# Proving Our Mettle: NIST and the Boltzmann Constant Olympics

*Michael Moldover,** Leader, **Fluid Metrology Group*

How tall are you? How old? How much do you weigh? Do you care? Is it important to you that the measurements for height, age and weight are accurate? What about the measurement of the gasoline that you pump into your tank? Is it important that the 12 gallons you pay for are truly 12 gallons? Measurement is part of our daily lives, and accurate measurements make our complex society run smoothly. At NIST, we measure many vital quantities, such as time, temperature, the flow of gases and liquids, as accurately as possible, using, whenever possible, standards relating to the unchanging properties of atoms.

Among these measurements, I played a significant role in measuring the Boltzmann constant. The Boltzmann constant is the fundamental constant that, in effect, converts the average energy-per-“atom” into temperature. Either the tiny Boltzmann constant (*k*) or its macroscopic cousin the universal gas constant (*R*) are used whenever a temperature-dependent property, such as pressure, magnetization, or heat capacity, is calculated from fundamental physics. Such calculations always start with tiny, atomic or molecular-scale energies, but their end results must be converted to temperatures for use in engineering applications such as calculating the cooling capacity of an air conditioner from the properties of its hardware and refrigerant.

Accurate measurements are harder to come by than you might think and, on occasion, mistakes are made.

# To err is human; to correct, divine

For instance, in 1976, a research team at the U.K.’s National Physical Laboratory made a measurement error that changed the course of my career. That year, the NPL team published a report that their new value of the Boltzmann constant was four times more accurate than the best prior measurement. Surprisingly, their new value of the Boltzmann constant was more than five times further from the previously accepted value than the “fuzziness” or “uncertainty” of these values. The fivefold difference could have been a one-in-million random event or the result of someone’s mistake.

I proposed to deduce the Boltzmann constant in the same way the NPL team had: by measuring the speed of sound in low-density argon gas and combining the results with other well-known physical constants. However, my new idea was to determine the speed of sound by measuring the volume of a hollow, argon-filled sphere and by measuring the frequencies (tones) at which the argon “sings” or “resonates” much like the tones generated by a pipe organ. By contrast, the NPL work determined the speed of sound by moving a piston in a cylindrical cavity to measure the wavelength of sound while keeping the frequency of the sound constant. A spherical cavity provides a more accurate measurement than a cylindrical cavity because the acoustic resonances (tones) are narrower in frequency and have smaller corrections from “wall effects,” which occur when sound waves are dampened by exchanging heat with and “rubbing” against the cavity’s wall.

When I proposed the spherical resonator project, I had never measured the speed of sound in anything, so my proposal to do so with best-in-the-world, part-per-million uncertainties was somewhat arrogant. I did take the precaution of asking “Izzy” Rudnick, a world-class expert in acoustics at UCLA, if a spherical resonator would work as I had hoped. After a moment’s thought, he replied: “Yes, it could work.” This encouragement was the first of three gifts that Rudnick gave to me, and their value to me cannot be overstated.

I had chosen a spherical cavity to minimize wall effects. Rudnick’s second gift was to guess that the resonance frequencies of the argon in the spherical cavity were only weakly affected by the smooth changes of the cavity’s shape. Because of this, the metal shell enclosing the cavity could be made with ordinary, part-per-thousand machine-shop techniques and still measure the speed of sound with part-per-million uncertainties. Later, we proved that Rudnick’s guess was correct and that a similar effect occurs for microwave resonance frequencies.

Rudnick’s third gift was to introduce me to Martin Greenspan, the retired chief of NIST’s Sound Section and past president of the Acoustical Society of America. Greenspan became a wise and generous collaborator. About the same time, I began a lifelong collaboration with Jim Mehl, a physics professor at the University of Delaware.

Two years after my spherical-resonator proposal was okayed, the NPL team discovered their error and published a correction. Their 1979 corrected values of the Boltzmann constant agreed with earlier values and had a small uncertainty of 8.4 parts-per-million. Their correction eliminated the strong motivation for completing our spherical-resonator project; however, Jim Mehl, Martin Greenspan and I continued to pursue accurate speed-of-sound measurements on a part-time basis. Between 1980 and 1986 we wrote papers describing our growing understanding of the problem. We calculated and tested how the acoustic resonance frequencies were affected by microphones that generated and detected sound and by the recoil of the spherical shell in response to the oscillating gas within it and by the wall effects. We mastered state-of-the-art methods for measuring temperature and purifying argon. Finally, we assembled our spherical resonator. Rich Davis of NIST’s Length and Mass Division and guest researcher Martin Trusler determined the volume of the 3-liter cavity inside the resonator with an uncertainty of less than one part-per-million by weighing the mercury required to fill it. Our final measurements of the speed of sound in argon took more than a year, and it took another year to analyze and write up the results.

Twelve years after NPL’s initial measurement, Martin Trusler, Terry Edwards, Jim Mehl, Rich Davis and I published “Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator.” Our 1988 values for the universal gas constant and its microscopic cousin, the Boltzmann constant, had an uncertainty of 1.7 parts-per-million, a factor of five improvement over the NPL measurement that had stimulated the project.

# On top of the sphere

For the next 23 years, our value of the Boltzmann gas constant remained the most accurate value — unusually long in the business of measuring fundamental constants. A cynic might say that the passage of 23 years showed that no one cared enough to repeat the measurements. A Pollyanna might say that our measurement was so impressive that no one could imagine improving it. But it turned out that we only had a temporary lead in what became the “Boltzmann Olympics,” an informal race to lower the uncertainties associated with measuring this fundamental constant.

Around 2007, the international community of standards laboratories developed a plan, including a deadline, to drastically revise the International System of Units so that the fundamental constants would be permanently assigned fixed values instead of being measured anew from time to time. The combination of an internationally accepted plan and a deadline encouraged funding agencies in many countries to support research teams. Each team competed to measure one of the fundamental constants so well that their result would become the permanent value.

Between 2011 and 2017, at least 15 teams measured the Boltzmann constant. Among the many new measurements, six teams had at least one with a small enough uncertainty to be included in the final average, which had an uncertainty of only 0.37 parts-per-million. The final average was consistent with the 1979 NPL results and with our 1988 result. It was great to know that our work had been so accurate.

While we did advise the teams using acoustic resonators, my collaborators and I did not compete in the Boltzmann Olympics. Instead, between 1988 and 2006, we worked part-time with guest researcher Laurent Pitre and other NIST scientists to use acoustic resonators to measure temperatures between 7 K (-266 °C or -447 °F) and 550 K (276 °C or 530 °F). Simultaneously, Jim Mehl and I improved the theory and the technologies needed to measure still higher and lower temperatures. Perhaps our greatest contribution was the invention and modeling of quasi-spherical cavities. This kind of cavity has specially designed distortions to enable accurate microwave measurements of the cavity’s volume while retaining the small wall corrections of a sphere. In the Boltzmann Olympics, the top three competitors used quasi-spherical cavities. Now, these top competitors are making highly accurate temperature measurements using NIST-generated innovations. I anticipated this would happen, but I never guessed how long it would take, nor did I recognize the practical value of the spherical resonator project.

# Going with the flow

You see, to determine the speed of sound in argon, we “corrected” the measured resonance frequencies to account for the exchange of heat between the gas and the cavity’s wall. This correction requires accurate values of the thermal conductivity of argon. In 1995, I recognized that the best measurements of the thermal conductivity of helium were less accurate than the best values calculated from the fundamental principles of quantum mechanics and statistical mechanics. Furthermore, the accuracy of the calculated values was rapidly improving and could be extended from helium to argon. I highlighted this challenge/opportunity for theorists in my most-frequently cited paper. Citing this paper, theorists obtained funding for improved calculations of gas properties to enable more accurate measurements of the universal gas constant and temperature. Today, calculations of the thermal conductivity of helium and argon are amazingly accurate. Furthermore, theorists are now using fundamental principles to accurately calculate properties of commercially important gases such as methane.

Now we understand how to measure temperature using the acoustic and microwave resonance frequencies of a gas-filled cavity. Can this understanding be used for anything else? Yes! Working with Keith Gillis and Jodie Pope, I came up with a standard for calibrating gas flowmeters at up to 70 times atmospheric pressure. The standard will collect gas in a nearly spherical pressure vessel affectionately known as the “big blue ball” or “BBB.” We used microwave resonance frequencies to determine the volume of the BBB as a function of the temperature and pressure with uncertainties smaller than 0.01 percent. Measurements of the pressure and the acoustic resonance frequencies determine the average temperature of gas in the BBB with similarly small uncertainties. These are remarkably accurate results for such a large pressure vessel, and I am confident that they can be scaled up to much larger pressure vessels. By measuring the pressure, volume and average temperature, we can determine the quantity of gas, and we can use that know-how to calibrate the meters used to buy and sell natural gas as it flows through pipelines.

So, an error, a guess, and years of experimentation and theory devoted to something as arcane as accurately measuring the Boltzmann constant will generate economic benefits that I didn’t and couldn’t have foreseen. When you pay your gas bill this month, consider that the accuracy of that bill is not accident: It is a result of measurement science made, in part, by curiosity-driven and/or challenge-driven scientists.

*This post originally appeared on **Taking Measure**, the official blog of the **National Institute of Standards and Technology (NIST)** on December 21, 2017.*

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*About the Author*

*For more than 50 years, Michael Moldover and his collaborators have measured the thermophysical properties of fluids to solve scientific and engineering problems and to improve standards of temperature, pressure and flow. Mike is the leader of NIST’s Fluid Metrology Group, a Fellow of NIST, the American Physical Society, and the Acoustical Society of America. From 2018 onward, the Boltzmann constant will be exactly 1.380649x10^–23 joule/kelvin. Is it any wonder that his cats are named “Joule” and “Kelvin”?*