James B. Mehl

Gas‐filled spherical resonators: Theory and experiment

Gas‐filled spherical resonators are excellent tools for routine measurement of thermophysical properties. The radially symmetric gas resonances are nondegenerate and have high Q’s (typically 2000–10 000). Thus they can be used with very simple instrumentation to measure the speed of sound in a gas with an accuracy of 0.02%. We have made a detailed study of a prototype resonator filled with argon (0.1–1.0 MPa) at 300 K, with the objective of discovering those phenomena which must be understood to use gas‐filled spherical resonators to measure the thermodynamic temperature and the universal gas constant R. The resonance frequencies fN and half‐widths gN were measured for nine radially symmetric modes and nine triply‐degenerate nonradial modes with a precision near 10−7 fN. The data were used to develop and test theoretical models for this geometrically simple oscillating system. The basic model treats the following phenomena exactly for the case of a geometrically perfect sphere: (1) the thermal boundary la...

Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations R smaller than 4 bohrs and for the distance R = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near R = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller R (all R in the case of a QED component) and optimized orbital bases up to the cardinal number X = 7 for larger R. Near-exact asymptotic constants were used to describe the large-R behavior of all components. The fitted potential, exhibiting the minimum of -10.996 ± 0.004 K at R = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound (4)He(2) dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 Å relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α(2) and α(3), respectively. The remaining retardation effect, of the order of α(4) and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.

Potential energy surface for interactions between two hydrogen molecules

Nonrelativistic clamped-nuclei energies of interaction between two ground-state hydrogen molecules with intramolecular distances fixed at their average value in the lowest rovibrational state have been computed. The calculations applied the supermolecular coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and very large orbital basis sets-up to augmented quintuple zeta size supplemented with bond functions. The same basis sets were used in symmetry-adapted perturbation theory calculations performed mainly for larger separations to provide an independent check of the supermolecular approach. The contributions beyond CCSD(T) were computed using the full configuration interaction method and basis sets up to augmented triple zeta plus midbond size. All the calculations were followed by extrapolations to complete basis set limits. For two representative points, calculations were also performed using basis sets with the cardinal number increased by one or two. For the same two points, we have also solved the Schrodinger equation directly using four-electron explicitly correlated Gaussian (ECG) functions. These additional calculations allowed us to estimate the uncertainty in the interaction energies used to fit the potential to be about 0.15 K or 0.3% at the minimum of the potential well. This accuracy is about an order of magnitude better than that achieved by earlier potentials for this system. For a near-minimum T-shaped configuration with the center-of-mass distance R=6.4 bohrs, the ECG calculations give the interaction energy of -56.91+/-0.06 K, whereas the orbital calculations in the basis set used for all the points give -56.96+/-0.16 K. The computed points were fitted by an analytic four-dimensional potential function. The uncertainties in the fit relative to the ab initio energies are almost always smaller than the estimated uncertainty in the latter energies. The global minimum of the fit is -57.12 K for the T-shaped configuration at R=6.34 bohrs. The fit was applied to compute the second virial coefficient using a path-integral Monte Carlo approach. The achieved agreement with experiment is substantially better than in any previous work.

Precision acoustic measurements with a spherical resonator: Ar and C2H4

The spherical acoustic resonator is a remarkably accurate and convenient tool for the measurement of thermophysical properties of gases at low and moderate densities. The speed of sound (c) in a gas of interest can be measured with an accuracy of 0.02% merely by measuring the frequencies of the radial resonances when the resonator is filled with the gas of interest and then repeating the frequency measurements with a reference gas such as argon. The resonance frequencies of the radial modes are easily measured because these modes have very high Q’s, typically 2000–10 000. In this work the precision and accuracy of speed of sound measurements have been substantially improved by including a detailed acoustic model of the resonator in the analysis. Many of the important parameters of the model can be determined from acoustic measurements: Painstaking mechanical measurements are not required. We have used a spherical resonator to measure the speed and attenuation of sound in C2H4 in the temperature range 0–10...

Spherical acoustic resonator: Effects of shell motion

The exact theory of classical elasticity is used to calculate the response of an isotropic spherical shell to an acoustic mode of the fluid enclosed by the shell. The results are used to calculate the shifts of the acoustic resonance frequencies from the values which correspond to perfectly rigid shell walls. Acoustic modes with pressure proportional to Ynm (θ, φ) excite shell vibrations with the radial displacement also proportional to Ynm. The shell response depends upon the mode index n, the ratio of the shell diameters, Poisson’s ratio for the shell material, and a dimensionless frequency parameter. Numerical results for a useful range of acoustic frequencies are presented for radial (n=0) modes and for nonradial modes with mode indices n between 1 and 3. Numerical calculations of the shell resonance frequencies are presented for a wide range of shell thicknesses.

Reentrant radio‐frequency resonator for automated phase‐equilibria and dielectric measurements in fluids

A reentrant rf cavity resonator has been developed for automated detection of phase separation of fluid mixtures contained within the cavity. Successful operation was demonstrated by redetermining the phase boundaries of a CO2+C2H6 mixture in the vicinity of its critical point. We developed an accurate electrical model for the resonator and used helium to determine the deformation of the resonator under pressure. With the model and pressure compensation, the resonator was capable of very accurate dielectric measurements. We confirmed this by remeasuring the molar dielectric polarizability Ae of argon and obtained the result Ae=(4.140±0.006) cm3/mol (standard uncertainty) in excellent agreement with published values. We exploited the capability for accurate dielectric measurements to determine the densities of the CO2+C2H6 mixture at the phase boundaries and to determine the dipole moment of 1,1,1,2,3,3‐hexafluoropropane, a candidate replacement refrigerant. Near the operating frequency of 375 MHz the capa...

Designing quasi-spherical resonators for acoustic thermometry

We describe a quasi-spherical, noble-gas-filled cavity designed to determine the thermodynamic temperature of the gas from measurements of the frequencies and the half-widths of microwave and acoustic resonances in the cavity. The quasi-spherical shape retains the advantages of spherical acoustic resonators (non-degenerate, radially symmetric acoustic modes, negligible viscous damping at the boundary) while simplifying the determination of the cavitys thermal expansion using microwave resonances. As a specific example, we consider a cavity bounded by four quadrants of a perfect sphere of radius a connected to each other by narrow cylindrical sections of thickness 21a and 22a, where 1 and 2 are of the order of 10−3. The cylindrical sections split the degenerate microwave triplets (TM11, TE11, TM12, etc) into three easily resolved components with predictable orientations. Preliminary measurements show that two judiciously located microwave probes excite and detect all three components of each microwave triplet with approximately equal amplitudes. We discuss the placement of acoustic transducers and the dimensions of ducts that admit gas from a manifold into the cavity. Corrections were made to this article on 14 June 2004. The corrected electronic version is identical to the print version. The corrections are in the first two paragraphs of section 5.2 and in table 6.

Precondensation phenomena in acoustic measurements

Theoretical and experimental studies of the behavior of acoustic resonators whose walls are coated with a film of condensed vapor are reported. As a sound wave is reflected from the resonator walls, further condensation and evaporation will alter the thickness of the condensed film during the course of an acoustic cycle. We have modeled this effect for smooth walls and have calculated the associated specific acoustic admittance, which we refer to as βfilm. Over a wide range of conditions, the magnitude of βfilm is governed by the derivative of the film thickness with respect to pressure. Thus, for typical adsorption isotherms, the magnitude of βfilm becomes large at very low pressures and at pressures just below the saturated vapor pressure. It seems possible that acoustic techiques could be used to study adsorption in both pressure regimes. If the resonator walls are rough (e.g., machined metal), a significant quantity of vapor will condense in the recesses of the walls at pressures well below the satura...

Analysis of resonance standing‐wave measurements

Linear and nonlinear numerical techniques for fitting measurements on resonant standing‐wave systems to theoretical resonance formulas are described. The fits are correct for background and enable the normal‐mode frequencies and widths to be precisely determined.

Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases

We evaluate a quasi-spherical, copper, microwave cavity resonator for accurately measuring the relative dielectric permittivity er(p,T) of helium and argon. In a simple, crude approximation the cavity’s shape is a triaxial ellipsoid with axes of length a,1.001a and 1.005a, with a=5 cm. The unequal axes of the quasi-sphere separated each of the triply degenerate microwave resonance frequencies of a sphere (f11TM,f12TM,…,f11TE,f12TE,…) into three nonoverlapping, easily measured, frequencies. The frequency splittings are consistent with the cavity’s shape, as determined from dimensional measurements. We deduced er(p,T) of helium and of argon at 289 K and up to 7 MPa from the resonance frequencies flnσ, the resonance half-widths glnσ, and the compressibility of copper. Simultaneous measurements of er(p,T) with the quasi-spherical resonator and a cross capacitor agreed within 1×10−6 for helium, and for argon they differed by an average of only 1.4×10−6. This small difference is within the stated uncertainty of...