Effects of 4 He film on quartz tuning forks in 3 He at ultra-low temperatures
T. S. Riekki, J. Rysti, J. T. Mäkinen, A. P. Sebedash, V. B. Eltsov, J. T. Tuoriniemi
NNoname manuscript No. (will be inserted by the editor)
Effects of He film on quartz tuning forks in He atultra-low temperatures
T. S. Riekki · J. Rysti · J. T. Mäkinen · A. P. Sebedash · V. B. Eltsov · J. T. Tuoriniemi
Received: date / Accepted: date
Abstract
In pure superfluid He-B at ultra-low temperatures, quartz tuningfork oscillator response is expected to saturate when the dissipation causedby the superfluid medium becomes substantially smaller than the internaldissipation of the oscillator. However, even with small amount of He cover-ing the surfaces, we have observed saturation already at significantly highertemperatures than anticipated, where we have other indicators to prove thatthe He liquid is still cooling. We found that this anomalous behavior hasa rather strong pressure dependence, and it practically disappears above thecrystallization pressure of He. We also observed a maximum in the fork res-onance frequency at temperatures where the transition in quasiparticle flowfrom the hydrodynamic to the ballistic regime is expected. We suggest thatsuch anomalous features derive from the superfluid He film on the oscillatorsurface.
Keywords
Quartz tuning fork · Helium-3 · Helium-3–Helium-4 mixture · Helium-4 film
T. S. Riekki † · J. Rysti · J. T. Mäkinen · V. B. Eltsov · J. T. TuoriniemiAalto University - School of ScienceDepartment of Applied PhysicsLow Temperature LaboratoryP.O. BOX 15100 FI-00076 Aalto, Finland † E-mail: tapio.riekki@aalto.fiA. P. SebedashP. L. Kapitza Institute for Physical Problems RASKosygina 2, 119334 Moscow, Russia a r X i v : . [ c ond - m a t . o t h e r] N ov T. S. Riekki et al.
Quartz tuning forks (QTFs) are used for temperature, pressure, viscosity andturbulence measurements in normal and superfluid helium [1,2,3,4]. Theseinfluence the width (full width at half maximum) and the frequency of thefork resonance. The characteristic dimensions of a typical QTF may also matchthe wavelength of first or second sound in pure helium or isotope mixtures,at certain temperature and pressure, resulting in acoustic phenomena [5,6,7]that are interesting in their own right, but can also make interpreting the forkdata more difficult.On cooling of He below the superfluid transition temperature T c , the dis-sipation caused by thermal excitations, or quasiparticles, becomes smaller, astheir number decreases, which is observed as reduction in the QTF resonancewidth. In the B-phase at the lowest temperatures, the quasiparticle densitydecreases exponentially with temperature, and eventually dissipation causedby the quasiparticles becomes smaller than the internal dissipation of the fork,giving typically a residual width 10-20 mHz, which poses the low-temperaturelimit for thermometry.When He is added to a He system, the fork analysis becomes more com-plex, as the surfaces become coated with He [8,9]. Below
100 mK , the Helayer becomes superfluid [10], and due to superfluid film flow it will spreadout to cover all the surfaces of the experimental cell. The film will change thequasiparticle reflection conditions [11] on the QTF surface affecting its res-onance response. At sufficiently high pressures, the He layer becomes solid,and is no longer mobile as a liquid layer would be. However, even the presenceof solid layer may affect the quasiparticle reflection conditions, provided thatthe layer is thick enough to affect the surface roughness.Boldarev et al. [12] observed in the He-rich phase of phase-separated He– He mixture, between and
350 mK , that the QTF deviated from thepredicted viscosity and density dependent response. They attributed this anoma-lous behavior to the He film covering the surface of the QTF, which theyestimated to have thickness of several microns. The non-trivial response madethe fork calibration more difficult, but in their experiment the fork still hadclear temperature sensitivity.We have studied the behavior of He-coated quartz tuning forks in Heat temperatures below , where we observed saturation in the QTFs’ re-sponse at higher temperatures than anticipated. We have two independent ex-periments that demonstrate similar saturation behavior: one is a nafen-filled He cell with surfaces coated with approximately 3 atomic layers of He, andthe other an adiabatic melting cell that contains saturated He– He mixtureat He crystallization pressure, where we expect a much thicker equilibriumfilm. In both experiments we have also observed a maximum in the resonancefrequency at temperatures where the quasiparticle flow regime changes fromthe hydrodynamic to the ballistic one. In this paper we focus on reporting ourexperimental observations, while the detailed explanation on the origin of theeffects remains a task for the future. ffects of He film on quartz tuning forks in He at ultra-low temperatures 3
QTF1 He fork
QTF2
Fig. 1 (color online) Schematic drawings of the experimental cells. The adiabatic meltingcell (a) consists of a large main volume and a sinter-filled heat exchanger volume separatedby a cold valve. The cell is monitored with two quartz tuning fork oscillators, one in the Hephase (QTF1) and another in the mixture phase (or frozen in solid He, depending on thestage of the experimental run). The nafen cell (b) has two separate samples with differentnafen densities. They are both connected to a volume of bulk He, where the thermometerquartz tuning fork (QTF2) is located. The cell is mounted on the nuclear stage of a rotatingcryostat. The surfaces in both systems are coated with a layer of He, although of differentthickness. He in phase-separated He– He mixture at He crystallization pressureIn the adiabatic melting experiment [13,14,15], sub- . temperatures in He– He mixtures are pursued at He crystallization pressure .
64 bar [16,17] by first precooling a system of solid He and liquid He with an adiabaticnuclear refrigerator, and then allowing the solid to melt by extracting He,mixing the two isotopes providing cooling [18].Fig. 1a shows a sketch of the melting experiment cell. More details ofthis setup can be found in Ref. [19]. The resonance width of the QTF inmixture is about
400 Hz , and the effects of the He coating on its behavior areindistinguishable. On the other hand, the resonance width of the QTF in He(QTF1), located at the top of the main cell volume, reaches approximately . at the end of the melting process, and then the superfluid He filmcauses it to have an anomalous response.Fig. 2 shows the QTF1 response during the melting process. The fork wasmeasured in the tracking mode which enables us to receive datapoints everyfew seconds even at very narrow widths by assuming a Lorentzian lineshapewith a constant area [2]. The melting was started at around 4 Hz resonancewidth, corresponding to about . T c ≈ . temperature. Initially theresonance width decreases rapidly as the cell cools down, and the narrowestwidths are already reached within the first few minutes of the process. Thetemperature calibration for the QTF1 was obtained using the self-calibrationmethod described in Ref. [20]. The observed
150 mHz resonance width wouldthen correspond to about . T c ≈ . . At this temperature with the He T. S. Riekki et al.
Fig. 2 (color online) (Main panel) Left y-axis: resonance width of the quartz tuning fork(QTF1) in the He phase during the melting process with zero time chosen to be at thebeginning of the melting. Right y-axis: He extraction rate. (Inset) QTF1 resonance widthversus resonance frequency during the coldest stage of the run, showing the saturation of thewidth value, and the anomalous features that occur during the melting. Red arrows indicatethe direction of time; at around 3 min the width backtracks slightly and then continues todecrease. extraction rate ˙ n ≈ µ mol / s , the cooling power of the melting process[18] is approximately . This is a much larger value than the heat leakto the cell . , which was estimated during the warm-up period, after themelting, when the QTF1 width started to have temperature sensitivity again.Also, the viscous heating effects during the melting are considered insignif-icant. With the estimated heat leak, the liquid should cool down to below . , suggesting that the resonance width is no longer proportional to thequasiparticle density in bulk. Even after the melting was stopped, the QTF1did not show any rapid change from the saturation value which indicates thatthe actual temperature of the liquid was lower than the value given by theresonance width. The fork’s self-calibration relies on the transition to the bal-listic flow regime, the point of which cannot be determined precisely. However,we do not believe that the uncertainty in the temperature calibration couldexplain the discrepancy between the temperature given by the QTF1 and thecooling power of the melting process. Even at . , with µ mol / s Heextraction rate, the cooling power . is still larger than the estimatedexternal heat leak.The inset in Fig. 2 shows that during the melting, there appears anomalousfeatures on the QTF1’s frequency-width plot. These resonance-like featuresonly occur while the melting is being carried out; they do not reproduce whenthe cell is slowly warming up after the melting is over. We point out thatduring the melting period the distance between the fork and the He–mixturephase-separation boundary is changing. As the melting is being carried out, He is dissolved into He released from the solid, decreasing the volume ofthe He phase, while increasing the volume of the mixture phase. When the ffects of He film on quartz tuning forks in He at ultra-low temperatures 5 mixture phase, containing He, approaches QTF1, it will increase the thicknessof the He film covering the fork due to the Rollin film effect [21,22]. Anotherobservation, that seems to corroborate the phase-separation boundary vicinityeffect, is the
30 mHz resonance frequency shift from the before-melting valuethat remained even after the melting had been stopped.2.2 He with small amount of He presentThe nafen experiment consists of two separate samples of He confined in thenematic nano-material nafen [23], which are connected to a volume of bulk He(Fig. 1b). The temperature of helium is controlled by changing the magneticfield applied to the nuclear demagnetization cooling stage. The properties of He in the two nafen samples are probed by means of nuclear magnetic reso-nance (NMR). A quartz tuning fork in the bulk He volume (QTF2) is used asa thermometer. In this experiment He is present only to coat the surfaces ofnafen to prevent the formation of paramagnetic solid He [24]. The thicknessof He on the nafen strands was determined to be approximately 2.5 atomiclayers [24]. The surfaces, including the quartz tuning fork, could adsorb more He, thus the He layer was not maximal [24]. This was clearly demonstratedafter the measurements presented in this paper, as adding more He into thesystem and repressurizing back to 29.5 bar changed the tuning fork width atthe bulk He superfluid transition from 800 Hz to 570 Hz.Fig. 3a shows the QTF2 resonance width and the NMR frequency shiftduring cooling and warming of the sample at 3 bar pressure. These two quan-tities give independent measurements of temperature. The tuning fork widthdisplays a resonance-like feature at 1.1 A current in the demagnetization mag-net, a minimum of about 4 Hz at 0.9 A, and an eventual saturation to 9 Hztoward the lowest temperatures. The NMR frequency shift, on the other hand,indicates continuous cooling of the sample all the way down to the lowest de-magnetization current. The NMR frequency of superfluid He, in the polarphase confined in nafen, is shifted from the Larmor value as a function of tem-perature in axial magnetic field [25]. The QTF2 was measured by continuouslysweeping over the resonance.Fig. 4 plots the QTF2 resonance width and frequency at 23 bar duringslow cooling and warming. Here the QTF2 was measured by applying pulseexcitation and recording the ring-down signal. This gives superior data acqui-sition rate and noise at small resonance widths compared to the continuoussweeping method. Multiple resonance-like features are seen together with ashallow minimum in the width and an eventual saturation to about 1 Hz. Inthe absence of anomalous behavior, QTF2 width of 1 Hz would correspond toabout . T c temperature, or 0.4 mK at 23 bars. The frequency of the oscilla-tor continues to change even after the width has saturated. The same patternis repeated during warming of the sample, but the QTF2 response does notreturn exactly along the same path. T. S. Riekki et al.
Fig. 3 (color online) (a): Quartz tuning fork width (solid lines) and NMR frequency shift(dots) from the Larmor frequency (363 kHz) at 3 bar pressure as functions of the demagne-tization magnet current, which controls the temperature of the sample. Blue color indicatescooling and red warming. (b): Narrowest QTF widths as a function of pressure. The dashedcurve is a guide to the eye. Open points are from the nafen experiment and the black pointfrom the adiabatic melting experiment.
The anomalous behavior of the tuning fork strongly depends on pressure. Itis present at 23 bars, where resonance-like features are observed and the tuningfork width would not go below 1 Hz. At 29 bar pressure, which is above thecrystallization pressure of He, there is no indication of any anomalies and thetuning fork width could be reduced to 190 mHz without evidence of saturation(inset of Fig. 4). Measurements have been performed at various pressures, butthe QTF2 was measured using the pulse method only at 23 and 29 bars. Thecontinuous sweeping method may not reveal small anomaly patterns of theoscillator or the saturation. At 3 bar pressure the anomaly is the strongest.The minimum attained resonance widths as a function of pressure are plottedin Fig. 3b. Small resonance-like features are visible at 16, 6, 5, and 2 barpressures, even with the sweeping QTF2 measurement.The resonance-like features could be attributed to the first sound reso-nances in the tuning fork cavity [26]. The diameter of our tuning fork volume(9 mm) and the oscillator frequency (32 kHz) match roughly the frequen-cies of radial acoustic modes, especially at lower pressures, where the speedof first sound is smaller. It is not clear without more detailed analysis if theresonance-like features seen at 23 bar can be explained by acoustic resonances.The absence of these anomalies at 29 bar pressure might be due to the largerspeed of sound. ffects of He film on quartz tuning forks in He at ultra-low temperatures 7
Fig. 4 (color online) Quartz tuning fork width and frequency at 23 bar during cooling( t < ) and warming ( t > ) of the nafen cell over a 10-hour period, demonstrating multipleresonance modes and saturation of the tuning fork width. A minimum of the width is reachedat approximately ± . h. The inset plots the QTF width in terms of frequency at 23 bar(red) and at 29 bar (black). T c . This is illustrated in the mainpanel of Fig. 5. The inset additionally shows the case of reduced He amountin the nafen cell. In this case, the maximum disappears, and the resonancefrequency instead saturates at the lowest temperatures, which is consistentwith observations in pure He [20].The appearance of the frequency maximum with the increasing He cover-age probably originates from the change in He quasiparticle scattering condi-tions on the QTF surface [11], as the thickness of the film grows. The maximumoccurs at around the temperatures, in which the quasiparticle flow is expectedto change from the hydrodynamic to the ballistic flow regime, and it couldpossibly be used as an indicator of such. Thus the maximum could be utilizedin the QTF self-calibration described in Ref. [20].
We have observed a saturation in the temperature dependence of the resonancewidth of quartz tuning fork oscillators in two independent experiments in He systems with surfaces coated by He. In the melting experiment, the
T. S. Riekki et al. -4 -3 -2 -1 0 1 2 f ! f (Hz) " f ( H z ) QTF1 QTF2
Adiabatic melting cell 25.64 bar, f = 32330 HzNafen cell 23.0 bar, f = 32024 Hz
120 130 140 150 160 170 f ! f (Hz) " f ( H z ) f = 32024 Hz A QTF2 ! < He, 0.5 bar > : He, 0.6 bar
Fig. 5 (color online) Main panel: Fork resonance width as a function of the resonancefrequency in the adiabatic melting experiment (blue, QTF1) and in the nafen experiment(orange, QTF2), both of which show a maximum in the resonance frequency. The resonancefrequencies have been shifted by f , given in the legend. The inset shows that the maximumdisappears if the He content of the cell is low enough. temperature indicated by the QTF resonance width saturates to a value atwhich the cooling power of the helium isotope mixing process would still besignificantly larger than the external heat leak. In the nafen experiment, on theother hand, we had an NMR-based thermometry that showed further coolingin the experimental cell, even after the QTF width had saturated. We alsoobserved strong pressure dependence in the value of the saturated width, withthe maximum being at . Both setups also displayed a maximum in theQTF resonance frequency, which appears only if there is enough He in thesystem.We suggest that this behavior originates from the He film covering theQTF, however, the detailed understanding of the phenomenon requires morework. In particular, studying the dependence on the fork size might be impor-tant. As a practical conclusion, forks covered by He remain relatively reliablethermometers only down to widths of about 1 Hz, provided that the geome-try of the fork volume excludes coupling to the acoustic modes. At pressuresnear the He crystallization pressure, this limit corresponds to approximately . T c . Acknowledgements
This work was supported in part by the Jenny and Antti WihuriFoundation (Grant No. 00170320), and it utilized the facilities provided by Aalto Universityat OtaNano - Low Temperature Laboratory infrastructure. We also acknowledge the sup-port by the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (Grant Agreement No. 694248).ffects of He film on quartz tuning forks in He at ultra-low temperatures 9 ———————References
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