Max Packing Fraction of Hexagonal Unit Cell Volume w/ Hard Spheres

[malawi_glenn]

Homework Statement

Calculate the maximum packing fraction of the unit cell volume that can be filled by hard spheres in the Hexagonal structure

Relevant eq: Volume of spheres is number of lattice points multiplied with the maximum volume of one sphere.

The Attempt at a Solution

I know maxium is obtained when c = a i.e when height of the cell is as high as one of the sides in the hexagon. Hence, maximum sphere radius is a/2 (I have shown geometrically that the spheres can touch each other).

Now I am to determine the number of lattice points in this structure, I know that one primitive cell contains totally one lattice point, and a unit cell of a hexagonal structure can be made up by exactly three primitve cells, so the number of lattice points is 3. Is that the correct way to do this?

The rest I can figure out by my self, just are unsure how to determine the number of lattice points.