Test of a virtual cylindrical acoustic resonator for determining the Boltzmann constant

Abstract

We report progress toward determining the Boltzmann constant kB using the concept of a virtual acoustic resonator, a hypothetical resonator that is mathematically equivalent to a cylindrical cavity with periodic boundary conditions. We derived the virtual resonator by combining the measured frequencies of the longitudinal acoustic modes of two argon-filled, cylindrical cavity resonators in such a way to minimize the effects of the cavities’ ends, including transducers and ducts attached to the ends. The cavities had lengths of 80 mm and 160 mm and were operated in their longitudinal (ℓ,0,0) modes. We explored virtual resonators that combine modes of the two resonators that have nearly the same frequencies. The virtual resonator formed from the (2,0,0) mode of the 80 mm resonator combined with the (4,0,0) mode of the 160 mm resonator yielded a value for kB that is, fractionally, only (0.2 ± 1.5) × 10−6 larger than the 2010 CODATA-recommended value of kB. (The estimated uncertainty is one standard uncertainty corresponding to a 68% confidence level.) The same virtual resonator yielded values of the pressure derivatives of the speed of sound c in argon, (∂c2/∂p)T and (∂c2/∂p2)T, that differed from literature values by 1% and 2%, respectively. By comparison, when each cavity was considered separately, the values of kB, (∂c2/∂p)T, and (∂c2/∂p2)T differed from literature values by up to 7 ppm, 10%, and 5%, respectively. However, combining the results from the (3,0,0) or (4,0,0) modes of shorter resonator with the results from the (6,0,0) or (8,0,0) modes of the longer resonator yielded incorrect values of kB that varied from run-to-run. We speculate that these puzzling results originated in an unmodeled coupling, either between the two cavities (that resonated at nearly identical resonance frequencies in the same pressure vessel) or between the cavities and modes of the pressure vessel.