Characterization of the volume and shape of quasi-spherical resonators using coordinate measurement machines

Acoustic thermometry using a gas-filled quasi-spherical resonator (QSR) is one of the most promising techniques for measuring the Boltzmann constant k B with low uncertainty. Dimensional metrology with coordinate measurement machines (CMMs) can be used to determine the resonator's volume, either directly or in combination with measurements of the resonator's microwave spectra. We assessed the uncertainty achievable when using a CMM to characterize the shape and volume of three QSRs. The resonators differed significantly in their design and construction: their inner volumes ranged between 524 cm 3 and 2225 cm 3 , while the QSR geometries ranged from a diamond-turned triaxial ellipsoid to the variable misalignment of spheroidal hemispheres. Comparative coordinate measurements of two solid spherical density standards were used to identify and estimate type B uncertainties. We tested the regression of the CMM data to spherical harmonic expansions and determined the volume of a QSR directly with a relative uncertainty u R < 30 parts in 10 6 . Additionally, spherical harmonic regression of the CMM data can place uncertainty bounds on the eccentricity parameters, ε 1 and ε 2 , typically with a relative uncertainty u R ≈ 0.02. This is sufficient to determine corrections to both the acoustic and the microwave resonance frequencies of the QSR with a relative uncertainty u R < 1 part in 10 6 for all resonances. These figures assume that the enclosed volume of an assembled QSR is equal to the sum of the volumes of its two component 'hemispheres'. In practice this cannot be strictly true and the additional uncertainties in the volume of the assembled QSR are discussed.