Friday, May 01, 2009

Counting Atoms in a Sphere

To measure means to count. We measure a length by counting marks on a ruler, and a time span by counting ticks of clock. We compare the quantity we want to measure to multiples of a standardised quantity, the unit of measurement, such as the metre, the inch, or the second.

While in earlier times, the unit of length may have been set by the foot of the king, the French Revolution originated the metric system, with units of length, time, and mass rooted in the natural world, independent of the contingencies of human history. Nowadays, the second is defined by counting 9192631770 beats of the valence electron of an isolated cesium-133 atom flipping its spin in the magnetic field of the cesium nucleus, and the metre is the distance travelled by light in a vacuum in 1/299792458 of a second.

It's just the unit of mass, the kilogram, which is still defined in an old-fashioned, 19th century style, by a prototype meticulously kept in a vault of the Pavillon de Breteuil near Paris, France.

 The International Prototype of the Kilogram (IPK) inside three nested bell jars at the Bureau International des Poids et Mesures (BIPM, International Bureau of Weights and Measures), Paris, France.

Using a material artefact such as the Prototype as the standard of mass has a few obvious disadvantages: It is not easy to reproduce exactly, and it is even subject to tiny changes over time, for example by accumulation of contaminants on the surface, which are not completely understood.

Thus, as Peter Becker and Arnold Nicolaus write in their feature article "The marathon race to an new atomic kilogram" (Europhysics News 40(1), 2009, pp 23-26), the Comité International des Poids et Mesures (CIMP, the International Committee for Weights and Measures) hopes to reach a redefinition of the kilogram by 2011, linking the unit of mass to fundamental constants of nature, in a way that is reproducible with a measurement uncertainty of the order 10−8.

There are now two main roads which are being explored to reach these goals, the Watt balance and the Avogadro Project. While the Watt balance tries to link the kilogram to the Planck constant h by electro-mechanical experiments and exploiting the Josephson and quantum hall effects, the Avogadro Project aims at counting as accurate as possible the atoms in sphere of pure silicon, thus linking the unit of mass to the mass of a single silicon atom.

I find the approach of the Avogadro Project very intriguing, because it is so elementary.

In the current definition of the Système International, the base unit mole is linked to the kilogram: It's the amount of a substance which contains as many particles as there are carbon atoms in 12 gram of monoisotopic carbon-12. This number is the famous Avogadro constant, NA = 6.02214179(30)×1023, a huge number which is currently known with an uncertainty of 5×10−8.

Now, if there was a way to reliably count the number of a atoms in a piece of matter to a higher accuracy, on could revert the relation of the mole and the kilogram: The kilogram then could be defined as the mass of a certain amount of atoms. This is the idea behind the Avogadro Project.

To put this idea in practice, one fabricates spheres out of silicon single crystals, whose diameters are measured to within parts of a nanometre. From the accurately known size of the silicon crystal unit cell, one can then calculate the number of atoms in the sphere to a very high precision. If one knows the isotopic composition of the silicon, or uses highly enriched, nearly monoisotopic silicon-28, one can calculate the mass of the sphere.

One of the "Silicon Avogadro Spheres" at the National Physical Laboratory (NPL), Teddington, UK. 

Peter Becker and Arnold Nicolaus, who are working on the Avogadro Project at the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig, the German national metrology institute, describe in their report the many complicated steps and technically challenges to be met to realise the elementary idea of the "counting of atoms".

And it is a truly international project: The highly enriched silicon-28 is produced in Russia, it is grown in defect-free single crystals in Berlin, and processed to the perfect spheres by the Australian Centre for Precision Optics (ACPO). The Australian laboratory also has the means to precisely measure the size of the sphere. Finally, the surface of the sphere, which is covered by a thin film of oxide, is characterized by the British National Physical Laboratory (NPL).

 A "Silicon Avogadro Sphere" as seen by the Australian Centre for Precision Optics: Colour encodes the diameter of the sphere along different directions, running from 93.634370 mm (blue) to 93.634415 mm (red). From Peter Becker and Arnold Nicolaus: The marathon race to an new atomic kilogram.

It is not clear yet whether the Avogadro Project can reach the accuracy requested by the International Committee for Weights and Measures, and if it can outdo the Watt balance project as the method of choice to define the kilogram. Anyway, it is a wonderfully elegant idea to define a unit of mass: Just link it to the mass of a silicon atom and count atoms.


  1. This is interesting. I wonder though why they take 1 kilogram and not something more handy, like a gram or so? Wouldn't that be easier to deal with?

  2. Hello Stefan,
    I understood that the Number of Atoms
    in that silicon sphere is calculated
    by division of the spere`s volume
    by the unit cell volume, the latter
    resulting from x-Ray diffraction .
    Is this really "counting"?

  3. Hi Stefan,

    A very interesting piece and as yourself I prefer the Avogadro approach better as it allows us to stay more with the physically realizable side of things. Although I must admit their choice of silicone as the element for me is still a bit unintuitive; for when I think of mass my mind conjures up how elemental mass is relative to each other. That is I think of the specific gravity relation as Archimedes first brought to light. That is a gram representing to be a cubic centimetre of pure water at a specific temperature/pressure with then all other elements being fractions or multiples of this. In this way I’ve always found it comforting that I could work out the mass of things in simply being able to measure its size and calculate the volume. I guess though counting silicone atoms closer to my gut feelings for things rather than the other approach you mention.



  4. Hi Stefan,

    Oh yes, as a follow up I do realize that water is not an element yet rather a combination of two. Yet as it is the bases of life as we know it I find it hard to imagine anything else when it comes down to imagining things. Besides it would be somewhat difficult to take a silicone bath :-)



  5. To measure means to count.Mild dissent - in the kitchen "measure" can mean - does the quantity seem approximately right. :)

    I think because of the microelectronics industry we have the most experience with growing perfect silicon crystals.

    It is also fitting that silicon is the most abundant solid element in the earth's crust (OSiAlFeCaNaKMg, if I remember correctly :))

  6. Hi Arun,

    Yes all those things you mention are true and I acknowledge that in our world silicon to be more significant at times then water. In fact what I do for a living is highly related to the importance and utilization of this most abundant element. Still though if one is required to build a scale with the simplest of materials and utilizing the most available means of measure, you can’t beat plain old H2O, as it’s still easier then piling sand and more pure in its natural found state. You might call it the standard of mass for the masses :-)



  7. Does mass exist? The Standard Model is massless. No gravitation theory contains composition, only geometry or fields. Perhaps mass is a spacetime anomaly that requires a more fundamental model - the reason why mass is tied to a physical artifact rather than a theoretical constuct.

  8. Hi Uncle Al,

    “Does mass exist? The Standard Model is massless.”

    This all boils down to QM for most being of a singular ontological nature rather a dualistic one. In as my metaphysical center has this ontology as being dual I have no problem with the standard model representing mass as being a measureable consequence in the responsive nature of the two.



  9. This spheres have to be single
    crystals of a element, hard enough to be ground
    and polished, stable enough to react
    (and change weight thereby) less
    than the old standards.
    (At least within the time needed to
    make the spheres and do the measurements)
    At the end of the project the spheres
    have to be weighed against the old
    kilogram prototypes, for that reason
    You need a kilogram, not a gram or
    a milligram.
    Eventually after some years one
    can decide, whether new secondary
    kilograms made from silicon should be
    introduced, maybe the old ones
    will persist, beeing more stable
    over long times, maybe.

  10. Hi Stefan,

    As a further follow up after reading the article you pointed to I discovered what is quoted here at the end. It seems that with all their x-rays and laser beams, Archimedes's Principle of Buoyancy cannot be negated in the overall consideration. It’s pleasing to know that with all our advancements we still are reliant upon many fundamental ideas or in this case one of the many original eureka moments.“Mass:
    The mass of the silicon sphere is linked to the prototype kilogram by weighing. Since the two materials have different densities, extensive buoyancy corrections are necessary. Volume and mass determinations must then still be adjusted in terms of the oxide layer.”



  11. I wonder why they chose to make silicon spheres when there is so much technology dedicated to making rectangular solids. Chip makers have been building circuits out of blocks of silicon with accuracy on the order of few atoms this way and that. They even have all sorts of clever imaging techniques for studying the surface layers to check for flaws in the well understood packing structure. Silicon forms crystals in layers. Why not take advantage of some of the more advanced technology that we have?

  12. What about melting one kg of
    silicon in an oven on board of
    Would the silicon make a sphere
    on cooling and crystallizing?

  13. "Kepler Conjecture is speaking about cannon balls. Tom Hales writes,"Nearly four hundred years ago, Kepler asserted that no packing of congruent spheres can have a density greater than the density of the face-centered cubic packing."

    13th Sphere of the GreenGrocer

  14. Dear all,

    thanks for the feedback, and sorry for the somewhat late response...

    Dear Bee,

    I wonder though why they take 1 kilogram and not something more handy, like a gram or so?

    Hm, I've been wondering about this too... Maybe it is technically not that much more easy to produce a sphere of just one gram. And, as Georg explains, they really want to reproduce the kilogram, as they have to compare it to the existing prototype. After all, that the rest of the SI system should be kept as it is now.

    Hi Georg,

    ... the number of Atoms in that silicon sphere is calculated by division of the spere's volume by the unit cell volume, the latter resulting from x-Ray diffraction. Is this really "counting"?Well, you have a point - it's not counting in the sense of enumerating all atoms one by one. That my be a bit difficult for that many ones ;-). So this method is some practical shortcut to actually counting them, in my view.

    Actually, I have seen on their web site that the PTB has been experimenting with another way to assemble a test mass of a kilogram using an ion beam. This is even closer to counting, as the atoms are indeed added one by one. But also in this method, a shortcut to true counting is used, by measuring the electrical current of the beam as a proxy.

    And thanks for your comment explaining why using kilogram spheres of silicon are a natural choice!

    Hi Phil,

    Although I must admit their choice of silicone as the element for me is still a bit unintuitiveOne essential point in this approach is to have at hand large defect-free single crystals, and I guess this excludes water, for example, at least for the purpose of the "Avogadro Project".

    Moreover, as Arun points out, the technology to grow such crystals of silicon exists already, for the fabrication of silicon wavers in the semiconductor industry. And then, Silicon-28 has 14 protons and 14 protons, and its mass can probably be better linked to the mass of Carbon-12, which is used in the current definition of the mol.

    Hi Arun,

    in the kitchen "measure" can mean - does the quantity seem approximately right. :)Well, one teaspoon or two teaspoons ;-) You often don't have to count much higher ;-)

    Hi Uncle,

    Does mass exist?I could try this argument next time I have to pay my food by the kilogram at the supermarket ;-)

    Hi Kaleberg,

    I wonder why they chose to make silicon spheres when there is so much technology dedicated to making rectangular solids.I've also been wondering why they do not produce a cube. I am not sure, but I think of two points why they do not:

    First, the techniques you mention, epitaxy and so on, usually do not produce big pieces of material. All these tricks of the chip industry are applied to tiny samples, and I do not know if and how they can be scaled up to produce a cube of 1 kilogram.

    And second, the cube has edges which probably wear down much more easily than a sphere would loose mass when handled...

    Cheers, Stefan

  15. Hi Georg,

    What about melting one kg of silicon in an oven on board of ISS? Would the silicon make a sphere on cooling and crystallizing?Sounds like a cool suggestion! Actually, how defect-free are the crystals grown in microgravity? And how exact would by the sphere?

    Cheers, Stefan

  16. Hi Phil

    It seems that with all their x-rays and laser beams, Archimedes's Principle of Buoyancy cannot be negated in the overall consideration.Exactly! I had a similar feeling when I read that: All high-tech to produce the sphere, and in the end, when it comes to compare to the prototype, you have to take care of old Archimedes ;-)

    Best, Stefan

  17. Hi Kaleberg,

    “ I wonder why they chose to make silicon spheres when there is so much technology dedicated to making rectangular solids?”

    I can certainly see your point in one respect and yet in another when it comes to physics where symmetry plays such an important role one can’t do better than a sphere as being your starting point. In a way the standard kilogram which is actually a right-circular cylinder (height = diameter) of 39.17 mm as to minimize its surface area addresses this, with an actual sphere being the limit, which brings us back to Archimedes. With a minimum surface to volume one minimizes the reaction of the exposed surface in relation to its volume to best minimize such effect.

    So as Archimedes proved a sphere contained within such a cylinder would be exactly 2/3 the volume and thereby the related mass of this object. Also as it’s composed of 90% platinum and 10% iridium, with corresponding specific gravities of 21.45 and 22.4 respectively we end up with an averaged specific gravity of 21.545. So calculating V=(4/3 pi R^3)x 3/2 one ends up with 47200.89 cubic millimetres or 47.20 cc’s of material. If you multiply that by the averaged specific gravity you get 1016.94 grams or 1.017 kilograms; which within the limits of this crude approximation checks out.

    This is why I related to Stefan that for me the ancient ways I still find to be more physically intuitive relating to constants that we have been long familiar with. To rephrase Archimedes, given a place to stand, a large enough bath tub, water to fill it to the top, another identical tub to catch the spill over, plus a long enough lever with an adjustable fulcrum point and he could have weighed the earth, well at least in theory:-)



  18. Hi Stefan,

    It’s nice to know there be some others who still give thanks to Archimedes as it rates to what he left for us all. Even more to the point, as it’s been more recently discovered that he also had imagined and utilized one of the core ideas of calculus some two millennia before being redinvented and expanded upon by Newton and Leibnitz. Thus I am often brought to wonder, what kind of world we would have now if things had proceeded without such loss and/or interruption? It also serves as a reminder for me of how important it is not just to promote discovery yet also in finding better ways to preserve what we learn if by some unfortunate set of circumstance we once go through another dark ages.



  19. Pt as a mass standard is historically faulty. Consider: Trace hydrogen in air will diffuse in after casting. Steel frame buildings have iron corrosion from plumbing, roof leakage, and condensing humidity (air conditioning). They vent hydrogen - as do mammal and termite posteriors.

    A kilogram of Pt (32.1507 troy oz) is $(US)35,140.

  20. Due to the binding energy, the mass of a crystal consisting of N atoms will be less than N times the mass of a single atom.

    This effect is almost of the same order as the accuracy that one demands for the new mass standard.

  21. Phil,
    We are always on the edge of a dark age.

  22. Hi Count,

    Due to the binding energy, the mass of a crystal consisting of N atoms will be less than N times the mass of a single atom.Good point!

    I am now just a bit confused... the mole, in SI, is defined as the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon12, but then, In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to. How is this supposed to be realized in a partical way? How can one determine a mass of 12 gram carbon-12 against the kilogram protoype unless the carbon is a solid, and the atoms bound?

  23. Fascinating. Any scoop from anyone on a related issue I've heard about, that the standard meter bar doesn't match up the way it should?

  24. OK, it makes sense to try for a sphere. Minimum surface and great symmetry are good arguments.

    Of course, with all the cosmologists here, I would expect someone to propose using a 1 kilogram black hole as the standard. After all, we have good measures for distance and time. Maybe we'll luck out, and a suitable black hole will pop out of the LHC. Of course, the real problems will start when we get out the old calipers to measure it.

  25. Hmmm...just thinking.

    Ancient ways of ingenuity flooding over, or, new robotic adventures in information transfer?

    Discrete measures, can then be counted by a wave, as the wave "passes through and over" as a distinction of any discrete measure?? An "emotive quality" that sends signals to "other parts" of the system?

    You fills all space? Has reductionism then runs it limit?

    Most certainly a measure of things come into question as remnants of "particulate" point of view are focused on.


  26. Hello Kaleberg,
    Stefan wrote:
    "And second, the cube has edges which probably wear down much more easily than a sphere would loose mass when handled..."
    This is right, and in addition this
    effects start their evil work already
    during grinding of prisms.
    Look into a catalog for optic
    materials, You will find prisms
    less precise, more expensive than
    lenses (sperical forms!) and
    always some text like :
    "minor faults at edges are
    unavoidable and do not impair the
    prisms in practical use"

  27. This comment has been removed by the author.

  28. Hi Arun,

    “We are always on the edge of a dark age.”

    True enough and better still would be to suggest that for the overwhelming majority, even if only the first world considered we have not yet entered an enlightened age. What I mean is if you stopped random people in the street to be given the problem of the king’s crown that Archimedes had been presented, what solution would be offered by most and could they execute it if asked? So are we truly living in an age of scientific understanding or rather simple one of scientific trust and general belief?



  29. I was at a conference in Bad Honnef a couple of years ago discussing (among others) precisely this question
    and we got a nice talk about the silicon spheres .. including how they were transported from Australia (or was it the other way).
    Apparently they went (in special boxes, of course) in the hand luggage and the guy just hoped that turbulence would not be too bad.

    I think spheres are also easier to machine and measure accurately as there is only one dimension to worry about... across.

  30. Hi everybody

    Might be worth to mention that, for the kilogram redefinition, also a third experiment is on the way, which is the GAMS experiment at ILL (Grenoble, France) where I'm currently working. I both work on the avogadro project, for the Si d220 determination, and the GAMS experiment, for the molar Planck constant determination.
    This latter is of main importance, since, if you fix the Avogadro number instead of the Planck constant (by Watt Balance), you still need a consistent tool to link the two. The molar Planck constant is determined by a gamma ray spectroscopy experiment.

    With the Avogadro project, the Watt Balance and the Gams experiment, you are able to close the so-said Mass Triangle.


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