determination of the boltzmann constant and new …physics.vniim.ru/si50/files/fischer.pdf ·...

Determination of the Boltzmann Constant and new Definition of the Kelvin

Joachim FischerJoachim Fischer

Physikalisch-Technische BundesanstaltPhysikalisch Technische BundesanstaltBerlin, Germany

1 50 years of efforts toward quantum SI, St. Petersburg, 6 Dec 10


Contents

i l di f CCT/10 34including concerns of CCT/10-34

The Kelvin Definition

Impact

Implications of Changing the DefinitionImplications of Changing the Definition

Determination of the Boltzmann Constant

Summary and Outlook


Standards and Scales

the scale:

triple point

the scale:additional fixed points and interpolating instrumentstemperatures from primary thermometers p p

of water

…H2 Ne O2 Ar Hg Ga In

primary thermometers

0273.16 × the unit

temperature intensive quantity

the unitthe unit

length extensive quantity


the scale

Definition of the Kelvin (1968)and triple point of water

“The Kelvin : fraction 1/273 16 of the1/273.16 of the thermodynamic temperature of triple

i t f t ”

(13. CGPM: Metrologia, 1968, 4, 43)

point of water”

T = 273 16 KTtpw = 273,16 K

p 611 66 PaWilliam Thomson,

definition = no uncertaintyptpw = 611,66 Pa the later

Lord Kelvin of Largs(1824-1907)


Effects of D and 18O on TP temperature

0 04

0.05

) D0.02

) 18O

0.03

0.04

ffere

nce

(K)

0.015

ffere

nce

(K) O

10 mK

10 mK

0.02

pera

ture

dif

0.01

erqa

ture

dif 10 mK

0

0.01Tem

p

0.005

Tem

pe00.000 0.004 0.008 0.012

Mole fraction

00.00 0.02 0.04 0.06

Mole fraction


Rod White, MSL N.Z.

Isotope effect, a material property

This substance is not available for filling of TPW cells


New definition based on Boltzmann constant k

system of “particles“thermal energy E per degree

of freedom

thermodynamic temperature T

E = ½kTk = conversion factor E = ½kTk = conversion factor

CODATA value of Boltzmann constant 2006:

k = R/NA = 1.3806504 (24) ·10-23 J/K u = 1.7·10-6 *)


A ( ) )*) Rev. Mod. Phys. 80 2008, 633

The Boltzmann Constant k and the kelvin

1/τ = dσ / dUBoltzmann: σ = ln P

entropymeasured in joulethermodynamic temperaturemeasured in kelvin

J1 38 10 20

Planck: S = k σ = k ln P 1/T = dS / dU

k = conversion factor 1.38x10-22

1.38x10-20

1.38x10-21

k conversion factor between energy and temperaturefixing the value of k :

1.38x10-23

1.38x10-24

1.38x10-25temperaturefixing the value of k : Boltzmann´s original i i ( 1)

1.38x10-26


intention (concern 1)P probabilityU internal energy

Wording

1. Reliable determination of k with different methods2 Fixing of the value of k2. Fixing of the value of k3. New definition of the kelvin like:

Explicit-unit definition

The kelvin is the change ofExplicit-constant definitionThe kelvin is the change of thermodynamic temperature Tthat results in a change of thermal

p f

The kelvin, unit of that results in a change of thermal energy kT by exactly 1.380 65X X × 10−23 joule, where

thermodynamic temperature, is such that the Boltzmann constant is exactly

Ludwig Boltzmann

k is the Boltzmann constantconstant is exactly1.380 65X X × 10−23 joule per kelvin.


g(1844 - 1906)


Contents


Impact



Summary and Outlook


Future uncertainties for high temperatures

0,35Best uncertainty

0,25

0,30Best uncertainty

ITS-90

At very low and very high temperatures there will

0,20n=1 (ITS-90

thermo-dynamic

At very low and very high temperatures there will be in future no need to reference back to the triple point of water

0,10

0,15 dynamicMain practical advantage of the new definition

0 00

0,05ITS-XX

0,001000 1500 2000 2500 3000 3500

Temperature / KBloembergen et al.


pTMCSI 2003, 291-296figure 2

The impact of a new definition of the kelvin

Short term:• the kelvin definition is independent• the kelvin definition is independent

of any material• no favoured fixed point • no favoured measurement method• no error propagation from TPW• thermodynamic measurementsthermodynamic measurements

and ITS-90 are coexisting • <20 K and >1300 K

thermodynamics are superiorthermodynamics are superior

Long term:

Ludwig Boltzmann

• With improvement of primary thermometry thermodynamic measurements may replace ITS-90


g(1844 - 1906)

y p


Contents


Impact



Summary and Outlook


CCT WG4 Task Group (TG-SI) membersp ( )

Joachim Fischer (PTB) chairman( )Anatoly Pokhodun (VNIIM)Ken Hill (NRC)G h M hi (NPL)Graham Machin (NPL)Mike Moldover (NIST)Laurent Pitre (LNE/CNAM)Laurent Pitre (LNE/CNAM)Andrea Merlone (INRIM)Richard Davis (BIPM) Executive Secretary CCT( ) yOsamu Tamura (NMIJ)Hüseyin Ugur (CCT) President CCTR d Whit (MSL)Rod White (MSL)Inseok Yang (KRISS)Jintao Zhang (NIM)


Jintao Zhang (NIM)

Status of ITS-90

For the foreseeable future :

Most temperature measurements in core temperature range (~ - 200 ºC … 960 ºC ) with SPRTs calibrated accord. to ITS-90( )

ITS-90 will remain intact, with defined values of T90 for all of the fixed points, including the TPW

Uncertainties in T will not changeUncertainties in T90 will not change

Dominated by uncertainties in the fixed-point realizations, y p ,

and the non-uniqueness of SPRTs, typically totalling < 1 mK


Fixed points of ITS-90 ( table 1.2 red book )

T90

u(T90) / mK u(T) / mK u(Tk fixed) / mK

Cu (fp) 1357 77 K 15 60 60 1Cu (fp) 1357.77 K 15 60 60.1Au (fp) 1337.33 K 10 50 50.1 Ag (fp) 1234.93 K 1 40 40.1 Al (fp) 933 473 K 0 3 25 25 1

no changeat all

Al (fp) 933.473 K 0.3 25 25.1Zn (fp) 692.677 K 0.1 13 13.1 Sn (fp) 505.078 K 0.1 5 5.10 In (fp) 429 7485 K 0 1 3 3 11In (fp) 429.7485 K 0.1 3 3.11Ga (mp) 302.9146 K 0.05 1 1.15 H2O (tp) 273.16 K 0.02 0 0.49 Hg (tp) 234 3156 K 0 05 1 5 1 55Hg (tp) 234.3156 K 0.05 1.5 1.55Ar (tp) 83.8058 K 0.1 1.5 1.50 O2 (tp) 54.3584 K 0.1 1 1.00 Ne (t ) 24 5561 K 0 2 0 5 0 50Ne (tp) 24.5561 K 0.2 0.5 0.50e-H2 (vp) ≈20.3 K 0.2 0.5 0.50 e-H2 (vp) ≈17.0 K 0.2 0.5 0.50

H 13 8033 K 0 1 0 5 0 50


e-H2 (tp) 13.8033 K 0.1 0.5 0.504He (vp) 4.2221 K 0.1 0.3 0.30

Uncertainties in thermodynamic temperature

If 2002 CODATA recommended value of k were taken to beIf 2002 CODATA recommended value of k were taken to be exact and used to define the kelvin :

Uncertainty of k would be transferred to the value of TTPW

B t ti t f th l f T till 273 16 KBest estimate of the value of TTPW still 273.16 K,

but instead of being exact as result of definition of the kelvin :but instead of being exact as result of definition of the kelvin :

Uncertainty associated with estimate would become :

ur(TTPW) = 1.8 × 10−6, corresponds to 0.49 mK


Error propagation from TPW

All thermodynamic measurements currently defined as ratios with respect to TPW :as ratios with respect to TPW :The 0.49 mK uncertainty propagates to all historicalthermodynamic temperature measurements

2,5

3,0 Temperatue range of SPRTs

1,5

2,0

t) /

mK

0,5

1,0

u (t

How well represents ITS-90 thermodynamic temperatures ?on

al

0,0

0,5

-250 0 250 500 750 1000

thermodynamic temperatures ?

addi

tio


temperature / °C

Fixed points of ITS-90 ( table 1.2 red book )

T90

u(T90) / mK u(T) / mK u(Tk fixed) / mK

Cu (f ) 1357 77 K 15 60 60 1Cu (fp) 1357.77 K 15 60 60.1Au (fp) 1337.33 K 10 50 50.1 Ag (fp) 1234.93 K 1 40 40.1 Al (f ) 933 473 K 0 3 25 25 1hAl (fp) 933.473 K 0.3 25 25.1Zn (fp) 692.677 K 0.1 13 13.1 Sn (fp) 505.078 K 0.1 5 5.10 I 429 7485 K 0 1 3 3 11

no changeat all

change > 0.1 mK

In (fp) 429.7485 K 0.1 3 3.11Ga (mp) 302.9146 K 0.05 1 1.15 H2O (tp) 273.16 K 0.02 0 0.49 H 234 3156 K 0 05 1 5 1 55Hg (tp) 234.3156 K 0.05 1.5 1.55Ar (tp) 83.8058 K 0.1 1.5 1.50 O2 (tp) 54.3584 K 0.1 1 1.00 N 24 5561 K 0 2 0 5 0 50Ne (tp) 24.5561 K 0.2 0.5 0.50e-H2 (vp) ≈20.3 K 0.2 0.5 0.50 e-H2 (vp) ≈17.0 K 0.2 0.5 0.50


e-H2 (tp) 13.8033 K 0.1 0.5 0.504He (vp) 4.2221 K 0.1 0.3 0.30

Uncertainties in thermodynamic temperature

⇒ TG-SI could not foresee any experiment where the slightly increased uncertainties of u(T ) would present a problemincreased uncertainties of u(Tk fixed) would present a problem

Any future changes in the temperature scale much smaller than y g ptolerances associated with current documentary standards for thermocouples and IPRTs :⇒ No requirement is anticipated for any future change in temperature scales to propagate to the documentary standards

Once k has been fixed in 2015 : TG-SI is not aware of any new technology for a primary thermometer providing a significantly gy p y p g g yimproved uncertainty u(TTPW) ⇒ no change of the assigned value of TTPW for the foreseeable f ( 2)


future (concern 2)

The Roles of the Mise en Pratique for the Definition of the Kelvin I

The mise en pratique (“practical realization”) for the definition of the kelvin (MeP-K) was created by the Consultative Committee for Thermometry(MeP K) was created by the Consultative Committee for Thermometry (CCT) in 2006 to give practitioners of thermometry a guide to the realization of the kelvin, i.e., measurement of temperature in kelvins, in accord with the International System of Units. y

The International Committee for Weights and Measures also foresaw that adoption of the proposed new definition of the kelvin would require a p p p qMeP-K

MeP-K will describe three categories of measurements:

- primary methods for measuring thermodynamic temperature T- formal approximations to T, in particular the International Temperature

S l f 1990 (ITS 90) d h P i i l T S l fScale of 1990 (ITS-90) and the Provisional Low Temperature Scale from0.9 mK to 1 K (PLTS-2000)

- indirect approximation methods that are neither primary nor defined ont t l t bl f ti ll l t i ti


a temperature scale, yet capable of exceptionally low uncertainties or increased reliability.

The Roles of the Mise en Pratique for the Definition of the Kelvin II (concern 3)( )

By providing a framework for primary methods and indirect methods, the MeP-K will foster development and application of new methods, such asMeP K will foster development and application of new methods, such as the use of absolute radiometry or high-temperature fixed points

MeP-K currently includes the text of the formal scales and a Technical Annex yof essential additional information

Next version will include recommended differences, T – T90, between the 90thermodynamic temperature and the temperature on the ITS-90 along with the associated uncertainty (ready for inclusion)

By documenting known ITS-90 biases, the MeP-K will support thermodynamically accurate measurements without mandating replacement of the ITS-90 in industry

In this way, the MeP-K provides the CCT with a mechanism to update and expand the thermometric methods in common use, without imposing on


industry the high costs of changing the International Temperature Scale

Schematic representation of relationship between MeP-K and other documents

SI D fi i iSI Definition

MeP-K

SI Brochure

MeP-KSection 3 Section 4 Section 5

Primary Methods Formal Approximations Indirect Approximations

Supplementary Guides

Supplementary Guides

ITS-90 Text

ITS-90 Technical

PLTS-2000 Text

Annex

ITS-90 Supplementary

I f ti

PLTS-2000 Supplementary

I f ti M P K 2006Information Information MeP-K 2006

on CCT webpagebox with solid border: prescriptive document


under preparationbox with solid border: prescriptive documentbox with dashed border: non-prescriptive guidance


Contents


Impact



Summary and Outlook


Determination of the Boltzmann constant for the redefinition of the kelvin

Coordinator:Physikalisch-TechnischeBundesanstalt

Partners:a t e sDanish FundamentalMetrology

Collaborators: Universidad de Valladolid


Acoustic gas thermometryg ym atomic mass (40Ar)γ ratio of heat capacities cp/ cV = 5/3 for gas of single atomsu speed of sound

u02 = γ kT / m

u0 speed of sound

AGTL. Pitre, C. Guianvarc'h, F. Sparasci, A Guillou D Truong Y HermierA. Guillou, D. Truong, Y. Hermier, M. Himbert, C. R. Physique 10 (2009) 835-848

R.M. Gavioso,G. Benedetto, P.A. Giuliano Albo,transducer

receiver

, , ,D. Madonna Ripa, A. Merlone, C. Guianvarc`h,F. Moro, and R. Cuccaro, Metrologia 47 (2010) 387-409

Measured quantites: d

g ( )

sound frequency ν at resonancedimension d of resonatori ld d f d

d


yields speed of sound u0

Quasi-spherical cavity resonators

J.B. Mehl, M.R. Moldover,L. Pitre Metrologia 41, 2004,295 (QSCR)

Refractive index gas thermometry295 (QSCR)E.F. May, L. Pitre, J.B. Mehl,M.R. Moldover, J.W. Schmidt,Re Sci Instr m 75 2004

p = kT (n2 - 1) ε0 / α0Rev. Sci. Instrum. 75, 2004, 3307 (QSCR)J.W. Schmidt, R.M. Gavioso, E F May and M R Moldover

n refractive indexε0 electric constant, definedα0 atomic polarizability, known by theory

E.F. May, and M.R. Moldover,PRL 98, 254504, 2007 (RIGT) RIGT

QSCR enablemicrowave measurementto determine dimension

or RIGT: measurement of ε (p,T)


or RIGT: measurement of ε (p,T)of helium and argon QSCR

AGT preliminary results of spring 2010

Gavioso et al., Metrologia 47 387–409 (2010)


AGT

Dielectric constant gas thermometryg y

p = k T ε0(εr - 1) / α0ε0 electric constant, definedεr ≈ C(p)/C(0)

t i l i bilitα0 atomic polarizability,known by theory DCGT

Measured quantites:pressure p

C. Gaiser, B. Fellmuth and N. Haft, I t J Th h 29 2008 18


pressure pcapacitance ratio C(p)/C(0)

Int. J. Thermophys. 29 2008, 18

DCGT: Critical parameters

pressure: capacitance:piston gauge:

present uncertainty: 10 MPa u (p)/p = 5 ppm10 MPa u (p)/p = 5 ppmmain problem: uncertainty of area determination

co-operation withpressure lab

i 1 ppmaim: 1 ppm

co-operation with electricity lab:

present uncertainty:absolute: u (CN) 20 ppbrelative: u (CX /CN) 3 ppb


( X N) pp

aim: u (ΔCX / CX) 1 ppb

Doppler broadening thermometry

ΔνD = [2 kT /(mc02)]1/2⋅ν0

m atomic massc0 speed of light

DBT0 p g

C. Daussy, M. Guinet, A. Amy-Klein, K. Djerroud, Y. Hermier, S. B i d Ch J B dé d CBriaudeau, Ch.J. Bordé, and C. Chardonnet, Phys. Rev. Lett. 982007, 250801G Casa A Castrillo GG. Casa, A. Castrillo, G. Galzerano, R. Wehr, A. Merlone, D. Di Serafino, P. Laporta and L. Gianfrani Phys Rev Lett 100

ΔνD

Gianfrani, Phys. Rev. Lett. 1002008, 200801J. Petersen, J. Hald, Danish Fundamental Metrology

Measured quantites:doppler line width Δν

Fundamental Metrology


ν0

doppler line width ΔνDcentre wavelength ν0

University Paris North/LNEUniversities Naples and Milan/INRiMp

DBT

Paris: CO2 laser at 10 µm and ammonia cellNaples: 1.4 µm diode laser and water-vapour cell

Problems in line fitting at 10-4 to 10-5 leveldue to pressure effects


Noise thermometry

<U2> = STΔf = 4 k T R Δf Nyquist´s Formula

bandwidth of detection systemswitched digital input correlator JNTvalid to order hf/2kT ~ 0.09 ppm at 1 MHz, 273 K

<U2>mean square noise voltage

resistance

R

relative method absolute method


Zn fixed point : u = 2•10-5 (k=1)

k/h from Noise Power Ratios

Spectral density of thermal noise powerQuantum-Hall Ohm

Spectral density of electrical noise powerAC Josephson VoltageAC Josephson Voltage

Jifeng Qu, S. P. Benz, H. Rogalla and D R White Metrologia 46

X 0 003874 resistance in units of R

and D. R. White, Metrologia 46 (2009) 512–524

XR 0.003874 resistance in units of RK-90

T = 273.16 K temperature as realized in TPW cell

D2 = 1 47 ×10-6/N2 dimensionless number from digital synthesisD = 1.47…×10 6/N dimensionless number from digital synthesis

fs= DAC sampling frequency, 10 GHz (f1= Δf = fs/M)

M = 24×106 bits pattern repetition length (memory)


M 24×10 bits, pattern repetition length (memory)

N = number of Josephson junctions JNT


Contents


Impact



Summary and Outlook


Present state of k determinationconfidence interval = 68.3 %1,38075

Review of methods in F ll th G i Fi h

1,38070) DBTLPL

Fellmuth, Gaiser, Fischer,Meas. Sci. Technol. 17 2006, R145 - R159

1 38065 ( J

K-1

AGT NIST

RIGTNIST

CODATA 2006

LPL

1,38065

k x

1023

35 ppm

AGT

DCGTPTB

1,38060

DBTk =1

AGTLNE

INRiMNPL

1,380551985 1995 2005 2015

UniNAk =1


Year

Development of the achieved and envisaged relative standard uncertainties Δk/krelative standard uncertainties Δk/k

Method 2nd WS 2006 4th WS 2009 2013 possibility institute

AGT > 20 ppm 3 ppm 1 ppm CEM (UVa), INRiM, LNE/CNAM,NPL, NIM

DCGT 15 ppm - 2 ppm PTB

JNT - 25 ppm 5 ppm NIST, INRiM

DBT 200 ppm 37 ppm only Type A 10 ppm DFM, LNE/CNAM (LPL), INRiM

(UniNA2, PoliMI), UWA


2007 : Report to the CIPM on the implications of changing the definition of the base unit kelvin g g

that within the next four years there exists the possibility of achieving a reliable…that within the next four years there exists the possibility of achieving a reliable uncertainty of the value of k of order one part in 106 based on measurements applying different methods of primary thermometry. Thus, an improved value of the Boltzmann constant proposed for defining the kelvin would ideally have beenBoltzmann constant proposed for defining the kelvin would ideally have been determined by at least the two fundamentally different methods AGT and DCGTand be corroborated by other – preferably optical − measurements as TRT and DBT with larger uncertaintyDBT with larger uncertainty.

The TG-SI appreciates the considerable progress of ongoing experiments to determine the Boltzmann constant in order to corroborate the present value. It isdetermine the Boltzmann constant in order to corroborate the present value. It is assumed that the experiments currently underway to measure R or k will achieve consistent results by the end of 2010, so that the CODATA group can recommend in its 2010 constants adjustment a new value for k with a relative standard juncertainty about a factor of two smaller than the current ur of approximately 2×10−6. With the new definition of the kelvin adopted, this would result in a value of ur(TTPW) of about 1×10−6, corresponding to about 0.25 mK.


POSSIBLE CONSEQUENCES OF THE REDEFINION OF KELVIN AND RECOMENDATION TO CCT AND CIPM

CCT/10-34 by A. Phokodun, VNIIM:

…In view of the above said, as a member of the CCT Task Group investigating the consequences of a new definition of Kelvin based on the Boltzmann constant, I would suggest the CCT not to be over-hasty in adopting the new definition for the temperature unit of SI and, by the example the last meeting of the CCM in March 2010, to adopt a resolution stipulating that the redefinition of Kelvin should be conditioned by a number of criteria removing the above mentioned preoccupations.


2010 : RECOMMENDATION OF THE CCT SUBMITTED TO THE CIPM

RECOMMENDATION T2 (2010) Considerations for a new definition of the kelvin

The Consultative Committee for Thermometry (CCT) recalling its previous Report to the CIPM in 2007, entitled “Report to the CIPM on the implications of changing the definition of the base unit kelvin” TG SI/docs05 andimplications of changing the definition of the base unit kelvin , TG-SI/docs05, and considering

• further discussion at its 24th and 25th meetings held in 2008 and 2010, • recent progress in experimental determinations of the Boltzmann constant k as reported at• recent progress in experimental determinations of the Boltzmann constant, k, as reported at

the 3rd and 4th International Workshops on Progress in Determining the Boltzmann Constant held in 2008 and 2009 and

• other experimental progress allowing a mise en pratique for the new definition of the kelvinalready established and presently extended to cover direct measurement of thermodynamic y p y ytemperature,

noting

• that various experiments, such as acoustic gas thermometry, dielectric constant gas thermometry, Johnson noise thermometry, total radiation thermometry and Doppler broadening thermometry represent distinct routes to determining the Boltzmann constant,

• that the experiments currently underway to measure k need another two years before CODATA can recommend a robust value for k with a relative standard uncertainty about a f t f t ll th th t f i t l 2 10−6


factor of two smaller than the current ur of approximately 2×10 6. • That a relative standard uncertainty of 1×10−6 in k corresponds to a standard uncertainty of

about 0.25 mK in the temperature of the triple point of water after the redefinition,

RECOMMENDATION OF THE CCT SUBMITTED TO THE CIPM (continued)( )

appreciating the considerable progress of ongoing experiments to determine the Boltzmann constant in order to improve confidence in the present value, recommendsrecommends

1. that before proceeding with the redefinition of the kelvin a relative standard uncertainty of the value of k of order one part in 106 be obtained, based on measurements applying different methods of primary thermometry,different methods of primary thermometry,

2. that these measurements ideally include at least two fundamentally different methods such

as acoustic gas thermometry and dielectric constant gas thermometry and be corroborated by other measurements such as Johnson noise thermometry, total radiation thermometry orby other measurements such as Johnson noise thermometry, total radiation thermometry orDoppler broadening thermometry,

3. that the CODATA recommended value be adopted for the Boltzmann constant.


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