Dielectric-constant Gas Thermometry Research Articles

Ab Initio Calculation of Fluid Properties for Precision Metrology

Recent advances regarding the interplay between ab initio calculations and metrology are reviewed, with particular emphasis on gas-based techniques used for temperature and pressure measurements. Since roughly 2010, several thermophysical quantities – in particular, virial and transport coefficients – can be computed from first principles without uncontrolled approximations and with rigorously propagated uncertainties. In the case of helium, computational results have accuracies that exceed the best experimental data by at least one order of magnitude and are suitable to be used in primary metrology. The availability of ab initio virial and transport coefficients contributed to the recent SI definition of temperature by facilitating measurements of the Boltzmann constant with unprecedented accuracy. Presently, they enable the development of primary standards of thermodynamic temperature in the range 2.5–552 K and pressure up to 7 MPa using acoustic gas thermometry, dielectric constant gas thermometry, and refractive index gas thermometry. These approaches will be reviewed, highlighting the effect of first-principles data on their accuracy. The recent advances in electronic structure calculations that enabled highly accurate solutions for the many-body interaction potentials and polarizabilities of atoms – particularly helium – will be described, together with the subsequent computational methods, most often based on quantum statistical mechanics and its path-integral formulation, that provide thermophysical properties and their uncertainties. Similar approaches for molecular systems, and their applications, are briefly discussed. Current limitations and expected future lines of research are assessed.

Abinitio determination of the polarizability of neon

The static electric-dipole polarizability $\ensuremath{\alpha}$ of the neon atom was determined with a relative uncertainty of only about 0.003% using state-of-the-art ab initio approaches. The new value, $\ensuremath{\alpha}=2.661\phantom{\rule{0.16em}{0ex}}067(77)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, is almost five times more accurate than the previous ab initio estimate, $\ensuremath{\alpha}=2.660\phantom{\rule{0.16em}{0ex}}80(36)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, by Lesiuk et al. [Phys. Rev. A 102, 052816 (2020)]. Similar to their work, we calculated $\ensuremath{\alpha}$ using ab initio methods up to full configuration interaction and added corrections for finite nuclear mass and size, relativistic, and quantum electrodynamics (QED) effects. The uncertainty reduction of this work was achieved in particular by employing extremely large basis sets, including newly developed ones of 11Z, 12Z, and 13Z quality. Moreover, the finite nuclear mass effects and most of the relativistic contributions were calculated at much higher levels of theory than in the work of Lesiuk et al. However, we adopted their values for the orbit-orbit part of the relativistic correction and for the Bethe logarithm needed to compute the QED correction. The uncertainty of our final value is still an order of magnitude larger than that of the experimental value recently measured by Gaiser and Fellmuth [Phys. Rev. Lett. 120, 123203 (2018)] with an uncertainty of only a few parts per million using dielectric-constant gas thermometry. Yet, our ab initio value agrees with their value, ${\ensuremath{\alpha}}_{\mathrm{exp}}=2.661\phantom{\rule{0.16em}{0ex}}057(7)\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$, almost within the experimental uncertainty. This could indicate that the higher-order relativistic corrections and QED contributions, which dominate our uncertainty budget, are more accurate than expected considering the uncontrolled approximations involved in their calculation.

Primary thermometry at 4 K, 14 K, and 25 K applying dielectric-constant gas thermometry

This short note is a supplement to the paper ‘Primary thermometry from 2.5 K to 140 K applying dielectric-constant gas thermometry’ (2017 Metrologia 54 141–7). It deals with thermodynamic temperature measurements at the boiling point of the heavy isotope of helium (4He, 4 K) and the triple points of hydrogen (14 K) and neon (25 K). This is of special interest because recent data published in this temperature range have an unexpectedly large spread. The results presented in this short note show that the International Temperature Scale of 1990, ITS-90, is thermodynamically correct at 4 K and 14 K, but too high by about half a millikelvin at 25 K. Furthermore, care is given to two aspects of dielectric-constant gas thermometry in the low-temperature range. First, dielectric-constant gas thermometry can be applied for practical primary thermometry. The measurement of only one isotherm in one day yields a temperature value with an uncertainty order of a few tenths of a millikelvin. Second, the use of recent ab initio values for the virial coefficients of helium as a measuring gas can reduce the efforts significantly. Even one data pair of pressure and dielectric constant can yield a thermodynamic temperature value of uncertainty less than a factor of two larger compared to the uncertainty of the usually used fit evaluation. For 4 K and 25 K, the achieved final uncertainty for T is still larger than the realization uncertainty of T 90. Thereby in the case of 14 K, both uncertainties are comparable.

Thermophysical properties of low-density neon gas from highly accurate first-principles calculations and dielectric-constant gas thermometry measurements.

New interatomic potential energy and interaction-induced polarizability curves for two ground-state neon atoms were developed and used to predict the second density, acoustic, and dielectric virial coefficients and the dilute gas shear viscosity and thermal conductivity of neon at temperatures up to 5000 K. The potential energy curve is based on supermolecular coupled-cluster (CC) calculations at very high levels up to CC with single, double, triple, quadruple, and perturbative pentuple excitations [CCSDTQ(P)]. Scalar and spin-orbit relativistic effects, the diagonal Born-Oppenheimer correction, and retardation of the dispersion interactions were taken into account. The interaction-induced polarizability curve, which in this work is only needed for the calculation of the second dielectric virial coefficient, is based on supermolecular calculations at levels up to CCSDT and includes a correction for scalar relativistic effects. In addition to these first-principles calculations, highly accurate dielectric-constant gas thermometry (DCGT) datasets measured at temperatures from 24.5 to 200 K were analyzed to obtain the difference between the second density and dielectric virial coefficients with previously unattained accuracy. The agreement of the DCGT values with the ones resulting from the first-principles calculations is, despite some small systematic deviations, very satisfactory. Apart from this combination of two virial coefficients, the calculated thermophysical property values of this work are significantly more accurate than any available experimental data.

Highly-accurate second-virial-coefficient values for helium from 3.7 K to 273 K determined by dielectric-constant gas thermometry

Since the redefinition of the base unit kelvin via fixing the Boltzmann constant in 2019, it has been possible to realise the unit applying different gas-thermometry methods in accordance with the ‘Mise en pratique for the definition of the kelvin in the SI’. For this application, the use of data for the gas properties resulting from ab initio calculations is of special interest because it makes primary thermometry much easier. But since a rigorous estimation of the uncertainty of theoretical calculations is at least very complex, if not impossible, a check of the results by a comparison with highly-accurate experimental values is mandatory. Such a check is performed for the second virial coefficient of helium, which is a widely used measuring gas, in the temperature range from 3.7 K to 273 K. For obtaining highly-accurate second-virial-coefficient values (relative uncertainty at a few tenth of a percent level or even better), isotherms were measured with the PTB dielectric-constant gas thermometer. The highly-accurate isotherm data were evaluated by fitting, applying an extended working equation for the dependence of the gas pressure on the dielectric constant. The comparison with the results of the latest ab initio calculations shows coincidence within the combined uncertainty estimates.

Thermodynamic-temperature data from 30 K to 200 K

New measurements of thermodynamic temperature T with Dielectric-Constant Gas Thermometry (DCGT) were performed at PTB from 50 K to 200 K. Particular care was taken to check for possible systematic sources of errors by performing experiments applying three working gases, namely helium, neon, and argon, the polarizability of which differs by a factor of up to eight. Together with former DCGT values of thermodynamic temperature the new results yield a consistent dataset in the range from 30 K to 200 K. This dataset is in good agreement with the newest results of Acoustic Gas Thermometry (AGT) and Refractive-Index Gas Thermometry (RIGT), which have quite different sources of uncertainty compared with DCGT. The combination of these DCGT, AGT, and RIGT data with the ‘Estimates of the differences between thermodynamic temperature and the ITS-90’, being as an appendix of the ‘Mise en pratique for the definition of the kelvin in the SI’ the present-day recommendation of the Consultative Committee for Thermometry, yields a new function T − T90 versus ITS-90 temperature T90 for the range from 35 K to 195 K, the uncertainty of which is reduced by a factor up to about four.

Thermodynamic temperature of the triple point of xenon measured by refractive index gas thermometry

Measurements of the refractive indices of helium and argon using a quasi-spherical microwave resonator and upgraded experimental protocol are reported at the temperatures of the triple points of water and xenon. The results at the triple point of water are used to determine the compressibility of the resonating cavity, which is bounded by the copper shell of the resonator. This experimentally-determined compressibility is consistent with the literature value for copper, but with much smaller measurement uncertainty. The measured compressibility is extrapolated to the triple point of xenon, and combined with the refractive index results at that temperature to determine the thermodynamic accuracy of the International Temperature Scale of 1990 (ITS-90): mK at ITS-90 temperature K, corresponding to a xenon triple point thermodynamic temperature of K. The experimental compressibility is further extrapolated to the triple points of argon, oxygen and neon, and used to re-analyze earlier refractive index gas thermometry measurements made using the same resonator, yielding updated values with reduced uncertainties: mK at T90 = 83.8058 K, mK at T90 = 54.3584 K, and mK at T90 = 24.5561 K. The ITS-90 thermodynamic accuracy results of the present refractive index gas thermometry study agree with those previously reported by acoustic gas thermometry and dielectric constant gas thermometry.

Highly-accurate density-virial-coefficient values for helium, neon, and argon at 0.01 ○C determined by dielectric-constant gas thermometry.

The dielectric-constant gas thermometer of Physikalisch-Technische Bundesanstalt (PTB) developed for measuring the Boltzmann constant with a relative uncertainty of 1.9 parts per million was used for determining the virial coefficients of the three noble gases, helium, neon, and argon, at the triple point of water (0.01 ○C). For this purpose, isotherms were measured up to a maximum pressure of 7 MPa. The evaluation of the highly accurate data by fitting is required to derive an extended working equation for the dependence of the gas pressure on the dielectric constant. The following values have been obtained for the second B and third C virial coefficients, with the standard uncertainties given in parentheses as a multiple of the last digit, considering literature data for the dielectric virial coefficients: helium: BDCGT He0.01 ○C=11.925715 cm3/mol, CDCGT He0.01 ○C=113.4958 cm6/mol2; neon: BDCGT Ne0.01 ○C=10.994528 cm3/mol, CDCGT Ne0.01 ○C=215.815 cm6/mol2; argon: BDCGT Ar0.01 ○C=-21.233144 cm3/mol, CDCGT Ar0.01 ○C=1143.339 cm6/mol2. These values are compared with the results of the latest ab initio calculations of the second and third virial coefficients.

Determinations of the Boltzmann constant

Abstract We review measurements of the Boltzmann constant, k, the value of which is soon to be fixed at exactly 1.380 649 × 10 − 23 J⋅K−1 for the future revised Systeme international of units. In addition to a description of the theoretical background and of diverse experimental techniques (acoustic thermometry, Johnson noise thermometry, dielectric constant gas thermometry, and Doppler broadened molecular spectroscopy), the article highlights the decisive role of ab initio calculations of the thermophysical properties of gases, especially helium-4. Perspectives for improvements in thermometry are outlined in the wake of the new definition.

Polarizability of Helium, Neon, and Argon: New Perspectives for Gas Metrology.

With dielectric-constant gas thermometry, the molar polarizability of helium, neon, and argon has been determined with relative standard uncertainties of about 2 parts per million. A series of isotherms measured with the three noble gases and two different experimental setups led to this unprecedented level of uncertainty. These data are crucial for scientists in the field of gas metrology, working on pressure and temperature standards. Furthermore, with the new benchmark values for neon and argon, theoretical calculations, today about 3 orders of magnitude larger in uncertainty, can be checked and improved.

An improved electronic determination of the Boltzmann constant by Johnson noise thermometry

Recent measurements using acoustic gas thermometry have determined the value of the Boltzmann constant, k, with a relative uncertainty less than 1 × 10−6. These results have been supported by a measurement with a relative uncertainty of 1.9 × 10−6 made with dielectric-constant gas thermometry. Together, the measurements meet the requirements of the International Committee for Weights and Measures and enable them to proceed with the redefinition of the kelvin in 2018. In further support, we provide a new determination of k using a purely electronic approach, Johnson noise thermometry, in which the thermal noise power generated by a sensing resistor immersed in a triple-point-of-water cell is compared to the noise power of a quantum-accurate pseudo-random noise waveform of nominally equal noise power. The experimental setup differs from that of the 2015 determination in several respects: a 100 Ω resistor is used as the thermal noise source, identical thin coaxial cables made of solid beryllium–copper conductors and foam dielectrics are used to connect the thermal and quantum-accurate noise sources to the correlator so as to minimize the temperature and frequency sensitivity of the impedances in the connecting leads, and no trimming capacitors or inductors are inserted into the connecting leads. The combination of reduced uncertainty due to spectral mismatches in the connecting leads and reduced statistical uncertainty due to a longer integration period of 100 d results in an improved determination of k = 1.380 649 7(37) × 10−23 J K−1 with a relative standard uncertainty of 2.7 × 10−6 and a relative offset of 0.89 × 10−6 from the CODATA 2014 recommended value. The most significant terms in the uncertainty budget, the statistical uncertainty and the spectral-mismatch uncertainty, are uncorrelated with the corresponding uncertainties in the 2015 measurements.

NRC Microwave Refractive Index Gas Thermometry Implementation Between 24.5 K and 84 K

The implementation of microwave refractive index gas thermometry at the National Research Council between 24.5 K and 84 K is reported. A new gas-handling system for accurate control and measurement of experimental gas pressure has been constructed, and primary thermometry measurements have been taken using a quasi-spherical copper resonator and helium gas at temperatures corresponding to three defining fixed points of the International Temperature Scale of 1990 (ITS-90). These measurements indicate differences between the thermodynamic temperature T and ITS-90 temperature $$T_{90}$$ of $$\left( T - T_{90} \right) = -0.60 \pm 0.56$$ mK at $$T_{90} = 24.5561$$ K, $$\left( T - T_{90} \right) = -2.0 \pm 1.3$$ mK at $$T_{90} = 54.3584$$ K, and $$\left( T - T_{90} \right) = -4.0 \pm 2.9$$ mK at $$T_{90} = 83.8058$$ K. The present results at $$T_{90} = 24.5561$$ K and $$T_{90} = 83.8058$$ K agree with previously reported measurements from other primary thermometry techniques of acoustic gas thermometry and dielectric constant gas thermometry, and the result at $$T_{90} = 54.3584$$ K provides new information in a temperature region where there is a gap in other recent data sets.

Final determination of the Boltzmann constant by dielectric-constant gas thermometry

Gaiser et al published in 2013 (Metrologia 50 L7–11) a second, improved value of the Boltzmann constant k determined by dielectric-constant gas thermometry at the triple point of water (k = 1.380 6509 × 10−23 J K−1, relative standard uncertainty 4.3 parts per million (4.3 ppm)). Subsequently, the uncertainty was able to be reduced to 4.0 ppm by reanalysing the pressure measurement. Since 2013, further progress regarding this primary-thermometry method has been achieved in terms of the design and the assembly of the measuring capacitors, the determination of their effective compressibility, the sensitivity of the capacitance bridge, and the scattering and the evaluation of the data. Based on a huge amount of data, two new k values have been obtained by applying two different capacitors. The combination of these two values with the 2013 result, fully taking into account the correlations, has yielded a final result of k = 1.380 6482 × 10−23 J K−1 with a relative standard uncertainty of 1.9 ppm. This value is about 0.2 ppm smaller than the CODATA 2014 one, which has a relative standard uncertainty of 0.57 ppm.

Primary thermometry from 2.5 K to 140 K applying dielectric-constant gas thermometry

Recent thermodynamic temperature measurements at PTB with dielectric-constant gas thermometry were performed up to 140 K. Particular care was taken to check for possible systematic sources of errors by repeating experiments applying two new measuring capacitors and both helium and neon as working gases. The development of a new method for determining the effective compressibility of the new capacitors as a function of temperature has decreased the uncertainty significantly. The combination of the recently obtained results with former values of thermodynamic temperature, corrected using new thermodynamic input data, has yielded a consistent dataset in the range from 2.5 K to 38 K as well as at 84 K, 120 K, 130 K, and 140 K. This dataset is in good agreement with the newest results of acoustic gas thermometry, which has quite different sources of uncertainty compared with dielectric-constant gas thermometry.

Concept of a new technique for the combined determination of dielectric and density virial coefficients3

For the combined determination of dielectric and density virial coefficients, a new measurement concept based on dielectric-constant gas thermometry (DCGT) has been elaborated. The developed apparatus design allows conventional DCGT measurements as well as gas expansion experiments to enable a distinction between density and dielectric virial coefficients. The new technique presented in this technical design note offers an independent method to obtain experimental data, which are highly desirable for the improvement of thermodynamic property models for pure gases and gas mixtures.

Linking the thermodynamic temperature to an optical frequency: recent advances in Doppler broadening thermometry.

Laser spectroscopy in the linear regime of radiation–matter interaction is a powerful tool for measuring thermodynamic quantities in a gas at thermodynamic equilibrium. In particular, the Doppler effect can be considered a gift of nature, linking the thermal energy to an optical frequency, namely the line centre frequency of an atomic or molecular spectral line. This is the basis of a relatively new method of primary gas thermometry, known as Doppler broadening thermometry (DBT). This paper reports on the efforts that have been carried out, in the last decade, worldwide, to the end of making DBT competitive with more consolidated and accurate methodologies, such as acoustic gas thermometry and dielectric constant gas thermometry. The main requirements for low-uncertainty DBT, of both theoretical and technical nature, will be discussed, with a special focus on those related to the line shape model and to the frequency scale. A deep comparison among the different molecules that have been selected in successful DBT implementations is also reported. Finally, for the first time, to the best of my knowledge, the influence of refractive index effects is discussed.

Progress towards the determination of thermodynamic temperature with ultra-low uncertainty.

Previous research effort towards the determination of the Boltzmann constant has significantly improved the supporting theory and the experimental practice of several primary thermometry methods based on the measurement of a thermodynamic property of a macroscopic system at the temperature of the triple point of water. Presently, experiments are under way to demonstrate their accuracy in the determination of the thermodynamic temperature T over an extended range spanning the interval between a few kelvin and the copper freezing point (1358 K). We discuss how these activities will improve the link between thermodynamic temperature and the temperature as measured using the International Temperature Scale of 1990 (ITS-90) and report some preliminary results obtained by dielectric constant gas thermometry and acoustic gas thermometry. We also provide information on the status of other primary methods, such as Doppler broadening thermometry, Johnson noise thermometry and refractive index gas thermometry. Finally, we briefly consider the implications of these advancements for the dissemination of calibrated temperature standards.

Method for extrapolating the compressibility data of solids from room to lower temperatures

Around room temperature, the elastic constants of solids can be measured comfortably applying commercial equipment, e.g., resonant ultrasound spectroscopy devices, whereas at low temperatures special efforts are necessary. To decrease the effort, a method has been developed and tested for extrapolating data on the compressibility of solids to lower temperatures. The method is based on well-known approximate relations between the thermodynamic properties of solids, the derivation of which was pioneered by Gruneisen, and the measurement of quantities, which are easier accessible experimentally than the elastic constants (thermal expansion, specific heat). In essence, the extrapolation uses data for the thermal expansion. The applicability of the method has been tested for copper, the properties of which are documented comprehensively in literature even at low temperatures. In a second step, the method has been applied to extrapolate the compressibility data of a copper-beryllium-cobalt alloy from near room temperature down to about 80 K with a relative uncertainty of order 0.1%. This extrapolation is needed for and verified by dielectric-constant gas thermometry, a special kind of primary thermometry for determining thermodynamic temperatures. For such gas-thermometry experiments cylindrical capacitors, the electrodes of which have been made from the investigated copper beryllium alloys, are used.

Progress towards a new definition of the kelvin

At its 24th meeting (2011) the General Conference on Weights and Measures noted the CIPM’s intention to express new definitions of the kilogram, ampere, kelvin, and mole in terms of fixed numerical values of the Planck constant, elementary charge, Boltzmann constant, and Avogadro constant, respectively. The changes proposed for the International System of Units will not actually be adopted until the experimental results on the new definitional constants that are proposed have reached a further stage of refinement. This paper provides an overview of the activities and progress of the research groups who are carrying out experiments to determine the Boltzmann constant. The most promising methods, acoustic gas thermometry, dielectric-constant gas thermometry, Johnson noise thermometry and Doppler-broadening thermometry, are reviewed. The prospects for meeting the requirements of the Consultative Committee for Thermometry for a new definition of the kelvin are discussed.

Dielectric-constant gas thermometry

The principles, techniques and results from dielectric-constant gas thermometry (DCGT) are reviewed. Primary DCGT with helium has been used for measuring T–T90 below the triple point of water (TPW), where T is the thermodynamic temperature and T90 is the temperature on the international temperature scale of 1990 (ITS-90), and, in an inverse regime with T as input quantity, for determining the Boltzmann constant at the TPW. Furthermore, DCGT allows the determination of several important material properties including the polarizability of neon and argon as well as the virial coefficients of helium, neon, and argon. With interpolating DCGT (IDCGT), the ITS-90 has been approximated in the temperature range from 4 K to 25 K. An overview and uncertainty budget for each of these applications of DCGT is provided, accompanied by corroborating evidence from the literature or, for IDCGT, a CIPM key comparison.