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# Boltzmann constant k, “What is temperature?” and the new definition of the SI system of physical units

(by Gerhard Fasol, CEO of Eurotechnology Japan KK. Served as Associate Professor of Tokyo University, Lecturer at Cambridge University, and Manger of Hitachi Cambridge R&D Lab.)

Keynote presented at the 6th Ludwig Boltzmann Symposium on February 20, 2014 at the Embassy of Austria in Tokyo.

(in preparing this talk, I am very grateful for several email discussions and telephone conversations, and for unpublished presentations and documents, to Dr Michael de Podesta MBE CPhys MInstP, Principal Research Scientist at the National Physical Laboratory NPL in Teddington, UK, who has greatly assisted me in understanding the current status of work on reforming the SI system of units, and also his very important work on high-precision measurements of Boltzmann’s constant. Dr Michael de Podesta’s measurements of Boltzmann’s constant are arguable among the most precise, of not the most precise measurements of Boltzmann’s constant today, and therefore a very important contribution to our system of physical units).

## Boltzmann constant k, the definition of the unit of temperature and energy

Temperature is one of the physics quantities we use most, and understanding all aspects of temperature is at the core of Ludwig Boltzmann’s work. People measured temperature long before anyone knew what temperature really is: temperature is a measurement of the average kinetic energy of the atoms of a substance. When we touch a body to “feel” its temperature, what we are really doing is to measure the “buzz”, the thermal vibrations of the atoms making up that body.

For an ideal gas, the kinetic energy per molecule is equal to 3/2 k.T, where k is Boltzmann’s constant. Therefore Boltzmann’s constant directly links energy and Temperature.

However, when we measure “Temperature” in real life, we are not really measuring the true thermodynamic temperature, what we are really measuring is T90, a temperature scale ITS-90 defined in 1990, which is anchored by the definition of temperature units in the System International, the SI system of defining a set of fundamental physical units. Our base units are of fundamental importance for example to transfer semiconductor production processes around the world. For example, when a semiconductor production process requires a temperature of 769.3 Kelvin or mass of 1.0000 Kilogram, then accurate definition and methods of measurement are necessary to achieve precisely the same temperature or mass in different laboratories or factories around the world.

## The SI system of physical units

The SI system consists of seven units, which at the moment are defined as follows:

• second: The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
• metre: The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
• kilogram: The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.
• Ampere: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.
• Kelvin: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
• mole:
1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12
2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
• candela: The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

The definitions of base units has long history, and are evolving over time. Today several of the definitions are particularly problematic, among the most problematic are temperature and mass.

## SI base units are closely linked to fundamental constants:

• second:
• metre: linked to c = speed of light in vacuum
• kilogram: linked to h = Planck constant.
• Ampere: linked to e = elementary charge (charge of an electron)
• Kelvin: linked to k = Boltzmann constnt
• candela:

## Switch to a new framework for the SI base units:

Each fundamental constant Q is a product of a number {Q} and a base unit [Q]:

Q = {Q} x [Q],

for example Boltzmann’s constant is:
k = 1.380650 x 10-23 JK-1.

Thus we have two ways to define the SI system of SI base units:

1. we can fix the units [Q], and then measure the numerical values {Q} of fundamental constants in terms of these units (method valid today to define the SI system)
2. we can fix the numbers {Q} of fundamental constants, and then define the units [Q] thus that the fundamental constants have the numerical values {Q} (future method of defining the SI system)

Over the next few years the SI system of units will be switched from the today’s method (1.) where units are fixed and numerical values of fundamental constants are “variable”, i.e. determined experimentally, to the new method (2.) where the numerical values of the set of fundamental constants is fixed, and the units are defined such, that their definition results in the fixed numerical values of the set of fundamental constants. This switch to a new definition of the SI system requires international agreements, and decisions by international organizations, and this process is expected to be completed by 2018.

Today’s method (1.) above is problematic: The SI unit of temperature, Kelvin is defined as the fraction 1/273.16 of the thermodynamic temperature at the triple point of water. The problem is that the triple point depends on many factors including pressure, and the precise composition of water, in terms of isotopes and impurities. In the current definition the water to be used is determined as “VSNOW” = Vienna Standard Mean Ocean Water. Of course this is highly problematic, and the new method (2.) will not depend on VSNOW any longer.

In the new system (2.) the Kelvin will be defined as:

Kelvin is defined such, that the numerical value of the Boltzmann constant k is equal to exactly 1.380650 x 10-23 JK-1.

## Measurement of the Boltzmann constant k:

In order to link the soon to be fixed numerical value of Boltzmann’s constant to currently valid definitions of the Kelvin, and in particular to determine the precision and errors, it is necessary to measure the value of Boltzmann’s current in terms of today’s units as accurately as possible, and also to understand and estimate all errors in the measurement. Several measurements of Boltzmann’s constants are being performed in laboratories around the world, particularly at several European and US laboratories. Arguably today’s best measurement has been performed by Dr Michael de Podesta MBE CPhys MInstP, Principal Research Scientist at the National Physical Laboratory NPL in Teddington, UK, who has kindly discussed his measurements and today’s status of the work on the system of SI units and its redefinition with me, and has greatly assisted in the preparation of this article. Dr Podesta’s measurements of Boltzmann’s constant have been published in:
Michael de Podesta et al. “A low-uncertainty measurement of the Boltzmann constant”, Metrologia 50 (2013) 354-376.

Dr Podesta’s measurements are extremely sophisticated, needed many years of work, and cooperations with several other laboratories. Dr. Podesta and collaborators constructed a highly precise resonant cavity filled with Argon gas. Dr. Podesta measured both the microwave resonance modes of the cavity to determine the precise radius and geometry, and determined the speed of sound in the Argon gas from acoustic resonance modes. Dr Podesta performed exceptionally accurate measurements of the speed of sound in this cavity, which can be said to be the most accurate thermometer globally today. The speed of sound can be directly related to 3/2 k.T, the mean molecular kinetic energy of the Argon molecules. In these measurements, Dr. Podesta very carefully considered many different types of influences on his measurements, such as surface gas layers, shape of microwave and acoustic sources and sensors etc. He achieved a relative standard uncertainty of 0.71. 10-6, which means that his measurements of Boltzmann’s constant are estimated to be accurate to within better than on millionth. Dr. Podesta’s measurements directly influences the precision with which we measure temperature in the new system of units.

Over the last 10 years there is intense effort in Europe and the USA to build rebuild the SI unit system. In particular NIST (USA), NPL (UK), several French institutions and Italian institutions, as well as the German PTB (Physikalische Technische Bundesanstalt) are undertaking this effort. To my knowledge there is only very small or no contribution from Japan to this effort, which was surprising for me.

## What is today’s best value for the Boltzmann constant k:

Today’s accepted best value of Boltzmann’s constant is the “2010 Codata value”:

k = 1.380 6488 . 10-23 JK-1, and the standard uncertainty is:
su = 0.000 0013 . 10-23 JK-1