Physics / Short Notes

Vol. 7, NO. 1 / June 2022

# The Kilogram

Jeremy Bernstein

#### Letters to the Editors

In response to “The Kilogram

When I took freshman physics in the late 1940s, I was told that mass was the quantity of matter. Even then I was struck by the circularity of this definition. When I taught physics some years later, we used a standard textbook, which I still have. It is so heavy that I can barely lift it. The authors did not attempt to define mass and wrote only that the mass of anything should be compared to a standard platinum alloy cylinder which was created in 1889 and is stored in Paris. This struck me as like defining an animal—say, an owl—by comparing its attributes to those of my neighbor’s cat.

In this brief note, I will not attempt to define mass. Instead, I will try to explain what has happened to the standard kilogram. This development touches on almost every area of physics—with the possible exception of the Higgs boson. I bring up the Higgs boson because the method used to define the kilogram involves something called the Kibble balance. When I first read about this instrument, I thought it might be named after Tom Kibble, a British theoretical physicist whom I knew and who was one of the first to describe the Higgs mechanism. In fact, the name refers to another British physicist, Bryan Kibble, who seems to be unrelated. Both are now deceased, having passed away within a few weeks of each other during the summer of 2016.

To determine the mass of an object using the classical approach, one might begin by making a series of replicas in varying sizes using the cylinder in Paris as a reference. A given object would then be placed on one side of a balance scale and a cylinder on the other, with further cylinders added until the scales balance. Putting aside the practicality of this approach, there is an accuracy question. Since 1889, the cylinder in Paris has lost about 50 micrograms of mass by surface ablation. The kilogram is not what it once was—albeit only fractionally. Still, this situation was clearly unsatisfactory for a standard measure. In the mid-1970s, Bryan Kibble decided to address this problem. The first step was to design a balance that only required classical physics, originally named the watt balance. This instrument was then related to quantum mechanics. In the standard approach to measuring mass, one gravitational force is balanced against another: from the example above, the gravitational force acting on the object being weighed is balanced against that acting on the cylinders. When the two match, a mass can be weighed. Rather than balancing gravitational forces, Kibble’s instrument matches a gravitational force against an electromagnetic force.

A watt balance is constructed using a coil of length L that can carry an electric current. If this coil is moved in a magnetic field B with a speed v, it can generate a voltage, BvL. A force is produced that can be expressed as BLI, where I is the current produced by moving the coil. This force can be adjusted so it matches the weight of a given object. By measuring these electromagnetic parameters—say, when the Paris cylinder is balanced—an immutable definition of a kilogram can then be obtained. The key limitation is how accurately these parameters can be measured. And this is where quantum mechanics comes in.

Max Planck was born in Kiel on April 23, 1858. His father was a professor of constitutional law. Throughout his life, Planck remained a staunch conservative with a firm belief in the rule of law. He was drawn to physics, in part, because it seemed to be governed by universal laws. When he was trying to decide his future profession he was warned not to choose physics because everything of interest had already been discovered. Planck first tried his hand as an experimental physicist, but he soon switched his attention to theory. Very early in his career he hit on a problem that exhibited a range of universal characteristics that he found captivating. This was the problem of blackbody radiation.

If the walls of any given container, regardless of its shape or composition, are heated, thermal radiation will be emitted. Such a container is termed a blackbody. The frequency distribution of the interior radiation can be measured to produce a curve that represents the intensity of the radiation at a given frequency. This curve depends only on the temperature of the walls. Planck set out to derive the distribution from classical physics and soon found that this simply could not be done. In the end, he was forced to make a radical assumption.

When the atoms in the walls of the container are heated, they begin to oscillate. In doing so, they emit radiation into the interior, which can, in turn, be absorbed by another oscillator and then re-emitted. This process continues until an equilibrium distribution of atoms and radiation is achieved. In order to account for this distribution, it is necessary to study how the oscillators emit and absorb radiation. From classical physics, it seemed that such radiation could be absorbed or emitted in any amount of energy—a continuum. But Planck found that he was only able to derive the correct distribution if he assumed that thermal radiation could only be emitted and absorbed in discrete amounts that he named quanta.

In deriving his distribution, Plank needed to relate the energy of quanta to their frequencies. While energy is measured in units like ergs or electron volts, a frequency is measured per second. The relationship between energy and frequency could thus be expressed as E = hv, where E is an energy and v is a frequency, if the constant h has the dimensions of energy seconds. Planck immediately realized the significance of his new constant. To represent it, he chose h for hohlraum, from the German for “cavity.” Due to the universal character of this radiation, Planck noted that h must be a new universal constant alongside the speed of light in a vacuum and the charge of the electron. In units of meters, kilograms, and seconds, the constant has the present experimental value

6.62607004 × 10–34m2kg/s.

The fact that h is so tiny when expressed in conventional units is the reason why we are not aware of quantum mechanics in our daily lives. Using techniques derived from the quantum theory of solids, the Kibble balance was originally used to measure the Planck constant. But it quickly became apparent that the Planck constant could also be used to define the kilogram. In practice, a kilogram is the amount of matter related to the experimentally determined Planck constant. This can be expressed as h = BW, where h is the Planck constant, B is a parameter that represents all the characteristics of the balance, and W is the weight in kilograms. The equation can be read in the following way. First, make a note of the known value of the Planck constant, as determined by other means. One kilogram can then be defined as the weight that produces this value for the Planck constant. The result is a new definition of a kilogram that will not deteriorate over time.

In closing, a word about Planck. I am not sure that he ever really accepted quantum theory, even though he and Albert Einstein became colleagues in Berlin. After the Nazis began expelling Jewish scholars from German universities in 1933, Planck met with Adolf Hitler in person and tried to persuade him to stop the purge. Hitler flew into a rage and Planck went into isolation, although he did make an effort to assist some of his Jewish colleagues. Planck’s son Erwin was one of the conspirators involved in the 20 July plot to kill Hitler. After the plot failed, Erwin was arrested and brutally executed. Max Planck survived the war. He passed away in October 1947 at the age of 89.

Jeremy Bernstein is Professor Emeritus of Physics at the Stevens Institute of Technology.

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