## Mass, Planck, and Einstein

How can the SI kilogram unit – embodied in a single lump of metal cast in the 19th Century – be re-defined in terms of an invariant of nature and scaled up or down conveniently, accurately, and repeatably?

After decades of debate, the international metrology community has chosen to answer those questions by defining the kilogram in terms of an invariant of nature called the Planck constant for its discoverer, Max Planck, one of the pioneers of quantum science.

For many observers, the connection between mass on the scale of a liter of water and a constant deriving from the very earliest days of quantum mechanics may not be immediately obvious. The scientific context for that connection is suggested by a deep underlying relationship between two of the most celebrated formulations in physics.

One is Einstein's famous *E =mc ^{2}*, where

*E*is energy,

*m*is mass,and

*c*is the speed of light. The other expression, less well known to the general public but fundamental to modern science, is

*E = hν*, the first "quantum" expression in history, stated by Max Planck in 1900. Here

*E*is energy,

*ν*is frequency, and

*h*is what is now known as the Planck constant.

Einstein's equation reveals that mass can be understood and even quantified in terms of energy. Planck's equation shows that energy, in turn, can be calculated in terms of the frequency (*ν*) of some entity such as a photon — or alternatively, with some mathematical substitutions, a significant mass — times an integer multiple of *h*. The integer aspect is what makes the relationship "quantized."

Taking the two equations together yields a counterintuitive but hugely valuable insight: Mass – even on the scale of everyday objects – is inherently related to *h*, which Planck first used to describe the vanishingly small energy content of individual photons emitted by the atoms in hot objects. The value of *h* is about 0.6 trillionths of a trillionth of a billionth of 1 joule-second. The joule is the SI unit of energy.

As a practical matter, experiments linking mass to *h* with extraordinary precision became possible in the late 20th century as the result of two separate discoveries which led to two different physical constants related to voltage and resistance respectively.*

Both constants are defined in terms of *h* and the elementary charge of the electron (*e*), and both are known to uncertainties of 1 part per billion or better. More to the point, those constants can be used to obtain extremely precise electrical measurements in a device called a watt balance, first developed at the UK's National Physical Laboratory (NPL) and shortly thereafter at NIST.

By the 1980s scientists at NPL and at NIST were heavily involved in using watt balances to determine the value of *h, *a process that requires an exactly known mass such as a kilogram prototype and electrical measurements made with the voltage and resistance constants. Soon *h* had been measured to about 1 part in 10 million. (Its value is now known to 44 parts in a billion. It will be assigned a specific fixed value at the time of the impending SI redefinition.)

## Watts or Spheres?

Meanwhile, debate was intensifying about possible ways to redefine the kilogram, thus finally eliminating the artifact standard. Two principal schools of thought emerged. One would define the kilogram in terms of the mass of a silicon atom by counting the number of atoms in a 1 kg sphere of ultra-pure silicon-28. (See Silicon Kilogram.)

The other was championed, among others, by NIST scientists Peter Mohr and Barry Taylor. In 1999, in a succinct letter published in the journal *Metrologia, *they proposed assigning a fixed value to the Planck constant as the basis for a new definition. Mohr and Taylor reasoned that if a watt balance could use an exactly defined mass to measure the unknown value of *h*, then the process could be reversed: By setting an exact fixed value of *h*, the same system could be used to measure an unknown mass.

The idea, which came to be known as the "electric" or "electronic" kilogram, was widely discussed and finally endorsed in principle in 2011 by the international General Conference on Weights and Measures (CGPM), with a few provisions. One of them was that, prior to re-definition, at least one instrument, and preferably more, would have to measure* h* to a benchmark uncertainty of 2 parts in a hundred million (10^{8}). NIST's most recent measurement has a stated relative standard uncertainty of 3.4 X 10^{8}. In addition, the values obtained by the watt balances should be in reasonable agreement with those from scientists using the atom-counting approach to defining the kilogram.

Half a dozen watt balances around the world are now in pursuit of CGPM's uncertainty target. NIST is deploying a new instrument, completed in 2014, for the task. The measured values from different groups will have to be in very good agreement in order to set an official fixed value for *h*.

* These are the Josephson constant (*K*_{J }= 2*e/h*) and the von Klitzing constant (*R*_{K }= *h*/*e ^{2}*).

The Josephson constant is related to the AC Josephson effect. This occurs when a voltage applied across a superconducting junction causes an alternating current with a frequency that is proportional to the voltage. Because frequency can be measured more precisely than any other quantity, *K*_{J }provides an extremely accurate way of measuring voltage. It is used in that manner in watt balances.

The von Klitzing constant describes the way electrical resistance is quantized in certain kinds of physical systems. Because of its extraordinarily high precision, *R*_{K} is employed around the world as a standard of electrical resistance. In a watt balance, it is used to measure the current applied to a coil of wire.

Both constants also involve *e*, the fundamental charge of the electron. Because of the way the watt balance measures electrical power (albeit indirectly), *e* cancels out of the equations. That leaves *h* as the sole quantity of interest.