Frequency-dependent Study of Solid Helium-4 Contained in a Rigid Double-torus Torsional Oscillator

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by  Jaewon Choi

Frequency-dependent Study of Solid Helium-4 Contained in a Rigid Double-torus
Torsional Oscillator
Jaewon Choi,∗ Jaeho Shin, and Eunseong Kim†
Department of Physics and Center for Supersolid and Quantum Matter Research,
Korea Advanced Institute of Science and Technology (KAIST),
291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
(Dated: January 27, 2017)
The rigid double-torus torsional oscillator (TO) is constructed to reduce any elastic effects in-
7 herent to complicate TO structures, allowing explicit probing for a genuine supersolid signature.
1 We investigated the frequency- and temperature-dependent response of the rigid double-torus TO
0 containing solid 4He with 0.6 ppb 3He and 300 ppb 3He. We did not find evidence to support the
2 frequency-independentcontributionproposedtobeapropertyofsupersolidhelium. Thefrequency-
dependentcontribution which comesfrom thesimpleelastic effect ofsolid helium coupled toTO is
n
essentiallyresponsiblefortheentireresponse. Themagnitudeoftheperioddropislinearly propor-
a
J tional to f2, indicating that theresponses observed in this TO are mostly caused bythe overshoot
of ‘soft’ solid helium against thewall of thetorus. Dissipation of therigid TO is vastly suppressed
6
compared to those of non-rigid TOs.
2

]
r I. INTRODUCTION frequency-independentresonantperiodchangeaftersub-
e
tracting the frequency-dependentterm andascribedthis
h
t to a possible supersolid signature. This interpretation
o The resonant period of an ideal torsional oscillator
can be questioned since the measurements reported rel-
. (TO) is proportional to the square root of its rotational
t atively large period drop which can be associated with
a inertia √I, and the superfluid decoupling can be de- the non-rigidity of TO.
m
tected by the reduction of the resonant period. The For this article, we measured the period drop and ac-
d- resonant period drop of a TO containing solid helium companieddissipationpeakusingadouble-frequencyTO
n was originally interpreted as the appearance of a super- thatwasconstructedtobehighlyrigidtominimizeshear
o solid phase1–17. Recently, a number of experimental and modulus effects. The frequency dependence of the rigid
c theoretical efforts have indicated that the anomaly in double-frequency TO was investigated in various modes
[ the TO response can be explained by the shear modulus of representation. As a result, we elucidate whether or
1 change18–27. The previous finite element method (FEM) not solid helium-4 exhibits true supersolidity.
v simulation suggested that the influence on the period of
1 TO due to the change in shear modulus of solid helium
2 was negligible28–30. Nevertheless, the effect can be sig- II. EXPERIMENTAL DETAILS
6 nificantly amplified resulting from non-ideal TO design.
7
Four mechanisms of non-ideal TO response have been
0 KAIST rigid double-torus torsional oscillator (TO)
suggested30–33. In order to minimize the contribution of
. (Fig. 1) was carefully designed to minimize the various
1 the elastic effect to the resonant period of TO, it should
0 elastic effects caused by: (1) shear-modulus-dependent
be meticulously constructed to be rigid. However, most
7 relative motion between TO parts (the glue effect)30,33,
of TOsused in the previoussupersolidexperiments were
1 (2) solid helium contained in the torsional rod (the tor-
not rigid.30–32. Recently, the Chan group reported that
: sion rod hole effect)31, and (3) solid helium layer grown
v theresonantperioddropwassubstantiallyreducedwhen
Xi therigidityofTOwassystematicallyincreased34–36. The on a thin TO base plate (Maris effect)32.
We first assembled every joint in the TO using
resonantperioddropofahighlyrigidTOwasonlyafew
r stainless-steel screws or hard-soldering to diminish elas-
a timesgreaterthanthatduetotheelasticeffectestimated
tic effects on the TO period due to the relative motion
by FEM simulation. They set the upper bound for the
between different parts. The torsion plate and the tor-
non-classical rotational inertia (NCRI) to be less than
sionrodwererigidly connectedby machiningthe assem-
approximately 4 ppm36.
bly from a single piece of Be-Cu. The torus-shaped TO
However, new evidence of a ‘true’ supersolid signa- cell for solid helium was constructed by hard-soldering
ture wassuggestedby Reppyet al.37,38 basedondouble- twopiecesofsemi-circularcopper tubing. The toruswas
frequency TO experiments. Superfluidity or the NCRI hard-soldered onto thick copper plate and the combined
is independent of frequency while non-superfluidity or structures were fastened down directly on the torsion
therelaxationphenomenonresultingfromshearmodulus plate by four screws.
change leads to a strong frequency dependence. Accord- Second, we used a thick torsion rod and a very thin
ingly, analysis on the frequency dependence can be used fill line to preventthe elastic effect of solidhelium in the
to differentiate the superfluid response from the non- fill line. The change of shear modulus of solid helium-4
superfluid response30,38. Reppy et al. observed a small contained in the torsion rod can induce the TO period
2
operated at four resonant frequencies. This is possible
due to the configurationofourTO whichconsists oftwo
torus-shaped solid helium containers attached to upper
and lower stages. This configuration enables modifica-
tion of the resonant frequency by loading solid helium
into the upper or both tori. We first placed the solid
sample in both tori. After collecting the first dataset,
the solid helium inside the lower torus was carefully re-
moved at low temperature to avoid damaging the solid
sample placed in the upper torus.
Finite element method (FEM) simulations indicates
thata30%changeintheshearmodulusofsolidhelium-4
in the upper torus leads to only a 0.93-ns period drop in
FIG. 1. KAISTrigid double-torustorsional oscillator. first(1st)mode (in-phase)anda 1.16-nsdropinthe sec-
ond (2nd) mode (out-of-phase). These simulations cor-
respond to 2.3 10−5 and 1.9 10−4 respectively in the
× ×
anomaly31. This effect may be responsible for the pe- framework of so-called NCRI fraction. The major con-
riod drop observed in the majority of TO experiments. tribution seemingly comes from the overshooteffect, the
In order to remove this effect, a 1-mm-diatmeter CuNi relative motion of solid helium with respect to the outer
filling capillary was directly installed on top of the TO wall.
cell instead of making a hole through a torsion rod. The empty cell has resonant frequencies of 449.57 Hz
Third, the TO cell containing solid helium was hard- and 1139.77 Hz for in-phase and out-of-phase modes re-
soldered to a 3-mm-thick copper plate and was designed spectively. Two pairs of electrodes attached to the lower
sothatsolidheliuminthetorusdidnothaveadirectcon- torus were used to drive and detect the TO response.
tactto the baseplate nearthe torsionrod. In acylindri- The additional electrodes installed on the upper plate
calTO,solidheliumwasgrowndirectlyonitsbaseplate. enables the detection of the amplitude and phase of the
If the thickness of the base plate is not sufficiently thick, upper torus. We confirmed that the phase difference be-
then the period drop due to the change in shear modu- tween the two tori was approximately 0 degrees for the
lus of solid helium can be significantly amplified32. The in-phase mode and 180 degrees for out-of-phase mode.
contribution of solid helium is greater in the proximity Themechanicalqualityfactorwasapproximately1 106
×
of the torsionrod andwhen the base plate is thinner. In at 4.2 K for both modes.
this study, solid helium confined in a rigid torus channel Bulk solid helium-4 samples containing an impurity
onthethickcopperplatewasnotexpectedtoexhibitthe concentration of 0.6 ppb and 300 ppb helium-3 were
strong Maris effect. grownby the blocked capillary method. The sample cell
was first pressurized to target pressures of 67-82 bar at
Finally, we carefully tuned the so-called ‘overshootef-
3.2 K and the mixing chamber was then cooled to the
fect’ caused by direct coupling of elastic properties of
base temperature. The resonant period shifted accord-
solid helium to the TO response. Since solid helium is
ing to solid growth in the upper torus: 39,800 ns for the
much softer than the wall of the container, it undergoes
in-phase mode and 6,040 ns for the out-of-phase mode.
additional displacement. The shear modulus change of
The large period shift due to solid helium and its solid
solidheliumnotmovingin phasewith the confining wall
stability in the first mode enables us to distinguish the
can induce the period anomaly without non-linear am-
change of the resonant period within about 2 ppm reso-
plification. The resonant period drop introduced by the
lution. This sensitivity is lower than the upper limit of
overshoot effect is known to be linearly proportional to
NCRI fraction reported by the PSU group (4 ppm)36.
the square of the TO frequency. For this experiment, we
optimized the overshooteffect so that the system was in
a regime where the elastic effect is not too small to be
detected and not too big to make the TO softer. In this III. TO PERIOD AND DISSIPATION
TO,solid helium was locatedin a torus with a relatively
large cross-section (diameter: 5 mm), which allows us The period and dissipation of the TO for both the in-
to have large mass loading and high resolution for the phaseandout-of-phasemodesweremeasuredoveratem-
so-called NCRI fraction. One can effectively reduce this peraturerangeof20-300mK.Figure2presentsthe reso-
effect by confining solid helium in a thick copper torus nantperiodoftherigiddouble-torusTOcontaininghigh-
with a small cross-section at the expense of diminishing purity solid helium-4 (0.6 ppb) and commercial-purity
the capability of frequency analysis. This fine-tuning al- solid helium-4 (300 ppb) as a function of temperatures
lows us to investigate the effect of the shear modulus on atdifferentrimvelocities. The perioddropanomalywas
thedouble-frequencyTOandpossiblesupersoliditymore observed for both modes of the rigid double-torus TO.
clearly. Compared with the empty cell data (dashed line), the
In addition, the KAIST rigid double-torus TO can be perioddropin0.6-ppbsolidsampleappearsinitiallyat80
3
e
ng (a) 47.2 m/s (b) 47.7 m/s (c) 52.6 m/s (d) 21.3 m/s
ha1.5 87.5 m/s 87.8 m/s 131.3 m/s 52.8 m/s
Period C1.0 248387801509...4881. 8 mmmm///sss/s 248196810552...342 7. 2 mmmm//ss/s/s 251154189256..0007.. 68 mmmm//ss//ss 125354240...237 mmm///sss
e
v0.5
ati
el
R
0.0
T)
-1 Q( 4 x10-7
n 3
o
ati 2
p
si 1
s
Di 0
0.02 0.04 0.06 0.080.1 0.02 0.04 0.06 0.080.1 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3

Temperature (K)
FIG. 2. Period and dissipation data of the KAIST rigid double-pendulum TO containing solid helium with 0.6-ppb 3He ((a)
in-phaseand(b)out-of-phase)and300-ppb3He((c)in-phaseand(d)out-of-phase). Theempty-cellbackgroundwassubtracted
in both period and dissipation data. The y-axis of upper panels represent absolute values of period changes, calculated from
theempty-cell backgrounds.
mK,andbecamesaturatedbelow25mKforbothmodes anonsettemperatureof160mKandreachesamaximum
(colored symbols). The magnitude of the period drop is of1.49ns(dP /∆P =3.7 10−5)at40mK.Intheout-
− −
×
suppressed by the increase in the rim velocity of the al- of-phasemode,theonsettemperatureishigherthanthat
ternating current (AC) oscillation for both modes. The of the in-phase mode, about 200 mK. The shifted onset
suppression of the TO anomaly appears at a rim veloc- temperature at higher frequency modes has been previ-
ity of 100 µm/s for both modes. Similar behaviors have ously reported from a double-pendulum TO by Rutgers
been observed in numerous previous experiments1,2,8–15, group9 and also in shear modulus measurements in an-
including the measurement of the rigid TOs34–36. The other experiment21. The magnitude of the period drop
lowonsettemperatureofabout80mK andsharpertem- saturatesto 1.55ns (dP /∆P =2.6 10−4) at 40mK.
− −
×
perature dependence of the TO anomaly were reported The maximum period dropobservedfor both modes has
for low helium-3 impurity concentrations18,39. The pe- a similar order of magnitude as those anticipated by the
riod values measured at different rim velocities follows FEM simulation. The dissipation in TO amplitude ap-
theemptycellbackgroundathightemperaturesforboth pearsoverthesametemperaturerange. Atthelowestrim
resonant modes. velocity of 50 µm/s, the dissipation peak was located at
The maximum period drop at the lowest rim veloc- approximately 105 mK.
ity is 1.43 ns and 1.66 ns for the in-phase (-) and the We investigated seven different solid helium samples.
out-of-phase (+) modes respectively, which are a similar The resonant period showed similar order of magnitude
order of magnitude to those expected from the contri- drops, approximately 1.3-1.7 ns for both modes. How-
bution by the shear modulus (0.93 ns for the in-phase ever, only two solid samples described above show dis-
mode and 1.16 ns for the out-of-phase mode with a 30% sipation in the TO response. In addition, the size of
shear modulus change). By subtracting the empty-cell these dissipation peaks is approximately 10−7 and is
background,wecalculateddP withrespecttothemass orders of magnitude smaller than those from typical
±
loading of solid helium ∆P , equivalent to the NCRI TO experiments. These minute dissipation features are
±
fraction, dP /∆P =3.5 10−5 for the in-phase mode; consistent with the results from the Chan group who
− −
dP /∆P =2.5 10−4 for×the out-of-phasemode. Con- showed no clear dissipation features in their rigid TO
+ +
sidering that the×maximum change in shear modulus at experiments35,36. The absence of and/or the significant
thelowesttemperaturecanvaryfrom8%18 to86%40,the reduction of dissipations can be connected to a rigidity
perioddropcanbereasonablyattributedtothestiffening of the TO.
effect of the shear modulus in solid helium. Dissipation
inthe TOresponseisalsoobservedforbothmodes. The
dissipationpeakappearsaround30mK atwhichthe pe- IV. FREQUENCY-DEPENDENT STUDY
riodofthe TOchangesmostdrastically. The dissipation
peak is also suppressed by increasing the rim velocity in The frequency dependence of the TO responses is
bothmodesagreeingwithpreviousmeasurements1,2,8–15. examined to clarify the origin of the marginal period
We observed essentially the same results for the drop. While dP/∆P is independent of the measure-
commercial-purity (300 ppb) solid helium-4 sample ex- ment frequency in the supersolid scenario, it is propor-
cept for the anomaly found at higher temperatures. In tional to the square of the measurement frequency in
thein-phasemode,theanomalousperioddropappearsat the shear-modulus effect scenario30. Reppy et al.37,38
4
provided a simple mathematical methods to decom-
pose the period drop observed in their TO experi- 4 (a) (b)
ment into both frequency-independent and frequency- 47.2m/s 47.2m/s
dependent parts explicitly. The measured period drop -5 10)3 8278.15.8mm/s/s 8278.15.8mm/s/s
consists of two tienrdms: (i) the frequency-independent P- (x2 487850..81mm//essxp 487850..81mm//ss
tseorlimd[sdiPgn±a/t∆urPe,±a]nd(T(i,i)Vt)h,eregfraerqdueednacsy-adpeputeantdivenetsutperemr- P-/ a(dt P4-7/.2P-m)/s
d1
dep
[dP /∆P ] (T,V,f), attributed to the elastic over-
± ±
shoot effect, where T is temperature and V is rim veloc-
0
ity. The total period drop observed in TO experiments
0.02 0.0 5 0.1 0.2 0.02 0.05 0.1 0.2
can be written as follows:
Temperature (K)
FIG. 3. (a) Frequency-independent dP−/∆P−ind and (b)
dP± exp = dP± ind(T,V)+ dP± dep(T,V,f) frequency-dependentdP−/∆P−dep asafunctionoftempera-
(cid:20)∆P (cid:21) (cid:20)∆P (cid:21) (cid:20)∆P (cid:21) ture. The same color-coded pair for each modewas obtained
± ± ±
(1) usingthesamerimvelocity. Nodistinguishablevelocityeffect
Since the period drop originating from the overshoot isseenindP−/∆P−ind (a),incontrastto(b)demonstrating
effect is proportional to f2, the last term can be substi- aclearrim-velocitydependence. dP−/∆P− atthelowestrim
velocity 47.2µm/s is shown for comparison.
tuted with [dP /∆P ]dep(T,V,f) = a(T,V)f2. Then,
± ±
ind
both the frequency-independent term [dP /∆P ]
− −
andthefrequency-dependentterm[dP /∆P ]dep ofthe frequency-independent drop. The frequency-dependent
− −
in-phase mode are decomposed as follows: TO response can be extrapolated to values less than
4 ppm (0.16 ns) when the stiffening of solid helium is
significantly suppressed, indicating that the entire TO
dP− ind = f+2 dP− exp f−2 dP+ exp (2) anomalycanbe ascribedto the shearmodulus changeof
(cid:20)∆P (cid:21) γ (cid:20)∆P (cid:21) − γ (cid:20)∆P (cid:21) solid helium at low temperatures.
− − +
Figure 4 shows the low temperature dP/∆P for four
resonant modes as a function of frequency. Two solid
dP− dep = 1 f+2 dP− exp+ f−2 dP+ exp triangles are measured for solid samples grown in both
(cid:20)∆P (cid:21) (cid:18) − γ (cid:19)(cid:20)∆P (cid:21) γ (cid:20)∆P (cid:21) toriand the other two solid circles correspondedto sam-
− − +
(3) ples grown only in the upper torus. In the log-log plot
where γ =f2 f2. of dP/∆P and f2, the data can be linearly fitted to
To measu+re−[dP− /∆P ]ind and [dP /∆P ]dep at a the equation log(dP/∆P) = ( 9.74)+(1.003)log(f2).
− − − − −
certainrimvelocity,thedrivingACvoltagewascarefully Theslopeofequationisnearly1,indicatingthatdP/∆P
adjusted to have the same rim velocity for the in-phase is linearly proportional to f2. Converting the log-scale
and out-of-phase modes. Accordingly, the same color- axis to a linear one, the y-intercept is equivalent to
coded pair of temperature scans for eachmode as shown 2.62 10−7 (-0.3 ppm) which indicates that the mea-
− ×
inFigure3wasobtainedatthesamerimvelocity,despite sured period drop originates from the shear modulus ef-
of different driving AC voltages. fect rather than supersolid mass decoupling.
The temperature dependence of (a) the frequency- The frequency dependence can be analyzed by other
independent and (b) the frequency-dependent term at methods. The ratio between the period shifts of the two
various TO rim velocities are plotted in Figure 3. The modes δP+/δP− can distinguish the origin of TO re-
datasets identified with the same color in both Figure sponse at low temperature: either supersolidity or shear
3-(a) and 3-(b) are obtained using the same rim veloc- modulus change30. The ratio for the supersolid scenario
ity. However, the two figures are significantly different. (δP+/δP−)SS would follow the mass-loading (or miss-
No sizable frequency-independent period drop at any ing) change which can be easily obtained by measuring
rim velocity can be identified in Figure 3-(a), while the themass-loading-inducedperiodshiftsofbothmodes. In
frequency-dependent period response at the rim velocity contrast, the ratio for the shear modulus effect required
of47.2µm/sisnearlythesameasthetotalperioddropin an additional frequency-dependent contribution as fol-
the measurements. The averagedfrequency-independent lows:
period drop is approximately 0 4 ppm, which is simi-
lar to the upper limit set in the±rigid TO study36. Ac- (δP /δP ) = f2/f2 (δP /δP ) (4)
cordingly, the majority of the period anomaly can be + − SM + − + − SS
(cid:0) (cid:1)
attributed not to the appearance of supersolidity but The mass-loading ratio (δP /δP ) in our experi-
+ − SS
to the stiffening of the shear modulus of solid helium- ment is measured to be approximately 0.152. The solid
4. Besides, the frequency-dependent term was strongly straight line shown in Figure 5 indicates mass decou-
suppressed with increasing TO rim velocity. In con- plingorthe supersolidscenarioestimatedbyFEMsimu-
trast,noapparentdrivedependencewasobservedforthe lation and experimental measurements. The effect from
5

3
2.5 SS scenario
loaded on both tori
1.5 SM scenario (analytic)
2 loaded on the upper torus
SM scenario (FEM)
linear fit (slope=1.00325)
experiment (0.6ppb)
)1.5
-4 x10 ns)1.0 experiment (300ppb)
P ( 1 + (
P
P/
d
0.5
0.5
0.0
105 2 3 4 5 6 8 106 2 0.0 0.5 1.0 1.5
2 2
Frequency (Hz ) P- (ns)
FIG. 4. Log-Log plot of dP/∆P measured in four differ- FIG.5. Plotoftheratiooftheperiodshiftsmeasuredinboth
ent frequencies as a function of the TO frequency squared modes. Thedotted line, thesupersolid mass decoupling(SS)
for 0.6-ppb 3He (solid triangle) and 300-ppb 3He (solid cir- scenario, is directly calculated by the ratio of mass-loading
cles). TheredsolidtrianglesrepresentdP/∆P measuredwith in both modes. Both solid and dashed lines indicate the
bothtorifilledwiththesolidheliumsample,whiletheorange shear modulus (SM) scenario estimated by using finite ele-
solidcirclesrepresentdP/∆P measuredwithonlyuppertorus ment method (FEM) simulation and an analytical solution,
filled. Thegreydashedlineindicatesthelinearlyfittedresult. respectively. The experimental results of solid helium with
The y-intercept can be converted to frequency-independent 0.6-ppb 3He (solid triangles) and 300-ppb 3He (solid circles)
dP/∆P of -0.3ppm. are consistent with theSM scenario.
the shear modulus change of solid helium is estimated
analytically (the dashed line) and with FEM simulation [dP /∆P ]ind =1.86 10−3. Theperioddropanddissi-
− −
(the solidline). The slope of δP+/δP− plot in the shear pation were analyzed b×y the complex response function.
modulus scenario is steeper than that in the supersolid
The TO response they observed was not in agreement
mass-loading scenario due to the additional frequency-
with the functional formof simple glassydynamics. The
dependentcontribution. The measuredδP /δP values
+ − authorsconcludedthatadifferentphysicalmechanismis
for both 0.6 ppb and 300 ppb solid helium-4 (Figure 5)
required for explaining the TO responses.
indicate that the ratio δP /δP collapses to the shear
+ −
modulus scenario. The ratios obtained in the previous
TheCornellgroupalsodesignedacompoundTOwith
studies lie between shear modulus expectations and su-
anannularsamplespaceandmeasuredtheTOresponses.
persolid expectations, suggesting the possible existence
After removing the overshoot effect by frequency anal-
of a putative supersolid9,37,38,41. However, we confirmed
ysis, they extracted a finite frequency-independent pe-
that the TO responses from the KAIST rigid double- riod drop [dP /∆P ]ind = 1 10−4, several orders of
− −
torusTO originatedfromthe non-supersolidorigin. The ×
magnitudelargerthanthe elasticcontributionestimated
discrepancywithpreviousdouble-frequencyTOobserva-
by their FEM simulation. Additional dissipation intro-
tions may arise from the rigidity of TO.
duced by solid helium was measured to be very small.
The height of the dissipation peak was reported to only
5 10−8inbothresonantmodes. Theyproposedthatthe
V. DISCUSSIONS ×
finite frequency-independent period drop could possible
be new evidence for supersolidity in bulk solid helium.
Recently, Both the Cornell37,38 and London41 groups
constructedadouble-frequencyTOtoinvestigatethefre- In our rigid TO, the frequency-independent period
quency dependence of the period anomaly. The London dropistwoorthreeordersofmagnitudesmallerthanthe
groupmeasuredtheperiodanddissipationofatwo-mode value reportedfrom the Cornelland London group. The
TO containing a poly-crystalline solid helium-4 sample. dissipationpeak is notobservedinmostsolidsamplesor
They observed a period drop and concomitant dissipa- the size of the dissipationpeak was verysmall ( 10−7).
tionfeatures,equivalenttodP /∆P =2.10 10−3and We believe that this discrepancy is presumably∼due to
− −
dP /∆P =8.04 10−3. Thetorsionrodhole×effectand TOrigidity. IfthetrueNCRI fractionisabout100ppm,
+ +
×
Maris effect was removed by analytical calculations. Af- thentheperioddropanomalyshouldhavebeenobserved
ter fitting a linear equation to those data in f2-domain, in highly rigid TO experiments at PSU and in our rigid
they found a sizable frequency-independent period drop double TO experiments.
6
VI. CONCLUSION be smaller than 4 ppm.
WestudiedthefrequencydependenceofTOresponses ACKNOWLEDGMENTS
ofsolidhelium-4usingtherigiddouble-torustorsionalos-
cillator. Theperioddropanomalyisobservedforbothin- The authors acknowledge Duk Y. Kim at Pennsylva-
phaseandout-of-phasemodes. Frequencyanalysisshows nia State University for fruitful discussions and advices
thatthefrequency-independentperiodshiftislessthan4 in the early stage of construction of the rigid double-
ppm, closeto the upper limit setby the PSU group,and pendulum torsional oscillator. This work is supported
thefrequency-dependentcontributionisalmostthesame by the National Research Foundation of Korea through
as the TO response. We conclude that the TO response the Creative Research Initiatives. J. Choi would like to
at low temperatures is not due to the appearance of su- thank the POSCO TJ Park Foundation for its financial
persoliditybutduetothechangeintheshearmodulusof supportandgenerositythroughtheTJParkScienceFel-
solid helium. The supersolid fraction, if it exists, should lowship.
∗ [email protected] 23 A. D. Fefferman, X. Rojas, A. Haziot, S. Balibar, J. T.
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