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### Graphene Mass Standard

February 16, 2012 Mass is of fundamental importance in physics. Our system of units, now called the International System of Units (SI, for Système international d'unités), was called the MKS system, for meter, kilogram, second. The primary mission of CERN's very expensive Large Hadron Collider (LHC) is the search for the Higgs boson, the particle that's conjectured to give all other elementary particles their mass. For all the importance of mass, the method for standardizing mass is a relic of a classical physics period long passed. The present mass standard is a physical object, a cylinder of a 90% platinum - 10% iridium (by weight) alloy, the international prototype kilogram. This object, maintained at the Bureau International des Poids et Mesures (International Bureau of Weights and Measures), is lovingly compared with copies that are distributed to international weights and measures organizations for comparison against their own physical mass standards. As you can imagine, all the care in the world, and even the use of such a durable and non-tarnishing alloy, can't eliminate all errors. In fact, the catalytic action of platinum might be responsible for reactions with polluted air that add a little more mass to this object, year after year. However, the change has been small, less than 100 parts per billion (ppb), as can be seen in the graph that shows the upwards increase in mass of eleven transfer mass standards.*Upwards drift of eleven international mass standards.(Via Wikimedia Commons).*

There have been recent efforts to replace the international prototype kilogram with a mass standard that is linked to fundamental physical constants. Aside from a possible advantage in this approach of improved future accuracy, such a standard would not be linked to a specific object in a specific location. Every national standards body could have its own very precise mass reference. I outlined two approaches to a mass standard in previous articles (Mass Standard, November 1, 2010 and Standard Kilogram, June 28, 2007. The Avogadro project is an international collaboration to replace the international prototype kilogram with an equivalent object made from isotopically pure silicon-28 (

^{28}Si). Natural silicon is composed of the isotopes,

^{28}Si, 92.23%,

^{29}Si, 4.67%, and

^{30}Si, 3.1%, so the isotope separation is an important step. After that, perfect crystals of

^{28}Si are made and formed into perfect spheres. These spheres would contain an Avogadro number of atoms, as determined by the volume. These spheres would have a reproducible weight of very nearly 28 grams, and they could be made in multiply exact copies at any place at any time. A more fundamental technique is the watt balance, which would link the kilogram to Planck's constant. The equipment required to realize such a measurement is quite complicated, but it is a more accessible measurement, since silicon isotope separation is not required. The technique is called a watt balance, since the weight of a test mass is balanced against the product of an electric current and a voltage, the units of which are related only to Planck's constant and the speed of light. The US National Institute of Standards and Technology (NIST) has achieved an error for this measurement of 36 ppb. It's agreed that when the errors are at the 10 ppb level, this will be a viable standard. Then there's graphene, our present wonder material. Phil Fraundorf of the Department of Physics and Astronomy and the Center for NanoScience, the University of Missouri (St. Louis, MO), has just posted a paper on arXiv in which he proposes a crystal of carbon (

^{12}C) formed from graphene sheets as a mass standard that contains a mole of carbon atoms. Fraundorf writes that

"Graphite is constructed from graphene sheets whose controlled synthesis at the atomic-scale is likely to see great progress by nanotechnologists in the years ahead. If any macroscopic object will be possible to assemble from a chosen number of atoms in the decades ahead, this may be it."Fraundorf examines the growth of graphite from a seed plane of twenty four carbon atoms, as shown in the figure. Lateral growth is accomplished by adding units to the periphery, and it can be seen that at every hexagonal edge of the ensemble there are two added atoms m for every added cell element n.

*Counting atoms in graphene sheets. (Illustration by author, rendered with Inkscape).*

When building an hexagonal prism of graphene sheets of m layers, the number of atoms N is given by the equation,

N = 3 mIt's possible to build a structure that comes extremely close to having an Avogadro's number of atoms, as shown in the table.^{2}+ (9/2) m^{3}

Such a graphite crystal would be about 1.71 cm high. A circumscribed cylinder around this crystal would have a diameter of about 1.45 cm. Of course, it would weigh twelve grams.

n m N 0 0 0 1 2 48 2 4 336 3 6 1080 4 8 2496 5 10 4800 6 12 8208 7 14 12936 8 16 19200 9 18 27216 10 20 37200 25,575,030 51,150,060 6.02214 x 10 ^{23}

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Linked Keywords: Mass; physics; International System of Units; meter; kilogram; second; CERN; Large Hadron Collider; Higgs boson; elementary particle; classical physics; platinum; iridium; alloy; international prototype kilogram; Bureau International des Poids et Mesures; durable; non-tarnishing; errors and residuals in statistics; catalysis; pollution; polluted air; parts per billion; Wikimedia Commons; physical constant; Avogadro project; isotopically pure; silicon-28; silicon; isotope separation; crystal; sphere; Avogadro number; volume; nuclear binding energy; mass defect; watt balance; Planck's constant; electric current; voltage; speed of light; US National Institute of Standards and Technology; graphene; Phil Fraundorf; Department of Physics and Astronomy; Center for NanoScience; arXiv; carbon; mole; atom; seed plane; Inkscape; hexagonal prism; Avogadro's number; centimeter; cm; circumscribe; cylinder; gram.

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