Journal of Low Temperature Physics. Vol. 113. Nos. 3/4, 1998
Excitations of Liquid 4He in Disorder
H.R. Glyde, B. Fak,* and O. Plantevin*
Department of Physics and Astronomy, University of Delaware, Newark,
DE 19716, USA
" Commissariat a I'Energie Atomique, Departement de Recherche Fondamentale
sur la Matiere Condensee, SPSMS/MDN, 38054 Grenoble, France
We discuss the effect of disorder and confinement on the excitations in superfluid and normal liquid 4He. Neutron scattering measurements of the
excitations to date are limited to helium in aerogel. There the phonon-roton
energy and width are slightly modified by disorder but there is no evidence
for additional excitations at low energy nor of a gap in the phonon energy at
long wavelengths. Experimental difficulties are discussed. In a recent pathintegral Monte-Carlo study, in which a high density of point impurities are
introduced at random positions, a significant broadening and energy shift are
found together with additional low-energy excitations.
PACS numbers: 67.40.-w.
1. INTRODUCTION
The impact of disorder and confinement on condensed Bose systems,
denoted "dirty bosons", is of great current interest. Liquid 4He in aerogel,
vycor, and other confining media are model dirty Bose systems. Magnetic
flux lines (bosons) in type II (dirty) superconductors, Josephson junction
arrays and disordered thin films and wires are other examples. Phase transitions (e.g. superconducting-normal, melting), the nature of the phases,
and the thermodynamic properties of dirty bosons differ significantly from
properties of pure systems and have been extensively studied both experimentally and theoretically. For example, the normal to superfluid transition,
the transition temperature T\, and the superfluid density ps(T) below T\ of
liquid 4He in disorder differ from the bulk and depend sensitively on the
nature of the disorder.1-3
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H. R. Glyde, B. Fak, and O. Plantevin
The impact of disorder on the elementary excitations of liquid 4He is of
equal interest but much less studied. Does the disorder significantly change
the energy and lifetime of the characteristic phonon-roton excitation and can
the changes be related to phases in disorder? For example, does disorder
introduce a gap in the phonon-roton energy at long wavelengths signalling
localization in the disordered phase?4 Are there additional excitations at
low energy as proposed in model studies?5 Is the sound velocity or excitation
energy lowered as predicted?4,6 A recent Path-Integral Monte-Carlo (PIMC)
study suggests that short range, quenched disorder can significantly change
the dynamic structure factor S(Q,w).7
In Sec. 2, we survey all neutron scattering measurements to date on
the excitations of liquid 4He in aerogel. Measurements in vycor and other
systems are in progress but results have not yet been reported. 4He confined
on surfaces are discussed in Ref. 8. PIMC calculations are discussed in Sec. 3,
and future prospects are outlined in Sec. 4.
2. NEUTRON SCATTERING WORK
Aerogel is a highly tenuous structure of irregularly connected silica
strands with porosities between 85 and 99.5% and with a large surface to
volume ratio. A typical spacing between strands is of the order of 100 A (for
95% porosity) and there is a wide distribution of length scales from a few to
a few hundred A. The mean free path in aerogel is typically 1000 A, which
is much longer than the wavelength of the excitations studied by neutron
scattering.
Three different groups have studied the excitations of liquid helium in
aerogel by inelastic neutron scattering, which measures directly S ( Q , w ) ,
We will first summarize the main results of these studies and then discuss
experimental difficulties encountered in these measurements and their consequences for the future.
2.1. Experimental Results
In general, the excitations of helium in aerogel observed to date are very
similar to those of bulk helium. In particular, no low-lying excitations are
seen below the phonon-roton energy. A complicating feature in aerogel is
multiple scattering. An incident neutron scatters inelastically from the liquid
4
He creating an excitation in the usual way, but on the way out (or in) the
neutron scatters again elastically from the aerogel. This second (multiple)
scattering changes the neutron wave vector so that the excitation appears to
have a modified Q value. As a consequence, a density-of-states-like feature
of the excitation spectrum will be seen at all Q values, with an intensity
Excitations of Liquid 4He in Disorder
539
Fig. 1. Temperature dependence of the energy shift and width of the roton excitation in aerogel and bulk helium. From Ref. 10.
which depends on the strength of the scattering from the aerogel.
The first measurements9 on superfluid helium in aerogel were performed
on the time-of-flight spectrometer Mibemol at the Laboratoire Leon Brillouin, using an energy resolution of 160 fieV. Both base-catalyzed and
neutral-reaction aerogel of a porosity of 96-96.5% were studied at temperatures of 1.6 and 1.8 K. The aerogel had been treated in a flow of deuterium
gas in order to replace the hydrogen adsorbed on the surface, thereby reducing the amount of multiple scattering. Due to the limited counting statistics
and the coarse resolution, no difference was found between the excitations
of helium in aerogel and bulk helium.
Considerably better statistics was obtained in a more recent
experiment10 performed at the high-flux reactor of the ILL, using the tripleaxis spectrometer IN 12 with an energy resolution of 120 ueV. The measurements were performed at temperatures between 0.5 and 2.25 K on a
base-catalyzed aerogel sample of 95% porosity prepared using fully deuterated products. The multiple scattering was negligible. The intrinsic width
of the phonon-roton excitation was found to have the same temperature dependence as in bulk helium, but for Q > 1 A- 1 , the width saturated at a
finite value of T ~ 5 /tteV at low temperatures (see Fig. 1b for the roton).
The temperature dependence of the shift of the excitation was similar to
that in bulk helium (see Fig. la), but the energy was lower by a few /^eV at
low temperatures. The energy of the phonon at Q = 0.2 A-1 was identical
to that in bulk helium, and no low-energy excitations were observed below
the phonon-roton excitation.
The most extensive measurements11,12 have been performed on the
time-of-flight backscattering spectrometer IRIS at the ISIS pulsed neutron
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H. R. Glyde, B. Fak, and O. Plantevin
source with an energy resolution of 15 fieV and a somewhat coarse Qresolution. Complementary measurements were also made on the SPINS
triple-axis spectrometer at NIST with a resolution of 30 //eV. Measurements
for temperatures between 1.3 and 2.2 K were done on non-deuterated aerogels with 90 and 95% porosity. The multiple scattering was an order of
magnitude higher than in the ILL experiments. The results for the excitation width are very similar to those of the ILL measurements, while the
temperature dependence of the roton energy is different. Between T - 1.3
and 1.9 K, the roton energy decreases with T more slowly in aerogel than in
the bulk (see Fig. la). Between T = 1.9 K and T\, the roton energy decreases
more rapidly in aerogel than in the bulk.12 The roton energy in aerogel is
found to lie 20 ^ueV above the bulk value at T = 1.9 K. No difference was
quoted between aerogels of 90 and 95% porosity.
2.2. Experimental problems
There are several experimental problems related to neutron scattering
measurements of the excitations of liquid helium in aerogel, which are chiefly
due to the smallness of the effects observed, when compared to bulk helium.
A major problem is multiple scattering, which gives an additional peak
with little Q-dependence that reflects the density of states of the phononroton excitation. This feature is peaked at the roton energy but has also
a rather strong contribution at the maxon energy. The multiple scattering
partly overlaps with the phonon-roton excitation at all Q values, and if the
intensity is sufficiently strong it could give an apparent shift and broadening of the excitations. In these cases, the analysis of the data is further
complicated by the fact that the density of states is temperature dependent.
A second difficulty is that since the broadening and shift of the excitations are so small, a very good energy resolution is required. The effects
observed in the measurements at the ILL10 are close to the limit of detection.
A related problem is that the resolution of a given spectrometer must be very
well known, both in energy and wave vector, and calibration measurements
on bulk helium performed under identical conditions are necessary.
In most experiments, the bulk helium calibration measurements were
performed only at low temperatures, in order to obtain the instrumental
resolution. This implies that the calibration of the thermometers is important, since data for bulk helium are then taken from literature. An error in
the absolute temperature scale of 0.1 K at T = 1.9 K give rise to an error in
the roton energy of 10 ^teV.
For temperatures above 1.8 K, the results are very sensitive to the way
the data is analyzed, and in particular how the multiphonon excitations
and the contribution from the normal fluid are treated. At a temperature
Excitations of Liquid 4He in Disorder
541
Fig. 2. S(Q,w) obtained from PIMC calculations (Ref. 7) at the phonon and
roton wave vectors for bulk (pure) liquid 4He (dotted line) and different disordering
potentials (there are M impurities with effective radius or).
of 1.9 K, different methods of subtracting the multiphonon excitations can
give roton energies that differ by 10 peV, as seen in e.g. Fig. 5 of Ref. 10. It
is important to compare energies and widths obtained by analyzing the data
with exactly the same functions in aerogel and bulk at each temperature.
3. PATH-INTEGRAL MONTE-CARLO CALCULATIONS
PIMC methods have been used to evaluate S(Q,w) of superfluid 4He in
disorder.7 The disorder was represented by randomly distributed static impurities interacting with the helium atoms via a simple attractive potential.
Although this potential does not resemble the structure of aerogel, it allows
to study the influence on the excitations of both the impurity concentration
and the correlation length of the disordering environment. New weight (i.e.
low-energy excitations) in S(Q,w) is induced by disorder, as predicted in
some earlier calculations. Assuming that S(Q,w) consists of a single peak,
the calculations suggest that this peak broadens and that its position shifts
(see Fig. 2). In the phonon region the shift is to lower energies in agreement
with dilute-gas and model studies, in the maxon region the shift is to higher
energies, and in the roton region to lower energies again. These calculations
suggest that disorder having a shorter correlation length and higher density
than aerogel may be needed to observe significant changes in S(Q,w).
4. DISCUSSION AND OUTLOOK
Neutron scattering experiments have shown that the excitations in superfluid 4He are only little effected by the disorder and confinement in 90-
542
H. R. Glyde, B. Fak, and O. Plantevin
95% porous aerogel. The effects are sufficiently small that new measurements
should be performed both in the bulk and in aerogel under identical conditions and the data analyzed in an identical manner. It seems particularly
important to reduce the multiple scattering. It would also be highly interesting to study the excitations in materials with smaller pore sizes, such as
denser aerogel, xerogel, vycor, or zeolites, where larger effects on the excitation spectrum are expected. Exciting and significant changes are predicted
for shorter range disorder.
PIMC calculations could be extended to study several important issues
concerning the excitations in 4 He, such as the separation of effects due to
confinement and those due to disorder, the difference between random and
ordered impurities, between point and line impurities (strands), and between
static and dynamic impurities (such as 3 He). Clearly, more theoretical and
experimental work is needed for a better understanding of the excitations of
liquid helium in disorder.
ACKNOWLEDGMENTS
This work was supported in part by the National Science Foundation
through research Grant Nos. INT-9314661 and DMR-9623961. We have
benefitted from discussions with J. R. Beamish, M. Boninsegni, J. Bossy,
and N. Mulders.
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