Effects of $^4$He film on quartz tuning forks in $^3$He at ultra-low temperatures

In pure superfluid $^3$He-B at ultra-low temperatures, quartz tuning fork oscillator response is expected to saturate when the dissipation caused by the superfluid medium becomes substantially smaller than the internal dissipation of the oscillator. However, even with small amount of $^4$He covering the surfaces, we have observed saturation already at significantly higher temperatures than anticipated, where we have other indicators to prove that the $^3$He liquid is still cooling. We found that this anomalous behavior has a rather strong pressure dependence, and it practically disappears above the crystallization pressure of $^4$He. We also observed a maximum in the fork resonance frequency at temperatures where the transition in quasiparticle flow from the hydrodynamic to the ballistic regime is expected. We suggest that such anomalous features derive from the superfluid $^4$He film on the oscillator surface.

for temperature, pressure, viscosity and turbulence measurements in normal and superfluid helium [1,2,3]. These quantities are usually derived from the width and the frequency of the fork resonance. The characteristic dimensions of a typical QTF also match the wavelength of first or second sound in pure helium or isotope mixtures under certain temperature and pressure resulting in acoustic phenomena [4,5,6] that are interesting in their own right, but can also make interpreting the fork data more difficult.
On cooling of pure 3 He, below the superfluid transition temperature T c , the dissipation caused by thermal excitations, or quasiparticles, becomes smaller, as their number decreases, which is observed as reduction in the QTF resonance width. In the B-phase, at the lowest temperatures, the quasiparticle density decreases exponentially with temperature, and eventually dissipation caused by the quasiparticles becomes smaller than the internal dissipation of the fork providing the low-temperature limit for thermometry.
When 4 He is added to a 3 He system, the fork analysis becomes more complex, as the surfaces become coated with 4 He. Below 100 mK, the 4 He layer becomes superfluid [7], and due to superfluid film flow it will spread out to cover all the surfaces of the experimental cell. The film will change the quasiparticle reflection conditions [8] on the QTF surface affecting its resonance response. At pressures above the 4 He crystallization pressure 25.64 bar [9], the 4 He layer becomes solid, and is no longer mobile as a liquid layer would be. However, even the presence of solid layer may affect the quasiparticle reflection conditions, as the coating will probably smooth the rough features of the QTF surface.
Boldarev et al. [10] observed in saturated 3 He-4 He mixture, between 15 and 350 mK, that the QTF deviated from the predicted viscosity and density dependent response. They attributed this anomalous behavior to the 4 He film covering the surface of the QTF, which they estimated to have thickness of about 8 µm. The non-trivial response makes the fork calibration more difficult, but in their experiment the fork still had clear temperature sensitivity.
We have studied the behavior of 4 He-coated quartz tuning forks in 3 He at temperatures below 1 mK, where we observed saturation in the QTF's temperature response at higher temperatures than anticipated. We have two independent experimental setups that have observed similar saturation behavior: one is a nafen filled 3 He cell with surfaces coated with approximately 3 atomic layers of 4 He, and the other an adiabatic melting cell that contains saturated 3 He-4 He mixture at 4 He crystallization pressure, where we expect a much thicker equilibrium film.    3 He phase during the adiabatic melting process with zero time chosen to be at the beginning of the melting. Right y-axis: melting rate. (Inset) QTF1 resonance width versus resonance frequency during the coldest stage of the run. Red arrows indicated direction of time; at around 3 min the width backtracks slightly and then continues to decrease.
ing a system of pure solid 4 He, and pure liquid 3 He with an adiabatic nuclear refrigerator, and then allowing the solid to melt, mixing the two isotopes. The mixing of 3 He and 4 He absorbs heat, cooling the helium sample to sub-0.1 mK temperatures [11,12,13]. Fig. 1 shows a sketch of the experimental cell. More details of the experimental setup can be found in Ref. [14]. The resonance width of the QTF in mixture is of order 400 Hz, and the effects of the 4 He film on its behavior are thus indistinguishable. On the other hand, the resonance width of the QTF in 3 He reaches 0.1 Hz at the end of the melting process, and the superfluid 4 He film has a noticeable effect on this fork. The pure 3 He QTF (QTF1) is located at the top of the main cell volume, and while the melting is carried out, the pure 3 He -mixture phase separation interface moves closer to it as 3 He is dissolved into 4 He released from the solid. The movement of the phase separation interface can introduce additional acoustic modes to be observed, or its proximity can affect the thickness of the QTF1's superfluid 4 He coating. Fig. 2 shows the QTF1 response during the adiabatic melting process. The QTF1 was measured in the tracking mode which enables us to receive datapoints every few seconds even at very narrow widths (See Ref. [2] for more information). The melting was started at around 0.5 mK temperature, corresponding to about 4 Hz resonance width. Initially the resonance width decreases rapidly as the cell cools down. The narrowest width is already reached within the first few minutes of the process. The temperature calibration for the QTF1 was obtained using the self-calibration method described in Ref. [15]. Following it, the temperature at about 150 mHz resonance width would be about 0.3 mK. The cooling power of the adiabatic melting process is given by [16] which at the melting rateṅ 4 ≈ 260 µmol/s, and the temperature T ≈ 0.3 mK gives approximately 2 nW of cooling power. This is larger than the heat leak to the cell 0.1 nW, which was estimated during the warm-up period, after the melting, when the width started to show temperature sensitivity again. The liquid should then be able to cool down to below 0.3 mK, which suggests that the resonance width is no longer proportional to the quasiparticle density in bulk. Even after the melting has been stopped, the QTF1 does not show any rapid change from the saturation value which indicates that the actual temperature of the liquid is lower than the value given by the resonance width. The temperature calibration is not by any means perfect, as it relies on the determination of the QTF1's transition to the ballistic flow regime, which has some inaccuracies related to it. However, we do not believe the temperature calibration to be off with such a large margin that it would explain the discrepancy between the temperature given by the QTF1 and the cooling power of the adiabatic melting process. Even at 0.1 mK, with 260 µmol/s melting rate, the cooling power 0.2 nW is still larger than the estimated external heat leak. The inset in Fig. 2 shows that as long as the melting is going on, there appears some anomalous features on the QTF1's frequency-width plot. We suggest that these resonance-like features could be dependent on the distance between between the fork and the phase-separation boundary, as it affects the thickness of the QTF1's 4 He coating. When the melting is stopped, the resonance frequency has shifted by about 30 mHz from the value at the start of the melt. During the warm-up period the QTF1 does not return following the original path, but rather the frequency shift remains. The cooling power of the melting process is produced at the interface between the pure 3 He phase and the mixture phase, so that during the melting the QTF1 can be slightly warmer than the fluid surrounding it causing the superfluid 4 He film on it to become thicker. Conversely, during the warm-up, as the external heat leak is coming mainly through the walls of the experimental volume, the QTF1 can be colder than 3 He and thus the 4 He film moves away from it to the warmer parts of the cell.

3 He with small amount of 4 He present
The nafen experiment consists of two separate samples of 3 He confined in the nematic nano-material nafen [17], which are connected to a volume of bulk 3 He (Fig. 1). Temperature of helium is controlled by changing magnetic field applied to the nuclear demagnetization cooling stage. The properties of 3 He in the two nafen samples are probed by means of NMR. Quartz tuning fork in the bulk 3 He volume (QTF2) is used as a thermometer. In this experiment 4 He is present only to coat the surfaces of nafen to prevent the formation of paramagnetic solid 3 He. The thickness of 4 He on the nafen strands was determined to be approximately 2.5 atomic layers [18]. The surfaces, including the quartz tuning fork, could adsorb more 4 He, thus the 4 He layer was not maximal. This was clearly demonstrated after the measurements presented in this paper, as adding more 4 He into the system changed the fork width at the bulk 3 He superfluid transition at 29.5 bar from 800 Hz to 570 Hz. Fig. 3 shows the QTF2 resonance width and the NMR frequency shift during cooling and warming of the sample at 3 bar pressure. These two quantities give independent measurements of temperature. The tuning fork width displays a resonance at 1.1 A current in the demagnetization magnet, a minimum of about 4 Hz at 0.9 A, and an eventual saturation to 9 Hz toward the lowest temperatures. The NMR frequency shift, on the other hand, verifies continuous cooling of the sample all the way to the lowest demagnetization current. The NMR frequency of superfluid 3 He is shifted from the Larmor value as a function of temperature in axial magnetic field [19]. The QTF2 was measured by continuously sweeping over the resonance. Fig. 4 plots the QTF2 resonance width and frequency at 23 bar during slow cooling and warming. The QTF2 was measured by applying pulse excitation and recording the ring-down signal. This gives superior data acquisition rate and noise at small resonance widths compared to the continuous sweeping method. Multiple resonances are seen together with a shallow minimum in the width and an eventual saturation to about 1 Hz. In the absence of anomalous behavior, QTF2 width of 1 Hz would correspond to about 0.16T c temperature, or 370 µK at 23 bars. The frequency of the oscillator continues to change even after the width has saturated. The same pattern is repeated during warming of the sample, but the QTF2 response does not return exactly along the same path. When the measurement of Fig. 4 was performed slightly faster (7 hours), the two routes differed more, corroborating the conclusion of 4 He movement due to temperature differences made from similar observations in the melting experiment.
The anomalous behavior of the tuning fork was strongly dependent on pressure. It was present at 23 bars, where resonances were observed and the tuning fork width would not go below 1 Hz. At 29 bar pressure, which is above the crystallization pressure of 4 He, there was no indication of any resonances and the tuning fork width could be reduced to 190 mHz without evidence of saturation (inset of Fig. 4). Measurements were performed at various pressures, but unfortunately the QTF2 was measured using the pulse method only at 23 and 29 bars. The continuous sweeping method may not reveal small resonance patterns of the oscillator or the saturation unless it is very strong. At 3 bar pressure the anomaly was the strongest, and is shown in Fig. 3, where the QTF2 width saturated at 9 Hz. The minimum attained resonance widths as a function of pressure is plotted in Fig. 3. Small resonances were visible at pressures 16, 6, 5, and 2 bar pressures, even with the sweeping QTF2 measurement.
The quartz tuning fork also responded to rotation of the cryostat. When the cryostat first started to rotate, the resonance frequency dropped by 140 mHz as the width was kept constant by a PID controller. The cryostat was then rotated at various rotation velocities for 20 hours, during which the frequency slowly returned toward its original value.

Conclusions
We have observed a saturation in the temperature dependence of quartz tuning fork oscillators in two independent experiments in 3 He systems with surfaces coated by 4 He. In the adiabatic melting experiment, the temperature indicated by the QTF resonance width saturates to a value at which the cooling power of the helium isotope mixing process is still significantly larger than the external heat leak. In the nafen experiment, on the other hand, we had an NMR thermometer that showed continuing cooling in the experimental cell, even after the QTF width had saturated. We also observed strong pressure dependence in the saturation width value, with maximum being at 3 bar. We suggest that the anomalous behavior is due to the 4 He film covering the QTF, however, the detailed understanding of the phenomenon requires more analysis.