Phil. Trans. R. Soc. A (2011) 369, 4014–4027doi:10.1098/rsta.2011.0197
Determination of the Boltzmann constant usinga quasi-spherical acoustic resonator
BY LAURENT PITRE*, FERNANDO SPARASCI, DANIEL TRUONG,ARNAUD GUILLOU, LARA RISEGARI AND MARC E. HIMBERT
Laboratoire Commun de Métrologie LNE-CNAM (LCM), 61 rue du LandyF 93210 La Plaine Saint-Denis, France
The paper reports a new experiment to determine the value of the Boltzmann constant,kB = 1.3806477(17) × 10−23 J K−1, with a relative standard uncertainty of 1.2 parts in 106.kB was deduced from measurements of the velocity of sound in argon, inside a closed quasi-spherical cavity at a temperature of the triple point of water. The shape of the cavitywas achieved using an extremely accurate diamond turning process. The traceability oftemperature measurements was ensured at the highest level of accuracy. The volumeof the resonator was calculated from measurements of the resonance frequencies ofmicrowave modes. The molar mass of the gas was determined by chemical and isotopiccomposition measurements with a mass spectrometer. Within combined uncertainties,our new value of kB is consistent with the 2006 Committee on Data for Science andTechnology (CODATA) value: (knew
B /kB_CODATA − 1) = −1.96 × 10−6, where the relativeuncertainties are ur(knew
B ) = 1.2 × 10−6 and ur(kB_CODATA) = 1.7 × 10−6. The new relativeuncertainty approaches the target value of 1 × 10−6 set by the Consultative Committeeon Thermometry as a precondition for redefining the unit of the thermodynamictemperature, the kelvin.
Keywords: acoustic resonance; Boltzmann constant; International System of Units; kelvinredefinition; microwave resonance; quasi-sphere
1. Introduction
This paper reports a new experimental determination of the Boltzmann constant(kB). The value is deduced from measurements of the velocity of sound in argon,as in the work performed by Moldover et al. at the National Institute of Standardsand Technology (NIST) in 1988 [1]. However, several fundamental modificationsand improvements have been made to measure and control the parameters thatinfluence the measurement of kB [2,3].
The aim of this work was to determine the value of kB with an improvedaccuracy. By obtaining this value, a definition of the kelvin based on thisfundamental constant is possible, instead of the temperature of the triple pointof water (TTPW) [4,5].
Figure 1. Comparison of the recent determinations of kB with the 2006 CODATA recommendedvalue kB_CODATA [6]. The shaded area spans the uncertainty u(kB_CODATA). The experimentaldeterminations by Pitre et al. [7] in 2009, by Sutton et al. [2] in 2009 and by Gavioso et al. [8] in 2010were, respectively, obtained at the Laboratoire National de Métrologie et d’Essai-ConservatoireNational des Arts et Métiers (LNE–CNAM), at the National Physical Laboratory (NPL) and atthe Istituto Nazionale di Ricerca Metrologica (INRiM), by measuring the speed of sound in Heand Ar with quasi-spherical resonators. The value from the study of Zhang et al. [9] was obtainedwith a cylindrical acoustic resonator.
The experimental result is
kB = 1.3806477(17) × 10−23 J K−1(1.2 ppm).
This value is consistent with both the previous determinations and the Committeeon Data for Science and Technology (CODATA) value [6], as represented infigure 1.
2. Principle of measurement
The present method is based on the determination of the universal gas constant,R, from a series of measurements of the speed of sound ca in a monatomic gas.Measurements were performed in a 0.5 l quasi-spherical cavity made of copper, atdifferent pressures, along an isotherm line close to the temperature of the triplepoint of water, TTPW = 273.16 K.
The measurements of ca depend on the pressure, and kB is calculated as follows:
kB = RNA
= Ar(Ar) · Mu
(5/3)TTPWNAlimp→0
∗c2a(p, TTPW), (2.1)
where c2a(p, TTPW) is the square of the speed of sound at pressure p and
temperature TTPW. Ar (Ar) is the relative atomic mass of argon (alternatively,we can use helium, and this term becomes Ar (4He)), Mu is the molar mass
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constant, NA is the Avogadro constant and the factor 5/3 is the ratio g = Cp/CVof the specific heat capacities for dilute monatomic gases. The term lim
p→0∗ indicates
that we estimate the terms independent from the pressure in c2a(p, TTPW).
The velocity of sound in a gas can be determined experimentally by measuringacoustic resonances in a spherical cavity of known radius. The method involvesthe measurement of the resonance frequencies of standing waves in cavities ofsimple geometry. The acoustic fields of high-quality, non-degenerate modes of suchcavities are experimentally well-defined, theoretically well understood [10,11], andcan be achieved with a mechanically simple apparatus [12]. The speed of soundis related to the following elements: resonance frequencies, f A
n,l , the correctionsDf A
n,l taking into account the effects of the cavity and the non-ideality of the gas;and the eigenvalues ZA
n,l for the wavenumbers (n, l) in the acoustic cavity. In theparticular case of a spherical cavity of radius a, we have
ca(p, TTPW) = f An,l(p, TTPW) + Df A
n,l(p, TTPW)
ZAn,l
2pa(p, TTPW). (2.2)
A microwave technique can be used in combination with the acousticmeasurements to determine the cavity dimensions as a function of temperatureand pressure [7,13]. This experimental technique allows the measurement ofthe thermal expansion of spherical cavities in acoustic thermometry, and hassuccessfully been applied in several experiments [14–19].
The radius a(p, TTPW) of the cavity can be obtained using
c(p, TTPW) = f EMn,l (p, TTPW) + Df EM
n,l (p, TTPW)
ZEMn,l
2pa(p, TTPW), (2.3)
where c(p, TTPW) is the speed of light in the gas at pressure p and 273.16 K,f EMn,l are the measured electromagnetic resonance frequencies and ZEM
n,l are theelectromagnetic eigenvalues for the wavenumbers (n, l) in the cavity. Df EM
n,l arecorrections related to the non-ideality of the resonator walls.
A quasi-spherical geometry, a triaxial ellipsoid, has been preferred in orderto remove the degeneracy associated with the microwave eigenfunctions [12,20].A second-order correction of microwave eigenvalues [21] has been computed todeduce the radius a(p, TTPW). Microwave and acoustic measurements were carriedout simultaneously, and kB was determined by combining equations (2.2) and (2.3)with equation (2.1),
kB =⟨35
mTTPW
· limp→0
∗⎛⎝(
2paZA
n,l
)2 (〈f A
n,l + Df An,l 〉
)2
⎞⎠⟩
=⟨35
m · c20
TTPW
(ZEM
n,l
ZAn,l
)2
limp→0
∗( 〈f A
n,l + Df An,l 〉
〈f EMn,l + Df EM
n,l 〉
)2⟩, (2.4)
where m is the atomic mass of the gas and c0 is the speed of light in vacuum.
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Figure 2. Schematic of the acoustic and electromagnetic acquisition systems.
Table 1. Overall relative standard uncertainty on the Boltzmann constant acoustic determinationperformed in argon at LNE-CNAM.
uncertainty contributionuncertainty source on kB (parts in 106)
temperature measurements 0.3volume electromagnetic measurements 0.6acoustic measurements 0.8molar mass and gas purity 0.6repeatability over two isotherms 0.3
combined uncertainty 1.2
Figure 2 shows a schematic of the complete measurement system, allowing therealization of simultaneous acoustic and electromagnetic measurements.
Table 1 lists the different contributions to the evaluation of the uncertainty inthe measured value of kB. Each item is related to the different quantities measuredin equation (2.4) and is described in detail in the subsequent sections of this paper.
The main improvements with respect to the previous determination [1] comefrom temperature measurements and molar mass of gas measurements. The mostinnovative technique is the measurement of the volume of the resonator usingelectromagnetic microwaves. The corresponding contribution to the uncertaintyis of a completely different nature from that of the pyknometry used by Moldoveret al. [1]. The microwave technique requires an accurate model involving thegeometry of the resonator, the perturbations from antennas and the penetrationof microwaves into the copper shell. Finally, this new determination of theBoltzmann constant has been made using a realization of the newly refineddefinition of the TPW that takes into account the influence of the isotopiccomposition.
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Figure 3. (a) Schematic of the resonator in the thermostat: stirring propeller (a), Dewar (b), heater(c), pressure vessel (d), radiation shield (e), vacuum vessel (f), Ar or He gas (g), water + ethyleneglycol bath (h), cooler (i) and heater (j). (b) Cut view of the thermostat head in the bath: corrugatedhose thermalizing tubes and wires (k), thermal link (l), vacuum vessel lid (m), thermal shield topplate (n), gold-plated copper pressure vessel (o), gold-plated copper thermal shield (p) and stainlesssteel vacuum vessel (q).
3. Temperature control on the acoustic resonator
Acoustic measurements for the determination of the Boltzmann constant arecarried out at the TTPW, which is the most accurately realized parameter upto now, according to the present definition of the kelvin.
The acoustic resonator (named BCU3) was installed in a temperature-controlled thermostat operating at temperatures near TTPW and was keptin isothermal conditions, so that thermal gradients over its surface wereminimized during the experiment. The system was conceived like a quasi-adiabatic calorimeter, with a thermally shielded experimental chamber kept undervacuum. This ensured that temperature drifts were small enough to generate onlynegligible thermal gradients. Figure 3 shows the layout of the thermostat anddetails of the experimental chamber.
Four capsule-type standard platinum resistance thermometers, calibrated atTTPW, were used to map the temperature distribution and to determine theaverage temperature of the resonator. Several configurations were tested andtemperature measurement results are shown in figure 4. The positions of twothermometers (1551 and HS135 in figure 4) were swapped in different runs of theexperiment (configurations named C1 and C2 in figure 4), while the two others(1825277 and 229073 in figure 4) were unchanged. This helped in discoveringpossible thermal gradients over the shell, but no gradients were detected. Different
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Figure 4. Plot of DT − DT values and the associated standard uncertainty bars u(DT ) for thefour thermometers. Nine thermal maps of the resonator are shown (Tm1–Tm9). Measurements arecarried out in configuration C1 or C2, with either argon (Ar) or helium (He), at different resonatoraverage temperatures, DT .
gas pressures were applied in the resonator (not explicitly mentioned in figure 4),and thermal maps were studied at several temperatures within a few tens ofmillikelvins from TTPW. Additional checks were performed in helium after theend of the experiments in argon (one point reported in figure 4).
The control of the temperature results in an overall relative standarduncertainty contribution of 0.3 × 10−6 on kB from the measurements performedwith BCU3 in argon. This is equivalent to one-third of the temperatureuncertainty contribution in the previous determination of kB [1].
4. Precise measurement of the volume of the resonator
In 1986, Mehl & Moldover [13] used first-order perturbation theory to prove thatthe mean eigenfrequency of a multiplet was independent of volume-preservingdeformations, and suggested that the volume of an imperfect spherical resonatorcould thus be determined from its microwave spectrum. Several of our earlierpapers have set the groundwork for this determination [2,3,12,17,22].
In this work, the average radius of BCU3 has been determined frommeasurements of the frequencies and half-widths of nine microwave triplets,corrected to take into account three effects: (i) the microwave penetration depthusing measured half-widths of the resonances, as described by Sutton et al. [2],(ii) the inlet and outlet gas ducts and the two microwave antennas using theextensive study of the effects of probes and holes in a quasi-spherical resonator(QSR) performed by Underwood et al. [22], and (iii) the shape of the QSR, usingsecond-order theory [21] and our measurements of the frequency splitting of themicrowave triplets.
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Figure 5. Radius measured with loop antennas (bottom). Circles, aeq loop TM; right-facingtriangles, aeq loop transverse electric (TE) mode; dotted lines, aeq mean loop antenna.
The determination of the equivalent radius and its uncertainty budget wasaccomplished in four steps.
— At 20◦C, with flowing argon and using straight microwave antennas(probes), we measured the frequencies of five transverse magnetic (TM)mode microwave triplets while the microphone ports were closed withblank copper plugs. We corrected these frequencies to account for shapeand probe perturbations using the models and the results in Underwoodet al. [22].
— With loop antennas, we determined the electrical conductivity of thecopper surface and established a bound on any dielectric layer on thesurface, in the same experimental conditions (20◦C and flowing argon)as those specified for the previous step.
— After having replaced the blank ports, we obtained the final equivalentradius of the QSR with the acoustic microphones and loop antennasinstalled on it (figure 5).
— We estimated the resonator’s compressibility on isotherms and measuredthe resonator’s thermal contraction upon cooling from 20◦C to TTPW.These determinations give access to the thermal expansion coefficient andto the compressibility of the resonator, which are useful to calculate thechange in the resonator radius as a function of temperature and pressure.
The agreement obtained in QSR radius measurements between straight andloop antennas was the order of 0.05 parts in 106.
Table 2 summarizes the budget of the uncertainties associated with thedetermination of the equivalent radius of the resonator, and hence that associatedwith the volume determination.
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Figure 6. Final values of kB determined by six acoustic modes in May 2009 (circles) and fourmodes in July 2009 (diamonds) are plotted as fractional deviations from the CODATA value.The uncertainty bars represent the uncertainty of the term independent from the pressure inca(p, TTPW).
Table 2. Uncertainty budget associated with resonator radius determination for determining theBoltzmann constant. Details on each component are provided by Pitre et al. [23].
effect on kBuncertainty source (parts in 106)
resonance fit 0.09scatter among microwave radii (includes
In 1988, Mehl & Moldover [13] suggested that the volume of an imperfectspherical resonator could be determined from its microwave spectrum, althoughthey could not achieve accurate measurements at that time. By applying
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Table 3. Uncertainty budget associated with acoustic measurements for the determination of theBoltzmann constant. Details on each component are provided by Pitre et al. [23].
effect on kBuncertainty source (parts in 106)
scatter among modes 0.35thermal conductivity of argon 0.02dispersion related to 1/p term in c2
this technique now in our experiment, we have reduced the uncertainty associatedwith the determination of the volume involved in the present experiment by nearlya factor of 1.7 with respect to Moldover et al. [1].
5. Measurements and analysis of the acoustic data
For the acoustic measurements, the experimental techniques and data analysismethods are built on foundations established over two decades [1,11], andsubsequently applied to temperature measurements from 7 to 552 K [17,24,25].
In our experiment, we measured the velocity of sound in argon inside ourcavity from the measurement of the acoustic resonance frequencies, f A
0,n . Thewavenumbers (0, n) refer to acoustic resonances having a radial symmetry.Nonlinear effects were negligible at the low sound pressure levels used inthe experiment. A detailed description of the experiment will be publishedelsewhere [23].
Figure 6 shows the results obtained in two runs of the same experiment: oneperformed in May 2009 and one in July 2009. Acoustic modes from (0,2) to (0,9)were measured (modes (0,6) and (0,7) were perturbed by the coupling betweenthe shell’s motion and acoustic resonances of the gas [25], and were not used inthe determination of kB; modes (0,8) and (0,9) were too noisy and unusable inthe measurement of July 2009, and were discarded as well).
The relative standard uncertainty of kB related to the acoustic measurementsis 0.80 × 10−6, as detailed in table 3. This uncertainty value is the same as thatobtained in [1], but with a 0.5 l cavity instead of a 3 l cavity.
The main uncertainty component in table 3 is the dispersion of the acousticmodes related to the term 1/p in the acoustic virial expansion of the squared speedof sound c2
a(p, TTPW) [23]. Physically, this effect is generally associated with theheat flux across the interface between the gas and the walls of the resonator [1],but it can also be related to other effects, such as impurities in the gas.
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6. Control of the purity and measurement of the molar mass of the gasin the resonator
The composition of the gas used in this experiment was analysed by massspectrometry at the Institute for Reference Material and Measurements (IRMM)of the European Joint Research Centre [26]. The analysis provided us withinformation on the isotopic ratio and the residual impurities in the gas.
Gas in the resonator is supplied by a controlled continuous flow from a gas-handling system (shown in figure 7). The continuous flow ensured that all theimpurities, trapped inside the inner surface of the resonator and continuouslydesorbed during the experiment, are promptly evacuated and did not affectthe gas composition. Flow and pressure control were regulated by mass flowcontrollers. Table 4 shows the properties of the gas.
The gas-handling system includes a purification step that consists of passingthe gas through a cold trap and a getter. The argon flows from the manufacturer’scylinder through the cold trap at a temperature near 100 K. The getter is a Valcohelium purifier (HP2). The manufacturer claimed that if the total concentrationof impurities at the inlet were less than 10 parts in 106 by mole fraction,then the concentration of impurities at the outlet would be less than 10 partsin 109 for H2O, H2, O2, N2, NO, NH3, CO, CO2 and CH4. Other impuritiesremoved include CF4, CCl4, SiH4 and light hydrocarbons. The 100 K cold trapis used to remove some noble gases such as Kr and Xe. This purification systemcannot remove neon and helium from the argon because these noble gases donot react with the getter and the temperature of the cold trap is too high tocondense them.
Helium and neon are elements lighter than argon and they contribute onlyto a diminution of the average molar mass. As a result, the upper and lowerbounds of the interval in which the best estimate of the molar mass lies are
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isotopic ratiosa standard uncertainty on isotopic ratiosa
r(36Ar/40Ar) 0.0033460 1.5 × 10−6
r(38Ar/40Ar) 0.00063477 0.26 × 10−6
atomic massb standard uncertainty on atomic massb
10−3 kg mol−1 10−3 kg mol−1
M40 39.962383123 0.003 × 10−6
M36 35.9675451 0.3 × 10−6
M38 37.9627324 0.4 × 10−6
total: M cAr 39.9478051 6.0 × 10−6
impuritiesa standard uncertainty on impuritiesa
parts in 106 parts in 106
N2 <2.00 2.00O2 <0.150 0.10H2O <0.50 0.50CO2 <0.50 0.50H2 not measured not measuredTHC not measured not measuredHe <1.00 1.00Ne <1.00 1.00Kr not measured not measuredXe not measured not measuredaMeasured by the IRMM.bValues and uncertainties from CODATA 2006 [6].cCalculated by combining isotopic ratio and atomic mass data.
not symmetric, and the evaluation of the final combined uncertainty cannot beperformed with conventional uncertainty propagation techniques. Nevertheless,tools are available to take these components into account [27], especially whenthere is not enough information to choose an appropriate probability distribution.To evaluate the uncertainty on kB related to the presence of neon and helium,we used approximate evaluation techniques, after having considered the followingelements.
— In similar measurements performed in the past [1,28], concentrations ofneon and helium as high as the bounds in table 4 were not observed. TheNPL group intends to measure the presence of neon and helium with adetection level below 0.3 × 10−6. To date, they have shown xNe ≤ 0.3 ×10−6 [28].
— Our experiments performed with the getter and the cold trap show thatthe impurity bounds provided by the IRMM are overestimated by almosta factor 10 [23].
— The argon is produced by an air-liquefaction process, where heliumand neon are present only in sub-part-per-million levels. During theproduction process, light impurities such as oxygen and nitrogen are
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Table 5. Uncertainty contributions from gas properties on the determination of the Boltzmannconstant. Details on each component are provided in Pitre et al. [23].
relative standard uncertaintyuncertainty source on kB (parts in 106)
isotopic composition 0.15Avogadro constant 0.05getter Allan deviation 0.02cold trap Allan deviation 0.06subtotal (combined expression) 0.16presence of He 0.52presence of Ne 0.29combined uncertainty 0.60
removed; therefore, it is extremely plausible that most of the helium andneon are removed at the same time.
Finally, we applied flow rates ranging from 0.05 to 1.33 cm3 s−1 in order todetect possible outgassing in the sphere. No effects related to outgassing or tothe presence of water were observed.
Table 5 summarizes the contributions of the gas properties with respect tothe uncertainty on the determination of kB. Controlled gas flow and gas filteringoperated with our gas-handling system led us to an uncertainty value nearlyidentical to Moldover et al. [1].
7. Conclusion
We have reported on isothermal acoustic measurements carried out at LNE-CNAM (LCM) in a copper triaxial ellipsoid resonator of 0.5 l filled with argon,yielding a new value of the Boltzmann constant. The average experimentalvalue for the Boltzmann constant, with its associated combined standarduncertainty is
kB = 1.3806477(17) × 10−23 J K−1.
The combined standard uncertainty corresponds to a relative combined standarduncertainty of 1.2 parts in 106, better than the previous 2006 CODATArecommended value [6]. Furthermore, this value of kB lies 1.9 parts in 106
below the CODATA value, and is consistent with it, as the fractional standarduncertainty of the CODATA value is 1.7 × 10−6.
The value for the universal gas constant, R, deduced from our measurement is8.3144562(99) J mol−1 K−1.
This measurement has been achieved using a 0.5 l resonator. Several newexperimental techniques were implemented, and we improved our knowledge andexperience on existing theories and technologies. We will apply this experience forfuture measurements using a 3.1 l QSR. As proved by the present result and by
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numerous pending works on the determination of kB, the final aim of reducing theuncertainty below 1 part in 106, which appears to be suitable for a new definitionof the kelvin, is likely to be achieved.
The authors gratefully acknowledge Michael R. Moldover and James B. Mehl for sharing theirlong experience on the subject and their constant advice. Numerous persons contributed tothis work. Michel Bruneau, Anne-Marie Bruneau and Cécile Guianvarc’h provided acousticexpertise. Simona Lago and Paolo Alberto Giuliano Albo measured the elastic properties ofa BCU3 copper sample. Gaël Obein, Patrick Ballereau, Michel Bruneau and Antoine Legaywere sources of constructive discussions about the gas flow effect on the speed of sound. Wegratefully acknowledge funding under the French National Research Agency ANR and the EMRPprogramme of the European Union. Last but not least, these programmes were made possiblethrough the very fruitful collaboration between our laboratory and other European laboratories,notably the NPL and the INRiM. In particular, we would like to thank Roberto Gavioso, PaoloAlberto Giuliano Albo, Michael de Podesta, Robin Underwood and Gavin Sutton for the manydiscussions, the shared project results and years of close collaboration. We would not have beenable to achieve these results without the support of Terry Quinn, Françoise Le Frious and YvesHermier.
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