Experiments on the rapid mechanical expansion of liquid ${}^4\mathrm{He}$ through its superfluid transition

Phenomena following a rapid mechanical quench of liquid ${}^4\mathrm{He}$ from its normal to its superfluid phase are reported and discussed. The mechanical expansion apparatus is an improved version of that described previously. It uses a double-cell geometry to effect a partial separation of the sample from the convolutions of the bellows that form the outer wall of the cell. Consistent with earlier work, no evidence is found for the production of quantized vortices via the Kibble-Zurek (KZ) mechanism. Although the expansion is complete within $15\mathrm{ms}$, the second-sound velocity and attenuation continue to increase for a further $\sim 60\mathrm{ms}$; correspondingly the temperature decreases. Subsequently, the temperature rises again toward its final value as the second-sound velocity and attenuation decrease. It is shown that this unexpected behavior is apparently associated with a large-amplitude second-sound oscillation produced by the expansion, and it is suggested that the observed vortices are created by the normal fluid–superfluid counterflow that constitutes the second-sound wave. If production of large-amplitude second sound is inherent to the mechanical expansion of liquid ${}^4\mathrm{He}$ through the superfluid transition, as appears to be the case for final temperatures more than $3\mathrm{mK}$ from the $\lambda$ transition, the phenomenon sets a lower bound on the density of KZ vortices that can be detected in this type of experiment.

Figure 1

Phase diagram of ${}^4\mathrm{He}$ showing schematically (dashed line) a trajectory from an initial temperature and pressure $(T_i,P_i)$ in He I to a final state $(T_f,P_f)$ in He II. The dotted line separates regions of positive and negative nonlinear coefficients of second-sound velocity.

Figure 2

Schematic diagrams of the expansion apparatus showing (a) the arrangements for filling and compressing the cell; (b) the outer cell; (c) the inner cell. The cylindrical cell was of quartz, closed by two plates holding the heater and bolometer, respectively. It was placed inside the expansion chamber, which had phosphor-bronze concertina walls. Slots in the cell walls allowed for equalization of the internal and external pressures. The sample filling capillary was closed off by means of a hydraulically operated needle valve after the chamber had been pressurized. A capacitance gauge was used to determine the pressure.

Figure 3

Time dependence of the chamber length during typical expansions: from $T_i=4.2\mathrm{K}$, $P_f=3\text{bar}$ (top curve) ending in He I; and from $T_i=2.1\mathrm{K}$, $P_f=4\text{bar}$ (lower curve).

Figure 4

Typical train of reflected second-sound pulses as detected by the bolometer, becoming more attenuated on successive passages through the vortex tangle. Note that the test pulses are superimposed on top of a slow oscillatory temperature oscillation.

Figure 5

Energy attenuation of successive second reflections, plotted as a function of time after the quench, providing a measure of the inverse vortex density. The relative attenuation in the absence of vortices is shown by the horizontal line at $Q_{i+1}∕Q_i=0.9$.

Figure 6

Time dependence of the second-sound velocity $v(t,T)$ (left hand scale) after the quench for different initial temperatures $T_i$. The right hand scale shows the temperature difference from the superfluid transition, $T_{\lambda }-T$.

Figure 7

Temperature oscillation in the cell after a quench. Vertical lines indicate the times when probe pulses were excited from the heater. The direct signal picked up by the bolometer, and some reflections, are indicated in each case. SS indicates second sound.

Figure 8

The parameter $q=\alpha ∕(1-{\alpha }^{\prime })$ (dimensionless units) plotted versus $T_{\lambda }-T$ for He II.