J Low Temp Phys (2012) 168:194–220 DOI 10.1007/s10909-012-0621-8
Elastic Properties of Solid 4 He John Beamish
Received: 20 December 2011 / Accepted: 20 March 2012 / Published online: 31 March 2012 © Springer Science+Business Media, LLC 2012
Abstract The shear modulus of solid 4 He increases below 200 mK, with the same dependence on temperature, amplitude and 3 He concentration as the frequency changes recently seen in torsional oscillator (TO) experiments. These have been interpreted as mass decoupling in a supersolid but the shear modulus behavior has a natural explanation in terms of dislocations. This paper summarizes early ultrasonic and elastic experiments which established the basic properties of dislocations in solid helium. It then describes the results of our experiments on the low temperature shear modulus of solid helium. The modulus changes can be explained in terms of dislocations which are mobile above 200 mK but are pinned by 3 He impurities at low temperature. The changes we observe when we anneal or stress our crystals confirm that defects are involved. They also make it clear that the shear modulus measured at the lowest temperatures is the intrinsic value—it is the high temperature modulus which is reduced by defects. By measuring the shear modulus at different frequencies, we show that the amplitude dependence depends on stress in the crystal, rather than reflecting a superfluid-like critical velocity. The shear modulus changes shift to lower temperatures as the frequency decreases, showing that they arise from a crossover in a thermally activated relaxation process rather than from a true phase transition. The activation energy for this process is about 0.7 K but a wide distribution of energies is needed to fit the broad crossover. Although the shear modulus behavior can be explained in terms of dislocations, it is clearly related to the TO behavior. However, we made measurements on hcp 3 He which show essentially the same modulus stiffening but there is no corresponding TO anomaly. This implies that the TO frequency changes are not simply due to mechanical stiffening of the oscillator—they only occur in the Bose solid. We conclude by pointing out some of the open questions involving the elastic and TO behavior of solid helium. J. Beamish () Department of Physics, University of Alberta, Edmonton, Canada e-mail:
[email protected]
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Keywords Supersolidity · Solid helium · Elasticity · Dislocations · Acoustics · Elastic constants · Defects · Plasticity
1 Introduction Helium’s small mass and weak interatomic forces make it a uniquely quantum system. It remains liquid at low pressures and the Bose isotope, 4 He, becomes a superfluid at temperatures below 2.17 K. Although 4 He can be crystallized by applying pressures above 25 bar, quantum effects remain important in the solid. For example, vacancies are delocalized and could have finite concentrations at zero temperature. It was suggested [1, 2] more than 40 years ago that such “zero point vacancies” would Bose condense, producing a solid in which superflow coexists with crystalline order. Zero point vacancies have not been seen but “supersolidity” is still possible— the most spectacular manifestation of the quantum nature of solid helium. Despite a number of searches, direct evidence of supersolidity was not found until 2004, when torsional oscillator (TO) experiments [3] with solid 4 He showed frequency increases below 200 mK. These were interpreted as mass decoupling—the “non-classical rotational inertia” (NCRI) which characterizes supersolidity [4]. It soon became clear that the TO NCRI was changed by annealing [5] and that the onset temperature was extremely sensitive to 3 He impurities [6] suggesting that defects were involved in the anomalous behavior. Shear stiffness is a fundamental property which distinguishes solids from liquids and allows transverse sound waves to propagate. In helium, zero point motion greatly reduces the solid’s elastic moduli but sound propagates normally. Elastic constants are usually determined from sound speeds but this assumes that the density is constant, which would not be true in a supersolid if some of the mass decouples from the lattice [7]. Elastic measurements also provide a sensitive probe of defects, especially of dislocations since these can move in response to shear stresses and soften the shear modulus. There have been many measurements of solid helium’s elastic moduli but most of these were indirect (from sound speeds), were made at high frequencies (∼10–50 MHz) and at temperatures well above the TO anomaly. We have directly measured the shear modulus of solid helium in the low frequency and low temperature regime where the TO anomaly was observed. We found [8] a large and unexpected shear stiffening below 200 mK, with the same dependence on temperature, amplitude, frequency and 3 He impurity concentration as the TO NCRI. The two anomalies are clearly related but it is not obvious exactly how. Both could be signatures of a new state, perhaps supersolid, but it is also possible that the TO behavior is caused by the shear modulus increase, which could mimic mass decoupling by stiffening the TO and raising its frequency. In this paper I describe the elastic behavior of solid helium at low temperatures, beginning with a brief review of earlier ultrasonic work which established the elastic constants and the basic properties of dislocations in solid 4 He. The main results of our shear modulus experiments are then summarized and I show how they can be understood in terms of dislocations which are mobile at high temperatures but pinned by bound 3 He impurities at low temperatures. I include a brief description of recent
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Fig. 1 Elastic constants of hcp 4 He as functions of pressure (from Ref. [10]). Lines are guides to the eye
elastic and acoustic experiments by other groups, which are reviewed in more detail elsewhere in this volume. The paper ends with some comments on open questions and possible new phenomena involving quantum dislocations. 2 Background Unlike liquids, solids resist shear. For example, an isotropic solid (e.g. a glass) has a shear modulus μ as well as a bulk modulus K. Many TO and other experiments on helium involve polycrystalline samples which, if the crystallites are small and randomly oriented, can be described in terms of effective (averaged) shear and bulk moduli [9]. In general, however, crystals are anisotropic and have a larger set of elastic constants Cij which depends on their symmetry. At low temperatures, 4 He forms a hexagonal (hcp) crystal with 5 independent elastic constants (C11 , C33 , C12 , C13 and C44 ). The elastic constants of single crystals of helium have been measured with an accuracy of a few percent using longitudinal and transverse ultrasound [10–13]. Their values at pressures below 65 bar are shown in Fig. 1. These measurements were made at temperatures close to melting. Subsequent ultrasonic measurements showed that the sound speeds increase at lower temperatures. Because measurements on solid helium are made at constant density, these changes are small (they correspond to changes of less than 1 % in the elastic constants). The phonon contribution to a solid’s thermal energy reduces sound speeds by an amount proportional to T4 . This behavior is observed in helium near its melting temperature and in impure crystals [14, 15]. Figure 2 shows the speed of 3 MHz longitudinal sound in a 110 bar 4 He crystal containing 0.14 % 3 He impurities. The solid line is a fit to the data showing the expected T4 variation. In higher purity 4 He crystals, a T4 velocity decrease is seen at high temperatures but frequency-dependent anomalies appear in the sound speed at low temperatures
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Fig. 2 Temperature dependence of the speed of 3 MHz longitudinal ultrasound in a 110 bar 4 He crystal containing 0.14 % 3 He. Solid line is a fit of the form v = vo − aT 4 (data from Ref. [15])
[16, 17]. An example is shown in Fig. 3, for 110 bar crystal containing 0.3 ppm 3 He. The anomalous velocity changes are due to dislocation lines moving in response to the ultrasonic stress wave and provide a useful probe of dislocation properties. Dislocations [18] are line defects, characterized by a lattice mismatch Burgers vector b, which is either perpendicular (edge) or parallel (screw) to the dislocation line. A shear stress in the direction of the Burgers vector exerts a sideways force on the dislocation which can respond by gliding in the plane defined by the line and its Burgers vector. The lattice creates a periodic Peierls barrier to this motion but dislocations can overcome it by creating one dimensional kinks which move along the dislocation more easily. Motion perpendicular to the glide plane (climb) is more difficult since it requires mass transport. It normally involves one dimensional steps perpendicular to the glide direction (jogs) and occurs only at temperatures near melting where thermally activated vacancies are plentiful. Most crystals have many dislocations, created during growth or by subsequent plastic deformation. These form an irregular network, strongly pinned at nodes where they intersect. Large dislocation densities produce more intersections and so reduce the average length of dislocation loops between nodes (the network length LN ). The motion of dislocations creates a strain which adds to the elastic strain and reduces the crystal’s effective shear modulus. The response of dislocations to sound waves has been described in terms of a vibrating string model [19]. A dislocation has an effective string tension ∝ μb2 due to elastic energy stored in its stress field and inertia per unit length ∝ ρb2 from the corresponding strain. In this model the periodic Peierls potential is neglected so dislocations move in response to even small stresses. A moving dislocation experiences a damping force B due to thermal excitations. The dominant contribution comes from phonon scattering [20] via “fluttering”, which gives B ∝ T 3 .
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Fig. 3 Velocity of 3 and 9 MHz longitudinal ultrasound in a 110 bar hcp 4 He single crystal (0.3 ppm 3 He). A background T4 dependence like that in Fig. 2 (obtained by fitting the 9 MHz data between 2 K and melting at 3.6 K) has been subtracted to show the anomaly due to dislocations. The lines are fits to the Granato Lucke “vibrating string” model (described in the text) with the dislocation parameters shown (from Ref. [17])
At low frequencies, the inertia and damping can be neglected and a dislocation responds to an applied stress by bowing out between pinning points, with a displacement proportional to its length L and to the component of the stress along the Burgers vector in its glide plane. This produces a dislocation strain in phase with the stress 2 and a corresponding decrease in the crystal’s shear modulus Δμ μo = −CRL , where μo is the crystal’s intrinsic shear modulus (in the absence of dislocations), R is an orientation factor (of order 0.1) which depends on the angle between the applied stress and the dislocation’s Burgers vector, and C is a constant which depends on the distribution of loop lengths (∼0.1 for a single length and ∼0.5 for an exponential distribution with average length L). The softening is proportional to L2 , so a few long dislocations can have the same effect as many short ones. This means that the dislocation density cannot be determined from low frequency shear modulus measurements alone. The dimensionless combination L2N depends on the geometry of the dislocation network. For example, a regular cubic network has L2N = 3 and so would reduce the shear modulus by about 30 %. Even larger changes are possible if the dislocations are preferentially aligned, e.g. in grain boundaries. Dislocations produce dissipation as well as changing the modulus. At low frequencies their motion is nearly in phase with the applied stress and the associated dissipation is small but in ultrasonic measurements at low temperatures the attenuation due to dislocations can be very large. However, above about half the melting temperature, dislocations are so strongly damped that they have almost no effect on the ultrasonic velocity or attenuation. This means that the elastic constants shown in Fig. 1 (determined from measurements near melting) are the intrinsic values for a
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perfect crystal. At lower temperatures, there is less damping and the moving dislocations reduce the sound speeds. At the lowest temperatures, a dislocation line vibrates like an undamped string. Its resonant frequency depends on its length L as well as its string tension and inertia: V 2 ω(L) = (1) 1−ν L where ν is Poisson’s ratio and V is the speed of transverse sound in the solid, i.e. the resonance occurs when the sound wavelength λ = 2πV /ω is comparable to the loop length L. In hcp 4 He, a resonance frequency f = ω/2π = 10 MHz corresponds to a loop of length L ∼ 10 µm. For sound frequencies below the resonance, dislocations soften the crystal and reduce sound speeds. Above the resonant frequency, they are out of phase with the stress and so stiffen the crystal and raise sound speeds. The parameters which characterize a dislocation network ( and LN ) can be determined by fitting the measured temperature and frequency dependence of the ultrasonic speed and attenuation. In real crystals, dislocation networks are random and a distribution of lengths is expected. Using an exponential distribution of lengths, fits of ultrasonic data [16] gave an average length L ∼ 6 µm and R ∼ 3 × 104 /cm2 for hcp 4 He single crystals grown at pressures near 35 bar. The relevant slip system in hexagonal metals involves edge dislocations gliding in the basal plane and measurements of the orientation dependence of ultrasonic attenuation [21] indicate that the same is true in hcp 4 He. Using a typical value for the orientation factor for this slip system, R = 0.1, gives a dislocation density ∼ 3 × 105 /cm2 . The data shown in Fig. 3 (from a 110 bar single crystal) were fit in the same way and gave a smaller density ∼ 3 × 104 /cm2 and slightly shorter loop length, L = 3.8 µm. However, these values depend on the choice of length distribution. The values of L come from the frequency dependence near the loops’ resonance and are reasonably well determined. For example, in Fig. 3 the dislocations reduce the sound speed at 3 MHz but increase it at 9 MHz, indicating that the dominant loops have a resonance frequency between 3 and 9 MHz. The dislocation densities, however, are much less certain. Short dislocations have little effect on the modulus and the fitted values of R can vary by an order of magnitude when different distributions are assumed. Also, screw dislocations and edge dislocations lying in other directions are probably not mobile and so would not contribute to the sound speeds. These dislocation densities should probably be regarded as lower limits. Much smaller dislocation densities are possible in single crystals grown from the superfluid. Ultrasonic measurements [22] gave densities below 700/cm2 in crystals grown at 0.9 K and growth rate measurements imply that crystals can be grown at very low temperatures (20 mK) with no screw dislocations at all [23]. One set of measurements on 4 He single crystals [21] gave much higher dislocation densities, ∼ 3 × 109 /cm2 , but these were determined less directly, from ultrasonic attenuation in the overdamped regime near melting. The authors assumed a very short loop length L ∼ 0.03 µm due to pinning by thermally excited jogs. However, shear modulus measurements made at 331 Hz using a torsional oscillator [24] did not see evidence of jog pinning but instead were shown to be consistent with the ultrasonic results if very different dislocation parameters were assumed. Crystals grown by the
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blocked capillary method are expected to be polycrystalline and have more defects, but there have not been any ultrasonic measurements from which dislocation densities could be extracted. The effect of dislocations is greatly reduced if they are pinned between network nodes, for example by impurities. An impurity atom with a volume different from atoms in the host crystal is attracted by the strain field around an edge dislocation and bound to it. The binding energy EB can be estimated [14] from elastic constants and molar volumes, giving a value of roughly 0.6 K for a 3 He impurity in hcp 4 He. These atoms provide additional pinning sites at low temperatures and result in an impurity pinning length Li which can be smaller than LN . At high 3 He concentrations the effects of dislocations on ultrasonic waves are completely suppressed, as seen in Fig. 2 for a 4 He crystal containing 0.14 % 3 He. Jogs, which can be created thermally or by plastic deformation, are usually less mobile than kinks and may also act as pinning sites. In either case, the additional pinning reduces L while leaving unchanged and so always stiffens the crystal at low frequencies. Impurity pinning is much weaker than that at network nodes (a force of roughly 10−14 N will tear a dislocation away from a bound 3 He atom [14]) so dislocations can be unpinned by large stresses. This stress-induced breakaway reduces the speed of high amplitude sound waves (and increases their attenuation). The binding energy of a 3 He impurity in hcp 4 He has been estimated [14] from the amplitude dependence of longitudinal ultrasound and of the low frequency shear modulus in a TO experiment [14], giving values EB of 0.3 K and 0.7 K, respectively.
3 Experimental Details In order to understand the origin of the decoupling inferred from TO experiments, we made measurements of the shear modulus of solid 4 He using a direct low frequency technique [8]. Crystals were grown in a narrow gap of thickness D (0.2 or 0.5 mm) between two rigidly mounted piezoelectric shear transducers (PZT 5A, with a piezoelectric coefficient d15 = 1.0 × 10−10 m/V at low temperature). An AC voltage V with frequency f was applied to one transducer and generated a shear displacement Δx = d15 V at its front surface. At frequencies well below the resonances of the transducers (∼500 kHz) and of the helium in the gap (∼800 kHz for D = 0.2 mm) this created a uniform shear strain = Δx/D in the helium and a stress σ = μ on the detecting transducer. This produced a charge q which was measured as a current I = 2πf q, giving the shear modulus as 1 D I σ (2) μ= = 2 V f 2πAd15 where A is the surface area of the transducers. For purely elastic deformations, stress is in phase with the applied strain but in general the stress lags the strain. The phase angle φ between stress and strain is related to the dissipation in the solid by 1/Q = tan φ (≈φ for small dissipation). The amplitude and phase of the current thus give the real and imaginary parts of the helium’s shear modulus. However, the absolute accuracy of the shear modulus measurements was limited to about 10 % by
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the calibration of the piezoelectric coefficient d15 and electronic phase shifts meant we could only measure relative values of 1/Q. In contrast to torsional oscillator measurements, this technique is non-resonant and so measurements can be made over a wide frequency range, from below 1 Hz (limited by noise) up to several kHz (limited by resonances which interfere with the shear modulus signal). In some cases we were able to extend the range of our measurements to 8 kHz by exciting and measuring an acoustic resonance in the solid helium surrounding the transducers. We could make shear modulus measurements at drive voltage as low as 2 mV (corresponding to a strain of 10−9 and a stress of ∼0.02 Pa) and as high as several volts (stresses ∼20 Pa). This is important because the NCRI measured in TO experiments decreases with increasing oscillation amplitude [3]. We grew crystals using the blocked capillary technique, starting with liquid helium at pressures above 50 bar. When freezing occurs at constant volume, the pressure drops by more than 20 bar. This plastically deforms the solid and creates polycrystals with many defects. We grew crystals from both commercial 4 He (containing about 0.3 ppm of 3 He) and from isotopically purified 4 He (with about 1 ppb of 3 He). We also studied 3 He crystals; these were grown from gas with relatively high isotopic purity (1.35 ppm 4 He). The solid helium filled the gap between transducers and the surrounding space in the cell (a much larger volume of ∼25 cm3 , which was reduced to ∼5 cm3 for the 3 He crystals). A 0.1 mm i.d. CuNi capillary was used to introduce 4 He to the cell and was thermally anchored at the fridge’s 4 K flange and 1 K pot and at its heat exchangers and mixing chamber, with a length of about 0.5 m between the anchoring points. Temperatures were measured with a calibrated germanium thermometer, supplemented with a 60 Co nuclear orientation thermometer for temperatures below 50 mK.
4 Results and Discussion Our first measurements [8] revealed a large and unexpected increase in the low frequency shear modulus of solid 4 He in the temperature range below 200 mK. Figure 4 shows the temperature dependence of the shear modulus in crystals at two different pressures (33.3 bar and 29.3 bar). The top (red) and bottom (orange) curves are the 2000 Hz data for the two crystals. Both change by about 10 %, with similar temperature dependences. The upper three curves (red, green and blue) are measurements at different frequencies (2000, 200 and 20 Hz) in the 33.3 bar crystal. The magnitude of the modulus change (8 %) is independent of frequency but its onset shifts to lower temperature as the frequency is reduced. The temperature dependence of the shear modulus closely resembles that of the torsional oscillator frequency shifts which have been interpreted as mass decoupling in a supersolid. Figure 5a compares the 200 Hz shear modulus data from Fig. 4 to the TO NCRI from the original measurements of Kim and Chan [3] (measured at 910 Hz in a 51 bar crystal). The corresponding dissipation in the elastic [25] and TO measurements is shown Fig. 5b. Each has a peak near the temperature where the modulus or NCRI is increasing most rapidly. The remarkable similarities between the shear modulus and TO anomalies suggest that they are closely related. It is natural to ask whether the shear modulus shows
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Fig. 4 Temperature dependence of the shear modulus (in experimental units of pA/Hz) in two hcp 4 He crystals. The top three curves are for a 33.3 bar crystal at measurement frequencies (from top to bottom) of 2000 Hz (red symbols), 200 Hz (green) and 20 Hz (blue). The bottom (orange curve) is for a 29.3 bar crystal at 2000 Hz (data from Ref. [8]) (Color figure online)
other features of the TO behavior, for example a critical velocity or a strong dependence on 3 He impurity concentration. Figure 6 shows the dependence of the shear modulus anomaly on the voltage applied to the driving transducer [25]. The shear stiffening is suppressed at high drive voltages, much as the TO NCRI decreases for large oscillations. Figure 7 shows the amplitude dependence in this crystal at low temperature (48 mK) and at temperatures (700 or 800 mK) well above the anomaly. The shear modulus measured during temperature sweeps (e.g. the open symbols in Figs. 6 and 7) agrees with that measured while reducing the drive voltage at constant temperature (solid symbols in Fig. 7). At 700 and 800 mK there is almost no amplitude dependence but at 48 mK the shear modulus begins to decrease when the drive voltage exceeds 30 mVpp (corresponding to strain ≈ 4 × 10−8 and stress σ ≈ 0.8 Pa). The data in Fig. 6 were taken during cooling from 700 mK at constant amplitudes; the constant temperature data in Fig. 7 were taken by cooling from 800 mK at the highest amplitude and then reducing the drive voltage Vpp . The details of this procedure are relevant because the shear modulus, like the TO NCRI, shows amplitude dependent hysteresis at low temperatures. This is shown in Fig. 8. After cooling to 36 mK at high drive, the modulus increased as the amplitude was reduced (open circles), becoming constant for drive voltages below 30 mVpp . When the amplitude was then raised at 36 mK, the modulus did not decrease until the drive voltage exceeded 1 Vpp . This hysteresis disappeared for temperatures above about 60 mK, as illustrated by the 120 mK data in Fig. 8. The amplitude dependence of the shear modulus also affects acoustic resonance measurements [26] which show nonlinearity, including hysteresis and bistability. It is clear that very low amplitudes must be used in order to ensure that elastic or acoustic measurements are in the linear regime. The shear modulus at low temperatures is also extremely sensitive to the concentration of 3 He impurities, x3 . Figure 9 shows the temperature dependence of the shear
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Fig. 5 (a) Shear modulus at 200 Hz in a 33.3 bar crystal (solid green symbols, left axis) and torsional oscillator NCRI at 910 Hz in a 51 bar crystal (open circles, right axis). (b) Dissipation corresponding to the shear modulus (solid green symbols, left axis) and TO (open circles, right axis) data (data from Refs. [3] and [25]) (Color figure online)
modulus changes Δμ, normalized by the low temperature values Δμ (18 mK), for crystals with 3 He concentrations of 300 ppb (0.3 ppm), 85 ppb and 1 ppb. Reducing x3 from 300 ppb (the nominal concentration in commercial 4 He gas) to 1 ppb shifts the midpoint of the modulus anomaly, T50 , (where 50 % of the stiffening has occurred) from 89 mK to 38 mK. This remarkable sensitivity to x3 is also seen in TO measurements [27]. The open symbols in Fig. 9 are TO NCRI data (also normalized to their low temperature values) for crystals with 300 ppb (red circles) and 1 ppb (blue squares). The effect of changing x3 is essentially the same as for the shear modulus (the differences between the modulus and NCRI curves are within the sample to sample variations in the two measurements). Another important feature of TO measurements is the effect of annealing. This can change the NCRI, sometimes reducing it dramatically [5], and shows that defects are involved. Annealing also affects the shear modulus anomaly [28] as shown in Fig. 10 for an isotopically pure (1 ppb 3 He) 33.4 bar 4 He crystal. The lower (black) curve is
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Fig. 6 Temperature dependence of the shear modulus for different drive amplitudes (measured at 2000 Hz in a 38 bar 4 He crystal). Values have been normalized to the value at low temperature and low amplitude but have not been shifted vertically. The curves correspond to drive voltages ranging from 10 mVpp (top) to 3 Vpp (bottom) (from Ref. [25]) (Color figure online)
Fig. 7 Amplitude dependence of the shear modulus at 2000 Hz (normalized to the low amplitude value at 48 mK) in the 38 bar crystal of Fig. 6. Open symbols are values from temperature sweeps at constant amplitude; closed symbols are from amplitude sweeps at constant temperature (from Ref. [25]) (Color figure online)
the shear modulus measured immediately after growing the crystal. The upper (red) curves shows the corresponding data after the crystal was annealed at 1.7 K (i.e.
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Fig. 8 Hysteresis in the amplitude dependence of the shear modulus (measured at 2000 Hz in the crystal of Figs. 6 and 7). Open symbols are data taken while reducing the drive voltage and solid symbols are data while increasing the drive (from Ref. [25]) (Color figure online)
Fig. 9 Effect of 3 He impurities on the shear modulus (measured at 2000 Hz). Solid symbols are shear modulus changes (normalized to unity) in 4 He crystals with 3 He concentrations of 300 ppb (right curve, red symbols), 85 ppb (center, green) and 1 ppb (left, blue). Open symbols show the analogous temperature dependence of the TO NCRI (also normalized to unity) for a standard 300 ppb 3 He crystal (red squares) and an isotopically pure, 1 ppb 3 He crystal (blue squares) (from Ref. [8]) (Color figure online)
200 mK below its melting temperature) for 11 hours. Annealing reduced the size of the anomaly Δμ/μo from 11.3 to 7.4 %. However, it was the high temperature value of μ which increased—the value of μ at 25 mK was almost unchanged. Similar behavior was seen [26] in crystals grown from standard purity (300 ppb 3 He) gas.
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Fig. 10 Effect of annealing on the shear modulus (normalized to the annealed crystal’s low temperature value μo ) of a 33.4 bar, isotopically pure (1 ppb 3 He) crystal (from Ref. [28]) (Color figure online)
Fig. 11 Effect of applying large low temperature stresses on the shear modulus (normalized to the annealed crystal’s low temperature value μo ) of a 33.4 bar, isotopically pure (1 ppb 3 He) crystal (from Ref. [28]) (Color figure online)
We could also change the shear modulus by applying large stresses at low temperatures [28]. Figure 11 shows the shear modulus of an annealed 29.3 bar crystal (300 ppb 3 He), before and after applying large stresses (∼700 Pa) at the lowest temperature. As in the case of annealing, the low temperature modulus is unaffected but
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the high temperature modulus increases. The changes introduced by stress were reversed by warming above 500 mK. Both annealing (which eliminates some defects) and stressing (which introduces new defects) change the modulus at high temperature but leave the low temperature value unchanged. This suggests that the low temperature value is the intrinsic modulus which would be measured in a defect-free crystal. The modulus anomaly should not be seen as stiffening at low temperatures but rather as defect-induced softening of the crystal’s intrinsic shear modulus at high temperature. The modulus changes we observe are much larger than the intrinsic temperature dependence shown in Fig. 2. Large changes in elastic constants are often associated with phase transitions but there is no evidence from other measurements of the large structural or thermodynamic changes that would be needed to explain modulus effects of this size. The annealing effects seen in both TO and shear modulus measurements indicate that defects are involved. Dislocations are the only type of defect known to produce such large shear modulus changes and they provide a natural explanation of the effects shown in Figs. 4 through 11. At the lowest temperatures, 3 He atoms are bound to dislocations and prevent them from moving in response to stress. The impurity pinning length Li is short so the dislocation softening (which proportional to L2i at low frequencies) is very small and the crystal’s intrinsic shear modulus is observed. As the crystal is warmed, the impurities thermally “evaporate” from the dislocations, increasing the pinning length and allowing the dislocations to move. At high enough temperatures, the only remaining pinning is at the fixed network length LN and so the shear modulus is independent of temperature. The total softening we have observed ranges from about 5 % to 20 %, consistent with the expected values of CRL2 in dislocation networks. The sample to sample variations are presumably due to differences in dislocation densities and loop lengths in different crystals. In single crystals we would also expect a dependence on crystal orientation but this would be averaged out in polycrystalline samples. This dislocation picture also predicts amplitude dependence like that shown in Figs. 6 and 7. Stresses exert forces on dislocations which, if they are large enough, will tear them away from the relatively weak impurity pinning points, even at low temperatures. The stress at which this breakaway occurs depends on the impurity binding energy and on the distance Li between pinning points. Ultrasonic measurements [14] also showed the effects of this breakaway in the form of amplitude dependences for stresses larger than about 4 Pa. This corresponds to a strain of about ≈ 2 × 10−7 , which is close to the midpoint of the amplitude dependence of the shear modulus shown in Fig. 7. For sufficiently large stresses, 3 He impurities are completely ineffective in pinning dislocations and the shear modulus has its lower, unpinned value at all temperatures. This model also provides a qualitative explanation of the amplitude hysteresis shown in Fig. 8. At high temperatures dislocations are unpinned and oscillate with large amplitudes when large AC stresses are applied. If 3 He atoms are not able to attach to rapidly moving dislocations, then they will not be pinned as the crystal is cooled at high stress amplitude. If the drive amplitude is then reduced at low temperatures, dislocations will move more slowly, allowing 3 He atoms to bind and pin them. When the amplitude is subsequently raised, the dislocations are difficult to unpin (since the impurity pinning length Li is small and the
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force on an pinning site is proportional to L−1 i ) and so the breakaway stress is high, leading to the sort of hysteresis shown in Fig. 8. If we identify the shear modulus softening with thermal unbinding of 3 He impurities which allows dislocations to move, we can also understand the dependence on 3 He concentration x shown in Fig. 9. The first impurity to bind to a dislocation has 3 the largest effect on the modulus (since it approximately halves the pinning length LN ) so the midpoint of the softening transition corresponds to a 3 He concentration xD on the dislocation roughly equal to a/LN , where a is the atomic spacing along the dislocation. The concentration xD is enhanced from the bulk 3 He density x3 by EB
a Boltzmann factor (xD = x3 e kB T ) so the most rapid shear modulus changes should occur at the “pinning temperature” TP = −
1 EB . kB ln( LN x3 ) a
(3)
Fitting the midpoints T50 (≈TP ) of the shear modulus curves in Fig. 9 to this expression gives a binding energy EB ≈ 385 mK and a network length LN ≈ 10 µm. The pinning temperatures predicted with these values are 37, 64 and 80 mK (for the 1, 85 and 300 ppb 3 He samples). The values of EB and LN are comparable to those from earlier ultrasonic [14] and TO [24] experiments but, given the uncertainties in 3 He concentrations, our precise values should not be taken too seriously (especially the estimate of LN ). The effects of annealing (Fig. 10) or stressing (Fig. 11) strongly support this explanation of the shear modulus behavior. At the lowest temperatures, dislocations are completely pinned by 3 He impurities. They have no effect on the intrinsic shear modulus and so removing (annealing) or adding (stressing) dislocations should not change the measured modulus. At high temperatures dislocations are not pinned and so do reduce the modulus. However, removing dislocations by annealing will not necessarily reduce the total shear modulus softening (proportional to L2N ), since the network length LN will increase as decreases. In one of the crystals we studied, annealing increased the size of the anomaly, i.e. it further reduced the high temperature shear modulus [26]. In all other crystals, annealing had the effect shown in Fig. 10, increasing the shear modulus at high temperatures and reducing the size of the anomaly. Applying large stresses at low temperature also stiffened the crystal at high temperature, as shown in Fig. 11. This implies that the additional defects introduced by the stresses were more effective in pinning (reducing L) than in increasing the density of dislocations. The new defects also annealed quickly and reproducibly when the crystal was heated above 0.5 K, in contrast to the initial annealing illustrated in Fig. 10, which occurred slowly and only at temperatures close to melting. This suggests that, rather than creating new dislocations, the stresses we applied somehow pinned existing dislocations. One possibility is jogs, which are created when dislocations cut through each other. Jogs cannot glide easily and so can provide additional pinning but they are relatively easily annealed via diffusion of thermal vacancies. If dislocations can be completely pinned, their effects on the shear modulus will be eliminated. We have attempted to do this by confining solid helium in a high porosity (95 %) aerogel [29]. The silica strands which make up the aerogel are about 3 nm
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in diameter and should provide strong pinning sites. The mean free path between strands is about 100 nm. This is much shorter than the typical network length in bulk helium crystals and so any dislocations in the confined helium should be thoroughly pinned. To make the measurements, a thin (0.8 mm) slice of aerogel was cut and placed between the transducers, filling the gap. Helium was frozen in its pores and in the bulk space surrounding the sample. The moduli of the aerogel itself were much smaller than those of solid helium so the measured shear modulus was essentially that of the helium. Figure 12 compares the modulus and dissipation for two crystals grown in the aerogel to those of bulk 4 He. The total modulus changes in the aerogel are much larger (about 50 %) than in the bulk sample. In the aerogel, the modulus decreased gradually over the entire temperature range up to melting while in bulk helium it occurs entirely below 200 mK. The dissipation peaks in the aerogel samples are also much larger and broader than the corresponding peak in bulk helium, as well as occurring at higher temperature (around 0.7 K, compared to 110 mK for the bulk peak). The modulus changes and dissipation peaks in the aerogel were independent of the stress amplitude and were unaffected by annealing. Strong dislocation pinning by aerogel strands would require large amplitudes for stress-induced depinning and high temperatures for thermal unpinning, consistent with the observed behavior. The very broad transition and the lack of annealing effects may reflect the high degree of disorder imposed by the aerogel. However, below 150 mK we observed an additional anomaly in our aerogel measurements. These modulus and dissipation changes were much smaller than those at high temperatures (and smaller than in bulk helium) but they had bulk-like temperature and amplitude dependences and were reduced by annealing. A small bulk-like anomaly has also been seen in TO experiments with solid helium in a similar aerogel [30] so our low temperature behavior could be associated with the confined helium. However, our sample and transducers are embedded in solid helium and we believe that the anomaly we see below 150 mK is probably due to elastic changes in the bulk helium at edges of the transducers. The essential features of all our elastic measurements can be explained in terms of dislocations which respond to stresses at high temperatures but are pinned by 3 He impurities at low temperatures. This interpretation of the shear modulus behavior does not invoke supersolidity but it is clear from their many similarities that the shear modulus and TO anomalies are closely related. One obvious possibility is that the TO frequency changes are caused by the shear modulus changes. The frequency of a TO depends on the elastic properties of all its parts, so any increase in the shear modulus of the helium will stiffen the oscillator [31]. This will raise its frequency and mimic mass decoupling. However, simply changing the shear modulus would not explain the dissipation peaks that are associated with TO frequency changes. More complicated models which include thermally activated relaxation or a glassy [32, 33] or viscoelastic [34] response do predict dissipation peak and could provide a mechanical explanation of the TO behavior. The magnitude and frequency dependence of the elastic effects depend on the design of the oscillator. If the oscillator head is not completely rigid, e.g. if it has a cylindrical helium space and relatively thin walls, then solid helium can raise the TO frequency by increasing the head’s torsional stiffness. Elastic effects can be even larger in heads with thin annular sample spaces where the solid helium can act as a
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Fig. 12 (a) Shear modulus and (b) dissipation for solid helium confined in a 95 % porosity silica aerogel. The upper (blue) and lower (red) data are for crystals at pressures of 39.9 and 29.7 bar, respectively. The (green) curves which end at 1.2 K are typical bulk 4 He data from a 33.3 bar crystal. All data were taken at 2000 Hz (from Ref. [29]) (Color figure online)
“glue” between the inner and outer walls of the annular channel [35, 36]. Even if the oscillator head is completely rigid, solid helium is elastically very soft and some of it will “elastically decouple” at finite frequency, i.e. will oscillate with larger amplitude than the walls of its container. Any stiffening of the helium reduces this overshoot, raising the TO frequency and mimicking mass decoupling. This effect is reduced when the helium is confined in a narrow annulus and has a characteristic f 2 frequency dependence which distinguishes it from a true change in inertia. Simple estimates, as
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well as detailed numerical modeling of particular TO geometries, suggest that this effect is too small to explain the apparent NCRI in most experiments [37]. Also, it is very difficult to see how modulus changes which arise from dislocation motion could occur in helium confined in very small pores but a large NCRI has been seen in TO experiments involving vycor [38] which has pores a few nm in diameter. Another very direct way in which solid helium can raise a TO’s frequency is through its contribution to the stiffness of the torsion rod [24]. Most oscillators introduce helium into the sample space via a hole through the torsion rod. When this helium freezes, its shear modulus will directly stiffen the rod, as will any subsequent increase in the solid helium’s shear modulus. This effect is independent of frequency and so is difficult to distinguish from mass decoupling. Estimates of the magnitude of this effect [39] indicate that it is significant in torsional oscillators with relatively large holes in the torsion rod. To clarify the relationship between the TO and shear modulus anomalies, and to see how they depend on quantum statistics and crystal structure, we studied the behavior of solid 3 He in combined shear modulus and TO experiments [40]. The shear modulus measurements were done in our lab and the TO experiments in Moses Chan’s laboratory. Given the sensitivity to isotopic purity, it was important to use 3 He with low 4 He concentration. Both sets of experiments used the same 3 He gas, which contained 1.35 ppm 4 He. Figure 13 compares the temperature dependence of the shear modulus of bcc and hcp 3 He to that of hcp 4 He. The upper curves (blue symbols) show the modulus in two hcp 4 He crystals (29 and 33 bar), measured at 2000 Hz and low strain ( ≈ 2 × 10−8 ). The middle (red) curves show the corresponding shear modulus for two hcp 3 He crystals (both at 119 bar). The stiffening seen in 4 He also occurs in hcp 3 He. The transition is broader in 3 He and begins at higher temperature (around 400 mK). This probably reflects the larger impurity concentration (1.35 ppm 4 He) compared to that of the 4 He crystals (0.3 ppm 3 He). The lowest curves (green symbols) are for bcc 3 He crystals (at 55 and 68 bar). In contrast to the hcp crystals, these curves do not show an obvious low-temperature anomaly. The distinction between hcp and bcc crystals is even clearer in the amplitude dependence shown in Fig. 14 (the color coding for different amplitudes is the same for the three samples). In hcp crystals of both 3 He and 4 He, the modulus anomaly decreases for strains > 4 × 10−8 . The behavior of bcc 3 He crystals is quite different, with no systematic amplitude dependence. The effect of annealing in hcp 3 He crystals is similar to that in 4 He. Annealing reduces the size of the anomaly by increasing the shear modulus at high temperature but leaves the low temperature modulus unchanged [28]. In bcc 3 He crystals, however, annealing did not affect the modulus at high temperatures and resulted in only small and non-systematic changes at low temperatures. The similarities between 3 He and 4 He in the hcp phase, and the different behavior in bcc 3 He, are consistent with a dislocation interpretation of the modulus changes since dislocation properties depend strongly on crystal structure (e.g. dislocations are usually less mobile in bcc crystals) but not directly on quantum statistics. The fact that the shear modulus shows the same anomaly in hcp 3 He as in 4 He allows us to distinguish effects in TO measurements due to elastic changes from
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Fig. 13 Shear modulus (at 2,000 Hz) in 3 He and 4 He crystals. The upper (blue) curves show the modulus in hcp 4 He crystals (29 and 33 bar). The middle (red) curves are for hcp 3 He crystals (both at 119 bar) and the lower (green) curves are for bcc 3 He crystals (55 and 68 bar) (data from Ref. [40]) (Color figure online)
those due to quantum statistics. If the TO anomaly is due to shear modulus changes which mimic decoupling by stiffening the oscillator, then it should also appear in hcp 3 He, but not in the bcc phase. However, the TO measurements [40] showed no anomaly in either hcp or bcc 3 He, despite the modulus changes in hcp 3 He. This indicates that the TO frequency changes are not due to elastic stiffening—quantum statistics are crucial. The TO anomaly only occurs in the Bose solid, 4 He, supporting its interpretation as evidence of mass decoupling. The TO behavior in 4 He is clearly related to that of the shear modulus, although not as a simple mechanical consequence of the shear stiffening. Our modulus measurements have a significant advantage over the TO technique—the ability to vary the frequency over a wide range—which allows us to probe the dynamics of solid 4 He. For example, measurements at different frequencies allow one to distinguish amplitude dependence which depends on stress/strain (as expected for our dislocation interpretation of the shear modulus behavior) from amplitude dependence coming from a superfluid-like critical velocity (as suggested by the NCRI interpretation of TO experiments). The TO behavior of solid helium has been studied [41] at two frequencies (496 and 1173 Hz) in a compound oscillator. Its amplitude dependence appeared to scale better with velocity than with acceleration or displacement but more recent
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Fig. 14 Amplitude dependence of the shear modulus in hcp 4 He and in the hcp and bcc phases of 3 He. The strain amplitudes for hcp 4 He vary from 2 × 10−9 (top, black symbols) to 1.7 × 10−6 (bottom, cyan symbols), with the same color coding for hcp and bcc 3 He (from Ref. [40]) (Color figure online)
measurements [42] suggest that it may be the acceleration which controls the amplitude dependence. We can address this issue more definitively for the shear modulus by exploiting our much wider frequency range [25]. Figure 15 shows the amplitude dependence of the shear modulus at 18 mK for three different frequencies (20, 200 and 2000 Hz). In Fig. 15a, the modulus is plotted vs. shear strain and in Fig. 15b, the same data are plotted versus the corresponding velocity v = 2πf D. It is clear that the amplitude dependence scales with strain (or stress, which is proportional to strain) rather than with velocity. This suggests that the amplitude dependence seen in TO experiments might have its origin in inertial stresses in the helium (which are proportional to the acceleration in the oscillator). These inertial stresses can be estimated for TO measurements and are roughly 0.01 Pa at the point where amplitude dependence begins. A recent experiment [43] in which the shear modulus and NCRI were measured simultaneously confirmed that the critical stress for NCRI is much smaller than for the shear modulus. However, our shear modulus data has also been analyzed together with other TO NCRI data and shown to have essentially the same scaling with amplitude [44]. We have also used the frequency range of our measurements to probe [25] the dissipation and dynamics of the anomalies in solid 4 He. For the shear modulus, the dependence on frequency is already apparent in the data shown in Fig. 4. The anomaly (e.g. the midpoint of the modulus change) shifts to lower temperature as the fre-
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Fig. 15 Scaling of the amplitude dependence of the shear modulus in solid 4 He with (a) strain and (b) velocity for three different frequencies: 20 Hz (red triangles), 200 Hz (blue squares), and 2000 Hz (black circles) (from Ref. [25]) (Color figure online)
quency decreases. This is consistent with the shift of the NCRI onset temperature between the two frequencies (496 and 1173 Hz) seen in the compound TO experiments described above [41]. This frequency dependence, the dissipation peaks that are associated with the shear modulus and TO anomalies (Fig. 5), and the broadness of the transitions all suggest that we are dealing with a crossover in a thermally activated relaxation process rather than a true phase transition. Measurements at different angular frequencies ω give a direct measure of the temperature dependence of the relaxation time τ in such a process since the crossover occurs at the temperature where ωτ = 1. Figure 16 shows the shear modulus and dissipation in a 38 bar 4 He crystal (300 ppb 3 He), at frequencies from 0.5 to 2000 Hz [25]. The midpoint of the modulus
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Fig. 16 Frequency dependence of (a) shear modulus and (b) dissipation for an hcp 4 He crystal at 38 bar. Curves have been vertically shifted for clarity (the modulus scale applies to the 2000 Hz data). Open circles are the midpoints of the modulus changes; solid circles are the dissipation peak positions (from Ref. [25]) (Color figure online)
change, T50 , shifts from 126 mK (at 2000 Hz) to 55 mK (at 0.5 Hz). An Arrhenius plot of the modulus midpoints (1/T50 vs. ln ω) confirms that this is a thermally activated process, with an activation energy E ≈ 0.77 K. The position of the dissipation peak shifts in essentially the same way. Analyzing the data from the 33 bar crystal of Fig. 4 gives a similar activation energy E ≈ 0.73 K. These values are larger than the 385 mK activation energy estimated above from the 3 He concentration dependence of the crossover at a fixed frequency of 2000 Hz. However, that estimate came from Eq. (3), which is based on the thermal equilibrium concentration of 3 He atoms on
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dislocations and does not predict any frequency dependence. Since the data clearly are frequency dependent, that value (385 mK) will not be the actual 3 He binding energy. The 0.77 K activation energy extracted from the frequency dependence, on the other hand, reflects the dynamics of the motion of dislocation in the presence of 3 He atoms. This could be the 3 He binding energy, e.g. if a dislocation moves by thermally exciting 3 He atoms out of the potential wells which bind them to dislocations. However, dislocations might also move by dragging 3 He atoms along with them, which could involve different energy barriers. We have tried to fit our modulus and dissipation data [25] using a single activaE
tion energy for the relaxation time: τ = τo e kB T . For such a “Debye relaxation”, the shear modulus μ and dissipation 1/Q are related to the real and imaginary parts of a complex modulus: 1 Δμ μ =1− , μo μo 1 + (ωτ )2
(4)
Δμ 1 ωτ = Q μo 1 + (ωτ )2
(5)
where μo is the “unrelaxed modulus” (for ωτ 1, e.g. at low temperature) and μo − Δμ is the “relaxed modulus” (for ωτ 1, e.g. at high temperatures). The crossover from unrelaxed to relaxed modulus occurs where ωτ = 1; at this point, the dissipation is maximum, and 50 % of the modulus change has occurred. If the frequency ω is small, this crossover occurs at large τ (i.e. at low temperature if the relaxation is thermally activated). The general features of such a Debye relaxation agree with our measurements but the details do not. Figure 17 shows the 200 Hz shear modulus and dissipation data from Fig. 5. The dashed (blue) line is a fit using the activation energy from the frequency dependence of the transition midpoint (E = 0.73 K) and an “attempt time” (τo = 25 ns) chosen to make the midpoints of the fit and the data agree. It is clear that a simple Debye relaxation using this value of E predicts a much sharper crossover than is observed. We can get a broader transition and a better fit to the modulus data at a single frequency by using a smaller activation energy and a larger attempt time. The dotted (black) curve is a Debye fit for E = 175 mK and τo = 65 µs. Despite the good fit to the 200 Hz shear modulus data, it cannot be correct. First, this value of E is inconsistent with the frequency dependence shown in Fig. 5. Second, the predicted dissipation peak (the dotted black curve in Fig. 16b) is too large by a factor of 2.5. This is also true for the fit with E = 0.73 K (dashed blue curve), which in addition is much narrower than the measured peak. No choice of E and τo can resolve this discrepancy, since Eqs. (4) and (5) imply that the total modulus change and the height of the dissipation peak are directly related by 1 1 Δμ = . (6) Q peak 2 μo There is a similar discrepancy between the magnitudes of the frequency changes (NCRI) and dissipation peaks in TO experiments [45] but TO measurements are usually made at a single frequency so the actual value of the activation energy cannot be determined.
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Fig. 17 Fits of (a) shear modulus and (b) dissipation to different thermally activated relaxation models (details given in text) (from Ref. [25]) (Color figure online)
If a relaxation process involves a distribution of activation energies rather than the single value, then the total modulus change Δμ is unchanged but the crossover is broadened and the height of the dissipation peak is reduced. The red curves in Fig. 17 are fits of μ and 1/Q using a distribution of activation energies with characteristic energy Δ, width W and normalization constant B: 2
n(E) = Be
Δ) −[ (ln E−ln ] 2 W
.
(7)
We used Δ = 0.73 K (to retain the correct frequency dependence) and τo = 9 ns. We found we needed a wide distribution, W = 0.45, to match the broad crossover in the modulus. The dissipation peak predicted using these values has approximately the right height and width. This distribution also gives a good fit to the high frequency (8 kHz) acoustic resonance frequency and dissipation [25]. These shear modulus data have also been discussed [46] in terms of a “glassy” component in solid helium (as have TO frequency and dissipation data [32, 33, 45]). The temperature and frequency dependence of the shear modulus can be fit using response functions characteristic of glasses, although the best fit gives a negative value of the glass temperature Tg (which suggests there is no actual glass transition). Our measurements show that a single activation energy cannot explain the behavior of the shear modulus—there must be some form of disorder which broadens the response in solid helium. Precise measurements over an even wider frequency range would be needed to distinguish between our distribution of activation energies and other response functions, e.g. those used for glasses. Glassy behavior has also been inferred from the thermodynamic behavior of solid helium. Heat capacity data were
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reanalyzed [47] in terms of a linear T dependence which was then interpreted as evidence of the two level tunneling systems (TLS) which dominate the low temperature behavior of conventional glasses. However, a linear heat capacity could come from any excitations with a constant density of states, for example one dimensional excitations like kinks on dislocations. Other properties associated with TLS at low temperatures (e.g. a thermal conductivity proportional to T 2 , elastic moduli which increase as ln(T ), saturable amplitude dependent dissipation) are not seen in solid helium. A number of other groups have recently studied the elastic properties of solid helium. Their results, which agree with the behavior described above, also show a number of important new features. One experiment used acoustic resonance techniques to study the elastic behavior of single crystals of 4 He with known orientation [48]. The crystals were grown from the superfluid at very low temperatures (21 mK). This not only produced high quality crystals with few dislocations but also essentially eliminated isotopic impurities from the solid [49]. In the absence of 3 He pinning sites, these crystals were extremely soft, even at the lowest temperatures. When such a sample was warmed to 300 mK, 3 He atoms diffused into the solid, stiffening the crystal and restoring its intrinsic modulus at the lowest temperatures. This directly confirms of the role of isotopic impurities in stiffening helium crystals. Changes in acoustic resonance frequency as large as 20 % were observed in some crystals. If only the elastic constant C44 changes (as expected if the changes are due to edge dislocations gliding in the basal plane of the hcp structure), this implies that C44 almost vanishes when the dislocations are mobile. Similar large modulus changes have been seen in other acoustic resonance measurements [50]. These measurements also showed that the stiffening occurs at lower temperatures in single crystals than in polycrystals grown by the blocked capillary technique. Another experiment combined shear modulus and TO measurements in a single cell [43]. Although the elastic and NCRI measurements probed different parts of the sample, the two properties showed almost identical behavior. An important aspect of these experiments was their ability to look for interactions between the shear modulus and NCRI measurements. For example, large amplitude TO oscillations were used to generate large inertial stresses. These suppressed the shear modulus anomaly but the critical stress was much larger than the inertial stress at the onset of amplitude dependence in the NCRI. This agrees with previous estimates [37] of the inertial stress in torsional oscillators and implies that inertial stresses cannot explain the TO critical velocity.
5 Open Questions It is clear from their remarkable similarities that the shear modulus and TO anomalies are closely related. It is also obvious that defects are involved in both. The essential features of the elastic behavior are well described in terms of dislocations which are mobile at high temperatures but pinned by 3 He impurities at low temperatures. However, this basic picture leaves many unanswered questions, including the connection between the two phenomena.
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Despite all the effort since 2004, it is still not certain that the TO behavior is really due to mass decoupling of a supersolid fraction (NCRI). It is tempting to suggest that the TO frequency increases are just due to shear modulus changes which mechanically stiffen the oscillator. Experiments in which plastic deformation was shown to change the high temperature TO frequency but leave its low temperature value unchanged [35, 36] support this point of view. However, a number of other experiments argue against an elastic origin of the apparent NCRI. These include our measurements [40] with hcp 3 He crystals (which show shear modulus stiffening but no TO NCRI), the observation of NCRI in TO experiments with porous media like vycor [38] (where elastic stiffening is unlikely) and calculations [37] which show that the effect of stiffening is too small to explain the observed TO frequency changes. Recent experiments with a rotating TO [51] provide the strongest support for a supersolid interpretation— there does not seem to be any way that DC rotation could affect the elastic properties of a conventional crystal. If the TO frequency changes are a signature of NCRI, how is supersolidity related to the shear modulus behavior? Does superflow occur along a network of superfluid dislocations [52–54]? How can NCRI values of several percent be reconciled with dislocation densities involving much smaller fractions of the atoms (and how can we directly measure dislocation densities in helium)? Does unpinning of the dislocations somehow disrupt superflow, e.g. through dissipation introduced by moving kinks? Separately from the issue of supersolidity, we want to understand the unique properties of dislocations in a quantum solid like helium. Is there a Peierls barrier or are kinks and dislocations completely delocalized? Are dislocations rough or smooth [55] at low temperatures? Does “superclimb” provide a new mechanism [56] by which dislocations can move? How do 3 He atoms, which are highly mobile in solid helium, pin dislocations [57]? What are the kinetics of this pinning in the strain fields around dislocations? To experimentally answer such questions we need to control and characterize the dislocation density and geometries, for example by crystal growth and plastic deformation. Acknowledgements The shear modulus experiments described in this paper were performed by James Day, Alex Syshchenko and Arif Rabbani and would not have been possible without their hard work, careful measurements and thoughtful analysis. I would particularly like to thank Moses Chan, Sebastien Balibar, John Reppy, Izumi Iwasa, Norbert Mulders, Yuri Mukharsky, Eunseong Kim, Josh West, and Tony Clark for valuable and enjoyable discussions about experiments. I would also like to thank the many theorists who have tried to help me understand solid helium. This work was funded by the Natural Sciences and Engineering Council (NSERC) of Canada.
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