Helmholtz-resonator measurements of the superfluid density of liquidHe4in submicrometer-diameter channels

Abstract

The dependence of the effective relative superfluid density $\frac{{\rho }_{\mathrm{sp}}}{\rho }$ upon temperature $T$ has been measured in 0.10, 0.08, 0.05, and 0.03-μm-diameter cylindrical pores in 5.0-μm-thick polycarbonate films (Nuclepore) using a superfluid Helmholtz resonator. We observe that $\frac{{\rho }_{\mathrm{sp}}}{\rho }$ lies below the bulk relative superfluid density $\frac{{\rho }_{\mathrm{sb}}}{\rho }$, and that near $T_{\lambda }$, $\frac{{\rho }_{\mathrm{sp}}}{\rho }$ for a given pore size is represented well by the relationship $\frac{{\rho }_{\mathrm{sp}}}{\rho }=C{(t-t_{01})}^{\zeta }$, where $t\equiv \frac{(T_{\lambda }-T)}{T_{\lambda }}$, $C\simeq 2.46\pm 0.07$, $\zeta =0.675\mathrm{}\pm 0.020$, and $t_{01}$ is an effective pore-size-dependent reduced onset temperature equal to (0.26, 0.60, 0.89, and 2.84) × ${10}^{-3\mathrm{}}$ for the four pore sizes, respectively. At the same time we find that over a broad range of temperature, including the region near $T_{\lambda }$, $\frac{{\rho }_{\mathrm{sp}}}{\rho }$ for a given pore size can be represented well by the relationship $\frac{{\rho }_{\mathrm{sp}}}{\rho }=D[\frac{{\rho }_{\mathrm{sb}}\mathrm{}\left(t\right)\mathrm{}}{\rho }-\frac{{\rho }_{\mathrm{sb}}\left(t_{02}\right)}{\rho }]$, where $D$ is a constant and $t_{02}$ is an alternative effective reduced onset temperature. However, this form, which is linear in $\frac{{\rho }_{\mathrm{sb}}}{\rho }$, has an asymptotic behavior different from that above as $\frac{{\rho }_{\mathrm{sp}}}{\rho }\to 0$ and yields for each pore size a value for $t_{02}$ which is more than a factor 2 smaller than the value for $t_{01}$ above.