The Boltzmann constant and the re-definition of the kelvin

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1 The Boltzmann constant and the re-definition of the kelvin Part 2 Michael de Podesta Varenna July 2012

2 The Boltzmann constant and the re-definition of the kelvin 1. Introduction 2. How do you know what the temperature is? 3. Why change the definition of the kelvin? 4. The Boltzmann constant 5. The NPL-Cranfield Resonator

3 k B The Boltzmann Constant 6.0 Delta CODATA 2010/ppm Date

4 k B The Boltzmann Constant 6.0 Delta CODATA 2010/ppm 4.0 NIST LNE-CNAM Date

5 The Boltzmann Constant kb joules per kelvin Current (2012) CODATA estimate: kb = (13) x J Ku (k = 1) =10.91 ppm

6 How does one measure the Boltzmann Constant? Involves a primary determination of the temperature of the triple point of water. Mmmm. A thermometer that only measures one temperature?

7 To measure the Boltzmann Constant kb joules per kelvin We need a physical system simple enough to calculate and measure the joules Each accessible degree of freedom possesses ½kBT joules Photon gas Molecular Gas Electron Gas

8 For example: count all the kinetic energy of the molecules in a gas On average, each molecule has 3 ½kBT joules of kinetic energy

9 How to measure kb Absolute radiometry Photon gas Spectral methods Electron Gas Total radiometry Johnson Noise Thermometry Molecular Gas Relative QVNS Doppler Broadening NH3 CO2 Rubidium The classical method of limiting density Dielectric constant gas thermometry The refractive index alternative The dielectric constant method in microwave cavities Acoustic thermometry Spherical Resonators Cylindrical Resonators (Fixed Path or Changeable Path)

closed, isothermal cavity depends only on the temperature Independent of the material properties of the

10 How to measure kb Absolute radiometry Spectral methods Total radiometry Photon gas Not enough emission at the triple point of water Photons do not interact with each other: ideal gas The radiation field inside a closed, isothermal cavity depends only on the temperature Independent of the material properties of the cavity Photon modes are coupled to atomic and electronic oscillator modes in the walls Lλ λ = c1 λ5 exp 1 λ c2 1 λt

11 Cryogenic Radiometer: Quinn and Martin Cryogenic Blackbody 2.2 K (uses electrical substitution heating) Radiation Trap 4.2 K ` (must stop all out-of-beam radiation) Beam of Radiation (only 1 part in 300 of total emission) Large Blackbody held at TPW K (novel design)

12 Pros and Cons Brave Idea Gave rise to improved standards in optical power measurement Very Difficult Not much emission from a blackbody at TPW Emission is broadband and long wavelength Apertures were critical Difficult to verify Difficult to assemble and disassemble. No redundancy

13 How to measure kb Absolute radiometry Photon gas Spectral methods Electron Gas Total radiometry Johnson Noise Thermometry Molecular Gas Relative QVNS Doppler Broadening NH3 CO2 Rubidium The classical method of limiting density Dielectric constant gas thermometry The refractive index alternative The dielectric constant method in microwave cavities Acoustic thermometry Spherical Resonators Cylindrical Resonators (Fixed Path or Changeable Path)

14 How to measure kb Johnson Noise Thermometry Electron Gas Relative QVNS (Quantum Voltage Noise Source) DSP Processing DSP Amplifiers Switch TTP W conventional noise thermometer Compares noise from resistors at two different temperatures VT2 = 4k B T R(T ) f T By Switching between noise sources, the effect of amplifier noise can be eliminated.

15 Johnson Noise Thermometry Electron Gas Relative QVNS (Quantum Voltage Noise Source) QVNS How to measure kb DSP Processing DSP Noise thermometer Amplifiers Switch TTP W T = 4.2 K Compares noise in a resistor at TPW with a known source of noise with calculable spectral characteristics

much noise from a resistor at TPW Slow Building a 4 channel instrument.

16 Pros and Cons Purely electronic technique Noise from QVNS calculable in terms of flux quantum (h/2e) Very Difficult Nyquist formula VT2 = 4k B T R (T ) f Not much noise from a resistor at TPW Slow Building a 4 channel instrument. Difficult to know that noise is from resistor and not some other source Picture from NIST

17 How to measure kb Absolute radiometry Photon gas Spectral methods Electron Gas Total radiometry Johnson Noise Thermometry Molecular Gas Relative QVNS Doppler Broadening NH3 CO2 Rubidium The classical method of limiting density Dielectric constant gas thermometry The refractive index alternative The dielectric constant method in microwave cavities Acoustic thermometry Spherical Resonators Cylindrical Resonators (Fixed Path or Changeable Path)

18 Molecular Gas Iin Iout detector Laser Direct measurement of the speed distribution of molecules f Doppler f0 = 2k BT mc 2 Transmission How to measure kb Doppler Broadening NH3 CO2 Rubidium Frequency

19 Results from Daussy et al Transmission 0.4 Pa 0.8 Pa 1.2 Pa 1.6 Pa 5.0 Pa 2.4 Pa 3.0 Pa Frequency (THz) e-fold Doppler half width(mhz) Doppler Width Pressure (Pa) 2.0 Pressure (Pa) 2.5

Pros and Cons Exploits fantastic traceable

20 Pros and Cons Exploits fantastic traceable frequency measurements Line-width is not only broadened by Doppler Effect Pressure Broadening Lamb-Dicke-Mossbauer narrowing Hyperfine structure Effect of other molecules? Is the pressure too low? Absorption cell Ice-water mixture Insulation foam

21 How to measure kb Absolute radiometry Photon gas Spectral methods Electron Gas Total radiometry Johnson Noise Thermometry Molecular Gas Relative QVNS Doppler Broadening NH3 CO2 Rubidium The classical method of limiting density Dielectric constant gas thermometry The refractive index alternative The dielectric constant method in microwave cavities Acoustic thermometry Spherical Resonators Cylindrical Resonators (Fixed Path or Changeable Path)

22 How to measure kb The classical method of limiting density Dielectric constant gas thermometry Molecular Gas The refractive index alternative The dielectric constant method in microwave cavities DCGT Measure change in capacitance between vacuum and 7 MPa He Very sensitive Polarizability of helium can be calculated rather than measured! Clausius-Mossotti equation combined with the ideal-gas law : εr 1 P α0 = ε r + 2 k BT 3ε 0

Pros and Cons of DCGT Development of PV = nrt Exploits fantastic measurement sensitivity of capacitance bridges Calculability of polarizability of helium

23 Pros and Cons of DCGT Development of PV = nrt Exploits fantastic measurement sensitivity of capacitance bridges Calculability of polarizability of helium Development of PV = nrt Polarizability of He is small Requires new level of perfection in pressure measurement Depends on knowledge of compressibility of copper

24 How to measure kb Absolute radiometry Photon gas Spectral methods Electron Gas Total radiometry Johnson Noise Thermometry Molecular Gas Relative QVNS Doppler Broadening NH3 CO2 Rubidium The classical method of limiting density Dielectric constant gas thermometry The refractive index alternative The dielectric constant method in microwave cavities Acoustic thermometry Spherical Resonators Cylindrical Resonators (Fixed Path or Changeable Path)

25 How to measure kb Acoustic thermometry Spherical Resonators Cylindrical Resonators (Fixed Path or Changeable Path) The dielectric constant method in microwave cavities Combined microwave and acoustic resonators To estimate kb Measure the speed of sound In a monatomic gas of known mass At the triple point of water In the limit of low pressure Need < 1 PPM uncertainty in all quantities γ kbrnat c = M 2

26 Boltzmann Constant: Choice of gas kb M = γ Tc NA2 Choose monatomic gas γ = 5/3 exactly in the limit of low pressure Helium or Argon Knowledge of properties Isotopes Signal to Noise Purity Helium Calculable Easy (3He or 4He) Poor Hard Argon Measured 36Ar 38Ar 40Ar Easy Hard

27 Boltzmann Constant: How to measure c2 kb M = γ Resonator T c2 NA Yields multiple estimates of the c2 Allows thorough investigation of systematic effects Estimating c2 requires A theory A frequency preferably of a high Q resonance A characteristic dimension Shape Cylinder- variable length Cylinder- fixed length Sphere Theory Calculable Frequency Low Q Dimension Calculable Exact Low Q High Q Easy Not so Easy

28 The Boltzmann constant and the re-definition of the kelvin 1. Introduction 2. How do you know what the temperature is? 3. Why change the definition of the kelvin? 4. The Boltzmann constant 5. The NPL-Cranfield Resonator

29 The NPL-Cranfield Acoustic Thermometer Measures the speed of sound in a spherical resonator

30 NPLC 2 Jim Mehl Michael Moldover Laurent Pitre Roberto Gavioso Robin Underwood Gavin Sutton Thank you

31 Ultra-Precision Manufacture Machining/metrology set-up Hemisphere Turning tool Interferometer 31

33 Resonator is placed inside a isothermal vessel and held inside a pressure vessel

34 Acoustic Spectrum Argon T = 30 C (0,3) Amplitude (V) (0,2) (2,1) (1,1) Frequency / Hz

35 Acoustic Resonance

36 Acoustic Resonance Signals / V Centre Frequency Hz ± Hz Temperature 20.0 C ± C Frequency / Hz

37 All required to u 1 ppm. a, radius of sphere To be measured c= Resonant Frequency To be measured 2π a f n,l Eigenvalues Calculable for known shapes 37 z n,l

38 2 things Thing 1: How to measure the radius Thing 2: The beauty of self-consistency

39 Thing 1 How to measure the radius

40 Microwaves

41 Microwave resonance in a perfect sphere 100 TM11 Resonance Signal 80 F0 is inversely proportional to the radius Frequency (MHz) 2112

42 Microwave resonance in a nearly perfect sphere 100 TM11 Resonance Signal 80 F0 is in error but not possible to say by how much! Frequency (MHz) 2112

43 Microwave resonance in a triaxial ellipsoid 100 TM11 Resonance Signal 80 Measuring F0, F1 and F2 Average radius Shape Uncertainty mm mm deviation Frequency (MHz) 2112

44 Microwave Radius Estimates 10 nanometres (a (21.5 C) ), nm 0 ±3.5 nm ±9 nm Mode (TM1n) 6 7 8

45 Radius versus Pressure Requires correction for dielectric constant of gas aeq /mm Look at the residuals of a straight-line fit to this data Pressure /kpa

46 Residuals nanometres Residuals /Pa Residuals, nm pascal Pressure, kpa ±6 Pa

47 CMM Microwaves Pyknometry

48 Comparative CMM CMM Position 1 Comparative CMM CMM Position 2

Procedure Weigh Empty ~7.5 kg (± 0.000 000 1 kg) About 0.

1 mm3 r = 2 nm Fill with Water less than full Weigh Full ~8.5 kg (± 0.

49 Procedure Weigh Empty ~7.5 kg (± kg) About ºC/mm V = 0.1 mm3 r = 2 nm Fill with Water less than full Weigh Full ~8.5 kg (± kg) Heat until water reaches mark Record temperature u(t) ~1 mk Weigh Empty ~7.5 kg (± kg)

50 Filling

51 Weighing ~1g Mass ~8 kg Volume ~1 litre Buoyancy ~1 g For u = 1 mg we need to correct for air density to 1 part in 104 Temperature Pressure Humidity ~ 8 kg

52 CMM Microwave Comparison Radius /mm Torque/ N m Metrologia XXX

53 Pyknometry Microwave Comparison Volume cm Mar 01 Jun 01 Apr 01 May Measurement Date Metrologia 49(2012) Jul

54 Thing 2 Self Consistency

55 Self Consistency#1 In the limit of low pressure How well do different acoustic modes agree on the speed of sound?

56 Data for c2 Speed of Sound Squared c2 (m2 s-2) 95,000 m2 s 2 94,950 94,900 94,850 94,800 (0,2) (0,3) (0,4) (0,5) (0,7) (0,8) (0,9) 94, Pressure (kpa)

57 Self Consistency#2 Thermal Boundary Layer Correction

58 Thermal Boundary Layer 800 parts in 106 For (0,2) at P = 100 kpa Affects resonance frequencies Affects resonance half-widths Compare theoretical half-widths experimental half-widths

59 Half-Width (Experiment Theory) 106 x g/ f(0,n) Parts per 10 million of resonance 8 frequency (0,7) (0,5) (0,4) (0,8) (0,2) (0,3) (0,9) P / kpa

60 Half-Width (Experiment Theory) x g/ f(0,n) (0,2) (0,3) (0,4) (0,5) (0,7) (0,8) (0,9) P / kpa

61 Thing 3 Why haven t I finished yet?

62 Argon Purity 0.98 ppm in f Isotopic analysis predicts this should be higher, not lower 1.96 ppm in f2

63 40Ar/36 Ar Argon 40Ar/36Ar~30 Isotopes 0 40Ar/36Ar~300 40Ar/38Ar~ Ar/36 Ar

64 Argon Isotopes

65 k B The Boltzmann Constant 6.0 Delta CODATA 2010/ppm 4.0 NIST ppm LNE-CNAM Date

66 Summary The definition of the unit of temperature is about to change. It will be defined in terms of the Boltzmann Constant kb Basic thermal physics underpins every temperature measurement

67 Thank you 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 Brown, Eric Bennet, Alan Turnbull, Gareth Hinds, Phil Cooling, Michael Parfitt and others

68 Thank you

69 Comparative CMM

70 How Good is ITS-90?

71 Differences Between ITS90 & T T T90 (K) Steur Edsinger & Schooley 0.15 Guildner & Edsinger Astrov et al, revised Kemp et al Quinn et al Fox et al Stock et al Taubert et al Noulkhow et al Goebel et al 0.05 T 90, K Yoon et al Fischer & Jung Moldover et al Ewing and Trusler T Strouse et al Ripple et al Benedetto et al Pitre et al Edler et al Labenski et al various (T68 - T90) -0.1 ITS-90, +u (k = 1) Temperature, K Temperature (K) ITS-90, -u (k = 1) mean smooth function

72 Differences Between ITS90 & T T T90 (K) Steur Astrov et al, revised Guildner & Edsinger Edsinger & Schooley Kemp Quinn et al Strouse et al 0.0 Ewing & Trusler Benedetto et al T T 90, K Moldover et al Pitre et al Ripple et al (T68 - T90) ITS-90, +u (k = 1) ITS-90, -u (k = 1) mean smooth function smooth function above Temperature, K Temperature (K) tpw mean

73 Historical Temperature Scales Usually based on two arbitrary fixed-point temperatures. Unjustified assumption of linearity of interpolating thermometer. No rationale for extrapolation. Standard 1??? Unknown Unknown Unknown Unknown Unknown Unknown Standard 2 Not based on physics The Invention of Temperature. Hasok Chang ISBN13:

74 Acoustic Thermometry Speed of sound in a gas related to average molecular speed depends on T γ kbrnat c = M 2 In the limit of low density

75 Acoustic Thermometry Speed of sound in a gas related to average molecular speed depends on T γ kbrnat c = M 2 In the limit of low density

76 Boltzmann Constant To estimate kb Measure the speed of sound In a monatomic gas of known mass At the triple point of water In the limit of low pressure Need < 1 PPM uncertainty in all quantities γ kbrnat c = M 2

77 Microwaves Acoustics Radius Frequency Corrections (Dielectric) MW f MW ξ Theoretical Af2 Aξ2 Frequency Corrections (Boundary Layer) Pressure Pressure TM12 Triplet TM13 Triplet SPL (0,2) (0,3) (0,4) S12 Frequency (GHz) Frequency (khz)

78 Microwaves 1 Acoustics Radius 2 Frequency Corrections (Dielectric) MW f MW ξ Theoretical Af2 Aξ2 3 Frequency Corrections 4 (Boundary Layer) Pressure Pressure TM12 Triplet TM13 Triplet SPL (0,2) (0,3) (0,4) S12 Frequency (GHz) Frequency (khz)

79 Water Problems The Solution 1. De-aerated water 2. Benzotriazole

80 Cryogenic Radiometer: Quinn and Martin Cryogenic Blackbody 2.2 K (uses electrical substitution heating) Radiation Trap 4.2 K ` (must stop all out-of-beam radiation) Beam of Radiation (only 1 part in 300 of total emission) Large Blackbody held at TPW K (novel design)

Fundamental Temperature Measurement: Re-defining the Boltzmann Constant How do you really know what the temperature is? Michael de Podesta

Fundamental Temperature Measurement: Re-defining the Boltzmann Constant How do you really know what the temperature is? Michael de Podesta NPL December 2014 Also my wonderful colleagues Gavin Sutton, Robin