# Planck, Avogadro and measuring mass using fundamental constants

## Authors

Within the next five years major changes will be made to the the world's measurement system: the SI. Four of the seven base units: the kilogram, the Ampere, the Kelvin and the mole, will be redefined in terms of fixed values of fundamental constants: the Planck constant h, the elementary charge e, the Boltzmann constant k and the Avogadro constant NA respectively. In their paper “Accurate measurements of the Avogadro and Planck constants by counting silicon atoms” Bettin et al. describe work which has contributed to the redefinition of the kilogram and mole [1].

## Atom mass or the Planck constant

The unit of mass, the kilogram, will be defined in terms of a fixed value of the Planck constant, which can be considered to link the frequency of electromagnetic radiation to the energy of a quantum of that radiation: the photon. As energy can be related to mass by the Einstein relation , and as the value of the speed of light c is fixed by the definition of the metre, the Planck constant can be considered to link mass and frequency.

However it might be asked why mass was not defined in terms of the mass of a chosen entity: an atom or fundamental particle. But this would have required that all mass measurements be related to the chosen entity, which could limit future progress in mass measurement. In addition the reference state of the entity must be defined; usually: unbound, at rest and in its ground state. This means that practical primary measurements (realisations) of mass, using such a definition, may have to correct the measured mass for any energy differences between the measured and defined states. This is not usually a major problem but complicates both the definition and future realisations.

The chosen alternative of fixing the numerical value of the Planck constant does not require any subsidiary conditions and does not fix the scale at which the definition is realised.

Bettin et al. show that the watt balance [2], pictured in figure 1, measures mass in terms of the Planck constant but they also remind us that the Avogadro and Planck constants are related by an expression which allows one to be determined from the other with an added uncertainty which is negligible at present. So, after redefinition, both techniques can be used to maintain the macroscopic mass scale.

A further advantage to fixing the value of the Planck constant arises from the work of Josephson [3] and von Klitzing [4]. In modern metrology the Josephson effect provides a voltage reference, in SI units, which is limited by our knowledge of the values of the Planck constant and the elementary charge. The Quantum Hall Effect, discovered by von Klitzing, provides an SI resistance standard which is similarly limited. The regular changes to the recommended SI values of these constants would have disrupted the dissemination of accurate electrical measurements; this led to the adoption, in 1990, of fixed conventional values of the Josephson constant () and von Klitzing constant () for making electrical measurements throughout the world. This has the disadvantage that such measurements are not fully coherent with the rest of the SI. Fixing the values of the Planck constant and the elementary charge will restore these measurements to the SI providing the world with extremely accurate SI units of voltage and resistance.

There will be a benefit to physicists engaged in high precision experiments as the value of the constants fixed within the SI and combinations of their values will have zero uncertainty and will not change with the regular least squares adjustments carried out by the CODATA task group [5].

Bettin et al. describe the work required to prepare for the redefinition. They describe the careful investigations which are being carried out to check for problems which may disturb the atom counting result to ensure that the chosen value of the Planck constant is the best that can be produced. Similar efforts are being made by the watt balance groups. The International Prototype of the kilogram (IPK) will also be used for the last time to ensure that there will be agreement between the old and new definitions at the time the definition is changed.

## After redefinition

We are approaching the point where the numerical value of the Planck constant will be fixed and the kilogram redefined. The atom counting and watt balance experiments then become methods of measuring mass which can either be used directly or can contribute to measuring the values and stability of an ensemble of 1 kg mass standards which will be maintained by the Bureau International des Poids et Measures (BIPM).

This ensemble will provide a “flywheel” to allow calibration of the existing 1 kg mass standards of those countries of the world which do not have access to a direct realisation of SI mass.

As new primary realisations become available they will be integrated into this system in a way that maintains the stability of the mass scale. This process is already carried out in the dissemination of time from atomic clocks and the mass community will benefit from this experience.

It can be argued that there may be fixed offsets within any one realisation of the mass unit but, as there will be a number of such realisations available of differing detailed design, these offsets can mostly be considered to be random. Under these circumstances the uncertainty provided by the BIPM mass ensemble could be less than that of any of the primary realisations contributing to its value, provided that the contributions are made often enough to eliminate the drift of the ensemble and that any correlations between the contributing measurements are dealt with correctly.

An extremely important aspect of the new definition is its effect at atomic level. This can be illustrated by the Compton clock [6] which is described in section 'Atom mass or the Planck constant' of Bettin et al. The Compton frequency depends on the mass of a particle, the Planck constant and the velocity of light. In the redefined SI the latter two will have zero uncertainty and so the realisation of SI mass at the atomic level will ultimately depend on the measurement of frequency: a quantity that can be measured extremely precisely in SI units. Under these circumstances it can be seen that the uncertainty of SI mass measurement at the atomic scale could be far lower that that at the macroscopic scale and there is no requirement to relate the scale to a kilogram mass. However, as described by Bettin et al., atom counting techniques would provide a link between such mass realisations at the atomic scale and those at the macroscopic scale.

## The future

Bettin et al. show that we are at the beginning of a new chapter in mass metrology. We will be free to realise mass, in terms of the Planck constant, at any desired scale without the need to trace our measurements to a 1 kg mass.

This advantage to physics does come with a requirement that both before and after the redefinition we explain to the wider community why we have made the decision to define mass in terms of the Planck constant. This gives us the opportunity to show that quantum physics is relevant to everyday standards and that mass can be linked to a constant of nature that is at the heart of quantum physics, rather than an artefact that was produced in the late 19 century. This will be a challenge but, for the reward of a worldwide measurement system which is integrated, accurate, stable, flexible and robust the effort will be worthwhile.