## Measuring the Boltzmann constant for the last time

The gardens of the International Bureau of Weights and Measures (BIPM) in Paris

If you were thinking of measuring the Boltzmann constant, you had better hurry up.

If your research paper reporting your result is not accepted for publication by the end of this Friday 30th June 2017 then you are out of time.

As I write this on the morning of Tuesday 27th June 2017, there are four days to go and one very significant measurement has yet to be published.

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UPDATE: It’s arrived! See the end of the article for details
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What’s going on?

The Boltzmann constant is the conversion factor between mechanical energy and temperature.

Setting to one side my compulsion to scientific exactitude, the Boltzmann constant tells us how many joules of energy we must give to a molecule in order to increase its temperature by one kelvin (or one degree Celsius).

At the moment we measure temperatures in terms of other temperatures: we measure how much hotter or colder something is than a special temperature called the Triple Point of Water.

And energy is measured quite separately in joules.

From May 2019 the world’s metrologists plan to change this. We plan to use our best estimate of the Boltzmann constant to define temperature in terms of the energy of molecules.

This represents a fundamental change in our conception of the unit of temperature and of what we mean by ‘one degree’.

In my view, it is a change which is long overdue.

How will this changeover be made?

For the last decade or so, research teams from different countries have been making measurements of the Boltzmann constant.

The aim has been to make measurements with low measurement uncertainty.

Establishing a robust estimate of the measurement uncertainty is difficult and time-consuming.

It involves considering every part of an experiment and then asking two questions. Firstly:

• “How wrong could this part of the experiment be?”

and secondly:

• “What effect could this have on the final estimate of the Boltzmann constant?”

Typically working out the effect of one part of an experiment on the overall estimate of the Boltzmann constant might involve auxiliary experiments that may themselves take years.

Finally one constructs a big table (or spreadsheet) in which one adds up all the possible sources of uncertainty to produce an overall uncertainty estimate.

Every four years, a committee of experts called CODATA critically reviews all the published estimates of fundamental constants made in the last four years and comes up with a set of recommended values.

The CODATA recommendations are a ‘weighted’ average of the published data giving more weight to estimates which have a low measurement uncertainty.

In order to make their consensus estimate of the value of the Boltzmann constant in good time for the redefinition of the kelvin in 2019, CODATA set a deadline of 1st July 2017 – this coming Saturday.

Only papers which have been accepted for publication – i.e. submitted and refereed by that date will be considered.

After this date, a new measurement of the link between temperature and molecular energy will be reflected as a change in our temperature scale, not a change in the Boltzmann constant, which will be fixed forever.

The NPL Boltzmann constant estimate.

Professionally and personally, I have spent a decent fraction of the last 10 years working on an estimate of the Boltzmann constant – the official NPL estimate.

To do this we worked out the energy of molecules in a two-step process.

• We inferred the average speed of argon molecules held at the temperature of the triple point of water using precision measurements of the speed of sound in argon gas.
• We then worked out the average mass of an argon atom from measurements of the isotopic composition of argon.

Bringing these results together we were able work out the kinetic energy of argon molecules at the temperature of the triple point of water.

When we published our Boltzmann constant estimate in 2013 we estimated that it had a fractional uncertainty of 0.7 parts per million.

Unfortunately it transpired that our estimate was just wrong. Colleagues from around the world helpfully highlighted my mistake. That led to a revised estimate in 2015 with a fractional uncertainty of 0.9 parts per million.

At the time I found this cripplingly humiliating, but as I look at it now, it seems like just a normal part of the scientific process.

The source of my error was in the estimate of the isotopic content of the argon gas we used in our experiment.

Since then I have worked with many colleagues inside and outside NPL to improve this part of the experiment.  And earlier this month we published our final NPL estimate of the Boltzmann constant with a fractional uncertainty of… 0.7 parts per million: back to where we were four years ago!

Our estimate is just one among many from laboratories in the USA, China, Japan, Spain, Italy, France, and Germany.

But at the moment (7:30 a.m. BST on 27th June 2017) the NPL-2017 estimate has the lowest uncertainty of any published value of the Boltzmann constant.

The history of NPL’s recent estimates of the Boltzmann constant. The NPL 2017 estimate of the Boltzmann constant is close to CODATA’s 2014 consensus estimate

The LNE-CNAM Boltzmann constant estimate.

However my Frieval – i.e.friendly rival – Dr. Laurent Pitre from LNE-CNAM in France reported at meeting at BIPM last month that he had made an estimate of the Boltzmann constant with a fractional uncertainty of just 0.6 parts per million.

WOW! That’s right. 0.1 parts per million more accurate than the NPL estimate.

Dr. Pitre is a brilliant experimenter and if he has achieved this, I take my hat off to him.

But I have been looking daily at this page on the website of the journal Metrologia to see if his paper is there. But as I write – the paper has not yet been accepted for publication!

So after working on this project for 10 years I still don’t know if I will have made the most accurate measurement of the Boltzmann constant ever. Or only the second most accurate.

But I will know for sure in just 4 days time.

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UPDATE
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The article arrived this lunchtime

New Measurement of the Boltzmann Constant by acoustic thermometry in helium-4 gas

The paper reports a measurement of the Boltzmann Constant with a fractional uncertainty of just 0.6 parts per million.

The  measurements are similar in overall quality to those we published four years ago, but the French team made a crucial advance: they used helium for the measurements rather than argon.

Overall measurements are technically more difficult in helium gas than in the argon. These difficulties arise from the fact that helium isn’t a very dense gas and so microphones don’t work so well. Additionally the speed of sound is high – around three times higher than in argon.

But they have put in a lot of work to overcome these difficulties. And there are two rewards.

Their first reward is that by using a liquid helium ‘trap’ they can ensure exceptional gas purity. Their ‘trap’ is a device cooled to 4.2 degrees above absolute zero at which temperature every other gas solidifies. This has allowed them to obtain an exceptionally low uncertainty in the determination of the molar mass of the gas.

Their second reward is the most astounding. Critical uncertainties in the experiment originate with measurements of properties of helium gas, such as its compressibility or thermal conductivity.

For helium gas, these properties can be calculated from first principles more accurately than they can measured. Let me explain.

These calculations assume the known properties of a helium nucleus and that a helium atom has two electrons. Then everything is calculated assuming that the Schrödinger Equation describes the dynamics of the electrons and that electrons and the nucleus interact with each other using Coulomb’s law. That’s it!

• First the basic properties of the helium atom are calculated.
• Then the way electric fields affect the atom is calculated.
• The the way two helium atoms interact is calculated.
• And then the way the interaction of two helium atoms is affected if a third atom is nearby.
• And so on

Finally, the numbers in the calculation are jiggled about a bit to see how wrong the calculation might be so that the uncertainty of the calculation can be estimated.

In this way, the physical properties of helium gas can be calculated more accurately than they can measured, and that is the reward that the French team could use to overcome some of their experimental difficulties.

### One Response to “Measuring the Boltzmann constant for the last time”

1. edhui Says:

I’ve forwarded this to the Science Dept as an example of science in progrress- the excitement of the race. By the way, I’ve learnt something- I’ve for years and without knowing why used the unit Celcius rather than degrees Celcius, perhaps confusing it with kelvin which doesn’t have degrees in front of it. Seeing you write degrees Celcius meant that I immediately knew I have been wrong all this time.
Ed