A Summary of Mass Flux Measurements in Solid 4 He

Transcription

1 J Low Temp Phys (2012) 169: DOI /s A Summary of Mass Flux Measurements in Solid 4 He R.B. Hallock M.W. Ray Y. Vekhov Received: 13 June 2012 / Accepted: 11 July 2012 / Published online: 3 August 2012 Springer Science+Business Media, LLC 2012 Abstract Here we provide a summary and brief review of some of the work done with solid 4 He at the University of Massachusetts Amherst below a sample pressure of 28 bar. The motivation for the work has been to attempt to pass 4 He atoms through solid 4 He without directly applying mechanical pressure to the solid itself. The specific technique chosen is limited to pressures near the melting curve and was initially designed to provide a yes/no answer to the question of whether or not it might be possible to observe such a mass flux. The thermo-mechanical effect and direct mass injection have been separately used to create chemical potential differences between two reservoirs of superfluid 4 He connected to each other through superfluid-filled Vycor rods in series with solid 4 He, which is in the hcp region of the phase diagram. The thermo-mechanical effect is a more versatile approach. And, in a particular symmetric application it is designed to provide a mass flux with little or no net increase in the density of the solid. Our observations, off but near the melting curve, have included: (1) the presence of an increasing DC flux of atoms through the solid-filled cell with decreasing temperature below 650 mk and no flux above this temperature; (2) the presence of a flux minimum and flux instability in the vicinity of mk, with a flux increase at lower temperatures; (3) the temperature dependence of the flux above 100 mk and the dependence of the flux on the net driving chemical potential difference provide interesting insights on the possible mechanism that leads to the flux above 100 mk. The most recent data suggest that whatever is responsible for the flux in solid 4 He, at least for T>100 mk, may be an example of a Bosonic Luttinger liquid. R.B. Hallock ( ) M.W. Ray Y. Vekhov Laboratory for Low Temperature Physics, Department of Physics, University of Massachusetts, Amherst, MA 01003, USA hallock@physics.umass.edu Present address: M.W. Ray Amherst College, Amherst, MA 01002, USA

2 J Low Temp Phys (2012) 169: Keywords Solid helium Supersolid Quantum solid 1 Introduction More than forty years ago it was predicted that solid helium might demonstrate unusual behavior at low temperatures [1 3], but several early searches for novel behavior did not reveal anything particularly unusual [4, 5]. It was not until Ho et al. [6] reported unusual behavior in solid 4 He doped with 3 He that interest in the subject was re-kindled. Kim and Chan [7] used a torsional oscillator to study solid helium and observed a period shift of the oscillator at temperatures below about 250 mk. This period shift was interpreted as apparent mass decoupling below a temperature that depended on the 3 He concentration in the experimental cell. This observation and the interpretation that it might signify the presence of a supersolid has stimulated much of the current emphasis on solid 4 He. The observations of Kim and Chan [7, 8] have been replicated in a number of laboratories, but with a wide range of period shifts seen, including some situations in which no resolvable shift was seen [9]. Measurements of the shear modulus show temperature- and 3 He concentration-dependent signatures that are very reminescent of the temperature and concentration dependencies seen in the torsional oscillator experiments [10]. Since elasticity effects in the material studied in a torsional oscillator can influence the period, the shear modulus effects greatly complicate the original interpretation of the torsional oscillator results. Reppy, as a result of various measurements with cells of different design, including multiple frequency cells, has urged caution in the interpretation of torsional oscillator results [11, 12]. In spite of rather intense effort, the solid 4 He community seems to have not yet achieved a full understanding of solid 4 He. Were a supersolid to be present, it might be expected that such a solid would be capable of supporting a DC superflow, but experiments in which the solid was directly squeezed failed to reveal any evidence for flow behavior [4, 13 15]. In our laboratory at the University of Massachusetts Amherst experiments of a conceptually different nature were envisioned and have been carried out. In these experiments, rather than directly squeezing the solid lattice, a sample of solid helium was subjected to an imposed chemical potential difference [16 19]. The result of these experiments was the presence of consistent evidence for a mass flux through a sample cell filled with solid 4 He below an apparently pressure-dependent characteristic temperature. And, the temperature dependence of the mass flux demonstrated interesting behavior. Some of these experiments were reported in detail elsewhere [16 19] but will be briefly reviewed here along with some data not previously presented and data from some of our more recent work [20]. Brief comments about a possible mechanism that might be responsible for the mass flux that is observed in our work are also included, but mention of the substantial variety of theoretical ideas is left for the overview of recent solid helium work that will be published and related to this volume. Given that overview, we have also limited the number of references that we have included here.

3 266 J Low Temp Phys (2012) 169: Experimental Details The conceptual idea behind the experiments we have been doing differs from others that have been carried out in the field, e.g. torsional oscillators, squeezing the solid, etc. We created an apparatus in which a sample of solid 4 He could simultaneously be off the melting curve, but still be in direct contact with two separate reservoirs of superfluid 4 He. This is accomplished by the creation of a sandwich-like arrangement (Fig. 1) in which two reservoirs of superfluid 4 He are in series with two superfluid liquid helium-filled porous Vycor rods, and the sample of solid 4 He. Three fill lines lead to the cell, capillaries 1 and 2 go to liquid reservoirs R1 and R2 above the Vycor rods V1 and V2. The third fill line (a long heat-sunk narrow capillary that can be plugged with solid 4 He) leads directly to the solid chamber, S, and is used to initially fill the sample cell with liquid helium. Two capacitance pressure gauges, C1 and C2, are located on either side of the cell for in situ pressure measurements of the solid 4 He. Pressures in the lines 1 and 2 are read by pressure transducers outside the cryostat. Each reservoir has a heater, H1, H2, which prevents the liquid in it from freezing and allows a temperature difference to be created between the reservoirs. The reservoir temperatures are read by calibrated carbon resistance thermometers T 1 and T 2. The cell temperature T is recorded by a third calibrated carbon resistance thermometer, TC. The melting curve in the porous Vycor is elevated so that for pressures below 37 bar at low temperature the Vycor contains superfluid and is a poor thermal conductor. With solid helium in the sample cell and an appropriate choice of pressure and temperatures of the two reservoirs, R1 and R2, it is possible to have superfluid helium present in the reservoirs and in the Vycor rods. By this means there can be a superfluid-solid interface at pressures above the normal melting curve for 4 He. This allows us to apply a chemical potential difference across the sample by either raising the pressure [16, 17] in one of the reservoirs or by application of a temperature difference T 1 T 2 between them [18 20]. Due to the need for superfluidity in all regions of the Vycor we are limited to regions of the phase diagram rather close to the melting curve. The sample cell is initially filled with ultra high purity 4 He ( %, assumed to have a 300 ppb 3 He impurity concentration) passed through a 4.2 K trap through capillary 3 (Fig. 1), which is present to allow the Vycor to be bypassed and speed the filling of the sample cell. Solid samples can be grown by means of the blocked capillary technique starting, for example, with liquid in region S at a pressure near 50 bar and a temperature near 2.8 K and cooling with a fixed number of atoms present [17]. More typically for our work, solid samples are grown from the superfluid at a constant temperature of 350 mk starting at a pressure of 25 bar by means of a simultaneous increase in the pressure in lines 1 and 2, which results in injection of atoms through the Vycor and changes the density in the sample cell. During such growth near the melting curve, positive transients in the cell temperature are accompanied by decreases in the cell pressure [17, 21] and are believed to be caused by the solidification of small metastable regions of liquid. With solid 4 He in the sample cell, if injection of 4 He atoms to, say, reservoir R1 takes place (by means of an increase in the pressure of R1, measured as P 1),

4 J Low Temp Phys (2012) 169: Fig. 1 (Left) Schematic diagram of the cell used for flow experiments [19]. (Right) Another rendition of the apparatus [20]. Line 3 is not shown in this rendition. The Vycor sections are relatively longer than they appear in these sketches and the reservoirs are relatively smaller than shown (Color figure online) one watches for any change in the pressure of reservoir R2, P 2. If an increase is observed, it could only have resulted from a flux of atoms through the solid-filled cell [16, 17]. In experiments of a different sort, a temperature difference, T 1 T 2, is imposed between reservoirs R1 and R2. This results in the growth of a pressure difference (consistent with the fountain effect) between the two reservoirs. If a pressure difference P 1 P 2 appears, it must be that atoms moved to restore chemical potential equilibrium in response to the imposed temperature difference [18 20, 22]. A measure of the time rate of change of these pressure changes yields a measurement proportional to the flux of atoms that passed through the cell filled with solid helium. In most of our experiments the pressures at each end of the cell, C1 and C2, were also monitored, which allows an additional window on the behavior of the solid helium. 3 Results 3.1 Flow by Direct Mass Injection In our first generation of experiments the goal was simply to ascertain whether or not it was possible to see evidence for mass migration through solid helium by the application of an increase in pressure to one of the reservoirs. An early example of evidence that this was possible is shown in Fig. 2 (left) where an increase in the pressure, P 1, of reservoir R1 resulted in a gradual increase in the pressure in R2, P 2,andariseinC1 and C2. This was taken as rather clear evidence that by some mechanism it was possible for atoms to (1) move through the solid-filled cell, and to (2) simultaneously increase the local density in the fixed-volume cell. Thus, of the atoms injected into R1, some caused an increase in P 2 and others joined the solid and increased its density. This increase in density by mass injection to a fixed volume has been described as a syringe effect [23]. A plausible mechanism to explain both the presence of mass flux and the density increase was proposed: If mass flux were to take place by conductance along the cores of edge dislocations [23], an increase in

5 268 J Low Temp Phys (2012) 169: Fig. 2 (Left) Sample AO, created from superfluid at 358 mk, showed a flow of mass through solid helium. The pressure in R1, P 1, was increased by 0.5 bar at t 30 minutes, the regulator feeding helium to line 1 was closed at t 90 minutes, and changes in pressure were observed for about 5 hours. From these data and other data like these it was concluded that dp2/dt was nearly constant for a substantial duration and independent of P 1 P 2. Note C1 = C2 [from: Ref. [17], Fig. 8]. (Right) Sample AG, created from superfluid and studied at 362 mk, showed a flow of mass through solid helium following a change in P 1of 0.4 bar. Note in this case that C1 andc2 were not equal and both increased, but C1 C2 was essentially unchanged during the course of the measurement. (Replicated from Ref. [17], Fig. 16) (Color figure online) solid density would result if some 4 He were to remain on the dislocation cores, thus causing the dislocations to climb (effectively inserting a growing plane of atoms into the lattice). It also was the case that the rate of increase of pressure P 2 was nearly independent of the pressure difference P 1 P 2 that created the flow. This is one of the characteristics of superflow limited by a rapidly rising dissipation with increasing velocity, i.e. the presence of a critical velocity. In cases where no flow was seen (e.g. at temperatures above 650 mk), it was also the case that no changes in C1 orc2 were observed. One of the interesting observations from measurements of this sort is illustrated in Fig. 2, right (from Ref. [17], Fig. 16), where it is seen, in contrast to what was seen in Fig. 2, left, that it is possible for there to be a stable pressure gradient across the solid-filled cell even in the presence of an increase in density (seen equally on C1 and C2) that accompanies the presence of a flux through the cell. This observation clearly has consequences for those who study solid helium: a single pressure gauge may not adequately characterize samples of solid helium [21, 24]. As discussed, for example in references [17, 19], if we assume the cores of edge dislocations carry the flux, then we can write the mass flux as dm/dt = ξvρna, where ξ is the superfluid density, v is the flow velocity, ρ is the density (assumed to be equal to the density of the solid), N is the number of effective continuous dislocation cores that conduct and A is the effective cross sectional area of one dislocation core. For a typical mass flux of g/sec, with A = 1nm 2 and ξ = 1, we find that vn = cm/sec. With no independent knowledge of the value of N it is not possible to determine v. We will return to this point later in the light of more recent experimental results. Various quantitative aspects are discussed more completely in Refs. [17, 19].

6 J Low Temp Phys (2012) 169: Flow by Use of the Thermomolecular Effect The creation of a substantial pressure difference between the two reservoirs by mass injection clearly results in an increase in the density of the solid, which likely changes the structure of the solid, particularly any disorder that may be present. An alternate approach was taken on our next generation of experiments [18, 19]. For these we utilized the fountain effect: the application of a temperature difference between R1 and R2 creates a chemical potential difference, which will drive a flow of atoms from one reservoir to the other with no net injection of atoms from outside the reservoirvycor-cell system. For the first experiments of this type, with a stable solid in the sample cell and T 1 = T 2, P 1 = P 2, the temperature of one (or both) of the reservoirs was increased, held stable and then decreased. An example of the behavior seen for this type of measurement is shown in Fig. 3, left. In the example shown, first T 1 was increased by 24 mk. After P 1 and P 2 changed and chemical potential equilibrium was restored, T 1 was decreased back to its original value, and the procedure was repeated with T 2. Then T 1 and T 2 were simultaneously increased and then decreased. The increase of T 1 resulted in an increase (decrease) in P 1(P 2), increase in P 1 P 2, and decreases in both C1 and C2. These measured equilibrium pressure differences were consistent with expectations based on the fountain effect using the known properties of superfluid 4 He. This showed that as a result of the initial applied chemical potential difference atoms are added to R1, and taken from both R2 and the solid. When both T 1 and T 2 were simultaneously increased, P 1 P 2 was much more modestly changed, and the solid supplied atoms rather uniformly (capacitive pressure gauges C1 and C2 behave similarly), with changes in C1 and C2 again consistent with changes expected from the fountain effect. A measure of the flux vs. time present between the two reservoirs was taken to be the average slope of P 1 P 2 with respect to time determined by straight line fits to the data. As evident in Figs. 3 and 4, this slope was nearly constant during the flow. Examples that show the rather strong dependence of this flux on the temperature of the solid are given in Fig. 3, right. Separate measurements with the apparatus with superfluid present (no solid) confirmed that the temperature dependence seen was not due to the behavior of the helium in the Vycor. More recent work to be described shortly here has explored the behavior of the flux in more detail by documenting its time evolution of behavior vs. the net chemical potential difference between the two reservoirs [20]. Another example of this strong dependence of the flux vs. time on temperature is shown in Fig. 4. Again here, the average slopes of the data for P 1 P 2 vs. time were taken as proportional to the flux through the solid-filled cell. Note the substantial temperature dependence, and note that while the flux rises with decreasing temperature, in the vicinity of 75 mk the flux falls to quite a low value and appears to be less stable (Fig. 5, from Ref. [19], Fig. 18). Full details of this behavior appear in Ref. [19]. Data of this sort were taken at a span of temperatures from our minimum (near 60 mk) to above where the flux stopped, near 650 mk, at several pressures. It was seen that for most samples there was a strong minimum in the flux in the vicinity of 75 mk (Fig. 5, from Ref. [19], Fig. 18), and while the flux was rather stable over most

7 270 J Low Temp Phys (2012) 169: Fig. 3 (Left) Data (sample GT; see Ref. [19] for details on each sample) that illustrates the procedure used to measure the flux through solid helium by our early utilizations of the fountain effect. Pressures, P 1, P 2, pressures C1, C2 and reservoir temperatures, T 1, T 2 are shown. First T 1 is increased by 24 mk. P 1andP 2 respond, and after equilibrium is achieved, T1 is decreased back to its original value, and the procedure is repeated with T 2. Then T 1 and T 2 are simultaneously increased and then decreased. (Right) ΔP R = P 1 P 2forsampleGTatT = 199 mk, and sample GU at T = 402 mk. The non-zero baseline values of P 1 P 2 result from small baseline differences between T 1andT 2. Note that the average slopes (the predominant slopes of P 1 P 2 vs. time; straight lines were used to fit the data) are defined as positive numbers, denoted by letters as shown (Similar to: Ref. [19], Fig. 12) (Color figure online) Fig. 4 (Left) Data that further illustrates the temperature dependence of the flux through the solid filled cell that results from an initial ΔT = 24 mk. (Right) P 1 P 2 for samples HS, HR and HQ, (sample codes are identified in Ref. [19]), and samples IB, IF and IG. The non-zero baseline values of P 1 P 2result from small baseline differences between T 1andT2 (Color figure online)

8 J Low Temp Phys (2012) 169: Fig. 5 Flux (average slope) behavior of a single sample (described in detail in Ref. [19]) of solid 4 He. Beginning with sample HG, the temperature of the sample cell was changed to new stable values after which the flux was measured and after each change the same solid sample was given a new designation (see Ref. [19]). Here the average slopes (see, e.g. Fig. 3) ofp 1 P 2 are themselves averaged, ( A + B + C + D )/4. Samples HG-IH. Solid (open) symbols represent data taken while warming (cooling) (Taken from: Ref. [19], Fig. 18) (Color figure online) of the temperature regime, it was unstable in the vicinity of the minimum and could remain near its maximum or apparently spontaneously decrease to zero and become non-responsive to the imposition of further chemical potential differences (until the density of the solid, and presumably also the disorder, was changed by a disturbance such as a helium transfer which might modestly change the cell pressure). We have not yet been able to reach temperatures below about 60 mk due to the heat flux from the reservoirs and limitations imposed by the cryostat we have available to us. 3.3 Recent Work Using the Thermomolecular Effect Our most recent sets of measurements are designed to be similar to the set just described, but designed explicitly to explore in some detail the behavior of the flux as a function of the net chemical potential that drives the flux in the temperature regime T>100 mk. So, rather than quantify the generally stable average slopes of P 1 P 2 we wanted to examine the explicit rate of change of P 1 P 2 with time as equilibration was restored after the imposition of an initial chemical potential difference caused by the application of a steady ΔT. And, the new protocol was designed to reduce the net addition of atoms to the solid to close to zero. To accomplish this, an increase in T 1, δt, was always accompanied by an equal decrease in T 2, so that the applied temperature difference T 1 T 2 = 2δT. Following the imposition of δt, the temperatures of T 1 and T 2 are interchanged, which results in a change in the imposed temperature difference between the two reservoirs of 4δT. This process continues, typically with two such T 1, T 2 interchanges. Then a change is made in the size of δt and the interchanges of T 1 and T 2 continue. This approach is described

9 272 J Low Temp Phys (2012) 169: Fig. 6 Response of pressures P 1andP2 to the application of three different δt. T = 350 mk. Use of heaters, H1 and H2, results in changes in T 1andT2. The resulting changes in P 1andP2are best seen as P 1 P 2, shown here (uppermost data). The small drift in P 1,P2 of the sort seen here is not uncommon and variable and appears to have no influence on P 1 P 2. (These data are similar to the T = 390 mk data shown in Ref. [20]) (Color figure online) Fig. 7 Data for one example of ΔP = P 1 P 2 vs. time following the interchange of the roles of T 1andT 2 for the case of ΔT = 2δT = 27 mk for several different solid 4 He temperatures. Note that ΔP ranges about zero because of the change in T 1 T 2 from +2δt to 2δt. Fromsuchdata d(p1 P 2)/dt is obtained and is shown vs. Δμ in Fig. 8 (Color figure online) in detail in Ref. [20]. An example of the response of P 1 and P 2 to this protocol is shown in Fig. 6. From data such as that shown in Fig. 6, for a given value of δt it is possible to make measurements at several solid 4 He temperatures and an example of the results of one such set of measurements is shown in Fig. 7. For these more recent experiments, we also reduced the reservoir temperatures to further reduce the possibility that the Vycor could limit the flux; the absence of such flux restriction was

10 J Low Temp Phys (2012) 169: Fig. 8 Values of F = d(p1 P 2)/dt for a solid sample at a cell pressure of (C1 + C2)/2 = bar shown as a function of Δμ determined for the case ΔT = 27 mk; ΔT =2δT. Small shifts in Δμ have been applied to align the data at Δμ = 0. When flux is present depends on Δμ, and can be represented by F = A(Δμ) b. Solid lines result from the fits to this functional form (from: arxiv: [cond-mat], Fig. 3) (Color figure online) confirmed by measurements as a function of reservoir temperature with liquid in the sample cell. The chemical potential difference between R1 and R2 depends on P 1 P 2 and T 1 T 2 and is given by Δμ = m 4 [ (dp /ρ) (sdt )], where m 4 is the 4 He mass, ρ is the density and s is the entropy per unit mass. As in Ref. [20], here we continue to report Δμ in units of J/g instead of J/atom. We also continue to report our flux values in mbar/s, where a typical value of 0.1 mbar/s corresponds to a mass flux through the cell of g/sec. So, the application of a temperature difference between R1 and R2, with P 1 initially equal to P 2, creates an initial chemical potential difference caused by T 1 T 2. The pressure difference P 1 P 2 responds due to the fountain effect and the initial chemical potential difference falls to zero as the fountain pressure increases and chemical potential equilibrium is restored. In our most recent set of experiments, detailed measurements of F = d(p1 P 2)/dt (determined by a local derivative at each point) were taken as a function of Δμ in an effort to document more carefully the behavior of the flux as a function of the net chemical potential difference between the two reservoirs. To accomplish this, we measure T 1, T 2, P 1 and P 2 vs. time, compute Δμ vs. time and then deduce F = d(p1 P 2)/dt vs. Δμ. We found that the flux is a function of the net chemical potential that can be reasonably well characterized by F = A(Δμ) b,fig.8 (from Ref. [20], Fig. 3), where b 0.32 is within our errors independent of the temperature of the solid helium. It also can be noticed in the data presented in Fig. 8 that there appear to be small oscillations in F as a function of Δμ. We have not yet explored these to establish their reproducibility and how dependent on the sample they may be. We have also not yet explored the detailed behavior of F in the vicinity of very small values of Δμ. Results for F for fixed values of the chemical potential as a function of the temperature of the solid revealed that the temperature dependence of the flux could be characterized rather well by the two-parameter functional form, F = B ln(t /τ), Figs. 9 (from Ref. [20], Fig. 5), with τ 630 mk. Indeed, in general the data could be reasonably represented by the single composite function F (Δμ) b ln(t /τ).

11 274 J Low Temp Phys (2012) 169: Fig. 9 F determined at (C1 + C2)/2 = bar as a function of T for different values of Δμ, interpolated from the data shown in Fig. 8, forthe case of an applied ΔT = 2δT = 27 mk. These and data for other imposed ΔT values can be represented by F ln(t /τ). Note that with this functional form, F is extinguished for T τ. (Adapted from: arxiv: , [cond-mat], Fig. 5) (Color figure online) 4 Possible Mechanisms A key question is: What is responsible for the flux through the solid-filled cell that we see? On the melting curve Balibar s group [25 27] showed that a flux could be present due to conduction along grain boundaries, or the liquid channels associated with them. On might imagine that such channels could be a candidate to explain our results, but we believe that too many differences exist between the behavior expected for such channels and what we observe for such channels to be the explanation for our results. For example, the channels seen in Refs. [25 27] continued to conduct well above the temperature at which the flux we see is extinguished, the temperature dependence and dissipative nature of the flux we see seems inconsistent with liquidfilled channels, and such channels certainly would not be expected to show the strong temperature dependence that we see near 75 mk. One candidate that does seem consistent with many of our observations is the possibility that the cores of edge dislocations carry the flux [23, 28, 29]. Soyler et al. [23] proposed that to be the case and to explain the isochoric compressibility seen in our experiments; i.e., the observation that addition of atoms to the solid-filled cell causes a global increase in the density in the cell. This would take place were some of the atoms that move along the cores of edge dislocations to remain on the core and thus cause the dislocation to climb. A test is proposed for this: the dislocation climb should be suppressed at low enough temperatures. As we pointed out in reference [18], we have evidence that this is the case; that is, we have been unable to freshly grow samples from the superfluid below 300 mk by injection through the Vycor, but, samples grown at higher temperature are able to accept an increase in density at any temperature we have studied [18]. Our interpretation of these most recent data with the dependence of F on Δμ we observed, discussed more fully in Ref. [20], is that whatever carries the flux in solid helium results in behavior that may be characterized as an example of a Bosonic Luttinger liquid, at least in the temperature regime above 100 mk. If this is the case, as we discuss in Ref. [20], if we were to have a single conduction channel with Luttinger liquid behavior, one expects such behavior for k B T/ J, where J is the

12 J Low Temp Phys (2012) 169: flux in atoms/s. For our work, e.g. at T 0.2 K, with Δμ 0.01 J/g, we have a flux of J atoms/sec. The use of a typical temperature at which we see flux, T 0.2 K, yields k B T/ = This indicates that for Luttinger liquid behavior to be relevant to our results, the effective number of conducting pathways that carry flux, N, should be With a reasonable estimate of the diameter of such a pathway, presuming it to be the core of an edge dislocation or something of similar size, we conclude that the flow velocity in a given pathway should be 200 cm/sec. Our results suggest that there is a broader range of temperatures of interest in solid 4 He than has been the focus of most (but not all, e.g. Ref. [30]) of the torsional oscillator work. That is, we see clear evidence for flux at temperatures well above those where most torsional oscillator work has focused. We have recently proposed that whatever carries the flux in solid helium above 100 mk may behave as a Bosonic Luttinger liquid. It may be that the conductance is quenched at low temperatures by the condensation of 3 He on the dislocation cores. It is not clear how this would explain the observed increase of conductance seen below about 75 mk, but perhaps new connections among dislocations are created by some means and these provide detours around blocked pathways. It has been suggested (on the basis of a single dislocation model) that the presence of a flux minimum could be caused by nonlinear effects brought about by the size of the chemical potential difference [31]. We have not yet had the opportunity to test this prediction near 75 mk with our new protocols. 5 Future Work Among various possibilities, our priorities for the near future include an effort (1) to carry these experiments to lower temperatures than available from the apparatus as it is currently configured, (2) to study the flux behavior as a function of the 3 He concentration of the solid sample and (3) to study the behavior of the flux in more detail at smaller and well controlled values of Δμ. Further in the future also would like to (4) add in situ low temperature pressure gauges to the reservoirs, R1 and R2, (5) study the flux in the presence of in situ deformation of the solid sample and perhaps (6) change the micro-porous sections of our apparatus to allow operation to somewhat higher pressures than have been available to us to date. The addition of capacitive pressure gauges directly on the reservoirs should enhance our ability to study the flow associated with small differences of chemical potential. Since some of our work has revealed an unusual temperature dependence below 100 mk [18, 19], we think that it is particularly important to reach lower temperatures. To do so will require apparatus modification or a different refrigerator. 6 Conclusions Our experiments at the University of Massachusetts Amherst have shown that it is possible to cause 4 He mass flux through a cell filled with solid 4 He above the melting curve for all pressures accessible to us with the experimental techniques we have

13 276 J Low Temp Phys (2012) 169: employed. But, the detailed way in which our measurements relate to others in the field, especially the torsional oscillator measurements, is not yet clear to us. We have established that for cell pressures 26 bar there is an increasing flux with decreasing temperature in response to a chemical potential difference applied between two reservoirs in series with superleaks and the solid helium sample. The flux rises with falling temperature until the vicinity of 75 mk at which point it falls abruptly (over 5 mk) and then apparently rises again at lower temperatures. The reasons for this temperature dependence are not clear [32]. We also observe what has been termed isochoric compressibility ; it is under some (but not all) circumstances possible to increase the density of the solid by the injection of atoms. A plausible explanation for this and for the passage of atoms through the solid has been proposed by Soyler et al. [23], i.e. the behavior of edge dislocations that can climb and whose cores can transport superflux. Recent work has documented the behavior of the flux as a function of the net chemical potential difference between the two reservoirs, Δμ, fort>100 mk and shown that (1) F = A(Δμ) b, where b is independent of temperature and that (2) at fixed δμ, F ln(t /τ). This latter work suggests that whatever carries the flux in solid helium may behave as a Bosonic Luttinger liquid [33 37]. Acknowledgements We thank our colleagues B. Svistunov and N. Prokof ef for many stimulating and illuminating discussions over the course of our work. We have also had helpful conversations with many others including D. Aleinikava, S. Balibar, J. Beamish, M.H.W. Chan, E. Kim, H. Kojima, A. Kuklov, J. Machta, W.M. Mullin and J.D. Reppy. This work was supported by the National Science Foundation primarily through National Science Foundation grant DMR , with some support from DMR and with initial support from DMR We have also benefitted from support from Research Trust Funds provided by the University of Massachusetts Amherst and occasional access to facilities provided by the MRSEC (NSF DMR ) at the University. References 1. A. Andreev, I. Lifshitz, Sov. Phys. JETP 29, 1107 (1969) 2. G.V. Chester, Phys. Rev. A 2, 256 (1970) 3. A.J. Leggett, Phys. Rev. Lett. 25, 1543 (1970) 4. D.S. Greywall, Phys. Rev. B 16, 1291 (1977) 5. D.J. Bishop, M.A. Paalanen, J.D. Reppy, Phys. Rev. B 24, 2844 (1981) 6. P. Ho, I. Bindloss, J. Goodkind, J. Low Temp. Phys. 109, 409 (1997) 7. E. Kim, M.H.W. Chan, Nature (London) 427, 225 (2004) 8. E. Kim, M.H.W. Chan, Science 305, 1941 (2004) 9. D.Y. Kim, J.T. West, T.A. Engstrom, N. Mulders, M.H.W. Chan, Phys. Rev. B 85, (2012) 10. J. Day, J. Beamish, Nature (London) 450, 853 (2007) 11. X. Mi, J.D. Reppy, Phys. Rev. Lett. 108, (2012) 12. J.D. Reppy, Phys. Rev. Lett. 104, (2010) 13. J. Day, T. Herman, J. Beamish, Phys. Rev. Lett. 95, (2005) 14. J. Day, J. Beamish, Phys. Rev. Lett. 96, (2006) 15. A.S.C. Rittner, W. Choi, E.J. Mueller, J.D. Reppy, Phys. Rev. B 80, (2009) 16. M.W. Ray, R.B. Hallock, Phys. Rev. Lett. 100, (2008) 17. M.W. Ray, R.B. Hallock, Phys. Rev. B 79, (2009) 18. M.W. Ray, R.B. Hallock, Phys. Rev. Lett. 105, (2010) 19. M.W. Ray, R.B. Hallock, Phys. Rev. B 84, (2011) 20. Ye. Vekhov, R.B. Hallock. Phys. Rev. Lett. (2012, in press) M.W. Ray, R.B. Hallock, Phys. Rev. B 81, (2010) 22. M.W. Ray, R.B. Hallock, Phys. Rev. B 82, (2010)

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