Boltzmann Constant RSpdf - BIPM · resonator, hand polishedinnersurface, ... Principle of the experiment. ... ⇒⇒⇒Boltzmann constant but linked to a volume

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


-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


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


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


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


Principle of the experiment



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


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


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


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


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)


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


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


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


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


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)




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





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



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)



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


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


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


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)

Documents

Boltzmann Constant RSpdf - BIPM · resonator, hand polishedinnersurface, ... Principle of the experiment. ... ⇒⇒⇒Boltzmann constant but linked to a volume