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 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

Measurement brings the world into focus

I measured the temperature of the antenna to be 148.4 K Is that correct? On NPL-SAT-1 wasn t it 148.9 K in 2007? Image: NASA

How do you really know what the temperature is? 1. Measurement 2. The International System of Units (The SI) 3. The change to the New SI 4. Measuring the speed of sound really accurately 5. How do we know we are right? 6. If we have time: one last thing

Measurement is Quantitative Comparison Unknown Standard of an unknown quantity with a standard quantity

The Challenge Molecular Motion A thermometer How do we relate the number produced by a thermometer (e.g. 20 º C) to the basic physics describing the jiggling of molecules?

The International System of Units s Sèvres: We have a problem K m A The definition of the kelvin is not connected to the units which define energy! kg The kelvin, the unit of thermodynamic temperature, is the fraction mol 1/273.16 of the thermodynamic temperature of the triple point of water Cd

Triple point of water Every temperature measurement is a quantitative comparison of T TPW the level of molecular jiggling in the target with the level of molecular jiggling at this point in a triple-point cell

The NEW International System of Units ν 133 Cs Energy (joule) e s A c m kg h K cd Cd k B K N A mol

The NEW International System ν of Units 133 Cs k B s c K m e A The definition of the kelvin is now connected to the units which define energy! kg h mol Cd N A K cd

The NEW International System ν of Units k B The kelvin, the 133 unit Cs of thermodynamic temperature, is the fraction 1/273.16 of s the thermodynamic temperature c of the triple point of water K m e A The kelvin, the unit of thermodynamic temperature, is such that the Boltzmann constant has the exact kg value h k B = 1.380 65XX 10-23 joules per kelvin mol Cd N A K cd

From 2018 T TPW will still be a useful reproducible temperature But it will not define what we mean by one degree Every temperature measurement will be a quantitative comparison of the energy of molecular jiggling in the target with one joule of energy

Changing from old to new We want the size of the kelvin to stay the same: T TPW will still be 273.16 K How many joules of energy does a molecule possess at T TPW? Value of the Boltzmann constant k B is fixed in the new SI. What is value of the Boltzmann constant k B in the current SI? New kelvin Absolute Zero Old kelvin

How can we estimate the Boltzmann Constant?

Heat Hot Object Cold Object When materials touch, fast atoms slow down, slow atoms speed up until Average energy per molecule is equal. [Actually what equalises is the average energy per accessible degree of freedom]

Warning! Next two slides contain equations!

We measure in argon gas Molecular motions are simple in a gas We can approach ideal gas conditions at low pressure In an ideal gas the internal energy is just the kinetic energy of the molecules 1 2 m v z 2 = 1 2 k BT Ar 1 2 m v y 2 = 1 2 k BT 1 2 m v x 2 = 1 2 k BT

The big idea Look up mass of an argon molecule Carry out experiment at T TPW 1 k B = m 3T v x 2 + v 2 y + v2 2 m v x 2 + v 2 y + v2 z = 3 k 2 k B = 3m z B T 2 speed of sound 5T 9 5 speed of sound 2 Measure the speed of sound

Speaking tube Microphone 1 Acoustic Thermometer Loudspeaker Microphone 2 Timer

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

All required to u 1 ppm. a, radius of sphere To be measured Resonant Frequency To be measured c 2 z a n, l f n, l Eigenvalues Calculable for known shapes 25

Machining/metrology Ultra-Precision Manufacture set-up Hemisphere Turning tool Interferometer 26

Comparative CMM

Comparative CMM Comparative CMM CMM Position 1 CMM Position 2

All required to u 1 ppm. a, radius of sphere To be measured Resonant Frequency To be measured c 2 z a n, l f n, l Eigenvalues Calculable for known shapes 29

Resonator Placed inside a isothermal vessel Held inside a pressure vessel Dunked in bucket Liquid Stirred

Temperature Observed ΔT = 91 μk µk

Thermometer Reading ( C) Temperature 0.01000 0.00995 0.00990 0.00985 bottom L&N bottom 0.00980 0 2 4 6 8 10 12 14 equator16 elapsed hours top L&N top

Amplitude (V) Acoustic Spectrum 0.006 Argon T = 30 C (0,3) 0.005 0.004 (0,2) 0.003 (2,1) 0.002 (1,1) 0.001 0 0 2000 4000 6000 8000 10000 Frequency / Hz

Acoustic Resonance

Acoustic Resonance Signals / V Centre Frequency 7532.512 234 Hz ± 0.000 070 Hz Temperature 20.000 000 C ±0.000 005 C 0.005 0.004 0.003 0.002 0.001 0 7526 7528 7530 7532 7534 7536 7538 Frequency / Hz

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

Microwave Resonance

Microwave Radius Estimates in Vacuum (a (21.5 C) - - 62 032 600), nm nanometres 10 0 ±3.5 nm -10-20 -30-40 ±9 nm -50-60 1 2 3 4 5 6 7 8 Mode (TM1n)

All required to u 1 ppm. a, radius of sphere To be measured Resonant Frequency To be measured c 2 z a n, l f n, l Eigenvalues Calculable for known shapes 40

Speed of Sound Squared c 2 (m 2 s -2 ) Amplitude (V) Data for c 2 c2 EXP 95,000 94,950 0.006 0.005 0.004 0.003 0.002 Argon T = 30 C (0,2) (2,1) (1,1) (0,3) 0.001 94,900 0 0 2000 4000 6000 8000 10000 Frequency / Hz c 0 2 94,850 94,800 (0,2) (0,3) (0,4) (0,5) (0,7) k B is inferred from (0,8) this intercept (0,9) 94,750 0 100 200 300 400 500 600 700 Pressure (kpa)

k B - NPL (parts in 10 6 ) k B - CODATA (parts in 10 6 ) Result 1 u(k =1) = 0.71 10-6 0 2-1 1-2 0-3 -4 This Work NIST 1988 LNE 2011-1 -2

We don t know we are right!

Data for c 2 Speed of Sound Squared c 2 (m 2 s -2 ) c2 EXP 95,000 94,950 94,900 c 0 2 94,850 94,800 (0,2) (0,3) (0,4) A factor 3 lower than (0,5) anyone else! (0,7) Spread of estimates (0,8) is just 0.18 parts per (0,9) million 94,750 0 100 200 300 400 500 600 700 Pressure (kpa)

Signals / V Central frequency changes with temperature 0.005 0.004 Half-Width should be exactly what we expect 0.003 When f 0 =3548.8095 Hz 0.002 expected width = 2.864 Hz 0.001 measured width = 2.858 Hz 0 Frequency / Hz

Half-Width (Experiment Theory) Parts per million of resonance frequency 10 8 (0,7) 10 8 (0,7) 10 6 x g/ f (0,n) 6 4 2 0 (0,5) (0,4) (0,8) (0,2) (0,3) (0,9) 10 6 x g/f (0,n) 6 4 2 0 (0,5) (0,4) (0,8) (0,2) (0,3) (0,9) -2 0 100 200 300 400 500 600 700 P / kpa f 0 =3548.8095 Hz expected width = 2.864 Hz measured width = 2.858 Hz -2 0 100 200 300 400 500 600 700 Pressure (kpa)

Half-Width 2.0 1.5 Parts per million of resonance frequency Experiment Theory Experiment New Theory 1.0 2.0 1.5 1.0 10 6 x g/ f (0,n) 0.5 0.0-0.5-1.0-1.5 (0,2) (0,3) (0,4) (0,5) (0,7) (0,8) (0,9) -2.0 0 50 100 150 200 P / kpa 10 6 x g/f (0,n) 0.5 0.0-0.5-1.0-1.5 (0,2) (0,3) (0,4) (0,5) (0,7) (0,8) (0,9) -2.0 0 50 100 150 200 Pressure (kpa) f 0 =3548.8095 Hz expected width = 2.864 Hz measured width = 2.858 Hz

How do you know your thermometer is correct?

User 1 User 2 User 3 Calibration Lab NIM NIST NPL PTB LNE-CNAM International Bureau of Weights and Measures, BIPM

The International Temperature Scale of 1990 Temperature Fixed Points Interpolating Devices

Temperature Fixed Points of ITS90 How do we know that? T(K) T (ºC) Triple point of hydrogen 13.8033 259.3467 Triple point of neon 24.5561 248.5939 Triple point of oxygen 54.3584 218.7916 Triple point of argon 83.8058 189.3442 Triple point of mercury 234.3156 38.8344 Triple point of water 273.16 0.01 Melting point 1 of gallium 302.9146 29.7646 Freezing point 1 of indium 429.7485 156.5985 Freezing point of tin 505.078 231.928 Freezing point of zinc 692.677 419.527 Freezing point of aluminum 933.473 660.323 Freezing point of silver 1234.93 961.78 Freezing point of gold 1337.33 1064.18 Freezing point of copper 1357.77 1084.62

How to work out the Freezing point of Tin 273.16 K Speed of Sound squared (Speed 2) 2 (Speed 1) 2 Basic physics tells us Speed of Sound 2 Temperature Absolute Triple Point Freezing Point Temperature Zero of Water of Tin

T- T 90 (mk) How wrong is ITS 90? 10 5 NPL Preliminary 2014 0 PRELIMINARY DATA -5-10 -250-200 -150-100 -50 0 50 t 90 ( C)

How do you really know what the temperature is? Inter-comparisons and calibrations according to ITS-90 Make sure everyone agrees Fundamental Measurements Measure the errors in ITS-90

The Challenge? Molecular Motion Primary Thermometer A thermometer How do we relate the number produced by a thermometer (e.g. 20 º C) to the basic physics describing the jiggling of molecules?

How do you really know what the temperature is? 1. Measurement The definition of the units of temperature (the kelvin and the degree Celsius) 2. The International is about System to change. of Units (The SI) 3. The From change 2018, to temperature the New SI measurements will be fundamentally linked to the units of energy. 4. Measuring the speed of sound really accurately Every temperature measurement you make 5. How is do linked we know to our we fundamental are right? understanding of the thermal properties of matter 6. If we have time:one last thing