Close-packing of equal spheres

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.

Illustration of the close-packing of equal spheres in both HCP (left) and FCC (right) lattices
FCC arrangement seen on 4-fold axis direction
Cannonballs piled on a triangular (front) and rectangular (back) base, both FCC lattices.
Collections of snowballs arranged in pyramid shape. The front pyramid is hexagonal close-packed and rear is face-centered cubic.
Figure 2 Thomas Harriot in ca. 1585 first pondered the mathematics of the cannonball arrangement or cannonball stack, which has an FCC lattice. Note how the two balls facing the viewer in the second tier from the top contact the same ball in the tier below. This does not occur in an HCP lattice (the left organization in Figure 1 above, and Figure 4 below).
Figure 3 Shown here is a modified form of the cannonball stack wherein three extra spheres have been added to show all eight spheres in the top three tiers of the FCC lattice diagramed in Figure 1 .
Figure 4 Shown here are all eleven spheres of the HCP lattice illustrated in Figure 1 . The difference between this stack and the top three tiers of the cannonball stack all occurs in the bottom tier, which is rotated half the pitch diameter of a sphere (60°). Note how the two balls facing the viewer in the second tier from the top do not contact the same ball in the tier below.
Figure 5 This animated view helps illustrate the three-sided pyramidal ( tetrahedral ) shape of the cannonball arrangement.
An animation of close-packing lattice generation. Note: If a third layer (not shown) is directly over the first layer, then the HCP lattice is built. If the third layer is placed over holes in the first layer, then the FCC lattice is created.
Miller–Bravais index for HCP lattice