Physical origin of the mode-dependent inconsistencies in cylindrical acoustic gas thermometry: even–odd $A_{0,l}$ dependence, nonlinear gas–shell coupling, and negative excess half-widths

The 2017 NIM/NIST determination of the Boltzmann constant with a fixed-length cylindrical acoustic gas thermometer (c-AGT) documented three reproducible anomalies that its perturbation model could not explain: mode-to-mode inconsistency of the fitted zero-pressure squared speed of sound $A_{0,l}$, with an unexplained even–odd longitudinal-mode pattern; pressure-dependent frequency perturbations of shell-coupled modes (A4, A5), in which the deviations of A4 show nonlinear negative curvature above about 400 kPa, beyond first-order gas–shell coupling theory; and non-physical negative excess half-widths of the A1 mode at low pressure, indicating omitted phenomena such as transducer–transducer coupling. Since the 2019 redefinition of the kelvin fixed the Boltzmann constant exactly, the surviving open problem is the physical origin and quantitative modeling of these mode-dependent perturbations, which still limit cylindrical-cavity primary thermometry at the few-parts-in-$10^6$ level. Identifying and modeling (or eliminating) them is the identified route to sub-few-ppm c-AGT for $(T-T_{90})$ determination and a future revision of ITS-90.

Background

Acoustic gas thermometry (AGT) is a method of primary thermometry: it determines the thermodynamic temperature $T$ (or, before 2019, the Boltzmann constant $k_B$) from the zero-pressure speed of sound $w_0$ in a gas of known molar mass, through the ideal-gas relation $w_0^2 = \gamma_0 k_B T/m = \gamma_0 R T/M$, where $\gamma_0$ is the ideal-gas heat-capacity ratio, $m$ the molecular mass, $M$ the molar mass, and $R$ the molar gas constant. The speed of sound is measured from the resonance frequencies of a gas-filled metal cavity whose dimensions are known from independent metrology. While the most precise determinations used spherical or quasi-spherical cavities, the National Institute of Metrology of China (NIM), in collaboration with the US National Institute of Standards and Technology (NIST), developed fixed-path-length cylindrical acoustic gas thermometry (c-AGT) over 2011–2017, culminating in a combined value of $k_B$ with relative standard uncertainty $2.0\times10^{-6}$.

In a cylinder of length $L$ and radius $a$, the purely longitudinal acoustic modes $A_l$ ($l=1,2,\dots$) have unperturbed frequencies $f_l = l\,w/(2L)$, where $w$ is the speed of sound in the gas. Along an isotherm, the corrected values of $w^2$ at several pressures are fitted by an acoustic-virial expansion, $w^2(p) = A_0 + A_1 p + A_2 p^2 + A_{-1}p^{-1}$, where $A_0 \equiv w_0^2$ should be the same for every mode $l$ if the experiment is correctly modeled; the per-mode estimates are denoted $A_{0,l}$. The corrections that must be applied to measured resonance frequencies account for the thermal and viscous boundary layers where the gas meets the cavity walls (computed to second order), gas inlet and outlet ducts, the motion of the piezoelectric source and detector transducers, and the elastic vibrations of the resonator shell coupled to the acoustic field. These perturbations are treated as independent and additive, to first order; in first-order models, the frequency shift from gas–shell coupling is a linear function of the gas pressure. Each resonance also has a half-width $g$ (its damping rate), and the excess half-width $\Delta g = g_{\mathrm{meas}} - g_{\mathrm{cal}}$ — the difference between measured and calculated damping — serves as a diagnostic for unmodeled dissipative or coupling phenomena. A negative $\Delta g$ is non-physical, since known loss mechanisms only increase damping; it signals that the measured resonance profile is inconsistent with the single-mode resonance model.

The 2017 combined result (Feng et al. 2017, Metrologia 54, 748) used an oxygen-free-copper cylindrical resonator whose length was measured by microwave rather than optical-interferometric methods, and documented the anomalies that are the subject of this problem. First, the values of $A_{0,l}$ disagree from mode to mode at the parts-in-$10^6$ level; the disagreement is reproducible for each resonator and is, in the authors' words, 'by far the largest uncertainty' in every NIM c-AGT determination, past and present. Within this scatter there is a systematic even–odd pattern: the values of $A_{0,l}$ for even $l$ (2 and 4) are larger than those for odd $l$ (3 and 5), which the authors state they cannot explain. (The paper notes that an error in the center-of-mass correction would produce an even–odd pattern in the first-pressure-virial terms $A_{1,l}$, not in $A_{0,l}$, so that hypothesis does not account for the observation.) Second, the vacuum vibration spectrum of the resonator shows elastic shell modes overlapping the acoustic modes A4 and A5; the A4 mode in particular has an anomalous pressure dependence, and its deviations from the fitted isotherm show a definite negative curvature above about 400 kPa, whereas first-order coupling models predict pressure-linear shifts — the paper states that the negative curvature 'requires a better theory for the coupling of a gas mode to an elastic shell mode'. Third, the A1 mode exhibits large negative excess half-widths at low pressures, which the authors take as evidence that some phenomenon (they suggest coupling between the two piezoelectric transducers, which was directly observed in the vacuum spectrum) is omitted from the resonance model even in the zero-pressure limit; the A1 mode was therefore excluded from the $k_B$ analysis.

The metrological context has since changed in one respect: the CODATA 2017 special adjustment of the fundamental constants (CODATA is the Committee on Data of the International Science Council) and the 2019 redefinition of the SI fixed the Boltzmann constant exactly at $1.380649\times10^{-23}$ J/K, completing the determination campaign that motivated the source paper. Primary acoustic thermometry was then redirected to measuring the difference $(T-T_{90})$ between thermodynamic temperature $T$ and the International Temperature Scale of 1990 (ITS-90), whose consensus values underpin temperature measurement and are periodically revised from primary-thermometry data. Cylindrical resonators are a leading candidate for this task, especially above 300 K, because long cavities fit into heat pipes (highly uniform isothermal enclosures) and admit long-stem transfer thermometers; the NIM/NIST collaboration has extended c-AGT to $(T-T_{90})$ over 234–303 K and to temperatures up to 809 K. Throughout this continuation, the c-AGT standard uncertainty has remained at the few-parts-in-$10^6$ level ($1.9\text{–}2.6\times10^{-6}$ over 234–303 K, and $3\text{–}5\times10^{-6}$ in the 580–809 K extension), with the mode-dependent inconsistencies still unexplained — that is, the three anomalies of the 2017 paper, not the value of any constant, are what stand between c-AGT and the sub-parts-per-million regime.

Problem Statement

Identify the physical origin of, and construct a validated quantitative description for, the reproducible mode-dependent inconsistencies of fixed-length cylindrical acoustic gas thermometry, and thereby remove the mode-to-mode inconsistency that presently limits the method at the few-parts-in-$10^6$ level. Concretely, a solution must account for the following, as exhibited in measurement data of cylindrical-cavity acoustic thermometry: (i) the mode-to-mode scatter of the fitted zero-pressure squared speed of sound $A_{0,l}$, including its systematic even–odd longitudinal-mode pattern (the values for even $l$ reproducibly exceeding those for odd $l$), which is the dominant contributor to the c-AGT uncertainty budget; (ii) the pressure-dependent frequency perturbations of acoustic modes coupled to elastic shell vibrations — the shell-coupled A4 and A5 modes, in which the deviations of A4 show a definite nonlinear, negative curvature above roughly 400 kPa — which first-order coupling models, predicting pressure-linear shifts, do not capture, and which therefore require a gas–shell coupling theory beyond first order; and (iii) the non-physical negative excess half-widths of the A1 mode at low pressure, which indicate a phenomenon omitted from the resonance model (for example coupling between the source and detector transducers, or another modal interaction), including why the standard single-mode resonance fit fails there. Each of (i)-(iii) is mandatory and is to be treated as an independent sub-problem: a single unified mechanism is not required, but no item may be left unaddressed. For each item, to account for it means to give a mechanistic explanation together with a quantitative description that reproduces the corresponding published NIM/NIST cylindrical-cavity measurements — agreeing in sign and in order of magnitude and, where the data allow, in the observed pressure and mode dependence. The demand to remove the mode-to-mode inconsistency refers to item (i), the dominant contributor to the uncertainty budget, so a solution must additionally show how its mechanism reduces the mode-to-mode scatter of $A_{0,l}$; items (ii) and (iii) must be explained and modeled but need not be separately eliminated. No answer is required to determine the Boltzmann constant, whose value has been fixed exactly by the 2019 SI redefinition.

References

1. Feng X J, Zhang J T, Lin H, Gillis K A, Mehl J B, Moldover M R, Zhang K, Duan Y N (2017). Determination of the Boltzmann constant with cylindrical acoustic gas thermometry: new and previous results combined. Metrologia 54, 748–762. DOI: 10.1088/1681-7575/aa7b4a. https://doi.org/10.1088/1681-7575/aa7b4a
2. Lin H, Feng X J, Gillis K A, Moldover M R, Zhang J T, Sun J P, Duan Y Y (2013). Improved determination of the Boltzmann constant using a single, fixed-length cylindrical cavity. Metrologia 50, 417–432. DOI: 10.1088/0026-1394/50/5/417. https://doi.org/10.1088/0026-1394/50/5/417
3. Zhang J T, Lin H, Feng X J, Sun J P, Gillis K A, Moldover M R, Duan Y Y (2011). Progress Toward Redetermining the Boltzmann Constant with a Fixed-Path-Length Cylindrical Resonator. International Journal of Thermophysics 32, 1297–1329. DOI: 10.1007/s10765-011-1001-3. https://doi.org/10.1007/s10765-011-1001-3
4. Feng X J, Lin H, Gillis K A, Moldover M R, Zhang J T (2015). Test of a virtual cylindrical acoustic resonator for determining the Boltzmann constant. Metrologia 52, S343–S352. DOI: 10.1088/0026-1394/52/5/S343. https://doi.org/10.1088/0026-1394/52/5/S343
5. Moldover M R, Gavioso R M, Mehl J B, Pitre L, de Podesta M, Zhang J T (2014). Acoustic gas thermometry. Metrologia 51, R1–R19. DOI: 10.1088/0026-1394/51/1/R1. https://doi.org/10.1088/0026-1394/51/1/R1
6. Newell D B, Cabiati F, Fischer J, Fujii K, Karshenboim S G, Margolis H S, de Mirandés E, Mohr P J, Nez F, Pachucki K, Quinn T J, Taylor B N, Wang M, Wood B M, Zhang Z (2018). The CODATA 2017 values of h, e, k, and N_A for the revision of the SI. Metrologia 55, L13–L16. DOI: 10.1088/1681-7575/aa950a. https://doi.org/10.1088/1681-7575/aa950a
7. Fischer J, Fellmuth B, Gaiser C, Zandt T, Pitre L, Sparasci F, Plimmer M D, de Podesta M, Underwood R, Sutton G, Machin G, Gavioso R M, Madonna Ripa D, Steur P P M, Qu J, Feng X J, Zhang J, Moldover M R, Benz S P, White D R, Gianfrani L, Castrillo A, Moretti L, Darquié B, Moufarej E, Daussy C, Briaudeau S, Kozlova O, Risegari L, Segovia J J, Martín M C, del Campo D (2018). The Boltzmann project. Metrologia 55, R1–R20. DOI: 10.1088/1681-7575/aaa790. https://doi.org/10.1088/1681-7575/aaa790
8. Machin G (2018). The kelvin redefined. Measurement Science and Technology 29, 022001. DOI: 10.1088/1361-6501/aa9ddb. https://doi.org/10.1088/1361-6501/aa9ddb
9. Zhang K, Feng X J, Zhang J T, Duan Y Y, Lin H, Duan Y N (2020). Determination of T–T90 from 234 K to 303 K by acoustic thermometry with a cylindrical resonator. Metrologia 57, 024004. DOI: 10.1088/1681-7575/ab7171. https://doi.org/10.1088/1681-7575/ab7171
10. Xing L, Feng X-J, Si M-H, Zhang J-T, Lin H, Gillis K A, Moldover M R (2023). Cylindrical Acoustic Gas Thermometry. Journal of Physical and Chemical Reference Data 52, 031501. DOI: 10.1063/5.0139385. https://doi.org/10.1063/5.0139385
11. Xu L, Feng X, Zhang J, Lu X, Xing L, Gillis K, Moldover M R (2026). Progress of acoustic gas thermometry using a cylindrical resonator at temperatures up to 809 K. Philosophical Transactions of the Royal Society A 384, 20240454. DOI: 10.1098/rsta.2024.0454. https://doi.org/10.1098/rsta.2024.0454
12. Gaiser C, Fellmuth B, Gavioso R M, Kalemci M, Kytin V, Nakano T, Pokhodun A, Rourke P M C, Rusby R, Sparasci F, Steur P P M, Tew W L, Underwood R, White R, Yang I, Zhang J (2022). 2022 Update for the Differences Between Thermodynamic Temperature and ITS-90 Below 335 K. Journal of Physical and Chemical Reference Data 51, 043105. DOI: 10.1063/5.0131026. https://doi.org/10.1063/5.0131026
13. Gaiser C, Fellmuth B (2026). Challenges of primary thermometry above 300 K. Philosophical Transactions of the Royal Society A 384, 20240449. DOI: 10.1098/rsta.2024.0449. https://doi.org/10.1098/rsta.2024.0449