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There likely exists an even-integer number of m graphene-hexagons in a graphite stack, such that m atoms on each hexagonal sheet-side yields a number of carbon atoms closest to our standard value for Avogadro's number. The even-integer choice of m for atoms on the side of a closed-shell sheet comes from their natural pairing along the sheet edges, while the even-integer choice of m for number of sheets arises from symmetry considerations because the unit cell is two-layers thick in the (002) direction as shown in the figure at right. If m = 0, number of atoms Ntotal = mNhex = 3m2+9m3/2 = 0. If m = 2, Ntotal = 2 × 24 = 48 atoms. If m = 4, Ntotal = 4 × 84 = 336 atoms, etc.
Although it may not be obvious from the equation above, Ntotal will always be divisible by 12. One can see this from our physical model by pointing out that any structure with even m can separated into two equal layer-sets, each of which in turn (by symmetry) can be divided into six equal pie-slices. Mathematically, this is true since Ntotal/12 = (m/2)2(1+3(m/2)) is an integer if m is even.
Of course we are looking for m such that Ntotal ⇒ Avogadro's number. For example a faceted graphite-crystal with a set of m (002) graphene sheets, each of which has m atoms along the sheet hexagonal edges that make up the 6 {110} facets, would for m = 51,150,060 = 223251131218591 have 602,214,158,510,196,804,982,800 atoms which compares nicely to current approximations. The result of this choice for m is closer to 6.0221415×1023 than is the best (but non-realizable simple-cubic) model compared to it in the Fox article about Carbon cubes (Fox2007). It is also closest of the proposed physical models to the value of 6.02214179×1023 recommended here (Mills2010). Subtracting 2 from this value of m will put it closer to the lower value of 6.02214078×1023 based on Si-28 measurements here (Andreas2011).
This crystal would be about 1.71 cm thick, and have a (circumscribed-cylinder) diameter (twice the length of one side) of about 1.45 2.18? cm. A hexagonal prism like this is already approximately one-mole of carbon. If the physical model discussed here is used to redefine Avogadro's number, to the extent that Carbon-12 has a mass at or near 12 Daltons then one gram would be near if not equal to the mass of NA/12 = 50,184,513,209,183,067,081,900 atoms of 12C as well.
The divisibility of this suggested integer is related to its high factorability. As shown in Figure \ref{fig3}, the factors include five factors of 2 and five factors of 3. The hex-prism layer/side integer m = 51,150,060 is also relatively factorable, with two factors of 2, two of 3, and one each of 5, 13 and 21859.
Related references:
Ronald Fox and Theodore Hill (2007) "An Exact Value for Avogadro's Number: Untangling this constant from Le Gran Kcould provide a new definition of the gram", American Scientist 95:2, 104 link.
Ian M. Mills (2010) What is a mole?: Old concepts and new (continued) Chem. Int. 32 (1) 12-13.
B. Andreas et al. (2011) "Determination of the Avogadro constant by counting the atoms in a 28Si crystal", Phys. Rev. Lett. 106, 030801 link.
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