Le roi est mort ! Vive le roi !

In France, this sentence symbolised the continuous reign of the monarchy as the crown was passed directly from the deceased king to his successor. However, this is not the case for Louis XVI. His reign ended with the short, swift fall of the guillotine, taking with him the French monarchy as the French Revolution took hold at the end of the 18th century.

What does that have to do with the kilogram? Well, another effect of the French Revolution was to initiate the beginning of our present International System of Units, abbreviated worldwide as SI. In the spirit of the revolution, 'freedom, equality, fraternity', even measurement was something that should become equal and accessible for all, independent of royal requirements. This was a significant change - before, units such as mass or length were defined differently from kingdom to kingdom and duchy to duchy. Often the arm length of the king or duke was quite literally what defined unit of length.

In 1789, the Cahiers de Doléances, a kind of written collection of the grievances of the French people, called for the creation of a universal system of mass as a departure from the arbitrary, feudal rule. The first new measure defined in 1791 was the metre as the unit of length. The units for area and volume were derived from the metre - square metre, cubic metre, litre. The definition for the kilogram followed in 1799 and the first prototype for the mass was created, a small platinum cylinder. In the same year the metric system of units was established by law in France as the authoritative system.

During the next century the idea of the uniform metric system of units spread worldwide. In 1875, 17 states joined together to form the so-called Metre Convention, with the aim of ensuring a globally uniform system of measurement. Delegates of the member states still meet regularly at the so-called General Conference on Weights and Measures (CGPM, Conférence Générale des Poids et Mesures) to discuss the further development of the system of units and to decide on innovations.

At the first CGPM, in 1889, a new prototype for the kilogram was created. Also a cylinder, the revised prototype was remade using an alloy of platinum and iridium, which is more stable than pure platinum. During the meeting delegates established its definition: "This prototype shall henceforth be considered to be the unit of mass." Because it is such a special piece of metal, it was not only called an 'international kilogram prototype', but also got its own name: Le grand k.

This inconspicuous object, safely stored in a safe in a small village near Paris, was the reference for any mass measurement around the world for 130 years, from 1889 to 2019. You may ask yourself, why inconspicuous? Platinum is a very heavy element, and so the original kilogram measures only 39 mm in diameter and 39 mm in height - so actually very small and rather modest!

The kilogram is dead! Long live the kilogram!

Yet, the reign of Le grand k is over. From 20 May 2019 this last artefact is no longer the reference for international mass measurement. It will be replaced by a strange letter – h, which physicists refer to as the Planck constant. But what does a constant usually found in quantum physics and a kilogram have in common? How can we use it for mass measurement? Moving from a platinum block, and a simple and tangible way of defining the kilogram, these questions are challenging for everyone! We are going try and answer them later in this article, but for now we will address another question that often comes up when talking about the kilogram redefinition…

What was wrong with the artefact?

Since 1889, the kilogram was defined as the mass of the International Prototype. Each country – signatory to Metre Convention – received its own copy and maintained it as their own National Protype. Each (in theory) identical national copies are periodically compared with international prototype. Over the years, measurements done in these comparisons showed that masses of the various artefacts are diverging. The last comparisons showed that the spread has grown to 50 μg – about the same mass as a fly’s wing. Although it seems very little to the ordinary person buying flour in a shop, this is a huge amount for science. In the field of medicine, for example, 50 μg is a daily dose of D-vitamin for a newborn.

The path to the new quantum mechanical kilogram

The Planck constant (h) is a fundamental physical constant, meaning that it has the same value everywhere in the universe. As the quantum of electromagnetic action, it appears in most quantum mechanical equations. 

In recent years, the value of these Planck constant has been determined in many laboratories worldwide using independent experiments with the best possible accuracy. Two experimentally different measuring principles were used. 

The watt, or kibble, balance was one way of determining the Planck constant. It links the mass and h by measuring an electric current. Alternatively, silicon spheres of single silicon crystals can be used to determine of h. This measuring principle links the mass of the silicon atom with h.

Watt balance

The essence of the watt balance is in its name: the basic principle of this experiment is based on weighing, and balancing forces. In simple, mechanical weight scales, the downwards, gravitational force of two masses is balanced.

The watt balance is a current balance. Here, the gravitational force of a piece of mass is balanced with an electromagnetic force. The electromagnetic force can be generated by placing an electrical conductor wound into a coil in a static magnetic field and allowing a current to flow through the electrical conductor.  The flow of current generates a further magnetic field (in physics language: induced). An electromagnetic force acts between the two magnetic fields. 

This force is used in the watt balance to counteract the weight of a piece of mass. To do so, the object is placed on a weighing plate attached to an electrical conductor wound into a coil, which is located within a static magnetic field. When a current flows through the electrical conductor, with the correct strength and direction, the electromagnetic force between the static and induced magnetic field compensates for the gravitational force (weight) of the object. The piece of mass is levitated on a scale. 

So where does the Planck constant appear? As you may have guessed, balancing these forces is only a small part of a very complicated process with many measurements, boundary conditions and technical challenges. Two critical measurements are the measurement of the current intensity and the induced voltage. Nowadays, electrical currents and voltages are determined very precisely using measuring devices based on quantum mechanical principles, which is where h comes in (you can read more about this at the unit of the month in January, the ampere).

Silicon sphere 

Determining the value of the Planck constant is a very complex process when producing perfect spheres is involved. Well, nearly perfect spheres. They are spheres. They are crystals. They are only made up of one type of atoms: silicon. The closer an object actually fulfils these three properties, the more attractive it is to science. This is because amazing things can then be measured, namely how many atoms it contains. In such a sphere of approx. 1 kg: approximately 25 quadrillion (25 followed by 24 zeros). 

The best ‘kilogram’ spheres consist of 99.999 % of the silicon isotope ²⁸Si, which is arranged as a single crystal in a highly symmetrical lattice structure. The spheres have a diameter of around 93.7 mm. The deviation from the perfect spherical shape is only a few 10s of nanometres
(1 nanometre = 1 nm = 1 billionth of a meter). 

The purity of the material, the distance between the atoms in the lattice structure and the "topography", and thus the volume of the sphere, are determined and controlled experimentally. In addition, the chemical surface of the spheres is investigated because they react with the air surrounding them which must be considered when counting the atoms. Together with weighing the sphere, the mass of the silicon atom is finally determined.

Once the mass of a silicon atom has been determined, quantum mechanics comes into play again, because the classical laws of physics don’t apply at these magnitudes. So, in the last steps to determine h, we are joined by quantum mechanical natural constants (e.g. the Rydberg constant and the fine structure constant) that are very precisely known.

One approach is crucial for both experiments: Whilst the value for h had not been finally determined and fixed, it instead had to be calculated in both experiments from the weighing of masses. This weighing took place in the "old system of units", i.e. they led back to the International kilogram prototype using the usual comparative measurements. What at first seems contradictory, however, is quite decisive for the consistency of the mass weighing at the transition from the "old SI" to the "new SI".

Now, the numerical value for the natural constant is fixed as a constant numerical value, the mass of pieces can be calculated using the watt balance or silicon spheres - finally allowing the new definitions to come into effect. Funnily enough, this means that for the very first time it is possible to independently observe how the mass of the International Kilogram Prototype changes over the years. Even if this does not really matter within the framework of the International System of Units, we’ll no doubt be checking out of curiosity.

The effect of this definition is to define the unit kg m² s^-1 (the unit of both the physical quantities action and angular momentum). Together with the definitions of the second and the metre this leads to a definition of the unit of mass expressed in terms of the Planck constant h.

The subject isn’t easy and requires understanding of many physical phenomenon and a bag of formulas that need to be used. But if you would like to explore more about redefinition of the kilogram, please use the links below to start your journey with the “new” kilogram.

https://www.youtube.com/watch?v=Oo0jm1PPRuo
https://www.youtube.com/watch?v=bMYvVgsotIk
https://www.youtube.com/watch?v=VlJSwb4i_uQ
https://en.wikipedia.org/wiki/Kibble_balance
https://www.nist.gov/si-redefinition/kilogram-kibble-balance
https://www.bipm.org/en/bipm/mass/watt-balance/