Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases

We evaluate a quasi-spherical, copper, microwave cavity resonator for accurately measuring the relative dielectric permittivity ${\epsilon }_{\mathrm{r}}\mathrm{(p,T)}$ of helium and argon. In a simple, crude approximation the cavity’s shape is a triaxial ellipsoid with axes of length $\text{a,1.001a}$ and $\text{1.005a}$ , with $\text{a=5\hspace{0.5em}}\mathrm{cm}$ . The unequal axes of the quasi-sphere separated each of the triply degenerate microwave resonance frequencies of a sphere ${\mathrm{(f}}_{11}^{\mathrm{TM}}{\mathrm{,f}}_{12}^{\mathrm{TM}}{\mathrm{,\dots ,f}}_{11}^{\mathrm{TE}}{\mathrm{,f}}_{12}^{\mathrm{TE}}\mathrm{,\dots )}$ into three nonoverlapping, easily measured, frequencies. The frequency splittings are consistent with the cavity’s shape, as determined from dimensional measurements. We deduced ${\epsilon }_{\mathrm{r}}\mathrm{(p,T)}$ of helium and of argon at $\text{289\hspace{0.5em}}\mathrm{K}$ and up to $\text{7\hspace{0.5em}}\mathrm{MPa}$ from the resonance frequencies $f_{\ln }^{\sigma }$ , the resonance half-widths $g_{\ln }^{\sigma }$ , and the compressibility of copper. Simultaneous measurements of ${\epsilon }_{\mathrm{r}}\mathrm{(p,T)}$ with the quasi-spherical resonator and a cross capacitor agreed within ${\text{1×10}}^{\text{−6}}$ for helium, and for argon they differed by an average of only ${\text{1.4×10}}^{\text{−6}}$ . This small difference is within the stated uncertainty of the capacitance measurements. For helium, the resonator results for ${\epsilon }_{\mathrm{r}}\mathrm{(p,T)}$ were reproducible over intervals of days with a standard uncertainty of ${\text{0.2×10}}^{\text{−6}}$ , consistent with a temperature irreproducibility of $\text{5\hspace{0.5em}}\mathrm{mK}$ . We demonstrate that several properties of quasi-spherical cavity resonators make them well suited to ${\epsilon }_{\mathrm{r}}\mathrm{(p,T)}$ determinations. Ultimately, a quasi-spherical resonator may improve dielectric constant gas thermometry and realize a proposed pressure standard based on ${\epsilon }_{\mathrm{r}}\mathrm{(p,T)}$ .