### The Planck Constant and the Kilogram

July 21, 2016

James Watt, perfecter of the steam engine, is remembered by the unit for power, the watt. The watt was named the unit of power by the Second Congress of the British Association for the Advancement of Science in 1889. It was adopted into the International System of Units as an "SI" unit in 1960 by the 11th General Conference on Weights and Measures. I wrote about steam engines in an earlier article (Steam Power, January 28, 2011).

In Watt's time, horses were the engines of many industries, and the unit of power was the horsepower, so it was natural that Watt would compare the power of his steam engines to the lifting power of draft horses. To do this with any measure of accuracy, you would need to test many horses. statistically, one percent accuracy would require 10,000 horses. Today's standards equate a horsepower with 745.7 watts, although the nameplate horsepower on electric motors uses a rounded value of 746 watts. I don't think any mechanical engineer would worry about this 0.04% error.

James Watt (1736-1736).

During Watt's time, conventional steam engines, such as the Newcomen steam engine, condensed the steam inside the work cylinder by a spray of cold water. Watt's engine had an external condenser, which increased efficiency dramatically.

(From Popular Science Monthly, 1877, via Wikimedia Commons.)

Electrical power P is easily measured as the product of voltage E and current I; that is, P = EI. Advances in metrology have given us absolute ways to define voltage and current. The AC Josephson effect has given us a fundamentally accurate voltage standard; and, since electric current is just the number of charge carriers that pass through a conductor in a given time, we can even count the carriers to deduce the current.[2] A more practical current measurement method involves the quantum resistance value of the von Klitzing constant.

While electrical power in this fashion can be defined by the Planck constant and the speed of light, both fundamental constants,[1] the kilogram is still maintained as an artifact standard. The US National Institute of Standards and Technology (NIST) is developing an apparatus called the Watt balance to relate the kilogram to the fundamental constants.

NIST has been developing this
apparatus for quite some time, since I wrote about their Watt balance in 2010 (Mass Standard, November 1, 2010). The concept of the Watt balance is simple. The apparatus equalizes the force exerted on a mass due to gravitational acceleration with a force created by the flow of current in a wire (usually wound as a coil) in a magnetic field.

While the measurement concept is simple, the apparatus needed to make an extremely precise measurement is quite complex, as can be seen in the photo. One problem is designing the apparatus to keep the the magnetic field as
homogeneous as possible. NIST is not alone in building a Watt balance. National standards laboratories in England, France, Switzerland, and other countries, are developing their own.

NIST-4 Watt Balance.

This
Watt balance has measured the Planck constant to within 34 parts per billion.

According to theory, mass can be expressed as a function of this constant and other fundamental constants.

(NIST image 16PML013 by J. L. Lee .)

The Watt balance design eliminates the need to know the precise geometry of the mass-lifting coil, since the apparatus is self-calibrating. Moving the mass-lifting coil through the magnetic field at a known velocity will give an induced voltage according to Faraday's law of induction. In a recent paper in the Review of Scientific Instruments, NIST scientists have described their Watt balance measurements of the Planck constant from late 2015 through early 2016. They've found the value,[3-4]
h = 6.626 069 83 x 10-34 joulesec ± 22 x 10-42 Joule⋅sec

The Watt balance work at NIST and other standards agencies has been done towards the aim of redefining the kilogram in 2018 in a way that separates this standard from a physical object.[4] As the graph below shows, there's a continuing problem in transferring the standard from the international prototype kilogram, maintained at the International Bureau of Weights and Measures near Paris, France.

Left, a replica of the prototype kilogram at the Cité des Sciences et de l'Industrie, Paris, France. The prototype kilogram is made from an alloy of 90% platinum and 10% iridium, so it resists oxidation. The problem with this artifact standard is that it is only useful through use of transfer standards calibrated to itself. As the graph on the right shows, these transfer standards have been changing over the course of time, probably through accumulation of airborne contamination. The average change has been 50 micrograms over the course of a century. (Left image, via Wikimedia Commons. Right image, from NIST.)[5])

This measurement of the Plank constant on the latest NIST Watt balance, NIST-4, is consistent with watt balance measurements from other countries, and the uncertainty of measurement is actually lower than expected.[4] Says NIST physicist, Stephan Schlamminger, "This measurement was essentially a dry run... We were hoping to achieve an uncertainty of within 200 parts per billion by this point, but we got better fast."[4]

For standard application, the results of three experiments must result in a relative uncertainty of not more than 50 parts per billion, with at least one experiment having less than 20 parts per billion, and all in agreement at a 95% statistical confidence level. The results must also match those of an alternative, but more expensive, method of "counting" atoms in an ultra-pure sphere of silicon.[4] The "counting" is done through an xray diffraction measurement of the atomic spacing.

### References:

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