Transport in very dilute solutions of ${}^3$He in superfluid ${}^4$He

Motivated by a proposed experimental search for the electric dipole moment of the neutron (nEDM) utilizing neutron-${}^3$He capture in a dilute solution of ${}^3$He in superfluid ${}^4$He, we derive the transport properties of dilute solutions in the regime where the ${}^3$He are classically distributed and rapid ${}^3$He-${}^3$He scatterings keep the ${}^3$He in equilibrium. Our microscopic framework takes into account phonon-phonon, phonon-${}^3$He, and ${}^3$He-${}^3$He scatterings. We then apply these calculations to measurements by Rosenbaum et al. [J. Low Temp. Phys. 16, 131 (1974)] and by Lamoreaux et al. [Europhys. Lett. 58, 718 (2002)] of dilute solutions in the presence of a heat flow. We find satisfactory agreement of theory with the data, serving to confirm our understanding of the microscopics of the helium in the future nEDM experiment.

Figure 1

Thermal conductivity of a dilute solution of ${}^3$He in superfluid ${}^4$He at $T=0.45$ K. The solid line shows the phonon thermal conductivity, in the limit in which the ${}^3$He are at rest in equilibrium, calculated with ${}^3$He recoil (see the text). The dashed line shows the phonon thermal conductivity [Eq. (27)] without recoil corrections. The open circles are three representative measurements at $T\sim 0.43$ K at the average $x_3$ reported in Ref. 3; the solid circles with the error bars are the same thermal conductivities, but shown at the corrected $x_3$, as discussed in Sec. 6.

Figure 2

Ratio of the contribution of the viscous term [see Eq. (4)] to the drag on the phonons due to scattering against the ${}^3$He at a temperature of 0.45 K. The magnitude of the drag is given by the ${}^3$He pressure gradient [see Eq. (25)] as a function of ${}^3$He concentration $x_3$ (the concentrations measured in Ref. 4 are $x_3=7\times {10}^{-5}$ and $3\times {10}^{-4}$). The viscous term is calculated for laminar flow in a circular pipe of radius $R=2.5$ cm, using the ${}^4$He viscosity measured by Greywall.[23]

Figure 3

An example of the ${}^3$He distribution for a cell 5.2 cm in diameter and 5.0 cm high, with refrigerator temperature 0.45 K, average $x_3=3\times {10}^{-4}$, and heater power 15 mW. The sections are (a) perpendicular to the cylinder axis and containing the neutron beam axis and (b) perpendicular to the neutron beam (the cross in the figure) and through the center of the heat source. We assume in this simulation that the cylindrical wall of the cell is at the same temperature as that of the base where the refrigerator is attached; the top surface is insulated. The ${}^3$He concentration contours, of constant spacing, are marked in units of ${10}^{-4}$; the minimum is $x_3=0.69\times {10}^{-4}$ at the surface of the heater and the maximum concentration is $3.05\times {10}^{-4}$, on the lower cell boundary.

Figure 4

Contribution of the thermoelectric transport term relative to the effective diffusion constant $D_L$ extracted in Ref. 4. The temperatures are 0.45 K (dotted), 0.55 K (dashed), and 0.65 K (solid).

Figure 5

Representative results of the finite element calculation, including ${}^3$He recoil, of the relative integrated ${}^3$He (column) density along the neutron beam as a function of the quantity ${\xi }_L=T^7(\mathcal{P}/S_{\text{ph}}T)$, where $\mathcal{P}$ is the heater power, along with data from Ref. 4 (circles, 0.45 K; diamonds, 0.55 K, and squares, 0.65 K). The curves are for a refrigerator (and barrel) temperature of $0.45$ K: solid, $x_3=3\times {10}^{-4}$; dotted, $x_3=7\times {10}^{-5}$; coincidentally, the results for $T=0.65$ K and $x_3=3\times {10}^{-4}$ are essentially the same as the dotted curve. The curves are plotted for the average temperature (greater than the boundary temperature) along the neutron beam path. For example, at the actual value $T^6\mathcal{P}/S_{\text{ph}}=2.5$, the average ${\xi }_L$ is shifted upward by $\Delta {\xi }_L\approx 0.05K^7$ cm${}^3$/s for $T=0.45$ K, and by $\Delta {\xi }_L\approx 0.5K^7$ cm${}^3$/s for $T=0.65$ K.