Reinvestigation of the rotation effect in solid ${}^4\mathrm{He}$ with a rigid torsional oscillator

We reexamined the rotation-induced effect observed in solid ${}^4\mathrm{He}$ by using a rigid two-frequency torsional oscillator (TO). The previous rotation experiments reported the rotation-induced suppression of the “nonclassical” TO response that was interpreted as evidence of irrotational bulk superfluidity in solid ${}^4\mathrm{He}$. However, the experiment employed a nonrigid TO that could amplify the elastic contribution in the TO response. Thus, it is important to clarify if the rotation-induced suppression of the TO response could be attributed to an unavoidable elastic effect. In our rigid TO, complicated nonlinear viscoelastic contributions are systematically eliminated. In addition, the TO operating at two different resonant frequencies allows us to decompose a possible superfluidlike frequency-independent contribution on period drop from that of the linear elastic overshoot effect. We found no substantial rotation-induced effect in the out-of-phase resonant mode unlike that found in the previous rotation experiments. It indicates that the previous rotation effect in the nonrigid TO cannot be attributed to the genuine supersolidity. According to the frequency analysis of the TO response, the frequency-dependent period drop, which can be attributed to the elastic overshoot effect, remains unaffected upon application of dc rotation. However, the frequency-independent superfluidlike contribution exhibits a strikingly different rotation effect that is currently inexplicable.

Figure 1

KAIST rigid double-torus torsional oscillator (TO). (a) The TO consists of two torus-shaped solid ${}^4\mathrm{He}$ containers and two torsional rods. A pair of main and supplementary capacitive electrodes are attached to lower and upper container, for both drive and detection of the TO responses. (b) Similar to a double mass-spring system, the TO has two different resonant eigenmodes. The displacement of the TO estimated by finite-element method (FEM) simulation is shown. The color on the figures shows displacement of each component. Two containers oscillate in same direction in the in-phase mode (448.65 Hz), while they oscillate in the opposite direction in the out-of-phase mode (1131.20 Hz). (c) The TO is driven on its self-resonance through a phase-locked measurement by using two PAR124 lock-in amplifiers.

Figure 2

Effect of dc rotation on rigid TO responses. Period drop $dP$ of rigid TO, with respect to the mass loading of solid growth $\mathrm{\Delta }P\left(T\right)$, in the (a) in-phase and (b) out-of-phase mode are plotted against the temperature under various dc rotation speeds in a range of 0 to 4 rad/s. Dissipation $Q^{-1}\left(T\right)$ in the (c) in-phase and (d) out-of-phase mode are also measured simultaneously.

Figure 3

Dependence of the rigid TO responses on dc rotation and ac oscillation speed. (a) Rotation velocity-sweep scans were performed at a constant temperature of 30 mK. The measured period values $dP$ were shifted by $P_0$, for more clear representation; the value of $P_0$ is 2,267,631.549 ns (884,013.648 ns) for the in-phase (out-of-phase) mode. (b) Suppression percentage of frequency-independent term ${\left[d{P}_{\mathrm{in}}/\mathrm{\Delta }{P}_{\mathrm{in}}\left(T\right)\right]}^{\mathrm{ind}}$ and -dependent term ${\left[d{P}_{\mathrm{in}}/\mathrm{\Delta }{P}_{\mathrm{in}}\left(T\right)\right]}^{\mathrm{dep}}$ was plotted against a dc rotation speed and ac oscillating speed, respectively.

Figure 4

Frequency-dependent analysis under various dc rotation speed. The temperature dependence of $dP/\mathrm{\Delta }P$ in the in-phase mode are decomposed into (a) the frequency-independent term ${\left[d{P}_{\mathrm{in}}/\mathrm{\Delta }{P}_{\mathrm{in}}\left(T\right)\right]}^{\mathrm{ind}}$ (superfluid contribution) and (b) the frequency-dependent term ${\left[d{P}_{\mathrm{in}}/\mathrm{\Delta }{P}_{\mathrm{in}}\left(T\right)\right]}^{\mathrm{dep}}$ (elastic overshoot contribution) while maintaining the lowest ac oscillating speed.

Figure 5

Frequency-dependent analysis under various ac oscillating speed. Both (a) frequency-independent term ${\left[d{P}_{\mathrm{in}}/\mathrm{\Delta }{P}_{\mathrm{in}}\left(T\right)\right]}^{\mathrm{ind}}$ and (b) frequency-dependent term ${\left[d{P}_{\mathrm{in}}/\mathrm{\Delta }{P}_{\mathrm{in}}\left(T\right)\right]}^{\mathrm{dep}}$ were decomposed explicitly, under constant dc rotation speed of 0 (open marks) and 3 rad/s (solid marks). The ac driving voltage was carefully adjusted so that the solid sample experienced the same oscillating speed for the in-phase and out-of-phase mode at each ac oscillating speed. The same color-coded pair for each mode indicates the data obtained at the same ac oscillation speed.