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1 Open Access Repository eprint Terms and Conditions: Users may access, download, store, search and print a hard copy of the article. Copying must be limited to making a single printed copy or electronic copies of a reasonable number of individual articles or abstracts. Access is granted for internal research, testing or training purposes or for personal use in accordance with these terms and conditions. Printing for a for-fee-service purpose is prohibited. Title: Determination of the Boltzmann constant by dielectric-constant gas thermometry Author(s): Fellmuth, Bernd; Fischer, Joachim; Gaiser, Christof; Jusko, Otto; Priruenrom, Tasanee; Sabuga, Wladimir; Zandt, Thorsten Journal: Metrologia Year: 2011, Volume: 48, Issue: 5 DOI: / /48/5/020 Funding programme: imera-plus: Call 2007 SI and Fundamental Metrology Project title: T1.J1.4: Boltzmann constant: Determination of the Boltzmann constant for the redefinition of the kelvin Copyright note: This is an author-created, un-copyedited version of an article accepted for publication in Metrologia. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher-authenticated version is available online at doi: / /48/5/020 EURAMET Secretariat Bundesallee Braunschweig, Germany Phone: Fax: secretariat@euramet.org

2 Determination of the Boltzmann constant by dielectric-constant gas thermometry Bernd Fellmuth 1, Joachim Fischer 1, Christof Gaiser 1, Otto Jusko 1, Tasanee Priruenrom 2, Wladimir Sabuga 1, and Thorsten Zandt 1 1 Physikalisch-Technische Bundesanstalt (PTB), Abbestrasse 2-12, Berlin, and Bundesallee 100, Braunschweig, Germany 2 National Institute of Metrology (Thailand) (NIMT), 3/4-5 Moo 3, Klong 5, Klong Luang, Pathumthani 12120, Thailand, guest researcher at PTB Abstract Within an international project directed to the new definition of the base unit kelvin, the Boltzmann constant k has been determined by dielectric-constant gas thermometry (DCGT) at PTB. In the pressure range from about 1 MPa to 7 MPa, 11 helium isotherms have been measured at the triple point of water (TPW) by applying a new special experimental setup consisting of a large-volume thermostat, a vacuum-isolated measuring system, stainless-steel 10 pf cylindrical capacitors, an autotransformer ratio capacitance bridge, a high-purity gashandling system including a mass spectrometer, and traceably calibrated special pressure balances with piston-cylinder assemblies having effective areas of 2 cm 2. The value of k has been deduced from the linear, ideal-gas term of an appropriate virial expansion fitted to the combined isotherms. A detailed uncertainty budget has been established by performing Monte-Carlo simulations. The main uncertainty components result from the measurement of pressure and capacitance as well as the influence of the effective compressibility of the measuring capacitor and impurities contained in the helium gas. The combination of the results obtained at the TPW (k TPW = J/K, relative standard uncertainty 9.2 parts per million) with data measured earlier at low temperatures (21 K to 27 K, k LT = J/K, 15.9 parts per million) has yielded a value of k = J/K with uncertainty of 7.9 parts per million. 1. Introduction In response to the CIPM proposal to give the Boltzmann constant k a fixed value for a redefinition of the base unit kelvin [1], many projects have been started to measure independently the value of k with a target relative uncertainty of order one part in 10 6 (one part per million, ppm). Promising methods are dielectric-constant gas thermometry (DCGT) [2, 3], acoustic gas thermometry [4, 5], noise thermometry [6], and Doppler-broadening 1

3 thermometry [7, 8]. An overview of these methods is given in [9]. Within the framework of the imera Joint Research Project Boltzmann constant (JRP No. T1.J1.4), PTB designed a new DCGT experimental setup. This decision has been taken in view of the excellent experimental DCGT results, which were obtained in the low-temperature range [3, 10, 11, 12, 13] and allowed to set up a thermodynamic temperature scale between 2.5 K and 36 K. The paper presents results of the determination of the Boltzmann constant via the measurement of DCGT isotherms at the triple point of water (TPW) applying the new experimental setup. It is organized as follows. In Section 2, the setup consisting of a largevolume thermostat, a vacuum-isolated measuring system, stainless-steel 10 pf cylindrical capacitors, an autotransformer ratio capacitance bridge, a high-purity gas-handling system including a mass spectrometer, and traceably calibrated special pressure balances with pistoncylinder assemblies having effective areas of 2 cm 2 is described. The experimental results including the uncertainty budget are discussed in detail in Section 3. Finally conclusions are drawn and an outlook is given. 2. Experimental setup 2.1. DCGT principle As illustrated schematically in Figure 1, DCGT is based on the idea to replace the density in the equation of state of a gas by the dielectric constant ε [2, 3, 9]. This idea is realized by measuring the relative change in capacitance C (C(p) C(0)) / C(0) = ε r 1 + ε r κ eff p γ (1) (with ε r = ε/ε 0 ) of a gas-filled capacitor, where κ eff is the effective compressibility of the capacitor, ε r and ε 0 are the relative dielectric constant of the gas and the electrical constant, respectively. The capacitance C(p) is measured at constant temperature with the space between its electrodes filled with the measuring gas at various pressures p and with the space evacuated so that p = 0 Pa (measurement of isotherms). The compressibility term accounts for the deformation of the capacitor electrodes due to the gas pressure and is in sufficient approximation linear to p. For a real gas used in DCGT, a combination of the Clausius- Mossotti expansion and the density virial expansion has to be considered [2, 3] that leads to an expansion in the dimensionless parameter µ = γ / (γ + 3) 2

4 with p = A 1 (µ + A 2 µ 2 + A 3 µ 3 ), (2) A 1 = (A ε / RT + κ eff / 3) -1, (3) where R is the molar gas constant and A ε = N A α 0 / (3ε 0 ) is the molar polarizability with the Avogadro number N A and the atomic dipole polarizability α 0. The non-linear terms with the coefficients A 2 and A 3 contain the second and third density and dielectric virial coefficients describing the two- and three-particle interaction. A fit of Expansion (2) to an isotherm measured at a known thermodynamic temperature T yields, therefore, the ratio A ε /R, which allows to deduce the Boltzmann constant applying the relation k = (α 0 / ε 0 ) / (3 A ε / R). (4) Figure 1: Schematic sketch of the DCGT setup used at PTB (reference capacitor on the left, measuring capacitor on the right). Three quantities have to be measured: pressure p, capacitance C(p) and temperature T via the thermometer resistance R. χ = ε r -1 is the dielectric susceptibility of the gas. The capsule-type platinum resistance thermometers are inserted into holes in the block or plate made of copper. The inner diameter of the holes is only 0.1 mm larger than the outer diameter of the capsule. The thermal contact is improved with the aid of grease, which fills the gap. Three dry pumps are necessary for pumping the two vessels containing the capacitors and the large vacuum chamber, which contains the whole measuring system. 3

5 The application of Equation (4) requires to know α 0 from fundamental principles. But only for the measuring gas helium, ab initio calculations, providing the necessary uncertainty below 1 ppm, are available [14, 15]. Extreme demands are caused by the need to measure the very small electric susceptibility χ = ε r 1 of helium (it amounts to only at the TPW and 0.1 MPa). The target relative uncertainty of order one part in 10 6 can be achieved only if pressures up to 7 MPa are used. But even at these high pressures, it is necessary to apply a capacitance bridge that allows to measure capacitance changes with an uncertainty relative to the capacitance value of order a few parts in 10 9, see Section 2.3. A further challenge of these measurements is due to the deformation of the capacitor electrodes under the gas pressure causing a disturbing additional capacitance change. This means, the effective compressibility κ eff has to be determined with the necessary small relative uncertainty of a few parts in The investigation of systematic error sources requires comparisons of κ eff values of quite different special 10 pf capacitors. For this reason, both cylindrical capacitors and very large multi-ring toroidal cross capacitors described in [16] have been developed Thermal conditions within a special thermostat The measurement of the Boltzmann constant by DCGT requires to determine the temperature of the gas and thus of the capacitor electrodes traceably to the definition of the base unit kelvin, i.e. to the temperature of the TPW, with an uncertainty of order one tenth of a millikelvin. To realise a thermal environment of sufficient quality, a three-level arrangement has been realised in the new DCGT experimental setup. Each capacitor is surrounded by a rigid, metallic pressure vessel, which is thermally anchored to the thick central copper plate of the measuring system. The temperature of the plate is measured with the aid of three capsuletype standard platinum resistance thermometers, which have been calibrated at the triple points of mercury and water as well as at the melting point of gallium. Besides the central plate at the bottom, the system consists of a top plate, four thick rods connecting the two plates and an isothermal shield, all made from copper. In turn, the measuring system is placed within a vacuum chamber for thermal isolation. Finally, the chamber is inserted in a huge liquid-bath thermostat. The thermal conditions within the experimental setup have been investigated in detail as described in three accompanying papers [16, 17, 18]. The liquid-bath thermostat has an overall volume of the liquid of about 800 l and a central working volume, in which the vacuum chamber is located, with a diameter of 500 mm and a 4

6 height of 650 mm. The temperature stability and the temperature field at the boundary of the working volume without and with the chamber have been investigated carefully under different experimental conditions. It could be verified that, under optimum conditions, both the instability and the inhomogeneity of the temperature in the working volume are well below 1 mk as necessary [17, 18]. Dedicated experiments described in [16, 18] have shown that the temperature of the central copper plate can be controlled well within one tenth of a millikelvin under steady-state conditions over long time periods of a few days. Furthermore, the uncertainty component due to static temperature-measurement errors can be decreased to the same sufficient level. Special problems are caused by two unavoidable features of the new DCGT experimental setup. First, during the experiments, the temperature of the capacitor electrodes inside the pressure vessels cannot be measured directly. It is not possible to place thermometers inside the vessels because of the requirement to guarantee a purity of the helium gas of %. Second, due to the huge dimensions of the capacitors and the surrounding pressure vessels, the mass of the measuring system and thus its heat capacity is very large. This makes the thermal recovery of the system very slow, which is of importance for the measurements of isotherms. During such measurements, the flow of the measuring gas for changing the pressure inside the pressure vessel causes warming or cooling and thus temporary temperature changes. Both features require to investigate the thermal recovery of the system in order to reduce dynamic temperature-measurement errors. Dedicated experiments have been performed by simulating the gas-flow-induced temperature changes via the application of heat pulses, as also discussed in [16, 18]. The experimental results have been compared with theoretical calculations based both on simple rough models and finite-element methods. The obtained time constants of the thermal recovery have an order of magnitude of one hour. The measurement of an isotherm lasts, therefore, at least several days. Most important is the fact that the investigations yielded an agreement between experiment and theory, which is sufficient for deducing the temperature of the capacitor electrodes inside the pressure vessels with an uncertainty of order one tenth of a millikelvin. Under the real experimental conditions, this theoretical description of the recovery is of course accompanied by a careful observation of the drift of the capacitance values with time. 5

7 2.3. Capacitance measurement Considering the extreme demands concerning the measurement of capacitance changes, a high-resolution and high-precision autotransformer ratio capacitance bridge has been built and tested. Its main component is a home-made high-precision 1:1 inductive voltage divider used in an autotransformer configuration. For balancing the bridge, adjustable in-phase and quadrature currents can be injected. A detailed uncertainty budget for measuring small capacitance changes is presented in [19]. Considering correlations between main terms in the mathematical model, it is shown that it is possible to measure capacitance changes of at most a few 0.1% with a relative standard uncertainty below one part per million, i.e. with an uncertainty relative to the capacitance value of order one part per billion. The performed consideration of correlations requires that the measuring circuit is fully symmetric. For this reason, the reference capacitor is also located in the measuring system within the vacuum chamber. For checking the reliability of the uncertainty estimates experimentally, the results obtained with the newly developed bridge have been compared with those obtained with the preceding bridge applied in the past for DCGT measurements in the low-temperature range, see [12] and the references cited therein. The preceding bridge, containing a variable home-made highprecision inductive voltage divider with nine decades, is described shortly in [3]. Within the upper ratio-error limits of the divider of ten parts per billion, no discrepancies have been found Gas-handling system and purity analysis The whole gas-handling system was designed as an ultra-high-purity system using metal gaskets (VCR metal gasket, Swagelok*)), ultra-high-purity stainless-steel tubing (EN standard number / X2CrNiMo ), and electro polished internal surfaces (mean roughness index R a 0.25 µm) in all parts (tubes, valves etc.). For precise and reliable tube connections, an orbital arc welding machine was applied. The manifold allows to connect helium, neon and argon gas bottles to the system. Before entering the measuring capacitor, the gas flows through a gas purifier (MicroTorr SP70-902, SAES Pure Gas). According to the specification of the manufacturer, the gas purifier reduces the content of the impurities H 2 O, O 2, CO, CO 2, H 2 and non-methane hydrocarbons from rare gases to less than one part per billion. 6

8 The outgassing of hydrogen within the stainless-steel manifold including the capacitor electrodes might be an essential problem in view of the long time periods necessary for measuring one isotherm. But gold layers lower the hydrogen permeability by more than one order of magnitude [20]. Consequently, all central parts in contact with the measuring gas have been coated with a gold layer of approximately 2 µm thickness. For analysing the composition of the gas both before entering and directly after leaving the measuring capacitor, a mass spectrometer (GAM 400, InProcess Instruments) is integrated in the gas-handling system. Its mass resolution ranges from 0.5 amu to 2 amu, with a detection limit smaller than 300 ppb for H 2, 100 ppb for N 2, 50 ppb for CO 2, 20 ppb for CH 4, and for the most other elements smaller than 10 ppb (ppb parts per billion). A typical impurity content in the measuring gas helium after being more than one week within the measuring system is: 370 ppb of H 2, 45 ppb of CH 4, 40 ppb of Ne, about 10 ppb of Kr and Xe. The found impurity content leads to an uncertainty estimate for the component Impurities in the budget for the determination of the Boltzmann constant, see Section 3.4, of about 2.4 ppm and will be reduced in the future by using getters and cold traps. The estimate considers two main facts. All impurities have a polarizability, which is larger than that of helium, i.e. without a correction, the treatment of the influence of impurities is not symmetric as prescribed by the Guide to the Expression of Uncertainty in Measurement [21]. On the other hand, the results of the analysis do not allow to apply a reliable correction (uncertainty too large, use of detection limits). Thus, an overall maximum estimate has been calculated by summing up the possible influences of all impurities and dividing the result by the square root of three, which corresponds to the application of a non-symmetric rectangular distribution Pressure measurement The goal to measure pressures up to 7 MPa with a relative uncertainty of order 1 ppm is a challenge because it requires to characterise pressure balances with a unprecedented quality and even to improve the national standard of PTB significantly. For absolute pressure measurements in helium up to 7 MPa, a system of special pressure balances, as outlined in [22, 23], was designed, constructed and evaluated [24]. The system includes two pressurebalance platforms, three piston-cylinder units (PCUs) with effective areas of 20 cm 2 and three 2 cm 2 PCUs. Each platform is equipped with automated mass-piece handlers and a 150 kg mass-piece set allowing cross-float and pressure measurements in absolute mode. The 11 main mass pieces, each of 12.5 kg, were manufactured in accordance with the requirements of 7

9 recommendation OIML R to masses of Class E1, see [25], concerning the mechanical and magnetic properties of their materials, density and surface condition. The design of the PCUs and their mounting was optimised to reduce the pressure distortion coefficients and mounting-induced deformations. The gap width of the PCUs was adjusted to reach a compromise between fall rate and sensitivity. In order to perform automated cross floats of pressure balances with the possibility of using different gases and a high level of cross-float sensitivity, differential pressure cells were applied to indicate pressure equilibrium. Traceability of the pressure measurements to the SI base units up to 7 MPa was realised in two steps. First, the zero pressure effective areas of the 20 cm 2 PCUs have been determined from dimensional measurements. Second, the 2 cm 2 PCUs have been calibrated against the 20 cm 2 PCUs by cross-float comparisons. The calibration of the mass pieces traceable to the national mass standards and the accurate determination of the local gravity acceleration did not cause special challenges. Dimensional measurements on the required level of uncertainty below 1 ppm were possible only on the sufficiently large 20 cm 2 PCUs, whose operation-pressure range is limited to 0.75 MPa. Enhanced dimensional measurement techniques [26, 27, 28] were applied to measure diameters, straightness and roundness of the PCUs. For the first time, uncertainties of three-dimensional data of order 8 nm (pistons) and 16 nm (cylinders) have been obtained. The calculation of the effective areas and the estimation of their uncertainties required several efforts, see the detailed discussion in [23]: Evaluation of the three-dimensional data applying a least-squares method [29], Use of a two-dimensional flow model to consider the axial non-symmetry [30], Consideration of the gas-flow conditions in the clearance [31], Measurement of the elastic constants of the PCU s materials [32], Calculation of the pressure distortion coefficient by FEM [33], Estimation of the uncertainty of the dimensional data as introduced in [26]. The calculations yielded a relative uncertainty of the effective areas of the three 20 cm 2 PCUs between 0.5 ppm and 0.7 ppm. The 2 cm 2 PCU No used for the DCGT measurements at the TPW has been calibrated by cross floating with the 20 cm 2 PCUs. The resulting overall relative uncertainty of the calibration amounts to 1.5 ppm for the zero-pressure effective area, 8

10 see the detailed budgets presented in [23]. Unfortunately, the cross-float comparison of PCU No with another 2 cm 2 PCU at 7 MPa yielded an unexpected difference in pressure dependence of the effective areas. This may be caused by an error in the calculation of the pressure distortion coefficient. Applying a rectangular distribution, it causes an additional relative uncertainty component of 0.16 (p / MPa) ppm. A complete uncertainty budget for the measurement of a pressure of 7 MPa in a DCGT experiment at the TPW is given in Table 1. The budget is of course dominated by the components connected with the zero-pressure effective area and the pressure-distortion coefficient. Table 1: Uncertainty budget for the measurement of a pressure of 7 MPa using pistoncylinder unit No with an effective area of 2 cm 2. Component u(p) / p 10 6 Zero-pressure effective area 1.5 Pressure-distortion coefficient 1.1 Mass measurement (piston, mass pieces) 0.1 Gravity acceleration 0.1 Temperature measurement 0.2 Verticality of the PCU 0.1 Combined standard uncertainty Experimental results 3.1. Determination of the effective compressibility of the measuring capacitors The changes of the dimensions of the measuring capacitor due to the gas pressure p are considered in Equations (1) and (3) via the effective compressibility κ eff. For a cylindrical capacitor as used up to now, the changes of the radii of the inner and outer cylinder due to the pressure cancel out in the formula for the capacitance. Therefore in an ideal design, only the relative change l(p)/l(0) of the length of the electrodes is relevant. For this ideal case, the effective compressibility would be one third of the bulk compressibility, which is the inverse of the bulk modulus. 9

11 Resonant ultrasound spectroscopy (RUS) [34, 35] is an excellent method for measuring the elastic properties of a material that determine the bulk compressibility. Only specimen s normal-mode frequencies of free vibration were used along with the shape and mass to determine its elastic properties. Measurements were performed on 10 parallelepipeds cut from the same rod (stainless steel EN standard number / X39CrMo17-1), from which the capacitor electrodes were manufactured. The dimensions of the samples were approximately 17 x 13 x 10 mm 3, 13 x 10 x 9 mm 3 and 12.5 x 11 x 8 mm 3, respectively. Care was taken to ensure that the mean roughness index R a of the sample faces is within 0.4 µm, and parallelism and squareness are within 1 µm. The comparison of the results obtained for samples, which are different with respect to their dimensions and preparation, allowed to estimate how representative they are for the material comprising the capacitor electrodes. The corresponding uncertainty component is not dominant at the level achieved up to now. The dimensions of the samples and their masses have been measured with uncertainties of 1 µm and 0.5 µg, respectively. The sample resonances were measured at t = 0 C using the system RUSpec from Magnaflux Quasar. The excitation frequency was scanned approximately from 80 to 460 khz, measuring the first 60 resonances in each case. The determination of the elastic constants from the measured resonance frequencies is an inverse problem, the uncertainty of which is primarily limited by that of the dimensions of the sample. It starts with calculating theoretically an approximate resonance spectrum (direct problem), which then is compared with the measured spectrum. In an iteration process, the input parameters have to be adjusted during each iteration step to minimize the error function, which is defined as the square sum of the differences between the calculated and measured frequencies. In order to estimate the uncertainty reliably, two different methods were used to solve the inverse problem. Firstly, the direct problem was solved with the aid of a computer code, which has been developed by Migliori et al. [35], and for minimizing the error function, the generalized reduced gradient method GRG2 [36] came into force. Secondly, the direct problem was treated applying a finite-element-method (FEM) eigenfrequencies analysis using the commercial COMSOL Multiphysics simulation software (version 4.1). For the minimization of the error function, the Nelder Mead algorithm (downhill simplex) [37] was implemented in the FEM software. For both methods, the influence of variations in the input parameters on the output elastic constants has been analysed. Applying both methods to at 10

12 least six measurements for each sample revealed a relative standard deviation for the bulk modulus of about 0.02%. The mean values for the bulk modulus, Poisson s ratio and Young s modulus amount to (19) GPa, (3) and (10) GPa, respectively. This result leads to a material-determined value of the effective compressibility of κ eff,mat = Pa -1 at 0 C with a relative standard uncertainty of 0.14%. Facing the fact that the capacitor within the pressure vessel is a relatively complicated geometrical object including electrically isolating pieces and stabilising screws, its effective compressibility κ eff surely deviates from the ideal value. (The gold plating of the capacitor electrodes has no significant influence on κ eff.) Thus, κ eff has to be estimated reliably by performing a FEM simulation. For this simulation, a three-dimensional capacitor model has been built applying the COMSOL Multiphysics structural mechanics module. For the Young s modulus and Poisson s ratio, the RUS results were used. The density has been obtained from dimensional and mass measurements. Hydrostatic pressures up to 7 MPa were considered to act on the inner surface of the closed pressure vessel and all parts of the cylindrical capacitor. Finally, to prevent rigid body motions of the overall model, two additional displacement constraints (one for the horizontal directions and one for the vertical direction) were applied. By changing the constraints, it was later confirmed that they do not induce reaction forces. The simulations have been carried out after the mesh was completed (80768 tetrahedral, triangular and 2990 edge elements). The FEM simulation yielded the displacement of the electrodes under pressure, i.e. l(p)/l(0). It has been found that due to the special design of the capacitor and its supporting plate, l(p)/l(0) is larger for the outer electrode than for the inner one. Therefore, to estimate the influence of the difference of the electrode lengths on the capacitance value, additional FEM simulations had to be performed with the electrostatic module of COMSOL Multiphysics. The estimation showed that the capacitance change is dominated by l(p)/l(0) of the inner electrode, but around 28.6% of the length difference between inner and outer electrode has to be taken into account. This leads to an effective length change of l eff (p) = l inner (p) (l outer (p)-l inner (p)). The combination of the two FEM simulations leads to a relative correction of the effective compressibility of 1.69% with an uncertainty of 0.12%, which corresponds to a relative uncertainty of the Boltzmann constant of about 4 ppm. For future improvement, the influence of a more detailed model of the capacitor, the number of elements and the use of other element types on the FEM results will be investigated. Finally, it has to be considered that the compressibility measured by RUS is the adiabatic one, whereas for the DCGT the isothermal compressibility 11

13 is needed. The conversion between both compressibility values based on fundamental thermodynamic relations [38] and thermodynamic properties of the stainless steel used, given by the manufacturer (Dörrenberg Edelstahl), amounts to 1.4% with an uncertainty of 0.07%. The resulting effective compressibility amounts to κ eff = Pa -1 with a relative uncertainty of 0.20%. This result has been corroborated by measurements with neon as measuring gas employing the polarizability value published in [13] (for details see section 3.3). For further improvement of the determination of the bulk compressibility, the influence of the surface quality on the RUS signal will be investigated in more detail. Also the thermodynamic properties of the steel used will be determined individually to reduce the uncertainty of the adiabatic-isotherm correction DCGT isotherms A total of eleven isotherms have been measured with helium (nominal purity %, supplier Linde AG) and four with neon ( %, Linde AG) using cylindrical capacitors as described in [18], but without a cylindrical ground-shield spacer at the top. The isotherms consist of up to fourteen triplets of temperature T, pressure p and µ (result of the capacitance measurement, i.e. in fact measure of the susceptibility, see Section 2.1). The overall number of triplets amounts to 121 for helium and 56 for neon. The pressure values range from MPa to 6.75 MPa with a spacing of MPa below 5 MPa and 0.75 MPa above. In most cases, the pressure was increased during the measurement of an isotherm, but for both gases, two isotherms were taken decreasing the pressure to check the hysteresis of the capacitor. During all measurements, the deviation of the temperature from K was smaller than 200 mk. The ratio of the two evacuated capacitors ( zero measurement ) has been determined regularly over the period of the experiments of four months. This allowed to observe a long-term drift, which amounts to 8 ppm over the whole period and can be described by a quadratic function. By the aid of this function it is possible to apply for each isotherm an individual drift correction with an uncertainty of a few ppb. The sequence of an isotherm measurement with increasing pressure was as follows. After the zero measurement, gas was poured slowly into the measuring capacitor until the lowest pressure was reached. For measuring the pressure continuously, a quartz-crystal-resonator sensor is connected to the pressure line. The gas movement caused an increase of the temperature by a few millikelvin. Due to the slow thermal recovery of the system, see Section 2.2, it was necessary to wait six to eight hours to reach thermal equilibrium. Then, the 12

14 pressure balance was connected to the pressure line. The final measurement of a triplet lasted about ten minutes, whereas the readings of the thermometer, quartz sensor and the null indicator of the capacitance bridge were averaged over one minute and the mean values stored on a computer. Figure 2 shows the typical relative scattering of the three quantities temperature, pressure and susceptibility around the respective mean value over the finalmeasurement period. The scatter of the susceptibility must be the largest one because it detects also the scatter of all other quantities. 4 3 T - Measurement (STD 0.06 ppm) p - Measurement (STD 0.6 ppm) χ - Measurement (STD 1.4 ppm) 2 x/x / ppm time / min Figure 2: Typical relative scattering x / x of the three measurands temperature T (red circles), pressure p (blue squares), and susceptibility χ (black triangles) around their respective mean values over the final period for measuring one triplet of T, p and χ on a DCGT isotherm. The standard deviations (STDs) are given in the legend. Since the amount of gas poured into the capacitor is smaller for the following pressure steps, the resulting temperature increase has an order of magnitude of one millikelvin. This caused recovery periods of about three hours, i.e. the measurement of an isotherm with fourteen triplets lasted one working week. To investigate the hysteresis of the measuring capacitor, isotherms have been measured with increasing and decreasing pressure, respectively. For each of the 10 pressure values, the mean of the two resulting µ values, which differ both due to hysteresis and non-repeatability, has 13

15 been calculated. Figure 3 illustrates the results obtained. For the isotherm measured with increasing pressure, it shows the deviation of the µ value from the mean µ value in dependence on pressure. The long-term drift of the zero-measurement has been corrected for. The maximum hysteresis amounts to about 20 ppm. Since the hysteresis is mostly smooth, it can be well described by the fit function Equation (2) µ /µ / ppm p / MPa Figure 3: Half relative hysteresis of the DCGT capacitor in the pressure range from about 3 MPa to 7 MPa: µ = µ up (µ up + µ down )/2 is the deviation of the value µ up obtained during the measurement of a DCGT isotherm with increasing pressure from the mean value (µ up + µ down )/2 resulting from both isotherm measurements with increasing and decreasing pressure, respectively. 14

16 3.3. Evaluation of the DCGT isotherm data The evaluation of the data started with the correction caused by the long-term drift of the zero-measurement and the transformation to K. Then for checking purposes, singleisotherm fits were made applying Equation (2) with orders from three to five. It has been found that the fits of third order are most stable in accordance with the magnitude of the virial coefficients of helium at the TPW. For the therefore preferred order three, Figure 4 shows the single-isotherm fit residuals of all triplets measured with helium. Below the preferred pressure range from about 3 MPa to 7 MPa, the scattering is larger due to the small susceptibility of the gas. 40 Fit residuals of single points 20 p/p / ppm p / MPa Figure 4: Relative single-isotherm fit residuals of all measured helium triplets for the preferred order three of the fit function Equation (2). Below the preferred pressure range from about 3 MPa to 7 MPa, the scattering is larger due to the small susceptibility of the measuring gas helium. The final fits were made considering all triplets together, either as individual data points or as mean µ values for the fourteen pressure values. The comparison of the results obtained on both paths for order three yielded an indication for their robustness in addition to the Monte- Carlo simulations, see below. As an additional, optional check, the virial coefficients resulting from the fit parameters of the second and third order have been compared with theoretical values for the volume virial coefficients given in [39, 40, 41]. The differences between 15

17 experiment and theory increase smoothly with increasing pressure. These differences have no influence on the value of A 1, representing the ideal-gas behaviour at low pressures, but will be investigated more in detail later on with respect to the influence of the dielectric virial coefficients and the effective compressibility. Finally, the robustness has been investigated by decreasing the number of the mean µ values included in the fits. This has been done beginning both at low and at high pressures. The results are shown in Figure 5. It can be seen that ten data points are sufficient for obtaining stable results for the fitting parameter A 1 with a standard deviation of 3 ppm. For ten data points, the covered pressure ranges are MPa to 4.50 MPa and MPa to 6.75 MPa, respectively. It is remarkable that the A 1 values obtained at lower pressures coincide well with the higher pressure ones. This means, the larger scattering of the data at low pressures is fully statistical. 150 p-range reduced from high p p-range reduced from low p 100 k / k / ppm Number of points in the fit Figure 5: Dependence of the fit results for the Boltzmann constant on the number n of data points if the mean µ values are used as input data for the isotherm fit applying Equation (2). Starting with the maximum number of 14 points, n has been reduced by excluding points at low (black circles) and high pressures (red squares). In both cases, k / k is the relative deviation from the result obtained for n = 14. From the fit to the mean µ values of helium at the TPW in dependence on pressure, a value of the Boltzmann constant of J/K has been deduced applying Equation (4) with ε 0 = F/m and He α 0 = C 2 m 2 /J [14, 15] (for a detailed discussion see [42]), which corresponds to He A ε = m 3 /mol. Using the molar 16

18 polarizability of neon Ne A ε = m 3 /mol published in [13], the data obtained with neon corroborate the applied value of the effective compressibility of the measuring capacitor, see Section 3.1, yielding an absolutely larger result of /Pa with an uncertainty of about 0.9 %, which is mainly limited by the relative standard uncertainty of Ne Aε of 11 ppm Uncertainty budget The complete uncertainty budget for the determination of the Boltzmann constant k by DCGT at the TPW applying Equation (4) is given in Table 2. It has been established in accordance with the Guide to the Expression of Uncertainty in Measurement [21]. For the DCGT measurement, the direct access to the relevant uncertainty estimates is hindered because the final result A ε /R contained in parameter A 1 of the fit function Equation (2), see Equation (3), is gained by fitting. Therefore, the statistical information for estimating the type A components has been deduced by Monte-Carlo simulations. This allows also to consider the pressure dependence of the uncertainty sources. The simulations have been performed with data sets randomized with the standard deviation of the specific quantity. The combined type A estimate is corroborated by the comparison of the results obtained applying different fits, see Section 3.3. The type B components consider uncertainty sources, which influence the value of k systematically. The main uncertainty components result from the determination of the dielectric susceptibility via capacitance changes, the instability of the capacitance of the measuring capacitor (longterm drift, influence of pressure cycling), calculation of the effective compressibility by FEM using the elastic constants obtained with RUS, see Section 3.1, and the pressure measurement, see Section 2.5. A complete budget for the measurement of capacitance changes is given in [19]. The component estimated for the temperature measurement includes static and dynamic errors within the measuring system, see Section 2.2. For the measuring gas helium, the head correction due to the gas column is very small. Its overall relative magnitude for the present DCGT setup amounts only to 8 ppm. Impurities cause errors since their polarizability differs from that of helium. Hydrogen having a polarizability, which is larger by a factor of about four, is especially dangerous. Surface layers on the electrodes of the measuring capacitor play a minor role because they are present both for the C(p) and the C(0) measurement. The uncertainty of the polarizability of helium is estimated in [42, 15]. A detailed discussion of the uncertainty components can be also found in [3, 42, 43] 17

19 Table 2: Uncertainty budget for the determination of the Boltzmann constant k by dielectricconstant gas thermometry at the triple point of water. Component u(k)/k 10 6 Monte-Carlo simulations (type A components) Susceptibility (scatter of capacitance bridge reading) 3.5 Pressure repeatability 1 Temperature instability 0.5 Capacitance instability 5 Type B estimates Susceptibility measurement (capacitance change) 1 Determination of effective compressibility (RUS, FEM) 5.8 Temperature (traceability to the TPW) 0.3 Pressure measurement (7 MPa) 1.9 Head correction (pressure of gas column) 0.2 Impurities (measuring gas) 2.4 Surface layers (impurities) 1 Polarizability ab initio calculation (theory) 0.2 Combined standard uncertainty Conclusions and outlook A value of k TPW = J/K has been determined for the Boltzmann constant. This has been done by performing dielectric-constant gas thermometry at the triple point of water applying a new special experimental setup and helium as the measuring gas. The experimental setup consists of a large-volume thermostat, a vacuum-isolated measuring system, stainlesssteel 10 pf cylindrical capacitors, an autotransformer ratio capacitance bridge, a high-purity gas-handling system including a mass spectrometer, and traceably calibrated special pressure balances with piston-cylinder assemblies having effective areas of 2 cm 2. The detailed analysis of the uncertainty sources including Monte-Carlo simulations yielded an estimate for the relative standard uncertainty of the obtained k TPW value of 9.2 ppm. In an accompanying paper [44], the measurement of the Boltzmann constant in the temperature range from 21 K to 27 K is described. The low-temperature DCGT experiments were performed using an apparatus, which has a special cryostat as central part for the realisation of highly stable thermal conditions. The same apparatus was applied for 18

20 establishing a thermodynamic temperature scale in the range from 2.5 K to 36 K [12]. The thermodynamic reference necessary for measuring k was obtained via the realisation of the International Temperature Scale of 1990, ITS-90 [45]. For estimating the deviation T = T 90 T, the careful investigation of T in [46] has been used. To avoid correlations between the new measurements and former DCGT measurements considered in [46], the deviation T in the range between 21 K and 27 K was analyzed for two different scenarios. The first includes the former DCGT data and the second one neglects them by setting their uncertainty to practically infinity. In both cases, the necessary continues (linear) function T(T 90 ) for the deviation in the range between 21 K and 27 K was determined by a fit to the T values given in [46] at selected temperatures. The resulting functions T(T 90 ) for the two scenarios agree at a level of order 0.1 mk, and their mean difference has been used for estimating an additional small uncertainty component. This component has been added in quadrature to the uncertainties given in [46]. The uncertainty of the realisation of the triple point of neon as temperature fixed point (T 90 = K) of crucial importance has been estimated on the basis of an international star intercomparison of sealed triple point cells [47]. Considering the uncertainties of T 90 T and the realisation of T 90, a standard uncertainty of the thermodynamic reference of 9.5 ppm has been estimated. Other main uncertainty components resulted from the measurement of the dielectric susceptibility via capacitance changes, the instability of the capacitance of the measuring capacitor, the determination of the effective compressibility from results obtained at higher temperatures and the pressure measurement. The overall relative standard uncertainty of the value k LT = J/K obtained at low temperatures is 15.9 ppm. The weighted mean of the two values k LT and k TPW of the Boltzmann constant determined at low-temperatures and the TPW, respectively, amounts to k = J/K and has a relative standard uncertainty of 7.9 ppm. This value differs from the CODATA value of J/K [48] by 3.2 ppm. In view of the overall combined standard uncertainty of 8.1 ppm, the difference between the two values is not significant. Looking at the state-of-theart uncertainty level of the most promising methods of primary thermometry [AGT, RIGT, NT, DBT], it can be stated that, at present, DCGT is the only method with which an uncertainty of the determination of the Boltzmann constant below 10 ppm has been obtained without applying acoustic and / or microwave resonators. Such resonators are used for acoustic gas thermometry [5] and refractive index gas thermometry [49]. This fact is of 19

21 special importance because DCGT is independent of the other methods. It has quite different error sources. For the first determination of the Boltzmann constant by DCGT at the TPW, the two main uncertainty components are connected with the properties of the measuring cylindrical capacitor, namely with its instability and effective compressibility. Progress in decreasing these components significantly is expected by comparing the parameters of capacitors having quite different designs, see [16], and by using alternative electrode materials, e.g. tungsten carbide, the compressibility of which is smaller than that of stainless steel by a factor of three. First stability tests of cross capacitors gave encouraging results. This is important because the calculation of their effective compressibility should be more accurate. Activities to decrease the uncertainty of the pressure measurement to a level of 1 ppm are in progress. First of all, this concerns the collection of more data on the stability of the dimensions of piston-cylinder assemblies of the pressure balances and the ratios of their effective areas measured by cross floating. For reducing the influence of impurities, the outgassing from the walls inside the pressure vessels has to be investigated in more detail. This requires to analyse better the content of the most dangerous impurity hydrogen by mass spectrometry. Considering the experience gained during the first DCGT experiments at the TPW, it seems to be realistic to decrease the relative uncertainty of the Boltzmann constant to a level of only about 2 ppm within the next two years. Acknowledgements The research within the EURAMET Joint Research Project receives funding from the European Community's Seventh Framework Programme, imeraplus, under Grant Agreement No which is gratefully acknowledged. The authors thank Bettina Thiele- Krivoi and Norbert Haft for assisting in the development of the experimental setup and performing the measurements. They are also grateful to Thomas Konczak, Steffen Scheppner and Helga Ahrendt, who took care of the technical part of the project to design, put into operation and characterise special pressure balances. Support of this research by the International Graduate School of Metrology (IGSM) Braunschweig is acknowledged. 20

22 References *) Identification of commercial equipment and materials in this paper does not imply recommendation or endorsement by PTB, nor does it imply that the equipment and materials identified are necessarily the best available for the purpose. [1] BIPM 2005 Recommendation T 2 to the CIPM: New Determinations of Thermodynamic Temperature and the Boltzmann Constant. Working Documents of the 23rd Meeting of the Consultative Committee for Thermometry (BIPM, Document CCT/05-31, 2005) [2] Gugan D and Michel G W 1980 Metrologia [3] Luther H et al 1996 Metrologia [4] Moldover M R et al 1988 Phys. Rev. Lett. 60, [5] Pitre L et al 2011 Int. J. Thermophys. (to be published) (2011) [6] Benz S P et al 2011 Metrologia [7] Lemarchand C et al 2010 Int. J. Thermophys [8] Castrillo A et al 2009 C. R. Physique [9] Fellmuth B et al 2006 Meas. Sci. Technol. 17 R145-R159 [10] Fellmuth B et al 2009 C. R. Physique [11] Gaiser C and Fellmuth B 2009 Metrologia [12] Gaiser C et al 2010 Int. J. Thermophys [13] Gaiser C and Fellmuth B 2010 Europhysics Letters p p5 [14] Łach G et al 2004 PRL [15] Puchalski M et al 2011 Phys. Rev. A [16] Zandt T et al 2010 Int. J. Thermophys [17] Merlone A et al 2010 Int. J. Thermophys [18] Zandt T et al 2011 Int. J. Thermophys. DOI: /s [19] Fellmuth B et al 2011 IEEE Trans. Instr. Meas. DOI /TIM

23 [20] Ishikawa Y and Nemanič V 2003 Vacuum [21] BIPM 1995 International Electrotechnical Commission (IEC), International Federation of Clinical Chemistry (IFCC), International Organization for Standardization (ISO), International Union of Pure and Applied Chemistry (IUPAC), International Union of Pure and Applied Physics (IUPAP) and International Organization of Legal Metrology (OIML), Guide to the Expression of Uncertainty in Measurement. (Geneva, Switzerland: International Organization for Standardization (ISO/IEC Guide 98:1995), ISBN ) [22] Sabuga W 2007 MAPAN J. Metrology Soc. India [23] Sabuga W 2011 Pressure measurements in gas media up to 7.5 MPa for the Boltzmann constant redetermination PTB-Mitteilungen (to be published) [24] Sabuga W et al 2011 Design and Evaluation of Pressure Balances with Uncertainty for the Boltzmann Constant Project PTB-Mitteilungen (to be published) [25] OIML 2004 OIML R 111-1, Weights of classes E1, E2, F1, F2, M1, M1 2, M2, M2 3 and M3 Part 1: Metrological and technical requirements, OIML [26] Jusko O et al 2008 Dimensional calibration techniques for pressure for pressure balances to be used in the new determination of the Boltzmann constant CIMMEC 1st International Congress on Mechanical Metrology [27] Jusko O et al Proc. ASPE Spring Topical Meeting Mechanical Metrology and Measurement Uncertainty, Albuquerque, NM, USA, p ISBN [28] Jusko O et al 2010 Key Engineering Materials [29] Sabuga W and Priruenrom T 2007 IMEKO 20th TC3, 3rd TC16 and 1st TC22 International Conference Cultivating metrological knowledge 27th to 30th November, 2007 [30] Sabuga W and Priruenrom T th APMP Pressure and Vacuum Symposium Two- Dimensional Flow Model for Calculation of the Effective Area of Axially Non- Symmetric Piston-Cylinder Units [31] Sabuga W et al 2011 Determination of the Effective Area of Piston-Cylinder Assemblies Using a Rarefied Gas Flow Model. PTB-Mitteilungen (to be published) 22