Boltzmann Constant RSpdf - BIPM · resonator, hand polishedinnersurface, ... Principle of the...
Transcript of Boltzmann Constant RSpdf - BIPM · resonator, hand polishedinnersurface, ... Principle of the...
1Laboratoire Commun de Métrologie LNE-CNAM
THE BOLTZMANN CONSTANT
FROM THE SPEED OF SOUND
IN A NON-QUITE SPHERICAL CAVITY
Laurent Pitre,
Fernando Sparasci, Arnaud Guillou,
Daniel Truong, Lara Risegari
Laboratoire Commun de Métrologie LNE-CNAM
Paris, France
2Laboratoire Commun de Métrologie LNE-CNAM
-50
-40
-30
-20
-10
0
10
20
30
40
1975 1980 1985 1990 1995 2000 2005 2010
year
(k-k
co
data
)/k
co
data
.10
6
Colclough,Quinn,
and Chandler.
Moldover et al.
He et al.
Schmidt et al.
Daussy et al.
The situation in 2008
Measurement of the Boltzmann Constant
Uncertainty is 7 times lower compared to the other results
3Laboratoire Commun de Métrologie LNE-CNAM
Uncertainty on the Boltzmann Constant
For the new definition of the kelvin, the relative uncertainty on the
Boltzmann constant has to be < 1�10-6
Codata 2006 Uncertainty on kB = 1.8 ppmUncertainty on kB, k=1, ppm of kB
April 2009 Dec 2010
(Volume)2/3 1.8 0.57
Temperature 1 0.3
Molecular weight 0.3 0.6
Zero-pressure limit of 1.5 0.80
(fn + ∆∆∆∆fn)2
Repetability 1.1 0.12
Root Sum Squares 2.7 1.2*
April 2008 and April 2009: LNE-CNAM determinations, 0.5 L quasi-spherical copperresonator, hand polished inner surface, filledwith helium
In 2009, the LNE-CNAM has set up a new 0.5 L quasi-spherical copper resonator with a diamond turned inner surface
* Pre print L. Pitre, F. Sparasci, D. Truong, A. Guillou, L. Risegari, M. E.
Himbert, »Measurement of the Boltzmann Constant kB using a Quasi-Spherical
Acoustic Resonator », the article have been sent to IJOT the 31/12/2010
4Laboratoire Commun de Métrologie LNE-CNAM
Uexp (ms-1)
Pressure (MPa)
Gas helium
m
kT
3
5=
For real gas: ( )...)(12 ++= pTm
kTU apgA βγ
specific heats ratio
atomic mass
acoustic virial coefficient
Boltzmann constant
Acoustic Determination of the Boltzmann Constant
Principle of the experiment
5Laboratoire Commun de Métrologie LNE-CNAM
Simultaneous acoustic (u) and microwave (c) resonances in a cavity
Gas flow minimizes the effects of impurities
2on-quite spherical simplify the microwave measurements
gas flow
Acoustic resonances measurementAcoustic resonances measurement
⇒⇒⇒⇒⇒⇒⇒⇒ Boltzmann constant but linked to a volumeBoltzmann constant but linked to a volume
Microwave resonances measurementMicrowave resonances measurement
⇒⇒⇒⇒⇒⇒⇒⇒ Volume measurementVolume measurement
Acoustic Determination of the Boltzmann Constant
Principle of the experiment
6Laboratoire Commun de Métrologie LNE-CNAM
Acoustic Determination of the Boltzmann Constant
Relationship between the Boltzmann constant
and acoustic/microwave measurements
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
Gas atomic mass
Speed of light in
vacuum (exact)Measured resonance
frequency
Correction (theory)
Quasi-sphere’s
eigenvaluesPolynomial
extrapolation to
Zero pressure limit
Average over measured acoustic
and electromagnetic modes
Ratio: removes artefact
effects at the first order
7Laboratoire Commun de Métrologie LNE-CNAM
A Non-Quite Spherical Cavity
Inner shape: the
difference between r,
R and H is 0.025 mm:
H = 50.000 mm
R = 49.975 mm
r = 49.950 mm
H R
r ( ) ( ) ( )1
000.50975.49950.492
2
2
2
2
2
=++zyx
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
0.5 L diamond turned
resonator
• The use of a slightly deformed spherical geometry, a triaxial ellipsoid, removes the degeneracy of
resonator modes
8Laboratoire Commun de Métrologie LNE-CNAM
TM11 BCU3
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
2617.5 2618.0 2618.5 2619.0 2619.5 2620.0 2620.5
frequency (MHz)
am
plitu
de
2 equal amplitude
2 time the amplitude
m=0m=±1 m=±1
A Non-Quite Spherical Cavity
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
Electromagnetic measurements
in very good agreement with the
theoretical model
• The use of a slightly deformed spherical geometry, a triaxial ellipsoid, removes the degeneracy of
resonator modes
Acoustic measurements in a
good agreement with the
theoretical model
9Laboratoire Commun de Métrologie LNE-CNAM
Measurement of the Volume
Cooperation between LNE-CNAM, NPL and Jim Mehl
R J Underwood, J B Mehl, L Pitre, G Edwards, G Sutton, M
de Podesta, Meas. Sci. Technol. 21 (2010)
radius obtained by measuring the TM1,1..5 and TE1,1..6 on BCU3
with 2 microphone and with duct
49975.00
49975.01
49975.02
49975.03
49975.04
0 5000 10000 15000 20000 25000
frequency (MHz)
rad
ius (
µm
)
0.5 PPM
10 nanometers
No fitted parameters
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
>∆+<=
EM
nl
EM
nl
EM
nl
ff
cZRadius
π2
Skin depth
Holes and
Antennas effect
Speed of lightEigenvalue
Resonance
frequency
10Laboratoire Commun de Métrologie LNE-CNAM
Measurement of the Volume
• Comparison of the microwave technique to CMM measurements
• Use of a CMM as a comparator
• Cooperation between LNE-CNAM, NPL, INRiM, UWA, Jim Mehl
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
M. de Podesta, E. F. May, J. B. Mehl, L. Pitre, R. M.
Gavioso, G. Benedetto, P. A. Giuliano Albo, D. Truong and
D. Flack. Metrologia, 47, 588-604, (2010)
11Laboratoire Commun de Métrologie LNE-CNAM
Acoustic Field Modeling
A new model for the acoustic field was developed with coupling effect.
Cooperation between LNE-CNAM and the Acoustics Laboratory LAUM
New model gives the same results as the acoustic model used by NIST for the
determination of kB
Pressure fieldC. Guianvarc’h, L. Pitre, M. Bruneau, A.-M. Bruneau, J.
Acoust. Soc. Am. 125 1416 (2009)
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
12Laboratoire Commun de Métrologie LNE-CNAM
Characterization of the microphones
• Acoustic measurements are performed with ¼”
condenser microphones
• Microphones are extensively modeled and
characterized in air at ambient temperature
and static pressure, ≠ from experimental
conditions of the Boltzmann project
• Microphone calibrations have been performed
in pure helium and argon gas at ~ 273 K,
between 10 kPa and 600 kPa
• A good agreement between microphone
literature model and experimental data has
been found
• Cooperation between LNE-CNAM and INRiM
C. Guianvarc’h, R. M. Gavioso, G. Benedetto, L. Pitre,
M. Bruneau, Review of Scientific Instruments 80 (2009) 2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
13Laboratoire Commun de Métrologie LNE-CNAM
Acoustic Field Modeling
Comparison between measured half-width and calculated from thermal physical propriety of helium gas and acoustic model
Mode 03
Mode 04
Mode 05
Mode 06
2.106 (gmeas-gcal)/f
Density (mol/m3)
at 273.16K helium gasNo fitted parameter
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
14Laboratoire Commun de Métrologie LNE-CNAM
Thermal Cartography of the Resonator
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
Quasi
adiabatic
cryostat
Four capsule thermometers
1551 1825277 HS135 229073 unstable
-0.70
-0.60
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
∆T-
∆̅T (
mK
)
C1, Ar, ∆̅T = 7.21 mK
C1, Ar, ∆̅T = 6.59 mK
C1, Ar, ∆̅T = 8.66 mK
C1, Ar, ∆̅T = -15.68 mK
C2, Ar, ∆̅T = 65.21 mK
C2, Ar, ∆̅T = 69.86 mK
C2, Ar, ∆̅T = 8.00 mK
C2, Ar, ∆̅̅T = 8.66 mK
C2, He, ∆̅T = -223.07 mK
15Laboratoire Commun de Métrologie LNE-CNAM
Ar Molar Mass Evaluation
We need to remove impurities from Ar ���� use of a getter and a cold trap
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
Impurities in a Ar sample (IRMM)
16Laboratoire Commun de Métrologie LNE-CNAM
Ar Molar Mass Evaluation
We need to remove impurities from Ar ���� use of a getter and a cold trap
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
Impurities in a Ar sample (IRMM)
17Laboratoire Commun de Métrologie LNE-CNAM
Ar Molar Mass Evaluation
We need to remove impurities from Ar ���� use of a getter and a cold trap
2
0
2
,
20 lim
5
3
>∆+<>∆+<
=
→ EM
nl
EM
nl
A
nl
A
nl
pA
nl
EM
nl
watertp
Bff
ff
Z
Z
T
mck
Impurities in a Ar sample (IRMM)
18Laboratoire Commun de Métrologie LNE-CNAM
effect flow at 6.5 bar with helium
TE12
-20
-15
-10
-5
0
5
10
15
20
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
time (s)
de
lta f
/f (
pp
b)
flow 90 sccm
flow 3 sccm
flow 3 sccm
0.1ppb !
The out gassing is less than 4.5 ppb of water
Or 700 picomol/mol
Ultra clean gas handling system and quasi sphere
(ppm)
(s)
Ar Molar Mass Evaluation
19Laboratoire Commun de Métrologie LNE-CNAM
Uncertainty Budget
Uncertainty budget on the Boltzmann constant with 0.5 liter cavity in argon
Uncertainty on kB, k=1, ppm of kB
1.19Root of Sum
of Squares
Thermophysical properties of argon, Scatter among modes, Accommodation coefficient dispersion,Flow, Tubing acoustic impedance, Shell
Two isotherms
0.8
0.12
Zero-pressure
limit of
(fn + ∆∆∆∆fn)2
Repeatability
Isotopic ratio, Cold trap experiment, Getter experiment, Impurity, Uncertainty on measurements
0.6Molecular
weight
Calibration, Dispersion over thermometers, Uncertainty on resistance measurements
0.3Temperature
Holes and antenna effect, Dispersion over mode-shape, Conductivity, Uncertainty on frequency measurements
0.57Volume
Dec 2010
20Laboratoire Commun de Métrologie LNE-CNAM
Boltzmann Constant Determinations at LNE-CNAM
kB=(1.3806495±0.0000037)*10-23J/K
Helium gas, 0.5 liter resonator,
surface polished by handArgon gas, 0.5 liter resonator,
surface diamond turned
kB=(1.3806476+0.0000016)*10-23J/K
uncertainty of kB, k =1, ppm of kB
-4
-2
0
2
4
(kb
meas-k
bco
data
)/k
bco
data
*10
00
00
0
kbcodata
Helium Argon
LNE
BCU2v2 2009
LNE
BCU3 2010
Argon
NPL-LNE
TCU1 2009
Argon
21Laboratoire Commun de Métrologie LNE-CNAM
A 3.1 L Non-Quite Spherical Cavity
3.1 L diamond turned copper triaxial ellipsoid
• The resonator should give even better acoustic, electromagnetic and CMM
measurements
• The thermostat is improved
• Experiment planned for this year
22Laboratoire Commun de Métrologie LNE-CNAM
The Quasi Spherical Resonator, a New Jackknife for the Metrology?
• Primary Thermometer (LNE-CNAM and NPL)
• Atomic Standard of Pressure (NIST)
• Humidity measurement (INRiM and LNE-CNAM)