The surface of a sphere scales with the square of its radius r,but its volume scales with r
3. The total number of atoms N in this sphere scales linearly with volume. The fraction of atoms
at the surface is called dispersion F, and it scales with surface area divided by volume, i.e. with the inverse radius or diameter, and thus also with N 21/3. Basically the same relation holds for long cylinders of radius r and for thin plates of thickness d