Effect of Rotation on Elastic Moduli of Solid 4 He
Tomoya Tsuiki, Daisuke Takahashi, Satoshi Murakawa, Yuichi Okuda, Kimitoshi Kono, Keiya Shirahama
aa r X i v : . [ c ond - m a t . o t h e r] N ov Effect of Rotation on Elastic Moduli of Solid He T. Tsuiki,
1, 2
D. Takahashi,
3, 2
S. Murakawa,
4, 2
Y. Okuda, K. Kono, and K. Shirahama
1, 2 Department of Physics, Keio University, Yokohama 223-8552, Japan RIKEN CEMS, Wako 351-0198, Japan Center for Liberal Arts and Sciences, Ashikaga Institute of Technology, Ashikaga 326-8558, Japan Cryogenic Research Center, University of Tokyo, Bunkyo-ku, Tokyo 113-0032, Japan (Dated: July 18, 2018)We report measurements of elastic moduli of hcp solid He down to 15 mK when the samples arerotated unidirectionally. Recent investigations have revealed that the elastic behavior of solid He isdominated by gliding of dislocations and pinning of them by He impurities, which move in the solidlike Bloch waves (impuritons). Motivated by the recent controversy of torsional oscillator studies, wehave preformed direct measurements of shear and Young’s moduli of annular solid He using pairsof quarter-circle shape piezoelectric transducers (PZTs) while the whole apparatus is rotated withangular velocity Ω up to 4 rad/s. We have found that shear modulus µ is suppressed by rotationbelow 80 mK, when shear strain applied by PZT exceeds a critical value, above which µ decreasesbecause the shear strain unbinds dislocations from He impurities. The rotation - induced decrementof µ at Ω = 4 rad/s is about 14.7(12.3) % of the total change of temperature dependent µ for solidsamples of pressure 3.6(5.4) MPa. The decrements indicate that the probability of pinning of Heon dislocation segment, G , decreases by several orders of magnitude. We propose that the motionof He impuritons under rotation becomes strongly anisotropic by the Coriolis force, resulting adecrease in G for dislocation lines aligning parallel to the rotation axis. PACS numbers: 62.20.de,66.30.J-,67.80.B-
I. INTRODUCTION
Solid He, a unique bosonic quantum solid, shows var-ious quantum effects . He atoms exchange frequentlytheir positions between the lattice sites. The exchangeleads to a dramatic effect in the motion of isotopic Heimpurity, which is inevitably contained in any He sam-ples. A He impurity can move as a Bloch wave in theperiodic potential formed by surrounding He atoms, asituation like electrons in crystal. Such a Bloch state iscalled impuriton or mass fluctuation waves . Sincethere are particle exchanges, bosonic solid He may ex-hibit superfluidity by Bose - Einstein condensation whileits crystalline structure is kept . Search for the coexis-tence of superfluid and crystalline orders, called superso-lidity, was sparked by the discovery of torsional oscillator(TO) anomaly in 2004 by Kim and Chan . It is nowa-days realized that the TO behavior found by Kim andChan, a drop of resonant period associated with dissipa-tion, turned out to be originated from the elastic changeof solid He. Even so, supersolid state of matter has beenone of the most fundamental problems in condensed mat-ter physics and bosonic He is still a candidate to realizesuch a intriguing state .Intensive studies of the putative supersolid state haverevealed anomalous behaviors in elastic properties ofsolid He at subkelvin temperatures . The shearmodulus of solid He, µ , shows a minimum at temper-atures 200 < T <
500 mK. Below 200 mK, it increaseswith decresing T and saturates below about 50 mK. Theanomalous increase in shear modulus below 0.2 K hasbeen attributed to pinning of dislocation network exist-ing in solid He by He impurity atoms . The anoma- lous elastic behavior owing to the dislocation motion andits pinning by He is a consequence of quantum naturein solid He. First, quantum fluctuation of He atomsaround lattice sites makes the Peierls potential so smallthat each dislocation segment between intersections caneasily vibrate like a string in which both ends are fixed.Second, He dilutely dispersed in solid He form a Blochstate with energy bands and hence can move as an im-puriton which has a group velocity and a band effectivemass determined by band structure, especially by band-width. The quantum motion of impurity He was stud-ied in 1970-80’s . Magnetic resonance experiments andtheoretical studies revealed that at low enough concen-tration He impurity behaves as impuritons, but the in-teraction between dislocations and impuritons was notdiscussed. In real solid He, dislocation networks arestrongly pinned at their intersections (nodes) so that theycannot move as a whole but only the segments betweennodes oscillate in particular gliding directions. It shouldbe emphasized that, He needs to move freely in solid Heto realize the pinning of dislocations and consequentlythe increase in shear modulus, and the movement of Henear 0 K can be realized only by the impuriton mecha-nism.In this paper, we report on new phenomenon relatedto the elasticity of solid He below 0.1 K, a decrementof shear modulus when solid sample is steadily rotated.We have observed decrements in two solid samples withdifferent pressures. The decrements suggest that pinningof dislocation network by He atoms is disturbed by rota-tion. We propose a possible mechanism for the suppres-sion of He pinning by considering the quantum nature ofsolid and impurity: The Coriolis force acting on He im-puritons makes their motion strongly anisotropic. Thisanisotropy of He diffusion can decrease the probabilityof attaching He to dislocations.One of our motivation for the present work is to solvethe controversy in the TO measurements under rotation.Choi et al . found that the apparent supersolid frac-tion, i.e. the magnitude of period drop, obtained from aTO containing annulus solid He sample is strongly sup-pressed by steady DC rotation . They attributed thesuppression of the period drop to the intrusion of quan-tum vortices in supersolid. As the TO period shift is notoriginated from supersolidity, the origin of the rotationeffect discovered by Choi et al . is now a most importantpuzzle: It has been established by theoretical considera-tions and by Finite-Element Method simulations that in-crease in shear modulus of solid He contained in the bobof a TO stiffens the TO bob and thus decrease the TOperiod. This stiffening - induced period drop was mis-taken as a superfluid response of solid He inside. Giventhat the TO period drop is entirely due to elastic changein solid He, shear modulus has to be influenced by rota-tion. Choi et al . measured shear modulus under rotationby a pair of piezoelectric transducers (PZTs) located inthe center of their TO, but they did not observe rotationeffect . In our present setup, we measure shear modulusin the direction of circumference of annular solid samplesso that it is possible to conclude the origin of rotationeffect observed by Choi et al ..We organize this paper as follows: In the followingSection, we describe elastic properties solid He and cur-rently argued interpretation. In Sec. III, we show theexperimental method to measure shear and Youngs mod-ulus for annulus solid He sample under rotation. In Sec.IV, experimental results with and without rotation areshown, followed by the interpretation for these results interms of impuriton model in Sec. V. In Sec. VI we givesome discussion and the paper is summarized in Sec. VII.
II. BACKGROUNDSA. Shear modulus anomaly and dislocation in solid He Since 1970’s, it has been well established that solid he-lium contains dislocations as a defect by measurementsof plastic deformation , ultrasound and X-ray . Asin dislocations in ordinary matters such as metals, dis-locations in helium compose a network structure. Thenodes of dislocation network are strongly fixed, while dis-location segments between nodes can vibrate like stringsin response to stress . Ultrasonic studies showed that He impurity atoms attach to dislocation segments atlow temperatures and disturb their oscillations respond-ing to ultrasound . It is suggested that, in hcp solid He, He impurity atoms are trapped to edge and mixeddislocations in which both the dislocation lines and theirBurgers vectors are aligned to the basal plane of hcp lat- tice ((0001) plane). This is because elastic deformationaround the core of dislocations forms an attractive po-tential on the side where atomic density is low. Notethat another side has higher atomic density, and exertsa repulsive force. Screw dislocations do not attract He.The elastic modulus of solid He at subkelvin tem-peratures is determined by the motion of dislocationsegments . As solid is cooled below 1 K, disloca-tion movement is damped by the collision with phononsso that the shear modulus is relatively high. As temper-ature decreases, the number of phonons decreases anddislocation segments become free to glide in the baselplane in response to applied shear, while the intersec-tions act as nodes and are immobile. The shear modulushas a minimum at temperatures 0.2 - 0.5 K, dependingon He concentration. At lower temperatures (e.g. be-low 0.2 K), shear modulus starts to increase because Heatoms start to stick to dislocation segments and disturbtheir motion. Shear modulus gradually approaches to itsintrinsic value below 50 mK.The temperature and He concentration dependen-cies of shear modulus were first measured by Day andBeamish for polycrystalline solid samples . Later stud-ies by Balibar and coworkers in which the orientations ofsingle crystals to PZTs were precisely controlled revealedthat only the elastic constant c is responsible for theincrease in shear modulus below 0.2 K . The situationis shown in Fig. 1. c corresponds to the elastic ten-sor component when the hcp basal plane glides to [1010],[1100] or [0110] directions (Fig. 1(a)), and when (1010),(1100) or (0110) planes glide to the direction of c ([0001])axis ((b)). The dislocations which mainly determine c are aligned perpendicularly to shear strain and in par-allel to the gliding plane, illustrated as vertical lines inFig. 1(c). The vibration of the vertical dislocations glidethe planes and can relax shear stress. It is remarkable inhcp He that the glide motion occurs at very low appliedshear strain ǫ ∼ − .Although the microscopic mechanism of trapping Heto dislocation core has not been understood, the effectof He impurity on shear modulus has been phenomeno-logically discussed and has successfully explained the de-pendencies of shear modulus on temperature and appliedshear stress . We will discuss the detail of the modelsin Sec. V. B. Impuriton: quantum impurity He in solid He Elastic anomaly below 0.2 K occurs even when imposedshear stress is so small that the displacement of disloca-tion is less than 10 nm. Since the average length of dislo-cation segments is on the order of 10 µ m and should becomparable to the average separation of segments, Heatoms move a long distance in solid He, in order to betrapped by dislocation core. Definitely, such mobility of He near 0 K can be realized only by quantum effects. He impurities are delocalized and behave as quasi-
FIG. 1. (Color online) Two cases of shear motions of hcp Hethat contribute to change in c . (a) The hcp basal planeglides to [1010], [1100] or [0110] directions. (b) (1010), (1100)or (0110) planes glide to the direction of c ([0001]) axis. (c)Schematic drawing of dislocation network (cubic shape is as-sumed for simplicity). The vibration of dislocation lines per-pendicular to glide direction and in parallel to the glidingplane (shown by dashed lines) only contributes to shear stress.The direction of sample rotation Ω is also shown by an arrowfor later discussion. particles like Bloch waves in periodic potential of the He matrix, hence the wavenumber k of the quasipar-ticles become a good quantum number and the motionis determined by energy bands. The quasiparticle wascalled impuriton by Richards or mass fluctuation waves by Guyer . The impuriton picture for He in solid Hewas theoretically proposed by Andreev and Lifshitz , inwhich they called it defecton , and by the above authors,and then experimentally confirmed by many NMR stud-ies for He concentration from 0.02 (2 % molar ratio of He) down to 10 − (10 ppm) . Note that the im-puriton picture becomes better as the concentration de-creases. Therefore, although no NMR experiments wereperformed for commercial He gas in which typical Heconcentration is less than 1 ppm, one may expect that He behaves as impuriton in such a low concentrationsample.The most important feature of He impuriton is thatthe energy bandwidth is extremely small and it stronglydepends on the molar volume (i.e. pressure) of solid He. Exchange energy between He and He, J , atthe highest molar volume (near the melting pressure) 21cm /mole is of the order of 10 − K, which corresponds to1 MHz in frequency . The corresponding energy band-width ∆ is determined by crystal structure and is about10 J for hcp lattice, hence of the order of 10 − K, thetemperature of which is far smaller than the tempera-ture we perform experiment. With increasing pressure(decreasing molar volume), the bandwidth rapidly de-creases: At pressures of the solid samples in the presentwork, P = 3 . v g of impuritons also becomes small: As shown in Sec. V, v g is of the order of1 mm/s near melting, but it decreases down to 100 nm/sat P = 5 . m ∗ , is extremely large: m ∗ is 10 m where m is the bare mass of a He atom (7 × − kg) at themelting pressure, and it exceeds 10 m at 5.4 MPa. Asdiscussed in Sec. VI, the large effective mass and smallgroup velocity of impuritons are essential for the rotationeffect on shear modulus.The validity of the impuriton model at high pressures(e.g. 3 ∼
15 MPa) is an important problem becausemany TO experiments have been done at such high pres-sures. According to the NMR experiment performed byGreenberg et al ., the exchange frequency of impuritons J was measured up to 4 MPa (20 cc/mole), and it wasfound to be less than 1 kHz at the largest pressure .Therefore, the impuriton model is valid at pressures upto 4 MPa. The pressure of one of our solid samples is3.6 MPa, which is well below this pressure. Whether theimpuriton model correctly describe the behavior of Heat pressures higher than 4 MPa is yet to be confirmed ex-perimentally. In a TO measurement by Kim and Chan,it was found that the apparent supersolid response wasobserved for many solid samples up to 15 MPa, where theonset temperature of the apparent supersolidity showed no pressure dependence . If the supersolid - like behav-iors observed in the high pressure samples are entirelycaused by the elastic anomaly of solid He, He impuri-ties have to run a macroscopic distance even in such highpressure samples, otherwise the He - dislocation pinningmodel would never work out. There is also possibilitythat He can hop site to site by quantum tunneling with-out forming impuriton band. It is therefore concludedthat the picture of ”dislocation pinning by He movingquantum mechanically” is valid for solid at pressures upto 15 MPa.
C. Effect of rotation on properties of solid He:previous studies
Far before the 2004 Kim - Chan experiment, Pushkarovtheoretically studied the effect of rotation on defectonquasiparticles, in which he focused on zero point vacancyin solid He . He discussed the motion of quasi-particles under sample rotation in terms of a Fokker -Planck equation, and found that quasiparticle diffusionis strongly suppressed in the direction perpendicular tothe rotation axis . When crystal is steadily rotated,the temperature dependence of diffusion coefficient per-pendicular to the rotation axis change dramatically from D ∝ T − , which is the behavior under no rotation andis determined by phonon scattering, to T dependenceat subkelvin temperatures, while the diffusion constantparallel to the rotation axis remains unchanged. Thisdramatic anisotropy of defecton diffusion is essentiallythe same conclusion as our consideration focussing oncircular motion of impuritons by Coriolis force, which isdiscussed in Secs. V and VI. Pushkarov also claimed thatthe centrifugal force tends to change the distribution ofdefectons in the radial direction of a cylindrical solid .It is essentially an ordinary phenomenon of centrifuge,and we find that it is negligibly small for rotation speedof the order of 1 rad/s, even compared to the effect ofgravity on the vertical distribution of He impuritons.In order to study the possible quantized vortices in su-persolid, TO experiments have been performed using sev-eral rotatable dilution refrigerators in the world. Choi etal . employed a TO containing annular solid He and a ro-tating dilution refrigerator at RIKEN, which is also usedby us in the present work . They observed that themagnitude of period drop, which was previously called”non-classical rotational inertia fraction” (NCRIF), issuppressed when unidirectional rotation is imposed to thetorsional oscillation. As the rotation speed increases, themagnitude of the period drop decreases and the accom-panying dissipation increases . The oscillator used byChoi et al . had a PZT pair at the center of the torsionalbob so that shear modulus of the solid other than theannular part was measured independently. In contrastto the TO period, no significant change in shear mod-ulus was observed when the apparatus was rotated .It was thus concluded that the rotation effect on TOfrequency is related to supersolidity. They further stud-ied TO response by changing rotation speed continuouslyand observed a step - like TO period change, which wasagain interpreted as a vortex intrusion . Since any TOresponses were revealed not to be originated from super-solidity, these experimental results of Choi et al shouldbe reconsidered on the basis of elastic behavior of solid He.Other TO experiments under rotation performed byYagi et al . and Fear et al . employed rotating cryostatsof ISSP, Univ. Tokyo and of Univ. Manchester, respec-tively. They claimed that no significant rotation effectwas observed in the period of TO containing cylindricalor annular solid He samples. Especially, Fear et al . employed annular solid samples, which were similar tothat used by Choi et al ., but they did not observe anyrotation effect. Fear et al . claimed that the rotation ef-fect observed by Choi et al . should be originated frommechanical noise caused by rotating cryostat.This experimental controversy has recently been stud-ied by two TO experiments using the rotating cryostat atRIKEN. One experiment done by Jaewon Choi et al . em-ployed a TO which has two annular tubes (donut shape)connected with a torsion rod . This oscillator was spe-cially arranged to make elastic contribution to the reso-nant period as small as possible (by no use of epoxy, andso on), and to have multiple resonant frequencies to de-tect the frequency - independent component in the periodchange, which can be originated from supersolidity .The change in period drop by rotation up to 4 rad/swas small and did not reproduce the result of Choi etal . . Another experiment based on completely oppositeidea by Tsuiki et al . used a multiple - frequency TO FIG. 2. (Color online) The experimental cell. (a) Schematicview. Capillary is used for supplying line of He. (b) In-ner structure. The gaps between the PZTs are 0.5 mm. (c)Movements of two PZT’s. Top : neutral position. The middleand the bottom express movements of shear and compressivePZTs, respectively. which is very sensitive to elasticity of solid He insidethe bob . This TO was firstly proposed by Reppy etal. and is called a floating - core TO , in which theannular solid sample is formed between the outer shelland the inner core, which is hung by a torsion rod fromthe outer shell. In this elasticity - sensitive TO study,suppressions of TO period drop have been observed atintermediate temperatures depending on TO amplitude(i.e. applied shear stress), but the suppression at lowesttemperatures is much smaller than that observed by Choi et al . . Since both the experiments by Jaewon Choi etal . and Tsuiki et al . were done with using the same rotat-ing cryostat at RIKEN , the possibility of the spuriouseffect by mechanical noise, which was claimed by Fear et al. , is denied. The large rotation effect observed byChoi et al is yet to be understood. In Sec. VI we willgive a possible interpretation for the discrepancy in TOstudies. III. EXPERIMENTAL SETUP
Our sample cell to measure elastic moduli is schemat-ically shown in Fig. 2(a) and (b). A thin annular chan-nel is formed by gluing a cylinder to a container, both ofwhich are made of BeCu alloy. The inner and outer radii,and the height of the channel are 8.0, 10.5, and 12.0 mm,respectively. Two pairs of quarter - circle shape PZTs(10 mm height and 0.5 mm thick) are glued to the in-ner and outer walls with inserting round shaped Macorplates for insulation. The arc lengths are 14.14 and 12.73mm for outer and inner PZT, respectively. The gaps be-tween the inner and outer PZTs, i.e. the thickness ofsolid He sample, are set to 0.5 mm.We show in Fig. 2(c) movements of two PZTs. Whena voltage is applied, one of the PZTs moves in the az-imuthal direction, as shown in the middle. This motiongives a shear to the solid He sample between the pairPZT, so the PZT pair is used to measure shear modulusin the azimuthal direction of the annular solid, i.e. thesame direction as the rotation - induced velocity. An-other PZT expands in the radial direction as shown inthe bottom of the cartoon, hence the PZT compressesthe solid sample. This PZT pair applies a strain to thesolid and detect the resultant stress in the radial direc-tion. It therefore measures Young’s modulus. We referto the former PZT pair as the ”shear” PZT, whereas thelatter as the ”compressive” PZT.With these PZTs, we measure elasticity of solid He asfollows: We applied an AC driving voltage V AC with fre-quency f = 1 . ǫ = V AC d PZT /D to solid He, where d PZT is the piezoelectric constant,and D the gap between the drive and detection PZTs, 0.5mm. We estimate d PZT at low temperature (below 0.5 K)to be 7 . . × − m/V for the shear (compressive)PZT, respectively. The strain causes stress σ = λǫ . Theelastic modulus of solid He λ (the shear modulus µ orthe Young’s modulus E ) is evaluated by measuring thepiezoelectric current I induced on the outer detection-PZT as λ = ( D/ πAd )( I/f V AC ), where A is overlap-ping area of drive and detection PZTs. Since the current I is typically an order of 1 pA, a current preamplifierwith a fixed gain of 5 × V/A and lock-in techniqueare used to measure the current. The effect of electricalcrosstalks between the drive and detection PZTs, whichaffects the phase difference between I and V AC , is care-fully removed using the crosstalks obtained by measuringthe cell filled with superfluid He.The whole cell is mounted on the center of a flangeconnected to the mixing chamber of the RIKEN rotat-ing dilution refrigerator , the same apparatus as TOmeasurements by Choi et al . . The refrigerator is ro-tated with angular velocity − ≤ Ω ≤ ≤ Ω ≤ Hesamples of pressure P = 3 . He gas with He concentration x = 0.3 ppm. IV. RESULTSA. Effect of rotation on shear modulus
Figure 3 shows the temperature dependence of shearmodulus µ measured at three different strains ( ǫ ) for twosolid He samples of 3.6 and 5.4 MPa. In Fig. 3, data fordifferent Ω’s are shifted to coincide at 300 mK on the as-sumption that all dislocation is free from pinning by Heimpurities so that they have essentially the same mini-mum modulus value at this temperature and the wholedatasets with the same strain results are systematicallyshifted for clarity.
FIG. 3. (Color online) Shear modulus of two solid samples asa function of temperature. Data are shifted so as to coincideat 300 mK for each strain. Scales of shear modulus for twosamples are different. (a) P = 3 . , , , , P = 5 . The data were taken as a time series (partly shown inFig. 4) in the cooling run from 0.5 K to the lowest T .In both samples, changes in µ depend on T and ǫ . Inthe case of the 3.6 MPa sample shown in Fig. 3 (a), at ǫ ≤ . × − , the saturation of change in µ completesat 40 mK. The modulus increases from 300 to 20 mK byabout 22 %. On the other hand, at ǫ = 3 . × − , µ doesnot saturate at 40 mK. This is also observed in the sampleof 5.4 MPa shown in Fig. 3(b), in which the magnitude ofthe change by temperature is about 40 % at two different ǫ ’s and at 0 rad/s. This ǫ dependence is essentially similarto the previous shear modulus measurements by othergroups using a pair of flat PZT. The trend thatthe saturation temperature shifts to lower T at large ǫ indicates that ǫ = 3 . × − exceeds, or is in the vicinityof the critical strain ǫ c , above which µ is suppressed.We examined the effect of rotation for the two samples.In the 3.6 MPa sample at ǫ = 2 . × − , no character-istic dependence of µ on Ω is observed, although dataare scattered. At ǫ = 7 . × − , µ does not changeby rotation, except above 0.36 K, where the slope of µ changes as Ω increases to 2 rad/s. Also in the 5.4 MPasample, at ǫ = 1 . × − , µ is almost identical for 0 and4 rad/s, except for a small difference between 0.1 and 0.2 FIG. 4. (Color online) (a) A sequence of measurement of Ωdependent shear modulus µ at ǫ = 3.96 × − . Brown linesshow temperature, while other colors show µ at different Ω’s.Black points indicate data at Ω = 0 rad/s between 1 and2 rad/s, 2 and 3 rad/s, after 4 rad/s measurements. Shearmodulus has a minimum around 300 mK and has a plateauas the lowest temperature approaches. (b) Ω dependence ofthe ”minimum” and the ”plateau” values of µ ( µ ave (300 mK)and µ ave (20 mK)). The lowermost (red) and middle (black)dashed lines are results of the linear least square fitting. Theuppermost (red) line is a parallel shift of the minimum line(indicated by red arrows), thus shows a ”supposed” changein µ at the lowest T if there were no suppression by rotation.Suppression of µ by rotation is described by blue arrow (seetext). K. Although the change is too small to identify as a ro-tation effect because of poor signal-to-noise ratio of thedata, we will mention in Sec. V that a similar decrementat intermediate temperatures (0.1 - 0.2 K) has been ob-served in the floating-core TO . At ǫ = 3 . × − , µ at two pressures tend to have smaller value below 0.1 Kas Ω increases from 0 to 4 rad/s. This tendency is moreprominently seen in the 5.4 MPa sample.The Ω - dependent shear modulus at ǫ = 3 . × − ismore clearly confirmed by looking at the time sequenceof the data acquisition. The time sequence of the 3.6MPa data is shown in Fig. 4 (a). We set Ω to a constantvalue (1, 2, 3 and 4 rad/s) before either cooling or warm-ing scan. After each scan at a constant Ω, we stoppedrotation (Ω = 0 rad/s) and then immediately increasedΩ to a new value. The open circles indicate the high - T shear modulus which is obtained by averaging the dataaround 300 mK, and the open squares indicate the shearmodulus averaged at the lowest T . With these averaged data µ ave , we define ∆ µ ≡ µ ave (20 mK) − µ ave (300 mK),the magnitude of total change in µ from 300 to 20 mK,which is indicated by an arrow in Fig. 4 (a) for the Ω =0 rad/s data. ∆ µ decreases progressively as Ω increases.There is another important feature in Fig. 4(a). Atthe lowest T , µ once has a plateau value indicated byopen square, then increases abruptly. One can clearlysee that this abrupt change occurs while keeping Ω = 4rad/s (shown in red data). We will discuss this behaviorin Sec. V.The Ω dependence of µ is obtained by plotting µ ave (300 mK) and µ ave (20 mK) (open circles and squaresin Fig. 4(a)) as a function of Ω. It is shown in Fig. 4(b).Shear moduli µ ave (300 mK) and µ ave (20 mK) are approx-imately linear in Ω, having different gradients. Fromthese averaged data we evaluate the rotation effect onshear modulus. We tentatively assume that the Ω de-pendence of µ at high T is caused by the drift of mea-surement by unknown origin. We perform a least squarelinear fitting to this high - T modulus, then utilize thefitting as a ”supposed” shear modulus µ supposed (20 mK)(shear modulus if there were no rotation effect) for thelowest T . These are shown in Fig. 4(b). The rotationeffect is evaluated by a ratio δ (Ω) defined as δ (Ω) ≡ µ supposed (20 mK) − µ ave (20 mK) µ supposed (20 mK) − µ ave (300 mK) , (1)which is the ratio of the length of green arrow to blue onein Fig. 4(b). At Ω = 4 rad/s, δ (4 rad / s) is 0.147; i.e. theshear modulus is suppressed to 85 % of the original mod-ulus without rotation. For the 5.4 MPa solid, δ (4 rad / s)is 0.123, slightly smaller than that of 3.6 MPa case. As tothe definition of δ (Ω), a different consideration is givenin Appendix. B. Effect of rotation on Young’s modulus
Figure 5 (a) shows Young’s modulus E measured bythe compressive PZT for 5.4 MPa sample at ǫ = 7 . × − and 3 . × − without rotation. E ( T ) graduallydecreases as T decreases, then shows a minimum, fol-lowed by a large increase. At ǫ = 3 . × − , the overallbehavior shifts to lower temperatures, and E does notsaturate. The change in E from 200 to 20 mK is about60 % of that of the lower strain case. By analogy withshear modulus, a critical strain for Young’s modulus pre-sumably exists between the two strains. In Fig. 5(b),we compare E ( T ) data above the critical strain with andwithout rotation. Unlike shear modulus, no significantdifference was found in E ( T ), except for a small down-ward shift at Ω = 4 rad/s above 0.3 K. FIG. 5. (Color online) Young’s modulus E measured in cool-ing scans. (a) E measured at ǫ = 7 . × − and 3 . × − without rotation. (b) E measured at ǫ = 3 . × − with andwithout rotation. Data at Ω = 4 rad/s are shifted to coincideat 300 mK. V. INTERPRETATION FOR THE ROTATIONEFFECT
In this Section, we discuss the possible mechanism forthe rotation effect observed in shear modulus when strainover ǫ c is applied. We firstly show that the observed ro-tation effect can occur when the probability of attaching He impurities to dislocations, G , decreases by three or-ders of magnitude. Next, we propose that such a changein G can be realized by the effect of the Coriolis forceexerted to He impuritons.
A. Pinning of He to Dislocation Network
Rotation produces centrifugal and Coriolis forces onthe solid He samples, and these forces can influenceboth the structure of dislocation network and the mo-tion of He impurities. Iwasa points out that dislocationsegments in the network can be stretched by the stress in-duced by rotation . However, the magnitude of stretch-ing by centrifugal force is estimated to be 10 − of thesegment length L , when L is set to 1 µ m. This stretch-ing is too small to explain the 15 % reduction in µ byrotation (the effect of the Coriolis force is even smaller).Next, we discuss the effect of rotation on the pinning of He to dislocations. In the dislocation pinning model ,shear modulus µ is given by µ = µ el (1 − ν )Λ2 π L = µ el κ − (cid:18) κ ≡ − ν )Λ2 π L (cid:19) (2)where µ el is the shear modulus when there is no pinningeffect by impurity He, ν Poisson’s ratio, Λ the dislo-cation density, and L the average length of dislocationbetween pinning points. Hereafter, we adopt averagesfor all the lengths concerning dislocations for simplicity,but there are broad distributions in reality. We mentionthe effect of distributions later.Equation (2) clearly indicates that the shear modulusdecreases as L increases. L is given by L = L NA L iA L NA + L iA (3)where L NA and L iA are the average of length of disloca-tion segment between network nodes, L N , and the aver-age of the dislocation length between the pinned pointsby He impurities, L i , respectively. Since the dislocationnodes are strongly fixed, the distribution of L N is deter-mined when the solid sample is formed. L i is determinedas L N divided by the number of pinned He on a dis-location, n i , which is evaluated by considering pinningprocess.When He impurities approach a dislocation segment,they pin the segment with pinning rate R given by R = x L NA G (4)where x and G are He concentration and the proba-bility of pinning of He on the dislocation, respectively.Pinned He is detached from the dislocation with a rateof unpinning rate R . R is obtained by assuming anArrhenius type rate equation with thermal activation en-ergy E A by the binding energy of a He to dislocation asfollows. R = n i τ e − E A /T (5)where τ is the relaxation time (1 /τ is the attempt fre-quency of the thermal activation of dislocation). Theequilibrium number of attached He on a dislocation n i0 is evaluated by balance of R and R so that we obtain n i0 given by n i0 = x L NA Gτ e E A /T . (6)Eventually, L iA is obtained: L iA = L NA n i0 = exp ( − E A /T ) x Gτ . (7)Taking Eqs. (2), (3) and (7) into account, we find thatthe quantity κ depends on the pinning probability G , κ = 1 + (1 − ν )Λ2 π (cid:18) L NA L iA L NA + L iA (cid:19) (8)= 1 + (1 − ν )Λ2 π L NA exp ( − E A /T ) x Gτ L NA + exp ( − E A /T ) x Gτ ! (9)We evaluate the magnitude of change in µ by changingtemperature, using δ µ , the same as Eq. (A1). Using Eq.(2), Eq. (A1) becomes δ µ = µ el κ (0 .
02 K) − µ el κ (0 . µ el κ (0 . (10)= κ (0 . κ (0 .
02 K) − . (11)According to Iwasa , we choose Λ = 1 . × m − , G = 1 . × m − s − , τ = 10 × − s and E A = 0 . δ µ = 23 .
18% (inorder to calculate this expected value, here µ ave (20 mK)in δ µ is replaced by µ supposed (20 mK) of the previoussection), when we assume L NA ≈ . × − m accordingto Iwasa .We now consider how rotation affects the change in δ µ . As previously mentioned, L NA (= 1 . × − m) willnot change by rotation. Using Eqs. (7), (8) and (11),it is deduced that the experimental value δ µ = 19 . G ≈ . × m − s − . That is, by rotation, G has todecrease 10 − times the value of G without rotation. B. Effect of rotation on the motion of Heimpurities
In the previous section, we have concluded that theprobability of sticking He to dislocations G should de-crease three orders of magnitude to explain the observedrotation effect on shear modulus. We propose that sucha huge decrease in G can be realized by the Coriolis forceexerted to He impuritons. The rotational motion ofthe impuritons by the Coriolis force makes He diffusionstrongly anisotropic between the direction of rotation andother directions.There are three types of forces exerted to rotating solidhelium with constant rotation speed: gravity, centrifugaland Coriolis forces. Gravity and centrifugal forces areexerted to both the He and He atoms, and the magni-tudes are mg and mr Ω , where r is the radial positionof the atom. Here the mass m is the bare mass of Heor He. The centrifugal force can induce a distributionof He impurities in the radial direction, but the magni-tude of the force at Ω = 4 rad/s is an order of magnitudesmaller than the gravitational force. Therefore, the Hedensity distribution caused by centrifugal force is negli-gible in the range of Ω in the present work.On the other hand, the Coriolis force is a velocity -dependent force exerted on an object moving with veloc-ity v in the rotating frame. Therefore, the Coriolis forceis exerted only on moving He impurities , not on He atoms. The Coriolis force is given by 2 m ∗ ~ Ω × ~v . We em-phasize that the mass in the formula of Coriolis force isan effective mass of He impuriton, and the velocity isthe impuriton group velocity v g , which is determined bythe energy bandwidth ∆. Since the effective mass is large(10 m < m ∗ < m ), the magnitude of the Coriolisforce 2 m ∗ Ω v exceed several orders of magnitude of thegravitational and centrifugal forces.The equation of motion is dominated by the Coriolisforce: m ∗ a = − m ∗ Ω × v . (12)Consequently, the impuriton motion becomes spiral ,i.e. when we take the cylindrical coordinate ( r, θ, z ) inwhich z is the rotation axis, the motion is circular in the r − θ plane, whereas the motion in the direction of ro-tation axis ( z ) is unaffected. The radius of circle R C isgiven by h v g i r / h v g i r is the projection of thegroup velocity vector to the r − θ plane. The magnitudeof group velocity of He impuriton in hcp He is approx-imately given by (cid:10) v (cid:11) = 18 J a / ¯ h . Here J is theexchange energy between He and He atoms, and a isthe lattice constant (of the hcp basal plane).In order to evaluate the rotation effect, we need toadopt values of J from past NMR measurements, butno data are available for He concentration less than 10ppm and for pressure higher than 4 MPa. We estimate J from NMR studies of Greenberg et al . , in which J was obtained experimentally for solid samples with He concentration 0.01 and 0.02, and at molar volumefrom 21 to 20 cc/mole, which corrensponds to the pres-sure range from 2.5 MPa (i.e. melting pressure) to 4MPa. Greenberg et al . found that log J is linearly pro-portional to the molar volume V m , i.e. J ∝ exp( V m ).Since no substantial He concentration dependence wasseen in J , it is reasonable to assume the same orderof magnitude for J in our much more dilute samples(Note that the impuriton picture becomes better as the He concentration decreases). J is estimated for ourtwo samples by assuming the same molar volume depen-dence. We estimate possible maximum and minimum of J , taking the scattering of the original data of Green-berg into account. The lattice constant a is estimatedfrom the data of a past x-ray measurement . Since a is proportional to the cube root of the molar volume,it is reasonable to estimate a at 3.6 and 5.4 MPa usingleast square fitting and using the data of pressure molarvolume relation obtained by Edwards and Pandorf .The estimated impuriton group velocity v g , the Cori-olis force F Cor and the radius of the impuriton circularmotion R C at Ω = 4 rad/s are summarized in Table Itogether with the physical parameters used in the esti-mations. Note that the Coriolis force is a few ordersof magnitude larger than gravitational and centrifugalforces ( ∼ − ∼− N). The consequent circular motionof impuritons has a radius R C about 1 and 10 µ m at P = 5 . He impuritons will becomposed of a circular motion in r − θ plane and a linearmotion in z direction. If the radius of circular motion R c is much smaller than the dislocation segment length L N , the probability of He encountering to the dislocation G will be suppressed. Consequently, the shear modulus willdecrease at low temperatures. P V m a f (kHz) J (Joule) v g (m/s) F Cor (N) R C (m)(MPa) (cc/mole) (nm) min max min max min max min max2.50 21.17 0.3677 86 100 5 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE I. Parameters and estimates of some quantities for three pressures: Pressure P , molar volume V m , lattice constant a . Exchange frequency f and bandwidth J are obtained from Greenberg et al . . Group velocity v g , the Coriolis force F Cor , and radius of circular motion R C are derived from J and a . F Cor and R C are the values at Ω = 4 rad/s. The minimaand maxima for some quantities are anticipated from the f data of Greenberg et al ., taking the scatters of the data intoaccount. Note that F Cor is same for the maximum and minimum of v g . Two experimental situations should be further consid-ered in this interpretation: (1) The suppression of µ isobserved only when the applied shear strain exceeds thecritical strain. (2) In any real crystals, there is a broaddistribution in the segment length L N and the length be-tween impurity pinning points L i . We discuss the rota-tion effect involving these facts, based on the dislocationpinning model extended by Iwasa , Fefferman et al . and Kang. et al . .When a stress σ is applied to crystal, dislocationstrings feel a force. The force f which needs to depina He from dislocation has a critical value f c = bσL c / b is the Burgers vector and L c is a critical seg-ment length which separates pinned and depinned dis-locations: i.e. dislocation segments shorter than L c arepinned, while those longer than L c are depinned. L c istherefore given by L c = 2 f c /bσ, (13)and is inversely proportional to applied stress σ . Forsimplicity we assume that L c is constant throughout onesolid sample, although there might be inhomogeneity inits magnitude caused by distribution in the magnitudeof activation energy E A . L c is obtained by using threeshear stresses obtained in the measurement. As to thecritical force, we adopt f c = 6 . × − N, which wasobtained by Fefferman et al . . This value is close toother estimation f c = 1 × − N by Iwasa et al . . Weassume b = a , a is shown in Table I. In Fig. 6 (b), weplot the measured shear modulus µ = I/f V (20mK) as afunction of L c .Once a solid sample is formed, L N has a fixed distribu-tion. It is appropriate to assume a lognormal distributionfor L N50,51 . According to Kang et al ., the distribution of L N is given by N ( L N ) dL N = Z exp " − (ln L N − ln = L N ) s dL N , (14)where s is the width of the distribution. Z and = L N are Z = Λ √ πsL N A exp (cid:16) s (cid:17) and = L N = L N A exp (cid:16) − s (cid:17) ,respectively. On the other hand, an exponential distri-bution is assumed for N ( L i ), N ( L i ) dL i = Λ d L i A exp (cid:18) − L i L i A (cid:19) dL i , (15)where we take thermal activation formula Eq. 7 withactivation enerby E A = 0 . N ( L i ) at50, 100 and 300 mK. Note that the choice of thevalue of E A , in which various authors claim differentvalues (0.2 - 0.7 K), does not give any influence toour discussion. In both formula, N ( L N(or i) ) dL N(or i) isthe number of dislocation segments having length be-tween L N(or i) and L N(or i) + dL N(or i) , and the integral R ∞ L N(or i) N (cid:0) L N(or i) (cid:1) dL N(or i) gives the total length ofdislocations per unit volume Λ. N ( L N ), and N ( L i ) for50, 100 and 300 mK are shown in Fig. 6 (b). In thecalculation we take the average L N A = 1 . × − m.Let us first consider the virtual case that neither shearnor rotation is applied. In this case the pinning of Heto dislocations are controlled only by temperature. InFig. 6, as temperature decreases, the distribution N ( L i )shifts to shorter L i (left-handed) side, while N ( L N ) is un-changed. This means that shorter dislocation segmentsare pinned by He at lower T . This is exactly what hap-pens when solid He stiffens.Next, we think of the case that some shear strain isapplied. N ( L N ) (red curved line shown in Fig. 6(b)) hasa peak at L N ∼ × − m, and decreases logarithmicallyas L N increases. When applied shear is weak and hence0 L c is large, e.g. L c > . × − m, there are very few dis-location segments which are depinned by applied shear.This is quite consistent with our observation that µ wasnot suppressed at shear strain ǫ ≤ . × − , which cor-responds to four data points of I/f V at L c ≥ × − m of Fig. 6(b).At the highest shear strain ǫ = 3 . × − , L c is lo-cated at 5 × − m, which is indicated by dashed lineacross Fig. 6(a) and (b). In this case, the dislocationshaving lengths L N > L c , which are shown by orange col-ored region, are all depinned from He by applied shear,even if the distribution N ( L i ) went to short L i side. Thisdepinning occuring in the long segments will result in thesuppression of shear modulus, which is actually observedin measurement.Let us consider the effect of rotation. We show inFig. 6(b) the radius of circular motion R C at Ω = 4rad/s. The ambiguity in the estimation of R C is ex-pressed by colored area (green and blue for 3.6 and 5.4MPa, respectively). R C is approximately and of L c at ǫ = 3 . × − (the dashed line) for pressure 3.6and 5.4 MPa, respectively. The probability of pinning of He to dislocation segments with lengths L N ≤ L c willbe suppressed by the spiral motion of impuritons. It istherefore expected that shear modulus is suppressed byrotation, even in the case that dislocation segments hasa distribution in lengths.In this explanation, the necessary condition for thesuppression of pinning probability G at a certain tem-perature T is that R C is much smaller than dislocationsegment length which mainly contribute to shear mod-ulus at T . Once the distribution N ( L N ) is fixed in aparticular solid sample, the magnitude relationship be-tween R C and L N is also fixed. If R C ≪ L N holds formost of the dislocation segments (as in the case of Fig.6(b)), the suppression of µ will occur irrespective of ap-plied shear stress. Nevertheless, why is the rotation effectis seen only at ǫ ≥ ǫ c ? We consider that the suppres-sion of µ exists also in the data of smaller shear strains, ǫ < − . In the µ data of the 5.4 MPa sample, µ atΩ = 4 rad/s is slightly smaller than µ at rest in thetemperature range between 0.1 and 0.2 K. It is currentlydifficult to conclude whether this decrease in µ is thetrue rotation effect or not, because the change is close tothe experimental resolution of shear modulus. However,we have observed similar changes in a floating-core TOmentioned in Sec. II, in which the TO resonant frequencydecreases accompanied with additional dissipation in thetemperature range between 50 and 300 mK, dependingon the TO amplitude, which is roughly proportional toapplied shear strain. The floating-core TO is so sensitiveto the change in shear modulus of solid He contained be-tween the inner core and outer shell that the rotationeffect in shear modulus could be detected even if it weresmall. The TO data are now under analysis and will bepublished elsewhere.Our interpretation for the rotation effect can be rein-forced by considering directional anisotropy in disloca- tion segments that contribute to shear modulus. Figure7 shows a cartoon of a part of dislocation network andcircular impuriton motion, which are viewed from top ofthe PZT pair, i.e. in the direction of rotation axis z k Ω.As explained in Sec. II, edge dislocation lines parallel to z mostly contribute to µ when shear is applied to the θ direction. For simplicity, four such dislocation lines areshown as ⊥ separated by distance L N : i.e. four lines areconnected by other dislocation segments running in thedirections of r and θ , which are not shown in the car-toon. It is easily seen that, when R c ≪ L N , impuritonsdo not encounter the dislocations. On the other hand,impuritons moving (spirally) in z direction can still betrapped to dislocation lines lying in r − θ plane. How-ever, pinnings of dislocation lines in r − θ plane do notcontribute significantly to the change in c measured byPZT driven in θ direction. Therefore, the z -directed im-puriton motion will not influence the rotation effect.As easily understood in Fig. 7, impuritons rotatingwith R c close to L N will be eventually captured by dis-location lines in z direction. The rotation effect willtherefore decline as molar volume (pressure) of solid Heincreases (decreases), and will finally disappear as thepressure approaches melting point, where R c exceeds1 × − m, within which the lengths L N of most of thedislocation segments lie. Such a molar volume depen-dence may also explain the abrupt changes in µ we haveobserved near the end of each cooling scan of the 3.6 MPasample . VI. DISCUSSION
Our observation of the suppression of shear modulus bysteady rotation can be consistently related to the resultsof TO experiments under rotation. The interpretationbased on circular motion of He impuritons is also con-sistent with a theory of defecton quasiparticles in rotat-ing quantum crystal. Here we compare our results withthese previous experiments and theories, and commenton some unsolved problems.
A. Absence of rotation effect on Young’s modulus
We have not observed significant change in Young’smodulus under rotation. Within the consideration ofelasticity of continuous media, the absence of the rota-tion effect in Young’s modulus is puzzling, since shearand Young’s moduli are mutually related by Poisson’sratio. The absence of rotation effect might arise frominhomogeneity in displacements occuring in compression(expansion) experiment. Our solid helium sample in mea-surement is long in azimuthal ( θ ) direction while it is thinin between two PZTs. Moreover, the solid near the edgeof PZT pair cannot freely expand because the solid existsalso outside the sample between the PZTs, and becausetop and bottom of the solid sample is enclosed by BeCu1 L N R C L C (a) Pinned Depinned by shear L i S hea r M odu l u s ( p A / H z V ) -7 -6 -5 -4 -3 -2 (3.6 MPa) (5.4 MPa) N ( L N ) lognormal N ( L i ) 50 mK N ( L i ) 100 mK N ( L i ) 300 mK μμ R c(5.4) R c(3.6) L N , L i and L C (m) N ( L N o r L i ) ( m - ) FIG. 6. (Color online) (a) Schematic illustration of distribu-tion in L N , which corresponds to the abscissa of graph (b). L i and R c are also shown. (b) Number density of disloca-tion lengths L N and L i , denoted as N ( L N ) and N ( L i ), as afunction of L N and L i (for temperatures at 50, 100, and 300mK), respectively. These are anticipated by Eqs. (14) and(15). Measured shear modulus µ for the 3.6 and 5.4 MPasamples is shown as a function of L c , which is given by Eq.(13). We focus on the location of µ at L c = 5 × − m( ǫ = 3 . × − ). The location is indicated by dashed line.The dislocation segment between nodes with lengths L N > L c is depinned from He by applied shear stress. This is indicatedas orange colored area of N ( L N ). R c shown by colored (greenand turquoise) area and double-headed arrows, which indicateuncertainty (see text), is much smaller than L c . Therefore,the pinning of He to dislocation segments at lengths L N ≤ L c will be suppressed, resulting in the decrement in µ . wall. In this situation, compression of solid in r direc-tion will not produce much expansion in both θ and z directions. As a result the response detacted by one ofthe compression PZTs may only be determined by purecompression or expansion in r direction but will not con-tain the effect of shear motion in θ direction.Simulation such as FEM will help to elucidate the va-lidity of this speculation. B. Comparison with rotating torsional oscillatorexperiments
The suppression of shear modulus by rotation ob-served in the present work has a resemblance to the re-sults of rotating TO experiments performed by variousgroups . We propose that the most of the TO be-haviors induced by steady rotation are essentially caused R C L N L N Ω r θ R C FIG. 7. (Color online) Schematic view of dislocation networkand circular impuriton motion viewed in the direction of rota-tion axis z k Ω. As discussed in Sec. II, edge dislocation linesparallel to z mostly contribute to µ when shear is applied to θ direction. For simplicity, four dislocation lines are shownas ⊥ keeping distance L N . When R c ≪ L N , impuritons donot encounter the dislocations, and sticking such impuritonsto dislocation lines in r and θ directions (not shown) do notcontribute significantly to the change in c . Impuritons ro-tating with R c ∼ L N will be captured by the z - dislocations,so the rotation effect will disappear as molar volume of solid He increases. by suppression of shear modulus.Yagi et al . employed a TO containing a disk-shapesolid He (formed from a commercial He gas, solid pres-sure not exactly known) and rotated it up to 1.26 rad/s(0.2 rps) with TO driving velocity up to 400 µ m/s .The shift of resonant frequency, which is now attributedto stiffening of solid He inside the TO bob, decreasedas the rotation was applied below 70 mK, although theauthors were not aware of the decrement. The decrementbecame more prominent as Ω increased, and as the os-cillation velocity of TO, v rim (velocity of rim of the solidsample) increased: at v rim = 400 µ m/s, which was theirhighest velocity, the frequency shift caused by stiffeningof He is suppressed by approximately 10 % of from theoriginal magnitude under no rotation.Jaewon Choi et al . studied the effect of rotation for aTO which has two annular tubes for solid He samples,and which has two resonant frequencies . When rotationwas applied, decrement of period shift was observed inlower resonant mode below 80 mK. The decrement wasroughly proportional to Ω and was about 15 % whenrotation Ω = 4 rad/s was applied. The high resonantmode had no significant rotation effect.The rotation effects observed by these two TOs areattributed to the change in shear modulus. As mentionedin Sec. II, the change in TO period are attributed to thechange in shear modulus of solid He inside the TO bob.Given that the low - T period shift from the period athigh T (e.g. 500 mK) is entirely originated from stiffeningof solid He, the period shift ∆ P ( T ) can be considered tobe proportional to the change in shear modulus ∆ µ ( T ).2The magnitude of the rotation - induced decrement of µ isclose to our result. This suggests that the distributionsof dislocation network length L N are identical in thesetwo TOs and in our shear modulus apparatus.In comparison of TO experiments with shear modu-lus measurements, the estimation of magnitude of shearstrain or stress produced by oscillation of torsion bobcould be a problem: A large discrepancy has been pointedout in the nonlinear behaviors of ∆ µ ( ǫ ) in direct shearmeasurements and ∆ P ( ǫ ) in TO studies. This might becaused by inaccuracy in the estimation of ǫ produced byTO, which is described in Appendix. In particular, themagnitude of the critical pinning length L c is not clearin TO.Another TO experiment by Fear et al . found no signif-icant effect in ∆ P at 28 mK when cryostat was rotatedup to 2 rad/s . This was probably due to the smallmagnitude of ∆ P ( T ) in their TO design, which madethe observation of even smaller rotation effect difficult.Tsuiki et al . have recently carried out a rotation exper-iment using a floating-core TO . This TO has a numberof advantages for studying shear stiffening of solid He:It is very sensitive to the change in shear modulus ofsolid He between inner metal core and outer shell, whichcan oscillate in phase and out of phase, and can produceextremely large frequency shift, which is about 0.3 Hzat low resonant mode ( f ∼
850 Hz) and about 50 Hzat higher modes ( f ∼ He with highest sensitivity we haveever had. In this floating-core TO, Tsuiki et al . have ob-served rotation - induced suppression of frequency shift attwo resonant frequencies (about 850 and 6300 Hz), butthe suppression most prominently occured at tempera-tures between 50 and 200 mK (depending on applied TOdrive), and the suppression below 50 mK was not as largeas the case of other TOs. We suppose that this differ-ence are attributed to the difference in the distributionof L N . In the floating-core TO L N might have broaderdistribution than other TOs and our present apparatus,and the distribution might incline to long L N side. Sucha distribution can shift the occurence of suppression of He pinning to high temperature side.These TO results are consistent with each other andwith our present work. However, they do not repro-duce the result of Choi et al . : In Choi’s experiment,when rotation Ω = 4 rad/s was applied, the decrementof ∆ P ( T ) reached 42 % of ∆ P ( T ) at Ω = 0. It is stilldifficult to explain such a large rotation effect by simplyapplying the suppression of shear modulus caused by cir-cular motion of He impuritons. Similar large rotationeffect was observed in a TO containing ring - shapedporous Vycor glass (8 mm ring radius, 2 mm thick inradial direction, and 12 mm in height)(Tsuiki et al. , inpreparation ). In this TO, narrow space of about 30 µ mthick was probably formed between the Vycor ring andthe metal outer wall, due to the difference of thermal con-traction in Vycor glass and BeCu metal, and solid He confined in the narrow space produced large period shift(note that solid He in Vycor nanopores does not showstiffening because of absence of dislocations in solid Heconfined in nanopores). This large period shift showeda large decrement by rotation below 300 mK. Surpris-ingly, the decrement at Ω = 4 rad/s reached 60 % of theoriginal period shift.We speculate that these large rotation - induced decre-ments are originated from solid He in narrow spaces suchas cracks or crevices in the torsion bob. Such narrowspaces, for example, could be unintentionally formed inepoxy resin (Stycast etc.) which glues metal parts. Ithas been proposed in several experiments that stiffeningof solid He existing inside such cracks gives nonnegli-gible contribution to the period shift of TO . In par-ticular, if layers of thin solid He with thickness largerthan 10 µ m (larger than L N A ) are located closely to theplace of torsion rod, their stiffening by He - disloca-tion pinning mechanism produces a large period shift atlow temperatures . If dislocations are formed morein narrow spaces than in larger (i.e. bulk) open spaces,enough He atoms necessary for completely pin the dis-locations may not be provided from larger spaces whenTO is rotated, because the diffusion of He is stronglydisturbed by the rotation - induced Coriolis force. Thisscenario of nonequilibrium He spatial distribution mayexplain the large rotation effect of Choi et al . and Tsuiki et al ., in which case was not taken for unintentional for-mation of narrow spaces.
C. Theory of rotation effect in quantum solids
As mentioned in Sec. II, Pushkarov theoretically stud-ied the effects of rotation on quantum properties of solid He . He proposed a Fokker - Planck type equa-tion for defectons in solid, and derived the temperaturedependence of diffusion coefficient tensor in the temper-ature regime that phonon scattering deterimines the de-fecton diffusion. He obtained a surprising conclusionthat diffusion coefficient D in the directions perpendic-ular to the rotation axis change from T − at rest andat small Ω to T at Ω ∼ rad/s, while D in z di-rection is unchanged at all. This change in the power-law exponent of D r,θ makes defecton diffusion extremelyanisotropic, and results in the same effect as we propose.In Pusukarov’s theory the effect of rotation is introducedby a term proportional to Ω in collision integral, which isa consequence of Coriolis force acting on moving defec-tons. Since Pushkarov focused on zero - point vacancies(vacancion) in solid He, in which the effective mass isnearly the same order of magnitude as bare He mass, theapplication of the theory to the case of He impuritonswith large effective mass was not explicitly discussed. Inorder to apply this theory to our experiment, the theoryshould be specified to the case of He impurities and beextended to the temperature regime where the effect ofphonon scattering is negligible.3We point out that future theories of quasiparticle dy-namics taking into account the features of He impuri-tons (narrow bandwidth and heavy effective mass) andinteraction between dislocations and impuritons will de-scribe quantitatively the rotation effect on dislocation - He pinning.
VII. CONCLUSIONS
We have examined the effect of steady rotation on theelastic properties of solid He, using specially designedquarter circle PZTs and a rotating dilution refrigerator.At low strains, no significant rotation effect was observedin shear modulus µ ( T ), except for a slight decrease inshear modulus around 200 mK in the sample of pres-sure P = 5 . µ decreases because the shear strain unbinds dislocationsfrom He impurities, µ is suppressed by rotation below80 mK when sample is rotated with angular velocity upto Ω = 4 rad/s. The decrement of µ at Ω = 4 rad/s isabout 14.7(12.3) % of the total change of µ ( T ) from 15 to500 mK, for solid samples of pressure 3.6(5.4) MPa. Inorder to explain the decrement, the probability of pinningof He on dislocation segment, G , must decrease aboutthree orders of magnitude. Such a decrease in G can berealized by the anisotropy of the motion (diffusion) of He impuritons by the Coriolis force. We propose thatthe radius of the spiral motion of He impuritons, R c , canbe much smaller than the length of dislocation segment L N . This will result in a decrease in pinning probabil-ity G for dislocation lines aligned parallel to the rotationaxis.Our experiment and interpretation may solve the con-troversy in the results of several rotating torsional oscil-lator experiments. Detailed analyses and discussion onseveral TO results including an analysis of the presentshear modulus result by considering the distribution in L N will be given in our future publication .The microscopic mechanism of the rotation effect on He pinning is to be further investigated. Concerning tothis problem, the dynamics of He impuritons is an inter-esting issue to pursuit. Since the Coriolis force acting on He impuritons has a close analogy to the Lorentz forceexerted to charged quantum particles , one may expectquantum phenomena as in rotating cold atoms and inelectrons in magnetic field. ACKNOWLEDGMENTS
We gratefully acknowledge fruitful discussions withIzumi Iwasa, Jaewon Choi and Eunseong Kim. D.T. ac-knowledges a financial support from Takahashi Industrialand Economic Research Foundation. This work is sup-ported by Grant-in-Aid for Scientific Research (S) (Grant No. 21224010) from JSPS and by RIKEN junior Re-search Associate Program.
Appendix A: Evaluation of rotation effect (i)Another consideration of the decrement ofshear modulus
In the evaluation of the rotation - induced decrement ofthe shear modulus described in the main text, we assumethat Ω dependence of µ at high T appears due to the driftof measurement. However, it is also possible to assumethat the change in µ at high T is a real rotation effect.In this case, it is not the absolute value of µ but the ratioof changes in µ from high to low T that should be takeninto consideration. In this regard, we define a reducedchange of shear modulus δ µ as follows: δ µ = µ ave (20 mK) − µ ave (300 mK) µ ave (300 mK) (A1)where µ ave (20 mK) and µ ave (300 mK) are defined in themain text and shown as open squares and circles in Fig.3 of the main text, respectively.In Table II, values of µ ave are summarized. Here weconsider shear modulus at Ω = 0 and 4 rad/s. The δ µ at 0 rad/s is evaluated to be 0.2245; i.e. 22.45 %. Ifthere were no rotation effect in δ µ , δ µ at 4 rad/s shouldbe also 0.2245. In reality, however, the results shown inTable II show that δ µ at 4 rad/s is 0.1976 (19.76 %),which is smaller than 0.2245. Therefore, at Ω = 4 rad/s δ µ decreases by 11.94 %. (ii)Uncertainty in shear modulus Here we confirm the rotation effect on shear modulusby discussing uncertainty in shear modulus under rota-tion. We discuss the 3.6 MPa solid sample. In Fig. 8, weshow µ which is averaged at 20 mK for 30 minutes. Stan-dard deviations are indicated by error bars. As shown inFig. 8, at strains ǫ ≤ . × − , averaged shear modulusare on the horizontal line within error bar. This indicatesthat µ at ǫ ≤ . × − does not depend on Ω. On theother hand, µ at ǫ = 3 . × − decreases far over theerror bars as Ω increases. The decrease in µ at Ω = 4rad/s is an order of magnitude larger than the standarddeviation. Thus, the suppression of shear modulus is notexperimental artifact but a significant rotation effect. TABLE II. Shear modulus values of 3.6 MPa sample at lowand high T , with and without rotation (unit : pA/HzV)0 rad/s 4 rad/s µ ave (20 mK) 0.09415 0.08982 µ ave (300 mK) 0.07689 0.07500 (cid:1) -9 (cid:1) -9 (cid:1) -8 S hea r m odu l u s ( p A / H z V ) Omega (rad/s)
FIG. 8. (Color online) The average shear modulus µ withstandard deviation as error bar for each Ω taken from Fig. 3in the main paper. Lines are guide to eye. Appendix B: Estimation of stress for annular solid He in a torsional oscillator
In the previous TO experiment by Choi et al . , sup-pression of TO period change by rotation was observedwhen they fixed AC oscillation speed at 6 µ m/s (at thecircumference of the annular sample). By using this oscil-lation speed and other relevant values, the stress appliedto solid He is calculated to be ∼ . × − Pa. Thedetails of the calculation is shown below.Considering the torque acting on solid He by the os-cillation of the TO cell, we obtain the stress at sample asfollows: σ = ρtωv ρ is the density of solid He (for 4.0 MPa sample, ρ ≈
160 kg/m ), t the thickness of the annular solid sam-ple (0.4 mm), ω the resonant frequency (2 π ×
911 rad/s),and v the oscillation velocity of the sample. Substitut-ing v = 6 µ m/s to Eq. (B1), σ is calculated to be about1 . × − Pa, which is two orders magnitude smaller than σ generated by the critical strain applied under presentrotation experiment. A. F. Andreev, in
Progress in Low Temperature Physics ,Vol. VIII, edited by D. F. Brewer (North-Holland, Ams-terdam, 1982) pp. 67–131. D. I. Pushkarov,
Quasiparticle Theory of Defects in Solids (World Scientific, Singapore, 1991). C. M. Varma and N. R. Werthamer, in
The Physics ofLiquid and Solid Helium , Part I (Wiley, New York, 1976)pp. 503–570. R. A. Guyer and L. I. Zane, Phys. Rev. Lett. , 660(1970). R. A. Guyer, R. C. Richardson, and L. I. Zane, Rev. Mod.Phys. , 532 (1971). M. G. Richards, J. Pope, and A. Widom, Phys. Rev. Lett. , 708 (1972). W. Huang, H. A. Goldberg, and R. A. Guyer, Phys. Rev.B , 3374 (1975). B. N. Esel’Son, V. A. Mikheev, V. N. Grigor’ev, and N. P.Mikhin, Sov. Phys. JETP , 2311 (1978). A. Andreev and I. Lifshits, Sov. Phys. JETP , 2057(1969). G. V. Chester, Phys. Rev. A , 256 (1970). A. J. Leggett, Phys. Rev. Lett. , 1543 (1970). E. Kim and M. H. W. Chan, Nature , 225 (2004). E. Kim and M. H. W. Chan, Science , 1941 (2004). J. Ny´eki, A. Phillis, A. Ho, D. Lee, P. Coleman, J. Parpia,B. Cowan, and J. Saunders, Nature Physics , 455(2017). J. Day and J. Beamish, Nature (London) , 853 (2007). J. Day, O. Syshchenko, and J. Beamish, Phys. Rev. B ,214524 (2009). J. Day, O. Syshchenko, and J. Beamish, Phys. Rev. Lett. , 075302 (2010). J. R. Beamish, A. D. Fefferman, A. Haziot, X. Rojas, and S. Balibar, Phys. Rev. B , 180501 (2012). A. Haziot, A. D. Fefferman, J. R. Beamish, and S. Balibar,Phys. Rev. B , 060509 (2013). I. Iwasa, J. Low Temp. Phys. , 30 (2013). H. Choi, D. Takahashi, K. Kono, and E. Kim, Science , 1512 (2010). H. Choi, D. Takahashi, W. Choi, K. Kono, and E. Kim,Phys. Rev. Lett. , 105302 (2012). W. Choi, D. Takahashi, D. Y. Kim, H. Choi, K. Kono, andE. Kim, Phys. Rev. B , 174505 (2012). H. Suzuki, J. Phys. Soc. Jpn. , 1472 (1973). R. Wanner, I. Iwasa, and S. Wales, Solid State Comm. , 853 (1976). I. Iwasa and H. Suzuki, J. Phys. Soc. Jpn. , 1722 (1980). I. Iwasa, H. Suzuki, T. Nakajima, S. Suzuki, M. Ando,I. Yonenaga, M. Takebe, and K. Sumino, J. Phys. Soc.Jpn. , 4225 (1987). A. V. Granato and K. L¨ucke, J. Appl. Phys. , 583 (1956). C. Zhou, J.-j. Su, M. J. Graf, C. Reichhardt, A. V. Bal-atsky, and I. J. Beyerlein, Phil. Mag. Lett. , 608 (2012). A. Haziot, X. Rojas, A. D. Fefferman, J. R. Beamish, andS. Balibar, Phys. Rev. Lett. , 035301 (2013). A. Fefferman, F. Souris, A. Haziot, J. Beamish, and S. Bal-ibar, Phys. Rev. B , 014105 (2014). A. S. Greenberg, W. C. Thomlinson, and R. C. Richard-son, Phys. Rev. Lett. , 179 (1971). E. Kim and M. H. W. Chan, Phys. Rev. Lett. , 115302(2006). D. I. Pushkarov, Phys. Rep. , 411 (2001). D. I. Pushkarov, Europhys. Lett. , 56002 (2012). M. Yagi, A. Kitamura, N. Shimizu, Y. Yasuta, andM. Kubota, J. Low Temp. Phys. , 492 (2011). M. J. Fear, P. M. Walmsley, D. E. Zmeev, J. T. M¨akinen, and A. I. Golov, J. Low Temp. Phys. , 106 (2016). J. Choi, T. Tsuiki, D. Takahashi, H. Choi, K. Kono, K. Shi-rahama, and E. Kim, arXiv:1701.07190 (2017). J. Choi, J. Shin, and E. Kim, Phys. Rev. B , 144505(2015). T. Tsuiki, D. Takahashi, Y. Okuda, K. Kono, and K. Shi-rahama, (in preparation). J. D. Reppy, X. Mi, A. Justin, and E. J. Mueller, J. LowTemp. Phys. , 175 (2012). D. Takahashi and K. Kono, in
AIP Conference Proceedings ,Vol. 850 (AIP, 2006) pp. 1567–1568. D. Y. Kim, H. Choi, W. Choi, S. Kwon, E. Kim, and H. C.Kim, Phys. Rev. B , 052503 (2011). I. Iwasa, (private communication). C. Huan, S. S. Kim, D. Candela, and N. S. Sullivan, J.Low Temp. Phys. , 354 (2016). A. S. Greenberg, W. C. Thomlinson, and R. C. Richard-son, J. Low Temp. Phys. , 3 (1972). R. S. Shah,
X-ray and pressure measurements of heliumsolids , Ph.D. thesis, Univ. Illinois at Urbana-Champaign(1999). D. O. Edwards and R. C. Pandorf, Phys. Rev. , A816(1965). E. S. Kang, H. Yoon, and E. Kim, J. Phys. Soc. Jpn. ,034602 (2015). I. Iwasa, K. Araki, and H. Suzuki, J. Phys. Soc. Jpn. ,1119 (1979). D. Y. Kim and M. H. W. Chan, Phys. Rev. B , 064503(2014). H. J. Maris, Phys. Rev. B , 020502 (2012). D. Y. Kim and M. H. W. Chan, Phys. Rev. Lett. ,155301 (2012). G. Dattoli and M. Quattromini, arXiv:1009.3788 (2010). N. R. Cooper, Adv. Phys.57