Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction.
The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular.
The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.
There are two simple regular lattices that achieve this highest average density.
[4] The problem of close-packing of spheres was first mathematically analyzed by Thomas Harriot around 1587, after a question on piling cannonballs on ships was posed to him by Sir Walter Raleigh on their expedition to America.
[5] Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid.
Both arrangements produce a face-centered cubic lattice – with different orientation to the ground.
Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.
In close packing all of the spheres share a common radius, r. Therefore, two centers would simply have a distance 2r.
For simplicity, say that the balls are the first row and that their y- and z-coordinates are simply r, so that their surfaces rest on the zero-planes.
Again, the centers will all lie on a straight line with x-coordinate differences of 2r, but there will be a shift of distance r in the x-direction so that the center of every sphere in this row aligns with the x-coordinate of where two spheres touch in the first row.
In an A-B-A-B-... stacking pattern, the odd numbered planes of spheres will have exactly the same coordinates save for a pitch difference in the z-coordinates and the even numbered planes of spheres will share the same x- and y-coordinates.
Both types of planes are formed using the pattern mentioned above, but the starting place for the first row's first sphere will be different.
[8] In general, the coordinates of sphere centers can be written as: where i, j and k are indices starting at 0 for the x-, y- and z-coordinates.
A packing density of 1, filling space completely, requires non-spherical shapes, such as honeycombs.
However, such FCC or HCP foams of very small liquid content are unstable, as they do not satisfy Plateau's laws.
[9] There are two types of interstitial holes left by hcp and fcc conformations; tetrahedral and octahedral void.
Layered structures are formed by alternating empty and filled octahedral planes.
Two octahedral layers usually allow for four structural arrangements that can either be filled by an hpc of fcc packing systems.
In unit cells, hole filling can sometimes lead to polyhedral arrays with a mix of hcp and fcc layering.