PROGRESS IN THE DETERMINATION OF THE BOLTZMANN CONSTANT WITH DIELECTRIC-CONSTANT GAS THERMOMETRY. C. Gaiser, T.Zandt, B. Fellmuth

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1 PROGRESS IN THE DETERMINATION OF THE BOLTZMANN CONSTANT WITH DIELECTRIC-CONSTANT GAS THERMOMETRY C. Gaiser, T.Zandt, B. Fellmuth

2 PROGRESS IN THE DETERMINATION OF THE BOLTZMANN CONSTANT WITH DIELECTRIC-CONSTANT GAS THERMOMETRY Dielectric-Constant Gas Thermometry Previous Results Improvements and Status Polarizability of 4 He Summary & Outlook 2

3 Dielectric-Constant Gas Thermometry (DCGT) Clausius-Mossotti equation combined with the ideal-gas law: r 1 p 0 2 kt 3 r 0 Measuring quantity : 3 C( p) C(0) C(0) r 1 r eff p r 0 dielectric constant electric constant 0 atomic polarizability eff effective compressibility electric susceptibility p pressure T temperature

4 Main Challenges Pressure Goal: u r (k) 2 ppm Temperature Change of capacitance ratio 4

5 5 Determination of the Boltzmann constant with DCGT Isotherm measurement ( T T TPW ) First series coefficient: 1 eff TPW kt A A A A p Linear regression: eff 1 TPW 0 0 A T k Result:

6 Determination of the Boltzmann constant with DCGT 2011 k DCGT = (22) J/K u(k)/k = 15.9 ppm 2 k DCGT = (13) J/K u(k)/k = 9.2 ppm 1 1 B. Fellmuth et al., Metrologia 48, (2011) 2 C. Gaiser and B. Fellmuth, Metrologia, 49, L4-L7 (2012) 6

7 Determination of the Boltzmann constant with DCGT 2011 k DCGT-2011 = (11) J/K u(k)/k = 7.9 ppm 1 1 B. Fellmuth et al., Metrologia 48, (2011) 7

8 Uncertainty budget DCGT 2011 (TPW) Typ A Component u(k)/k 10 6 Overall weighted mean of A 1 values 6.3 Typ B Component u(k)/k 10 6 Susceptibility measurement (capacitance change) 1.0 B. Fellmuth et al., Metrologia 48, (2011) 8 Pressure measurement 1.9 Temperature 0.3 Determination of the effective compressibility 5.8 Head correction 0.2 Impurities (measuring gas) 2.4 Surface layers (impurities) 1.0 Polarizability ab initio calculation (theory) Combined standard uncertainty: 9.2 ppm 0.2

9 Reducing the uncertainty of the effective compressibility Improvements in RUS measurements (e.g. temperaturedependent measurements) Resonant ultrasound spectroscopy (RUS): Refinement of evaluation models (FEM, Monte-Carlo simulation) Test samples also for the insulation materials (Al 2 O 3 ) Determination of the thermal expansion coefficient and the molar specific heat capacity adiabatic isothermal 9

10 Reducing the uncertainty of the effective compressibility FEM calculations of effective compressibility for specific capacitor geometry optimization of capacitor design Switch from stainless steel to tungsten carbide as capacitor material Tungsten carbide eff Pa -1 Result (2013) Composite isothermal compressibility: x Pa -1 (u rel = 0.17%) Stainless steel eff Pa -1 10

11 Lock-in voltage (nv) Reducing the uncertainty of the susceptibility Careful analysis of the capacitance measuring network (chokes) and switch to low noise cable Only one cable for the zero detector (increase of the sensitivity by nearly a factor two) time (min) 11

12 Purity of the measuring gas % Helium (Linde AG) Gas purifier (adsorber) (Micro Torr SP70, SAES Pure Gase, Inc.) Helium purifier (getter) (HP2, Valco Instruments, Co. Inc.) Component Certificate Gas (ppb) Specification Getter (ppb) Specification Adsorber (ppb) H 2 < 30 < 10 < 0.1 H 2 O < 50 < 10 < 0.1 O 2 < 30 < 10 < 0.1 CO < 30 < 10 < 0.1 CO 2 < 30 < 10 < 0.1 N 2 < 10 Hydro-carbons < 1 < 10 < 0.1 Noble gases 12

13 Purity of the measuring gas Mass-spectrometry measurements Experiment (UHP gas tubing) Mass-spectrometry measurements Component Dectection limit (ppb) H 2 < 300 H 2 O < 20 O 2 < 10 CO < 100 CO 2 < 50 N 2 < 100 CH 4 < 20 Ne < 10 Ar < 10 Kr < 10 Xe < 10 13

14 Purity of the measuring gas % Helium (Linde AG) Gas purifier (adsorber) (Micro Torr SP70, SAES Pure Gase, Inc.) Helium purifier (getter) (HP2, Valco Instruments, Co. Inc.) Mass-spectrometry measurements Experiment (UHP gas tubing) Mass-spectrometry measurements Component Mass-Spec and Getter Spec. (ppb) Sensitivity in He Uncertainty (ppm)* H 2 < H 2 O < O 2 < CO 2 < N 2 & CO < Ne < Ar < Kr < Xe < Combined uncertainty 1.0 *(asymmetric rectangular distribution) 14

15 Results 2013 (10 isotherms) 15

16 Uncertainty budget DCGT 2013 (TPW) Typ A Component u(k)/k 10 6 Overall weighted mean of A 1 values 2.6 (6.3) Typ B Component u(k)/k 10 6 Susceptibility measurement (capacitance change) 1.0 (1.0) Pressure measurement 1.9 (1.9) Temperature 0.3 (0.3) Determination of the effective compressibility 2.4 (5.8) 16 Head correction 0.2 (0.2) Impurities (measuring gas) 1.0 (2.4) Surface layers (impurities) 0.5 (1.0) Polarizability ab initio calculation (theory) 0.2 (0.2) Combined standard uncertainty: 4.3 ppm (9.2 ppm) B. Fellmuth et al., Metrologia 48, (2011), C. Gaiser et al., Metrologia, 50, L7-L11 (2013)

17 Determination of the Boltzmann constant with DCGT (TPW) k DCGT = J/K u(k)/k = 9.2 ppm 1 k DCGT = J/K u(k)/k = 4.3 ppm 2 1 B. Fellmuth et al., Metrologia 48, (2011), 2 C. Gaiser et al., Metrologia, 50, L7-L11 (2013) 17

18 Improved pressure standard 20 cm 2 Systems 0.7 MPa cm 2 Systems 7 MPa 18

19 Improved pressure standard / / / / / / / / /1343 A 1163/ / / / / /1343 u 2013 (p)=1.9 ppm u 2015 (p)=1.0 ppm T. Zandt et al., Metrologia, Special-issue on k (2015) 19

20 Determination of the Boltzmann constant with DCGT (TPW) k DCGT = J/K u(k)/k = 4.3 ppm 2 k DCGT = J/K u(k)/k = 9.2 ppm 1 k DCGT = J/K u(k)/k = 4.0 ppm 3 1 B. Fellmuth et al., Metrologia 48, (2011), 2 C. Gaiser et al., Metrologia, 50, L7-L11 (2013) 3 C. Gaiser et al., Metrologia, Special-issue on k (2015) 20

21 Polarizability of 4 He (Theory) Calculations [1-3] u r (ppm) Non-relativistic Term (Cm 2 /V) Mass-polarization Corr. (ppm) Relativistic Corr. (ppm) First order-qed Corr. (ppm) Bethe Log; Higher order QED Est. (ppm) [1] Pachucki K. and Sapirstein 2000 PRA [2] Cencek W et al PRL [3] Łach G et al PRL [4] Puchalski M et al PRA [5]Jentschura U D et al 2011 PRA

22 Polarizability of 4 He (Experiment) k 3 T 0 TPW 1 A1 3 1 A1 0 eff 0 3 0TTPW eff 3 k A 1, eff k DCGT data (same uncertainty as for k-determination) Boltzmann constant derived from NIST AGT value (to avoid correlations and discussions on the actual discrepancies (see you on Friday)) k = J/K with u r = 1.8 ppm 0 0 DCGT = (98) Cm 2 /V with u r = 4.3 ppm 22

23 Polarizability of 4 He (Theory Experiment) [RIGT] Schmidt J W et al. PRL

24 Outlook / Decrease of the uncertainty Type A Component u(k)/k 10 6 Overall weighted mean of A 1 values 2.6 What is next: DCGT measurements with two different tungsten carbide cylindrical capacitors and test measurements with a 24-ring cross-capacitor Hope: Extremely stable results reduction in the Type A uncertainty. Consistent results for at least two or even three different capacitors reduction for u( eff ) Component u(k)/k 10 6 Susceptibility measurement (capacitance change) 1.0 Pressure measurement 1.0 Temperature 0.3 Determination of the effective compressibility 2.4 Head correction 0.2 Impurities (measuring gas) 1.0 Surface layers (impurities) 0.5 Polarizability ab initio calculation (theory) Type B

25 Summary A new value for k has been determined applying DCGT k DCGT = x J/K with a relative uncertainty of 4.0 ppm. 1.5 ppm larger than the 2010 CODATA value. Most accurate experimental value for 0. In excellent agreement with the ab initio theory. Improvements compared to 2011 Improved determination of the effective compressibility eff Use of tungsten carbide cylindrical capacitors Increase of the signal to noise ratio Better dealing with the purity of the measuring gas Extended piston gauge calibration. 25

26 Thank for your attention!