Search for supersolidity in solid 4He usingmultiple-mode torsional oscillatorsAnna Eyala,1, Xiao Mia,2, Artem V. Talanova,3, and John D. Reppya,1

aDepartment of Physics, Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853

Contributed by John D. Reppy, April 8, 2016 (sent for review March 3, 2016; reviewed by John Beamish and Robert B. Hallock)

In 2004, Kim and Chan (KC) reported a decrease in the period oftorsional oscillators (TO) containing samples of solid 4He, as thetemperature was lowered below 0.2 K [Kim E, Chan MHW (2004)Science 305(5692):1941–1944]. These unexpected results consti-tuted the first experimental evidence that the long-predictedsupersolid state of solid 4He may exist in nature. The KC resultswere quickly confirmed in a number of other laboratories andcreated great excitement in the low-temperature condensed-mat-ter community. Since that time, however, it has become clear thatthe period shifts seen in the early experiments can in large part beexplained by an increase in the shear modulus of the 4He solididentified by Day and Beamish [Day J, Beamish J (2007) Nature450(7171):853–856]. Using multiple-frequency torsional oscillators,we can separate frequency-dependent period shifts arising fromchanges in the elastic properties of the solid 4He from possiblesupersolid signals, which are expected to be independent of fre-quency. We find in our measurements that as the temperature islowered below 0.2 K, a clear frequency-dependent contribution tothe period shift arising from changes in the 4He elastic propertiesis always present. For all of the cells reported in this paper, how-ever, there is always an additional small frequency-independentcontribution to the total period shift, such as would be expected inthe case of a transition to a supersolid state.

supersolid | helium 4 | torsional oscillators

The idea of the existence of a supersolid state, a new phase ofmatter, stimulated a large number of experimental and the-

oretical works over the past 45 y, especially in the past decade.Such a state is assumed to present properties of superflow in a,most likely bosonic, solid. It was first suggested to exist based ontheoretical grounds by Andreev and Lifshitz (1), Chester (2), andLeggett (3) in the late 1960s and early 1970s. Because 4Heexhibits superfluidity, and because it is a bosonic material witha low mass and weak interatomic interactions, solid 4He was anatural candidate in the search for supersolidity.Torsional oscillators (TOs) have been the chief instrument in

the investigation of this possible supersolid state of 4He. The useof TOs in the search for this state follows a suggestion by Leggettin 1970 (3) that the appearance of the supersolid state wouldresult in an anomalous superfluid-like decoupling of a portion ofthe moment of inertia of the solid 4He sample from the TO in amanner analogous to the decoupling of the superfluid fraction ofliquid 4He as observed by Andronikashvili (4) in his early TOexperiments. Such a reduction of the moment of inertia uponentering the superfluid or supersolid state was termed by Leggettto be a nonclassical rotational inertia (NCRI). The NCRI re-duction in the moment of inertia is then expected to begin toincrease with decreasing temperature below an onset tempera-ture and lead to an observable decrease in the period of a TOcontaining a supersolid sample.The first attempt to observe the proposed supersolid state

using a TO, at Bell Labs (5), gave negative results and it wasconcluded that a supersolid signal, if present, was below thedetection level of few parts in 105. However, subsequent TOexperiments by Kim and Chan (KC) gave positive results and theanomalous decrease in the TO period or NCRI expected for the

supersolid state was observed. The first observations were obtainedfor solid 4He samples contained in porous Vycor glass (6). Later,the NCRI effect was observed by KC (7) for bulk samples of solid4He. KC demonstrated in a series of experiments that the magni-tude of the NCRI was velocity-dependent, being reduced at highwall velocities, and also sensitive to the impurity concentration of3He. These results for the bulk solid were soon replicated in anumber of different laboratories (8–11) and were initially inter-preted as evidence for a supersolid phase in solid 4He.Over time, the supersolid interpretation of the early TO data

has been brought into question by the increasing awareness ofthe importance of the significant role that an anomaly in the elasticconstants of solid 4He can play in contributing to the period shiftsobserved in the TO experiments. Following the earlier work at BellLabs by Paalanen et al. (12), Day and Beamish (DB) (13) at theUniversity of Alberta made a detailed study of the temperaturedependence of the shear modulus of solid 4He. DB found ananomalous decrease in the shear modulus of the solid as thetemperature was raised from temperatures below 40 mK toabove 250 mK. This is the same temperature range over whichthe NCRI phenomenon has been observed in TO experiments.In their early work, DB report fractional changes in the shearmodulus of up to 10%. Depending on the design of a TO, suchvariations in the helium shear modulus with temperature canlead to period shift signals with a temperature dependence andmagnitude similar to those observed by KC and others. Oneexample for this is the careful repetition (14) of the early solid4He in Vycor (6) experiment by the Pennsylvania State Uni-versity group, in which an upper limit of 2× 10−5 was set for

Significance

The concept of a supersolid state in solid 4He was first suggestedin 1968, yet it was only in 2004 that experimental evidence forthe existence of this new state of matter was presented. Thepossible discovery of the long-sought supersolid state createdmuch excitement and was followed, however, by experimentaland theoretical work that suggested that the original in-terpretation was not satisfactory. It is now clear that many of theoriginally observed signals can be explained by changes in theshear modulus of the solid in the experimental cells. In the ex-periments reported here, we have used an experimental tech-nique that enables the distinction between the effects of theshear modulus and those of a possible supersolid state.

Author contributions: A.E., X.M., A.V.T., and J.D.R. designed research, performed re-search, analyzed data, and wrote the paper.

Reviewers: J.B., University of Alberta; and R.B.H., University of Massachusetts Amherst.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.1To whom correspondence may be addressed. Email: anna.eyal@gmail.com or jdr13@cornell.edu.

2Present address: Department of Physics, Princeton University, Princeton, NJ 08544.3Present address: John A. Paulson School of Engineering and Applied Sciences, HarvardUniversity, Cambridge, MA 02138.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1605738113/-/DCSupplemental.

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supersolid in Vycor. The period shifts in the original experimentprobably arose from an increase of shear modulus μ of a thin layerof bulk solid 4He in the TO cells. These results were also confirmedby double TOmeasurements (15) of solid helium in Vycor, where allof the apparent signal was attributed to elastic effects. Therefore, thecurrent consensus in the field is that a supersolid phase has probablynot been observed experimentally in solid 4He so far. However, bothrecent theoretical predictions (16) and simulations (17, 18) do notrule out the existence of some type of superflow in the solid.In an attempt to overcome the problems raised by the interplay

between the mechanics of TOs and the shear modulus anomaly,we have used multiple-frequency TOs in a strategy to discriminatebetween period shifts arising from the elastic anomaly and thosedue to the existence of a possible supersolid. This approach hasbeen successful in that in our most recent experiments with threeTOs of different design we have observed the existence of a sampledependent frequency-independent contribution to the observedperiod shifts. The magnitude of the observed frequency-independentcontributions, when normalized by the period shift measured uponfreezing of the sample, ranges from 0.4 × 10−4 to 1.5 × 10−4. Suchfrequency-independent signals are possible candidates of the pe-riod shifts expected for the supersolid phase. In this paper wegive a detailed discussion of these results including a carefulconsideration of the influence of the elastic anomaly on the me-chanics of our individual TOs. We shall also include a discussionof previous multiple-frequency TO experiments (9, 19) of Koji-ma’s group at Rutgers University and the more recent experi-ment (20) by Cowan’s group at Royal Holloway, University ofLondon. In the discussion of the interplay between temperature-dependent changes in the solid 4He elastic modulus and the TOs wehave used both analytic calculations and a finite element methods(FEM) approach for our analysis. An important conclusionfrom our analysis is that with careful design many contributionsarising from the elastic anomaly can be strongly suppressed, and theremaining elastic contribution will be frequency-dependent,chiefly varying as the square of the TO frequency.

Extended Details of the SearchBrief Early History. The early measurements of KC (6, 7) showedboth the period shift and the velocity dependence that wereexpected to be present. Their signals ranged between 0.75% and2.5% of the total moment of inertia of the solid sample. Thesenumbers, ΔP=ΔPF, where ΔP is the period change attributed tothe supersolid transition and ΔPF the change in period uponfreezing of the solid sample, can be defined as fractional periodshifts (FPS). That is analogous to the superfluid density ratio, orwhat is called NCRIF in many of the papers on the subject. Thesignals observed varied from sample to sample, with wide vari-ation in signal size from the several percent level to the detectionlevel of the measurements. This extreme dependence on sam-ples, in addition to dependence on cell geometry, gave earlyindications, which were not immediately fully appreciated, thatthe experimental situation was not simple.

Unexpected Results. As other groups started repeating and ex-tending the KC results, more confusing results were observed.One of the first of these were Rittner and Reppy’s annealingexperiments (8). Annealing the samples for which signals wereobserved would result in significant reduction of the signals.However, quench-cooling of the samples, which generated stressand disorder, resulted in increased signal size. Double-modeoscillator experiments (9) revealed frequency-dependent signals,which were also inconsistent with a simple supersolid scenario.Flow measurements, performed at first by Hallock’s group (21)and later by Chan’s group (unpublished) and Beamish’s group(22) revealed mass flow that decreased substantially below about80 mK with a strong dependence on 3He concentration. Rotatingoscillator experiments, performed by the groups of Kubota (23),

Shirahama (poster presentation at QFS2012), and Kim (24)showed a rotation dependence of the signals, with a velocitydependence different from that obtained by the TO velocitymeasurements. However, recent DC rotation measurements bythe Golov group (25) show no such dependence. On the theo-retical level, Path Integral Monte Carlo simulations performed byPollet et al. (17) show no superflow in defect-free crystals but doshow superfluidity of grain boundaries, and a Luttinger liquid inthe cores of some dislocations. Anderson (26), however, demon-strated superflow when investigating the Bose–Hubbard modeland suggested that 4He would exhibit the same behavior. All theseresults were inconsistent with the original supersolid scenario.

Elastic Properties of Solid 4He. In 2007, DB (13) discovered ananomalous softening with increasing temperature of the shearmodulus of solid helium over the same temperature range as thesupposed supersolid. The anomalous period shifts observed inTO experiments can be partially explained by the elastic anom-aly, and therefore are not necessarily a signature of a supersolidstate. Given the importance of elastic effects in the temperature-dependent period of TOs used in the supersolid search, we willstart with a brief summary of the elastic properties of solid 4He.In Fig. 1A the shear modulus data, supplied by Beamish’s group,for a polycrystalline sample are plotted as a function of tem-perature. The 4He sample in these measurements was containedbetween two closely spaced parallel piezoelectric plates. One platewas excited in low-frequency transverse motion. The second pie-zoelectric plate served to detect the strain induced in the solid bythe motion of the active plate. There are three distinguishabletemperature regions seen in these data. There is a high-temperatureregion, above roughly 200–300 mK, where the shear modulus showsonly a weak variation with temperature. In the region between 200mK and 50 mK, however, there is a remarkable increase in themagnitude of the shear modulus to a maximum value achievedbelow 50 mK, in another temperature-independent region. Al-though the decrease in the shear modulus for a polycrystallinesample as shown in Fig. 1A is only 8% of the low-temperaturevalue, much larger reductions have been observed in other ex-periments with single crystal samples. For example, in the giantplasticity experiments performed by Balibar’s group at the ÉcoleNormale Supérieure in Paris (27), reductions in the shear

0.1 1708090100110120130140150

140 Hz40 Hz6 Hz2 Hz

16000 Hz6500 Hz3000 Hz1500 Hz600 Hz

Shearmodulus[bar]

Temperature [K]

0.0 0.1 0.2 0.3 0.4 0.50.900.920.940.960.981.00

B

2000 Hz200 Hz20 Hz

μ/μ 0

Temperature [K]

A

Fig. 1. (A) Normalized shear modulus vs. temperature of a polycrystallinesample of 4He, for three different excitation frequencies. μ0 = 1.5×108 dyn/cm2.Data from ref. 13. (B) Shear modulus vs. temperature for several different ex-citation frequencies for a single crystal of 4He. Data from ref. 28.

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modulus in excess of 50% were observed for single crystals be-tween the low- and high-temperature regions.DB demonstrated a sensitivity to 3He concentration similar to

that observed by KC. They also investigated the frequency de-pendence of the anomaly as well as the dependence on the strainlevel determined by the amplitude of the active plate. Fig. 1B showsrecent 4He shear modulus data, from the Paris group of Balibar(28), as a function of temperature for frequencies ranging from 2 to1.6× 104 Hz. At the lowest temperature, and also in the region be-tween 400 mK and 500 mK, there is little dependence on frequencycompared with the region where the shear modulus is decliningrapidly with increasing temperature, where a frequency dependenceis clearly evident.The current understanding of the phenomena of this elastic

anomaly is based on the temperature-dependent pinning of dis-location lines by the 3He impurities. Given a sufficient concen-tration of 3He atoms in the 4He solid, the dislocation lines will bestrongly pinned at low temperatures; however, as the temperatureis raised, the dislocations become unpinned and mobile as the pin-ning 3He atoms evaporate from the lines. The increased mobility ofthe dislocation lines with unpinning leads to a reduction in theshear modulus of the solid 4He. Once the 3He has completelyevaporated, the shear modulus becomes again, as in the low-temperature region, relatively independent of temperature. Atconsiderably higher temperatures the mobility of the dislocationsis restricted by damping from thermal excitations in the solid.Increasing the stress on the samples also results in tearing the 3Heatoms away from the dislocation lines, hence creating an apparentvelocity dependence of the signals. For a comprehensive discus-sion of the rather complex problem of the motion of dislocationlines in solid 4He, see the review article by Balibar et al. (29). Inour work, the frequency range of the multiple frequency TOs isrestricted to a range between 600–2,500 Hz, much smaller thanthat shown in Fig. 1. In the analysis of the data, we will be empha-sizing the changes in TO period for changes in temperatures be-tween the low-temperature region below 50 mK and the high-temperature region between 300 mK and 500 mK, thus avoiding theregion where the shear modulus is strongly frequency-dependent.Importance of the elastic effects. Over the past several years therehas been a gradual emergence of an appreciation of the signifi-cance of the interplay between elastic properties of the solid andthe mechanics of TOs. These explain much, if not all, of whathad been seen in the TO experiments. As shown in the DB plot(Fig. 1A), μ decreases with increasing temperature, and thisshear modulus decrease leads to an increase in the period of aTO when the shear modulus plays a role in the stiffness of theTO, either in the fill line or in the body of the sample cell, thusmimicking the reduction of the period with increasing temper-ature expected for a supersolid. These TO period shifts can arisefrom any one of the following:

i) Acceleration of the solid helium in the cell (30).ii) The Maris effect: the counterstress of solid on the cell

walls (31).iii) Twisting of the cell walls.iv) The fill line effect: stiffening of the solid 4He in the torsion

rods (32).v) The dissipative dynamics of the solid (33).vi) Coupling different mobile parts of the cell via the solid helium.

There are two main approaches to calculating these differentcontributions:

i) The analytic approach, in which simple geometries are usedto give approximations and upper limits of the observed con-tributions. These show a frequency squared dependence ofthe elastic contributions inside the cell, suggesting the use ofmultiple-frequency TOs as a tool to distinguish betweenthese effects and a possible supersolid signal.

ii) FEM modeling can give more accurate results for specificcell geometries. In this approach as well, where only elasticeffects are being considered, plots of FEM FPS vs. f2 showno frequency-independent contributions, within the uncer-tainties of the calculation, for cells in which the torsion rodeffect is not present or significant.

In Supporting Information, we provide a more detailed discussionand calculations of the known elastic effects of solid 4He for one ofour TOs, using both an analytical approach and an FEM.A main concern is that in single-frequency oscillators it is not

possible to distinguish between a possible real supersolid and anelastic contribution. In recent work by the Pennsylvania State Uni-versity group (34) a very rigid cell was used in attempt to decrease asmuch as possible all elastic effects. Two different rigid oscillatordesigns were used to examine samples at pressures between 34 barand 60 bar with rim velocities between 11 μm/s and 19 μm/s. AnNCRI signal of 4× 10−6 was obtained for one cell and the othershowed a value of 2× 10−5.Solution: Multiple-mode TOs. The effect of the helium’s shear mod-ulus change on the TO’s period leads to the problem of how todistinguish a genuine supersolid signal from one induced bychanges in the elastic modulus of the 4He sample. One needs toconsider the dynamics of the TO and the interplay between the TOmechanics and the elastic properties of the sample. One conclusionto be emphasized is that if one hopes to identify a true supersolidsignal then any potential candidate should be put to the multiple-frequency test to eliminate elastic contributions. It is critical in thedesign of a TO that the significant elastic effects should be un-derstood and reduced to a minimum to facilitate the analysis of theobserved FPS. In such a case there will be only two contributions tothe observed signal: a frequency squared elastic term and a fre-quency-independent supersolid signal. A double-frequency TOwould then be enough to distinguish between the two.For the simple case of a double oscillator with cell and dummy

moments of inertia, placed on a vibration isolator (VI) satisfyingthe condition IVI � Icell and IVI � Idummy, for example, the influ-ence of the VI on the resonance periods of the TO is small and onecan approximate the eigenmodes of the system by solving�

−ω2Ic + kc −kc−kc −ω2Id + kc + kd

��θcθd

�=�00

�. [1]

Here kc and kd are the torsion constants of the rods of the celland dummy piece, I are the moments of inertia, and θ are theangular displacements.Therefore, under the approximation that the VI has an infinite

moment of inertia, the two resonance angular frequencies aregiven by

ω±2 =�Icðkc + kdÞ+ Idkc

2IcId

�×

1±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

4IcIdkckdðIcðkc + kdÞ+ IdkcÞ2

s !.

[2]

The sensitivity to changes in moment of inertia for a TO canbe determined experimentally from the period shifts observedupon freezing of the sample or estimated analytically from theequations of motion for the TO. In the case of a single-mode TO,when the moment of inertia of the solid sample, IHe, is much lessthan Ic, the moment of inertia of the sample cell, the sensitivity isgiven by the approximate expression

Δpp

=−12IHe

Ic. [3]

In the case of a two-mode TO, the expression for the sensitivityhas an additional factor and is given by the expression

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�Δpp

�±=−

12IHe

Ic

ω2c −ω2

∓

ω±2 −ω2

∓, [4]

where ω2c = kc=Ic. For our open cylindrical cell we get a reduction

in the sensitivity of about a factor of two compared with the caseof a single-mode oscillator. Even with that reduction, our sensi-tivity to changes in the moment of inertia of the helium is aroundΔpF=p= 0.01, comparable to the numbers reported by the Changroup for their high-sensitivity long-path-length experiments (35).For the case of triple-mode oscillators, or when taking the VI

into account, there is a closed form for the frequencies, and theycan be easily determined using root-finding algorithms.

ExperimentsGeneral Apparatus and TO Designs. In the work reported here, wehave used TOs of two different designs: an open-cylinder samplegeometry and an annular sample geometry. There have also beenseveral iterations of our basic design making minor changes in anattempt to improve the sensitivity of our data. In Fig. 2 we showschematics of three of our TOs. Fig. 2 A and B show an open-cylinder design and in Fig. 2C the solid 4He sample is containedin an annular region. The frequencies and mass loading sensi-tivities for these cells are shown in Table 1.The cells were driven and detected using parallel plate ca-

pacitors. The detection was mostly done with two capacitorsconnected in parallel, placed on opposite sides on the cell, toreduce floppy mode vibrations. In some of our later designs onecapacitor plate was placed on the dummy piece and the otherwas placed on the vibration isolator, to further reduce vibrationaleffects on the detection. The moving electrode was biased with200 VDC. For most of our measurements the modes were drivensimultaneously with identical maximum rim velocities, usuallyaround 4 μm/s for each mode. We also made several measure-ments with different velocities and with constant driving force,rather than constant oscillation amplitude, as well as severalmeasurements where only a single mode was driven.The general course of measurements was the following. Be-

fore the formation of a solid 4He sample we made carefulbackground measurements of the period p of the different modesof the oscillator with vacuum in the cell. This measurement wasdone either by slowly sweeping the temperature or by heating/

cooling in steps and waiting for the period to relax to a constantvalue at each temperature step. Measurements of the de-pendence of the period of the empty cell on the velocity/accel-eration were then also made. The relaxation times observed ateach temperature step are strongly temperature-dependent, asshown in Fig. 3, where the relaxation times observed for ourtriple TO are shown. The large increase in the relaxation timewith decreasing temperature is suggestive of thermal activation.After the empty cell data were obtained, we formed poly-

crystalline samples by the blocked-capillary method from com-mercial 4He gas, having a nominal 0.3 ppm 3He impurity level.We then repeated the temperature and velocity altering mea-surements that we had performed for the empty cell. The periodchange ΔpðTÞ is the deviation of the period of the TO from theempty cell background, which is matched at 300 mK to the fullcell’s values. The 300-mK temperature was chosen because atthis temperature the shear modulus is relatively independent ofboth frequency and temperature.

Expectations from a Supersolid State. The supersolid state is as-sumed to be a Bose-condensed state in a solid sample of 4He andas such it should present properties resembling those of super-fluids. One of these expected basic properties is a clear phasetransition temperature. Specifically, if a solid sample containedin a TO is cooled below the supersolid transition temperature,the period of the oscillator is expected to decrease, depending onthe size of the supersolid fraction. That period drop should beindependent of the frequency of the oscillator, assuming thevelocity of the cell is below the critical velocity. Critical velocity isyet another basic property of the supersolid. Increasing the ve-locity of the cell below the transition temperature should cause areduction in the period change. These properties should alsohold in the case of a lower dimensional superflow in the solid,such as superflow of grain boundaries or dislocation cores (17).

A B C

Cent imeters

Fig. 2. (A) A double TO design with a cylindrical sample space. (B) A tripleTO with an open-cylinder sample space. (C) A double TO design with anannular sample space. All TOs are attached to VIs.

0.00 0.05 0.10 0.15 0.200

1000

2000

3000

4000

5000

6000

7000

τ[s]

T [K]

Empty cell28 bar sampleEmpty cell fitSample fit

Fig. 3. Relaxation times vs. temperature for both an empty cell and a cellfull of solid 4He. Data taken from the high mode of the triple mode TOpresented in this paper. The lines are fits to exponential decay functions.

Table 1. Details of the TOs shown in Fig. 2

A B C

f+, Hz 2,073.6 2,129.3 1,543.0fmid, Hz 1,336.9f−, Hz 758.6 706.1 646.0ΔpF+, μs 5.18 2.68 1.69ΔpF  mid, μs 1.80ΔpF−, μs 13.48 8.96 2.75

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Our Results. We performed experiments using both open cylin-drical cells and annular cells, with both double- and triple-modeoscillators made of several materials and observed a qualitativelysimilar behavior in all cases. All our samples were grown usingthe blocked-capillary method, with sample pressures rangingbetween 26 bar and 34 bar. The mass loading—the periodchange upon filling the cell with solid—was measured both upongrowing the samples and when melting them at a constanttemperature, usually of 0.5 K. The pressure dependence ofthe cell was taken into account when comparing the two.Double-mode oscillators. We used both an open cylinder and anannular sample cell geometry in our double-mode oscillatormeasurements. One reason we used an open cylinder was toincrease the size of the small expected signal. The annular cell,however, had a smaller signal but was much less sensitive tochanges in the elastic modulus of the helium. Cross-sections ofthe double TOs used for the measurements presented here ap-pear in Fig. 2 A and C. The open-cylinder cell was made out ofMg and the torsion rods were BeCu, as was the dummy piece.The TO was mounted on a double VI with Cu torsion rods andPb moments of inertia. The outer Mg wall of the cell was tightlyconnected to an Al base with a 36-threads-per-inch screw joint.The threads were coated with TRA bond epoxy to provide aleak-tight seal. The torsion rod, in turn, was attached to the Albase. The inner radius of the cell was 0.794 cm, the outer radius0.953 cm, the inner height 3.251 cm, and the thickness of thebottom 0.159 cm. The Al plate epoxied to the cell was 0.794 cmthick. The torsion rods had a diameter around 0.37 cm. The rodclosest to the cell was 1.05 cm long, and the one between thedummy moment of inertia and the VI was 1.71 cm long. Steelinserts were epoxied or soldered into holes in the copper VI toprovide a hard surface for the lead O-ring seals between the TOand the VI.The annular cell, its dummy piece, and torsion rod were all

constructed of Al 6061. The torsion rod had been heat-treated.The VI for that cell was made of Ti alloy. In this cell most of thesample was confined in a long annular channel with inner radiusof 0.635 cm, outer radius of 0.794 cm and a height of 3.28 cm.The height of the cylindrical space at the bottom of the cell was0.127 cm.In the design of our TOs we have taken care to minimize the

inner diameter of the cell fill line to reduce the influence ofchanges in the shear modulus of the solid in the fill line on thefrequency of the TO. In the open-cylinder design, the cell wasfilled through a 0.1 mm i.d. × 0.25 mm o.d. CuNi tube, whichpassed through a 0.71-mm-diameter hole drilled along the cen-tral axis of the torsion rods. The void between the outer diameterof the fill line and the hole drilled in the torsion rod was filledwith 1266 Stycast epoxy to exclude the possibility of solid 4He inthis region. For the annular cell the torsion rods had inner filllines with a radius rfill = 0.017 cm and an outer radius rrod = 0.256 cm,for a ratio of rfill=rrod = 0.067. The different moments of inertiaand torsion constants are shown in Tables 2 and 3.For the open cylinder TO detection electrodes were mounted

on the VI, and their capacitances were 2.3 pF and 2.5 pF. For theempty cell, the low mode frequency was 758.60 Hz and the highmode frequency was 2073.61 Hz. The corresponding frequency

shifts upon solidification of the sample were 7.68 Hz and 22.04 Hz,respectively. These give periods shifts of ΔpF− = 13.48 μs andΔpF+ = 5.18 μs of mass loading for the different modes, whichtranslate to ΔpF=p = 0.01 for both modes. These numbers re-quire an accuracy of only a few parts of a millihertz to detect 10−4

signals. The frequency stability for our modes was less than0.06 mHz for the low mode and less than 0.02 mHz for thehigh one. The Qs were 105 for the high mode and 2.5× 105 forthe low mode at low temperatures. We also measured thepressure dependence of the period of the cell. The periodchanged by 1 ns for the high mode and 2.2 ns for the low modefor each 1-bar change. These give corrections around 0.5% forthe mass loadings.For the annular cell at T = 0.5 K, the resonance frequencies of

the empty TO were f− = 646.0 Hz for the low-frequency modeand f+ = 1,543.0 Hz for the high-frequency mode. The total pe-riod shifts for the two modes, between the liquid and the solidphases, were ΔpF− = 2.75 μs and ΔpF+ = 1.69 μs.Fig. 4 shows the period vs. temperature of the two modes of

the open-cylinder TO both upon cooling and heating for asample with P = 31 bar. The sample pressure was estimated fromthe melting temperature of 1.8 K, taken upon slow heating of thesample. Uncertainties in determining the exact melting temper-ature would suggest an uncertainty of 0.5 bar at most in the

Table 2. Double mode TO - open cylinder

Torsion constant, dyn·cm Moment of inertia, g·cm2

Kcell = 9.29E+08 Icell = 9.0Kdummy = 5.26E+08 Idummy = 13.7KVI1 = 3.59E+08 IVI1 = 957.8KVI2 = 8.62E+08 IVI2 = 886.4

IHe = 0.41

Table 3. Double mode TO - annular cell

Torsion constant, dyn·cm Moment of inertia, g·cm2

Kcell = 1.69E+09 Icell = 26.0Kdummy = 1.35E+09 Idummy = 58.7KVI = 1.03E+9 IVI = 547.0

IHe = 0.25

0.0 0.1 0.2 0.3 0.4 0.5

487776

487780

487784

487788

0 3 6 9 12 150.4820.4830.4840.4850.4860.4870.488

p +[ms]

Time [hours]

1.3201.3241.3281.3321.336

p -[ms]

p +[ns]

T [K]

0.0 0.1 0.2 0.3 0.4 0.5

1331926

1331928

1331930

1331932

1331934

Empty cell heating - shiftedEmpty cell cooling - shifted31 bar Sample heating31 bar Sample cooling

p -[ns]

T [K]

Cylindrical cell

Fig. 4. Period vs. temperature for a 31-bar sample. (Top) The low 750-Hzmode and (Bottom) the high 2,050-Hz mode. Empty cell data points wereshifted to match the full cell at 300 mK. Both modes were driven with aconstant maximal velocity of v = 4 μm/s. (Inset) The period of the modesupon forming the sample.

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pressure estimation. The full markers show the equilibrium val-ues of the period of the TO containing the sample at the dif-ferent temperatures, and the empty markers are the period ofthe empty cell, shifted to match the values of the full cell at300 mK. The difference between those two datasets is the periodshift Δp. All points were taken using a constant maximal velocityof 4 μm/s for each mode. Fig. 4, Inset shows the period shifts ΔpFupon growing a sample.For this particular sample the mass loadings were ΔpF− =

13,741.19 ns and ΔpF+ = 5,294.43 ns, from reducing the cellpressure at 500 mK. Dividing Δp taken from the data of Fig. 4 bythese mass loadings gives the temperature-dependent FPS.These are plotted in Fig. 5 for both modes. If the observed sig-nals were attributed entirely to supersolidity, we would expectthe FPS to be frequency-independent and depend only on tem-perature. If, however, there is no supersolid and the signal comesentirely from the elastic effect arising chiefly from the acceler-ation of the 4He solid (because our cell is designed to suppressother elastic effects), then the signal would depend on the fre-quency of the mode with an f 2 dependence. The results actuallyshow both.

We can therefore decompose the measured Δp=ΔpF of the twodifferent modes into these contributions: For each mode, theFPS can be written as Δp±ðTÞ=ΔpF± = aðTÞ+ bðTÞf 2± , where aðTÞis the temperature-dependent supersolid (or other frequency-independent) contribution to the FPS. The elastic contributionto the FPS is bðTÞf 2± . In Fig. 6, we plot both the elastic contri-butions to the FPS and the frequency-independent one as afunction of temperature. It can be seen that the elastic contri-bution continues to change above T = 0.2 K. This feature isconsistent with the continuing change in μ above 0.2 K and isseen in previous TO experiments that are dominated by elasticeffects (36, 37). The frequency-independent term, however,seems to level out above around 0.1 K.Another way to present the data is shown in Fig. 7, which

shows a plot of the FPSs vs. f2. Here Δp is taken as the periodshift between 30 mK and 300 mK, where the empty cell datamatching was done. These temperatures were also chosen be-cause their shear modulus is only slowly varying with tem-perature and has a relatively small frequency dependence.Specifically, 300 mK usually gives the smallest zero-frequencyintercept, and matching at other temperatures between 200 mKand 500 mK can increase the intercept by up to 30%. The zero-frequency intercept in this case is the part of the signal that doesnot depend on frequency, and therefore is not an elastic con-tribution, but possibly a supersolid term. In the annular cell, thesame as in the open-cylinder cells, we measured a frequency-independent contribution to the data, on the order of 10−4 of thetotal moment of inertia of the helium in the cell at the lowtemperature limit. The main difference in this cell is the differ-ence in the elastic effects size. For the annular cell the elasticeffects were about an order of magnitude smaller than in theopen cylinder.The modes were usually driven simultaneously for our exper-

iments, but we repeated the measurements with the modesdriven separately upon heating and cooling and got the sameresults, with a small 0.67 ns difference for the high mode. Thehigh mode’s error bar in Fig. 7 arises from both that differenceand the difference between the heating and cooling curves. Theacceleration or velocity dependence of the sample was carefullymeasured. We measured both the empty and full cell’s velocitydependence ranging over more than two orders of magnitude,with the modes driven separately, at two different temperatures,

0.0 0.1 0.2 0.3 0.4 0.5-2.0x10-4

0.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

Cylindrical cell

Elasticcontributionsand

zerofrequencyintercept

T [K]

f+=2050Hz elastic part - bf2

f- =750Hz elastic part- bf2

zero frequency intercept - a

Fig. 6. The different contributions to the signal, aðTÞ and bðTÞf±2, defined inthe relation Δp±ðTÞ=ΔPF± = aðTÞ+bðTÞf±2, for a 31-bar sample.

0.0 0.1 0.2 0.3 0.4 0.5

-2.0x10-40.0

2.0x10-44.0x10-46.0x10-48.0x10-41.0x10-31.2x10-31.4x10-3

Cylindrical cell

f- =750Hz modef+=2050Hz mode

Δp/Δp F

T [K]

Fig. 5. Fractional period change vs. temperature for a 31-bar sample. T =300 mK was set as zero.

0 1 2 3 40.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

Annular cellCylindrical cell

Zero frequency intercepts:Cylindrical cell:Δp/ΔpF (f=0) = 9.2x10

-5±1.2x10-5

Annular cell:Δp/ΔpF (f=0)=1.2x10

-4±2x10-5

Δp/Δp F

f2 [KHz2]

Fig. 7. The fractional period shifts for the two modes vs. their frequencysquared. Results from both the annular cell and cylindrical cell are presented.The lines are linear fits.

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the first one being the low temperature of 30 mK and the secondone 300 mK. The sample’s 300-mK data’s acceleration de-pendence is the same as that of the empty cell both at 30 mK andat 300 mK— which agrees well with choosing the 300-mK tem-perature as the point where the low temperature effects nolonger appear. Fig. 8 shows the acceleration dependence of thetwo modes at these two temperatures, as well as the empty celldata at those temperatures.As can be seen in Fig. 9, the zero-frequency intercept seems

not to depend on the acceleration up to 15 rad/s2, where it startsdropping, and reaches zero around 60 rad/s2. The elastic parts ofthe signal, however, show a continuous drop in the signal sizewith acceleration. In terms of shear stress, 15 rad/s2 correspondto 0.05 Pa, or a strain of 3× 10−9, which, comparing to shearmodulus measurement experiments (38), is below the 10−7 crit-ical strain reported, and is well into the fully pinned region forthe 3He atoms on the dislocations at 30 mK (39). We did notmeasure above 100 rad/s2 because that made irreversible changesto the frequency; 100 rad/s2 correspond to 0.3 Pa, which,according to a recent paper by Kim and coworkers (39), is in thehysteretic region for the dislocation behavior. At high enoughaccelerations even the frequency of the empty cell was altered,and the temperature of the cell could be affected.Triple-mode open cylinder. To test the validity of our assumption,that in our double-mode TOs there are only two significantcontributions—the frequency-independent one and an f2-dependentone—we constructed a triple compound oscillator. A schematiccross-section of the TO can be seen in Fig. 2B. The entire TO wasmade out of aluminum 6061, and the torsion rods Kcell and Kdummy1were heat-treated at 413 °C for 3 h, followed by a slow cool-downto room temperature. The triple oscillator was placed on a VI witha brass piece as a moment of inertia and an aluminum torsion rod.The Al rod, Kdummy2, connecting the TO to the VI was 1.55 cmlong with a 6.35-mm diameter and had a 0.343-mm-diameter holedrilled through its center for filling the cell, and the two other rods(both 1.27 cm long with a 4.57-mm diameter) had a 0.71-mm-diameter hole filled with Stycast 1266 through which a tube

with an inner diameter of 0.1 mm was inserted and served tofill the cell. The bottom and top torsion rods were connectedby screws, and so was the brass piece. The cell itself was anopen cylinder that was sealed with a TRA-BOND 2151 on thethreads connecting it to the end of the torsion rod. The height ofthe inner part of the cell was 3.3 cm, the inner radius was 0.79 cm,the wall’s and bottom part’s thickness was 0.13 cm, and the top ofthe cell was 0.95 cm thick. The torsion constants and moments ofinertia of the different parts of the cell are shown in Table 4.The measured empty cell frequencies at 40 mK were f1 =

2,129.54 Hz, f2 = 1,336.99 Hz, f3 = 706.20 Hz, and the Qs were allin the several 105 range. Our frequency stability for the differentmodes was less than 0.008 ns for the high mode, 0.034 ns for themiddle mode, and 0.14 ns for the low mode. The pressure de-pendence of the cell, as measured at 5 K by changing the cellpressure between 0 and 60 bar was Δp=ΔpF = 2× 10−4/bar for allthree modes. For one set of data, for example for a 28-barsample, the mass loading sensitivities were measured to beΔp1F = 2,682 ns, Δp2F = 1,802 ns, and Δp3F = 8,956 ns. Cor-recting these values by the pressure dependence changes themonly by about 0.5% of the ΔpF of the different modes. Themoving electrode for this oscillator was attached to Idummy  2. Weused two detection capacitors, both of 2.5 pF.We grew five different samples, four of which had pressures

ranging between 26 and 28 bar, and one that had around 17%liquid in the cell. All solid samples deviated from the empty cellbackground below 0.2 K. The sample containing 17% liquid didnot show this deviation. However, it did not follow the empty cellbackground above 0.2 K, but rather had a curved profile with theperiod increasing with temperature. For this particular experimentwe were not able to measure stable data points at 30 mK, butbecause the period hardly changes below 50 mK, we take the 40-mK data point as being close enough to a lower temperature valueand give a reasonable estimate of the signal size.The values of Δp=ΔpF for the 28-bar sample are shown in Fig.

10. The error bars come from the small difference between theslopes of the empty and full cell data points. We matched theempty cell and full cell points at 0.3 K.All our measurements were done with outer cell wall maximal

velocities below 20 μm/s, at a regime where Δp=p changes by lessthan 10−7 by changing the velocity. The empty cell Δp=p valuesfor all three modes had almost the same dependence on thevelocity of the cell, changing by about 4× 10−7 when increasingthe velocities to 100 μm/s. The full cell period starts to deviatefrom the empty cell background at around 20 μm/s. The plot ofthe lowest temperature values (40 mK) from our triple TO

10 1000

5

10

Cylindrical cell

Sample - 300mKSample - 30mKEmpty - 300mKEmpty - 30mK

Δp +[ns]

Acceleration [rad/s2]

10 100

0

2

4

Sample - 30mKSample - 300mKEmpty - 30mKEmpty - 300mK

Δp -[ns]

Fig. 8. Acceleration dependence of the period of the different modesfor a 31-bar sample. The data points were shifted to give the measured FPSat the accelerations used for the temperature-dependent data shown inFigs. 4–7. The empty cell data points were shifted to match those of thefull cell at 300 mK to enable a clearer comparison of their accelerationdependence.

2 20100.0

2.0x10-54.0x10-56.0x10-58.0x10-51.0x10-41.2x10-41.4x10-41.6x10-41.8x10-42.0x10-4 v- [μm/s]

v+ [μm/s]3.4

1.2

34

12

17

Zero frequency interceptLow mode elastic part

a(T=30mK)andb(T=30mK)f -2

Acceleration [rad/s2]

6

0.0

2.0x10-44.0x10-46.0x10-48.0x10-41.0x10-31.2x10-31.4x10-3

High mode elastic part

b(T=30mK)f +

2

Fig. 9. The zero-frequency intercept and the elastic terms vs. acceleration.The error bars come from matching the sample signal with the empty cellacceleration dependence. The top axis shows the velocities of the two dif-ferent modes at these accelerations.

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shown in Fig. 10, vs. the frequency squared of each mode can beseen in Fig. 11. The line in the plot is a fit to a straight line. Itgives a nonzero intercept of 7.3× 10−5 ± 2.8× 10−5 with the y axis.The different intercepts for the rest of the samples varied be-tween 3.55× 10−5 ± 0.32× 10−5 and 1.33× 10−4 ± 0.39× 10−4.We performed FEM simulations on our triple TO design as well,

changing the shear modulus of the helium inside the cell. All shearmodulus changes produced plots with zero y axis intercepts for theΔp=ΔpF vs. f 2 plots, within the uncertainties of the calculation.From these simulations, we find that a shear modulus change ofaround 20% can account for the elastic part of our observed signal,but not for the frequency-independent term. The only otherfrequency-independent contribution which can exist in ourcell is the fill line effect, which for our design for a 100%change of the shear modulus of the helium in the fill linesgives Δp=ΔpF of less than 3× 10−8 for all modes, three ordersof magnitude smaller than our smallest nonzero intercept.

The Validity of Our Assumptions. There are several assumptionsunderlying the analysis of our data. The main one is that theelastic contribution to the signal can be well approximated by anf2 dependence. That term is actually just the leading term in theseries expansion of the changes in the effective moment of in-ertia of the cell due to elastic effects (30).For our annular cell, using an approximation of an infinite

cylinder with a radius r0 � Δr, where Δr is the width of the an-nulus, we get

IeffIHe

≈2

ωΔr

ffiffiffiμ

ρ

rtan�12ωΔr

ffiffiffiρ

μ

r �

= 1+Δr2

12ρ

μω2 +

Δr4

120ρ2

μ2ω4 + . . . .

[5]

Here ρ and μ are the density and shear modulus of solid 4He.We are interested in the change in this expression due

to variations in the shear modulus Δμ= μ1ðT = 30 mKÞ−μ2ðT = 300 mKÞ.

ΔIeffIHe

=2

ωΔr

� ffiffiffiffiffiμ1ρ

rtan�ωΔr2

ffiffiffiffiffiρ

μ1

r �−

ffiffiffiffiffiμ2ρ

rtan�ωΔr2

ffiffiffiffiffiρ

μ2

r ��

=Δr2ω2ρ

12

�1μ1

−1μ2

�+Δr4ρ2ω4

120

�1μ21

−1μ22

�+ . . . .

[6]

The error introduced by considering only the first ω2 term inour analysis is

Error�ΔIeffIHe

�= 1−

Δr2ω2ρ

12

�Δμμ1μ2

��ΔIeffIHe

. [7]

This gives, for shear modulus values of μ1 = 1.5× 108 dyn·cm−2

and μ2 = 1.0× 108 dyn·cm−2, a correction of 0.08% for the highmode and of 0.01% to the low mode.

For our open-cylinder cell, with the infinite cylinder approximation

IeffIHe

≈4ωr0

ffiffiffiμ

ρ

r 

J2�r0ω

ffiffiρμ

q �

J1�r0ω

ffiffiρμ

q �

= 1+r2024

ρ

μω2 +

r40384

ρ2

μ2ω4 + . . . ,

[8]

where Jn(x) are the Bessel functions of the first kind. Again, inour analysis, we consider only the ω2 term in the series expansionfor ΔIeff=IHe. Here, the errors we get are 1.9% for the high modeand 0.3% for the low mode. The shear modulus difference wasevaluated from the size of the measured elastic contribution. Herewe used μ1 = 1.5× 108 dyn·cm−2 and μ2 = 1.24× 108 dyn·cm−2. It isimportant to point out that the influence of the following terms,such as the ω4 correction, would be in expanding the horizontalaxis of figures such as Fig. 7. That, in turn, would slightly increasethe zero-frequency intercept.Another assumption of our model is that the shear modulus of

the helium is frequency-independent. As can be seen in Fig. 1,there is actually a strong frequency dependence of the shearmodulus in the intermediate range of 50 to 200 mK, where theshear modulus itself changes with temperature; however, for thefrequencies used in our experiments, in the temperature rangesof either below about 40 mK and above around 250 mK theshear modulus has only a small dependence on frequency. Fromthe polycrystalline data by Beamish’s group (shown in Fig. 1A), at300 mK, the difference in shear modulus for the frequencies usedin our experiment is less than 1% of the total shear modulus.Hence, the correction to the frequency-independent term in theplots of FPS vs. f2, where the signals are taken as the difference inperiod between 300 mK and 30 mK (or 40 mK for the triple TO),would be less than 1% of its value. It is worth mentioning thateven in the intermediate range where the shear modulus changesthe most with frequency the correction to the data shown in Fig. 6would be less than 2.7% at most (around 0.1 K) in that range.Another issue is the strong dependence of the zero-frequency

intercept on the measured value of the low mode. Therefore, theaccuracy of the measurement of that mode is crucial. Large massloading sensitivities, as the ones in our experiments, alongsidelow vibrational noise at the low mode’s frequency enable high-sensitivity measurements with small errors.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2.0x10-4

0.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

Cylindrical cell

Δp/Δp F

T [K]

High mode, f=2129.3 HzMiddle mode, f=1336.9 HzLow mode, f=706.1 Hz

Fig. 10. Fractional period change for a sample with a pressure around 28 bar.

Table 4. Triple mode TO - open cylinder

Torsion constant, dyn·cm Moment of inertia, g·cm2

Kcell = 9.11E+08 Icell = 11.4Kdummy1 = 9.11E+08 Idummy1 = 18.3Kdummy2 = 2.77E+09 Idummy2 = 54.6KVI = 4.03E+08 IVI = 805.8

IHe = 0.41

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Finally, we assumed a homogenous sample where the shearmodulus of the helium in the cell would change by the sameamount all over the cell. That does not necessarily need to be thecase, but because it will not change the general frequency de-pendence it will not have an effect on the zero frequency termfor our cells, where non-f 2 contributions, such as the fill lineeffect, were reduced to a minimum.

Comparison with Previous ResultsPrevious Multiple TOs. Two other groups previously performedand reported the results of measurements of solid helium con-fined in a double TO, the Royal Holloway (RH) group (20) andthe Rutgers group (9, 19). In these experiments, the distinctionbetween the elastic and a possible supersolid effect is unclear,because the FPS signals contain significant contributions due tothe stiffening of solid 4He in the torsion rods of both experiments.In the Rutgers TO, the fill line drilled down the axis of the

torsion rod to fill the cell had a diameter of 0.8 mm, whereasthe torsion rod itself had a diameter of 1.9 mm. A 20% change inthe shear modulus of the helium in the torsion rod would giveΔp=ΔpF values of 1.1× 10−3 for the low mode and 0.86× 10−3 forthe high mode. These numbers are comparable to the valuesreported in their 2007 paper (9). In that case the fill line effectalone could account for the entire observed signal in the ex-periment, so any supersolid signal, should it exist, would be veryhard to discern from the data.For the design of their cell the RH group chose a “pancake”

geometry. This design is open to several elastic effects thatcontribute to the period shift signal. The cell had a relatively thinbase plate, only 1 mm thick, for a radius of 7 mm. The holesdrilled for filling their cells gave ratios of 0.17 and 0.37 for theinner to outer radii of the torsion rods. These enable elasticsignals, such as the filling line effect and the Maris effect, on theorder of their measured signals, making the distinction between afrequency-independent contribution arising from elastic effectsto one arising from a different origin difficult.In a previous work, Mi and Reppy (37) reported the pre-

liminary result from a double-frequency TO that was designed tobe chiefly sensitive to elastic effect due to acceleration of solid4He. In that experiment, a frequency-independent contributionto the FPS was observed that was equivalent to an inertial massdecoupling of the cylindrical sample corresponding to an NCRI/supersolid fraction of 1.2× 10−4. That number is comparable tothe signals reported in this paper.

Other Relevant Experiments. There are only two other experimentsshowing signals smaller than the ones reported here. These arethe rigid TO experiment by the Chan group (34) and a double-oscillator experiment by the Kim group (40). In both these TOexperiments, in which they report an upper limit of 4× 10−6 to apossible supersolid signal, the experimental cell had a nonsimplyconnected geometry, and the samples had pressures all above 34bar. It is possible that either the sample pressure has an effect onthe size of the signal, or, should there be a percolating networkallowing a flow to take place, then the path length and cross-section of the sample might play a role.Recent flow measurements by three groups (21, 22) (and

Haziot, unpublished) report flow in the solid that has a differenttemperature dependence than the signals observed with TOs. Inthese flow experiments, the signals decrease with increasingsample pressure, and no flow is apparent above 31 bar. Hallock’sgroup has recently reported (41) flow through solid 4He in acylindrical cell with radius of 3.175 mm. The observed mass fluxrate peaked at 5.3× 10−8 g/s before falling to 2.4× 10−8 g/s attemperatures below 80 mK. The maximum flow corresponds to1.7× 10−7 g/s cm2 through the cross-section of the cylindrical sample.This flow can be compared with our TO results. For example, with asupersolid fraction of 10−4 and a flow velocity of 25 μm/s (Fig. 9) thecorresponding current would be 5× 10−8 g/s cm2 for our annular cell.

Comparison with Theory. The original theoretical calculations (3)regarding the superfluid fraction of solid helium at low tem-peratures give an estimation of around 10−4. Our numbers,ranging between 0.4× 10−4 and 1.3× 10−4 of the total moment ofinertia of the solid helium in the cell, are comparable to the onesassumed by theoretical modeling of a supersolid (42). Path in-tegral Monte Carlo simulations results perhaps show no vacancy-based supersolid in perfect single crystals containing severalthousand atoms (43) yet do not rule out other mechanisms forsuperflow in solid 4He, such as flow along percolated networks ofdislocations or grain boundaries (17).

ConclusionFor all of the cells reported in this paper, for all samples grown,there was always a frequency-independent contribution to thesignal, of the same order of magnitude. We did, however, useother cells of different geometries, which, as opposed to the cellsin this paper, were all not simply connected. For those cellgeometries there was a significant reduction in the size of thefrequency-independent contribution. The results of these ex-periments have not yet been published.Our results, therefore, indicate the existence, for some, but not

all, solid samples contained in a variety of sample geometries, ofa frequency-independent contribution to the period. The exis-tence of such a signal can be taken, in the absence of anotherexplanation, as evidence for the supersolid state.

Future TestsThere are yet some tests left to perform to pin down the exactnature of the apparent phenomenon—which will enable statingfor certain whether it is a supersolid or not. One of them is re-peating previous blocked annulus experiments (37) with a mul-tiple-mode TO. A supersolid phase would only give a periodchange in a TO if the condensed phase has a closed circular path,so blocking its path should cause almost the entire signal todisappear. Elastic effects, however, should not decrease by thesame amount.Recent TO experiments (34) with solid 3He in the HCP

phase have shown period shift signals similar to those seen insolid 4He. Again, the shear modulus of the hcp phase of solid3He has an anomaly similar to that of solid 4He, which suggeststhat the 3He period shift signal must arise as a consequence of theelastic anomaly. Solid 3He is not Bose system and therefore is not

0 1 2 3 40.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

1.4x10-3

Cylindrical cell

Δp/ΔpF (f=0)=7.3x10-5±2.8x10-5

Δp/Δp F

f2 [KHz2]

SampleLinear fit

Fig. 11. FPS vs. f 2 for a 28-bar sample.

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expected to exhibit a Bose-condensed supersolid phase. Therefore, if afrequency-independent contribution were to be observed in the case ofsolid 3He, this would provide a strong argument against a supersolidorigin for the observed zero-frequency contribution seen in our 4Heexperiments.

ACKNOWLEDGMENTS. We thank P. W. Anderson, J. Beamish, W. F. Brinkman,M. H. W. Chan, J. C. Davis, D. A. Huse, and E. Mueller for useful and encouragingdiscussions. We thank J. Beamish and A. Haziot for sharing their data with us.This work was supported by National Science Foundation Grant DMR-060586and Cornell Center for Materials Research Grant DMR-1120296 and was partiallyfunded by the New England Foundation, Technion.

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