Michael de Podesta - BIPMAlso my wonderful NPL colleagues Gavin Sutton, Robin Underwood, Gordon Edwards, Graham Machin, Richard Rusby, David Flack, Andrew Lewis, Michael Perkin,1

Reflections on the NPL-2013

Estimate of the Boltzmann Constant

Michael de Podesta

Fundamental Constants

February 2015

Also my wonderful NPL colleagues

Gavin Sutton, Robin Underwood,

Gordon Edwards, Graham Machin,

Richard Rusby, David Flack,

Andrew Lewis, Michael Perkin,

Stuart Davidson, Kevin Douglas,

Rob Ferguson, David Putland,

Anthony Evenden, Louise Wright,

Eric Bennett, Alan Turnbull,

Gareth Hinds, Phil Cooling,

Gergely Vargha, Martin Milton,

Michael Parfitt, Peter Harris, Leigh Stanger

and others

Also my colleagues outside NPL

Paul Morantz, Cranfield

Darren Mark and Fin Stuart, SUERC

Laurent Pitre, LNE-CNAM

Roberto Gavioso, INRiM

Inseok Yang, KRISS

1. Background

2. History

3. The NPL uncertainty estimate

4. NPL Update February 2015

5. The NPL Analysis

6. Summary



The NEW International System of Units

c h

kB

∆𝜈

Cs133

K

m kg

s

Energy

(joule)

Molecular

Motion Thermometer

How do we relate the number produced by a

thermometer (e.g. 20 ºC) to the basic physics

describing the jiggling of molecules?

The Challenge

Primary thermometers are based on gases • Molecular motions are simple

• We can approach ‘ideal gas’ conditions at low pressure

• In an ideal gas the internal energy is just the kinetic energy of

the molecules

• Per molecule 𝟏

𝟐𝒎 𝒗𝒙

𝟐 + 𝒗𝒚𝟐 + 𝒗𝒛

𝟐 = 𝟑 ×𝟏

𝟐𝒌𝐁𝑻

𝟏

𝟐𝒎 𝒗𝒙

𝟐

𝟏

𝟐𝒎 𝒗𝒚

𝟐

𝟏

𝟐𝒎 𝒗𝒛

𝟐

Ar

=𝟏

𝟐𝒌𝐁𝑻

=𝟏

𝟐𝒌𝐁𝑻

=𝟏

𝟐𝒌𝐁𝑻

He

𝟗

𝟓𝒔𝒑𝒆𝒆𝒅 𝒐𝒇 𝒔𝒐𝒖𝒏𝒅 𝟐

𝒌𝐁 =𝒎

𝟑𝑻 𝒗𝒙

𝟐 + 𝒗𝒚𝟐 + 𝒗𝒛

𝟐

The big idea…

𝟏

𝟐𝒎 𝒗𝒙

𝟐 + 𝒗𝒚𝟐 + 𝒗𝒛

𝟐 =𝟑

𝟐𝒌𝐁 𝑻 𝒌𝐁 =

𝟑𝒎

𝟓𝑻 𝒔𝒑𝒆𝒆𝒅 𝒐𝒇 𝒔𝒐𝒖𝒏𝒅 𝟐

Find mass of

molecule

Measure the

speed of sound Carry out experiment

at TTPW

Measure the speed of sound in a spherical resonator

Speed of Sound =𝑅𝑒𝑠𝑜𝑛𝑎𝑛𝑡 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 × 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑎𝑑𝑖𝑢𝑠

constant

0

0.001

0.002

0.003

0.004

0.005

7526 7530 7534 7538

Sig

na

ls /

V

Frequency / Hz

Centre Frequency

7532.512 234 Hz

± 0.000 070 Hz

Temperature

20.000 000°C

±0.000 005 °C

?

Measure the Average Radius using Microwaves

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑎𝑑𝑖𝑢𝑠 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 × 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑒𝑠𝑜𝑛𝑎𝑛𝑡 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦

Microwave resonance

in a perfect sphere

0

20

40

60

80

100

2109 2110 2111 2112

Sig

na

l

Frequency (MHz)

TM11 Resonance

F0 is inversely proportional

to the radius

Microwave resonance

in a nearly perfect sphere

0

20

40

60

80

100

2109 2110 2111 2112

Sig

na

l

Frequency (MHz)

TM11 Resonance

F0 is in error – but not

possible to say by how much!

Microwave resonance

in a triaxial ellipsoid

0

20

40

60

80

100

2109 2110 2111 2112

Sig

na

l

Frequency (MHz)

TM11 Resonance

Measuring F0, F1 and F2

•Average radius

•Shape

•Uncertainty

Radius 62 mm

0.031 mm eccentrcity

Infers kB from measurements of the speed of sound in a

combined microwave and acoustic resonator

Acoustic Gas Thermometery Microwaves Acoustics

SPL

Frequency (kHz)

(0,2) (0,3) (0,4)

Frequency

Corrections

(Boundary Layer)

Theoretical

S12

Frequency (GHz)

TM12 Triplet

Radius

Frequency

Corrections

(Dielectric)

TM13 Triplet

Pressure

MWf

MWξ

Pressure

Af2

Aξ2

Microwaves Acoustics

SPL

Frequency (kHz)

(0,2) (0,3) (0,4)

Frequency

Corrections

(Boundary Layer)

Theoretical

S12

Frequency (GHz)

TM12 Triplet

Radius

Frequency

Corrections

(Dielectric)

TM13 Triplet

Pressure

MWf

MWξ

Pressure

Af2

Aξ2 Radius

Pressure

ξ

Δf0

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,5)

(0,7)

(0,8)

(0,9)

Spe

ed o

f S

oun

d S

qua

red

c2

(m2 s

-2)

Pressure (kPa)

Isothermal data for c2

𝑐𝐸𝑋𝑃2

𝑐02

0

0.001

0.002

0.003

0.004

0.005

0.006

0 2000 4000 6000 8000 10000

Argon T = 30 C

Am

plit

ud

e (

V)

Frequency / Hz

(1,1)

(2,1)

(0,3)

(0,2)

kB is inferred from 𝒄𝟎𝟐

1. Background

2. History



5. The NPL Analysis

6. Summary



2005:

…and just 4 years later…

2009:

2010:

2010:

𝑢𝑅 = 3.1 × 10−6

Highlighted Issues

• Variability of Molar Mass

• Need for precise pressure measurement

• ‘Negative’ Excess Half-Widths

• Line narrowing

2010:

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6 7 8 9 10 11

Measurement

Ch

an

ge i

n M

(Ar)

/10

6

Repeat

Repeat Repeat

-2%

-1%

0%

1%

2%

3%

4%

5%

6%

1 2 3 4 5 6 7 8 9 10

Measurement

Ch

an

ge

Fro

m M

old

over'

s S

tan

da

rd

Change in R36/40

Change in R38/40

Repeat

Repeat

Repeat

2011:

2011:

2010:

CMM

Microwaves Pyknometry

2011:

Microwaves u(k = 1) = 11.7 nm,

uR(k = 1) = 0.17× 10-6.

Pyknometry

CMM u(k = 1) = 114 nm,

uR(k = 1) = 1.8× 10-6.

2012:

Pyknometry u(k = 1) = 37 nm,

uR(k = 1) = 0.60 × 10-6.

Microwaves u(k = 1) = 11.7 nm,

uR(k = 1) = 0.17× 10-6.

CMM u(k = 1) = 114 nm,

uR(k = 1) = 1.8× 10-6.

2013:

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,5)

(0,7)

(0,8)

(0,9)

Spe

ed o

f S

oun

d S

qua

red

c2

(m2 s

-2)

Pressure (kPa)

Isothermal data for c2

𝑐𝐸𝑋𝑃2

𝑐02

kB is inferred from 𝒄𝟎𝟐

𝑢𝑅 = 0.71 × 10−6

Highlighted Issues

• Variability of Molar Mass

• Need for precise pressure measurement

• ‘Negative’ Excess Half-Widths

• Line narrowing

Since 2013

Acoustic Thermometry

c2 kB = g

M

T NA

-10

-5

0

5

10

-250 -200 -150 -100 -50 0 50

T-

T9

0 (

mK

)

t90

(°C)

-10

-5

0

5

10

-250 -200 -150 -100 -50 0 50

T-

T9

0 (

mK

)

t90

(°C)

WG4-

WG4+

-10

-5

0

5

10

-250 -200 -150 -100 -50 0 50

T-

T9

0 (

mK

)

t90

(°C)

PRELIMINARY

NPL High-Temperature Cylindrical Resonator

Hea

ter

He

ate

r

Insu

lati

on

Air

Gap

1. Background

2. History



5. The NPL Analysis

6. Summary



Working Equation

𝑘𝐵 =𝑀/𝛾

𝑁A𝑇TPW×

2𝜋𝑎 𝑓 0,𝑛 + Δ𝑓 0,𝑛

𝜉 0,𝑛

2

• Measure f(0,n) in the limit of low pressure

1. Resonator radius, 𝑎

2. Frequency Corrections Δ𝑓 0,𝑛

3. Eigenvalues 𝜉 0,𝑛

4. Pressure

5. Temperature

6. Molar Mass

1: How wrong could the radius estimate be?

1. Resonator radius, 𝒂

2. Pressure



5. Temperature

6. Molar Mass

Microwave Radius Estimates in Vacuum

-60

-50

-40

-30

-20

-10

0

10

1 2 3 4 5 6 7 8

(a (

21

.5 °

C)

- 6

2 0

32

60

0),

nm

Mode (TM1n)

-60

-50

-40

-30

-20

-10

0

10

1 2 3 4 5 6 7 8

(a (

21

.5 °

C)

- 6

2 0

32

60

0),

nm

Mode (TM1n)

±3.5 nm

±9 nm

nanometres

With Microphones

No microphones

2: How wrong could the pressure be?


2. Pressure



5. Temperature

6. Molar Mass

11.7 nm 0.38 ppm

u(k=1) u(kB)

Low Pressure Measurements

• Affects thermal boundary layer correction

• Affects estimate of p-1 term

94,750

94,755

94,760

94,765

94,770

94,775

94,780

0 20 40 60 80 100

-50

0

50

100

150

200

2500 10 20 30 40

Spe

ed o

f S

oun

d S

qua

red

(m

2 s

-2)

Pressure (kPa)

Data including

the p-1 term

App

roxim

ate

Fra

ctio

nal C

han

ge (p

arts

in 1

06)

Molar density (mol m-3)

Microwave Radius Estimates at Pressure

62.0104

62.0105

0 100 200 300 400 500 600 700

aeq /m

m

Pressure /kPa

62.0104

62.0105

0 100 200 300 400 500 600 700

aeq /m

m

Pressure /kPa

Data has been corrected for

dielectric constant of gas

(which depends on pressure)

Look at the residuals of a

straight-line fit to this data for

systematic pressure errors

100 nm

Pressure

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

0 100 200 300 400 500 600 700

Isotherm 3Isotherm 4Isotherm 5

-30

-20

-10

0

10

20

30

Re

sid

uals

(nm

)

Pressure (kPa)

Re

sid

uals

(Pa)

Traceable pressure

meter calibration

(before and after)

3: How wrong could the eigenvalues be?


2. Pressure

3. Eigenvalues 𝝃 𝟎,𝒏


5. Temperature

6. Molar Mass

11.7 nm 0.38 ppm

6.3 Pa 0.11 ppm

u(k=1) u(kB)

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,5)

(0,7)

(0,8)

(0,9)

Spe

ed o

f S

oun

d S

qua

red

c2

(m2 s

-2)

Pressure (kPa)

Data for c2

𝑐𝐸𝑋𝑃2

𝑐02

From high pressure

studies

Common to all modes

Low Pressure Speed of

Sound Squared

Common to all modes

‘Accommodation’

Correction to Boundary

Layer

Common to all modes

Experimental

Estimates

Function of

pressure P

and mode, n

Virial Correction

Common to all modes

‘Shell’ Correction

Varies with mode

Virial Correction

Common to all modes

Data Model

𝑐𝐸𝑋𝑃2 −𝐴3𝑃3= 𝑐0

2 + 𝐴−1𝑃−1 + 𝐴1𝑛𝑃 + 𝐴2𝑃2

After correction for Boundary Layer

-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

l

Pressure (kPa)

96% of data

within ±2uR

Residuals of all data to fits

expressed in terms of standard uncertainty


Sound Squared

Common to all modes

Data Model


2 + 𝐴−1𝑃−1 + 𝐴1𝑛𝑃 + 𝐴2𝑃2

𝑢(𝑐02) = 0.017 m2 s−2

𝑢𝑅(𝑐02) = 0.18 × 10−6

4: How wrong could the Frequency Corrections be?


2. Pressure


4. Frequency Corrections 𝚫𝒇 𝟎,𝒏

5. Temperature

6. Molar Mass

11.7 nm 0.38 ppm

6.3 Pa 0.11 ppm

0.18 ppm

u(k=1) u(kB)

Central frequency

0

0.001

0.002

0.003

0.004

0.005

Sig

na

ls /

V

Frequency / Hz

Half-Width should be

exactly what we expect

When f0 =3548.8095 Hz

expected width = 2.864 Hz

measured width = 2.858 Hz

-2

0

2

4

6

8

10

0 100 200 300 400 500 600 700

10

6 x

g/

f (0,n

)

P / kPa

(0,5)

(0,2)

(0,3)

(0,4)

(0,7)

(0,9)

(0,8)

-2

0

2

4

6

8

10

0 100 200 300 400 500 600 700

10

6 x

g/f

(0,n

)

Pressure (kPa)

(0,7)

(0,5)

(0,4)

(0,8)

(0,2)

(0,3)

(0,9)

Half-Width (Experiment – Theory) Parts per million of resonance frequency

f0 =3548.8095 Hz



Half-Width (Experiment – Theory) Shell Interaction

Shows the effect of mode and pressure

0

2

4

6

8

10

2 3 4 5 6 7 8 9

Excess H

alf-W

idth

(

g/f

) (p

pm

)

Radial Mode Index

Pre

ssure

(0,2) (0,9)

(0,6)

-2

0

2

4

6

8

10

0 100 200 300 400 500 600 700

10

6 x

g/

f (0,n

)

P / kPa

(0,5)

(0,2)

(0,3)

(0,4)

(0,7)

(0,9)

(0,8)

-2

0

2

4

6

8

10

0 100 200 300 400 500 600 700

10

6 x

g/f

(0,n

)

Pressure (kPa)

(0,7)

(0,5)

(0,4)

(0,8)

(0,2)

(0,3)

(0,9)

Half-Width (Experiment – Theory) Parts per million of resonance frequency

f0 =3548.8095 Hz



Half-Width

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 50 100 150 200

(0,2)(0,3)(0,4)(0,5)(0,7)(0,8)(0,9)

10

6 x

g/

f (0,n

)

P / kPa

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 50 100 150 200

(0,2)(0,3)(0,4)(0,5)(0,7)(0,8)(0,9)

10

6 x

g/f

(0,n

)

Pressure (kPa)

f0 =3548.8095 Hz



Experiment – Theory

Part

s p

er

mill

ion o

f

reso

nance f

requency

Experiment – New Theory

5: How wrong could the Temperature be?


2. Pressure



5. Temperature

6. Molar Mass

11.7 nm 0.38 ppm

6.3 Pa 0.11 ppm

0.18 ppm

~0

u(k=1) u(kB)

Temperature Gradient

Temperature gradient was

±91 µK about equator

Temperature

0

2

4

6

8

10

-150 -100 -50 0 50 100 150

Perc

en

tag

e o

f S

urf

ace

Are

a o

r V

olu

me

Deviation from Mean Temperature (K)

Volume

Surface

Average of Equatorial

Thermometers

We modelled the

temperature at the inner

surface of the sphere, and

then in the gas.

6: How wrong could the Molar Mass be?


2. Pressure



5. Temperature

6. Molar Mass

11.7 nm 0.38 ppm

6.3 Pa 0.11 ppm

0.18 ppm

~0

0.364 ppm 0.099 mK

u(k=1) u(kB)

Relative Molar Mass determined acoustically

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

2

f/f

wrt

Isoth

erm

5 g

as /10

-6

Gas Sample

Iso

the

rm 5

Iso

the

rm 3

&4

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

2

f/f

wrt

Isoth

erm

5 g

as /10

-6

Gas Sample

Iso

the

rm 5

Iso

the

rm 3

&4

Iso

the

rm 6

&7

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

2

f/f

wrt

Isoth

erm

5 g

as /10

-6

Gas Sample

Iso

the

rm 5

Iso

the

rm 3

&4

Iso

the

rm 6

&7

Iso

the

rm 8 B

OC

Ar 4

19

3

Sp

ectra

Ga

se

s

Ar 6

27

1

Ar 6

27

2

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

2

f/f

wrt

Isoth

erm

5 g

as /10

-6

Gas Sample

Iso

the

rm 5

Iso

the

rm 3

&4

Iso

the

rm 6

&7

Iso

the

rm 8 B

OC

Ar 4

19

3

Sp

ectra

Ga

se

s

Ar 6

27

1

Ar 6

27

2

Molar Mass Differences from Isotherm 5 Gas

kB gas Further Investigations

6: How wrong could the Molar Mass be?


2. Pressure



5. Temperature

6. Molar Mass

11.7 nm 0.38 ppm

6.3 Pa 0.11 ppm

0.18 ppm

~0

0.364 ppm 0.099 mK

0.390 ppm 0.390 ppm

Traceable to isotopic composition

of atmospheric argon

u(k=1) u(kB)

Uncertainty

-4

-3

-2

-1

0

1

-2

-1

0

1

2

k

B -

NP

L (

part

s in 1

06)

k

B - CO

DA

TA

(parts

in 1

06)

LNE2011

NIST1988

ThisWork

uR(k =1) = 0.71 ×10-6

Have we learned anything since 2013

that could shed light on the LNE-CNAM-

NPL discrepancy?

1. Background

2. History



5. The NPL Analysis

6. Summary



CODATA November 2015

NPL Update

February 2015

1. New estimate for thermal conductivity

2. Temperature Gradients

3. Argon Isotopic Molar Mass

16.30

16.40

16.50

16.60

16.70

0 100 200 300 400 500 600 700

(m

W m

-1 K

-1)

P (kPa)

New estimate for thermal conductivity of argon

• Δ𝜆 (-0.11% ) is close to

estimated uncertainty

• (u = 0.1% ) NPL paper

New uncertainty in 𝜆

(u = 0.02% ) is a factor 5

lower than we estimated

16.30

16.40

16.50

16.60

16.70

0 100 200 300 400 500 600 700

(m

W m

-1 K

-1)

P (kPa)

16.30

16.40

16.50

16.60

16.70

0 100 200 300 400 500 600 700

(m

W m

-1 K

-1)

P (kPa)

LNE

New Results

94,680

94,700

94,720

94,740

94,760

94,780

94,800

0 20 40 60 80 100-800

-600

-400

-200

0

200

400

0 10 20 30 40

Sp

ee

d o

f S

oun

d S

qu

are

d (

m2 s

-2)

Pressure (kPa)

(0,2)

As Measured

(0,3) (0,4)

(0,5)

(0,6)

(0,7)

(0,8)

(0,9)

Ap

pro

xim

ate

Fra

ctio

na

l Ch

an

ge

(parts

in 1

06)

Molar density (mol m-3)

94,680

94,700

94,720

94,740

94,760

94,780

94,800

0 20 40 60 80 100-800

-600

-400

-200

0

200

400

0 10 20 30 40

Sp

ee

d o

f S

oun

d S

qu

are

d (

m2 s

-2)

Pressure (kPa)

(0,2)

As Measured

Corrected

(0,3) (0,4)

(0,5)

(0,6)

(0,7)

(0,8)

(0,9)

Ap

pro

xim

ate

Fra

ctio

na

l Ch

an

ge

(parts

in 1

06)

Molar density (mol m-3)•ΔkB = -0.19 ppm.

•uR ~ 0.69 x 10-6

• Excess half-widths

increased

• Δg/f ~ + 0.1 x 10-6

@100 kPa


-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 50 100 150 200

(0,2)(0,3)(0,4)(0,5)(0,7)(0,8)(0,9)

10

6 x

g/f

(0,n

)

Pressure (kPa)

Part

s p

er

mill

ion o

f

resona

nce f

requency

Experiment – New Theory

• Thermal conductivity

revision will increase the

low-pressure end of these

curves by ~0.1 parts in 106.


NPL Update

February 2015





• Temperature gradient was

±91 µK about equator

Since 2013

• Replaced microphones

• Moved the pre-amplifier.

• Two additional thermometers

added to sphere (6 in all).

• No systematic gradient:

(max – min) is ±58 µK

• No ‘change in kB’ (u~0.3 ppm).

0.00980

0.00985

0.00990

0.00995

0.01000

0 2 4 6 8 10 12 14 16

bottom L&Nbottomequatorequatortop L&Ntop

Therm

om

ete

r R

ead

ing (

°C)

elapsed hours


NPL Update

February 2015




NPL update#3: Molar Mass

• IRMM and KRISS have made gravimetrically traceable isotopic analyses

• Comparison between KRISS (2014) and IRMM (2009) analysis of the same samples

Figure 7. The observed variability of the speed-of-sound squared in argon compared with gas used in

Isotherm 5. The data were estimated by measuring the resonant frequency of the (0,3) resonance with

the gas close to the temperature of the triple-point of water and at pressure of 200 kPa. The graph

shows twice the fractional shift in the resonant frequency expressed as parts in 106. Plotting the data

in this way means the vertical axis is equivalent to the fractional change in molar mass of the gas.

• No clear pattern of agreement or disagreement between KRISS and IRMM

1. Background

2. History



5. The NPL Analysis

6. Summary



Self-Consistent Analysis

•Makes the significance of fits to data meaningful.

•Data

•Type A uncertainty of Data

•Model

•Fit the model to the data

•Look at residuals (data - model)

•Show data and model are self-consistent

•We published our data and analysis scripts.

•These have been independently checked

Data

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,7)

(0,8)

(0,9)

(c0)2

(m

2 s

-2)

Pressure (kPa)

Isotherm 3

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,7)

(0,8)

(0,9)

(c0)2

(m

2 s

-2)

Pressure (kPa)

Isotherm 5

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,7)

(0,8)

(0,9)

(c0)2

(m

2 s

-2)

Pressure (kPa)

Isotherm 4

From high pressure

studies

Common to all modes


Sound Squared

Common to all modes

‘Accommodation’

Correction to Boundary

Layer

Common to all modes

Experimental

Estimates

Function of

pressure P

and mode, n

Virial Correction

Common to all modes

‘Shell’ Correction

Varies with mode

Virial Correction

Common to all modes


2 + 𝐴−1𝑃−1 + 𝐴1𝑛𝑃 + 𝐴2𝑃2

Type A Uncertainty

• Estimated from repeats of a resonance acquisition.

• Use pooled uncertainty to weight the data used in the fit.

• Inflate uncertainty estimate

0.00

0.01

0.10

0 100 200 300 400 500 600 700

Typ

e A

Unce

rtain

ty u

(c2)

(m2 s

-2)

All modes0.00

0.01

0.10

0 100 200 300 400 500 600 700

Typ

e A

Unce

rtain

ty u

(c2)

(m2 s

-2)

All modes

0.00

0.01

0.10

uPooledInflated

0 100 200 300 400 500 600 700

Typ

e A

Unce

rtain

ty u

(c2)

(m2 s

-2)

All modes

Normalised Residuals from Global Fit: Isotherm 3

• 96% of residuals fall within ±2u of best fit.

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,2)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,9)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,8)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,7)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,4)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,3)

Pressure (kPa)



-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,2)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,3)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,4)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,7)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,8)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed R

esid

uals

(0,9)

Pressure (kPa)

Isotherm 4



Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed R

esid

uals

(0,9)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,8)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,7)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,4)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,3)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rmalis

ed R

esid

uals

(0,2)

Pressure (kPa)

Isotherm 5

Residuals by Mode

Normalised Residuals from Global Fit: (0,2)


-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rmalis

ed R

esid

uals

(0,2)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,2)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,2)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rma

lised

Resid

ua

ls

(0,3)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,3)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Resid

ua

ls

(0,3)

Pressure (kPa)

Isotherm 5





-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rma

lised

Resid

ua

ls

(0,4)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Resid

ua

ls

(0,4)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,4)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rma

lised

Resid

ua

ls

(0,7)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,7)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Resid

ua

ls

(0,7)

Pressure (kPa)

Isotherm 5




-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rma

lised

Resid

ua

ls

(0,8)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,8)

Pressure (kPa)

Isotherm 4

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,8)

Pressure (kPa)

Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700

Isotherm 3

No

rma

lised

Resid

ua

ls

(0,9)

Pressure (kPa)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lised

Resid

ua

ls

(0,9)

Pressure (kPa)

Isotherm 4Isotherm 5

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Resid

ua

ls

(0,9)

Pressure (kPa)

Alternative Analysis

Data

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,7)

(0,8)

(0,9)

(c0)2

(m

2 s

-2)

Pressure (kPa)

Isotherm 3

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,7)

(0,8)

(0,9)

(c0)2

(m

2 s

-2)

Pressure (kPa)

Isotherm 5

94,750

94,800

94,850

94,900

94,950

95,000

0 100 200 300 400 500 600 700

(0,2)

(0,3)

(0,4)

(0,7)

(0,8)

(0,9)

(c0)2

(m

2 s

-2)

Pressure (kPa)

Isotherm 4


2 + 𝐴−1𝑃−1 + 𝐴1𝑃 + 𝐴2𝑃2

• Do this 18 times

• Average the 18 estimates of 𝑐02

• Treat each of 18 isotherm/modes independently

• Gives 18 estimates for 𝑐02 instead of 1

• Gives 18 estimates for 𝐴1 instead of 6

• Gives 18 estimates for 𝐴2 instead of 1

Normalised Residuals of Individual Fits: Isotherm 3

• Type A uncertainty estimated from 10 – or more – repeats of a resonance acquisition.

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 3 (0,2)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 3 (0,9)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 3 (0,8)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 3 (0,7)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 3 (0,4)

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Re

sid

ua

lsPressure (kPa)

Isotherm 3 (0,3)



-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 4 (0,9)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 4 (0,8)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 4 (0,7)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 4 (0,4)

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Re

sid

ua

ls

Pressure (kPa)

Isotherm 4 (0,3)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 4 (0,2)



-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 5 (0,7)

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Re

sid

ua

ls

Pressure (kPa)

Isotherm 5 (0,9)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 5 (0,8)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 5 (0,4)

-4

-2

0

2

4

0 100 200 300 400 500 600 700

No

rma

lise

d R

esid

ua

ls

Pressure (kPa)

Isotherm 5 (0,2)

-4

-2

0

2

4

0 100 200 300 400 500 600 700N

orm

alis

ed

Re

sid

ua

ls

Pressure (kPa)

Isotherm 5 (0,3)

NPL Strengths#3: Alternative Analysis

• Fit each isotherm individually and produce 18 estimates for 𝑐02.

• Average value is ~+ 0.25 ppm higher than ‘global’ estimate (+1.4u)

94755.9

94756.0

94756.1

94756.2

94756.3

94756.4

94756.5

94756.6

-3

-2

-1

0

1

2

3(c

0)2

(m

2 s

-2)

Mode/Isotherm

Sh

ift from

glo

ba

l estim

ate

(pp

m)

Isotherm 3 Isotherm 4 Isotherm 5

1. Background

2. History



5. The NPL Analysis

6. Summary



Summary

1. NPL-2013 estimate of kB has uR = 0.71 x 10-6

2. Differs significantly from LNE-CNAM-2011

3. Significant differences in analytical

assumptions and estimated sensitivity to errors

4. Possible reconciliation is through a molar mass

error by either NPL or LNE-CNAM.

5. Time will tell.

Documents

Michael de Podesta - BIPMAlso my wonderful NPL colleagues Gavin Sutton, Robin Underwood, Gordon Edwards, Graham Machin, Richard Rusby, David Flack, Andrew Lewis, Michael Perkin,1