Cylindrical Acoustic Resonator for the Re-determination of the Boltzmann Constant

Abstract

The progress towards re-determining the Boltzmann constant k B using two fixed-path, gas-filled, cylindrical, acoustic cavity resonators is described. The difference in the lengths of the cavities is measured using optical interferometry. Thus, a literature value for the density of mercury is not used, in contrast with the presently accepted determination of k B . The longitudinal acoustic resonance modes of a cylindrical cavity have lower quality factors Q than the radial modes of gas-filled, spherical cavities, of equal volume. The lower Qs result in lower signal-to-noise ratios and wider, asymmetric resonances. To improve signal-to-noise ratios, conventional capacitance microphones were replaced with 6.3 mm diameter piezoelectric transducers (PZTs) installed on the outer surfaces of each resonator and coupled to the cavity by diaphragms. This arrangement preserved the shape of the cylindrical cavity, prevented contamination of the gas inside the cavity, and enabled us to measure the longitudinal resonance frequencies with a relative standard uncertainty of 0.2 × 10 −6 . The lengths of the cavities and the modes studied will be chosen to reduce the acoustic perturbations due to non-zero boundary admittances at the endplates, e.g., from endplate bending and ducts and/or transducers installed in the endplates. Alternatively, the acoustic perturbations generated by the viscous and thermal boundary layers at the gas-solid boundary can be reduced. Using the techniques outlined here, k B can be re-determined with an estimated relative standard uncertainty of 1.5 × 10 −6 .

Progress Toward Redetermining the Boltzmann Constant with a Fixed-Path-Length Cylindrical Resonator

“…The perturbations to the ideal resonance frequencies f 0 l caused by imperfect geometry and other well-understood physical phenomena have been described in our previous publications [9,15]. For completeness, we review these perturbations here.…”

“…. is the longitudinal mode index [9]. (Since this work focuses on pure longitudinal modes, we refer to these modes using only the longitudinal index l in subscripts for convenience.…”

“…In general, the mechanical admittance associated with these deformations is difficult to calculate. In a prior publication [9], we calculated perturbations for three deformations: (1) radial ( f sh,1 ) and (2) axial ( f sh,2 ) deformations of a finite-length cylindrical shell, and (3) bending of the end plates ( f sh,3 ). These perturbations have similar functional forms:…”

“…Since the 1970s, acoustic resonators have been the preferred method for determining k B [2][3][4][5][6][7][8][9], thermodynamic temperatures [10][11][12][13][14], and the thermophysical properties of gases. For determining k B , acoustic resonators rely on kinetic theory to relate the speed of sound in a dilute monatomic gas to the kinetic energy of the gas atoms and, therefore, to the thermodynamic temperature.…”

“…(Part of this work was published previously [9,[15][16][17].) We also describe the corrections to the acoustic frequencies and the determination of k B from fits of the frequencies by a physically motivated function of the pressure.…”