Acoustic damping of quartz tuning forks in normal and superfluid ${}^3\mathrm{He}$

We investigate the damping experienced by quartz tuning fork resonators in normal and superfluid ${}^3\mathrm{He}$ as a function of their resonance frequency from 22 to 250 kHz and contrast it with the behavior of the forks in ${}^4\mathrm{He}$. For our set of tuning forks the low frequency damping in both fluids is well described by the existing hydrodynamic models. We find that the acoustic emission becomes the dominating dissipation mechanism at resonator frequencies exceeding approximately 100 kHz. Our results show that the acoustic emission model used in ${}^4\mathrm{He}$ fluid also describes acoustic damping in superfluid ${}^3\mathrm{He}$ and normal ${}^3\mathrm{He}$ at low temperatures using the same geometrical prefactor. The high temperature acoustic damping in normal ${}^3\mathrm{He}$ does not exceed prediction of this model and thus the acoustic damping of moderate frequency devices measured in ${}^4\mathrm{He}$ should be similar or smaller in ${}^3\mathrm{He}$ liquid.

Figure 1

(a) A schematic of the electronic measurement setup incorporating a photograph of a fork array comprising five tuning forks. (b) A picture of the quasiparticle camera with five tuning fork arrays used in the ${}^3\mathrm{He}$ experiments. Each fork is placed in an individual cylindrical cavity.

Figure 2

Log-log plot of frequency dependence of the tuning fork damping in ${}^4\mathrm{He}$ liquid for the fundamental (open symbols) and overtone (filled symbols) modes of the forks. The 4.2 and 1.5 K data [22] was taken at saturated vapor pressure, while the 450 mK measurements were carried out at a pressure of 22 bars. The dashed line corresponds to the hydrodynamic contribution described by Eq. (3). The dotted lines correspond to the 3D model of acoustic damping with ballistic or hydrodynamic contributions.

Figure 3

Log-log plot of the measured damping versus resonance frequency of the forks at the fundamental (open symbols) and overtone modes (filled symbols) in superfluid ${}^3\mathrm{He}$-B and vacuum (top); and normal fluid ${}^3\mathrm{He}$ (bottom). The dashed line corresponds to the hydrodynamic damping experienced by the forks. The dotted line shows the acoustic damping model described by Eq. (6) with hydrodynamic or ballistic contributions.

Figure 4

Log-log plot of the fork damping in ${}^4\mathrm{He}$ (top), normal ${}^3\mathrm{He}$ (center), and superfluid ${}^3\mathrm{He}$ (bottom) versus the sound wavelength corresponding to the fork's fundamental resonance (open symbols) and overtone resonance (filled symbols). The top panel shows bulk measurements in ${}^4\mathrm{He}$ using faded colors along with the measurements performed in a cylindrical cavity in bold colors. The shaded area to the left highlights the wavelengths corresponding to unsuppressed sound emission. The dotted and dashed lines correspond to the total and hydrodynamic damping models, respectively.