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August 10, 2015

There's a big difference between mathematical constants and physical constants. Most mathematical constants are specified by a series expansion, and they are known to arbitrary precision. You just need to be willing to take the time to do the calculation. In this way, pi, the ratio of the circumference to the diameter of a circle also known as Archimedes' constant, is known to 1013 digits.

These many digits of pi are over-kill, since you need just sixty-two digits of pi to calculate the circumference (2.8 x 1027 meters) of the observable universe from its diameter to the precision of a Planck length (about 1.6162 x 10−35 meters).

Bailey–Borwein–Plouffe formula

The Bailey–Borwein–Plouffe formula allows calculation of an arbitrary digit of pi without calculating the preceding digits, as required in a series expansion calculation. Although this formula calculates a pi hexadecimal digit (base-16), there's an easy way to convert the result to a decimal digit. (Equation rendered using Inkscape.)

Most physical constants are known to far fewer digits, since measurement is required. One physical constant is far more precise, since it's defined, rather than measured. The vacuum permeability, symbolized by μ0, is defined as 4π x 10-7 newtons/ampere2; so, that's known, too, to 1013 digits.

Precise values of the physical constants are important beyond calculations in homework assignments. Since they're essential to technology and the commerce that it enables, governments have agencies tasked with keeping track of their best values, and doing experiments to refine their values further. The National Institute of Standards and Technology is the US agency tasked with such metrology. I wrote about the science of mass standards in a previous article (Mass Standard, November 1, 2010).

Since 1973, the Committee on Data for Science and Technology (CODATA) has organized and published measurements of the physical constants, and it's decided on a recommended value for each. The last such compendium of data available though December 31, 2014, was published on June 25, 2015. The present values are available on the NIST web site.[1]

Earlier this year, metrologists from around the world convened in Eltville, Germany, for a workshop on the determination of the fundamental constants (February 1-6, 2015). Papers from this workshop have just appeared in the Journal of Physical and Chemical Reference Data. One of these is an overview of the workshop by scientists from the Max-Planck-Institut für Quantenoptik (Garching, Germany), the Pulkovo Observatory (Saint Petersburg, Russia), and the National Institute of Standards and Technology (Gaithersburg, Maryland).[2-3]

Another is a summary of measurement of the value of the Avogadro constant obtained by "counting" the atoms in silicon spheres by scientists from the Istituto Nazionale di Ricerca Metrologica (Torino, Italy), the Bureau International des Poids et Mesures BIPM (Sèvres Cedex, France), the National Metrology Institute of Japan (Tsukuba, Japan), and the Physikalisch-Technische Bundesanstalt (Braunschweig, Germany).[4-5]

Artifact mass and length standards
Old school metrology. This is a replica of the international prototype kilogram (left), and an illustration of the former International Prototype Metre (right). The meter bar, made from a 90% platinum - 10% iridium alloy, was the standard of length until 1960, when the meter was redefined in terms of the wavelength of light from a krypton-86 lamp. (left image and right image via Wikimedia Commons.)

Metrology is advancing beyond the need to maintain standards as artifacts (see photos, above), as the 1960 change of the meter definition from an alloy bar to a wavelength of light illustrates. Even these modern artifacts are an advance over the attempted standards of centuries past. At about 1300, the legal length standard in England was specified as follows:
"It is ordained that 3 grains of barley dry and round do make an inch, 12 inches make 1 foot, 3 feet make 1 yard, 5 yards and a half make a perch, and 40 perches in length and 4 in breadth make an acre."[6]

There are many simultaneous experiments conducted for precision measurement of various physical constants, and these generate values that disagree slightly. The February workshop revealed that the measurements of the Boltzmann constant, which converts particle energy to temperature, are converging on the same value. In the future, the kelvin temperature unit will be defined by the Boltzmann constant.[3]

Also converging are measurements of Planck's Constant, which will eventually help to define a new kilogram standard.[3] Says NIST's Peter Mohr, coauthor of the summary paper about the workshop,[2]
"The Planck constant was problematic in the past, as there were disagreeing values obtained by different experiments. However, the values seem to be converging to a sufficiently reliable value for the redefinition of the SI to move forward... The new definitions will make many of the physical constants that are measured now exact in the future. Others, although not exact, will be more accurate.. This will stabilize the values of the constants and provide accurate measurement standards."[3]

The metrologists' goal is to define all SI units in terms of fundamental constants by 2018, thereby replacing the artifact standards.[3] A major hurdle in this is the kilogram definition, and there are efforts underway to define the kilogram in terms of fundamental constants.[3-5] One method for this is the "watt balance" that relates mass to electric current and voltage (see figure). It derives its name from the fact that the unit of electrical power, the watt, is the product of voltage and current.

NIST watt balanceA watt balance at NIST

In a watt balance, the magnetic attraction between current-carrying coils balances the weight of a kilogram.

Knowledge of the local gravitational acceleration allows conversion of the weight to a mass.

(NIST image.)

As mentioned earlier, a team of researchers from Germany, France, Italy, and Japan has been working to measure the value of the Avogadro constant by manufacture of precise kilogram spheres of isotopically-pure silicon (silicon-28). The Avogadro constant, the number of particles in a mole, is a number known to all students of science - about 6.02 x 1023. Defining the kilogram using fundamental constants would require a good value of Planck's constant, which can be derived from the Avogadro constant.[5]

This method is enabled by the technology for the routine growth of huge, perfect crystals of silicon, and the spacing of the silicon atoms can be determined to high precision using X-ray diffraction techniques. The difficult part is the etching and polishing to produce the spheres of nearly perfect roundness and no metal contamination. At this time, the best value of the Avogadro constant obtained by this method is 6.02214082(11) x 1023, where the number in parentheses represents the uncertainty of the last digit.[5]

Sphere of silicon-28A silicon kilogram.

The number of atoms in this nearly perfect sphere of silicon-28 is known to about 20 per billion.

(Photograph by Enrico Massa and Carlo Sasso.)

It's anticipated that the standard kilogram will be defined in terms of Planck's constant in 2018.[5] Says Giovanni Mana of the Istituto Nazionale di Ricerca Metrologica, a coauthor of the silicon sphere paper[4],
"Prior to redefining the kilogram, we must demonstrate that the new realization is indistinguishable from the present one, to within the accuracy of the world's best balances... Otherwise, when changing from the present definition to the new one, all users in science, industry, and commerce must change the mass value of all the existing artefacts."[5]


  1. CODATA Internationally recommended 2014 values of the Fundamental Physical Constants at the NIST web site.
  2. Savely G. Karshenboim, Peter J. Mohr, and David B. Newell, "Advances in Determination of Fundamental Constants," Journal of Physical and Chemical Reference Data, vol. 44, no. 3 (September, 2015), article no. 031101, DOI:http://dx.doi.org/10.1063/1.4926575.
  3. Constant change - Advances in determination of fundamental constants to guide redefinition of scientific units to rely on constants of nature instead of physical standards, American Institute of Physics Press Release, July 14, 2015.
  4. G. Mana, E. Massa, C. P. Sasso, M. Stock, K. Fujii, N. Kuramoto, S. Mizushima, T. Narukawa, M. Borys, I. Busch, A. Nicolaus, and A. Pramann, "The Correlation of the NA Measurements by Counting 28Si Atoms," Journal of Physical and Chemical Reference Data, vol. 44, no. 3 (September, 2015), article no. 031209, DOI:http://dx.doi.org/10.1063/1.4921240.
  5. More precise estimate of Avogadro's number to help redefine kilogram, American Institute of Physics Press Release, July 14, 2015.
  6. Supposedly in volume nine of Owen Ruffhead, "The statutes at large: from Magna Carta to the end of the last parliament," (M. Baskett, 1765), but there was too much there to scan for verification.

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