Dimensional characterization of a quasispherical resonator by microwave and coordinate measurement techniques

Abstract

We describe the dimensional characterization of copper quasisphere NPL-Cranfield 2. The quasisphere is assembled from two hemispheres such that the internal shape is a triaxial ellipsoid, the major axes of which have nominal radii 62.000 mm, 62.031 mm and 62.062 mm. The artefact has been manufactured using diamond-turning technology and shows a deviation from design form of less than ±1 µm over most of its surface. Our characterization involves both coordinate measuring machine (CMM) experiments and microwave resonance spectroscopy.We have sought to reduce the dimensional uncertainty below the maximum permissible error of the CMM by comparative measurements with silicon and Zerodur spheres of known volume. Using this technique we determined the equivalent radius with an uncertainty of u(k = 1) = 114 nm, a fractional uncertainty of 1.8 parts in 106. Due to anisotropy of the probe response, we could only determine the eccentricities of the quasihemispheres with a fractional uncertainty of approximately 2%.Our microwave characterization uses the TM11 to TM18 resonances. We find the equivalent radius inferred from analysis of these modes to be consistent within ±4 nm with an overall uncertainty u(k = 1) = 11 nm. We discuss corrections for surface conductivity, waveguide perturbations and dielectric surface layers.We find that the CMM radius estimates derived from each hemisphere cannot be used to accurately predict the equivalent radius of the assembled resonator for two reasons. Firstly, the equatorial flanges are flat only to within ±1 µm, leading to an equatorial ‘gap’ whose dimension cannot be reliably estimated. Secondly, the resonator undergoes significant elastic distortion when the bolts connecting the hemispheres are tightened. We provide CMM and microwave measurements to support these conclusions in addition to finite-element modelling.Finally, we consider the implications of this work on a forthcoming experiment to determine the Boltzmann constant with a relative uncertainty below 1 part in 106.