors and all pressures [18]. It turns out that m * ( Q ) / m 3 depends only weakly on wave vector and pressure, taking a value close to 1.8. However, it is not clear whether the introduction of a Q-dependent mass (or the bare mass as in the paramagnon model) in the spin-fluctuation response is compatible with calculations of the zero-sound dispersion.
3. Liquid 3He-4He mixtures The introduction of small quantities of 3He (up to 6%) in 4He alters only slightly the superfluid properties. The mixture can still be considered as a Bose liquid, where the 3He impurities increase the normal-fluid component ,On, in a way similar to an increased temperature. On the other hand, the thermodynamic properties change drastically. The specific heat is dominated by the low-energy 3He excitations. The 3He atoms form a weakly interacting Fermi liquid, where the quasiparticles move rather freely in the surrounding superfluid 4He, the backflow of which introduces an effective mass of the 3He quasiparticles. By varying the 3He concentration, the behavior can be changed from a nearly free Fermi gas to a correlated Fermi liquid. As in 4He, the role of the interaction can be probed by changing the density via an external pressure. Neutron scattering observes excitations from both 3He and 4He (plus an interference term), as shown by Eq. (1). The measured spectrum, Fig. 3, shows the phonon-roton branch, which is only slightly modified by the presence of 3He impurities. The low-energy 3He particle-hole exci-
B. Fttk, R. Scherm / Physica B 197 (1994) 206-214
<-2.5
5
I0
15
/
'
I
t
'
211
>30 10"-3
20
30 70 rnK_
25
20
t,
t5
tO Ii
5
,
0 0
1
Momentum
I 2
Q /
,,~-'
Fig. 3. Dynamical structure factor S*(Q, E) in a liquid 3He-"He mixture (p = 1 bar, T = 0.07 K, x 3 = 4.4%). The intensity is represented by the greyscale shown at the top of the figure. The lower branch is the particle-hole excitations from 3He, the upper branch is the phonon-roton excitation, and at even higher energies is a broad contribution from multiparticle excitations, similar to that observed in pure 4He.
tations are seen below the phonon-roton curve, they resemble to what is expected from a noninteracting Fermi gas. The small shift and broadening of the phononroton mode in dilute mixtures have been studied in detail with neutron scattering as a function of pressure, temperature, and 3He concentration [7]. When 3He atoms are added to 4He, the density decreases because of the larger zeropoint motion of 3He. This leads to a decrease in
the position of the roton minimum Q0, hence a shift of the excitation energy at a given Q. To eliminate this trivial effect, which obscures other more interesting phenomena, the measurements in the mixture and in pure 4He were performed at the same number density, obtained by increasing the pressure of the mixture by about 1 bar. The shift and broadening of the phonon-roton excitation at constant density (Fig. 4) show anomalies in the region where the 3He excita-
B. Fdk, R. Scherm / Physica B 197 (1994) 206-214
212
0.3
capacity measurements [19-20]. ek, which deviates from the quadratic dispersion expected for a noninteracting Fermi gas, is often parameterized as
(b) x=5% SVP T=O.07K/ /
"~-'~0.2 "-r "1- 0.t oeU
o
' ~ J ,o dA \Po ' Q ~ j x,,,/'~.,~.......
1 e k = 80 + 2 ~ - h2k2(1 + 3'k 2 + 8k 4 + . . . ) .
0
0.0
(o)
,•0.2
o
I
1
I
/0--2.46 A ,. . ~ ....,,'!/o---2.30
u_
o-iioi
"1- 0.1 (.,0 r
0.0
o.s
i
,.o
i
2.0 Q
Fig. 4. Shift (a) and broadening (b) of the p h o n o n - r o t o n excitation in liquid 3He-4He mixtures. Circles are experimental data, solid and dashed lines are results from a model calculation using different ranges of the effective interaction between the hybridizing modes; the dotted line is a constant contribution attributed to phonon-scattering processes.
tions are about to cross the p h o n o n - r o t o n curve. Calculations of the level repulsion based on Landau damping of the p h o n o n - r o t o n excitation in this region reproduce quite well the Q dependences of the shift and the broadening. However, different potentials had to be used for these two quantities, which suggests a possible shortcoming of these calculations. The strong temperature dependences of the shift and the broadening have not yet been satisfactorily described by any theory. The 3He quasiparticle dispersion e k in 3He4He mixtures have been obtained from heat Table 1 Parameters of the 3He quasiparticle dispersion Eq. (3), extrapolated to p = 0, x 3 = 0 and T = 0
Neutron scattering [21] Specific heat [23] Specific heat [19] Viscosity [24]
m*/m3
3' [nm 2]
2.244 (15) 2.272 (8) 2.30 2.21
-7.59 -7.39 -6.68 -9.80
(3) x 10 4 (46) x 10 -4 x 10 4 × 10 -4
(3)
Another method to determine 6k is neutron inelastic scattering, which measures directly the quasiparticle-quasihole excitations. The average energy O)q of the particle-hole band coincides with Ek only if the 3He quasiparticles have a quadratic dispersion. From neutron scattering measurements [7] we determined (~)q and evaluated 6k according to Eq. (3), keeping terms up to k 4, i.e. including y [21]. It turns out that both m* and 3' are temperature dependent. This came initially as a surprise, since in heat capacity measurements, m* and 3" are normally taken to be temperature independent. However, since e k is obtained from the temperature dependence of the heat capacity, it is not possible from such measurements to infer a unique temperature dependence of the parameters in the expansion (3). On the other hand, neutron scattering observes the quasiparticle dispersion e k at any temperature, making it possible to investigate whether it depends on temperature. In fact, a temperature dependent quasiparticle dispersion is in agreement with Landau Fermi liquid theory [22], where the quasiparticle energy depends on the distribution of the excited quasiparticles in the system, and hence on temperature. We have nevertheless verified that the temperature dependence of m* and 3" is not an artefact of terminating the expansion at the k 4 term, by repeating the analysis keeping terms up to order k 6, hence including 6. The result is that the temperature dependence of m* and 3' would become even stronger. Finally, we have calculated the specific heat directly from e k ( T ). The results [21] are in excellent agreement with specific-heat measurements [19-20]. The parameters m* and % when extrapolated to zero pressure, zero 3He concentration, and zero temperature, compare well with those obtained from the specific heat, second sound, and viscosity measurements [19,23,24], as shown in Table 1.
B. Fdk, R. Scherm / Physica B 197 (1994) 206-214
4. Outlook Improvements of the resolution and the precision in recent neutron scattering experiments on the quantum liquids 3He, 4He, and their mixtures have revealed new physics in these model systems. We will now discuss further possible experimental improvements. In liquid 3He, the improved energy resolution (three times sharper) of the present data has shown that none of the models used up to date for the spin fluctuations agrees with experimental data. These experiments were done in a pressure cell with a thick front window that gives rise to strong elastic scattering, thus obscuring the lowenergy part of the spin-fluctuation p e a k - only the high-energy tail is visible. Using a new lowpressure cell with a very thin window in a different geometry [25], we aim to measure the spin fluctuation scattering for smaller wave vectors and energies than possible hitherto. By studying the temperature dependence of the spin-fluctuation scattering, more information on the effective mass and the interactions is expected. The temperature dependence of the zero-sound mode may also reveal new information about the damping mechanisms responsible for the large width observed. In liquid 3He-nile mixtures, better estimates of the quasiparticle dispersion, and in particular the effective mass, can be obtained by extending these measurements to lower wave vectors and lower energies. To assess the role of the 3He3He interaction, lower 3He concentrations will have to be measured. Since the Fermi temperature decreases with decreasing 3He concentration, these measurements need to be performed at even lower temperatures, an experimental challenge in view of the large heat production caused by the neutron absorption in 3He. The intensity observed in a neutron inelastic scattering experiment is proportional to the convolution of the dynamic structure factor S(Q, E) with the instrumental resolution function R(Q, E). Better methods for estimating this effect, hopefully leading to a procedure that allows a deconvolution of the resolution function, or at least its energy and wave vector
213
dependence, even with noisy data, would make extractions of line widths and line shapes, and hence comparisons with theories, much easier.
Acknowledgements This work represents a collaboration over many years with a large number of people. In particular, we want to express our gratitude to A.J. Dianoux, H.R. Glyde, H. Godfrin, K. Guckelsberger, M. K6rfer, K. Sk61d, W.G. Stirling, A. Stunault, A. Szprynger and M. Weyrauch.
References [1] For a comprehensive review, see H.R. Glyde and E.C. Svensson, in: Methods of Experimental Physics, Vol. 23, Part B, eds. D.L. Price and K. Sk61d (Academic Press, New York, 1987) p. 303. [2] H.R. Glyde and A. Griffin, Phys. Rev. Lett. 65 (1990) 1454; H.R. Glyde, Phys. Rev. B 45 (1992) 7321. [3] J.M. Rowe, D.L. Price and G.E. Ostrowski, Phys. Rev. Lett. 31 (1973) 510. [4] R. Scherm et al., J. Phys. C 7 (1974) L341; P.A. Hilton, R.A. Cowley, R. Scherm and W.G. Stirling, J. Phys. C 13 (1980) L295. [5] K. Sk61d, C.A. Pelizzari, R. Kleb and H.R. Ostrowski, Phys. Rev. Lett. 37 (1976) 842; K. Sk61d and C.A. Pelizzari, Phil. Trans. Roy. Soc. London B 290 (1980) 605. [6] R. Scherm, K. Guckelsberger, B. F~k, K. Sk61d, A.J. Dianoux, H. Godfrin and W.G. Stirling, Phys. Rev. Lett. 59 (1987) 217. [7] B. Fhk, K. Guckelsberger, M. K6rfer, R. Scherm and A.J. Dianoux, Phys. Rev. B 41 (1990) 8732. [8] D. Pines, Can. J. Phys. 65 (1987) 1357, and references therein. [9] F.J. Bermejo et al., Phys. Lett. A 158 (1991) 253; F.J. Mompe~in et al., J. Phys.: Condens. Matter 5 (1993) 5743. [10] D.W. Hess and D. Pines, J. Low Temp. Phys. 72 (1988) 247. [11] C.H. Aldrich III, C.J. Pethick and D. Pines, J. Low Temp. Phys. 25 (1976) 691. [12] C.H. Aldrich III, C.J. Pethick and D. Pines, Phys. Rev. Lett. 37 (1976) 845; C.H. Aldrich III and D. Pines, J. Low Temp. Phys. 32 (1978) 689. [13] H.R. Glyde and F.C. Khanna, Can. J. Phys. 58 (1980) 343. [14] A. Holas and K.S. Singwi, Phys. Rev. B 40 (1989) 167.
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[15] K. Stierstorfer, PhD thesis, University of ErlangenNiirnberg, 1991 (unpublished). [16] C.W. Greeff, H.R. Glyde and B.E. Clements, Phys. Rev. B 45 (1992) 7951. [17] M.T. B6al-Monod, J. Low Temp. Phys. 37 (1979) 123; 39 (1980) 231. [18] B. F~k, K. Guckelsberger, R. Scherm and A. Stunault, Physica B 194-196 (1994) 739. [19] D.S. Greywall, Phys. Rev. Lett. 41 (1978) 177; Phys. Rev. B 20 (1979) 2643. [20] J.R. Owers-Bradley et al., J. Low Temp. Phys. 72 (1988) 201.
[21] R. Scherm, K. Guckelsberger, A. Szprynger and B. Fhk, J. Low Temp. Phys., in press. [22] D. Pines and P. Nozi~res, The Theory of Quantum Liquids, Vol. 1 (Benjamin, New York, 1966) ch. 1. [23] R.M. Bowley, J. Low Temp. Phys. 71 (1988) 319. [24] L.A. Pogorelov, B.N. Esel'son, O.S. Nosovitskaya and V.I. Sobolev, Fiz. Nizk. Temp. 5 (1979) 83 [Soy. J. Low Temp. Phys. 5 (1) (1979) 40]. [25] K. Guckelsberger, to be published.
Physica B 197 (1994) 206-214
Neutron scattering studies of liquid 3He and 3He-4He mixtures B. Ffik a'*, R. Scherm b aCEA, DOpartement de Recherche Fondamentale sur la MatiOre Condensde, SPSMS/ MDN, Grenoble Cedex, France blnstitut Laue-Langevin, Grenoble Cedex, France
Abstract
Density and spin fluctuations in liquid 3He and 3He-4He mixtures have been studied by neutron inelastic scattering. We discuss the similarities between the density fluctuations in normal liquid 3He and superfluid 4He (anomalous dispersion and the 'roton minimum' that reflects the local order) as well as damping mechanisms that could explain the large width of the 3He zero-sound mode. A line-shape analysis shows that existing theories cannot describe the spin fluctuations observed in liquid 3He. In 3He-4He mixtures, the shift and broadening of the phonon-roton mode is critically analyzed in terms of Landau damping. The 3He quasiparticle dispersion, and therefore the effective mass, obtained from neutron scattering measurements on 3He-4He mixtures, shows a clear temperature dependence. Although this is in contrast to what is normally assumed, the specific heat calculated from this dispersion is in excellent agreement with thermodynamic measurements.
1. Introduction
T h e q u a n t u m liquids 3He and 4He, which are m o d e l systems of strongly interacting fermions and bosons, continue to reveal new fascinating physics, and attract a broad interest among condensed matter physicists [1]. The properties of these liquids are best understood in terms of e l e m e n t a r y excitations. Inelastic scattering of t h e r m a l (or cold) neutrons is a unique tool for studies of q u a n t u m liquids, since the m o m e n t u m and energy of the excitations can be measured simultaneously. In a neutron scattering experim e n t , neutrons with a well defined incident wave
* Corresponding author.
vector k i are scattered by the sample, and the wave vector kf of the scattered neutrons is determined. Hence, the neutron transfers a m o m e n t u m Q = k i - k f and an energy E = ( k ~ k ~ ) h 2 / 2 m n to the sample. In this way, one (or several) excitation(s) with wave vector Q and energy E can be created. At nonzero t e m p e r a tures, an excitation can also be annihilated, thereby increasing the energy of the scattered neutron. Since neutrons interact with the nuclei via the strong interaction, it is the density fluctuations of the system that are observed. If the nuclei studied have a spin, as for 3He, nuclear spin fluctuations are observed in addition to the density fluctuations, the latter being p r o b e d by the spin-averaged coherent scattering. The intensity of the scattered neutrons is proportional to the dynamic structure factor S * ( Q , E ) , which for a mixture of 3He and 4He can be written as
0921-4526/94/$07.00 O 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0470-2
B. Fdk, R. Scherm / Physica B 197 (1994) 206-214
o'S*( Q, E) = (1 - x3)o'4S44(Q , E) + x 3 o r 3c S 3 3 ( Q ,
E) + x3o'3S33(Q i I , E)
-[- ~ / X 3 ( 1 -- x 3 ) o 3 4 8 3 4 ( O , E )
(1)
where x 3 is the molar concentration of 3He, the o-'s are scattering cross-sections for the different nuclei, Sj~(Q, E) describes density correlations between atoms of type j and k, and S~3(Q , E) describes spin correlations between the 3He nuclear spins. Equation (1) includes the limits of pure 3He and 4He as well. Note that there is a contribution from both density and spin fluctuations from 3He, whereas only density fluctuations are possible in 4He, since its nuclear spin is zero. Although neutron scattering is an ideal probe to study both density and spin fluctuations of 3He, the enormous absorption cross-section makes these experiments very difficult. In addition, the heat produced by the absorption of neutrons by 3He nuclei limits the lowest temperature that can be achieved. The properties of superfluid 4He was at large explained by Landau in 1947, who introduced the concept of elementary excitations. The dispersion of these excitations, the phonon-roton curve, was obtained phenomenologically by Landau. The first microscopic calculation of the dispersion was made by Feynman in 1954. The dispersion was measured a few years later for the first time, using neutron inelastic scattering. However, the nature of the excitations proposed by Landau and by Feynman remained an enigma, and the temperature dependence of the excitations was never well understood. Recently, Glyde and Griffin [2] proposed a new interpretation, where the phonon part corresponds to collective density fluctuations, and the roton part to single-particle excitations that are coupled to the density fluctuation spectrum through the Bose condensate. Hence, it is the Bose-broken symmetry that is responsible for the fact that a sharp mode is observed in superfluid 4He at low temperatures (--1K). The interpretation by Glyde and Griffin is the first to address the temperature dependence of the scattering, as well as providing a microscopic basis for the nature of the excitations. The effective interac-
207
tions in liquid 4He can be probed by varying the density of the system, through the application of a pressure of up to 24 bars. This provides a unique opportunity to test theoretical models. The first experiment on a strongly absorbing 3He-aHe mixture in 1973 [3] was made possible by improvements in neutron scattering techniques. A few years later, the first experiments were done on pure 3He by groups at the Institut Laue-Langevin (ILL) [4] and at Argonne National Laboratory [5]. In this paper, we will summarize the results obtained from the secondgeneration neutron scattering experiments on liquid 3He [6] and 3He-nile mixtures [7], performed in 1986 and 1987 at the ILL, and still producing new results. We will also compare these results with recent theories, where appropriate. Finally, we will discuss along which lines future experiments and data analysis might develop.
2. Liquid 3He Figure 1 shows the measured total dynamic structure factor S*(Q, E) as a function of wave vector Q and energy E for normal liquid 3He at saturated vapor pressure (SVP) and a temperature of T = 0.12 K (which is well below the Fermi temperature -1 of --1.5 K). At wave vectors above 2k F = 1.57 A (k F is the Fermi wave vector) the scattering consists of a broad peak resembling that from weakly interacting particle-hole pairs. For wave vectors smaller than 1 A - l , two excitations are distinguished: at low energies a continuum of spin-fluctuations, and at higher energies the zero-sound mode. The latter is a collective density fluctuation similar to phonons in solids. In fact, the resemblance between the dispersion of the broadened zero-sound mode in 3He and the sharp phonon-roton excitations in superfluid 4He is striking (Fig. 2). In the figure, the wave vectors have been rescaled with density, so that the first Bragg reflection would occur at Q* = 1, if the liquids were BCC solids. Since the zero-sound mode is heavily damped for wave vectors larger than 1 A-~, its 'dispersion' was
208
B. Fdk, R. Scherm / Phvsica B 197 (1994) 206-214
I <=2.5
30
J~
2.5
5
'
7.5
lO
I
'
I
,
>~2.5 " -2
I
12." sv 25
20
QJ C bJ
15
10
,
1
I 2
Momentum
0 / A-'
Fig. 1. Dynamical structure factor S*(Q E) in liquid 3He (p = 0 bar T = 0.12 K). Low intensity is represented by light regions, higher intensity with darker regions. For small wave vectors (Q < 1/~ ), the low-energy scatterin_gz(E- 1 K) corresponds to spin fluctuations, while the high-energy scattering (E = 5-15 K) to the zero-sound mode. At Q = 1.3 ,~- the two modes merge, and at even higher wave vectors the scattering resembles that expected for particle-hole pairs.
taken as the 'mountain ridge' of the broad scattering, obtained by finding the maximum of S*(Q, E) as a function of energy for each Q. According to the polarization potential theory by Pines and c o w o r k e r s [8], this agreement is maybe not so surprising, since the collective density fluctuations in 3He and 4He have the same origin, namely the restoring forces, which depend on the strong particle interaction, being similar in both liquids. On the other hand, the
'roton minimum' at a wave vector Q0 reflects the local order in the liquid, which is related to the maximum of the static structure factor S(Q). The quantitative agreement between the two liquids, shown in Fig. 2, is spoiled at higher pressures, although the position of the minimum of the dispersion curves occurs at the same wave vector Q*. Even nearly classical liquids like D 2 have collective excitations whose dispersion follow a p h o n o n - m a x o n - r o t o n curve [9].
B. Ftik, R. Scherm / Physica B 197 (1994) 206-214
i
31te
I
o SVP [] 20 bars
4H e
SVP
. . . . 18 bars
o
Z0 [] rl / f . . . . ".
[]
OCt#
07
t. O
'O
[]
.09 °
// o j t
I 1
Q* = Qdtlo/2~ Fig. 2. Comparison of the position of the maximum of the broad zero-sound and particle-hole scattering from 3He with the sharp p h o n o n - r o t o n mode in superfluid 4He. The wave vector Q* is scaled with respect to density.
For small wave vectors where the zero-sound mode is well defined, the dispersion has anomalous character (upward dispersion), the phase velocity exceeding the ultrasonic zero-sound velocity by more than 10% at SVP. At higher pressures, the anomalous character diminishes, and disappears completely at pressures somewhat smaller than 20 bars. A similar behavior is observed in 4He, where the anomalous character of the dispersion (4% at SVP) disappears at about the same pressure. Pines and co-workers have shown that the reduction of the anomalous character of the dispersion in both 3He [10] and 4He [11] is a natural result of the polarization potential theory. Their calculations are based on the random-phase approximation, in which the dynamic susceptibility x(Q, E) of 3He is given by
Xsc(Q,E)
x(Q, E) - 1 - V(Q, E)Xsc(Q, E)
(2)
where Xsc(Q, E) is the screened susceptibility of
209
the quasiparticles, and V(Q, E) is the effective quasiparticle-quasihole interaction. The density response is obtained by using the spin-symmetric interaction VS, while the spin fluctuations are described by using the spin-antisymmetric interaction Va. The dynamic structure factor S(Q, E) is proportional to the imaginary part of X(Q, E). In the Landau picture, the effective mass m*/ m3, which enters the screened susceptibility, is 2.8 at SVP. The above approximation was first introduced by Pines and co-workers [12], who reproduced the zero-sound dispersion obtained by Sk61d et al. [5], and later [10] the pressure dependence observed by Scherm et al. [6]. More simplified calculations [13], using the Landau interaction in Eq. (2), did not reproduce the anomalous dispersion and the flattening of the dispersion at large Q, thereby showing the importance of the specific form of the polarization potentials used by Pines and coworkers. The main disadvantage with the RPA model is that the damping of the zero-sound mode cannot be included in a straightforward way. The zero-sound mode in 3He is strongly damped, its line width is of the order of 0.2 meV at T = 0 . 1 2 K . This is in contrast to 4He, where the phonon-roton mode is extremely sharp (its width at T = 1 K is only 2 ixeV). There are three possible damping mechanisms: (1) the zerosound can decay into two other zero-sound modes, i.e. a three-phonon process; (2) the zerosound can decay by exciting a particle-hole pair (so-called Landau damping): (3) the zero-sound can decay by exciting several particle-hole pairs, i.e. damping due to multipairs. Process 1 is kinematically allowed only in the absence of anomalous dispersion, and can hence not explain the broadening observed at low pressures. For wave vectors below 1 A-~, process 2 is excluded since no overlap between the zero-sound mode and the particle-hole continuum is visible in neutron scattering data. Consequently, it was concluded in Ref. [6] that the damping due to multipairs (process 3) was the dominating decay mechanism for the zero-sound mode. There has been several calculations of the damping of the zero-sound mode due to multipairs. Glyde and Khanna [13] predicted a zero-sound damping
210
B. Fgtk, R. Scherm / Physica B 197 (1994) 206-214
much larger than the one observed. Holas and Singwi [14] made an extension to the RPA model to calculate the zero-sound width due to multipair damping, using the polarization potentials of Hess and Pines [10]. They found that the inclusion of the damping spoiled the good agreement with the zero-sound dispersion obtained in Ref. [10], although the agreement could be restored by modifying the polarization potentials. It is not clear whether these new polarization potentials also correctly describe the transport properties of liquid He. The calculated zero-sound width reproduces quite well the Q dependence of the measured width, but not its pressure dependence. Most recently, Stierstorfer [15] in a more exact approach obtained similar results for the Q dependence at SVP. An alternative explanation of the large zero-sound width has been forwarded by Glyde and collaborators [16]. Their calculations based on Green function formalism suggest that the Lindhard function describing the particle-hole pairs in strongly interacting 3He may develop a tail well outside the 'traditional' particle-hole band. The overlap of this tail with the zero-sound mode would then make Landau damping possible (process 2). However, the calculations in Ref. [16] were performed on spin-polarized 3He, so any quantitative estimates of the damping have to await that these complicated calculations can be performed on normal liquid 3He. The spin fluctuations can be treated on an equal footing to the density fluctuations, using the RPA approximation (2) with the spin-antisymmetric particle-hole interaction. Existing theories use either an effective mass m * / m 3 of 2.8 (Landau picture) or 1 (paramagnon model [17]) in the Lindhard function Xsc. However, it was shown recently that none of these models agrees with recent high-resolution neutron scattering data [18]. Landau-type models (e.g. Refs. [12-14]) using an effective mass of about 2.8 cannot describe the tail of the spin-fluctuation scattering. This failure is independent of the interaction potential used in (2); it is related to the upper edge of the particle-hole band, which is inversely proportional to the effective mass. The paramagnon model [17], which uses the bare
mass ( m * / m 3 = 1) in Eq. (2) and a single-parameter interaction V ( Q , E ) = J , is in fair agreement with the experimental data at low pressures. At higher pressures, the model clearly fails to describe the low-energy enhancement of the spin fluctuations. In fact, fits of the model to the data give a value of J > 1; this would correspond to a ferromagnetic instability, which is not observed in liquid 3He. By introducing a Qdependent effective mass m * ( Q ) , a good description of the spin-fluctuation scattering is obtained for all wave vectors and all pressures [18]. It turns out that m * ( Q ) / m 3 depends only weakly on wave vector and pressure, taking a value close to 1.8. However, it is not clear whether the introduction of a Q-dependent mass (or the bare mass as in the paramagnon model) in the spin-fluctuation response is compatible with calculations of the zero-sound dispersion.
3. Liquid 3He-4He mixtures The introduction of small quantities of 3He (up to 6%) in 4He alters only slightly the superfluid properties. The mixture can still be considered as a Bose liquid, where the 3He impurities increase the normal-fluid component ,On, in a way similar to an increased temperature. On the other hand, the thermodynamic properties change drastically. The specific heat is dominated by the low-energy 3He excitations. The 3He atoms form a weakly interacting Fermi liquid, where the quasiparticles move rather freely in the surrounding superfluid 4He, the backflow of which introduces an effective mass of the 3He quasiparticles. By varying the 3He concentration, the behavior can be changed from a nearly free Fermi gas to a correlated Fermi liquid. As in 4He, the role of the interaction can be probed by changing the density via an external pressure. Neutron scattering observes excitations from both 3He and 4He (plus an interference term), as shown by Eq. (1). The measured spectrum, Fig. 3, shows the phonon-roton branch, which is only slightly modified by the presence of 3He impurities. The low-energy 3He particle-hole exci-
B. Fttk, R. Scherm / Physica B 197 (1994) 206-214
<-2.5
5
I0
15
/
'
I
t
'
211
>30 10"-3
20
30 70 rnK_
25
20
t,
t5
tO Ii
5
,
0 0
1
Momentum
I 2
Q /
,,~-'
Fig. 3. Dynamical structure factor S*(Q, E) in a liquid 3He-"He mixture (p = 1 bar, T = 0.07 K, x 3 = 4.4%). The intensity is represented by the greyscale shown at the top of the figure. The lower branch is the particle-hole excitations from 3He, the upper branch is the phonon-roton excitation, and at even higher energies is a broad contribution from multiparticle excitations, similar to that observed in pure 4He.
tations are seen below the phonon-roton curve, they resemble to what is expected from a noninteracting Fermi gas. The small shift and broadening of the phononroton mode in dilute mixtures have been studied in detail with neutron scattering as a function of pressure, temperature, and 3He concentration [7]. When 3He atoms are added to 4He, the density decreases because of the larger zeropoint motion of 3He. This leads to a decrease in
the position of the roton minimum Q0, hence a shift of the excitation energy at a given Q. To eliminate this trivial effect, which obscures other more interesting phenomena, the measurements in the mixture and in pure 4He were performed at the same number density, obtained by increasing the pressure of the mixture by about 1 bar. The shift and broadening of the phonon-roton excitation at constant density (Fig. 4) show anomalies in the region where the 3He excita-
B. Fdk, R. Scherm / Physica B 197 (1994) 206-214
212
0.3
capacity measurements [19-20]. ek, which deviates from the quadratic dispersion expected for a noninteracting Fermi gas, is often parameterized as
(b) x=5% SVP T=O.07K/ /
"~-'~0.2 "-r "1- 0.t oeU
o
' ~ J ,o dA \Po ' Q ~ j x,,,/'~.,~.......
1 e k = 80 + 2 ~ - h2k2(1 + 3'k 2 + 8k 4 + . . . ) .
0
0.0
(o)
,•0.2
o
I
1
I
/0--2.46 A ,. . ~ ....,,'!/o---2.30
u_
o-iioi
"1- 0.1 (.,0 r
0.0
o.s
i
,.o
i
2.0 Q
Fig. 4. Shift (a) and broadening (b) of the p h o n o n - r o t o n excitation in liquid 3He-4He mixtures. Circles are experimental data, solid and dashed lines are results from a model calculation using different ranges of the effective interaction between the hybridizing modes; the dotted line is a constant contribution attributed to phonon-scattering processes.
tions are about to cross the p h o n o n - r o t o n curve. Calculations of the level repulsion based on Landau damping of the p h o n o n - r o t o n excitation in this region reproduce quite well the Q dependences of the shift and the broadening. However, different potentials had to be used for these two quantities, which suggests a possible shortcoming of these calculations. The strong temperature dependences of the shift and the broadening have not yet been satisfactorily described by any theory. The 3He quasiparticle dispersion e k in 3He4He mixtures have been obtained from heat Table 1 Parameters of the 3He quasiparticle dispersion Eq. (3), extrapolated to p = 0, x 3 = 0 and T = 0
Neutron scattering [21] Specific heat [23] Specific heat [19] Viscosity [24]
m*/m3
3' [nm 2]
2.244 (15) 2.272 (8) 2.30 2.21
-7.59 -7.39 -6.68 -9.80
(3) x 10 4 (46) x 10 -4 x 10 4 × 10 -4
(3)
Another method to determine 6k is neutron inelastic scattering, which measures directly the quasiparticle-quasihole excitations. The average energy O)q of the particle-hole band coincides with Ek only if the 3He quasiparticles have a quadratic dispersion. From neutron scattering measurements [7] we determined (~)q and evaluated 6k according to Eq. (3), keeping terms up to k 4, i.e. including y [21]. It turns out that both m* and 3' are temperature dependent. This came initially as a surprise, since in heat capacity measurements, m* and 3" are normally taken to be temperature independent. However, since e k is obtained from the temperature dependence of the heat capacity, it is not possible from such measurements to infer a unique temperature dependence of the parameters in the expansion (3). On the other hand, neutron scattering observes the quasiparticle dispersion e k at any temperature, making it possible to investigate whether it depends on temperature. In fact, a temperature dependent quasiparticle dispersion is in agreement with Landau Fermi liquid theory [22], where the quasiparticle energy depends on the distribution of the excited quasiparticles in the system, and hence on temperature. We have nevertheless verified that the temperature dependence of m* and 3" is not an artefact of terminating the expansion at the k 4 term, by repeating the analysis keeping terms up to order k 6, hence including 6. The result is that the temperature dependence of m* and 3' would become even stronger. Finally, we have calculated the specific heat directly from e k ( T ). The results [21] are in excellent agreement with specific-heat measurements [19-20]. The parameters m* and % when extrapolated to zero pressure, zero 3He concentration, and zero temperature, compare well with those obtained from the specific heat, second sound, and viscosity measurements [19,23,24], as shown in Table 1.
B. Fdk, R. Scherm / Physica B 197 (1994) 206-214
4. Outlook Improvements of the resolution and the precision in recent neutron scattering experiments on the quantum liquids 3He, 4He, and their mixtures have revealed new physics in these model systems. We will now discuss further possible experimental improvements. In liquid 3He, the improved energy resolution (three times sharper) of the present data has shown that none of the models used up to date for the spin fluctuations agrees with experimental data. These experiments were done in a pressure cell with a thick front window that gives rise to strong elastic scattering, thus obscuring the lowenergy part of the spin-fluctuation p e a k - only the high-energy tail is visible. Using a new lowpressure cell with a very thin window in a different geometry [25], we aim to measure the spin fluctuation scattering for smaller wave vectors and energies than possible hitherto. By studying the temperature dependence of the spin-fluctuation scattering, more information on the effective mass and the interactions is expected. The temperature dependence of the zero-sound mode may also reveal new information about the damping mechanisms responsible for the large width observed. In liquid 3He-nile mixtures, better estimates of the quasiparticle dispersion, and in particular the effective mass, can be obtained by extending these measurements to lower wave vectors and lower energies. To assess the role of the 3He3He interaction, lower 3He concentrations will have to be measured. Since the Fermi temperature decreases with decreasing 3He concentration, these measurements need to be performed at even lower temperatures, an experimental challenge in view of the large heat production caused by the neutron absorption in 3He. The intensity observed in a neutron inelastic scattering experiment is proportional to the convolution of the dynamic structure factor S(Q, E) with the instrumental resolution function R(Q, E). Better methods for estimating this effect, hopefully leading to a procedure that allows a deconvolution of the resolution function, or at least its energy and wave vector
213
dependence, even with noisy data, would make extractions of line widths and line shapes, and hence comparisons with theories, much easier.
Acknowledgements This work represents a collaboration over many years with a large number of people. In particular, we want to express our gratitude to A.J. Dianoux, H.R. Glyde, H. Godfrin, K. Guckelsberger, M. K6rfer, K. Sk61d, W.G. Stirling, A. Stunault, A. Szprynger and M. Weyrauch.
References [1] For a comprehensive review, see H.R. Glyde and E.C. Svensson, in: Methods of Experimental Physics, Vol. 23, Part B, eds. D.L. Price and K. Sk61d (Academic Press, New York, 1987) p. 303. [2] H.R. Glyde and A. Griffin, Phys. Rev. Lett. 65 (1990) 1454; H.R. Glyde, Phys. Rev. B 45 (1992) 7321. [3] J.M. Rowe, D.L. Price and G.E. Ostrowski, Phys. Rev. Lett. 31 (1973) 510. [4] R. Scherm et al., J. Phys. C 7 (1974) L341; P.A. Hilton, R.A. Cowley, R. Scherm and W.G. Stirling, J. Phys. C 13 (1980) L295. [5] K. Sk61d, C.A. Pelizzari, R. Kleb and H.R. Ostrowski, Phys. Rev. Lett. 37 (1976) 842; K. Sk61d and C.A. Pelizzari, Phil. Trans. Roy. Soc. London B 290 (1980) 605. [6] R. Scherm, K. Guckelsberger, B. F~k, K. Sk61d, A.J. Dianoux, H. Godfrin and W.G. Stirling, Phys. Rev. Lett. 59 (1987) 217. [7] B. Fhk, K. Guckelsberger, M. K6rfer, R. Scherm and A.J. Dianoux, Phys. Rev. B 41 (1990) 8732. [8] D. Pines, Can. J. Phys. 65 (1987) 1357, and references therein. [9] F.J. Bermejo et al., Phys. Lett. A 158 (1991) 253; F.J. Mompe~in et al., J. Phys.: Condens. Matter 5 (1993) 5743. [10] D.W. Hess and D. Pines, J. Low Temp. Phys. 72 (1988) 247. [11] C.H. Aldrich III, C.J. Pethick and D. Pines, J. Low Temp. Phys. 25 (1976) 691. [12] C.H. Aldrich III, C.J. Pethick and D. Pines, Phys. Rev. Lett. 37 (1976) 845; C.H. Aldrich III and D. Pines, J. Low Temp. Phys. 32 (1978) 689. [13] H.R. Glyde and F.C. Khanna, Can. J. Phys. 58 (1980) 343. [14] A. Holas and K.S. Singwi, Phys. Rev. B 40 (1989) 167.
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[15] K. Stierstorfer, PhD thesis, University of ErlangenNiirnberg, 1991 (unpublished). [16] C.W. Greeff, H.R. Glyde and B.E. Clements, Phys. Rev. B 45 (1992) 7951. [17] M.T. B6al-Monod, J. Low Temp. Phys. 37 (1979) 123; 39 (1980) 231. [18] B. F~k, K. Guckelsberger, R. Scherm and A. Stunault, Physica B 194-196 (1994) 739. [19] D.S. Greywall, Phys. Rev. Lett. 41 (1978) 177; Phys. Rev. B 20 (1979) 2643. [20] J.R. Owers-Bradley et al., J. Low Temp. Phys. 72 (1988) 201.
[21] R. Scherm, K. Guckelsberger, A. Szprynger and B. Fhk, J. Low Temp. Phys., in press. [22] D. Pines and P. Nozi~res, The Theory of Quantum Liquids, Vol. 1 (Benjamin, New York, 1966) ch. 1. [23] R.M. Bowley, J. Low Temp. Phys. 71 (1988) 319. [24] L.A. Pogorelov, B.N. Esel'son, O.S. Nosovitskaya and V.I. Sobolev, Fiz. Nizk. Temp. 5 (1979) 83 [Soy. J. Low Temp. Phys. 5 (1) (1979) 40]. [25] K. Guckelsberger, to be published.