Frequency-dependent Study of Solid Helium-4 Contained in a Rigid Double-torus Torsional Oscillator
aa r X i v : . [ c ond - m a t . o t h e r] J a n Frequency-dependent Study of Solid Helium-4 Contained in a Rigid Double-torusTorsional 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: March 16, 2018)The rigid double-torus torsional oscillator (TO) is constructed to reduce any elastic effects in-herent to complicate TO structures, allowing explicit probing for a genuine supersolid signature.We investigated the frequency- and temperature-dependent response of the rigid double-torus TOcontaining solid 4He with 0.6 ppb 3He and 300 ppb 3He. We did not find evidence to support thefrequency-independent contribution proposed to be a property of supersolid helium. The frequency-dependent contribution which comes from the simple elastic effect of solid helium coupled to TO isessentially responsible for the entire response. The magnitude of the period drop is linearly propor-tional to f , indicating that the responses observed in this TO are mostly caused by the overshootof ‘soft’ solid helium against the wall of the torus. Dissipation of the rigid TO is vastly suppressedcompared to those of non-rigid TOs. I. INTRODUCTION
The resonant period of an ideal torsional oscillator(TO) is proportional to the square root of its rotationalinertia √ I , and the superfluid decoupling can be de-tected by the reduction of the resonant period. Theresonant period drop of a TO containing solid heliumwas originally interpreted as the appearance of a super-solid phase . Recently, a number of experimental andtheoretical efforts have indicated that the anomaly inthe TO response can be explained by the shear moduluschange . The previous finite element method (FEM)simulation suggested that the influence on the period ofTO due to the change in shear modulus of solid heliumwas negligible . Nevertheless, the effect can be sig-nificantly amplified resulting from non-ideal TO design.Four mechanisms of non-ideal TO response have beensuggested . In order to minimize the contribution ofthe elastic effect to the resonant period of TO, it shouldbe meticulously constructed to be rigid. However, mostof TOs used in the previous supersolid experiments werenot rigid. . Recently, the Chan group reported thatthe resonant period drop was substantially reduced whenthe rigidity of TO was systematically increased . Theresonant period drop of a highly rigid TO was only a fewtimes greater than that due to the elastic effect estimatedby FEM simulation. They set the upper bound for thenon-classical rotational inertia (NCRI) to be less thanapproximately 4 ppm .However, new evidence of a ‘true’ supersolid signa-ture was suggested by Reppy et al. based on double-frequency TO experiments. Superfluidity or the NCRIis independent of frequency while non-superfluidity orthe relaxation phenomenon resulting from shear moduluschange leads to a strong frequency dependence. Accord-ingly, analysis on the frequency dependence can be usedto differentiate the superfluid response from the non-superfluid response . Reppy et al. observed a small frequency-independent resonant period change after sub-tracting the frequency-dependent term and ascribed thisto a possible supersolid signature. This interpretationcan be questioned since the measurements reported rel-atively large period drop which can be associated withthe non-rigidity of TO.For this article, we measured the period drop and ac-companied dissipation peak using a double-frequency TOthat was constructed to be highly rigid to minimize shearmodulus effects. The frequency dependence of the rigiddouble-frequency TO was investigated in various modesof representation. As a result, we elucidate whether ornot solid helium-4 exhibits true supersolidity. II. EXPERIMENTAL DETAILS
KAIST rigid double-torus torsional oscillator (TO)(Fig. 1) was carefully designed to minimize the variouselastic effects caused by: (1) shear-modulus-dependentrelative motion between TO parts (the glue effect) ,(2) solid helium contained in the torsional rod (the tor-sion rod hole effect) , and (3) solid helium layer grownon a thin TO base plate (Maris effect) .We first assembled every joint in the TO usingstainless-steel screws or hard-soldering to diminish elas-tic effects on the TO period due to the relative motionbetween different parts. The torsion plate and the tor-sion rod were rigidly connected by machining the assem-bly from a single piece of Be-Cu. The torus-shaped TOcell for solid helium was constructed by hard-solderingtwo pieces of semi-circular copper tubing. The torus washard-soldered onto thick copper plate and the combinedstructures were fastened down directly on the torsionplate by four screws.Second, we used a thick torsion rod and a very thinfill line to prevent the elastic effect of solid helium in thefill line. The change of shear modulus of solid helium-4contained in the torsion rod can induce the TO period FIG. 1. KAIST rigid double-torus torsional oscillator. anomaly . This effect may be responsible for the pe-riod drop observed in the majority of TO experiments.In order to remove this effect, a 1-mm-diatmeter CuNifilling capillary was directly installed on top of the TOcell instead of making a hole through a torsion rod.Third, the TO cell containing solid helium was hard-soldered to a 3-mm-thick copper plate and was designedso that solid helium in the torus did not have a direct con-tact to the base plate near the torsion rod. In a cylindri-cal TO, solid helium was grown directly on its base plate.If the thickness of the base plate is not sufficiently thick,then the period drop due to the change in shear modu-lus of solid helium can be significantly amplified . Thecontribution of solid helium is greater in the proximityof the torsion rod and when the base plate is thinner. Inthis study, solid helium confined in a rigid torus channelon the thick copper plate was not expected to exhibit thestrong Maris effect.Finally, we carefully tuned the so-called ‘overshoot ef-fect’ caused by direct coupling of elastic properties ofsolid helium to the TO response. Since solid helium ismuch softer than the wall of the container, it undergoesadditional displacement. The shear modulus change ofsolid helium not moving in phase with the confining wallcan induce the period anomaly without non-linear am-plification. The resonant period drop introduced by theovershoot effect is known to be linearly proportional tothe square of the TO frequency. For this experiment, weoptimized the overshoot effect so that the system was ina regime where the elastic effect is not too small to bedetected and not too big to make the TO softer. In thisTO, solid helium was located in a torus with a relativelylarge cross-section (diameter: 5 mm), which allows usto have large mass loading and high resolution for theso-called NCRI fraction. One can effectively reduce thiseffect by confining solid helium in a thick copper toruswith a small cross-section at the expense of diminishingthe capability of frequency analysis. This fine-tuning al-lows us to investigate the effect of the shear modulus onthe double-frequency TO and possible supersolidity moreclearly.In addition, the KAIST rigid double-torus TO can be operated at four resonant frequencies. This is possibledue to the configuration of our TO which consists of twotorus-shaped solid helium containers attached to upperand lower stages. This configuration enables modifica-tion of the resonant frequency by loading solid heliuminto the upper or both tori. We first placed the solidsample 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 solidsample placed in the upper torus.Finite element method (FEM) simulations indicatesthat a 30% change in the shear modulus of solid helium-4in the upper torus leads to only a 0.93-ns period drop infirst (1st) mode (in-phase) and a 1.16-ns drop in the sec-ond (2nd) mode (out-of-phase). These simulations cor-respond to 2 . × − and 1 . × − respectively in theframework of so-called NCRI fraction. The major con-tribution seemingly comes from the overshoot effect, therelative motion of solid helium with respect to the outerwall.The empty cell has resonant frequencies of 449.57 Hzand 1139.77 Hz for in-phase and out-of-phase modes re-spectively. Two pairs of electrodes attached to the lowertorus were used to drive and detect the TO response.The additional electrodes installed on the upper plateenables the detection of the amplitude and phase of theupper torus. We confirmed that the phase difference be-tween the two tori was approximately 0 degrees for thein-phase mode and 180 degrees for out-of-phase mode.The mechanical quality factor was approximately 1 × at 4.2 K for both modes.Bulk solid helium-4 samples containing an impurityconcentration of 0.6 ppb and 300 ppb helium-3 weregrown by the blocked capillary method. The sample cellwas first pressurized to target pressures of 67-82 bar at3.2 K and the mixing chamber was then cooled to thebase temperature. The resonant period shifted accord-ing to solid growth in the upper torus: 39,800 ns for thein-phase mode and 6,040 ns for the out-of-phase mode.The large period shift due to solid helium and its solidstability in the first mode enables us to distinguish thechange of the resonant period within about 2 ppm reso-lution. This sensitivity is lower than the upper limit ofNCRI fraction reported by the PSU group (4 ppm) . III. TO PERIOD AND DISSIPATION
The period and dissipation of the TO for both the in-phase and out-of-phase modes were measured over a tem-perature range of 20-300 mK. Figure 2 presents the reso-nant period of the rigid double-torus TO containing high-purity solid helium-4 (0.6 ppb) and commercial-puritysolid helium-4 (300 ppb) as a function of temperaturesat different rim velocities. The period drop anomaly wasobserved for both modes of the rigid double-torus TO.Compared with the empty cell data (dashed line), theperiod drop in 0.6-ppb solid sample appears initially at 80 (d)(b) (c) R e l a t i v e P e r i od C hange (a) x10 -7 D i ss i pa t i on Q - ( T ) 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-phase and (b) out-of-phase) and 300-ppb 3He ((c) in-phase and (d) out-of-phase). The empty-cell background was subtractedin both period and dissipation data. The y-axis of upper panels represent absolute values of period changes, calculated fromthe empty-cell backgrounds. mK, and became saturated below 25 mK for both modes(colored symbols). The magnitude of the period drop issuppressed by the increase in the rim velocity of the al-ternating current (AC) oscillation for both modes. Thesuppression of the TO anomaly appears at a rim veloc-ity of 100 µ m/s for both modes. Similar behaviors havebeen observed in numerous previous experiments ,including the measurement of the rigid TOs . Thelow onset temperature of about 80 mK and sharper tem-perature dependence of the TO anomaly were reportedfor low helium-3 impurity concentrations . The pe-riod values measured at different rim velocities followsthe empty cell background at high temperatures for bothresonant modes.The maximum period drop at the lowest rim veloc-ity is 1.43 ns and 1.66 ns for the in-phase (-) and theout-of-phase (+) modes respectively, which are a similarorder of magnitude to those expected from the contri-bution by the shear modulus (0.93 ns for the in-phasemode and 1.16 ns for the out-of-phase mode with a 30%shear modulus change). By subtracting the empty-cellbackground, we calculated dP ± with respect to the massloading of solid helium ∆ P ± , equivalent to the NCRIfraction, dP − / ∆ P − = 3 . × − for the in-phase mode; dP + / ∆ P + = 2 . × − for the out-of-phase mode. Con-sidering that the maximum change in shear modulus atthe lowest temperature can vary from 8% to 86% , theperiod drop can be reasonably attributed to the stiffeningeffect of the shear modulus in solid helium. Dissipationin the TO response is also observed for both modes. Thedissipation peak appears around 30 mK at which the pe-riod of the TO changes most drastically. The dissipationpeak is also suppressed by increasing the rim velocity inboth modes agreeing with previous measurements .We observed essentially the same results for thecommercial-purity (300 ppb) solid helium-4 sample ex-cept for the anomaly found at higher temperatures. Inthe in-phase mode, the anomalous period drop appears at an onset temperature of 160 mK and reaches a maximumof 1.49 ns ( dP − / ∆ P − = 3 . × − ) at 40 mK. In the out-of-phase mode, the onset temperature is higher than thatof the in-phase mode, about 200 mK. The shifted onsettemperature at higher frequency modes has been previ-ously reported from a double-pendulum TO by Rutgersgroup and also in shear modulus measurements in an-other experiment . The magnitude of the period dropsaturates to 1.55 ns ( dP − / ∆ P − = 2 . × − ) at 40 mK.The maximum period drop observed for both modes hasa similar order of magnitude as those anticipated by theFEM simulation. The dissipation in TO amplitude ap-pears over the same temperature range. At the lowest rimvelocity of 50 µ m/s, the dissipation peak was located atapproximately 105 mK.We investigated seven different solid helium samples.The resonant period showed similar order of magnitudedrops, approximately 1.3-1.7 ns for both modes. How-ever, only two solid samples described above show dis-sipation in the TO response. In addition, the size ofthese dissipation peaks is approximately 10 − and isorders of magnitude smaller than those from typicalTO experiments. These minute dissipation features areconsistent with the results from the Chan group whoshowed no clear dissipation features in their rigid TOexperiments . The absence of and/or the significantreduction of dissipations can be connected to a rigidityof the TO. IV. FREQUENCY-DEPENDENT STUDY
The frequency dependence of the TO responses isexamined to clarify the origin of the marginal perioddrop. While dP/ ∆ P is independent of the measure-ment frequency in the supersolid scenario, it is propor-tional to the square of the measurement frequency inthe shear-modulus effect scenario . Reppy et al. provided a simple mathematical methods to decom-pose the period drop observed in their TO experi-ment into both frequency-independent and frequency-dependent parts explicitly. The measured period dropconsists of two terms: (i) the frequency-independentterm [ dP ± / ∆ P ± ] ind ( T, V ), regarded as a putative super-solid signature, and (ii) the frequency-dependent term[ dP ± / ∆ P ± ] dep ( T, V, f ), attributed to the elastic over-shoot effect, where T is temperature and V is rim veloc-ity. The total period drop observed in TO experimentscan be written as follows: (cid:20) dP ± ∆ P ± (cid:21) exp = (cid:20) dP ± ∆ P ± (cid:21) ind ( T, V ) + (cid:20) dP ± ∆ P ± (cid:21) dep ( T, V, f )(1)Since the period drop originating from the overshooteffect is proportional to f , the last term can be substi-tuted with [ dP ± / ∆ P ± ] dep ( T, V, f ) = a ( T, V ) f . Then,both the frequency-independent term [ dP − / ∆ P − ] ind and the frequency-dependent term [ dP − / ∆ P − ] dep of thein-phase mode are decomposed as follows: (cid:20) dP − ∆ P − (cid:21) ind = f γ (cid:20) dP − ∆ P − (cid:21) exp − f − γ (cid:20) dP + ∆ P + (cid:21) exp (2) (cid:20) dP − ∆ P − (cid:21) dep = (cid:18) − f γ (cid:19) (cid:20) dP − ∆ P − (cid:21) exp + f − γ (cid:20) dP + ∆ P + (cid:21) exp (3)where γ = f − f − .To measure [ dP − / ∆ P − ] ind and [ dP − / ∆ P − ] dep at acertain rim velocity, the driving AC voltage was carefullyadjusted to have the same rim velocity for the in-phaseand out-of-phase modes. Accordingly, the same color-coded pair of temperature scans for each mode as shownin Figure 3 was obtained at the same rim velocity, despiteof different driving AC voltages.The temperature dependence of (a) the frequency-independent and (b) the frequency-dependent term atvarious TO rim velocities are plotted in Figure 3. Thedatasets identified with the same color in both Figure3-(a) and 3-(b) are obtained using the same rim veloc-ity. However, the two figures are significantly different.No sizable frequency-independent period drop at anyrim velocity can be identified in Figure 3-(a), while thefrequency-dependent period response at the rim velocityof 47.2 µ m/s is nearly the same as the total period drop inthe measurements. The averaged frequency-independentperiod drop is approximately 0 ± . Ac-cordingly, the majority of the period anomaly can beattributed not to the appearance of supersolidity butto the stiffening of the shear modulus of solid helium-4. Besides, the frequency-dependent term was stronglysuppressed with increasing TO rim velocity. In con-trast, no apparent drive dependence was observed for the exp at 47.2 m/s d P - / P - ( x - ) (a) Temperature (K) (b)
FIG. 3. (a) Frequency-independent dP − / ∆ P − ind and (b)frequency-dependent dP − / ∆ P − dep as a function of tempera-ture. The same color-coded pair for each mode was obtainedusing the same rim velocity. No distinguishable velocity effectis seen in dP − / ∆ P − ind (a), in contrast to (b) demonstratinga clear rim-velocity dependence. dP − / ∆ P − at the lowest rimvelocity 47.2 µ m/s is shown for comparison. frequency-independent drop. The frequency-dependentTO response can be extrapolated to values less than4 ppm (0.16 ns) when the stiffening of solid helium issignificantly suppressed, indicating that the entire TOanomaly can be ascribed to the shear modulus change ofsolid helium at low temperatures.Figure 4 shows the low temperature dP/ ∆ P for fourresonant modes as a function of frequency. Two solidtriangles are measured for solid samples grown in bothtori and the other two solid circles corresponded to sam-ples grown only in the upper torus. In the log-log plotof dP/ ∆ P and f , the data can be linearly fitted tothe equation log ( dP/ ∆ P ) = ( − .
74) + (1 . f ).The slope of equation is nearly 1, indicating that dP/ ∆ P is linearly proportional to f . Converting the log-scaleaxis to a linear one, the y-intercept is equivalent to − . × − (-0.3 ppm) which indicates that the mea-sured period drop originates from the shear modulus ef-fect rather than supersolid mass decoupling.The frequency dependence can be analyzed by othermethods. The ratio between the period shifts of the twomodes δP + /δP − can distinguish the origin of TO re-sponse at low temperature: either supersolidity or shearmodulus change . The ratio for the supersolid scenario( δP + /δP − ) SS would follow the mass-loading (or miss-ing) change which can be easily obtained by measuringthe mass-loading-induced period shifts of both modes. Incontrast, the ratio for the shear modulus effect requiredan additional frequency-dependent contribution as fol-lows: ( δP + /δP − ) SM = (cid:0) f /f − (cid:1) ( δP + /δP − ) SS (4)The mass-loading ratio ( δP + /δP − ) SS in our experi-ment is measured to be approximately 0.152. The solidstraight line shown in Figure 5 indicates mass decou-pling or the supersolid scenario estimated by FEM simu-lation and experimental measurements. The effect from loaded on both tori loaded on the upper torus linear fit (slope=1.00325) d P / P ( x - ) Frequency (Hz ) FIG. 4. Log-Log plot of dP/ ∆ P measured in four differ-ent frequencies as a function of the TO frequency squaredfor 0.6-ppb 3He (solid triangle) and 300-ppb 3He (solid cir-cles). The red solid triangles represent dP/ ∆ P measured withboth tori filled with the solid helium sample, while the orangesolid circles represent dP/ ∆ P measured with only upper torusfilled. The grey dashed line indicates the linearly fitted result.The y-intercept can be converted to frequency-independent dP/ ∆ P of -0.3ppm. the shear modulus change of solid helium is estimatedanalytically (the dashed line) and with FEM simulation(the solid line). The slope of δP + /δP − plot in the shearmodulus scenario is steeper than that in the supersolidmass-loading scenario due to the additional frequency-dependent contribution. The measured δP + /δP − valuesfor both 0.6 ppb and 300 ppb solid helium-4 (Figure 5)indicate that the ratio δP + /δP − collapses to the shearmodulus scenario. The ratios obtained in the previousstudies lie between shear modulus expectations and su-persolid expectations, suggesting the possible existenceof a putative supersolid . However, we confirmedthat the TO responses from the KAIST rigid double-torus TO originated from the non-supersolid origin. Thediscrepancy with previous double-frequency TO observa-tions may arise from the rigidity of TO. V. DISCUSSIONS
Recently, Both the Cornell and London groupsconstructed a double-frequency TO to investigate the fre-quency dependence of the period anomaly. The Londongroup measured the period and dissipation of a two-modeTO containing a poly-crystalline solid helium-4 sample.They observed a period drop and concomitant dissipa-tion features, equivalent to dP − / ∆ P − = 2 . × − and dP + / ∆ P + = 8 . × − . The torsion rod hole effect andMaris effect was removed by analytical calculations. Af-ter fitting a linear equation to those data in f -domain,they found a sizable frequency-independent period drop SS scenario SM scenario (analytic) SM scenario (FEM) experiment (0.6ppb) experiment (300ppb) P + ( n s ) P- (ns)
FIG. 5. Plot of the ratio of the period shifts measured in bothmodes. The dotted line, the supersolid mass decoupling (SS)scenario, is directly calculated by the ratio of mass-loadingin both modes. Both solid and dashed lines indicate theshear modulus (SM) scenario estimated by using finite ele-ment method (FEM) simulation and an analytical solution,respectively. The experimental results of solid helium with0.6-ppb 3He (solid triangles) and 300-ppb 3He (solid circles)are consistent with the SM scenario. [ dP − / ∆ P − ] ind = 1 . × − . The period drop and dissi-pation were analyzed by the complex response function.The TO response they observed was not in agreementwith the functional form of simple glassy dynamics. Theauthors concluded that a different physical mechanism isrequired for explaining the TO responses.The Cornell group also designed a compound TO withan annular sample space and measured the TO responses.After removing the overshoot effect by frequency anal-ysis, they extracted a finite frequency-independent pe-riod drop [ dP − / ∆ P − ] ind = 1 × − , several orders ofmagnitude larger than the elastic contribution estimatedby their FEM simulation. Additional dissipation intro-duced by solid helium was measured to be very small.The height of the dissipation peak was reported to only5 × − in both resonant modes. They proposed that thefinite frequency-independent period drop could possiblebe new evidence for supersolidity in bulk solid helium.In our rigid TO, the frequency-independent perioddrop is two or three orders of magnitude smaller than thevalue reported from the Cornell and London group. Thedissipation peak is not observed in most solid samples orthe size of the dissipation peak was very small ( ∼ − ).We believe that this discrepancy is presumably due toTO rigidity. If the true NCRI fraction is about 100 ppm,then the period drop anomaly should have been observedin highly rigid TO experiments at PSU and in our rigiddouble TO experiments. VI. CONCLUSION
We studied the frequency dependence of TO responsesof solid helium-4 using the rigid double-torus torsional os-cillator. The period drop anomaly is observed for both in-phase and out-of-phase modes. Frequency analysis showsthat the frequency-independent period shift is less than 4ppm, close to the upper limit set by the PSU group, andthe frequency-dependent contribution is almost the sameas the TO response. We conclude that the TO responseat low temperatures is not due to the appearance of su-persolidity but due to the change in the shear modulus ofsolid helium. The supersolid fraction, if it exists, should be smaller than 4 ppm.
ACKNOWLEDGMENTS
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