Decoupling of first sound from second sound in dilute ${}^3\mathrm{He}–\mathrm{superfluid}{}^4\mathrm{He}$ mixtures

Bulk superfluid helium supports two sound modes: first sound is an ordinary pressure wave, while second sound is a temperature wave, unique to superfluid systems. These sound modes do not usually exist independently, but rather variations in pressure are accompanied by variations in temperature, and vice versa. We studied the coupling between first and second sound in dilute ${}^3\mathrm{He}$–superfluid ${}^4\mathrm{He}$ mixtures, between 1.6 and 2.2 K, at ${}^3\mathrm{He}$ concentrations ranging from 0% to 11%, under saturated vapor pressure, using a quartz tuning fork oscillator. Second sound coupled to first sound can create anomalies in the resonance response of the fork, which disappear only at very specific temperatures and concentrations, where two terms governing the coupling cancel each other, and second sound and first sound become decoupled.

Figure 1

(a) Calculated temperature [relative to ${}^4\mathrm{He}$ superfluid transition temperature $T_{\lambda }=T_{\lambda }\left(x_3\right)$], where first sound decouples from second sound ($\alpha =0$), plotted as a function of molar ${}^3\mathrm{He}$ concentration $x_3$. (b) Values of the bracketed term of $\alpha$ in Eq. (5) as a function of relative temperature at different ${}^3\mathrm{He}$ concentrations. To clarify the relation between the plots, a few examples of equivalent points in the figures are indicated by $▵,\square$, and $◯$.

Figure 2

Schematic drawing of the ${}^3\mathrm{He}–{}^4\mathrm{He}$ mixture cell. The buffer volume above the horizontal quartz tuning fork volume helps maintain the liquid under saturated vapor pressure.

Figure 3

Resonance width ($\mathrm{\Delta }f$) of the 32 kHz quartz tuning fork versus its resonance frequency ($f$) with number of ${}^3\mathrm{He}$ concentrations. The figure contains temperature sweeps both down and up, falling basically on top of each other. The $\lambda$ point appears as tilted V shape near 20 Hz width, on each curve, and below that the linear decline is due to change in temperature as the viscous drag diminishes rapidly. Second sound anomalies appear as loops, whose magnitude is proportional to the coupling strength between second and first sound. There are two pure ${}^4\mathrm{He}$ measurement sets to demonstrate the reproducibility (see the inset). The 11.0% data set does not include the lowest temperature anomalies as its main purpose was to be a reference point for our ${}^3\mathrm{He}$ concentration analysis.

Figure 4

4.2% ${}^3\mathrm{He}$ concentration fork trace with key features highlighted. Temperature values at different points are shown next to the curve. Arrows indicate the direction of the temperature sweep: first down from 2.53 to 1.71 K, and then back up to 2.36 K. Dashed line represents the linear fit to the decline below $T_{\lambda }$, which was used in concluding the concentration value.

Figure 5

Closer view of one set of second sound anomalies (the ones shown in the inset of Fig. 3) followed through the decoupling region. The color of the line changes according to temperature, showing that the anomalies move to a higher relative temperature as the ${}^3\mathrm{He}$ concentration increases. Even though the shape of the features changes, we can still identify the three biggest anomalies of pure ${}^4\mathrm{He}$ also in 4.2% mixture, labeled as 1–3.

Figure 6

Locations of the second sound anomalies followed through the decoupling region in ${}^3\mathrm{He}$ concentration–relative temperature plane, as well as their amplitude (represented by the size of the circle). The set of anomalies that were presented in Fig. 5, up to 4.2% concentration, are indicated by the arrow. The dashed line corresponds to the empirical fit of Fig. 8, which separates the regions before and after the decoupling.

Figure 7

Amplitude $a$ of a typical second sound anomaly as a function of ${}^3\mathrm{He}$ concentration. Amplitudes were normalized to the value at 9% concentration. Fits at either side of the minimum were used to evaluate the decoupling concentration. Solid circles indicate the points where we could say with confidence that the anomalies had not yet disappeared, which were then used as the error bars of Fig. 8.

Figure 8

Decoupling temperatures and concentrations obtained from experimental data compared to the calculated behavior shown by the solid line [cf. Fig. 1]. The dashed line is an empirical fit to the evaluated decoupling points. Datapoint obtained from Fig. 7 is distinguished as the solid symbol.

Figure 9

Constant second sound velocity curves as a function of ${}^3\mathrm{He}$ concentration and relative temperature (cf. Fig. 6). The velocity values were obtained from the ${}^4\mathrm{He}$ second sound velocity data by Donnelly and Barenghi [21] when the anomalies were visible in pure ${}^4\mathrm{He}$. Otherwise (indicated by prefix $\sim$), they were evaluated by using the sources presented at the end of Sec. 2.