Dissipative superfluid mass flux through solid ${}^4\mathrm{He}$

The thermomechanical effect in superfluid helium is used to create an initial chemical potential difference $\Delta {\mu }_0$ across a solid ${}^4\mathrm{He}$ sample. This $\Delta {\mu }_0$ causes a flow of helium atoms from one reservoir filled with superfluid helium, through a sample cell filled with solid helium, to another superfluid-filled reservoir until chemical potential equilibrium between the reservoirs is restored. The solid helium sample is separated from each of the reservoirs by Vycor rods that allow only the superfluid component to flow. With an improved technique, measurements of the flow $F$ at several fixed solid helium temperatures $T$ have been made as a function of $\Delta \mu$ in the pressure range 25.5–26.1 bar, and measurements of $F$ have been made as a function of temperature in the range $180<T<545$ mK for several fixed values of $\Delta \mu$. The temperature dependence of the flow above 100 mK shows a reduction of the flux with an increase in temperature that is well described by $F=F_0^{*}[1-Kexp(-E/T)]$. The nonlinear functional dependence $F\sim {\left(\Delta \mu \right)}^b$, with $b<0.5$ independent of temperature but dependent on pressure, documents in some detail the dissipative nature of the flow and suggests that this system demonstrates Luttinger liquid-like one-dimensional behavior. The mechanism that causes this flow behavior is not certain, but is consistent with superflow on the cores of edge dislocations.

Figure 1

Schematic rendition of the sample cell (not to scale), which consists of two Vycor rods $V1$ and $V2$, reservoirs $R1$ and $R2$, their heaters $H1$ and $H2$, and a cylindrical space for the helium sample ($1.84$ ${\mathrm{cm}}^3$) with a Straty-Adams pressure gauge (Ref. [31]), $C1$ and $C2$, at each end of the solid helium region. A chemical potential difference can be applied between the two reservoirs. $TC$, $T1$, and $T2$ are calibrated resistance thermometers for the helium sample cell and for the two liquid helium reservoirs, respectively. Filling capillaries 1 and 2 lead to reservoirs $R1$ and $R2$, and capillary 3 is thermally connected to the 1 K pot and still and enters from the side and provides direct access to the helium sample space for efficient initial filling of the cell.

Figure 2

Dependence of changes in pressures that accompany changes in $T1$ temperatures for a cell temperature $TC=233$ mK. Changes in pressures $P1$ and $P2$ are seen, which result from stepwise increases of $T1$ in increments of 10 mK, with $T2$ constant. Short dashed tilted lines each with the same constant slope 0.126 mbar/s are guides to the eye to help reveal the presence of sequential changes in the slope of $\Delta P$. These representative data have been selected from a long sequence of similar data, and we have started the time axis at zero for this portrayal.

Figure 3

Dependence of the maximum flow rate $F$ resulting from 10 mK changes in the reservoir temperature $T1$ for fixed $T2$ with the pressure of the solid in the cell $26.11\pm 0.06$ bar (as measured by the gauges $C1$ and $C2$). Here $T1$ is the resulting temperature of reservoir R1 following each step of $\Delta T=10$ mK, and $F$ is the resulting maximum value of the flux observed for each step in $T1$. The data for $TC=233$ mK are obtained from the data for $TC=233$ mK shown in Fig. 2.

Figure 4

Response of pressures $P1$ and $P2$ to the application of sequential $\Delta T$ reversals (see text) with $\delta T$ = 5 mK for a sequence of solid helium temperatures, here $0.25\lesssim TC\lesssim 0.55$ K. Use of heaters $H1$ and $H2$ results in changes in $T1$ and $T2$. The resulting changes in $P1$ and $P2$ are best seen as $P1-P2$, shown here (b, red circles). The small drift in $P1$ and $P2$ of the sort seen here is typical and variable and appears to have no significant influence on $P1-P2$. These data have been selected from a longer sequence of similar data, and we have reset the time axis to zero for this portrayal.

Figure 5

Behavior of the pressure difference $\Delta P$ after applying a positive $\Delta T$ ($\delta T=5$ mK) between the two liquid helium reservoirs at different solid helium temperatures. These data are extracted from data sets like that shown in Fig. 4 with $P1-P2$ increasing. The temperatures shown are the cell temperatures $TC$. Similar behavior is present for $P1-P2$ decreasing.

Figure 6

The flow rate $F$ vs $\Delta \mu$ after application of $\Delta T>0$ between the two liquid helium reservoirs for different solid helium temperatures determined from the data shown in Fig. 5. Solid lines here are fits to the data by $F=a{\left(\Delta \mu \right)}^b$.

Figure 7

Temperature dependence of the fit parameters $a$ and $b$ for three solid ${}^4\mathrm{He}$ samples each with different cell pressure. Lines are guides to the eye.

Figure 8

Pressure dependence of the fit parameter $b$ determined for samples with six different solid ${}^4\mathrm{He}$ sample pressures; $Pmc=25.34$ bar. The line is a linear fit (see text).

Figure 9

Temperature dependence of the mass flow $F$ through solid helium for different fixed $\Delta \mu$ values at a cell pressure of 26.11 $\pm 0.1$ bar. Solid lines are the result of fits to Eq. (5) with $B/A=1.21$ and $E=117\pm 2,117\pm 2$ and $112\pm 4$ mK for $\Delta \mu =5,3$ and 1 mJ/g, respectively. Dashed lines are the result of an alternate fit described in the text.

Figure 10

Pressure dependence of the Luttinger parameter $g$ in terms of the distance from the melting curve presuming that $g=\left[\right(1/p)+1]/2$, with $p=b$; again here, $Pmc=25.34$ bar.

Figure 11

Response of pressures $P1$ and $P2$ to the application of several steps in $\Delta T$ for the case of liquid helium in the experimental cell at 23.2 bar and a cell temperature of $T=105$ mK. Use of heater H1 with H2 fixed results in changes to $T1$ ($\approx 10$ mK each) with $T2$ fixed. The resulting $\Delta T$ and changes in $P1$ and $P2$ are shown and best seen as $\Delta P=P1-P2$. Note that when the reservoir temperatures exceed $\approx 1.50$ K the response of $P1$ and $P2$ slows as shown by the decrease in $d\Delta p/dt$ above 1.5 K (lines of equal constant slope, 0.150 mbar/sec, are shown as guides to the eye); i.e., the flux is limited by the Vycor. As with an earlier figure, these representative data have been selected from a long sequence of similar data, and we have reset the time axis to zero for this portrayal.

Figure 12

Maximum flux (calculated from the data shown in the previous figure) following the application of a temperature step $\Delta T\approx 10$ mK for the case of liquid helium in the experimental cell at 23.2 bar for $T$ = 105 mK. Also shown for comparison are the data for the case of solid helium in the cell (from Fig. 3). Use of heater H1 or H2 results in changes to $T1$ and $T2$. In each case $Ti$ is changed with $Tj$ fixed at 1.462 K. The time resolution of our data limits our ability to accurately measure the flux to about 0.4 mbar/sec. For the $T1$ and $T2$ values used in our paper, the resulting values for the solid-limited flux fall well below the values achieved when only superfluid ${}^4\mathrm{He}$ is in the cell.

Figure 13

Response of pressures $P1$ and $P2$ to the application of a temperature step $\Delta T$ for the case of liquid helium in the experimental cell at 22.0 bar and $T=450$ mK. Use of heater H1 and H2 results in changes to $T1$ and $T2$. The resulting $\Delta T$ and changes in $P1$ and $P2$ are shown as is $\Delta P=P1-P2$. Lines through the data for $\Delta T$ and $\Delta P$ on this and the next figure are guides to the eye.

Figure 14

Response of pressures $P1$ and $P2$ to the application of a temperature step $\Delta T$ for the case of liquid helium in the experimental cell at 22.0 bar and $T=450$ mK. Use of heater H1 and H2, in this case, results in changes to $T1$ and $T2$, which here are the reverse of those shown in the previous figure. The resulting $\Delta T$ and changes in $P1$ and $P2$ are shown as is $\Delta P=P1-P2$.