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.

Measurement of the Boltzmann Constant k B Using a Quasi-Spherical Acoustic Resonator

“…One of the hypothesis is that when the two hemisphere are assembled and bolted, the pressure of the bolts may compress the hemispheres reducing the volume perceived by the electromagnetic field. This was first suggested by Albo [24], and it was confirmed by Underwood et al [25]. When we assemble resonators in the future, we will use the microwave technique to measure the changes in volume while we gradually tighten the bolts.…”

“…In contrast, the ratios were 1:1.25:1.82 in Fig. 6 from [25]. In [25], the amplitude ratio TM11y:TM11x = 1.25 corresponded to a misalignment of 6 µm.…”

“…6 from [25]. In [25], the amplitude ratio TM11y:TM11x = 1.25 corresponded to a misalignment of 6 µm. In our case, the ratio TM11y:TM11x = 1.05 corresponds to a misalignment of approximately 1 µm.…”

“…The difference between the average radius estimated using only TE modes and that estimated using only TM modes provides an estimation of the presence of such a layer. According to [25], we estimated a 10 nm copper oxide layer because a eq.TM11 − a eq.TE = 3.4 nm, where a eq.TM11 is the equivalent radius calculated with the TM11 mode and a eq.TE is the average of the radii calculated with the TE modes.…”

A determination of the molar gas constantRby acoustic thermometry in helium

“…For these results, the reported uncertainties are statistically determined from the regression, and are not representative of the overall uncertainty which is dominated by much larger type B contributions from the CMM instrumentation and procedure. Though these type of contributions may in principle be significantly reduced [21], their assessment requires an estimate of the elastic deformation of the assembled cavity upon tightening of the equatorial bolts, and the amplitude of the equatorial gap caused by the imperfect flatness of the mating equatorial flanges. A careful evaluation of these effects was not attempted here.…”