Effect of the boundary condition on the Kapitza resistance between superfluid ${}^3\mathrm{He}\text{−}B$ and sintered metal

Understanding the temperature dependence of thermal boundary resistance, or Kapitza resistance, between liquid helium and sintered metal has posed a problem in low-temperature physics for decades. In the ballistic regime of superfluid ${}^3\mathrm{He}\text{−}B$, we find the Kapitza resistance can be described via scattering of thermal excitations (quasiparticles) with a macroscopic geometric area, rather than the sintered metal's microscopic area. We estimate that a quasiparticle needs on the order of 1000 collisions to successfully thermalize with the sinter. Finally, we find that the Kapitza resistance is approximately doubled with the addition of two monolayers of solid ${}^4\mathrm{He}$ on the sinter surface, which we attribute to an extra magnetic channel of heat transfer being closed as the nonmagnetic solid ${}^4\mathrm{He}$ replaces the magnetic solid ${}^3\mathrm{He}$.

Figure 1

The experimental cell. The inner cell contains eighty $0.2\text{−}\mathrm{mm}$-thick and eight $1.1\text{−}\mathrm{mm}$-thick copper plates with dimensions $49\mathrm{mm}\times 28\mathrm{mm}$. The plates are sintered with a silver powder, giving a total microscopic sinter area of $80{\mathrm{m}}^2$. A $2\mathrm{cm}\times 2\mathrm{cm}\times 1\mathrm{cm}$ geometric volume is formed by cutting the middle plates. The cuts from these plates are in an alternating pattern of a $1.9\mathrm{cm}\times 1.1\mathrm{cm}$ rectangle followed by a $2.0\mathrm{cm}\times 1.0\mathrm{cm}$ rectangle to create $1\text{−}\mathrm{mm}$ protrusions that increase the surface area without changing the volume. The geometric volume contains the vibrating loop wire thermometer. A cylindrical volume containing the large wire and quartz tuning fork is connected via a circular cutout to the bottom of the geometric volume. The cylindrical volume also has a tail piece separated by a small hole.

Figure 2

Comparison of the response of the vibrating loop wire with diameter $4.5\mu \mathrm{m}$ for pure ${}^3\mathrm{He}$ and two layers of solid ${}^4\mathrm{He}$ coverage on the cell surface at temperatures of $197\mu \mathrm{K}$ ($\mathrm{\Delta }{f}_{\text{base}}=30.4\mathrm{Hz}$) and $193\mu \mathrm{K}$ ($\mathrm{\Delta }{f}_{\text{base}}=25.9\mathrm{Hz}$), respectively. The resonance width is related to temperature by the equation $\mathrm{\Delta }f=\mathrm{\Delta }{f}_{\mathrm{vac}}+\gamma exp\left(-{\mathrm{\Delta }}_B/k_{\mathrm{B}}T\right)$, where $\mathrm{\Delta }{f}_{\mathrm{vac}}=92\mathrm{mHz}$ is the resonance width in vacuum and $\gamma$ is a fitting constant [22]. The corresponding width change for the ${}^4\mathrm{He}$ data is given by the right $y$ axis. The ${}^3\mathrm{He}$ data were measured at a slightly different temperature, and hence the width change is $5\%$ larger for a given temperature change. Solid lines represent the fits from Eq. (1).

Figure 3

Comparison of quasiparticle relaxation time constant ${\tau }_b$ as measured by the vibrating loop thermometer. The average ${\tau }_b$ is $0.5\pm 0.1$ and $1.0\pm 0.1$ s for solid ${}^3\mathrm{He}$ and solid ${}^4\mathrm{He}$ coverage, respectively. Absence of discernible variation over temperature is consistent with an exponential dependence of $R_{\mathrm{K}}$.

Figure 4

The effective thermal boundary resistance $R_{\mathrm{K}}$ multiplied by the “geometric” sinter scattering area $A$ for quasiparticles created by the wire as a function of inverse temperature. $T_c=929\mu \mathrm{K}$ is the critical temperature of superfluid ${}^3\mathrm{He}$ at zero pressure. The gold and red lines are fits for data from Refs. [3, 14], respectively, and converted into thermal boundary resistance. For comparison, our data shown are only in pure ${}^3\mathrm{He}$. They lie mostly on the gold line, or between the two lines if the tailpiece volume is included.