An Exact Value for Avogadro's Number
Untangling this constant from Le Gran K could provide a new definition of the gram
Squaring N A with N A *
Adoption of N A * as the value for Avogadro's number would offer several advantages. With today's definition of Avogadro's number being the number of atoms in one mole of a particular element, this new fixed value for it would simply mean that the mass of a simple cube of carbon-12 atoms, exactly 84,446,888 atoms on a side, is exactly 12 grams by definition.
Practically speaking, however, carbon does not admit an extended simple cubic structure but does have a face-centered cubic (FCC) crystal structure in three dimensions, the same as diamond and silicon. This means that in a real physical cubic array of carbon atoms, the atoms are located not only in a simple cubic array but also at the centers of faces made by a square of four adjacent planar atoms and at certain interior tetrahedral centers of cubes made of eight adjacent corner atoms. The number of atoms in such an actual FCC array with k atoms on each edge can easily be calculated to be 8k 3-18k 2+15k-4.
Carbon-12 is special in the context of fundamental constants since, by convention, NIST uses carbon-12 to define both Avogadro's number and the basic atomic mass unit, amu. Thus, if one wanted a definition of Avogadro's number specifically tied to the actual physical FCC lattice structure of carbon-12, one could replace the earlier formula n3 = N A by 8k 3-18k 2+15k-4 = N A . This means that a physical FCC lattice of carbon-12 containing 42,223,444 atoms on each edge, exactly half the number of atoms on the edge of the hypothetical cube defining N A * above, would contain exactly 602,214,108,979,663,699,470,280 atoms (8k 3-18k 2+15k-4 with k = 42,223,444), which is also within the currently accepted range of values for Avogadro's number but not as close to the best estimate as is N A *.
Of course, the instant a fixed value for Avogadro's number is chosen, there no longer would be scientific interest in constructing an exact such cube anyway, just as there has been no scientific quest to construct the perfect meter stick since 1983. Building meter-stick and gram prototypes would be left to manufacturers of precision surveying and scales equipment.
Numerically, N A * is describable in nine digits (eight digits plus the exponent), and in that sense contains roughly the same order of magnitude of information as the fixed integers that define the speed of light and the second. Moreover, 84,446,888 (or 42,223,444) is easy to remember. Since N A * is almost dead center within the current known range of values for N A , it is consistent with current experimentally obtained results.